High-accuracy measurements of the x-ray mass attenuation coefficients of molybdenum and tin: testing theories of photoabsorption

Transcription

1 High-accuracy measurements of the x-ray mass attenuation coefficients of molybdenum and tin: testing theories of photoabsorption by Martin D de Jonge Submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy School of Physics The University of Melbourne Australia August 25, 2005 Produced on acid-free paper

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3 High-accuracy measurements of the x-ray mass attenuation coefficients of molybdenum and tin: testing theories of photoabsorption by Martin D de Jonge Supervised by Assoc. Prof. Christopher T Chantler The x-ray atomic form-factor determines the x-ray optical properties of materials and is a fundamental parameter for critical x-ray investigations. However, despite uncertainty estimates of order 1%, differences of 2 10% between x-ray mass attenuation measurements render comparison with the various theoretical tabulations meaningless. Moreover, such uncertainties impose limits on the accuracy of various quantitative investigations. We determine the imaginary component of the atomic form-factor from measurements of the x-ray mass attenuation coefficient. With the exception of the measurements of Tran et al. [Phys. Rev. A 64, ; 67, ; J. Phys. B 38, 89 with a 0.3% accuracy, previous work has been unable to achieve accuracies below 1%, and differences between results claiming this accuracy often exceed 6%. We have developed a full-foil mapping technique which has improved the measurement accuracy by an order of magnitude. This technique overcomes limitations arising from absorber thickness variations, using the average integrated column density and attenuation measurements across the entire surface of the absorber. We have examined measurements obtained over a wide range of parameter space for systematic deviations indicative of experimental error. Among others, this has led to the identification and correction of a 1% discrepancy arising from the x-ray bandwidth. Resulting measurement accuracies for molybdenum are %. Preliminary results for tin suggest a final accuracy of 0.1 1%. We compare these measurements with several commonly-used tabulations and identify a number of systematic discrepancies whose causes are discussed.

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5 This is to certify that (i) the thesis comprises only my original work towards the PhD, (ii) due acknowledgement has been made in the text to all other material used, (iii) the thesis is less than 100,000 words in length, exclusive of tables, bibliographies, and appendices. I authorise the Head of the School of Physics to make a copy of this thesis available to any person judged to have an acceptable reason for access to the information. Martin D de Jonge August 25, 2005

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7 Responses to the referee s comments I thank the referees for their careful and close reading of this work. Their comments have prompted several improvements, each of which is detailed here. The reviewers have pointed out some typographical errors. In particular: reference [1 was incomplete; the upper limit of the energy range of the XCOM database was reported as 1 GeV instead of 100 GeV (page 33), and; Delbrück scattering dominates the atomic scattering cross sections at energies above 100 MeV (page 27). In addition, it was pointed out that the term fluorescent radiation is somewhat redundant, and so it is replaced by the term fluorescence radiation (section 8.4). A short discussion of the nuclear resonant contribution to the atomic scattering cross section (page 28) implied that giant dipole resonances are limited to boundbound nuclear transitions. However, these resonances include the nuclear photoeffect, which appears as a broad peak several MeV wide in the 10 MeV 40 MeV region. The text has been updated, and references [2, 3, 4 have been added. Section 2.8 discusses theoretical approaches to the calculation of mass attenuation coefficients, observing that one approach [5 uses the Hartree-Fock renormalised values of Scofield s calculation of the photoelectric absorption cross sections [6, extrapolating them over a further five orders of magnitude in energy, to 100 GeV (page 31). We now mention in the text that the extrapolation was guided by the theoretical high-energy results of Pratt [7. While the extrapolation procedure enabled estimation of the photoelectric contribution to the total atomic cross section (dominated at high energy by other processes), the extrapolated photoelectric cross sections were not of high accuracy. vii

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9 Preface The work reported in this thesis has resulted in a number of written and oral presentations. These are detailed here for reference. Publications resulting directly from this thesis The following publications have arisen directly from this work. They are reprinted in full in Appendix D. M D de Jonge, C Q Tran, C T Chantler, Z Barnea, B B Dhal, D J Cookson, W-K Lee, and A Mashayekhi. Measurement of the x-ray mass attenuation coefficient and determination of the imaginary component of the atomic form factor of molybdenum over the keV energy range. Physical Review A 71(3), M D de Jonge, Z Barnea, C T Chantler, and C Q Tran. Bandwidth: determination by on-edge absorption and effect on various experiments. Physical Review A 69(2), M D de Jonge, Z Barnea, C Q Tran, and C T Chantler. Full-foil x-ray mapping of integrated column density applied to the absolute determination of mass attenuation coefficients. Measurement Science and Technology 15, , Publications related to this thesis The author has made significant contributions to the writing, development, and/or analysis presented in the following works. C Q Tran, M D de Jonge, Z Barnea, and C T Chantler. Absolute determination of the effect of scattering and fluorescence on x-ray attenuation measurements. Journal of Physics B 37, , ix

10 C Q Tran, Z Barnea, M D de Jonge, B B Dhal, Paterson D, D J Cookson, and C T Chantler. Quantitative determination of major systematics in synchrotron x-ray experiments: seeing through harmonic components X-Ray Spectrometry 32, 69 74, C Q Tran, Z Barnea, C T Chantler, and M D de Jonge. Accurate determination of the thickness or mass per unit area of thin foils and single-crystal wafers for x-ray attenuation measurements. Review of Scientific Instruments 75(9), , C Q Tran, C T Chantler, Z Barnea, M D de Jonge, B B Dhal, C T Y Chung, D Paterson, and J Wang. Measurement of the x-ray mass attenuation coefficient of silver using the x-ray-extended range technique. Journal of Physics B: Atomic, Molecular, and Optical Physics 38(1), , C Q Tran, M D de Jonge, Z Barnea, B B Dhal, and Chantler C T. Quantitative determination of the effect of the harmonic component in monochromatised synchrotron x-ray beam experiments. Developments in Quantum Physics, Nova Science. Eds. F Columbus and V Krasnoholovets, Conference proceedings M D de Jonge, C Q Tran, C T Chantler, and Z Barnea. Improved techniques for measuring x-ray mass attenuation coefficients. Proc. SPIE: Optical constants of materials for UV to x-ray wavelengths, Eds. R Soufli and J F Seely, 5538, , Oct C Q Tran, M D de Jonge, Z Barnea, B B Dhal, and Chantler C T. Quantitative determination of the effect of the harmonic component in monochromatised synchrotron x-ray beam experiments. 19th International conference on x-ray and inner-shell processes, Frascati Physics Series, Vol. XXXII, Eds. S Bianconi, A Marcelli, and N L Santini, , Conference abstracts M D de Jonge, C Q Tran, Z Barnea, B B Dhal, D J Cookson, and C T Chantler. X-ray extended-range technique for precision measurement of the x-ray mass attenuation coefficient and Im(f) for molybdenum using synchrotron radiation. Acta Crystallographica, Section A 58, section 1, p. C82, x

11 Oral presentations 2004: Accurate determination of mass attenuation coefficients [ µ with particular reference to molybdenum. Optical constants of materials for UV to x-ray wavelengths, SPIE 49 th annual meeting, Denver, CO, USA. 2002: Developments in the precise determination of Im(f) for molybdenum and tin. User science seminar series, Advanced Photon Source, Chicago, IL, USA. 2002: Developments in the precise determination of Im(f) for medium-z metals: molybdenum. 15th congress of the Australian Institute of Physics: Australian Optical Society, Sydney, Australia. Awarded OSA / SPIE prize for best student oral presentation. 2000: Proposed measurement of the imaginary component of atomic form factors for medium Z-elements, 14th congress of the Australian Institute of Physics: Australian Optical Society, Adelaide, Australia. Poster presentations 2005: L F Smale, C T Chantler, E C Cosgriff, M D de Jonge, Z Barnea, and C Q Tran. Failure of XAFS interpretation for ab initio investigations a new way forward. 16th congress of the Australian Institute of Physics, Canberra, Australia. 2005: C Q Tran, C T Chantler, M D de Jonge, Z Barnea, and N Rae. The x-ray extended range technique for high accuracy atomic structure in simple systems. 16th congress of the Australian Institute of Physics, Canberra, Australia. 2005: C T Chantler, Z Barnea, C Q Tran, and M D de Jonge. Latest results for silver atomic form factors in the relativistic regime a new frontier. 16th congress of the Australian Institute of Physics, Canberra, Australia. 2005: C T Chantler, M D de Jonge, Z Barnea, and C Q Tran. X-ray extended range technique for precision measurement of the x-ray mass attenuation coefficient and Im(f) for molybdenum using synchrotron radiation. 16th congress of the Australian Institute of Physics, Canberra, Australia. 2005: C T Chantler, E C Cosgriff, C Witte, L F Smale, C Q Tran, M D de Jonge, and Z Barnea. X-ray absorption near-edge structure calculations for silver. 16th congress of the Australian Institute of Physics, Canberra, Australia. 2003: M D de Jonge, C T Chantler, Z Barnea, C Q Tran, B B Dhal, and D J Cookson. X-ray extended range technique for precision measurement of xi

12 the x-ray mass attenuation coefficient and Im(f) for molybdenum using synchrotron radiation. Australian conference on optics, lasers and spectroscopy, Melbourne, Australia. 2002: M D de Jonge, C T Chantler, Z Barnea, C Q Tran, B B Dhal, and D J Cookson. X-ray extended range technique for precision measurement of the x- ray mass attenuation coefficient and Im(f) for molybdenum using synchrotron radiation. XIX Congress and general assembly of the International Union of Crystallography, Geneva, Switzerland. 2002: M D de Jonge, D J Paganin, C Q Tran, C T Chantler, and B B Dhal. Application of interferometry to a determination of the thickness of metallic foils. 15th congress of the Australian Institute of Physics: Australian Optical Society, Sydney, Australia. 2002: M N Kinnane, C T Chantler, M D de Jonge, A G Peele, C Q Tran, D J Paterson, and B B Dhal. Upgrades to x-ray data collection system, leading to absolute measurements of mass attenuation coefficients and sub-micron thickness variation detection at a local source. 15th congress of the Australian Institute of Physics: Australian Optical Society, Sydney, Australia. 2002: G Christodoulou, C T Chantler, D J Paterson, and M D de Jonge. Detector and spectrometer development for QED measurements using an EBIT. 15th congress of the Australian Institute of Physics: Australian Optical Society, Sydney, Australia. 2000: M D de Jonge, B B Dhal, and C T Chantler. A broad range channel-cut monochromating crystal for laboratory x-ray experiments between 5 30 kev. 14th congress of the Australian Institute of Physics: Australian Optical Society, Adelaide, Australia. 2000: M D de Jonge, C Q Tran, B B Dhal, Z Barnea, and C T Chantler. Absolute energy calibration of kev x-rays at the Advanced Photon Source. 14th congress of the Australian Institute of Physics: Australian Optical Society, Adelaide, Australia. 2000: I Blajer, B B Dhal, M D de Jonge, C Q Tran, D Paterson, and C T Chantler. Precision Measurement of x-ray complex atomic form-factor using rotating anode generator. 14th congress of the Australian Institute of Physics: Australian Optical Society, Adelaide, Australia. xii

13 Acknowledgments I wish to express gratitude to those who assisted my journey through this thesis. In the interests of brevity I mention only a few names and isolate only one or two qualities in recognition of each. I sincerely hope that all involved are aware of my appreciation of their particular contributions. I thank Chris for his optimism, Zwi for his (somewhat wicked) joie de vivre, and Chanh for his courage. Nick and Colin I thank for the space they provided prior to embarking on this journey. Fellow staff and students I thank for the diversity and richness of our interactions, and the university for providing a field for all of the above. The support of friendship is diverse, and it is precisely that diversity that makes for interesting times. My parents I thank for their kindness and unfailing trust. I thank David and Corinna for being friend and family during the Chicago era of this work. I thank my wife Amelia for her love. xiii

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15 Contents Abstract iii Responses to the referee s comments vii Preface ix Acknowledgements xiii Contents xv List of figures xix List of tables xxvii 1 Motivation and outline of this work 1 I Current status 7 2 Calculation of the mass attenuation coefficient Approximating the solid-state The atom: electronic structure calculations Background: history of x-rays and diffraction The x-ray atomic form-factor The modified form-factor The S-matrix approach X-ray interactions with matter Results of various theoretical approaches Summary and conclusion Measurement techniques Discussion of previous studies Principles of our experimental configuration X-ray production and monochromation X-ray detectors Correlation and statistics Sample attenuation levels Secondary photons: beam collimation Beam harmonic components Sample purity Beam size Determining attenuation from count rates xv

16 II Molybdenum 61 4 Experimental details 63 5 Determining the foil attenuations Filtering the data Measurements made during shutter motion Aperture vibration settling Dark current Determination of the intensity ratios Recovery of partially saturated data Interpolating the incident intensity ratios The foil attenuations Foil metrology Introduction Micrometer measurements of the foil thickness Determining the average integrated column density Measuring the foil masses Measuring the foil areas The determined average integrated column densities The full-foil mapping technique The attenuation profile Removal of the holder attenuation Determining the absolute value of [ µ Results of a low-energy full-foil mapping Discussion Further modelling of the holder attenuation profile Microstructure in the integrated column density Rotational alignment of the samples Improving the full-foil mapping technique Conclusions Determining the photon energies Requirements of the energy determination Previous measurements This measurement The measured rocking-curves Detector saturation Analyser crystal imperfections Implications of peak shaping Detector saturation Crystal imperfections Determination of peak positions Technique 1: fitting the leading-edge Technique 2: centre of mass Determining the photon energies xvi

17 7.8 Interpolating the photon energies Conclusions Determining [ [ µ, µ pe 2 of molybdenum The treatment of systematic contributions Determining the local integrated column densities The effect of x-ray bandwidth The effect of fluorescence radiation The effect of roughness The effect of harmonic components The effect of a dark-current error Incorporation of rapid measurements Consistency of the measured values [ µ pe The experimental error budget Summary and Conclusions III Tin and Silver Tin Experimental details Attenuation Full-foil mapping Determination of the photon energies Scaling the local integrated column densities Resolution of the 3 -energy points Conclusions, current status, and further work Silver: determining the photon energies Experimental details for 1-BM XOR Determination of peak positions Determining the photon energies Recovering lost encoder readings Interpolating the photon energies Discussion Conclusion Conclusions Molybdenum Tin Silver Informing form-factor calculations X-ray absorption fine structure Conclusion xvii

18 Bibliography 283 A The weighted mean 297 B Interferometric thickness measurement 299 B.1 Recording the Interferogram B.2 Interferometric information B.3 Decoding an interferogram B.4 The finite-area Fourier transform technique B.5 Application of the Fourier technique B.6 Discussion and conclusion C Experimental design modifications 309 C.1 Foil holder C.2 Modified design for daisy-wheels: wedges D Resulting publications 317 xviii

19 List of Figures 2.1 Percentage difference between the mass attenuation coefficients tabulated in FFAST and XCOM for Z = 1 92, E = kev Comparison of values for the attenuation of molybdenum tabulated in XCOM, CXRO, and Brennan and Cowan against those of FFAST Measurements of the mass attenuation coefficient of molybdenum, presented as a percentage difference from the FFAST calculation Schematic of experimental design Relative uncertainty of the mass attenuation coefficient as a function of the absorber attenuation, as per Nordfors Relative uncertainty of the mass attenuation coefficient as a function of the efficiency of the ion chamber used for the measurement Ion chamber efficiencies across a range of photon energies for a variety of gases Relative efficiency for the detection of the fundamental and the third harmonic energies for a variety of gases Rocking curves for the detuning of the second crystal of the silicon monochromator Experimental configuration used at XOR beamline 1-ID to determine the mass attenuation coefficients of molybdenum between 13.5 and 41.5 kev Study of the count rates recorded using the upstream ion chamber, showing the removal of measurements affected by the incomplete shutter motion Number of outliers identified using the shutter motion filter, plotted as a function of m, the number of standard-deviations required for rejection Effect of the aperture clipping on the ratio of the count rates I d I u measured with no sample in the path of the beam Number of outliers identified using the aperture clipping filter, plotted as a function of m, the number of standard-deviations required for rejection Dark currents measured in the second downstream ion chamber, plotted as a function of the time at which the measurement was made Correlation between the counts recorded in the upstream and downstream ion chambers for the measurements made with a sample in the path of the beam xix

20 5.7 Correlation between the counts recorded in the upstream and downstream ion chambers for the measurements made with a sample in the path of the beam, presented as a function of the approximate sample attenuation Contribution to the uncertainty in the intensity ratio arising from the spread in the ratios of the intensities and the dark current uncertainties Incident intensity ratios determined from the counts recorded in the second downstream ion chamber Study of the unsaturated measurements used to inform the rejection of saturated measurements shown in Fig Application of the cut to the saturated measurements Further study of the unsaturated data to be used to apply a second cut to the saturated data Application of the second cut to the saturated data Interpolation of the determined incident intensity ratios to the moment at which the attenuated intensity ratio was measured Attenuations [ µ [t determined using Eq. (5.10) The percentage uncertainties of the attenuations Uncertainties for the determined foil masses Definition of foil geometry used for the calculation of the foil area Average integrated column densities [t and their associated uncertainties Schematic of the holder construction Attenuation profile ([ ) µ [txy of the sample mounted in the holder109 S+H 6.6 Uncertainties σ( [ µ [t)s+h,mea in the measured attenuation at every point in the x-ray scan, determined from the standard error of ten repeated measurements Illustration of the seven classes of location on the foil-plus-holder through which the x-ray beam may pass Results of fitting the attenuation profile presented in Fig Residuals of the fit to the attenuation profile Recovered absorber attenuation profile with the edge omitted Residuals for the low-energy full-foil mapping Counts recorded by the upstream ion chamber and the first downstream ion chamber during the full-foil mapping at 13.5 kev Attenuation [ µ [t as a function of measurement time for the full-foil mapping at 13.5 kev Results of an x-ray scan taken at 13.5 kev, processed to determine values of ln ( I I 0 ) xy = [ µ [txy in the neighborhood of the centre of the foil Atomic-force microscope scan of an 80 µm 80 µm region on the surface of one of the absorbers Ratio of the intensity recorded downstream of the sample to that recorded upstream when the sample was rotated about the ˆx (outboard) axis xx

21 6.17 Ratio of the intensity recorded downstream of the sample to that recorded upstream when the sample was rotated about the ŷ (vertical) axis Conventional axis designations of a six-circle diffractometer Examples of rocking-curves measured with saturation in the scintillator Rocking-curves of reflections from the (111) planes of the analyser crystal Rocking-curves of reflections from the (333) and (444) planes showing the presence of the secondary peaks of reduced intensity and separation compared to those observed in Fig Illustration of the influence of imperfections in the analyser crystal on the diffraction profile Predicted rocking-curves for the reflection of 25-keV x-rays from the (111), (333), and (444) planes of a germanium crystal Results of fitting to determine the location of the leading-edge of two rocking-curves measured with a fine interval Results of fitting to determine the location of the leading-edge of two rocking-curves measured with a coarse interval χ 2 r resulting from fitting the location of the leading edge of the rockingcurve profiles with a Lorentzian-slit profile, presented separately for the coarsely and finely-measured profiles One-σ error estimates resulting from fitting the location of the leading edge of the rocking-curve profiles with a Lorentzian-slit profile, presented separately for the coarsely and finely-measured profiles σ χ 2 r measure of uncertainty resulting from fitting the location of the leading edge of the rocking-curve profiles with a Lorentzian-slit profile, presented separately for the coarsely and finely-measured profiles Determination of the centres of mass of two finely-measured rockingcurves χ 2 r uncertainties determined by fitting the rocking curves with a Gaussian beam profile, presented separately for the coarsely and finelymeasured profiles Results of a linear extrapolation to determine a single energy Fitted offset parameters determined from the leading-edge locations and the centre of mass locations The difference between the fitted offset parameters shown in Fig is approximately constant at around 5 millidegrees Comparison of the energy determined using the leading edge and the centre-of-mass locations Results of the interpolation of the photon energies Uncertainties associated with the determined photon energies, plotted against the nominal synchrotron energy The mass attenuation coefficients [ µ calculated by dividing the measured attenuations by the average integrated column densities xxi

22 8.2 The percentage difference between the measured mass attenuation coefficients and the weighted mean value at each energy Percentage difference of the mass attenuation coefficients from the weighted mean at each energy after determining the local integrated column density of each foil Significance of deviations of the mass attenuation coefficients from the weighted mean after scaling. Significance is defined in Eq. (8.6) Measured values of the mass attenuation coefficient in the near-edge region Percentage differences between the mass attenuation coefficients [ µ and the weighted mean value at each energy in the region of the absorption edge Percentage differences between the mass attenuation coefficients [ µ and the weighted mean values after correcting for the effect of the finite bandwidth of the x-ray beam Sketch of discretely sampled Gaussian and Lorentzian beam energy profiles used to evaluate the effect of the bandwidth The percentage effect of the x-ray bandwidth on the measured mass attenuation coefficients: a 1.5 ev bandwidth of a 20 kev beam can affect the measured attenuation coefficient by one percent Modelling of the effect of fluorescence on the measured mass attenuation coefficients, assuming a 100% fluorescence yield Modelling of the effect of sample roughness on the determined mass attenuation coefficients The results of daisy-wheel measurements, showing a clear deviation from linearity of the measured attenuation as a function of foil thickness The χ 2 r statistic resulting from fitting for a fraction of harmonic components in the beam and for an error in the determined dark current Illustration of a variety of causes for the recording of an incorrect value of the dark current and the impact of the use of this incorrect value on the determined mass attenuation coefficient Simulation of the effect of an incorrect dark current value on the mass attenuation coefficients The percentage discrepancies of measurements from the weighted mean, after the dark current correction The percentage change of the determined mass attenuation coefficients resulting from the correction to the dark current values Incorporation of rapid or single-foil measurements Percentage difference of the measured values from the weighted mean value within a small range of the zero line Plots of the measured values showing a high degree of consistency over a wide range of energies Percentage difference from the FFAST tabulation of the mass attenuation coefficients measured in the region of the XAFS Percentage difference from the FFAST tabulated values of: previous experimental work; the XCOM tabulated values, and; this work xxii

23 8.23 Percentage discrepancy between various tabulated values of [ µ and pe this work The x-ray spectrum expected from the 12-BM beamline, calculated using the XOP package Simulation of the efficiency of a 180-mm long ion chamber with a variety of gases over a range of photon energies, including a comparison of the relative efficiencies of detection of the first-order and fourth-order multiples of the desired beam energy Schematic of the experimental layout Attenuations of tin absorbers determined using the counts recorded by the first downstream ion chamber Uncertainties of the attenuations determined from counts recorded by the first downstream ion chamber Attenuations of tin absorbers determined using the counts recorded by the second downstream ion chamber Tuning curves for the downstream crystal over the energy range from kev Difference between the nominal energy determined from the upstream (444) monochromator (using the Bragg equation, assuming no angular offsets) and that of the downstream (333) monochromator The attenuation profile measured using x-rays of energy kev The residuals of the fit of the 4-mm-wide x-ray beam to the attenuation profile measured at kev The attenuation profile measured using x-rays of energy 42.8 kev The determined photon energies The mass attenuation coefficients determined by use of the average integrated column density Percentage deviation of the mass attenuation coefficients from their weighted mean at each energy, after determining the local integrated column densities Percentage deviation from the FFAST tabulated values of the mass attenuation coefficient after determining the local integrated column densities Determined values of the mass attenuation coefficient of tin within a short range of the absorption-edge at 29.2 kev Incorporation of the 3 -energy measurements. The measurements agree 4 to within their % uncertainties Percentage difference from FFAST tabulated values for [ µ of: our values; the XCOM tabulated values, and; various previous experimental values Significance of deviations from the weighted mean. Measurements at higher energies show no systematic trends in this figure, indicating that the spread of the values (Fig. 9.14) is indeed due to reduced statistical precision Schematic of the experimental layout xxiii

24 10.2 Measured profiles of the x-ray reflections of nominal energy 35.2 kev from the silicon (SRM 640b) powder specimen The energy determination plot of d cos (δ/2) versus d sin (δ/2) for x- rays of nominal energy 35.2 kev Effect of a constant angular offset of the δ scale of the diffractometer axis Effect of a bimodal offset δ = 600 µrad, bimodal δ = 100 µrad in the angular scale of the diffractometer axis The effect of a misalignment of the powder capillary with respect to the centre of rotation of the δ axis Effect of a misalignment of the powder capillary by a distance z = 600 µm downstream of the centre of rotation of the δ axis Schematic of the effect of a powder offset in the vertical or y direction Effect on the energy determination plot of a misalignment of the powder capillary by a distance y = 600 µm above the centre of rotation of the δ axis Determining the energies from the measured reflections Determined values of the gradient of the energy determination plot at each measured energy Nominal x-ray energy presented as a function of the recorded monochromator encoder angles Interpolating the encoder readings Calibrating the monochromator encoder values to determine the x-ray energies Angular locations of the measurements, as a function of energy Measured values of the mass attenuation coefficient of silver Percentage difference from the FFAST tabulated values of previous experimental work, XCOM tabulated values, and this work Percentage difference between various tabulated values of [ µ for pe molybdenum and this work Percentage discrepancy between the FFAST tabulated values and the preliminary results of measurements of the mass attenuation coefficients of tin Percentage discrepancy between the FFAST tabulated values and the measurements of the mass attenuation coefficients of silver Comparison of the difference between the FFAST tabulated values and our measurements for molybdenum, tin, and silver Difference between the XCOM tabulation and our measurements of the mass attenuation coefficients of molybdenum, tin, and silver Measured values of the mass attenuation coefficients of molybdenum in the region of the XAFS B.1 Schematic of apparatus and geometry used to record interferograms. 300 B.2 Air-gap interferogram taken using one of the molybdenum foils B.3 Simulated interferogram with constant fringe spacing. The noise level is Gaussian distributed with variance equal to 5 times that of the signal303 xxiv

25 B.4 Absolute value of the Fourier transform of the simulated interferogram (Fig. B.3), showing two spikes whose positions determine the fringe density and direction B.5 Angular recovery from an interferogram obtained from a molybdenum foil C.1 New sample holder design used in a number of recent experiments C.2 Modelled attenuation measurements taken along the length of two wedge-shaped absorbers C.3 Design of the 7075 aluminium wedge and the 316 (stainless) steel wedge313 C.4 Design of a wedge holder to enable the wedge to be mounted on the sample stage C.5 Attenuation measurements made along the length of the two wedges using a beam of 28 kev x-rays xxv

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27 List of Tables 2.1 Evaluated limits for the scattering powers of the atomic electrons. For cases where the limiting value is zero the asymptotic form of the convergence is indicated in parentheses Number of outliers identified from the shutter motion filter, determined from the signature presented to each of the ion chambers Rejections identified using the aperture clipping filter determined for each of the downstream ion chambers Comparison of the energies determined using the leading-edge and the centre of mass techniques Reduction to the χ 2 r resulting from the correction of the bandwidth effect E 8.2 The fwhm bandwidth, E 105, when the systematic due to the finite x-ray bandwidth is corrected using Gaussian and Lorentzian profiles with a variety of parameter choices Improvement to the χ 2 r resulting from the dark current correction Tabulation of the determined values for the x-ray energy E, the mass attenuation coefficient [ µ, and the imaginary component of the atomic form-factor f 2 of molybdenum, with uncertainties Contributions to uncertainties in Table 8.4, with source specified. Further established limits for the systematic uncertainty are provided Values of [ µ, with 1-σ uncertainties, and χ 2 r determined by fitting the attenuation profile using various (integer-valued) beam widths Tabulation of the determined values for the x-ray energy E and the mass attenuation coefficient [ µ of tin, with uncertainties xxvii

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29 Chapter 1 Motivation and outline of this work The x-ray atomic form-factor describes the transform of the spatial and energetic structure of the atomic electrons and determines the optical properties of atoms, allowing the calculation of refractive indices, attenuation coefficients, and scattering cross-sections. The atomic form-factor is used to describe the operation of lenses, multilayer mirrors, diffractive crystals, and filters. Accurate knowledge of the formfactor can result in improved designs for these optical components and more accurate modelling for analysis in fields as diverse as medical physics [8, astronomy [9, planetology [10, 11, high-energy physics, crystallography, charge density studies, and x-ray imaging [12, in fact, any investigation involving the interaction of x-rays with matter. The imaginary component of the atomic form-factor f 2 determines the photoabsorption of x-rays by atoms. In the x-ray energy region (from about 1 to 100 kev) photoabsorption dominates the atomic cross section for medium and high-z atoms, typically representing over 90% of the total attenuation. Other significant components are Compton and Rayleigh scattering and, for crystalline materials, Laue- Bragg and thermal-diffuse scattering. Where photoabsorption is dominant, f 2 can be determined accurately from measurements of the mass attenuation coefficient [ µ, and used to test theoretical predictions of photoelectric absorption using bound-state electron wavefunctions [13, 14. Measurements of the attenuation of x-rays by materials provide a wide variety of other information about the fundamental properties of matter. In particular, relative and absolute measurements of mass attenuation coefficients are used to investigate the dynamics of atomic processes, including shake-up, shake-off, and Auger transitions [15, 16, 17, 18, and to provide information on the density of 1

30 2 Chapter 1. MOTIVATION AND OUTLINE OF THIS WORK electronic states [19, molecular bonding, and other solid-state properties [20. The diversity of these studies is evidence of the wide variety of processes that influence the attenuation of x-rays. In order to develop a deeper understanding of the interactions between x-rays and matter it is necessary to make accurate measurements, so that each attendant process may be studied and compared with theoretical models. While relative measurements are adequate for some applications, absolute attenuation measurements provide additional, crucial, and demanding tests of theoretical predictions. example, while finite-difference calculations have recently had significant success in predicting EXAFS on a relative scale [21, they are in relatively poor agreement with the results of absolute measurements [22. Measurement inaccuracy coupled with discrepancies between theoretical calculations seriously impede the understanding of x-ray interactions with matter. This work reports high-accuracy measurements of the mass attenuation coefficients [ µ of molybdenum, tin, and silver over a wide range of energies. These measurements are examined in detail for effects of systematic errors to ensure the accuracy of the determined values. X-ray atomic form-factors are calculated using atomic theory and quantum mechanics. Chapter 2 introduces the x-ray form-factor from a historical perspective. A necessary diversion is then made into the theory of the calculation of the atomic structure. Classical and quantum-mechanical form-factor theory is described, followed by a discussion of the S-matrix approach. We conclude with a description of the development of theoretic and semi-empirical tabulations from the late 1960 s to the present. Results of current tabulations are compared. Chapter 3 provides a detailed discussion of the principles and practical application of the measurement techniques employed in this work, and in reference to relevant literature. Particular discussions concern the production and monochromation of the x-ray beam, the use of matched ionisation chamber detectors to optimise correlation and statistics, sample attenuation levels, and steps taken both to minimise and to quantify systematic effects resulting from x-ray beam harmonic components and sample fluorescence. The mass attenuation coefficients are determined in For

31 3 a system using matched ion chambers and discuss some possible limitations of this approach. Chapters 4 through 8 are concerned with the determination of the mass attenuation coefficients of molybdenum using a synchrotron-produced x-ray beam. The analysis is reported in detail to enable proper assessment of the employed methods and the determined uncertainty estimates. The analyses presented in these chapters have resulted in three publications [23, 24, 25 which are included in Appendix D for convenience. General principles of the measurement technique are presented in chapter 3. Chapter 4 discusses those details particular to the adopted experimental configuration. Chapter 5 describes our use of the recorded x-ray count rates to determine the attenuation [ µ [t of a number of absorbers at over 500 photon energies spanning the range from 13.5 to 41.5 kev. Recent reports have observed that, at 0.5 2%, the dominant and limiting source of error in the measurement of mass attenuation coefficients is due to inaccuracies in the determination of the thickness of the absorber along the path traversed by the x-ray beam [26, 27, 28, 29, 30, 31, 32. Chapter 6 develops a full-foil mapping technique to overcome these limitations. The technique is employed to determine the value of the mass attenuation coefficient which is accurate to 0.028%. Chapter 7 reports the determination of the photon energies to an accuracy of better than 0.01% (equivalent to 1 ev at 13.5 kev and 4 ev at 41.5 kev) from measurements of the angular locations of x-rays reflected from a germanium single crystal. The photon energies are required to this high level of accuracy in order that the energy uncertainty not limit the measurement accuracy in regions where the mass attenuation coefficient varies smoothly with energy. The rapid energy variation of the mass attenuation coefficient near the absorption edge and in the region of the EXAFS results in the unavoidable dominance of the energy uncertainty. these regions our measurements retain their precision and provide detailed structural information. Chapter 8 examines the mass attenuation coefficients obtained over an extended range of the measurement parameter space for the effects of systematic errors. Equa- In

32 4 Chapter 1. MOTIVATION AND OUTLINE OF THIS WORK tions predicting the effects of various systematic errors are derived. In each case a parameter describes the magnitude of a systematic error, such as the fraction of harmonic components in the x-ray beam. A fitting routine is used to determine the presence of the systematic error and, if present, a corresponding correction to the observed values. In this manner we treat effects relating to foil thickness variations (requiring -0.4 to +0.8% corrections), the finite bandwidth of the x-ray beam ( % corrections), and the dark current value (0.01 to 0.1% correction). With these corrections we have been able to determine mass attenuation coefficients of molybdenum with accuracies of 0.02 to 0.15%. The measured values are presented and the error-budget is discussed. Chapter 9 presents preliminary results of the determination of the mass attenuation coefficients of tin over the energy range from 29 to 60 kev. We use techniques developed in chapters 5 through 8, and therefore present only the main results of this analysis. The precision of the determined values is 0.1 1%. Chapter 10 reports the determination of the x-ray energies for a measurement of the mass attenuation coefficients of silver over the range of energies from 15 to 50 kev. For this measurement a powder-diffraction technique was employed using a six-circle diffractometer. The presence of a systematic offset in the measured values has required the use of a qualitative fitting model to determine the photon energies. The photon energies have been determined to an accuracy of 0.1%. We identify a number of limitations of the experimental technique and use these to modify the methodology. The improved methodology is used for measurements reported in chapters 7 and 9. Chapter 11 discusses our measurements of the mass attenuation coefficients of molybdenum, tin, and silver with relation to the tabulated values. We observe a consistent pattern of discrepancies from the values of the Form-Factor, Attenuation, and Scattering Tables (FFAST) tabulation [33, 34, 35 in the neighbourhood of the absorption edge, which indicates a systematic error in the calculation. A cause for the discrepancy is explored. The stability of the discrepancies indicates that the FFAST tabulation is well converged in the edge region. Other tabulations exhibit qualitatively similar but less consistent discrepancies, possibly indicating an

33 5 inconsistent application of convergence criteria. Several appendices present frequently-used formulae, the development of an algorithm and computer program to decode interference fringes, and modifications to the experimental design, which we expect to improve the accuracy of the full-foil mapping technique and to enable the linearity of the detection system to be probed. Appendix D contains reprints of several papers written by the author and arising directly from the work of this thesis. This work reports highly accurate measurements of the mass attenuation coefficients [ µ of molybdenum, tin, and silver over energies covering their K-shell absorption edges and regions of the EXAFS, XAFS, and XANES. The measurements are compared with various commonly-used tabulations of mass attenuation coefficients to investigate differences between tabulated and measured values. The measured values are found to be systematically higher than the tabulations in the region immediately above the K-shell absorption edges, in agreement with the observations of Kerr Del Grande [36, 37. However, similar systematic discrepancies observed below the K-shell absorption edges provide further clues as to the cause of the discrepancy. The systematic pattern of the discrepancies is most apparent from comparisons with the FFAST tabulation [34, 35, which exhibits improved stability in comparison to other tabulations. The measurements described herein have accuracies of up to 0.02% representing an improvement of one to two orders of magnitude over previous work. These accuracies have been achieved through our development of a number of techniques for overcoming a range of random and systematic sources of error in the measurement. Each of these techniques has been guided in the first instance by the application of the X-ray Extended-Range Technique (XERT) [22, which recommends making measurements over an extended range of every available dimension of the measurement parameter space. Our use of the technique aims to make measurements under optimal conditions but also extends these measurements out past the point where the conditions for good measurement break down. The nature of the breakdown provides information about the measurements made under optimal conditions, enabling corrections to be identified where necessary, and rigorously justifies the claimed measurement uncertainty.

34

35 Part I Current status 7

36

37 Chapter 2 Calculation of the mass attenuation coefficient It is possible to use the theory of quantum mechanics, atomic physics, and quantum electrodynamics to calculate values for the atomic form-factors as a function of photon energy. The imaginary component of the atomic form-factor can be related directly to the photoelectric absorption coefficient. Comparison of calculated and measured values of the mass attenuation coefficients can therefore test our understanding of theoretical approaches. In this chapter we provide an introduction to the theoretical basis for the prediction of mass attenuation coefficients. We discuss in detail modelling of the atomic structure which is required by all approaches. The atomic form-factor approach is discussed in some detail. Other approaches are outlined. We discuss a number of recent developments in the implementation of the form-factor approach and compare some commonly-used tabulations of mass attenuation coefficients. 2.1 Approximating the solid-state The interaction of a photon with an atom embedded in an environment is described by Ψ, γ, env I Ψ, γ, env, (2.1) where Ψ, γ, env represents the initial state of the atom Ψ, the incident photon γ, and the environment env, I represents the interaction, and Ψ, γ, env represents the final (possibly excited or ionised) state of the atom Ψ, the final photon γ, if it exists, and the environment env. When the environment is not changed by the interaction, Eq. (2.1) can be written Ψ env, γ I Ψ env, γ, (2.2) 9

38 10Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT where Ψ env and Ψ env represent the initial and final state of the embedded atom. Embedded in this case may refer to molecular bonding (of any sort), ionisation, or the presence of an external electric or magnetic field. The simplification represented by Eq. (2.2) certainly cannot be applied to photoelectric absorption in the EXAFS region, where an ejected photoelectron can interact with the environment to produce significant oscillatory structure in the interaction cross section. This process is discussed later in this thesis. It is difficult to calculate the embedded-atom state vector appearing in Eq. (2.2), although band-structure calculations and quantum chemical methods can treat such situations to various degrees of approximation. The influence of the atomic environment is often restricted to the valence electron arrangement which can be difficult to predict. The high density of valence energy states, difficulties in accurately modelling the environment, and errors inherent to the calculation of valence electron wavefunctions, are sensitive to approximations made for the inner shell orbitals. Environmental effects give rise to chemical shifts, which have many qualitative applications in the fields of Raman and XANES spectroscopy. When the environmental effects can be completely neglected, the photon-atom cross section is proportional to Ψ, γ I Ψ, γ. (2.3) Neglect of the effect of the environment on the atomic state vector is formally referred to as the isolated atom approximation. All tabulations employ the isolated atom approximation: when this approximation is not valid, poor agreement with the predicted cross section is to be expected. An example of the isolated atom approximation is the assumption that the absorbing power of a bulk material is a multiple of the isolated atom value. However, incorporation into a bulk material involves the redistribution of valence electrons into the conduction band. This assumption may invalidate detailed comparisons with the measured values in some (particularly optical) energy regions. All methods for describing x-ray-atom interactions require a description of the

39 2.2. THE ATOM: ELECTRONIC STRUCTURE CALCULATIONS 11 (atomic) wavefunction. calculations. In section 2.2 we describe methods for performing these 2.2 The atom: electronic structure calculations The total atomic wavefunction Ψ is an energy eigenstate and is therefore a solution of H Ψ( R 1,..., R N, r 1,..., r n ) = E Ψ( R 1,..., R N, r 1,..., r n ), (2.4) where the R I are the nucleon coordinates, r i are the electron coordinates, and H is the Hamiltonian operator, which can be written as a sum of the nucleonic, electronic, and mixed nucleon-electron interaction terms H = T n + V n n + V n e + T e + V e e (2.5) where T n = N I=1 T e = 2 I 2M I, V n n = V n e = n i=1 N I=1 n i=1 2 i 2, V e e = N I<J Q I Q J R IJ + V strong, Q I R I r i, (2.6) n i<j 1 r ij + V L.S. T n and T e describe the kinetic energy of the nucleons and of the electrons, respectively. V n e describes the nucleon-electron Coulomb interaction, where Q I nucleon charge, +1 for protons and 0 for neutrons. inter-nucleon and the inter-electron forces respectively. is the V n n and V e e describe the For the purposes of this introduction we neglect all interactions except Coulomb interactions between the point nucleus and the electrons, and electron-electron terms. Nuclear structure and transitions (T n and V n n ) can result in significant interactions in the MeV energy regime [38. Treating the nuclear mass as infinite relative to the electron mass and photon energy ignores reduced-mass effects which

40 12Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT can be significant for low-z atoms. Spin-orbit coupling (V L.S ) and further small terms can be treated as perturbations once an approximate solution is obtained. We therefore consider the solution of the eigenvalue equation for the electrons only H Ψ( r 1,..., r n ) = E Ψ( r 1,..., r n ), (2.7) with the Hamiltonian H = n i=1 2 n i 2 Z + r i i=1 n i<j 1 r ij, (2.8) which operates only on the position coordinates of the electrons. The terms in this Hamiltonian represent the electron kinetic energy, the nuclear attraction, and the inter-electron repulsion, respectively. The first two of these terms are single particle terms, depending (in this approximation) only on the electron wavefunctions. A Hamiltonian involving only these terms can be solved exactly. However, the third term is an interaction term, coupling each electron to all other electrons in the system, which prohibits Eqs (2.7) and (2.8) from being solved exactly. When the interaction term is sufficiently small it can be treated perturbatively, by expanding Eqs (2.7) and (2.8) as H Ψ = H 0 Ψ + λh Ψ ( n 2 n i = 2 i=1 i=1 ) ( n Z + V ( r i ) Ψ + r i i<j ) 1 V ( r i ) Ψ r ij (2.9a) (2.9b) = E Ψ, (2.9c) where V ( r i ) is a single-particle potential chosen to make each of the [ 1 r ij V ( r i ) small. Perturbation theory can be used to solve Eq. (2.9) by choosing V ( r i ) so that H is small. The choice V ( r i ) = V (r i ), i.e., where V does not depend on the angular coordinates of the electrons, is known as the central field approximation. The independent particle approximation (IPA) involves writing the n-electron

41 2.2. THE ATOM: ELECTRONIC STRUCTURE CALCULATIONS 13 wavefunction as a product of one-electron wavefunctions, i.e., Ψ = ψ 1... ψ n. (2.10) This separation of the wavefunction allows the perturbation term in Eq. (2.9) to be evaluated using single-electron operators. Hartree s method of the self-consistent field obtains a solution to Eqs (2.7) and (2.8) by iteratively evaluating a static interaction Hamiltonian (where the interaction term is fixed by the current electron locations) until the wavefunction converges. The original method of Hartree did not include the fermionic statistics appropriate for electrons. These are included in the Hartree-Fock (HF) method, by writing the electron wavefunctions as an antisymmetrised product of one-electron spinorbital wavefunctions. In particular, such an antisymmetrised product is known as a Slater determinant and is written as Ψ(q 1, q 2,..., q n, ) = 1 n! u α (q 1 ) u β (q 1 ) u ν (q 1 ) u α (q 2 ) u β (q 2 ) u ν (q 2 ), (2.11)... u α (q n ) u β (q n ) u ν (q n ) where the q i represent the spin and space coordinates of the i th electron, and α, β,..., ν each represent a set of four quantum numbers (n, l, m l, m s ). Solving the Hartree-Fock equations requires ν times more computation than solving the Hartree equations, because the self-consistent field must be calculated for each term in the Slater product The Hartree-Slater (HS) approximation reduces the amount of computation considerably by replacing the electron-electron interaction term with its average radial value [39. That is, V (r) = 1 V ( r) sin θ dθ dφ. (2.12) 4π φ θ We emphasise that HS wavefunctions are not more accurate than HF wavefunctions,

42 14Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT but that they are far simpler to compute. When computational power is limited, HS wavefunctions are an attractive solution to the atomic structure. Variational techniques can be used to refine estimates of the ground-state wavefunctions. The variational principle states that, for a ground-state energy E 0 and normalised trial wavefunction Φ, E 0 E[Φ = Φ H Φ. (2.13) Applying these techniques to Eq. (2.9), the energy of the electron-electron interaction can be written as a sum of two terms [40 Φ 1 Φ = 1 [ u λ (q i )u µ (q j ) 1 r ij 2 r λ µ ij u λ(q i )u µ (q j ) u λ (q i )u µ (q j ) 1 r ij u µ(q i )u λ (q j ) (2.14) = 1 [J λµ K λµ, (2.15) 2 λ µ where J λµ and K λµ are the direct and exchange energies, respectively. Here the direct term describes the energy associated with interactions between the electrons, and the exchange term describes that associated with the antisymmetrisation of the wavefunction. We have so far described approximate solutions to the atomic Schrödinger equation using a combination of Hartree, Hartree-Fock, Hartree-Slater, variational, and perturbation techniques. In these approaches, the wavefunction ψ describes the electrons. An alternate calculational scheme, the density functional theory (dft), employs instead the electron density ( r) to perform this function. Further, the electron density is considered to determine the properties of the system, just as much as the properties of the system determine the electron density. Such an approach is perhaps explained by considering that the electron interaction term 1 r ij appearing in Eq. (2.8) is both determined by and yet determines the electron wavefunctions. The study of such relationships is referred to as a functional theory, and such a study with respect to electron density, density functional theory.

43 2.2. THE ATOM: ELECTRONIC STRUCTURE CALCULATIONS 15 Some time before the development of dft proper, Thomas and Fermi developed a toy atomic model in which bound atomic electrons were treated as an electron gas [41. Using this model they recast the discrete many-body problem of the atomic structure as a problem involving only continuous distributions. Although the Thomas-Fermi model ignores the discrete properties of the system, it can be used to determine the electron energy levels using only the (local) electron density (that is, ignoring interaction terms) to modest accuracy. This local density approximation (LDA) is the continuous version of the independent particle approximation presented in Eq. (2.10). In 1964 Hohenberg and Kohn [42 proved that the ground-state atomic properties of an inhomogeneous electron gas could be described exactly as a functional of the electron density, and developed a variational principle for determining the ground state of the system. This promising theorem was very simple to prove and renewed interest in the Thomas-Fermi model. The form of the density functionals is not provided by the theory. However, the dft variational principle involves minimisation of [42 E v [ = T [ + V e e [ + ( r)v( r) d r, (2.16) and can be used to determine the ground-state energy and electron density of the system. In this equation, E v [, T [, and V e e [ are the ground state energy, the kinetic energy, and the (interacting particle) potential functionals, respectively. v( r) is an external potential, which is used to describe the nuclear locations and the structural framework of the system, but which can also contain non-universal components of the energy functionals. T [ and V e e [ are universal functionals of the density, as they depend only on the local value of the density and not on the external potential v( r). Kohn and Sham developed a dft version of the self-consistent field [43 and showed that these could describe electron-electron interactions in terms of local single-particle (non-interacting) density functionals [44. They demonstrate that the method can be used to describe exchange exactly. The central assertion of

44 16Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT the Kohn-Sham scheme is that there exists a non-interacting potential v n ( r) whose associated density n is identical to any density i resulting from an interacting particle potential v i ( r). The Kohn-Sham theorem states that exact solutions to the atomic structure can be found by allowing the density functionals to vary from their interacting particle values. Using this formalism a number of interactions have been calculated exactly [45. Interestingly, despite the clear physical interpretation given to the non-interacting density n = i, the non-interacting potential v n ( r) and the noninteracting wavefunction Ψ n have no corresponding interpretation. The density functional approach promises that many-body interactions can be reduced to a single particle problem. Density functional techniques are routinely applied to the description of atomic and molecular systems consisting of many (several hundred) atoms [46. Of particular ongoing concern and research for the dft is the restriction imposed by the use of the local density approximation, the inclusion of further interactions (correlation, spin, spin-orbit coupling), and relativistic effects [45, 46, Background: history of x-rays and diffraction The history of the atomic form-factor reflects the development of our understanding of x-rays and the structure of atoms, molecules, and crystals. This rich history has left the atomic form-factor with a variety of definitions, nomenclatures and notations. Here we present some of these developments so as to contextualise the terminology used to describe the atomic form-factor. The history presented here is intended to be neither complete nor rigorous and in no way represents a search for origins. The early diffraction experiments of M von Laue and W H Bragg were explained using two very different models of the interaction of x-rays with crystals. Laue conceived of x-ray diffraction as a scattering of x-ray waves from a 3-dimensional diffraction grating. Viewing the crystal as a volume of point scatterers, he employed a transmission or Laue geometry for his investigations. By contrast, Bragg conceived of this interaction in terms of the reflection of x-rays from planes within

45 2.4. THE X-RAY ATOMIC FORM-FACTOR 17 the crystal, and accordingly adopted a reflection or Bragg geometry. These very different models of the fundamental diffraction process were reconciled by Darwin, who showed that a rigorous treatment of the Laue scattering model of diffraction leads naturally to Bragg s reflection model nλ = 2d sin θ LB = 2a 0 sin θ LB h2 + k 2 + l 2 (2.17) which describes the angular positions θ LB of the diffraction maxima reflected from a cubic crystal. Equation (2.17) allows the wavelength λ of the diffracted x-rays to be determined in relation to the lattice spacing d. Further structure in the diffraction patterns, in the form of the observed relative intensities of the diffraction maxima, is not explained by Laue-Bragg diffraction. The early theory of the atomic form-factor was developed to describe the distribution of diffracted intensities. Other influences include the Debye-Waller factor and the Lorentz polarisation factor. The merits of various models explaining the diffracted intensities were hotly debated in the literature of the time as can be seen, for example, in the 1933 review article of Blake [48. W L Bragg recognised that the diffracted intensities could be explained by allocating to each atom a scattering power equal to its atomic (or electronic) number. Simple as it may seem, Bragg s observation represents the lowest-order approximation to the atomic form-factor, and is formally stated as f = Z. (2.18) Equation (2.18) implicitly describes the atomic form-factor in units of the equivalent scattering from a free, Thomson electron. 2.4 The x-ray atomic form-factor From the outset the atomic form-factor is defined by f = scattering power of an atom scattering power of a free electron, (2.19)

46 18Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT and is expressed in units of [electrons / atom. Bragg s point atom approximation [Eq. (2.18) was extended to include effects of the spatial distribution of electron scatterers around the atom. In crystallography it is natural to suppose that this spatial distribution will be associated with a further angular dependence of the intensities of the diffracted x-rays. By 1925 atomic theory had developed sufficiently for D R Hartree to calculate f-curves using the Bohr theory of the electron orbits [49, 50, 51. These f-curves describe the effective number of electron scatterers in the atom for a given scattering momentum transfer. Hartree considered an atom with purely radial electron density ( r) = (r), and so calculated his f-curves using f = V ( r)e i α. r dv = 0 4πr 2 sin αr (r) dr, (2.20) αr with α = k k 0 = 4π sin θ, (2.21) λ where k 0 and k are the incident and scattered photon momenta, respectively, α is the momentum transfer, θ = φ/2, where φ is the angle through which the x-ray is scattered, and λ is the wavelength of the radiation. Equation (2.18) is obtained in the zero-angle limit because of the absence of path-difference effects in the forwardscattering direction. This model of the atom provided insight into the phenomena of diffraction, but was unable to describe other interactions between light and matter. The electromagnetic theory describes a wide range of phenomena by treating the atom as an harmonic oscillator, with photon interactions modelled as: (a) forced oscillations in the atomic electron resulting from the interaction of the incident (electromagnetic) wave with the charge of the electron, accompanied by; (b) the emission of an electromagnetic wave by the oscillating electron, in accordance with the theory of accelerating charges. The properties of the total resultant field, being the sum of the incident and radiated fields, successfully described refractive indices, dielectric properties, and emission in the optical regime. Normal dispersion refers to the situation where the refractive index of a material

47 2.4. THE X-RAY ATOMIC FORM-FACTOR 19 decreases with the frequency of light. This effect gives rise to the splitting of light into its component colours as it passes into a glass prism. However, the dispersion function behaves anomalously within a narrow range of frequencies close to the natural frequency of the atom, and in this region the refractive index increases as the frequency of the light decreases. Normal and anomalous dispersion are satisfactorily described by classical theory in terms of the resonance of atomic, electronic harmonic oscillators. The classical, damped harmonic oscillator theory describes the scattering power of each atomic electron, f e, by f e = ω 2 ω 2 ω 2 e iκ e ω 1 1 ( ω e ω ) 2 i κ e ω, (2.22) where ω e and κ e are the natural frequency and the (radiative) damping associated with the electron. This description of the electron scattering powers is very different from the fixed and unitary value allocated to each of the electrons in the form-factor model presented in Eq. (2.20). The real and imaginary components of the scattering power can be written Re(f e ) = ω 2 (ω 2 ωe) 2 (ω 2 ωe) (κ e ω) 1 ( ωe 2 ( 1 ( ω e ω ) 2) 2 + ( κ e ω ), (2.23) 2 ω ) 2 and Im(f e ) = κ e ω 3 κe (ω 2 ωe) (κ e ω) ω 2 ( 1 ( ω e ω ) 2) 2 + ( κ e ω ). (2.24) 2 We have evaluated limits for the real and imaginary components of the electron scattering powers [Eqs (2.22), (2.23) and (2.24), and present these in Table 2.1. In the high energy limit ω ω e the real component of the scattering power of the electron is equal to that of a free electron. In this limit the imaginary component is zero. The high-energy limit returns the Thomson scattering condition as the electrons are effectively free when the photon energy is far greater than the binding energy of the electron. In the low energy limit ω ω e both components of the scattering power are zero, and the electrons are so tightly bound to the atom that they do not respond to the incident photon. In the region where ω ω e the imaginary component of the

48 20Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT Table 2.1: Evaluated limits for the scattering powers of the atomic electrons. For cases where the limiting value is zero the asymptotic form of the convergence is indicated in parentheses. in the limit as... Re(f) Im(f) ω 0 0 ( ω 2 ) 0 ( ω 3 ) ω 1 0 ( ω 1 ) ω ω e 0 ( (ω ω e )) ω e κ e scattering power dominates and the electron becomes highly absorbing. The Lorentzian line-shape of the characteristic emission lines is in good agreement with the predictions of the classical, harmonic oscillator model. In the quantum model, these resonances are reinterpreted as bound-bound and continuum-bound transitions within the quantum-mechanical atom, describing anomalous dispersion and characteristic emission lines. Absorption edges, which were not explained in the classical model, are naturally incorporated within the quantum model as boundcontinuum transitions. The transformation to a (semi-) quantum-mechanical description of the atom necessitates the rewriting of Eq. (2.22). Following [52 we define the oscillator function g(ω s ) to represent the number of oscillators with characteristic frequency ω s, and the oscillator density by dg. The scattering power of a single atomic electron dω [Eq. (2.22) becomes f e = ω e ( dg ) dω e 1 ( ω e ω ) 2 i κ e ω dω (2.25) where the integral extends up from the energy level of the electron ω = ω e the dissociation limit, ω =, corresponding to the allowed atomic absorption resonances. Equations (2.20) and (2.25) treat the interaction of x-rays with atoms in two particular situations, describing the influence of the spatial and energetic structure respectively. These complementary treatments can be combined to provide an expression for the atomic form-factor as the sum of the (spatial) contribution from each of the electrons weighted by their (energetic) scattering powers [53 to f = e { V e e i α. re dv ω e ( dg ) dω e 1 ( ω e ) 2 ω i κ e ω } dω. (2.26)

49 2.4. THE X-RAY ATOMIC FORM-FACTOR 21 e in this equation is the spatial density of each electron, which we shall discuss in great detail in the next section. The (nonrelativistic) Thomas-Reich-Kuhn sum rule e ω e ( dg ) dω e 1 ( ω e ω ) 2 i κ e ω dω = Z (2.27) informs us that the sum of the scattering powers of all atomic electrons is equal to the number of electrons, even though the scattering powers of the individual electrons may be very different from unity. In the high-energy limit the electronic scattering powers each tend to unity [Eq. (2.25), Table 2.1 and the atomic scattering factor reduces to the spatial form-factor component described by Eq. (2.20). Definitions & notation The atomic form-factor is a single parameter describing the strength of atomic transitions and the structure of the atomic electron orbitals. However, it can be broken down into several components which isolate these different properties. The total complex form-factor is written as f = f + if, (2.28) where f and f represent the real and imaginary components, respectively. However, the scattering powers and the spatial influences on the form-factor are still combined in the real component. In order to further isolate the real part of the anomalous component of the form-factor, the high-energy limit of the form-factor is isolated into the quantity f 0 given by { ( dg ) } f 0 = lim e e i α. re dω e dv E e V ω e 1 ( ω e ) 2 dω (2.29) ω i κ e ω = e e i α. re dv. (2.30) e V We have obtained Eq. (2.30) from Eq. (2.29) by use of the high energy limits of the atomic scattering factor shown in Table 2.1. In the high energy limit the atomic

50 22Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT electrons are effectively free, and we recover the non-resonant structure of the atom, leaving only spatial components. The non-resonant f 0 can be isolated from the atomic form-factor f to determine the resonant (or anomalous ) components f = f 0 + f + i f. (2.31) The four quantities f, f 0, f, and f defined here have many uses. Where the atomic structure is not relevant, the anomalous components f and f describe the optical properties of materials. This includes most cases of the reflection, refraction and absorption of x-rays. However, when the spatial distribution of the atomic electrons is relevant to the investigation, such as in the case of diffraction already discussed, the total form-factor f is required. The notation used to represent the real and imaginary components of the formfactor [Eq. (2.28) and the real and imaginary components of the anomalous scattering factor [Eq. (2.31) was altered in a 1974 meeting on Anomalous Scattering, organised by the International Union of Crystallography (IUCr) and reported in [54. At this meeting it was decided that the definitions of Eq. (2.28) would be discarded and that Eq. (2.31) would become f = f 0 + f + if. (2.32) In a follow-up meeting by the same group [55, the term anomalous scattering was dropped in the interests of accuracy, as the process is in fact well understood. It was agreed that resonant scattering was a more appropriate term for describing the resonance behaviour of f and f in the vicinity of the absorption edges. 2.5 The modified form-factor Investigators of high-energy scattering have observed that the form-factor f does not produce the correct result in the high energy limit [56. In particular, the zero-angle, infinite energy limit of the Rayleigh scattering cross-section determined by Levinger and Rustgi [57 from the limits of the dispersion equation was inconsistent with the

51 2.6. THE S-MATRIX APPROACH 23 limits of Rayleigh scattering calculations. Brown and Mayers [56 solved the problem by way of a semi-empirical investigation, introducing the reduction factor into the Rayleigh scattering expression. correction to the electron binding. The Brown-Mayers factor ( mc2 ) is a relativistic E+V Such relativistic binding effects were included analytically by Goldberger and Low [58 and Florescu and Gavrila [59 within a variety of formalisms. Schaupp et al. [1 have used the Brown-Mayers formulation to calculate relativistic modified atomic form-factors (MFF) for momentum transfers α = Å 1. Their work was concerned solely with elastic scattering factors. Applying their analysis to our concern, the expression for the complex atomic form-factor [Eq. (2.26), would require the inclusion of the Brown-Mayers factor, giving where E e g(α) = e { V e e i α. re m e c 2 E e V ( r) dv ω e ( dg ) dω e 1 ( ω e ) 2 ω i κ e ω } dω, (2.33) = ω e + m e c 2 is the (relativistic) electron binding energy and V ( r) is the potential in which the electron moves. Kahane has employed Eq. (2.33) to evaluate MFFs for a number of heavy elements (70 < Z < 100) using improved multiconfiguration Dirac-Fock wavefunctions [60. No current tabulation of formfactors in the x-ray regime uses the MFF as relativistic effects for photon energies significantly below the electron rest mass energy (about 500 kev) are very small. 2.6 The S-matrix approach An alternative formalism for describing the photon-atom interaction is known as the S-matrix approach. This approach uses the same atomic wavefunctions but describes the photon-atom interactions using bound-state QED. M nκmj In the S-matrix approach of Kissel et al. [61, 62, the second-order matrix element for Rayleigh scattering of a photon of energy ω scattering from a bound electron of energy E nκ in the state nκm j is evaluated using [61 M nκmj = p ( nκmj A p p A nκm j E nκ E p + ω + nκm ) j A p p A nκm j, (2.34) E nκ E p + ω

52 24Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT where A and A represent the photon creation and annihilation operators, respectively, and is a sum-integral over all intermediate atomic states p of energy E p p. Creagh has observed that Eq. (2.34) is qualitatively very similar to Eq. (2.25), despite the difference of the S-matrix approach [36, 37, 63. Equation (2.34) describes the polarisation states of the incident and scattered photons which are not isolated in the form-factor theories. The consequences of such investigations have been reported, and show new angular dependences of the elastic scattering factor [61, 64, 65. However, Creagh [36 has observed that this is of limited utility as experiments are generally not sensitive to x-ray polarisation states, and that the calculated polarisation dependence must therefore be averaged for comparison with experiment. The evaluation of Eq. (2.34) is computationally expensive. Accordingly, recent efforts by Kissel et al. [61, 62 to predict Rayleigh scattering amplitudes have used the S-matrix formalism only for inner-shell electrons: modified form-factor calculations [1 have provided outer-orbit contributions [61, 64. The difficulty of evaluating these matrix elements presents a serious obstacle to the application of the S-matrix approach which, if surmounted, may lead to an improved understanding and a more exact description of x-ray scattering and absorption. However, it is likely that the accuracy of the S-matrix formalism will be limited by the accuracy of the atomic wavefunctions input into the calculation. Current implementations of the S-matrix formalism introduce additional simplifications, even for K-shell electrons, of the nature of those used in the form-factor approach. S-matrix investigations have been used to study the assumptions of the formfactor approach [66, including the angular dependence of the elastic scattering factor [61, 64, 65. These investigations have revealed limitations of the form-factor construction. However, such limitations have little impact on the present concern, the prediction of mass attenuation coefficients and photoelectric absorption coefficients. With the current difficulties associated with the evaluation of S-matrix predictions it is not likely that an S-matrix tabulation of mass attenuation coefficients will appear within the near future.

53 2.7. X-RAY INTERACTIONS WITH MATTER X-ray interactions with matter The probability P that a single photon of energy E will be transmitted through the material of integrated column density [t is described by the Beer-Lambert absorption law P = I I 0 = exp { [ µ [t }, (2.35) where the ratio of the transmitted I and incident I 0 intensities expresses the transmission probability when many photons are incident upon the material. The mass attenuation coefficient [ µ describes all processes whereby a photon is absorbed ( destroyed ) or scattered (has its momentum altered). The mass attenuation coefficient can be written as a sum of absorbing and attenuating processes. In the kev photon energy range, the significant atomic cross sections are represented by photoelectric absorption [ µ, Compton scattering pe [ µ C, and Rayleigh scattering [ µ. Pair and triplet production, nuclear Thomson R scattering and Delbrück scattering contribute insignificantly in this energy range, and so we write the mass attenuation coefficient as [ µ [ = µ pe + [ µ C + [ µ (2.36) R for a disordered material. When the material is ordered, the phases of coherently scattered photons will interfere. The appropriate formula to use in this circumstance depends on whether the interference is constructive or destructive. When constructive interference occurs, the Rayleigh cross-section must be replaced by the Laue-Bragg scattering cross-section [ µ [ µ LB [ = µ pe + [ µ C + [ µ. (2.37) LB When the phases add destructively, the appropriate Rayleigh component of the crosssection is thermal diffuse scattering (tds). tds results from the thermal motions of the atoms about their crystalline positions, and can be thought of as scattering from the thermal disorder in the crystal. The mass attenuation coefficient is then

54 26Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT written [ µ [ = µ pe + [ µ C + [ µ. (2.38) tds The photoelectric absorption coefficient The photoelectric absorption coefficient [ µ is related to the imaginary component pe of the atomic form-factor f 2 by (see, fore.g., [67) [ µ pe = 2hcr e EuA f 2, (2.39) where E is the photon energy in ev, u the atomic mass unit, A the relative atomic mass of the absorbing atom, h the Planck constant, c the speed of light, and r e the classical electron radius. Inelastic scattering Incoherent or Compton scattering can be calculated by integrating the Klein-Nishina formula [ 1 µ C = πr2 e ua cos 2 θ + k2 (1 cos θ) 2 1+k(1 cos θ) [ 1 + k(1 cos θ) 2 S(q, Z) d(cos θ), (2.40) where q is the momentum transfer, Z the atomic number, k = ħω mc 2, and S(q, Z) is the incoherent scattering function which includes electron binding effects. Rayleigh scattering In the form-factor approximation the Rayleigh scattering cross-section is [63 [ 1 µ R = πr2 e (1 + cos 2 θ)f 2 (q, Z) d(cos θ), (2.41) ua 1 where r e is the classical electron radius, φ the angle through which the photon is scattered, f(q, Z) is the atomic scattering factor whose arguments are the momentum transfer q = sin (φ/2) λ and the atomic number of the target atom Z. The reader is referred to a review article by Kane et al. [64 for a discussion of the limitations of Eq. (2.41), and in particular the use of the modified form-factor to include relativistic effects at high photon energies.

55 2.7. X-RAY INTERACTIONS WITH MATTER 27 Laue-Bragg scattering Laue-Bragg scattering describes the interference of photons elastically scattered from a crystalline lattice. The Laue-Bragg scattering cross-section for the reflection of photons of wavelength λ from the h,k,lplanes of a crystal with multiplicity m hkl is given by [ µ LB = r2 eλ 2 [ 1 + cos 2 θ m hkl d hkl F hkl, 2 (2.42) 2NV c ua 2 hkl where N is the number of atoms in a crystal with unit-cell volume V c and N F hkl = f n exp { 2πi(hu n + kv n + lw n ) }, (2.43) n=1 with (u n, v n, w n ) describing the locations of each atom within the unit cell [68. Thermal-diffuse scattering When the scatterer is an ordered, crystalline substance and is explicitly misaligned from the Laue-Bragg scattering orientation, the dominant elastic scattering mode is thermal diffuse scattering (tds) σ tds = r2 eλ 2 [ 1 + cos 2 θ m hkl d hkl F hkl 2( 1 e 2M), (2.44) 2NV c 2 hkl where M is the Debye-Waller temperature of the crystal. Equation (2.44) is identical to Eq. (2.42) except for the inclusion of the Debye temperature factor, indicating that tds is qualitatively the same as Laue-Bragg scattering except that the thermal disorder dominates the intensity contributions. High-energy processes: Delbrück and nuclear scattering Delbrück scattering refers to the scattering of the photon from virtual electronpositron pairs formed in the atomic Coulomb field. Delbrück scattering can occur when the incident photon energy rises above the rest-mass energy of the positronelectron pair, 2m e c 2, and dominates the atomic scattering cross-section at energies above 100 MeV.

56 28Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT X-ray-electron interactions dominate the observed cross sections at lower energies. However, nuclear resonant scattering can lead to dramatic increases in the cross-section. These giant dipole resonances [2, 3, 4 are due to bound-bound and bound-free nuclear transitions, and generally occur at photon energies in excess of 10 MeV. Nuclear charge scattering or nuclear Thompson scattering occurs by virtue of the charge distribution of the nucleus, and is often modelled using a simple form-factor description of the nuclear structure with σ NT = ( ) 2 ( ) m r 0 M f nucleus 1 + cos 2 φ, (2.45) where m is the electron mass, M the nuclear mass, f nucleus the nuclear form-factor and φ the scattering angle. Further effects can be included to better describe the nuclear scattering factor. We do not discuss them here as they are insignificant in the x-ray energy region. Near-edge structure Structure in the attenuation coefficient is observed near and above absorption edges. Edge Structure (XANES) and the X-ray Absorption Fine Structure (XAFS). XAFS results from the self-interference of the ejected photoelectron. This structure results from a number of effects including the X-ray Absorption Near- Semiclassically, the photoelectron is ejected with energy E e = E x-ray E ionisation and an associated de Broglie wavelength λ e = hc E x E ionisation. (2.46) Elastic scattering of the electron from the neighbouring atoms results in a partially reflected electron wave which interferes with the outgoing electron wavefunction, thereby modifying its amplitude and leading to changes in the photo-ionisation probability. The dependence of this effect on the electron wavelength leads to sharp oscillations of the mass attenuation coefficient above the absorption edge when the

57 2.8. RESULTS OF VARIOUS THEORETICAL APPROACHES 29 distribution of the reflecting sites about the target atom is uniform throughout the material. The energy dependence of the XAFS has been interpreted to determine the approximate distance and orientation of scattering atoms located around a target atom [20, and has since been used in many quantitative studies. The interpretation of the XAFS spectrum involves a variety of normalisation procedures which enables the technique to be used with a variety of measurement strategies. However, these normalisation procedures prevent critical assessment of the quality of XAFS results and the further development of the interpretation. The quantified accuracy of the measurements reported in this work provide an experimental reference by which XAFS techniques might be tested. Joly has recently published the details of a finite-difference method for the calculation of near-edge structure (FDMNES) [21, which can be used for predicting the XAFS. Absolute XAFS measurements for silver [69 have been compared to the results of the FDMNES calculation [70, showing some progress toward the resolution of these issues. XANES results from bound-bound transitions in the atomic electron wavefunction, and accordingly is observed when the x-ray energy is slightly below the ionisation or K-edge energy. The XANES amplitude is strongly influenced by the density of the unoccupied higher-shell orbitals, and is used in qualitative studies to probe bonding rearrangements and annealing relaxation [71. Strong XANES amplitudes ( white-line transitions) have been observed in noble metals where partially occupied d 3 2 and d 1 2 levels lie above the Fermi energy [72. Recent XANES studies have been able to resolve the effects of bonding and of low-energy photoelectron scattering observed in the near-edge region [73. These investigations report accurate modelling of the observed spectra within a large (40 ev) energy range of the absorption edge. 2.8 Results of various theoretical approaches The primary interest of this thesis is mass attenuation coefficients and photoelectric absorption coefficients in the x-ray regime. Accordingly, we discuss developments

58 30Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT in the evaluation of the atomic wavefunctions and photoelectric absorption. Many developments of the prediction of atomic wavefunctions appear in other contexts, and we will pay attention to these as appropriate to our main goal. With the development of computational techniques and computers, it was possible to determine solutions to the atomic wavefunctions over a wide range of Z and energy in a reasonable amount of time. These calculations in turn enabled the calculation of the photoelectric absorption cross-sections. For a thorough discussion of the practical calculation of atomic structure using the methods of the self-consistent field we commend Herman and Skillman s, Atomic structure calculations [74. They discuss numerical methods required at each stage of the calculation, including the practical application of Hartree s self-consistency, calculational convergence, and starting values used in the self-consistent field calculation. Their work pays attention to nuts and bolts issues and provides a good foundation for understanding issues raised in the subsequent literature in this field. Atomic scattering factors calculated by Herman and Skillman using HS (Hartree- Slater) wavefunctions [75 were published next to calculations of Cromer et al. employing Hartree wavefunctions [76. The results of these two tabulations differ by 1-5% over the range of momentum transfer from 0 < α < 2. Early contributions to the theory of the calculation include Rakavy and Ron [77, who employed a relativistic Thomas-Fermi wavefunction, and Schmickley and Pratt [78, who used relativistic multipole expansion and included Coulomb correlation, or screening. Hubbell [79, 80 provides some discussion of a number of other contributions to the literature. In 1965 Liberman et al. [81 calculated wavefunctions using a relativistic DHS approach. Their approach was based on that of Herman and Skillman. Gavrila [82 showed how to include relativistic effects in the calculation of a fully-relativistic photoelectric absorption cross section. Brysk and Zerby [83 developed a procedure for evaluating the photoelectric cross-sections at lower energies, and applied their method to the relativistic wavefunctions calculated by [81. In 1970 Cromer and Liberman [84 evaluated self-consistent Dirac-Slater wavefunctions with Kohn-Sham exchange potential, 2/3[( r) 1/3. Experimental energy

59 2.8. RESULTS OF VARIOUS THEORETICAL APPROACHES 31 levels were used in preference to calculated energy levels [85. The computer code of [83 was used to evaluate the atomic scattering factors from the determined atomic wavefunctions. Relativistic and non-relativistic versions of this code have been made available through concurrent publications of Liberman [86 and Zangwill [87. Storm and Israel (1970) [88 use the (modified) Brysk-Zerby computer code to calculate the photon-atom cross-sections. These calculated values were extended above 200 kev by joining them to Schmickley-Pratt [78 and Rakavy-Ron [77 calculations in response to systematic divergence of the Brysk-Zerby values. Reported values covered energies from 1 kev to 100 MeV for elements Z = Their article provided detailed comparisons with measurements, and these comparisons were used to estimate typical uncertainties for the tabulation at less than 10% between 1 and 6 kev, and less than 3% at higher x-ray energies. Veigele s 1973 tabulation [89 presents a semiempirical combination of calculated and measured absorption coefficients over energies from 0.1 kev to 1 MeV. Calculated values used the non-relativistic bound-state wavefunctions of Herman and Skillman. Measurements were used to determine smooth absorption curves by repeated application of a least-squares fitting routine with judicious rejection of anomalous experimental values. Hubbell et al. tabulated atomic scattering factors F (α, Z) and incoherent scattering functions S(α, Z) for Z = over an extremely wide range of momentum transfer [90. Their tabulation aimed to produce internally consistent sets of F (α, Z) and S(α, Z), derived (at each energy) using the one formalism, and so used non-relativistic HF wavefunctions. A later publication by Hubbell and Øverbø [91 provided updated values calculated using relativistic DHF wavefunctions. A later work by Hubbell, Gimm, and Øverbø presented a tabulation of cross sections and attenuation coefficients for all elements (Z = 1 100) in the γ-ray energy range, from 1 MeV to 100 GeV [5. The 1973 calculation of Scofield [6 was extended and used to provide photoelectric absorption cross sections from 1 to 1.5 MeV. The Scofield calculations were performed using HS wavefunctions, with renormalisation constants provided to determine presumably more accurate Hartree-Fock atomic model values [5. The Scofield values were extrapolated, guided by the theoretical

60 32Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT high-energy results of Pratt [7, over five orders of magnitude so as to cover the energies up to 100 GeV. A semiempirical tabulation of Henke et al. [92 provides optical data in the low-energy region, from 0.03 to 10 kev. They have determined f 1 and f 2 from experimental data and from calculations using HS wavefunctions. Calculated values were scaled (by least-squares fitting) to match the experimental data, and so to determine the photoelectric absorption coefficients. Where insufficient experimental data existed, interpolations and extrapolations in both Z and E were employed. In principle their procedure aims to determine relative structures from the calculation, and to scale these to the level of the experimental data. However, the results of such a calculation are likely to exhibit significant errors as random and systematic errors in the experimental values range from %. Schaupp et al. [1 used the modified form-factor (MFF) formalism (see section 2.5) to evaluate atomic scattering factors [F (α, Z) in the high-momentumtransfer regime, E 1 MeV. Values were provided up to 100Å 1 despite serious errors in the tabulated values above α 20 50Å 1. A 1987 study by Saloman and Hubbell indicated that experimental values were in better agreement with Scofield s unrenormalised (HS) values than with his renormalised (HF) values [6. Accordingly, a 1988 tabulation of attenuation cross sections from 0.1 to 100 kev by Saloman, Hubbell, and Scofield [93 presents unrenormalised Scofield values % discrepancies of the tabulated values from Henke s semiempirical values [92 could not be resolved by comparison with experimental values indicating, perhaps, the paucity of good experimental data across the entire elemental range but particularly in the high-z region. Henke, Gullikson, and Davis [94 updated the Henke et al. [92 earlier tabulation, providing semi-empirical values for the anomalous scattering factors, f 1 and f 2, determined from measured mass attenuation coefficients by use of a relativistic Kramers-Kronig dispersion relation. The high-energy limit of their work was extended to 30 kev by use of the semiempirical tabulation of Biggs and Lighthill (unpublished). Semiempirical values were determined by fitting (the exact procedure is not described) a combination of the available measured data with the theoretical

61 2.8. RESULTS OF VARIOUS THEORETICAL APPROACHES 33 values of Doolen and Liberman, calculated using the code of Liberman and Zangwill [86, 87, modified for use at low photon energies. These values are now available on the Centre for X-Ray Optics web site [95, and are referred to in this thesis as CXRO [94, 95. The XCOM program and database of Berger, Hubbell, and Seltzer [96, 97, 98 provides absorption and scattering cross sections from 1 kev to 100 GeV. Photoelectric absorption is calculated using the unrenormalised HS wavefunctions of Scofield [6. The history of the Scofield calculation is interesting. The original tabulation provided exact HS values and approximate renormalised HF values. As the HF solution is in principle more accurate, the renormalised values were recommended. Over the next decade or so various measurements reported better agreement with either the renormalised or unrenormalised values for various elements and energies: most tabulations provided renormalised values. Gerward has observed that the Scofield values provide lower and upper bounds for the experimental values [99. Saloman and Hubbell (soon to collaborate with Scofield in the production of updated tables) declare that unrenormalised Scofield values are to be preferred. Finally Kumar et al. demonstrated that low-energy measurements were in substantially better agreement with the unrenormalised values [100. This demonstrates that improved physical models do not necessarily lead to improved agreement with experiment. However, in order to produce science at this juncture it is necessary to question and criticise the models and their implementations to determine the causes of such observations. Empirical observation provides justification for the use of the less sophisticated model: however, unquestioning acceptance of the conclusions of such empirical observations denies insight into the underlying physics and its implementation. The 1995 tabulation of form-factor, attenuation and scattering tables (FFAST) by Chantler [33 used as its starting point the Brennan and Cowan [101 revision of the Cromer-Liberman [84, 102 approach with the Brysk-Zerby [83 relativistic evaluation of the atomic form-factors. Instead of using a small number of characteristic energies, Chantler performed the calculation over a wide-range of (logarithmically-

62 34Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT spaced) energies. Extending the calculation towards a continuum evaluation in this way, and sampling the calculation parameter-space, Chantler identified a number of significant problems that had hitherto been hidden within the Cromer-Liberman formalism [103. Chantler s tabulation concentrates on improvements in the prediction of outer orbitals of high-z elements, the use of improved transform equations for the evaluation of numerical integrals, both near and away from absorption edges, and the removal of a number of singularities in f 1 above absorption edges. In the range of energies covered by his tabulation, relativistic correction factors are not considered and are insignificant in comparison with theoretical differences. Attention is also paid to a variety of issues affecting the various numerical integrals. An update of these tables [34 further improves the stability of the calculation in the neighbourhood of L and M-shell absorption edges below a few kev. In these regions, small errors in the wavefunction had accumulated to result in false structure in the mass attenuation coefficient of up to 200%. The formulation of Chantler claims significantly improved numerical stability and consistency of the reported values over a number of energy regions. The National Institute of Standards and Technology (NIST) currently provides both the XCOM and the FFAST tabulations, and explicitly acknowledges the significant discrepancies between the tabulated values and the inadequacy of their uncertainty estimations. Figure 2.1 presents the NIST comparison between these tabulations [104, showing the absolute value of the percentage difference of the first FFAST tabulation [33 from the XCOM values [96, 97, 98. At x-ray energies of interest in this thesis the difference between the XCOM and FFAST Rayleigh plus Compton scattering scattering components does not significantly alter the qualitative structure of this plot. The % differences occurring about the L and M-shells at low energies (1 3 kev) were addressed in the 2000 update of the tables discussed above [34. Significant differences occurring above the K absorption edge of molybdenum and tin are the subjects of investigation of this thesis. Figure 2.1 presents in a clear manner the difference between the XCOM and FFAST tabulations across a wide range of energies and elements. Unfortunately,

63 2.8. RESULTS OF VARIOUS THEORETICAL APPROACHES 35 Sn Mo Figure 2.1: Percentage difference between the mass attenuation coefficients tabulated in FFAST [33, 35 and XCOM [96, 97, 98. The K, L, and M-shell absorption edges appear as a series of curves in the differences. Major discrepancies near to the L and M absorption edges at lower energies were corrected by a later update of the FFAST tabulation [34. Discrepancies of 2 10% are common on the low-energy side of absorption edges, due to the low attenuation immediately beneath these edges. In this thesis we report measurements made over regions of increased difference immediately above the K-shell absorption edge of molybdenum (Z = 42, 10 15%) and tin (Z = 50, 2 5%). Taken from [104. such plots are not available for other tabulations. Since a significant component of this thesis concerns a measurement of the mass attenuation coefficients of molybdenum over the energy range from 13.5 to 41.5 kev, we present comparisons between various tabulations for molybdenum over this energy range. Figure 2.2 presents the percentage difference between the tabulated values of XCOM [96, 97, 98, CXRO [94, 95, and Brennan and Cowan [101, 105 and the FFAST values [33, 34, 35 for molybdenum, over the energy range from kev. Above the K-shell absorption edge of molybdenum at 20 kev the XCOM- FFAST discrepancy rises to about 17%. Below the absorption edge all of the predictions are within 5%, and show no significant structural differences. The Brennan- Cowan result shows the results of the Cromer-Liberman, Brysk-Zerby code prior to the modifications of Chantler, with differences of less than 2% below the absorption

64 36Chapter 2. CALCULATION OF THE MASS ATTENUATION COEFFICIENT Figure 2.2: Comparison of the predicted attenuation of molybdenum resulting from the calculations of XCOM, CXRO, and Brennan and Cowan against those of FFAST. Tabulations of form-factors have been converted into mass attenuation by use of Eq. (2.39) and have had the FFAST scattering cross-section added for consistency. The XCOM values diverge immediately above the K-shell absorption edge at about 20 kev. The tabulations due to CXRO and Brennan and Cowan do not provide values for the entire energy range presented here. There is a significant 2 5% variation of the predicted values below the K-edge. edge and of order 1% above the edge, indicating the change to the values arising from Chantler s modifications. 2.9 Summary and conclusion We have discussed several commonly-used approaches for calculating mass attenuation coefficients. Of these, atomic form-factor calculations employing HF and HFS wavefunctions have been used to compile extensive tabulations of the mass attenuation coefficients. Calculations employing the S-matrix formalism promise improved theoretical rigour, but such approaches have yet to overcome a variety of theoretical and computational obstacles. Commonly-used tabulations of mass attenuation coefficients differ by % for many elements over significant energy ranges. Improved (i.e., more sophisticated) theoretical models do not necessarily produce better agreement with experimental values, presumably due to implementation issues. Major differences between the tabulations result from the various theoretical frameworks employed for calculating

65 2.9. SUMMARY AND CONCLUSION 37 the atomic wavefunctions, each of which treats exchange, correlation, and overlap effects in a different manner. Further differences stem from the diverse application of approximate methods and convergence criteria. Confidence intervals are generally estimated from the requirements placed on the convergence and consistency: however, although such error estimates may be appropriate for the internal logic of the computation, they do not represent the model error or applicability. Small errors in calculated inner-orbital wavefunctions may accumulate, affecting the calculation of outer orbits, and resulting in errors greater than those predicted from convergence criteria alone. Various approximations may be valid in different energy regions, so it may be that no tabulation can be preferred across all energies. The results of measurements have been tabulated at various times to determine whether the body of the data may be more conclusive than individual measurements (e.g. [106, 107, 108, 93, 109). However, these tabulations exhibit unexplained differences of % in various regimes, and are useless for resolving differences between tabulations of calculated values. It is difficult to compare these tables globally. The best initial approach may well be to concentrate on particular representative elements and energy ranges, obtain good experimental results, and compare these with the various available tabulations. Over time, systematic discrepancies may suggest ways to proceed theoretically. In this work we measure mass attenuation coefficients of two medium-z metals, molybdenum and tin, in the kev energy range, and aim to investigate discrepancies between major tabulations near to and above absorption edges.

66

67 Chapter 3 Measurement techniques Atomic scattering factors are required as input data for a wide variety of techniques. The accuracy of these techniques will therefore be enhanced by improvements to the accuracy of this input data. Both absolute and relative measurements of the x-ray absorption of materials can test models of the atomic structure. While relative measurements can test structural details predicted by such models, absolute measurements provide an additional, crucial, and demanding test of these predictions and the understanding of the processes contributing to the interactions of x-rays with matter. 3.1 Discussion of previous studies Compilations of measured values of the x-ray mass attenuation coefficients show consistent discrepancies between the results of independent measurements of about 10%, often rising to 25% [106, 107, 108. The magnitude of the discrepancies between the measured attenuations represents a failing of current x-ray science and computation and provides a serious impediment to quantitative x-ray investigations. Figure 3.1 presents results of measurements of the mass attenuation coefficient of molybdenum in the range of energies from 1 to 100 kev as compiled in Hubbell [106, 107, 108. Measured values are presented as a percentage difference from the FFAST calculation. Multiple values arising from individual experiments are marked with the same symbol, establishing the trend of each set of measurements. The author code appearing in the key is that used by Hubbell [107. Despite individual quoted measurement uncertainties of 0.5 2% we see consistent differences of 10%, rising to about 20%, equivalent to standard deviations. These differences represent an unacceptable level of measurement variation, but are typical of all elements across all x-ray energies. 39

68 40 Chapter 3. MEASUREMENT TECHNIQUES Figure 3.1: Measured mass attenuation coefficients for molybdenum, presented as a percentage difference from the FFAST calculation [34, 35, (from [106, 107, 108). Measured values have typical quoted uncertainties of 0.5 2% but differ by 10 20%, indicating significant unidentified systematic errors. Author code as per [107. Inconsistencies between the measured mass attenuation coefficients have been the subject of many investigations, most recently under the auspices of Lawrence Livermore National Laboratory [109, the NIST [79, 80, 106, 107, 110, 111, and the IUCr [112, 113. Compilations of measurements have been used to determine semi-empirical tabulations of x-ray data over a wide range of energies [92, 94. These tabulations employ weighting schemes and data rejection criteria which implicitly and explicitly recognise the significant inconsistencies between the measured values and require subjective treatment, as claimed uncertainties cannot be trusted. Inaccuracies in tabulations of mass attenuation coefficients led the IUCr to devote a multi-laboratory project to the investigation of their causes [112, 113. The primary goal of the IUCr project was to critically examine measurement techniques and to establish protocols for making good measurements. The method involved critical comparison of measurements using standardised samples made in various laboratories. A second aim of the project was to identify an accurate theoretical tabulation for general use. Laboratories from around the world were invited to participate in the project [114 and were provided with samples of silicon and carbon (pyrolytic graphite)

69 3.1. DISCUSSION OF PREVIOUS STUDIES 41 for measurement. The results of the investigations were compared critically in two reports [112, 113. Eight different measurement techniques were employed, using four different types of x-ray sources (radioactive, fluorescent, characteristic, and bremsstrahlung synchrotron radiation was not used), with several counter configurations (solid-state energy-dispersive detectors, proportional counters, and scintillator detectors). We employ a technique very similar to arrangement [9 of [113, which unfortunately is not discussed in that work. Major conclusions of the project included the identification of likely sources of systematic error affecting some configurations, and recommendations that the physical nature and properties of the absorbers such as density, porosity, crystallinity, et cetera should be measured and considered by experimenters as these provide clues to the state of the sample itself. It was observed that a number of systematic errors could be detected by varying the absorber thickness, and by confirming the linearity of the absorption with thickness, so as to ensure that a unique value of the attenuation exists. This particular observation is in direct accord with the modus operandi of the XERT and the approach adopted in this thesis. Other notable conclusions of the IUCr project include the following: there is no compelling reason for preferring any one tabulation of mass attenuation coefficients over others; previous attenuation measurements have been compromised by the effects of a range of poorly-understood systematic errors; and previous experiments have generally been reported with insufficient detail for the likely presence and impact of these systematics to be properly evaluated. The provision of samples to experimenters enabled certain systematic effects to be tracked and controlled: however, the same approach prevented others from being exposed. Accordingly, it is not known whether sample quality may be partly responsible for the significant differences between earlier measured values. While the measurements reported for the IUCr project are consistent within 1% (after normalisation for known errors), we do not know how consistency would have been affected if the laboratories had provided their own specimens. It is unfortunate that the IUCr project did not report measurements of copper in detail. Such measurements would be of great interest, as copper includes an

70 42 Chapter 3. MEASUREMENT TECHNIQUES absorption edge at about 9 kev, and represents a significant step away from the low-z region of carbon (Z = 6) and silicon (Z = 14). Further, copper is metallic and may be used to investigate different (and more common) materials issues than either silicon or carbon. In particular, the problems associated with measurements of single-crystal silicon (Laue-Bragg scattering and tds) and pyrolytic graphite (void scattering) are quite specific to these elements. The IUCr project has precipitated a significant improvement in the quality of reporting and has publicised problems associated with these fundamental x-ray parameters; however, the conclusions and approach must be further generalised and developed. Balthazar-Rodrigues and Cusatis have recently reported a diffraction technique for measuring the attenuation of silicon and germanium precise to % [115. Their technique is based on the diffraction of x-rays from large crystals and therefore can only be applied to materials for which large crystalline samples are obtainable. They have made measurements at five energies only, and it is therefore difficult to assess their data for the effects of a variety of possible systematic errors. The measurements for silicon by Balthazar-Rodrigues and Cusatis [115 are consistent to within 2-σ with recent measurements of Tran et al. [14, 116, which may be interpreted as a good check for their combined uncertainty estimates. The measurements of Tran, Chantler, et al. were made using the x-ray extended-range technique (XERT). They have applied this technique to measure the attenuation of silicon [14, copper [13, silver [69, and gold [117 accurate to 0.3% using synchrotron radiation. Such measurements are of sufficient accuracy to resolve theoretical issues [118, 116. The consistent scatter of the discrepancies presented in Fig. 3.1 confuses assessment of any one set of measurements. Discrepancies between calculated values are of a similar magnitude (cf. Fig. 2.2), and cannot be used to assess various experimental techniques. Neither measured, calculated, nor any combination of measured and calculated values are in satisfactory agreement. These outstanding discrepancies may have resulted from an inadequate appreciation of experimental parameters and their effects on measurements, and from a failure of theory in describing x-ray interactions. Whatever the reason for the discrepancies, its resolution will improve experimental data sets and will develop understanding of experimental techniques.

71 3.2. PRINCIPLES OF OUR EXPERIMENTAL CONFIGURATION 43 The XERT of Chantler, Tran, Barnea, et al., and of this thesis, uses measurements made over a wide range of the measurement parameter space. When differences between measured values are identified, the cause and effect of the systematic errors is determined from measurements spanning the parameter space. Such investigations have been used to report the effects of beam harmonic components [119, 120, 121 and secondary photons [122 on measurements of the mass attenuation coefficients. 3.2 Principles of our experimental configuration The current thesis applies the principles of the XERT as the basis for the experimental design, and develops the application of the technique for the identification and correction of systematic errors where their effect on the measured mass attenuation coefficients is significant. Figure 3.2 shows the main components of the configuration used for measuring the mass attenuation coefficients of thin metallic foils. The x-ray beam is produced using a bending-magnet, undulator, or wiggler insertion device at a synchrotron facility. The beam is monochromated, usually by reflection from two or more silicon crystals. Following monochromation the beam is collimated and shaped to define an appropriate cross section. Ion chambers are located upstream and downstream of a construction of rotation and translation stages. These stages are used to place a number of high quality absorbers in the path of the x-ray beam so as to illuminate them at various locations and at various orientations to the beam. Between the Figure 3.2: Schematic of experimental design.

72 44 Chapter 3. MEASUREMENT TECHNIQUES sample stage and the upstream and downstream ion chambers are two wheels with three differently-sized apertures and foils of various thicknesses located around their perimeters. These daisy wheels are used to present various beam collimation, and to measure absorbers spanning a very wide range of attenuation. The second downstream ion chamber shown in this figure is a complementary counter used to improve statistical precision where it is most likely to be limited. A diffractometer located further downstream is used to record a number of Bragg reflections from a crystalline specimen and thereby to determine directly the energy of the x-ray beam employed for the measurement. The following sections discuss each of these components in detail, and explain how they are used to extend the parameter-space probed by the measurement so as to enable quantification of any error contributions. 3.3 X-ray production and monochromation Synchrotron sources can produce x-ray beams several orders of magnitude more brilliant than laboratory sources, and enable extremely high statistical accuracies to be attained within short counting times. The tunability of synchrotron sources allows the production of x-rays over a range of energies, representing a distinct advantage over radioactive and characteristic sources. This tunability is exploited in the XERT by making measurements at fine energy intervals to identify discontinuities between neighbouring measured values, should they exist. The use of synchrotron sources requires the application of specialised techniques to control the energy spectrum of the x-ray beam. Monochromation can lead to systematic errors due to the presence of harmonic components [123, 124. We employ a double-crystal monochromator which selects a harmonic series of energies from the x-ray beam. The intensities of the higher harmonics in this beam are reduced by detuning the second crystal of the pair [125, 126. We defer further discussion of the monochromation and conditioning of the synchrotron beam because these depend on the exact nature of the beamline. Such details are reported with the results of each experiment.

73 3.4. X-RAY DETECTORS 45 We have developed a technique to determine the effective fraction of harmonic x-rays in the beam, and the effect of these x-rays on the measurement [119. The technique is highly sensitive, and has been used to determine effective beam harmonic components of 0.1% of the primary, fundamental beam. Further issues, relating to the monochromator resolution function [127, 128, 129, 130, are investigated in the current work in terms of the bandwidth of the beam [ X-ray detectors The intensity stability of an x-ray source may be determined by the stability of electrical, geometrical, or physical parameters, or by statistical considerations. In most cases (with the notable exception of radioactive sources) intensity variations are not statistically limited. These variations constitute noise in an experiment which does not quantify their presence, and may lead to the dominance of uncertainty due to apparently incoherent scatter of results (e.g. [31). Monitoring the incident x-ray beam intensity allows the ratio of the attenuated and unattenuated x-ray beam intensities to be determined to a precision which is independent of the x-ray beam intensity fluctuations. This monitoring is sometimes achieved by measuring the x-rays scattered from a thin sample located in the path of the beam [113 or scattered from the x-ray beam monochromator [30. However, in such measurements the different response functions of the detectors may lead to uncorrelated variations and a compromised statistical precision. Further, the detection of x-rays scattered from the crystal monochromator [30 could result in extremely large uncorrelated components due to the strong angular dependence of Laue-Bragg scattering if there are any mechanical or alignment instabilities in the monochromator or the x-ray beam. Beam intensity monitoring has been also effected for laboratory sources by pointing an additional detector at the x-ray source [131, 132, 26. However, such measurements do not measure the same x-ray beam, and so cannot guarantee strong positive correlation between the monitor and detector intensities, particularly as the incident beam has a gaussian-like profile and meanders with time.

74 46 Chapter 3. MEASUREMENT TECHNIQUES Figure 3.3: Relative uncertainty of the mass attenuation coefficient as a function of the absorber attenuation, as per Nordfors [135. The absorber attenuation required for optimum statistical precision lies between 2 and 4. When counting statistics do not dominate the measurement uncertainty, the detailed conclusions of this study are not valid. However, conditions under which statistical inaccuracies may be minimised remain valid. We have instead employed matched serial-flow ionisation chambers (ion chambers) located upstream and downstream of the absorbing specimen, as shown in Fig The upstream ion chamber monitors the incident x-ray intensity, and the downstream ion chamber measures the x-ray intensity with and without an absorber in the path of the beam. Matched ion chambers have been used to resolve different contributions to synchrotron beam intensity fluctuations [133, 134. By having the gas flow through the ion chambers in a serial configuration we ensure strong positive correlation between any gas pressure variations in the ion chambers and avoid the formation of dead regions within the detector volumes. Ion chamber detectors have a number of adjustable parameters which can be selected to optimise their operation. These include the length of the detector and the detector gas, but also the detector windows and electrode spacing. The selection of these parameters depends mainly on the photon energy range, but includes considerations of the peak flux, the peak flux density, and the desired level of statistical accuracy. Nordfors has studied the sample attenuation [ µ [t required to measure the mass attenuation coefficient with optimum statistical precision for a measurement

75 3.4. X-RAY DETECTORS 47 using a single detector [135. Figure 3.3 shows the results of the Nordfors calculation, plotting the relative uncertainty in the mass attenuation coefficient against the attenuation of the absorber used to make the measurement. The curve of the relative error in the measured attenuation has a broad minimum between 2 and 4, which has been used by a number of experimenters (see, for example, [112 and references therein). When a single detector is employed for the measurement, that detector should be of the highest efficiency. However, for a measurement employing matched upstream and downstream ion chambers, the transparency of the detectors is crucial and the detector efficiency can be optimised. We have extended the Nordfors approach to develop a model to determine the relative uncertainty in the mass attenuation coefficient for a system employing matched upstream and downstream ion chambers. Figure 3.4 presents modelling of the uncertainty in the mass attenuation coefficient as a function of the ion chamber efficiency (1 exp { [ } µ [t ) for measurements employing a variety of sample attenuations. We assume no air-path attenuation, Poisson (counting) statistics throughout, and no dark-current uncertainty. While ic gas Figure 3.4: Relative uncertainty of the mass attenuation coefficient as a function of the efficiency of the ion chamber used for the measurement, showing an extremely broad minimum for detector efficiencies between 20 and 80%. Uncertainties are shown for different sample attenuations: the minimum uncertainty occurs for samples of attenuation [ µ [t 2.6, in agreement with Nordfors conclusion for ideal detectors (100% detection efficiency, no intensity loss) [135.

76 48 Chapter 3. MEASUREMENT TECHNIQUES Figure 3.5: Ion chamber efficiencies across a range of photon energies for a variety of gases. The modelled ion chamber length is 100 mm. Absorptive efficiency is the proportion of photons removed from the beam by the ion chamber, and is given by ( 1 exp { [ µ [t } gas). Argon gas is appropriate for x-ray energies between 13.5 and 41.5 kev. the impact of these assumptions can be significant, the form of this curve accurately represents the impact of the ion chamber efficiency on the statistical measurement accuracy. The curves in Fig. 3.4 show an extremely broad minimum for efficiencies between 20 and 80%, and describe a significant decrease in the count-limited statistical precision when the detector efficiency decreases below 2%, where the ion chamber is almost blind, or increases above 98%, where it no longer transmits sufficient intensity. Figure 3.5 presents the calculated detection efficiency of a 100-mm ion chamber employing various gases at STP. Argon is appropriate for use with x-ray energies between 13.5 and 41.5 kev. Above 41.5 kev a longer ion chamber could be used to increase detection efficiency, or krypton or xenon could be used. However, these gases are not appropriate for use at lower x-ray energies and are significantly more expensive than argon.

77 3.5. CORRELATION AND STATISTICS Correlation and statistics When a synchrotron operates in decay mode the loss of electron current in the storage ring leads to an exponential decay of the x-ray intensity. In addition to this relatively gradual decrease, the x-ray intensity wanders on a variety of time scales due to mechanical factors affecting critical components such as monochromators, bending magnets, and insertion devices, as well as the operation of dynamical components in the beam such as the RF kicker (used to maintain the ring energy) and the steering components of the beam. In an experiment which does not utilise matched upstream and downstream detection, these variations lead to apparent noise, whereas in our configuration these variations quantify the noise level of the detection system without assumption of the specific causes of this error. When the counting-time, the number of repetitions, and the detector gas are optimised, it is possible to resolve various contributions to the fluctuation of the measured count rates [133, 134. We record count rates in the ion chambers multiple times for each sample, allowing the determination of correlations. For measurements recorded with a correlation coefficient R the uncertainty for the mean of the ratio I d /I u is [133, 134 [ σ(id /I u) I d /I u 2 = ( σid I d ) 2 + ( σiu I u ) 2 2R ( )( σid σiu I d I u ), (3.1) whereas the uncertainty for the ratio of the means I d /I u is [ σ(id /I u) I d /I u 2 = ( σid I d ) 2 + ( σiu I u ) 2 + 2R ( )( σid σiu I d I u ). (3.2) The subscripts u and d denote the count rates recorded using the upstream and downstream ion chambers respectively. We aim to achieve strong positive correlations between the upstream and downstream ion chambers, allowing us to determine the ratio from Eq. (3.1) to high accuracy. There are three types of correlations in any detection system [133, 134. Positive correlations result, for instance, from an increase in the beam intensity, which increases the count rate recorded by both ion chambers. Negative correlations may

78 50 Chapter 3. MEASUREMENT TECHNIQUES result from an increased efficiency of the upstream ion chamber relative to the downstream ion chamber, possibly due to a local increase in the gas pressure. Such a pressure increase results in an increased count rate in the upstream ion chamber, leaving fewer photons in the beam for detection in the downstream ion chamber. Uncorrelated components may result from intrinsic statistical or counting limits, or from any event which results in an independent variation of the counts recorded by the two detectors. We aim to reduce all uncorrelated and negatively correlated components, and aim for strong positive correlations between the counts recorded in the ion chambers. The two most significant sources of correlation are likely to be beam intensity fluctuations and ion chamber gas pressure fluctuations. As we use the ion chambers in a serial gas-flow configuration both of these result in positive signal correlations. This work utilises two ion chambers in the downstream location to increase the statistical accuracy of the measurement where it is least reliable and to provide a backup measurement for the count rates recorded by the other downstream ion chamber. 3.6 Sample attenuation levels Several reports derive conditions for optimum precision in [ µ based on statistically limited measurements [136, 135. In particular, Nordfors [135 demonstrates that optimum precision is obtained for measurements of absorbers with attenuation in the range 2 < [ µ [t < 4 [135, 112. When low-brilliance sources are employed, statistical limitations and background contributions can limit overall measurement precision. However, such treatments are not appropriate when high-brilliance sources are used and when statistics do not limit measurement precision. Despite the changing nature of x-ray sources, few investigators have challenged these early conclusions. Most measurements reported between 1983 and 1993 and appearing in [106, 107 claim that the determination of the local foil thickness, that is, the amount of absorber traversed by the beam, is a major limitation to the measurement accuracy at about 0.5%. The fractional error in the determined thickness decreases linearly

79 3.6. SAMPLE ATTENUATION LEVELS 51 with increasing foil thickness. An experiment therefore reaches optimal precision when the uncertainty contributions from the foil thickness and the counting statistics are approximately equal. Therefore, these experiments may not have been optimised. The term thickness effect is used to refer to any systematic effect on the measured value of the attenuation that can be probed by varying the thickness of the absorber used for the measurement [127, 129, 137, 138. Some authors, concerned with XAFS measurements, have observed decreased thickness effects for measurements using thin samples, and have recommended the use of absorbers with [ µ [t 1 [137. Many experiments between 1983 and 1993 report measurements using absorbers with attenuations in the range 0.5 [ µ [t 4. The lower bound of the measured attenuation range has been extended down from the Nordfors minimum of [ µ [t = 2 to [ µ [t = 0.5 to avoid possible thickness effects. Investigators using thinner absorbers to reduce the impact of thickness effects on measured values [139, 140, 28 have not quantified residual effects on their measurements. Measurements clearly exhibiting thickness effects have been reported, with no investigation or discussion of implications for reported values [141. Some authors confuse counting statistical accuracy with accuracy in [ µ, observing (for instance) that counting statistics would greatly improve if the sample thickness is chosen such that µt < 1 [139. The avoidance of thickness effects by use of less attenuating absorbers is in direct conflict with stated measurement limitations, namely, the accuracy of the absorber thickness determination. In recent work of Tran, Chantler, Barnea, et al., measurements have been made over an extended Nordfors range, from 0.8 [ µ [t 5 [13, 14. Measurements over this extended range have been used to identify and correct for the effect of harmonic components present in the x-ray beam and to reduce absorber thickness uncertainties without significant degradation to statistical precision. Using the XERT we obtain measurements under optimal conditions and at the same time explore the measurement parameter space out to the point where the conditions for good measurement break down. In particular, we obtain measurements over an extremely wide range of sample attenuations, making measurements between 0.1 < [ µ [t < 8 using select high-quality absorbers, and covering an ex-

80 52 Chapter 3. MEASUREMENT TECHNIQUES tended range of attenuation between 0.05 [ µ [t using the daisy-wheel absorbers. Our use of a wide range of sample attenuations has enabled us to observe the effect of harmonic energy components in the x-ray beam [119, of the use of an incorrect value of the dark current [119, of contributions from sample fluorescence and scattering [122, and of the bandwidth of the x-ray beam [ Secondary photons: beam collimation Secondary photons are produced in the sample and along the air-path traversed by the beam. These secondary photons result mainly from Compton and Rayleigh scattering and fluorescence. As we aim to measure the intensities of the incident and transmitted beam, it is crucial that these secondary photons do not contribute to the measured count rates. Secondary photons are usually excluded by placing apertures or collimators before and after the absorbing specimen, which may minimise the effect of scattering but does not quantify the degree to which this minimisation has succeeded. An alternate technique places the absorber between diffracting crystals to remove all modified photons from the transmitted beam [131, 112. Reference [142 has examined the use of diffracting crystals to remove rays that are not parallel to the direct beam, finding serious error when double scattering within the absorber is significant. We were also reluctant to introduce a diffracting crystal between the ion chambers because of the likelihood of a significant effect on ion chamber correlations, leading to reduced statistical precision. A previous experiment employing the XERT has observed changes in the upstream count rate, identified as resulting from Rayleigh scattering and fluorescence [118. We employ a number of apertures of various diameters to collimate the beam. These apertures are mounted on the perimeter of the daisy-wheels shown in Fig Our use of multiple apertures allows us to vary the collimation of the beam and thus to determine its effect. Values of the mass attenuation coefficient determined using the various aperture diameters can be extrapolated to an ideal limit, providing perfect collimation and removing the secondary-photon count rate from the detectors.

81 3.8. BEAM HARMONIC COMPONENTS 53 We have recently used such measurements to quantify the effect of fluorescence on the measurement of the mass attenuation coefficient of silver [122, 69. Extrapolation to the perfect collimation limit presumes that the scattered photons have a simple angular dependence over the range of solid angles probed by the apertures. We employ apertures typically subtending a small range of solid-angles at the scattering sample. Fluorescent and Compton and Rayleigh scattered photons have smooth angular dependencies in the forward and backward directions, so that this technique is appropriate for their treatment. In contrast, small angle x-ray scattering (SAXS) may exhibit significant angular variation in the neighbourhood of the forward scattering angles. SAXS is commonly restricted to an angular range of 1, and so may pass through all apertures, and we may not be sensitive to SAXS. However, we perform measurements using a transmission geometry with the absorber oriented perpendicular to the beam, and so we expect that the contribution of SAXS will be negligible. If present, energy, thickness, and sample dependences would provide further signatures for the identification of the effects of SAXS. 3.8 Beam harmonic components Harmonic multiples of the fundamental beam energy may be present in the x-ray beam due to monochromation by diffraction from single crystals. The effect of these harmonic components on the measured attenuation additionally depends on the relative detection efficiency of the ion chambers for the fundamental and harmonic energies. Figure 3.6 shows the relative efficiency of detection of the photons of the fundamental energy compared to that of the third harmonic energy. This plot shows that the use of argon gas suppresses the effect of the harmonic photons by a factor of between 6 and 18. Further suppression of the harmonic energy components is effected by detuning the monochromator, as mentioned earlier. We make measurements using foils whose attenuations range from 0.05 [ µ [t at the fundamental x-ray energy so as to quantify the effect of the harmonic components. Foils are mounted on the perimeter of the daisy wheels as in Fig. 3.2, and are introduced into the path of the beam by suitable rotation of the wheels.

82 54 Chapter 3. MEASUREMENT TECHNIQUES Figure 3.6: Relative efficiency for the detection of the fundamental and the third harmonic energies for a variety of gases. Argon gas can be used to suppress the third-order harmonic by a factor of between 6 and 18 at energies between 10 and 40 kev. Ion chamber parameters as for Fig The extremely wide range of attenuation of these foils can been used to isolate and distinguish thickness-dependent systematic shifts to the measured attenuations. 3.9 Sample purity To determine the mass attenuation coefficient of an elemental material the absorber must be of sufficient purity. For the measurement of the mass attenuation coefficient of molybdenum, we have used foils with nominal purity of 99.98% [143. According to a typical assay provided by the supplier, ESPI [144, likely impurities are; K : 40 ppm, Cu : < 16 ppm, Ni : 25 ppm, Cr : < 32 ppm, and Fe : 52 ppm. The effect of impurities can be calculated using the sum rule for mixtures and compounds [63 [ µ mix = [ W µ i species i (3.3) where W i is the fraction by weight of each constituent. We have used the FFAST tabulated values of the mass attenuation coefficients [33, 34, 35 to evaluate Eq. (3.3) for the quoted assay of molybdenum to determine that the effect of impurities on the mass attenuation coefficient is likely to be less than 0.01% between 13.5 and

83 3.10. BEAM SIZE kev. The nominal impurities have no absorption edges in this energy range and so yield no discontinuities in the measured values (as a function of energy) which might have been used to further quantify the impurity levels Beam size The beam size and its footprint on the absorber is defined by the opening size of a pair of orthogonal slits located upstream of the first ion chamber. The adjustment and selection of the beam footprint is made with a number of factors in mind, including the x-ray flux density (photons/mm 2 ) and uniformity, the length scales over which the absorber is to be sampled, the desired degree of beam collimation, and the application of a full-foil mapping technique (developed in this thesis). The reduction of the beam footprint results in a directly proportional decrease in the x-ray flux used to make the measurement, and thus in a decrease of the statistical precision of the measurement. The dimensions of the beam footprint should be sufficiently large so that the measurement is neither statistically nor noise limited. The measured attenuation depends on the intensity distribution of the x-ray beam and on the thickness profile of the absorber under the beam footprint. We expect the absorbers to have thickness nonuniformities on every length scale, ranging from microstructure [13 to long-range variations [145. The reproducibility of the location of the sample in the path of the beam is of order 20 µm, due to our use of high-precision translation stages and accurately machined, robust sample holders. The reproducibility of the sample location is comparable with the length scales of scratches and microstructure. We therefore require a beam footprint that is much larger than these features so that the illuminated thickness is not highly sensitive to the sample location. Chapter 6 of this work describes the development of a full-foil mapping technique. This technique requires attenuation measurements to be made across the entire surface of the absorbing foil. As our absorbers are approximately mm 2 in size (sufficient to enable accurate mass and area determination), we require 625 beam area

84 56 Chapter 3. MEASUREMENT TECHNIQUES measurements to perform this mapping, with neither overlap of nor gaps between measurements. While the resolution of the attenuation profile is improved by reducing the beam footprint, this improvement is countered by the increased time taken to perform the measurement. We estimate that 625 measurements is an appropriate upper limit for the full-foil mapping. This lies between the number of measurements made at each energy (18) and the total number of measurements (200 energies * 18 = 3600), and is estimated to take between one and two hours, depending on the speed of the translation stage. Accordingly, we have planned to use a beam footprint no smaller than 1 mm 2 for the measurements, and increase these dimensions when the x-ray intensity is too low Determining attenuation from count rates Ion chambers, located upstream and downstream of the sample, gauge the incident, attenuated, and unattenuated beam intensities. In this section we show that it is possible to use these ion chambers to determine the transmission probability of an absorber so that the specific settings and properties of the ion chambers do not feature in the determined value. By performing this analysis explicitly the assumptions and possible sources of error in this approach are revealed. Consider the count rates recorded by a perfectly linear detector. Photoionisation events caused by the interaction of the x-ray beam with the detector gas and occurring within the active volume of the ion chamber produce electron-ion pairs. The probability or efficiency of detection ɛ of a single x-ray can be written { ɛ = 1 exp [ µ [tic }, (3.4) and the number of interactions per second occurring within the detector volume is scaled by the x-ray flux entering the detector, Φ Interaction rate = ɛ Φ (3.5) The charged particles created by the photo-ionisation events are accelerated to two

85 3.11. DETERMINING ATTENUATION FROM COUNT RATES 57 collector plates by a potential difference applied to the plates, and give rise to ionisation cascades. Assuming that every photoionisation event results in a certain yield Y of electron-ion pairs at the collector plates (determined to a large degree by the high-voltage level), we write the current produced in the ion chamber as Current = Y ɛ Φ (3.6) (3.7) This current is measured using an ammeter and a scaler. The (combined) operation of these two electronic devices can be parameterised (in this approximation) as a single amplification, so that Count rate = Amplification Y ɛ Φ + Offset (3.8) I = A Y ɛ Φ + dc (3.9) where the Offset represents the addition of any constant current or voltage to the signal in the course of amplification. The count rate, amplification, and offset are represented by I, A, and dc, respectively. The ion chamber removes ɛφ photons from the beam in recording the count rate. In Eq. (3.9) we have assumed that the combination of all terms results in a perfectly linear variation of the count rate with the photon flux. When two of these ion chambers are situated upstream and downstream of the sample, recording count rates I u and I d respectively, the counts detected in each ion chamber are I u = A u Y u ɛ u Φ 0 + dc u { [ [ I d = A d Y d ɛ d Φ 0 exp µ [t IC [ [ µ [t sam [ [ µ } [t + dc d, (3.10) air where the subscripts u and d refer to the upstream and downstream ion chambers, respectively, and Φ 0 refers to the incident photon flux. From these counts we wish to extract the sample attenuation ( [ µ [t)sam.

86 58 Chapter 3. MEASUREMENT TECHNIQUES The value of dc is determined by taking a dark current measurement, that is, by turning the beam off or shuttering it completely and recording the detector readings. The dark-current levels are therefore estimated by setting Φ 0 = 0 in Eq. (3.10). Once the dark currents are known I u dc u = A u Y u ɛ u Φ 0 (3.11) { [ [ I d dc d = A d Y d ɛ d Φ 0 exp µ [ [ [t µ [ [ [t µ } [t. (3.12) IC sam air We make readings with and without a sample in the path of the beam. These measurements necessarily occur at different times, and so we include possible temporal dependencies by attaching the subscripts 1 and 2 to all variables. The ratio of the upstream and downstream count rates give ( ) Id,1 dc d,1 = I u,1 dc u,1 sam+air,1 { [ [ A d,1 Y d,1 ɛ d,1 Φ 0,1 exp µ [t IC,1 [ [ µ [t sam [ [ µ } [t air,1 ( ) Id,2 dc d,2 I u,2 dc u,2 air,2 = A u,1 Y u,1 ɛ u,1 Φ 0,1 { [ [ A d,2 Y d,2 ɛ d,2 Φ 0,2 exp µ [ [ [t µ } [t IC,2 air,2. (3.13) A u,2 Y u,2 ɛ u,2 Φ 0,2 The attenuation is determined from the ratio of these quantities. Assuming temporal stability gives ( Id dc d I u dc u )sam+air ( Id dc d I u dc u )air { = exp [ } µ [t sam, (3.14) identically, as required. The assumption of temporal stability is equivalent to assuming that { [ [ A d,1 Y d,1 ɛ d,1 exp µ [t { [ [ A d,2 Y d,2 ɛ d,2 exp µ [t IC,1 IC,2 [ [ µ [t [ [ µ [t air,1 air,2 } A u,2 Y u,2 ɛ u,2 = 1. (3.15) }A u,1 Y u,1 ɛ u,1 Investigations employing radioactive sources have generally employed particular measurement sequences to reduce the effect of the decreasing source intensity [146,

87 3.11. DETERMINING ATTENUATION FROM COUNT RATES With a monitor detector Eqs (3.14) and (3.15) no longer refer to the beam intensity. However, we employ a similar measurement sequence to reduce any effects of detector and environmental variations. In the above calculation we have explicitly assumed the linearity of the ion chamber response. Many studies have been undertaken to assess the linearity of ion chamber detectors (see for example [147, 148, 67), but these have often focused on molecular dynamics and recombination. Our experience suggests that a welladjusted ion chamber is as linear as any device that has been used to compare with it. However, as long as 25% discrepancies between measurements of the mass attenuation coefficients remain unexplained, it is not easy to assess this matter in finer detail. The ion chamber linearity has been quantified quite effectively using the multiplefoil daisy wheel technique [119, but may be the least-well quantified potential source of systematic error for our determination of the mass attenuation coefficients. In Appendix C we discuss a technique we have developed which has the potential to probe the linearity of the detector response more closely than the current multiplefoil method.

88

89 Part II Molybdenum 61

90

91 Chapter 4 Experimental details Synchrotron measurements of the mass attenuation coefficient of molybdenum were performed using beamline 1-ID of the the advanced photon source (APS) facility in May, This chapter reports details of the particular experimental configuration employed for the molybdenum measurement and presents a number of operational details which are not mentioned elsewhere in this work. X-ray production and monochromation The x-ray beam was produced using the undulator insertion device at the 1-ID XOR beamline of the Advanced Photon Source facility at the Argonne National Laboratory [149. The (311) planes of a silicon double reflection monochromator were used to select a narrow range of energies from the undulator spectrum. The x-ray energy range covered by this investigation, from 13.5 to 41.5 kev, includes the K-shell absorption edge of molybdenum at around 20 kev, and extends over a wide range of energies above and below the edge. The energy range was limited primarily by the operational characteristics of the synchrotron beamline facility. The energy steps were varied in accordance with the expected structure in the mass attenuation coefficient of molybdenum: down to 0.5 ev within 100 ev of the absorption edge, and increased to 500 ev at energies far from the absorption edge. The monochromator angle was initially set to a low value with the intention of rotating it monotonically in one direction so as to reduce the impact of any hysteresis. The energy was adjusted by rotating the monochromator to the nominal angle corresponding to the new energy, as determined by the approximate (beamlineprogrammed) monochromator dispersion function. The undulator was then scanned through a range of gap settings about the setting corresponding to this energy, and was set at the position which resulted in the maximum count rate in an ion 63

92 64 Chapter 4. EXPERIMENTAL DETAILS Figure 4.1: Rocking curves for the detuning of the second crystal of the silicon monochromator. The scale of the ordinate is relative, representing a voltage applied to a piezo-electric crystal, used to adjust the angular detuning of the second monochromator crystal. The abscissa is the count rate recorded by our upstream ion chamber. The narrow peak on the left-hand side of the broad main peak is the harmonic component. The vertical line indicates the selected detuning setting. Note also the shoulder observed on the right-hand side of the main peak. Occasional (numerical) saturation of the counts recorded by the ion chamber is observed around the position of the maximum intensity, but has not significantly affected the chosen detuning setting. chamber located downstream of the monochromator. The fifth-order component of the undulator spectrum was selected to provide x-rays with energies between 41.5 and 25 kev, and the third-order component was selected for x-ray energies below 25 kev. To reduce the passage of harmonic components into the beam, the second crystal in the monochromator was detuned slightly from its position parallel to the first crystal [125, 126. Figure 4.1 presents a sample of the curves used to assess and set the detuning of the second crystal of the monochromator. On the left-hand side of each of the detuning curves we see a narrow secondary peak which is due to harmonic energy components. The selected detuning position is located well away from the passed harmonic components, and was set so that between 35 and 55% of the peak intensity was passed by the monochromator. Experimental geometry After monochromation, the x-ray beam travelled approximately 30 m down an evacuated pipe, through a thin beryllium window, and into the experimental hutch (see

93 65 beamdefining slits daisy wheel Bicron NaI detector from undulator silicon (3,1,1) monochromator, detuned, with angle encoder 'upstream' argon-gas ion-chamber sample stage with two translational and two rotational degrees of freedom daisy wheel 'downstream' argon-gas ion-chambers germanium single-crystal Figure 4.2: Experimental configuration used at XOR beamline 1-ID to determine the mass attenuation coefficients of molybdenum between 13.5 and 41.5 kev. Fig. 4.2). A pair of orthogonal adjustable slits were used to define the x-ray beam cross-section to be approximately 1 mm 1 mm. The x-ray beam then passed through the first of three 95-mm long, argon gas ion chambers. The ion chambers were of identical construction, and argon gas flowed through the detectors in a serial configuration at a rate of around 1 l/min. Two downstream ion chambers were employed to improve the counting statistics, to investigate the linearity of the ion chambers and their associated electronics, and to provide a cross-check of the measured attenuated beam intensity.

94

95 Chapter 5 Determining the foil attenuations The Beer-Lambert absorption law relates the attenuation [ µ [t to the transmission probability P = I I 0 by [ µ [t = ln I I 0 (5.1) = ln P. (5.2) In this chapter we describe the determination of the attenuation from the count rates recorded using the three ion chambers. 5.1 Filtering the data Upon completion of an experiment it is often necessary to assess the quality of the recorded data and to reject some of the data from the data set. In this section we undertake a symptomatic study of the recorded count rates to identify inconsistent data, likely causes of such inconsistency, and criteria for determining a consistent set of data free of such errors Measurements made during shutter motion The uppermost plot of Fig. 5.1 presents the count rates recorded by the upstream ion chamber as a function of the nominal x-ray energy. The upstream ion chamber is located upstream of apparatus used to move attenuators and apertures into the path of the beam. It is therefore expected that this ion chamber will report only two levels of count rate: those recorded with the x-ray beam on, and those with the x-ray beam off. This plot shows clearly the beam-off measurements (recording between 20 and 67

96 68 Chapter 5. DETERMINING THE FOIL ATTENUATIONS Figure 5.1: Study of the count rates recorded using the upstream ion chamber, showing the removal of measurements affected by the incomplete shutter motion. The measurements recording high ( 10 5 ) and low ( ) count rates relate to measurements recorded with the x-ray beam on and shuttered, respectively. The measurements of intermediate value are anomalous. The central plot presents the same data after the anomalous measurements have been removed by the application of a filter with m = 3σ, n = 2. The lower plot shows the measurements rejected by application of the filter.

97 5.1. FILTERING THE DATA cps) and the beam-on measurements (recording (2 8) 10 5 cps). Small variations of the beam-off and beam-on count rates are due to changes in the ion chamber settings, to the decay and refill cycle of the synchrotron ring current, and to the variation of the ion chamber efficiency with the photon energy. However, this plot also shows a number of measurements recording a count rate of intermediate value. Here we postulate the cause of these intermediate count rates and justify their subsequent rejection from the data. A time-sequence analysis shows that the intermediate count rates presented in Fig. 5.1 are restricted to the first few measurements in each series of ten. This correlation suggests a transient error somewhere in the system, i.e., a latent period at the beginning of each series of repeated measurements. We rule out a transient associated only with the upstream ion chamber by noting that corresponding intermediate count rates are also recorded by the downstream ion chambers. A likely cause of these intermediate count rates is that the beam shutter had not completed its motion prior to the taking of these measurements. In this experiment the x-ray beam was turned off by using a filter-bank to insert highly-attenuating foils into the path of the beam. The filter-bank is a permanent beamline component and is located far upstream of our hutch. As the filters located in the filter bank were very thick (upwards of 2 cm of molybdenum), we will refer to the filter bank as a shutter. We have examined the first few measurements in each series of ten, and have rejected those which have been affected by the incomplete shutter motion. This filtration was effected by testing the first n measurements to determine their difference (in standard-deviations) from the mean of the remaining (10 n) measurements. This is equivalent to a t-test of the consistency of the individual measurement with the mean of the remaining distribution. Individual outliers would still be correctly normalised on a point-by-point basis, but the second-order effect of the temporal variation may yield a significant ( 0.1%) systematic effect which is only detected by this procedure. If the first measurements are not within m-standard-deviations of the mean then they are rejected. We check successive measurements for differences only if previous

98 70 Chapter 5. DETERMINING THE FOIL ATTENUATIONS measurements have failed the filter criteria, consistent with the imputed incomplete shutter motion. That is, for parameters n and m: (1) Determine the mean (x) and population standard deviation (σ) of all measurements in the series (nominally 10) except the first n. (2) If the first measurement in the series differs by more than mσ from x then it is considered to be an outlier and is rejected on this basis. Only if this measurement was rejected, examine the next. Apply this filtration to the first n measurements in the series. Figure 5.2 presents the results of a study of the number of rejections as a function of the rejection deviation m, for values of n between one and five. This figure shows that the measurements consist of three normally-distributed populations, the first representing the statistical count-rate fluctuations with half-width at m 1σ, and Figure 5.2: Number of outliers identified using the shutter motion filter, plotted as a function of m, the number of standard-deviations required for rejection. The curves show the rejections as the number of points examined for discrepancies n varies from 1 through 5. Data is plotted for measurements using the upstream ion chamber only. The plot shows the presence of three separate distributions in the data, the first representing the statistical count-rate fluctuations with half-width at m 1σ, and the second and third resulting from the effect of the shutter motion on the beam-on and beam-off measurements, respectively, with half-widths at m 10σ and m 2000σ, respectively. The plot shows that the use of m 3σ with n = 2 discriminates the affected measurements.

99 5.1. FILTERING THE DATA 71 the second and third resulting from the effect of the shutter motion on the beamon and beam-off measurements, respectively, with half-widths at m 10σ and m 2000σ, respectively. We are indeed able to filter the measurements affected by the incomplete motion of the shutter due to the good separation between the halfwidths of these populations. This figure indicates that the filter can discriminate the affected measurements using m = 3σ, n = 1, to filter the first affected measurement. Alternately, m = 6σ, n = 3 can be used if the filter is required to discriminate more strongly the points located at m = 50σ. Examination of the equivalent of Fig. 5.1 for these choices of m and n indicates that m = 3σ, n = 2 represents a good compromise between these extremes. This data filter was applied to the measurements recorded by each of the ion chambers separately. Table 5.1 presents the number of measurements rejected after examining the count rates recorded by each of the ion chambers. The rejections identified for the first and second measurement out of each series of ten are indicated. Over one quarter of all first measurements are rejected by the filter, indicating the existence of an additional transient affecting the first measurement, possibly due to the ammeter readout or the timing of the gate signal. The number of second measurements rejected by the filter is around the 7% level, of which 4% are highly correlated due to their common transient cause, and the remaining 3% are statistical. We note that there are slightly more rejections for the measurements made by the downstream ion chambers, and will discuss this later (in section 5.1.2). The central plot of Fig. 5.1 shows the recorded count rates after application of this filter. Measurements recording intermediate (and rapidly fluctuating) count # of rejections Ion chamber upstream first second all measurement downstream downstream together first second either Table 5.1: Number of outliers identified from the shutter motion filter, determined from the signature presented to each of the ion chambers. The total rejections figure represents the union of the sets of rejections for each of the ion chambers.

100 72 Chapter 5. DETERMINING THE FOIL ATTENUATIONS rates have been successfully isolated. The very few measurements in the intermediate count-rate region exhibiting good self-consistency (low spread) have resulted from some other cause, and are not rejected. The lower plot of Fig. 5.1 presents the rejected measurements. Measurements of intermediate count rate have been discriminated. However, a number of measurements of both high and low count rates have also been rejected by the filter. Many of the rejected measurements in the low count-rate region are slightly higher than the base count rate, possibly due to a settling effect in the ion chambers, where the ion chamber takes some time to reach low count rates following a period of high counting activity. The few measurements rejected at high count rates are statistical, and represent less than 2% of all tested measurements, and less than 0.2% of the entire data-pool. The over-discrimination represented by these measurements is of little concern as only 2909 out of a total of measurements have been discarded by application of this data filter. Hence there is no significant loss of statistic but rather a clear rejection of an inconsistent subset Aperture vibration settling Figure 5.3 presents the incident intensity ratio, I d I u, defined as the ratio of the count rate recorded in the downstream ion chambers to the count rate recorded in the upstream ion chamber for the measurements made with no sample in the beam path. Figures (a), (b), and (c) show the values of this intensity ratio for measurements recorded with the large, medium, and small apertures located in the path of the beam, respectively. These plots indicate the smooth variation of the ratio punctuated by a discontinuity between 20 and 21 kev, which has resulted from changing the ion chamber gain and offset settings. The measurements using the smallest apertures [Fig. (c) exhibit clear deviations towards lower values of the ratio. Examination of the time-sequence of the measurements shows that the deviations are also restricted to the first couple of measurements in each series, indicating that the cause of the deviation is transient. The deviations are explained by a residual oscillation of the apertures following a rotation of the daisy-wheel. This residual motion has caused the smallest aperture

101 5.1. FILTERING THE DATA 73 to interact with the straight-through beam, causing it initially to clip the x-ray beam. This clipping has resulted in a reduction of the count rate recorded by the downstream ion chamber compared to that recorded by the upstream ion chamber. The apertures were always used in a symmetrical configuration (identical apertures in the upstream and downstream locations), with one of three pairs of apertures bracketing the sample at any given time. These apertures were in the shape of a 9 mm diameter circle, a 9 mm by 1.8 mm rectangle and a 1.8 mm diameter circle. When using the smallest aperture with an x-ray beam of 1 mm by 1 mm Figure 5.3: Effect of the aperture clipping on the ratio of the count rates I d I u measured with no sample in the path of the beam. Measurements made using the (a) large, (b) medium, and (c) small apertures are presented separately. The 1 15% deviations towards lower values of the ratio for measurements made using the small aperture, clearly observed in (c), are consistent with the imputed clipping of the beam by the aperture. The change in the level of the ratio between kev has resulted from the adjustment of the ion chamber electronics settings, and will be normalised by the corresponding measurements with the sample located in the path of the beam. The spread of the ratios in this region is also of some concern, and will be dealt with later in this chapter. In figure (d) we present the ratios calculated from the measurements using all three apertures after the application of a n = 2 point, m = 6σ filter described in the text.

102 74 Chapter 5. DETERMINING THE FOIL ATTENUATIONS cross-section, a residual oscillation of 250 µm of the aperture was sufficient to cause clipping of the beam. One might expect that the clipped measurements would have been removed by the imposition of the filter described in section Indeed, aperture clipping is likely to be responsible for the greater number of rejections in the downstream ion chambers identified using the shutter motion filter. However, as those rejections were performed on the basis of count rate, their sensitivity is limited to the level of the beam intensity variations. Here we consider the ratio I d I u level of the beam intensity variations. to probe below the A method similar to the filter described in section was used to reject the clipped measurements. An investigation of the choice of the number of measurements to test and the deviation from the average of the remaining measurements required for rejection was undertaken. As might be expected, the clipping of the beam presents a strong numerical signature in the ratio of the counts. In Fig. 5.4 we present the number of rejections identified in the measurements recorded by the first downstream ion chamber for measurements made with no absorber in the path of the beam, as a function of m, the number of standard-deviations required for rejection, and n, the number of measurements examined for differences from the remaining pooled distribution. The black line on Fig. 5.4, resulting from the examination of only the first point of each series, shows clearly that the ratio consists of two distributions with different standard-deviations. One of these distributions corresponds to the statistical variation of the count-rate ratios, and the half-width of this distribution is located at m 3 4 σ. The second distribution corresponds to the clipping of the beam by the smallest aperture and has a half-width of m 130σ. Accordingly, it is appropriate to choose m between 3 and 20σ, depending on the relative merits of discriminating the data. We use m = 6σ as this does not discard much of the data in the statistical variation pool, whilst discarding most of the data forming the clipped distribution. There is little evidence in Fig. 5.4 to suggest that testing more than the first two points in each scan improves the rejection of the clipped measurements. Accordingly, we test the ratio of the downstream to upstream counts for m = 6σ deviations in the first n = 2 measurements of each scan.

103 5.1. FILTERING THE DATA 75 Figure 5.4: Number of outliers identified using the aperture clipping filter, plotted as a function of m, the number of standard-deviations required for rejection. The curves show the rejections as the number of points examined for discrepancies n varies from 1 through 3. Data is plotted for measurements using the smallest aperture only. The plot shows clearly the presence of two normal distributions in the data, one representing the statistical count-rate fluctuations with half-width at m = 1σ, the second due to aperture clipping, with half-width at about m = 100σ. The plot shows that the use of m = 4σ optimises the discrimination of the clipped data for the first-point examination. About 300 of the first points are rejected out of a total of about 600. A doubling of the number of statistical rejections is observed when the first two points from each series are examined. However, the number of rejections in the clipped population does not increase proportionately, indicating that the filter is identifying less aperture clipping in the second measurement. The disappearance of the sharp boundary between the statistical population and the clipped population with increasing n is due to the reduced power of the filter as the number of points in the parent distribution decreases. Plot (d) of Fig. 5.3 presents all of the data in the above three plots (i.e., measurements made using all apertures) after the measurements have been filtered to remove those affected by the aperture clipping. This plot shows a significant reduction in the number of deviations, especially in the energy regions between kev and kev. The persistent deviations in the kev region will be discussed in section 5.4. While Fig. 5.3 presents only those measurements made with no absorber located in the path of the beam, the aperture clipping filter has also been applied to the measurements made with an absorber located in the path of the beam as this effect is pernicious.

104 76 Chapter 5. DETERMINING THE FOIL ATTENUATIONS Aperture first downstream second downstream ion chamber ion chamber Large 4 6 Medium 7 1 Small Table 5.2: Rejections identified using the aperture clipping filter determined for each of the downstream ion chambers. This filter was required to remove the effect of residual motion in the apertures which caused the apertures to clip the x-ray beam, giving rise to a reduction in the recorded downstream count rate which was not due to the attenuation of the foil sample. The union of the rejections identified from the measurements recorded using the first and second downstream ion chambers was 497, indicating the very high correlation between the measurements recorded using the two downstream ion chambers, consistent with the effect of the beam clipping. Table 5.2 presents the number of rejections resulting from the application of this filter. The rejections are presented for the measurements made using each of the apertures. This table confirms that the effect is predominantly seen with the measurements made using the smallest aperture. The rejection of measurements recorded using the medium and large apertures is consistent with statistical variation. 5.2 Dark current The dark current count rate is measured by shuttering the incident beam and measuring the count rate in the absence of the beam. Such a measurement should accurately reflect the desired zero-beam count rate in the detectors, subject to the low-count-rate linearity of the detectors, as discussed in chapter 3. We have recorded the dark current count rate reported by each of the ion chambers on many occasions during the course of the measurement. These dark current count rates are subtracted from the count rates recorded when the beam is on, in order to properly determine the count rate which has resulted from the beam intensity. Figure 5.5(a) presents the dark current count rates recorded in the second downstream ion chamber. The sharp discontinuities in the dark current count rate correspond with necessary changes to the ion chamber settings, required to ensure that the ion chamber settings are optimised. The times at which the ion chamber settings were altered are indicated on the plot by the vertical dashed lines. We refer to a period in which the ion chamber settings were unchanged as an ion-chamber epoch.

105 5.2. DARK CURRENT 77 The measured dark currents are remarkably stable within each of the ion-chamber epochs. Figures 5.5(b) (k) show the dark current count rates recorded during each of the ion-chamber epochs on separate plots. For clarity we have used as the marker for the recorded count rate either a cross ( ) or a dot ( ), depending only on the number of data points presented. Consider the dark current measurements recorded during ion-chamber epoch #1, shown in Fig. 5.5(c). This plot shows recorded dark current count rates varying from about 180 cps to 230 cps over the s duration of this epoch. The beamon measurements made within this epoch are not shown on the plot: this plot shows the dark current measurements at exactly those times at which the beam-on measurements were not made. Hence, it is necessary to interpolate the dark current count rates to those times at which the beam-on measurements were in fact made. The lines presented on Figs 5.5(b) (k) are the result of a linear fit to the measured dark currents for each of the ion-chamber epochs. The curves above and below the fitted lines represent the uncertainty for the interpolated values, determined from the covariant error matrix. These fitted linear trends are appropriate for the interpolation of the dark currents. Alternative methods for interpolating the dark currents (by using the dark current measured closest to the desired time or using a linear interpolation of dark currents measured at neighbouring times) are both subject to errors for the measurement distribution shown in Fig. 5.5, which is dominated by noise. Furthermore, these alternate methods are unable to provide a robust uncertainty estimate for the interpolated dark current. When the variation in the measurements of the dark current count rate is dominated by noise, the dark current count rate is best determined by averaging the measurements close to the time at which we desire to know the dark current count rate. However, when the dark current variations exhibit long period oscillations, the dark current count rate is best determined by interpolating the measured dark current count rates on a time-scale shorter than the oscillation period. As the dark current count rates in Fig. 5.5 exhibit no significant oscillatory structure, we treat the measurements recorded in each of the ion-chamber epochs as a single, time-sequenced data pool and fit the measurements with a linear trend. The

106 78 Chapter 5. DETERMINING THE FOIL ATTENUATIONS Figure 5.5: Dark currents measured in the second downstream ion chamber, plotted as a function of the time at which the measurement was made. The marker for the recorded counts is either a point ( ) or a cross ( ) for the sake of clarity, depending only on the number of points presented. (a): all measured dark currents. The vertical lines on this plot indicate the times at which the ion chamber settings were altered. (b) (k): dark currents measured within each ion-chamber epoch. The straight line on these plots is the result of a linear fit to the measured values, and is used to interpolate the dark currents for use with measurements made at other times. The curves around the straight line are the estimated interpolation uncertainty, determined from the χ 2 r multiplied by the population standard deviation, determined at each point using the covariant error matrix returned by the fitting routine.

107 5.2. DARK CURRENT 79 fitted linear trend may describe long-period temperature or pressure variations (e.g. diurnal variation) or long-period amplifier fluctuations. We assume that counting statistics do not apply to the measured dark currents and accordingly we have applied an equal weighting to each measurement in the fit. In the absence of known uncertainties in the measured dark currents, we have scaled the covariant error returned by the fitting routine by the square root of the reduced-chi-squared χ 2 r of the fit. We use the population standard deviation to determine the uncertainty of the interpolated dark currents to quantify the observed variation of the dark current count rates. The sample variance (the one-sigma covariant error divided by the number of independent measurements in the fit) is not used as it is not clear whether the observed dark current variations are real (in which case the full extent of the variation is the appropriate uncertainty) or whether they result from statistical variations around a well-defined value (in which case the uncertainty should be estimated from the variance). In view of the difficulty of determining the signal and noise components of the measured dark currents, we have overestimated the uncertainty in the interpolated dark currents, corresponding to the assumption that the dark current variation is dominated by signal. The estimated uncertainties shown on Figs 5.5(b) (k) adequately describe the linear trend and the variation of the measured dark currents. However, in ionchamber epochs 6 and 9 [Figs 5.5(f) and (h), the measured dark current values often reach count rates of 10 cps, indicating that no counts have been recorded (1 is the minimum readout of the electronics) in the 0.1 s count time. As a result, the correct value of the dark current over these periods may thus be lower than our fitted value. The effect is more pronounced in ion-chamber epoch 9, where the correct value may be 1σ below our fitted values. The implications of using an incorrect value for the dark current for the determination of the mass attenuation coefficient will be addressed in section 8.7.

108 80 Chapter 5. DETERMINING THE FOIL ATTENUATIONS 5.3 Determination of the intensity ratios The intensity ratios are evaluated using I = I d dc d I u dc u (5.3) for the count rates recorded with no absorber in the path of the beam and an absorber in the path of the beam. The intensities I d and I u refer to count rates recorded by the downstream and upstream ion chambers, respectively. The count rates I d and I u have been measured simultaneously in this experiment, and thus record the intensity of the same x-ray beam. Figure 5.6 presents the coefficient of correlation between the counts recorded in the upstream and downstream ion chambers with a sample in the path of the beam. The overwhelming majority of the points presented in this figure indicate very strong positive correlations, very close to r = 1. However, at energies below around kev this figure includes a number of measurements with correlation ranging down to about 0.5. The density of these points increases towards the absorption edge. The cause of the decreased positive correlation is demonstrated in Fig. 5.7, where we present the same correlations as a function of the (approximate) value of the sample attenuation. The loss of positive correlation is associated with the measurement of absorbers with attenuation above about 6, and is due to the increasing influence of the counting statistical precision compared to the positively correlated components for these measurements. In view of the strong positive correlations between I d and I u, we use the average of the ratio of the measurements recorded by the upstream and downstream ion chambers, as discussed in section 3.5. The error in the intensity ratios is taken to be the spread of the ratios added in quadrature with an error contribution resulting from the dark-current interpolation. The spread of the ratios is determined using ( ) Id dc d σ(i) = std.dev., (5.4) I u dc u and the uncertainty component arising from the uncertainty of the interpolated dark

109 5.3. DETERMINATION OF THE INTENSITY RATIOS 81 Figure 5.6: Correlation between the counts recorded in the upstream and downstream ion chambers for the measurements made with a sample in the path of the beam. Strong positive correlations are recorded across all measured energies, although uncorrelated measurements are also observed on the high-energy side of the absorption edge at 20 kev. Figure 5.7: Correlation between the counts recorded in the upstream and downstream ion chambers for the measurements made with a sample in the path of the beam, presented as a function of the approximate sample attenuation. The uncorrelated measurements observed in Fig. 5.6 are associated with the measurement of absorbers with attenuation above about 6, and result from the reduced statistical precision associated with those measurements. Measurements made using absorbers with attenuation below 6 have very strong positive correlation.

110 82 Chapter 5. DETERMINING THE FOIL ATTENUATIONS currents is determined using [ ( ) 2 [ Id dc d Idc = dcu + dc u I u dc u dc d [( ) 2 [( Id dc d dcu Id dc d = + I u dc u I u dc u I u dc u ( Id dc d I u dc u ) dcu ) dcd 2 (5.5) I u dc u 2. (5.6) This additional dark current uncertainty is not completely independent of the uncertainty given by Eq. (5.4), as the use of an incorrect dark current results in a small increase in the spread of the ratios. However, the correlation between these error components is very weak and accordingly we assume that they are totally uncorrelated. Figure 5.8 shows the contributions to the uncertainty in the intensity ratios arising from: (a) the spread in the intensity ratios [Eq. (5.4); (b) the uncertainty in the upstream dark current and; (c) the uncertainty in the downstream dark current. Plot (b) shows clearly that the uncertainty in the upstream dark current does not limit the measurement accuracy. However, plot (c) shows that the uncertainty in the downstream dark current can dominate when highly attenuating absorbers are used, rising over 100% in extreme cases. This dominance is a result of the treatment of the dark current uncertainty by use of Eq. (5.6) and the intensity ratio uncertainty by use of Eq. (5.4). For measurements using highly attenuating specimens, the counts recorded downstream are essentially identical to a dark current measurement. The dark currents are interpolated using a long-period linear trend (Fig. 5.5), which may determine a larger uncertainty than that determined from the spread of repeated measurements. It is therefore consistent that the uncertainty is dominated by the interpolated dark current uncertainty for measurements using highly attenuating specimens. The uncertainty of the downstream dark current is significant when the counts recorded in the downstream detector are of the same order as the dark current measured in this ion chamber, i.e., when I d dc d. The use of an increased range of absorbers with ln( I 0 I ) 6 shows the importance of including the dark current uncertainties.

111 5.3. DETERMINATION OF THE INTENSITY RATIOS 83 Figure 5.8: Contribution to the uncertainty in the intensity ratio arising from: (a) the spread in the ratios of the intensities [Eq. (5.4) and the uncertainty in the upstream (b) and downstream (c) dark currents [Eq. (5.6). The upstream dark current uncertainty never dominates the measurement uncertainty, as expected. In contrast, the downstream dark current uncertainty dominates for measurements using highly absorbing specimens.

112 84 Chapter 5. DETERMINING THE FOIL ATTENUATIONS 5.4 Recovery of partially saturated data The ion chamber gain settings have been adjusted regularly to ensure that the counting chain operates at optimum sensitivity. These adjustments are necessitated by the changing efficiencies of the undulator, the monochromator, and the ion chambers across the energy range of the experiment. As these parameters change gradually with energy, the optimisation needs to be performed only occasionally. Sometimes, however, the settings are not optimally adjusted. This section deals with the rejection and the partial recovery of the measurements which have been affected in this manner. Figure 5.9 presents the intensity ratios for the blank (no sample) measurements recorded in the second downstream ion chamber only. This figure shows the smooth variation of the incident intensity ratios across most of the energy range. There are two major discontinuities in these values at about 20 and about 21 kev, corresponding to large adjustments to the ion chamber settings. These discontinuities were discussed briefly in section 5.1.2, and are not themselves indicative of measurement breakdown. However, the large spread of the measurements made between and kev is of concern, as it indicates the use of a poorly optimised counting-chain for these measurements. The measurements made between and kev are presented in greater detail on the right side of this figure. Here we see that the Figure 5.9: Incident intensity ratios determined from the counts recorded in the second downstream ion chamber. The discontinuities at about 20 and 21 kev are the result of an adjustment to the ion chamber electronics settings. The high variability in the incident intensity ratios between and kev indicates the poor optimisation of the ion-chamber settings over these energies.

113 5.4. RECOVERY OF PARTIALLY SATURATED DATA 85 measurement variation is about 5 10%. This variation will result in a corresponding imprecision in the determined attenuations unless corrected. This instability was caused by setting the gain too high on the downstream ion chambers. As a result some of the measurements recorded by the downstream ion chambers were saturated. This saturation has affected only the measurement of the unattenuated intensity as these measurements result in the greatest intensity downstream of the absorber. Saturation can result from one of a number of mechanisms in the detection, scaling and counting chains. It can relate to processes within the active volume of the ion chamber, and have very serious consequences for the interpretation of the measured counts both at saturation and leading up to the saturation event. We do not have such a situation here: in our case the saturation resulted from a numerical saturation of the scaling device used to record the measured counts. This device, consisting of an analogue-to-digital converter, was only capable of representing counts with six decimal digits, that is, counts of less than However, the amplifier gain was such that the output of the analogue-to-digital converter was sometimes greater than this value. As the synchrotron x-ray beam intensity naturally fluctuates within some range, and as the amplifier gains were only a little too high, we have counts recorded which were not saturated. One might choose to reject all data thus affected, especially where there is an option to use data from an auxiliary downstream ion chamber, as employed in this experiment. However, both downstream ion chambers had their gain settings too high in this region and the measurements in question cover an important energy range, from to kev. A procedure for filtering the data was developed to recover this valuable data. The characteristics of the unsaturated data are used to inform the rejection of data from the saturated data pool. Numerical saturation results in the loss of correlation between the count rate measured in the upstream and downstream detectors. Peaks in the beam intensity cannot be tracked by the saturating downstream detector and return a lower value of the ratio I d I u. As we have ten measurements recorded in rapid succession, we can use

114 86 Chapter 5. DETERMINING THE FOIL ATTENUATIONS the spread of this ratio to determine whether there are saturated measurements in the pool and, furthermore, can identify the saturated measurements as those which return a lower value of the ratio. The ratio I d I u is not constant for all measurements due to the energy dependences of the absorption of the air path and the detectors, and further due to the varying detector gain settings. We therefore normalise each series of measurements against the maximum value of the ratio (which is most likely to represent an unsaturated measurement), and evaluate the normalised intensity ratio ( Id I u )/( ) Id. (5.7) I u max in a scan Figure 5.10 presents the results of this investigation for the unsaturated measure- Figure 5.10: Study of the unsaturated measurements, used to inform the rejection of saturated measurements shown in Fig Plots (a) and (b) show the intensity ratios determined from the unsaturated data for the counts recorded in the first and second downstream ion chambers, respectively. Plots (c) and (d) show the histogram for the normalised intensity ratios defined in Eq. (5.7). These plots show that the normalised intensity ratios for the unsaturated measurements follow a smooth distribution. The location of the 95-percentile cut-off is shown on each of the histograms.

115 5.4. RECOVERY OF PARTIALLY SATURATED DATA 87 ments recorded using the first downstream ion chamber on the left, and using the second downstream ion chamber on the right. Plots (a) and (b) show intensity ratios I d I u for the unsaturated measurements. Plots (c) and (d) show the distributions of the normalised intensity ratios [Eq. (5.7), indicating the location of the 95-percentile, which determines reasonable limits of variation of the normalised intensity ratios. The 95-percentile cutoff identified from the unsaturated pool is used to inform the rejection of measurements from the saturated data. Figure 5.11 presents again the count ratios and the distribution of the normalised count ratios, this time for the saturated measurements. The spread in the normalised count ratios is far greater for the saturated measurements. Comparison of the histogram of the normalised ratios for the measurements recorded by the first downstream ion chamber [plot (c) with its unsaturated counterpart (Fig. 5.10) indicates that there is little similarity between these distributions. However, the histogram of the normalised ratios for the second downstream ion chamber [plot (d) displays a significant peak at about 1, indicating some similarity between these distributions. The location of the 95-percentile determined from the unsaturated measurements is indicated on these plots. Plots 5.11(e) and (f) present the saturated measurements after rejection of the measurements outside the 95-percentile. While the intensity ratios recorded by the first downstream ion chamber [plot (e) are improved by the filter, the persistent spread of the intensity ratios indicates that the saturated measurements have not been completely discriminated. The intensity ratios for the second downstream ion chamber [plot (f) are significantly improved by the filter, and show the correct tendency towards their maximum value. We therefore concentrate our discussion on the measurements recorded using the second downstream ion chamber. Several anomalously low values of the ratio persist in the results of the second downstream ion chamber. These have not been removed by the filter because they come from a series of measurements in which every measurement is saturated. When every measurement in a series is saturated, the only measurement to pass through the filter is that which gives the highest value of the ratio: the normalisation of this measurement to the maximum value of the ratio returns unity, which is certainly above the cut-off point determined from the unsaturated measurements. That is, at

116 88 Chapter 5. DETERMINING THE FOIL ATTENUATIONS Figure 5.11: Application of the cut to the saturated measurements. Plots (a) and (b) show the intensity ratios for the saturated measurements recorded by the first and second downstream ion chambers, respectively. Plots (c) and (d) show the histogram of the normalised intensity ratios [Eq. (5.7). The histogram for the first downstream ion chamber is flat, indicating little correspondence to the unsaturated distribution of Fig. 5.10(c). The histogram for the second downstream ion chamber shows a significant peak at about 1, in good correspondence with the distribution of Fig. 5.10(d). The 95-percentile determined in Fig. 5.10(c) and (d) is shown on these plots. Plots (e) and (f) present the intensity ratios after rejection of all measurements occurring below this 95-percentile. The measurements for both of the downstream ion chambers are significantly improved by the application of this cut.

117 5.4. RECOVERY OF PARTIALLY SATURATED DATA 89 least one measurement from every series must pass through the filter: namely, the measurement which has returned the highest value of the ratio. We have reapplied the filtering technique, but this time normalising the measurements to the highest value of the ratio at each energy. Accordingly, we examine the distribution of ( Id I u )/( ) Id. (5.8) I u max at each energy By examining this ratio we are able to discriminate against a series in which all of the measurements were saturated, provided that there is at least one unsaturated measurement at each energy. Figure 5.12 presents the distribution of the intensity ratios normalised at each energy for the unsaturated measurements. The distribution of the unsaturated measurements shows a double or multiply-peaked structure for the normalised ratios. This multiple-peaking may be the result of slight differences in the ratio resulting from measurements made using different aperture combinations, as these allow vary- Figure 5.12: Further study of the unsaturated data to be used to apply a second cut to the saturated data. Plots (a) and (b) present the unsaturated measurements. Plots (c) and (d) present the histogram of the intensity ratio normalised to the maximum value of the ratio at each energy according to Eq. (5.8). There is a slight lumpiness of these histograms on the scale of 0.3% of the normalised intensity ratio.

118 90 Chapter 5. DETERMINING THE FOIL ATTENUATIONS Figure 5.13: Application of the second cut to the saturated data. Despite the poor match between the sample distributions shown in plots (c) and (d) with the parent distributions of Fig. 5.12(c) and (d), we see that the application of a more restrictive 90-percentile cut to the saturated data has enabled us to recover the measurements recorded by the second downstream ion chamber. The uncertainty of the second downstream ion chamber intensity ratios has been reduced to about 0.5%. The intensity ratios recovered from the first downstream ion chamber are very sporadic. Accordingly, these measurements have not been used in the evaluation of the mass attenuation coefficients: all further analysis in this energy range uses measurements recorded using the second downstream ion chamber.

119 5.5. INTERPOLATING THE INCIDENT INTENSITY RATIOS 91 ing amounts of x-rays scattered by the air-path to reach the detectors. It is not the result of making measurements at different times, (during which air pressure or ion chamber characteristics may have changed), because the lumpiness of the distributions presented in Fig does not support the continuous variation of parameters expected by this hypothesis. Geometrical differences arising from the location of the first and second downstream ion chambers might be responsible for the significant difference between the distributions of the ratios. To increase the size and statistical reliability of the saturated pool, we have pooled measurements made using all aperture combinations. These data have not been pooled for the unsaturated measurements. Figure 5.13 presents the results of applying this second cut to the saturated measurements. The histograms for the downstream ion chambers [plot (c) and (d) show no peak structure similar to those of the parent distributions [Figs 5.12(c) and (d). In view of the absence of a significant match between the parent and sample distribution we have used a more-restrictive 90-percentile cut-off. This cut-off leads to a further significant reduction in the spread of the filtered ratios. Plot 5.13(f) shows that the sample distribution has been markedly improved by the second round of filtering, with the variation of the measured ratios fairly consistent at about 0.5%. This represents a significant improvement on the 2 20% variation of the saturated measurements presented in Fig The intensity ratios recovered from the first downstream ion chamber [5.13(e) are very sporadic, indicating that the integrity of this pool cannot be restored by application of this filter. These measurements have not been used in further evaluation of the mass attenuation coefficients. 5.5 Interpolating the incident intensity ratios The transmission probability P is determined from the ratio of the attenuated and incident intensity ratios according to ( ) / ( ) Id dc d Id dc d P =, (5.9) I u dc u s I u dc u b

120 92 Chapter 5. DETERMINING THE FOIL ATTENUATIONS where the subscript s refers to the attenuated intensity ratio, determined from measurements made with a sample in the path of the beam and the subscript b refers to the incident intensity ratio, determined from measurements made without a sample in the path of the beam (no sample = blank = b). The incident intensity ratio is not recorded at the same time as the attenuated intensity ratio. The incident intensity ratio normalises the effects of the air-path attenuation and detector-specific settings, characteristics and offsets. We expect each of these variables to change smoothly with time, so the time-based interpolation and extrapolation of the incident intensity ratio is appropriate. Figure 5.14 presents incident intensity ratios determined at a number of energies. The determined values are plotted using three marker symbols indicating the size of the aperture used for the determination: + small, medium, and large. The ordinate represents the time in seconds. The zero of this axis is arbitrary. A linear trend has been fitted to the incident intensity ratios, and is shown as a solid line. The dashed lines around this trend indicate the σ χ 2 r uncertainty. The small plus symbols appearing along the fitted trend indicate the interpolated values of the incident intensity ratio. The 41.5 kev plot shows that we have been required to interpolate the incident intensity ratio on many occasions between s. This high measurement density relates to the a large number of measurements made in rapid succession over the entire surface of one of the foils. These full-foil mapping measurements will be interpreted further in chapter 6. The 37.5 kev plot shows a good distribution of measured incident and attenuated intensity ratios. It is clear from this plot that the time-interpolation can be used for these ratios. However, a number of alternate hypotheses can also explain the trends depicted here. While the 37.5 kev data are consistent with a constant value of the incident intensity ratio, measurements made over a longer time period at 24.4 kev indicate clearly that the incident intensity ratio can have significant temporal dependence. As an further alternate hypothesis, the intensity ratios can be tested for correlations with the diameter of the aperture used to make the measurement (due to air scattering, for example). However, the 24.4 kev plot demonstrates that temporal

121 5.5. INTERPOLATING THE INCIDENT INTENSITY RATIOS 93 Figure 5.14: Interpolation of the determined incident intensity ratios to the moment at which the attenuated intensity ratio was measured. The three marker symbols indicate the size of the aperture used for the determination of the incident intensity ratio: + small, medium, and large. The uncertainties associated with the determination are indicated by the error-bars. The ordinate represents the time in seconds. The zero of this axis is arbitrary. A linear trend has been fitted to the incident intensity ratios, and is shown as a solid line. The dashed lines around this trend indicate its σ χ 2 r uncertainty. The small plus symbols appearing along the fitted trend indicate interpolated values of the incident intensity ratios used for the determination of the transmission probability.

122 94 Chapter 5. DETERMINING THE FOIL ATTENUATIONS variations are far greater than those due to the aperture diameter. An alternate approach to the time-weighted interpolation would be to interpolate the incident intensity ratios as a function of energy for the measurements made within each ion-chamber epoch. However, while the first-order variation of the intensity ratios due to the energy-dependence of the air attenuation and the ion chamber efficiencies would be readily described using an energy interpolation, higher-order variations such as those correlating with time would not be probed using such an alternate approach. The time interpolation of the intensity ratios is therefore most appropriate for these measurements. In the absence of variations of the intensity ratio with the diameter of the aperture employed for the measurement, we have pooled the intensity ratios determined with all aperture combinations and have observed a consequent improvement in the interpolated intensity ratio. The uncertainty for the interpolated intensity ratios was determined using the population standard deviation, σ χ 2 r, which treats variations in the incident intensity ratio as real temporal variations, and which estimates the maximum uncertainty for the interpolated values. The kev plot shows the results of fitting the linear trend at an energy where only six values of the normalised incident intensity have been determined. We have set six to be the minimum number of measurements required for the interpolation. Uncertainties for the extrapolated points are magnified, as is appropriate for the small amount of information supplied by the measured values and the large time difference between the measured and extrapolated incident intensity ratios. The kev plot shows a case where only five measurements of the incident intensity ratio have been taken. In this case we do not fit a linear trend but use instead the weighted mean value for the interpolation and extrapolation. The uncertainty is determined from the maximum of the weighted error [Eq. (A.2) and the weighted mean standard error [Eq. (A.3). At kev there has been only one measurement of the normalised incident intensity. In this case we use the measured value and its uncertainty to determine all extrapolated intensity ratios.

123 5.6. THE FOIL ATTENUATIONS 95 These few examples clearly demonstrate that the time-weighted interpolation of the intensity ratios is appropriate for the measurements that we have presented in Fig Our use of the time-weighted interpolation accounts for possible changes in the characteristics of the ion chamber and variations in atmospheric conditions [ The foil attenuations The attenuation [ µ [t of each of the foils is calculated from the negative of the logarithm of the transmission probability according to { ( ) / ( ) } [ µ Id dc d Id dc d [t = ln. (5.10) I u dc u s I u dc u b Values of the attenuation as a function of energy are shown in Fig On the scale of this plot the measured attenuations are consistent, with only a few exceptions apparent for the 150 µm (light blue) foil between 25 and 29 kev and the 250 µm (orange) foil between 15 and 17 kev, where their attenuations rise above about 5 6. The uncertainty in the attenuation is determined by propagating the uncertainties in the incident and attenuated intensity ratios (subscripts b and s, respectively) determined from Eqs (5.4) and (5.6) through Eq. (5.10) according to [[ µ { = [t [ ( Id ) dc d Iu dcu s ( ) } 2 { / Id dc d + I u dc u s [ ( Id ) dc d Iu dcu b ( ) } 2 / Id dc d, (5.11) I u dc u Figure 5.16 presents the uncertainties for the foil attenuations presented in Fig This figure shows that there is at least one foil at every energy whose directlyevaluated statistical precision is better than 0.5% (with the exception of the measurement at 30 kev), and that the directly evaluated precision in [ µ [t is generally stable at about 0.03%. b

124 96 Chapter 5. DETERMINING THE FOIL ATTENUATIONS Figure 5.15: Attenuations [ µ [t determined using Eq. (5.10). The absorbers span a wide range of attenuations at each measured energy, allowing attenuation-dependent systematic errors to be detected. The colours represent foils of nominal thickness: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm; light blue 150 µm; maroon 200 µm; orange 250 µm. Six different symbols represent measurements using three different aperture combinations using the two downstream ion chambers: closed symbols (,, ) and line symbols (, +, ) indicate measurements recorded using the first and second downstream ion chambers with the small, medium, and large aperture combinations, respectively. Point symbols ( ) occurring between and kev represent measurements affected by numerical saturation.

125 5.6. THE FOIL ATTENUATIONS 97 Figure 5.16: The percentage uncertainties of the attenuations determined using Eq. (5.11). The colours represent foils of nominal thickness: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm; light blue 150 µm; maroon 200 µm; orange 250 µm. Six different symbols represent measurements using three different aperture combinations using the two downstream ion chambers: closed symbols (,, ) and line symbols (, +, ) indicate measurements recorded using the first and second downstream ion chambers with the small, medium, and large aperture combinations, respectively. Point symbols ( ) occurring between and kev represent measurements affected by numerical saturation. In the next chapter we determine the integrated column density [t for each of the measured foils using a variety of techniques, and develop a new technique for determining an absolute value for the mass attenuation coefficient which uses measurements of the attenuation made across the full surface of one of the foils. In principle the mass attenuation coefficient [ µ can be determined by dividing the foil attenuation by the appropriate integrated column density. However, in chapter 8 we adopt an alternate technique which allows us to probe the measurements for the presence of a range of systematic errors which can affect the determination of the mass attenuation coefficient. The results presented in Figs 5.15 and 5.16 show precisions better than 0.03%.

126

127 Chapter 6 Foil metrology In the previous chapter we determined the attenuations [ µ [t of a number of thicknesses of molybdenum foil. In order to determine the mass attenuation coefficient [ µ from these attenuations we must determine the integrated column density [t of each of the measured foils. Here we consider the information that we can obtain about the thickness and the integrated column density of each of the foils and describe our use of a full-foil mapping technique to determine the mass attenuation coefficient at a single energy on an absolute scale. 6.1 Introduction We have determined the thickness t and the integrated column density [t of the foils using a variety of independent techniques including (1) Micrometer measurements of the foil thickness. (2) Mass and area measurements to determine the average integrated column density. (3) Measurement of the attenuation of x-rays at locations across the entire surface of one foil to determine the average attenuation [ µ [t of this foil. We have combined the information provided by these measurements to report a highly-accurate determination of the mass attenuation coefficient in the publication Full-foil x-ray mapping of integrated column density applied to the absolute determination of mass attenuation coefficients, a copy of which appears in Appendix D. 99

128 100 Chapter 6. FOIL METROLOGY 6.2 Micrometer measurements of the foil thickness The thickness of each foil was measured at 5, 9 or 25 locations across its surface using a Mitutoyo 1-µm-precision digital micrometer. These measurements determine a thickness profile for each of the foils, the number of measurements reflecting the role played by the foil in the attenuation measurement. Previous experimenters [13, 14, 145 have used such a thickness profile in conjunction with a measurement of the average integrated column density to determine the distribution of the absorbing material at all locations across its surface. However, we have developed a more accurate technique for determining the distribution of the absorbing material, and so the micrometer measurements are only used in comparison with the information obtained by other methods. 6.3 Determining the average integrated column density The Beer-Lambert equation describes the attenuation of x-rays of a given energy passing through an absorber by ( ) I ln = [ µ [txy, (6.1) I 0 xy where I and I 0 represent the attenuated and unattenuated beam intensities respectively, [ µ the mass attenuation coefficient of the absorbing material at a given energy, and [t xy the integrated column density along the path taken by the x-ray beam through the location (x, y) on the absorber. The integrated column density represents the path integral of the density through the absorber according to [t xy = xyz dz = txy 0 xyz dz, (6.2)

129 6.3. DETERMINING THE AVERAGE INTEGRATED COLUMN DENSITY 101 where xyz represents the three-dimensional variation of the density within the absorber and t xy the thickness of the absorber at the point (x, y). The integrated column density provides the best macroscopic measure of the total amount of absorbing material in the path of the beam. We use [t to represent the integrated column density in order to maintain connection with the traditional quantities density and thickness t, but employ the square brackets to indicate that [t (and likewise [ µ ) are directly measured quantities and not combinations of µ,, and t. Traditionally the local value of the integrated column density has been determined as the product of the density and the thickness. Since the local density of a sample is difficult to determine, such techniques have assumed homogeneity, and thus that the bulk density accurately reflects the local density of the sample. Subject to this assumption, the problem of determining the integrated column density was reduced to one of determining the local thickness of the specimen. This problem had been studied in detail, and the local thickness had been determined by a variety of techniques using micrometry [13, 14, 31, 151, 152, profilometry [13, optical microscopy [115, step-profilometry [153, and x-ray scanning techniques [13, 14, 151. The author has also attempted to address this problem by use of interferometric measurements on the surface of foil absorbers to determine relative thickness variations on each surface of the absorber. The results of that investigation are reported in Appendix B. The average integrated column density [t has the advantage that it can be determined from measurements of the mass and of the area of the foil. That is, [t = mass Area = m A. (6.3) Accordingly, each foil has had its mass m determined by weighing on a microgram balance and area A by use of a shadow projection device with an x-y translatable stage. We report below the results of these mass and area determinations.

130 102 Chapter 6. FOIL METROLOGY Measuring the foil masses The masses of the absorbers have been determined by weighing using a Mettler 0.1-µg balance. The samples were weighed a number of times (usually three) to determine the repeatability and stability of the mass measurements: when the measurements were found to be suitably repeatable, the number of repetitions was decreased (usually to two). Measurements were made with no sample on the balance to determine the mass offset. These offset measurements were made frequently at the beginning of a measurement session to determine the stability of the mass offset. Once the mass offset had stabilised the measurement frequency was reduced. These sample and offset measurements can be used to determine the masses of the absorbers with a given uncertainty. However, in the course of making the measurements we noticed an inconsistency in the operation of the balance which increased the mass uncertainty. The operation of the balance involves three distinct mechanisms. A coarse range requires the operator to select an appropriate combination of counterweights by way of a number of levers. Once the correct counterweight is selected, a 10-mg full-scaledeflection, 100-µg minimum-division spring balance is employed. The precision of this spring balance is enhanced by adjusting the linear offset of the spring balance, enabling the mass to be determined to within a fraction of a microgram. It is possible to weigh some of the foils using two distinct coarse scales of the balance, as there is some overlap between their ranges. It was found that the mass determined using the different ranges differed by about 100 µg. This inconsistency may be due to a nonlinearity in the spring balance or to a degradation of the counterweights, leading to a systematic change in their incremental mass (e.g. as a result of oxidation). However, it is most likely that one of the measurements was not operating in the conventionally used range and hence was less accurate. We have tested the operation of the balance by use of a second weighing device. Weighing one sample (code mo 23) with the balance determined m = g, using an electronic scale determined m = g. The difference between these measurements (about 100 µg) is in good agreement with that observed from the

131 6.3. DETERMINING THE AVERAGE INTEGRATED COLUMN DENSITY 103 Figure 6.1: Uncertainties for the determined foil masses. The 1 form of the percentage uncertainties is due to the constant and dominant mass uncertainty determined m from the reproducibility tests. Mass uncertainties smaller than 0.01% have been achieved for all foils heavier than 1 g. inconsistency of the balance measurements. We have used this difference to set a lower bound for the uncertainties of the determined foil masses. This 100-µg uncertainty dominates the uncertainties for all measured foil masses. This limitation could be reduced dramatically by re-calibrating the balance scale. However, as the total uncertainty of [t is dominated by that of the area determination, this was unnecessary. The Mettler balance is buoyancy compensated for a sample of density = 8.4 g/cm 3. As the density of molybdenum is 10.2 g/cm 3, buoyancy effects yield an additional residual correction of around % to the measured masses which is insignificant in comparison with other uncertainties in the measurement. Figure 6.1 shows the percentage uncertainties in the determined masses. This figure shows that mass uncertainties of less than 0.01% have been achieved for all foils of mass greater than 1 g Measuring the foil areas The area was determined using a Mituotogyo travelling-stage optical comparator (shadow profiler) by measuring the (x, y) locations of all extremities of the absorber. The area measurement schedule was chosen to economise the time involved in the

132 104 Chapter 6. FOIL METROLOGY C D O B A E Figure 6.2: Definition of foil geometry used for the calculation of the foil area. measurement while producing reasonable estimates of the measurement error. Five corner or vertex locations were measured for each foil, one on each corner with a repeated measurement of the first corner to ensure that the foil had not moved during the measurement. Further measurements were made along any edges judged to be insufficiently straight. about one edge of every second foil. Such edge location measurements were required for The area was calculated from the measured foil extremities. We have developed an algorithm for the calculation of the sample areas which we summarise with reference to Fig. 6.2 (1) Choose an interior point near the centre of the foil. The location of this point is not critical for the algorithm. Call this interior point O. (2) Check that the perimeter locations (A, B... ) occur in order of increasing angle about the point O to avoid double-counting of area. (3) Calculate the angle subtended at O by each pair of neighbouring perimeter locations ( AOB, BOC... ) and also the distance from each perimeter location to O ( OA, OB... ). (4) The area of each triangular segment is evaluated from one-half of the crossproduct of each pair of neighbouring vectors, Area(AOB) = 1 OA 2 OB = 1 OA 2 OB sin ( AOB). (5) The sum of these triangular contributions determines the foil area. The uncertainty in the measured locations was determined from the variance of repeated measurements, which was found to be δ x = δ y = 5 µm δ on both measurement axes.

133 6.3. DETERMINING THE AVERAGE INTEGRATED COLUMN DENSITY 105 The area uncertainty is estimated by considering the consequence of the measurement uncertainty at the vertex locations. The position uncertainty on one axis at a vertex location results in an additional triangular area contribution A of A = 1 2 (side length) (position uncertainty). (6.4) As all of the foils are close to square and of approximately the same area (nominally 625 mm 2 ), we write this as 1 2 = 1 2 Area δ (6.5) A δ. (6.6) The area uncertainties arise from the positional uncertainty along both measurement axes at each corner, giving eight triangular contributions in all. These uncertainties are independent: accordingly, we sum each contribution in quadrature to determine the area uncertainty as A = 2 ( 1 ) 2 A δ 2 corners = 2A δ. The determined area uncertainty is foil-independent and is used for all measured foil areas. In this manner, the foil areas were determined with an uncertainty of A A = 2Aδ A = 2δ A, or around 0.03% The determined average integrated column densities The upper half of Fig. 6.3 shows the average integrated column density of each foil, determined by use of Eq. (6.3). Below are the uncertainties for these average integrated column densities. The accuracy of these average integrated column densities is limited to 0.07% by the mass determination for light samples and to 0.03% by the area measurement for heavier samples.

134 106 Chapter 6. FOIL METROLOGY Figure 6.3: Average integrated column densities [t of the different foils (upper), and their associated uncertainties (lower). The foil codes are indicated on the plot. The unusually large uncertainty associated with the foil code mo 402 (measurement 36) is due to an unresolved difference between two area measurements. This foil was not used for determining the mass attenuation coefficient. The uncertainties of the other foils converge to about 0.03% as the foil thickness increases, limited by the accuracy of the area measurement.

135 6.4. THE FULL-FOIL MAPPING TECHNIQUE The full-foil mapping technique The mass attenuation coefficient of a foil absorber can be determined accurately by using attenuation measurements made across the entire surface of the absorber. The average of the attenuation measurements made at a number of (x, y) locations on a foil [from Eq. (6.1) is ( ) I ln = [ µ [txy. I 0 xy As the mass attenuation coefficient [ µ is constant at a given energy, we re-cast this as ( ) I ln = [ µ [txy. I 0 xy When the entire surface of the absorber is probed by the (x, y) x-ray mapping, [t xy can be identified with the average integrated column density of the specimen [t. Thus ( ) I ln = [ µ [t, (6.7) I 0 xy and by mapping the attenuation across the entire sample we can determine the mass attenuation coefficient without directly determining the local integrated column density at any point of the absorber. The local value of the integrated column density does not appear in Eq. (6.7). Variations of this quantity across the surface of the absorber have limited other non-local measurements [13, 14, 26, 28, 29, 30, 32, 154, 155, 156, but do not limit the current technique The attenuation profile The nominally mm 3 molybdenum foil used for this measurement was mounted in a holder, which was in turn mounted on a combination of translatable and rotatable stages, so that the sample could be accurately positioned in the path of the beam. The sample holder was machined from two sheets of mm 3 Perspex. These were constructed by drilling a hole of approximately 13 mm diameter through the Perspex. This hole was bevelled, meeting the full thickness of the holder at a diameter of approximately 24 mm, as shown in Fig Through-holes and

136 108 Chapter 6. FOIL METROLOGY Back section of the holder Foil mounted in holders Figure 6.4: Schematic of the holder construction. The inner circular region represents a hole through the holder. The thickness of the holder increases radially between the circular regions, reaching the full thickness of the holder at the outer circle. Eight through-holes for the mounting screws are located around the perimeter of the holder. Two such pieces are used to hold the foil as seen on the right. The foil is indicated by the grey region. threads for eight screws were drilled and tapped around the perimeter of the holder. The sample foil was placed between two of these sheets and the screws tightened so that sample motion was prevented with minimal stress applied to the sample. This design minimised any motion of the sample within the holder so that subsequent attenuation measurements could be made reproducibly through the same location on the absorber. Figure 6.5 presents the result of the two-dimensional x-ray scan which has been processed (in chapter 5) to determine a value for ln ( ) I = ([ ) µ [txy I 0 xy S+H,mea at each measured (subscript mea ) (x, y) location on the sample mounted in the holder (subscript S + H), i.e., an attenuation profile of the sample-plus-holder. The measurements were made with a 1 1 mm 2 x-ray beam at 1 mm intervals in the x and y directions indicated in the figure. The attenuation profile clearly exhibits a number of features which have resulted from the attenuation of the x-ray beam by the sample and the holder. In the central region we see values resulting from measurements where the beam has passed through the molybdenum sample only.

137 6.4. THE FULL-FOIL MAPPING TECHNIQUE 109 Figure 6.5: Attenuation profile ([ ) µ [txy of the sample mounted in the holder. S+H The attenuation profile was produced from the x-ray scan, processed to determine a value of ln ( ) I = ([ ) µ [txy I 0 at every (x, y) location across the surface xy S+H,mea of the foil. The x-ray beam used to make the measurements was 1 1 mm 2 and measurements were taken at 1 mm intervals across the foil.

138 110 Chapter 6. FOIL METROLOGY Figure 6.6: Uncertainties σ( [ µ [t)s+h,mea in the measured attenuation at every point in the x-ray scan, determined from the standard error of ten repeated measurements. The directly-determined uncertainty is relatively constant at around [units of ln ( I I 0 ). Surrounding these points is a conical ramp in the measured attenuation resulting from the increasing thickness of the (bevelled) Perspex holder in the path of the beam. These measurements plateau at a value corresponding to the attenuation of the sample plus the full thickness of the holder. The skirt surrounding this plateau corresponds to measurements that have been made with the x-ray beam either fully or partially by-passing the sample. Thus, the values around the edge drop sharply from the sample-plus-holder value to that of the holder alone. The several sharp spikes in the measured attenuation occurring near the corners and mid-way between the corners of the sample are the result of the x-ray beam hitting the steel screws which have been used to mount the sample in the holder. Measurements at each (x, y) location on the absorber were repeated ten times in rapid succession to yield a direct measure of precision and reproducibility and to

139 6.4. THE FULL-FOIL MAPPING TECHNIQUE 111 optimise the treatment of correlations in the counting chain [133, 134 as reported in chapter 5. In Fig. 6.6 we present the directly-quantified uncertainties in the measured data σ( [ µ [t)s+h,mea, evaluated as the standard error of the results obtained from the ten repeated measurements. This figure shows that the uncertainty is relatively constant for all of the measurements at about [units of ln ( I I 0 ) Removal of the holder attenuation In order to use Eq. (6.7) to determine the mass attenuation coefficient on an absolute scale, we need to remove the effect of the holder attenuation on the measured attenuation profile. The simple and uniform geometrical shape of the holder and its clear signature in the attenuation profile allow the holder component of the attenuation profile to be modelled and then subtracted from the total measured profile. The sample can be removed from the holder and the holder attenuation profile measured in isolation [145. This alternate approach is useful but the holder and the sample-plus-holder attenuation profiles must be in exact register prior to subtraction. This registration is of similar complexity as the approach adopted here, and we preferred to leave the sample undisturbed in the holder. We have constructed a program to fit the total attenuation profile using a standard Levenberg-Marquardt least-squares fitting routine. The fitting function takes as input parameters describing the geometrical properties of the holder and of the sample, and calculates the resulting attenuation at any given (x, y) location on the sample-plus-holder. The measured values in Fig. 6.5 result from the interaction of an x-ray beam of finite cross-sectional area A with the sample and holder, and are thus predicted from ( [ µ ) [txy S+H,mod [ { ( [ I Beam 0,xy exp µ ) [txy S,mod = ln I Beam 0,xy da ( [ µ ) } [txy H,mod da, (6.8) where the subscript mod refers to the modelled attenuation profiles. For a beam of

140 112 Chapter 6. FOIL METROLOGY Beam size to scale: Figure 6.7: Illustration of the seven classes of location on the foil-plus-holder through which the x-ray beam may pass. Each of these classes of location requires a different algorithm for calculating the attenuation. The circles represent the bevelled region on the holder. The large grey square represents the foil. The dimensions of the x-ray beam used to make the measurements ( 1 mm 1 mm) is indicated by the small grey square. uniform intensity I 0,xy = I 0 this reduces to ( [ µ ) [txy [ 1 = ln A [ = ln S+H,mod exp Beam { { ( [ exp µ ) [txy ( [ µ ) [txy S,mod S,mod ( [ µ ) } [txy da H,mod ( [ µ ) } [txy. (6.9) H,mod In practice the average of the exponential of the attenuation over the illuminated region only needs to be calculated at those locations where the modelled attenuation varies significantly over the beam footprint. This averaging has been undertaken only for those measurements made around the edge of the foil sample. The attenuation profile is determined in units of [ µ [t. In these units the attenuation of multiple absorbers is calculated by adding the component attenuations. Accordingly, we evaluate the attenuation of the foil-plus-holder as the sum of the contributions from the foil and the holder separately. The geometrical construction of the foil-plus-holder results in seven distinct

141 6.4. THE FULL-FOIL MAPPING TECHNIQUE 113 classes of location at which attenuation measurements can be made. Indicated in Fig. 6.7, these are where the x-ray beam passes: (1) through the foil only, through the central region of the holder, (2) partially through the central region and partially through the bevel of the holder, (3) fully through the bevel of the holder, (4) partially through the holder bevel and partially through the holder plateau, (5) fully through the holder plateau, (6) partially through the holder plateau with foil in-between and partially through the holder plateau without foil in-between, and (7) through the holder plateau with no foil in-between. At each of these locations we require a different algorithm for predicting the attenuation of the foil-plus-holder. Although the foil is nominally uniform, we allow for some wedge-like character, describing a linear increase in thickness across the surface of the foil. We parameterise the foil geometry by use of eleven parameters: the locations of the four corners of the foil (eight parameters); the base attenuation of the foil and; two parameters describing the wedge-like character of the foil in the x and y directions. When the beam passes fully through the foil [i.e., all regions except (6) and (7), the contribution of the foil to the attenuation profile is evaluated using [ µ [txy = [ µ [tf + [ µ [tfwx (x x 0 ) + [ µ [tfwy (y y 0 ), (6.10) where [ µ [tf is the parameter describing the attenuation of the foil, and [ µ [tfwx and [ µ [tfwy describe the wedge-like character of the foil in the x and y directions, respectively. We orthogonalise the parameters by evaluating the wedge contribution from the measurement at (x 0, y 0 ), nearest to the central point of the foil. The correction to the foil attenuation resulting from its wedge-like character should be evaluated by use of the logarithm of the average of the exponential of the attenuation. However, in the cases treated here the wedge-like character of the foil is so small that our use of the average of the attenuation is sufficiently accurate. When the beam passes through the full thickness of the holder [i.e., regions (6) and (7), the contribution of the holder to the attenuation profile is evaluated using [ µ [txy = [ µ [th + [ µ [thwx (x x 0 ) + [ µ [thwy (y y 0 ), (6.11)

142 114 Chapter 6. FOIL METROLOGY where the symbols are defined as for Eq. (6.10). The attenuation of the beam when passing through the holder bevel [i.e., region (3) is evaluated by treating the beam as if it were concentrated at the centre-of-mass of the illuminated bevel region according to [ µ ( ) (x x0 ) [txy = 2 + (y y 0 ) 2 R 1 [ µ [th, (6.12) R 2 R 1 where R 1 and R 2 are the inner and outer radii of the bevel and [ µ [th is the attenuation of the full thickness of the holder. This treatment ignores the obvious change in the thickness of the holder along the bevel. The error arising from this approximation increases with the attenuation of the holder and decreases with the dimensions of the beam. We have used a small beam and, for the measurement at 41.5 kev, the holder attenuation is slight, so this approximation is justified. However, for the measurement at 13.5 kev, the approximation may not be valid due to the increased holder attenuation. When the beam footprint falls on the boundary between the regions described above [i.e., regions (2), (4), and (6), the attenuation is calculated by determining the fraction of the beam footprint falling into the two regions. The attenuation in the two regions is evaluated by adding the fractional contributions of the attenuations in these two regions. For example, when the beam passes partly through the foil [i.e., region (6), the foil contribution to the attenuation profile is evaluated using [ µ [txy [ { = ln (1 f) + f exp [ µ [tf [ µ [tfwx (x x 0 ) [ µ [tfwy (y y 0 )}, (6.13) where f is the fraction of the beam footprint interacting with the foil. The result of fitting the combined sample and holder attenuation profile is shown in Fig We note the very good agreement with the general form of the measured attenuation profile of Fig A more detailed investigation of the quality of this

143 6.4. THE FULL-FOIL MAPPING TECHNIQUE 115 Figure 6.8: Results of fitting the attenuation profile presented in Fig The fitted profile has been produced by calculation whose inputs are the fitted geometrical dimensions of the sample and of the holder scaled by their fitted attenuations. The function has been evaluated at each (x, y) location by summing the attenuation of the sample and holder calculated for measurement with a 1 1 mm 2 beam.

144 116 Chapter 6. FOIL METROLOGY Figure 6.9: Residuals of the fit to the attenuation profile. The random appearance of the distribution of the residuals implies the absence of any additional significant systematic or geometrical correction. The grey scale is arbitrary. Measurements around the foil edges (extreme bottom and right, second row from top, second column from left) exhibit variations resulting from small displacements of the foil in the beam. These variations are not described by the model function and provide no information for fitting the holder. To enable the holder attenuation profile to be more properly isolated the weighting of these measurements has been decreased in the fit. fit can be undertaken by examining the distribution of the residuals, defined as residual = ([ µ [t)s+h,mea ( [ µ [t)s+h,fit σ( [ µ, (6.14) [t)s+h,mea and presented in Fig The grey scale in this figure does not show the residual magnitudes but displays their distribution. Any structure in the residual pattern would indicate an inadequate description of the modelled sample or holder. The pattern of residuals shows no significant structure and, in particular, shows no structure reminiscent of the shape of the holder. This indicates that the holder component of the attenuation profile has been successfully modelled. It also shows a good normal distribution of values, with all levels of the grey scale well-represented.

145 6.4. THE FULL-FOIL MAPPING TECHNIQUE 117 To ensure that the holder component of the attenuation profile is properly determined it is necessary for the measured data to be correctly modelled by the fitting program. However, measurements taken with the beam overlapping the edge of the foil are subject to significant variation, resulting either from a tiny displacement of the foil in the beam or from a small change in the intensity distribution over the beam area. These variations depend only on the properties of the beam and the foil, and thus provide no information for the fitting of the holder component of the attenuation profile. Furthermore, as these variations are not quantified in the directly determined input error estimates, they can confuse the fitting of the holder component. Accordingly, the uncertainty of these measurements has been increased by a factor of 100. Measurements made where the x-ray beam interacted with the screws were discarded. The frame of middle-grey on the edges of Fig. 6.9, with a residual of approximately zero, results directly from the increased error bars applied to these measurements. We have modelled a sample which is perfectly flat but which may have a wedgelike shape, becoming linearly thicker as one traverses the surface of the absorber. By employing such a model the fitting program is able to resolve these features in the attenuation profiles of the sample and of the holder. The need to include such geometrical features of the sample and holder in the fitting program has been determined empirically by examining the distribution of the residuals of the fit. For instance, the wedge-like shape of the foil has been included in the fitting program in response to an observed systematic left-right pattern in the residuals of the fit when the foil was modelled as a perfectly flat object with parallel surfaces. Any secondorder variation in the integrated column density of the absorber (i.e., curvature) would show up in the residuals as a series of rings of alternating positive and negative deviation from zero. There is no such correlation in the residuals, so such curvature is not significant. The holder has also been allowed to have a wedge-like shape. The wedge-like features of the sample and holder are not degenerate due to the large number of sampled points where the foil and the holder are probed in isolation. The number and distribution of these points is sufficient to allow these parameters to be well

146 118 Chapter 6. FOIL METROLOGY resolved by the fitting routine, with low correlation. While the recovered sample attenuation profiles may differ when further higherorder contributions to the model are included, the average of these attenuations required to evaluate [ µ from Eq. (6.7) is insignificantly affected. In particular, the difference between the average sample attenuation obtained with and without the assumed wedge-like character of the foil and holder is significantly less than the uncertainty associated with each of the fitting schemes. Similarly, errors in the fit resulting from the beam size and non-uniform intensity profile have negligible effect on the recovered average sample attenuation. The reduced-chi-squared χ 2 r of the fit is χ 2 r 1 because the model is not intended to describe the attenuation profile of the sample. We carry out the fit in order to determine the holder contribution from its strong attenuation profile signature and therefore to determine the sample profile. After fitting, the true sample attenuation profile is determined by subtracting the fitted holder attenuation profile from the total measured attenuation profile according to ( [ µ [txy ) S,rec = ( [ µ [txy ) S+H,mea ( [ µ [txy ) H,fit, (6.15) where the subscripts rec, mea, and fit refer to the recovered, measured and fitted attenuation profiles respectively. In Fig we present the recovered sample attenuation profile after subtraction of the fitted holder contribution. The attenuation in the central region has no holder component, and thus is unchanged in the process of the recovery. This central region is completely consistent with the attenuation profile in the region where the holder contribution has been subtracted. The two diagnostics therefore confirm the appropriateness of the fitting model and the quality of the result. However, the aperiodic variation between neighboring measurements in Fig has a standard deviation of approximately This is greater than the determined uncertainty of approximately attributed to the points by means of the ten repeated measurements at each point (Fig. 6.6). Therefore, either the structure in the attenuation depicted in Fig is real (corresponding to randomly-distributed

147 6.4. THE FULL-FOIL MAPPING TECHNIQUE 119 Figure 6.10: Recovered absorber attenuation profile with the edge omitted. The recovered attenuation profile is the measured attenuation profile of the (sample plus holder) minus the fitted holder profile. The variations of order in the attenuation are partially explained by the statistical uncertainty in the measured attenuation. The statistical uncertainty is approximately for all measurements across the foil. The remaining variation could be explained by long-range, aperiodic deviations of order 0.3 µm in the foil thickness, but is more likely due to the statistical uncertainty requiring scaling by a factor of χ 2 r.

148 120 Chapter 6. FOIL METROLOGY thickness variations of about 0.3 µm) or the input uncertainties are under-estimated. Other work [133 has noted that ten consecutive measurements of a very short period of time (0.1 s each) repeated in rapid succession may not fully probe the random variation in intensities when compared with measurements made over a longer time interval (the full-foil mapping takes about one hour). determined uncertainty in the attenuation may be under-estimated. Accordingly, the directly The absence of any artifacts of the measurement sequence in Figs 6.9 and 6.10 is consistent with the claimed measurement reproducibility. The dominant component of χ 2 r appears to be due to the under-estimation of the input measurement uncertainties by a factor of χ 2 r. The χ 2 r of the fit to the combined sample and holder is 3.8. Hence the scaled uncertainties are used for the remainder of the calculation. The uncertainty at each point of the recovered sample attenuation profile is evaluated by adding the measurement and fitting errors in quadrature, assuming independence of the corresponding contributions, according to σ( [ µ [txy ) S,rec = [ ( σ( [ ) 2 ( µ [txy ) S+H,mea + σ( [ µ [txy ) H,fit ) 2 1 2, (6.16) where σ( [ µ [txy ) H,fit is the fitting uncertainty in the holder contribution to the attenuation profile, evaluated at each (x, y) location on the holder by use of the covariant error matrix returned from the fitting program. Once the sample attenuation has been recovered, we evaluate the average of the sample attenuation to be with an uncertainty of ( I ) ln = 1 ([ µ ) [txy = , (6.17) I 0 xy N S,rec xy x,y [ ( I ) ln I 0 xy [ = 1 σ ([ 1 2 ) µ 2 [txy N S,rec xy (x,y) = , (6.18) where N xy is the number of measurements included in the summation. The uncertainty is thus %, dominated by the statistical counting uncertainty.

149 6.5. RESULTS OF A LOW-ENERGY FULL-FOIL MAPPING Determining the absolute value of [ µ The mass attenuation coefficient is determined from the average of the attenuation profile according to Eq. (6.7), using [t = m. For the fully-mapped foil, the average A integrated column density was determined to be [t = ± g/cm 2. The uncertainty in the mass attenuation coefficient is determined by adding the propagated relative uncertainties in quadrature, yielding [ µ = ± cm 2 /g, accurate to 0.028%. In the measurement reported here the uncertainty in the mass attenuation coefficient is thus dominated by the accuracy of the determination of the area of the foil. 6.5 Results of a low-energy full-foil mapping The accuracy of the full-foil mapping technique can be further tested by performing a second mapping. We have performed this second mapping at a photon energy of 13.5 kev. The second mapped sample was nominally 50 µm thick, with an attenuation of ln ( I I 0 ) 2. At 13.5 kev the Perspex holders attenuate the x-ray beam by a factor of ln ( I I 0 ) 0.8. The high holder attenuation presents a strong signature which can assist in the identification and correction of any limitations in the fitting procedure. However, the accurate fitting and subtraction of this strong profile is more difficult due to the high sensitivity of the recovered attenuation profile to the fitted holder attenuation profile. We found it difficult to obtain a satisfactory fit to the attenuation profile measured at 13.5 kev. Figure 6.11 presents the residuals resulting from the fit to the low-energy full-foil mapping. This figure shows significant residuals distributed in a circular pattern around the holder bevel. This circular pattern is superimposed with a saddle (or quadrupole) component of the distribution, indicating an up-down, left-right asymmetry in the residuals. We have examined and tested modifications to the attenuation profile based on the properties of the mapping sequence (the order in which the attenuation profile was measured) and the geometries of the the holder and of the foil in order to explain this distribution. These are reported in

150 122 Chapter 6. FOIL METROLOGY section as they represent legitimate extensions to the modelling of the attenuation profile. However, none of these were found to be responsible for the poor fit to the measured attenuation profile shown here. The beam intensity had changed significantly during the course of this mapping. Figure 6.12 presents this beam intensity shift, showing the counts recorded in the upstream ion chamber (left) and the downstream ion chamber (right) over the course of the full-foil mapping. Here we see clearly a significant (50%) shift in the incident beam intensity during the scan. One might normally use this beam intensity shift to reject these measurements out-of-hand. However, our procedure for normalising the intensities should remove the effect of the beam intensity shift on the measured attenuations. On the left of Fig we present the attenuations measured for the full-foil mapping. On the scale of this plot we cannot detect the effect of the beam intensity shift. On the right we present a detail of the attenuations measured through the full thickness of the foil and holder. On this expanded scale we see that the determined attenuation has been affected by the beam intensity shift by 2 3 parts in 288, or about 0.8%. We believe that the blank, unattenuated intensities have been undersampled over the course of the full-foil mapping in the presence of this shift. The first- Figure 6.11: Residuals for the low-energy full-foil mapping. The distribution of the residuals is in strong correspondence with the holder features, indicating a significant problem with the measured data or the fitting routine.

151 6.5. RESULTS OF A LOW-ENERGY FULL-FOIL MAPPING 123 order correction for the beam intensity shift has been implemented by using the upstream ion chamber to normalise the downstream ion chamber readings. The second-order correction for the beam intensity shift requires measurement of the blank, unattenuated intensities to normalise additional experimental drifts such as those due to the environment and the detector response. The signature of the effect on the determined attenuations does not exactly match the form of the beam intensity shift presented in Fig This is because the residual signature is not due to the beam intensity shift per se, but to the use of interpolated unattenuated intensities which do not accurately reflect the experimental conditions during the beam intensity shift. The beam intensity variation has resulted in an apparent structure in the attenuation profile of order 0.8%. As this structure does not correspond with any feature Figure 6.12: Counts recorded by the upstream ion chamber (left) and the first downstream ion chamber (right) during the full-foil mapping at 13.5 kev. The beam intensity shift of concern to us here can be seen clearly in the measurements recorded by the upstream ion chamber, occurring at about 1400 s into the measurement. The regular structure observed in the downstream measurements is a direct result of the measurement sequence, and represents the transmission profile measured along linear sections across the foil mounted in the holder. Accordingly, the transmission repeatedly varies from a low level, when the beam passes through the full thickness of the foil-plus-holder, to a high level, when the beam passes through the foil only. The beam intensity shift is also observed in the measurements recorded by the downstream ion chamber, particularly in the measurements made through the full thickness of the foil and holder, with transmitted intensities of about ( ) The triangular markers on the plots represent blank measurements, recorded with the absorber removed from the path of the beam. The blank measurements have been made prior to and following the full-foil mapping. It is clear that these blank measurements do not probe changes in the detector response over the event of the significant beam intensity shift.

152 124 Chapter 6. FOIL METROLOGY of the modelled foil and holder, the fitting technique cannot resolve the problem. Nevertheless, the results of this measurement still provide interesting information about the measured foil. Figure 6.14 presents the attenuation profile in the region of the centre of the foil. Here we see clearly the manifestation of a regular and periodic variation in the integrated column density of the foil, of fractional amplitude %. The periodicity of this structure in the integrated column density lends itself to interpretation as a result of the foil preparation by rolling. There is also a clear and systematic increase in the attenuation of the foil, the integrated column density increasing with the x ordinate of the plot. Few previous measurement schemes have been able to reveal such structure in the thickness of the foil and, where such variations in the attenuation have been observed (in the course of a random probe of the consistency of the measured attenuation across the surface of a sample, for example), they have been used to quantify the likely error in the attenuation measurement [151. By using the full-foil mapping technique these variations can now be quantified and Figure 6.13: Attenuation [ µ [t as a function of measurement time for the full-foil mapping at 13.5 kev. Attenuations of about 1.86 have resulted from measurements made through the foil only: those with attenuations of about 2.85 correspond to measurements made through the full thickness of the foil-plus-holder. On the scale of the left-hand plot the determined attenuations show no sign of instability between s into the measurement. On the right we present a detail of the measured attenuations, showing only values with attenuation within a small range of 2.85, made through the full thickness of the foil and holder. On this scale the effect of the beam instability is clear. The signature of the instability appears to have been delayed and smoothed somewhat by our normalisation, but is significant, causing an estimated change in the mass attenuation coefficient of 2 3 parts in 288, equivalent to about 0.8%.

153 6.5. RESULTS OF A LOW-ENERGY FULL-FOIL MAPPING 125 Figure 6.14: Results of an x-ray scan taken at 13.5 kev, processed to determine values of ln ( ) I = [ µ [txy I 0 in the neighborhood of the centre of the foil. The x- xy ray beam used to make the measurements was 1 1 mm 2. The periodic variation in the attenuation is due to a real variation in the integrated column density, probably relating to preparation by rolling. The variation in the integrated column density depicted here corresponds to a thickness variation of µm 0.1 µm. 1.87

154 126 Chapter 6. FOIL METROLOGY understood, and the mass attenuation coefficient can be determined to an accuracy which is significantly better than any fluctuation of the integrated column density across the sample. Assuming that the mass attenuation coefficient [ µ and the density of the sample have been determined, we can interpret the variations presented in Fig as a variation in the thickness of the foil of around 0.1 µm, occurring over a characteristic length scale of around 5 mm. The length scales and the absolute nature and accuracy of this measurement are quite remarkable, and difficult to match using any other currently available technique. 6.6 Discussion Further modelling of the holder attenuation profile We have explored the inclusion of a number of additional features in the model of the attenuation profile, and report them here for reference. Modelled features include A minimum thickness for the holder, where the bevel does not taper to zero thickness. Separate parameters describing the geometry of the two holder components. Inclusion of these parameters allows us to describe the attenuation profile of the holders when they are out of alignment (i.e., when they do not share the same central position) and allows the bevel diameters to differ. As this dramatically increases the number of fitted parameters (doubling the holder parameters), we have often restrained those parameters most likely to be degenerate for the two holders. In order to describe hysteresis in the x-y translation stages we have included a parameter to shift the location of every second row by a small amount. An elongation variable was included to describe a possible difference between the movement scale of the x and y translation axes. Such an elongation would appear as an eccentricity of the circular holder features.

155 6.6. DISCUSSION 127 Figure 6.15: Atomic-force microscope scan of an 80 µm 80 µm region on the surface of one of the absorbers. The long vertical structures seen here are possibly due to the method by which this thick foil was prepared, either by rolling or sawing. Such structure was common on the surfaces of the thicker foils but is of little concern as it represents a σ rms thickness variation of less than about 0.1 µm. We have trialed the inclusion of these features in the modelling of the present attenuation profiles but have found that their inclusion is not supported by the data Microstructure in the integrated column density Our use of the full-foil x-ray mapping technique directly quantifies variations of the integrated column density on the millimetre scale. Here we consider the effect of nonuniformities of the integrated column density on length scales smaller than the beam dimensions. We have used an atomic force microscope (AFM) to measure the surface variations at multiple locations on both surfaces of eight of the foils. As the maximum range of the AFM scanning region was 80 µm 80 µm we were able to use these measurements to provide an estimate of the surface uniformity on this length scale. We present in Fig the results of such a scan. The range of the scan is appropriate for quantifying the surface variations resulting from the numerous linear scrub marks on the foil surface. These features may have been produced during manufacture, possibly during the rolling, cutting, or polishing of the foil. The features observed in Fig have been observed on most of the absorbers.

156 128 Chapter 6. FOIL METROLOGY The root-mean-square height variations σ rms shown in Fig are about 100 nm. We have found that the observed σ rms variations often increase with foil thickness, and that the fractional surface variation σrms t thinner foils. evaluated using is observed to be a maximum for the The effect of small roughness on the measured attenuation can be [ µ σrms [ µ [ µ smooth smooth [ µ 1 smooth [t ( [ µ 2 smooth σ2 rms 2! ). (6.19) This expression for the effect of roughness is derived in section 8.5. Accordingly, we estimate the effect of the roughness on the mass attenuation coefficient determined from the 41.5-keV full-foil map to be less than % Rotational alignment of the samples The apparent integrated column density is influenced by the perpendicular alignment of the sample in the path of the beam, increasing by an amount proportional to 1 cos θ, where θ is the angular misalignment from perpendicular. The stages on which the samples were mounted could be rotated about two directions orthogonal to one another and to the beam. By measuring the attenuation of the samples through a wide range of angles Figure 6.16: Ratio of the intensity recorded downstream of the sample to that recorded upstream when the sample was rotated about the ˆx (outboard) axis. These measurements were not repeated ten times and so we cannot estimate their uncertainties directly. The angular dependence of the attenuation dominates the plot. The measured ratios have been fitted to determine the angular alignment of the sample relative to the beam.

157 6.6. DISCUSSION 129 Figure 6.17: Ratio of the intensity recorded downstream of the sample to that recorded upstream when the sample was rotated about the ŷ (vertical) axis. The angular dependence of the attenuation dominates the plot. The fitted function is shown as a continuous curve. we have determined the angular alignment of the sample relative to the beam. We present in Fig the intensity ratio I d I u measured at selected angles during this rotation. A symmetrical structure is apparent within 5 of the zero angle. We have not been able to identify the cause of this structure, and note that it does not appear in the other rotational measurements. These intensity ratios have been fitted with a function describing the geometrical increase of the integrated column density with rotation angle, { } I d b = a exp I u cos (θ + δ θ ) + c, (6.20) where a, b, and c are parameters describing the attenuation of the air path and the ion chamber normalisation. The use of these parameters is appropriate as we only aim to determine a value for the angular offset δ θ from the rotational scan. We have fitted this function to the measurements presented in Fig and also separately to two subsets of these measurements, which have excluded (a) the central pimple region and (b) the measurements above 25 (to quantify any possible influence of the asymmetrical data set), respectively. These three fits determine the same angular offset δ θ within their uncertainties. The rotational misalignment about the horizontal axis was thus determined to be 0.8 ± Measurements made while the same sample was rotated about the vertical rotation axis are presented in Fig These measurements have been fitted using

158 130 Chapter 6. FOIL METROLOGY Eq. (6.20) to determine the vertical misalignment angle to be 0.38 ± The vertical and horizontal misalignment angles are added in quadrature to deduce that the samples were presented to the beam at an angle of θ 0.89 ±0.05 from their preferred normal orientation. The result of this misalignment is to increase the integrated column density of the sample presented to the beam by 0.012%. This systematic error in [ µ is well below the uncertainty in the foil area determination and is therefore insignificant in the current context Improving the full-foil mapping technique A number of extensions to the full-foil mapping technique may improve the accuracy of the determined mass attenuation coefficient. The accuracy of the determination of the mass attenuation coefficient is limited by the accuracy of the foil area determination. The relative effect of this uncertainty can be decreased to an arbitrary level by increasing the area of the foil used in the measurement. However, such an increase in the foil area also increases the time taken to perform the full-foil x-ray mapping. An alternative method for improving both the accuracy and precision of the determination of the area of an absorbing specimen is based on the convenience of circular geometry. When an absorber can be machined into the shape of a disc by use of a lathe, its circularity can be quite perfect. The area of such an absorber can be determined by using an optical comparator to measure the locations of a large number of points around the perimeter of the foil, or by using a travelling microscope or a micrometer to measured the diameter of the absorber [157. We have used the full-foil mapping measurements to determine the locations of the foil corners. It may be possible to determine the foil area by use of such measurements. However, in order to determine the foil area from the full-foil measurements a number of technical issues would need to be overcome, including The motion of the translation stages would need to be both reproducible and accurate to within a few microns. The beam intensity profile would need to be known. The beam intensity profile

159 6.7. CONCLUSIONS 131 could be determined by scanning a highly absorbing knife edge in the path of the beam in two directions (x and y). The determination of the foil area by x-ray mapping would be greatly facilitated by use of a beam of uniform intensity. One would need either to assume that the integrated column density of the foil around its perimeter was the same as the average of its value over the surface, or to extrapolate the near-perimeter integrated column densities to the perimeter locations. The removal of the holder attenuation profile by way of the fitting routine can be improved by redesigning the holder so that it is of simpler geometrical construction and affects fewer of the measurements. This modification has been implemented and is reported in Appendix C Conclusions We have used the full-foil mapping to determine an attenuation profile of an absorber and have combined this profile with measurements of the average integrated column density to determine an absolute value for the mass attenuation coefficient. We have overcome problems associated with other techniques for determining the amount of absorbing material placed into the path of an x-ray beam. The technique has been used to determine the mass attenuation coefficient of molybdenum at a nominal x-ray energy of 41.5 kev to be [ µ = ± cm 2 /g, accurate to 0.028% reported in [24 and [25. This extremely accurate full-foil mapping determination of the mass attenuation coefficient will be used in chapter 8 to provide an absolute scale for the integrated column densities of the other foils. The technique has a demonstrated sensitivity of better than 0.2% in the integrated column density [t, and has been used to quantify a corresponding 0.1 µm thickness variation of a 50-µm-thick specimen. The variations have occurred over a length scale of 5 mm. The technique can provide absolutely-scaled information for use in a number of metrological situations.

160

161 Chapter 7 Determining the photon energies 7.1 Requirements of the energy determination Away from the absorption edges the mass attenuation coefficient varies approximately linearly on a log-log scale with photon energy [34 according to ln [ µ k ln E + c. (7.1) An expression relating the fractional uncertainty of the photon energy to the fractional uncertainty of the mass attenuation coefficient can be derived by differentiating Eq. (7.1), giving d [ µ [ µ k de E. (7.2) The FFAST tabulation can be used to estimate the magnitude of the log-log linear gradient k to be between 2.7 and 2.8 at energies between 13.5 and 41.5 kev. Equation (7.2) shows that the photon energies must be determined to within a fractional uncertainty which is 1 k times better than that obtained for [ µ in order that the energy uncertainty not limit the measurement accuracy. As the uncertainty in our determination of [ µ is nowhere better than about 0.03%, we set the goal for the accuracy of the determination of the photon energy at around 0.01%, equivalent to 1 ev at 13.5 kev and 4 ev at 41.5 kev. Near the absorption edge and in the region of the XAFS the rapid variation of the mass attenuation coefficient as a function of the photon energy results in the need for the photon energies to be determined to better than around % in order that they not limit the measurement accuracy. However, photon energies cannot be determined readily to this high accuracy (within 0.02 ev at 20 kev) by use of diffrac- 133

162 134 Chapter 7. DETERMINING THE PHOTON ENERGIES tion techniques due to lattice parameter uncertainties and experimental difficulties. Near the absorption edge and in the region of the EXAFS we therefore expect the measurement uncertainty to be dominated by photon energy uncertainties. 7.2 Previous measurements The majority of measurements of the mass attenuation coefficients reported in the literature have utilised radioactive sources or characteristic emission spectra to produce a relatively monochromatic x-ray beam of well-defined energy. These investigations have, therefore, not required a determination of the photon energies as the quoted literature values of characteristic x-ray and γ-ray energies have typically low uncertainties. Measurement at energies provided by radioactive and characteristic sources was also seen to be sufficient as most other experimental work was performed using these sources. In a measurement using Bremsstrahlung radiation [151 the photon energies were determined to 0.1%. The authors measured the dispersion function of their monochromator by using radioactive sources to provide standard reference energies. Their method required replacement of the Bremsstrahlung x-ray source with the radioactive standards. The technique requires great care as the replacement of the source may also inadvertently alter the diffraction geometry. More recent trends towards the use of synchrotron sources have enabled experimenters to select the photon energy without compromising photon intensity. Recent attenuation measurements using synchrotron sources [13, 14 have determined the photon energy to high accuracy by measuring the angular locations of x-rays diffracted from the NIST standard reference material SRM 640b (silicon) and the secondary LaB 6 standard using a powder diffractometer. In this setup the BigDiff [158 apparatus available at ANBF 20B in Tsukuba was used to obtain the powder pattern. The use of BigDiff is very efficient as the locations of many diffraction peaks can be measured simultaneously, and this apparatus has recently been used to determine the lattice parameter of LaB 6 to extremely high accuracy [159. However, the quality of those measurements requires the use of BigDiff which was not

163 7.3. THIS MEASUREMENT 135 available for the experiments reported in this work. In another experiment [69, a Huber six-circle diffractometer was used to measure reflections from a silicon powder sample (SRM 640b). Those measurements were analysed by myself and are reported in chapter 10. The accuracy of that determination was good but limited by a lack of measurements of high-index reflections, which restricted the examination of systematic errors arising from the alignment of the powder sample in the diffractometer. High-order reflections also provide more accurate information for determining the photon energies [68. A well-prepared powder can be used to present crystals at every orientation to the beam simultaneously. This is particularly useful when the locations of all of the diffracted beams can be recorded simultaneously, as in the Debye-Scherrer camera, and removes the need to set the orientation of the diffracting crystal with respect to the beam axis. However, the advantages of the powder technique are diminished when it is not possible to record the angular positions of the diffracted beams simultaneously. Additionally, the low interaction coefficient usually associated with the use of narrow powder capillaries of low-z powders results in reflected intensities several orders of magnitude lower than those achievable by the use of a single-crystal specimen. 7.3 This measurement The APS beamline 1-ID facility is equipped with an in-hutch six-circle Huber diffractometer. As the diffractometer recorded the locations of the reflections sequentially, we have used a single-crystal diffraction technique to determine the photon energies. A monolithic, channel-cut double-reflection germanium 111-oriented monochromator was mounted on the inner φ axis of the Huber six-circle diffractometer (see Fig. 7.1). The crystal was located and aligned so that the x-ray beam was incident on one of the outer surfaces of the monochromator. The dispersion axis of the system lay in the vertical plane. The inner φ axis of the six-circle goniometer was chosen so as to expedite the measurement and to minimise the stresses placed on the diffractometer by rotations of the outer axes.

164 136 Chapter 7. DETERMINING THE PHOTON ENERGIES Figure 7.1: Conventional axes designations when using a six-circle diffractometer. A sodium-iodide scintillator counter was mounted on the δ axis of the diffractometer. This detector was used in a wide open geometry, without any apertures between the analyser crystal and the detector face. A rocking-curve for each reflection was measured by locating the scintillator at twice the Bragg angle 2θ B for the reflection and by scanning the φ axis through the Bragg angle of the reflection. The scintillator remained stationary during the measurement of each reflection. We measured only reflections from the (hhh) lattice planes, which are (approximately) parallel to the crystal surface, in order to reduce the measurement time and the impact of systematic errors associated with asymmetric diffraction geometries. The peak intensity of reflections from (hhh) lattice planes varies due to the structure factor and to the intrinsic reflection widths. In order to measure each reflection at an optimised statistical count rate we have used the foils mounted on the daisy wheels (see Fig. 4.2) when it was necessary to attenuate the x-ray beam. This optimisation was implemented by first measuring each reflection at low angular resolution and then adjusting the thickness of the filter by rotation of the daisy wheel.

165 7.4. THE MEASURED ROCKING-CURVES The measured rocking-curves The expected form of the measured rocking-curves is a single peak with its shape determined by the reflection profile and experimental broadening effects. In modest resolution studies such as described here, broadening effects dominate and the observed peak can be described by a convolution of the Gaussian, Lorentzian, and slit profiles associated with different broadening mechanisms. The rocking-curves measured in this experiment conformed broadly to these expectations. However, a significant proportion of the measured rocking-curves exhibit some additional structure which, as we shall show, resulted from two different effects: detector saturation and imperfections of the analyser crystal. In this section we describe a study of the imputed causes of this peak shaping. We then discuss consequences of this shaping for our determination of the photon energies in section Detector saturation Section 7.3 detailed our use of attenuators to enable all rocking-curves to be measured at optimum counting-rates required for high statistical accuracy. Nevertheless, Figure 7.2: Examples of rocking-curves measured with an insufficient thickness of filter in the path of the beam. The scintillator is clearly saturated around the peak intensity location. The peaks are not used to determine the photon energies as the level of the saturation is too great. However, the observed saturation occurs at count-rates of the order of cps which is close to the count rates observed during the measurement of other less obviously saturated rocking-curves.

166 138 Chapter 7. DETERMINING THE PHOTON ENERGIES a number of the rocking-curves were measured with some saturation in the sodiumiodide scintillator counter. Figure 7.2 shows two examples of rocking-curves affected by saturation in the scintillator. The saturation has lead to an inversion of the measured profile, where an increase in the number of photons incident on the detector gives rise to a decrease in the measured count rate. This inversion is readily identifiable and is clearly observed in Fig While the inversion at the peaks of these curves is readily identifiable, there is further concern regarding the likely non-linearity of the detector at countrates slightly below the saturation threshold Analyser crystal imperfections The analyser crystal used in this experiment was one of the outer surfaces of a germanium double-reflection monochromator. Germanium was chosen in preference to silicon because of the higher reflection intensities which were appropriate at the high energies probed in this and in a subsequent experiment. A silicon wafer was considered for use as the analyser crystal in this experiment. However, a consideration of the DuMond [160, 161 diagram describing the double reflection from the (detuned) silicon monochromator followed by a silicon analyser crystal (in [+,,+ configuration) suggested that such a crystal would be very difficult to align due to the very narrow reflection widths. These considerations provided further justification for using the thick germanium analyser. In all crystals there may be long, medium, or short range lattice imperfections. These depend on the method by which the crystal has been produced and its history. Short-range disorder can be a direct result of high-energy radiation damage, as in the formation of amorphous silicon by ion-implantation [162. Medium range disorder, including mosaicity, may likewise be created in perfect crystalline specimens by tailored ion-implantation [163, 164. Long-range disorder in large crystals generally results from an imperfection at the growth or regrowth stage of manufacture, or as a result of mechanical shock and treatment. Twinning, for example, involves a growth fault which gives rise to regions of crystal misaligned with respect to one another. The outer surface of our germanium crystal was pitted in a manner that suggested

167 7.4. THE MEASURED ROCKING-CURVES 139 Figure 7.3: Rocking-curves of reflections from the (111) planes of the analyser crystal. These rocking-curves were measured in complementary geometries, as indicated in Fig. 7.5 and discussed in the text. that it had been etched. A number of the measured diffraction peaks exhibit peak shaping which is not consistent with the saturation of the detector. Figure 7.3 shows rocking-curves of reflections from the (111) planes of the analyser crystal. This figure shows a broad secondary peak structure accompanying the high-intensity reflection. This structure results from a shaping effect whose origin can be traced back to the x-ray beam, the analyser crystal, or the scintillator. Figure 7.4 presents two rocking-curves measured shortly after those presented in Fig. 7.3, using the same x-ray beam. These rocking-curves resulted from reflection from the (333) and (444) planes of the germanium crystal. The reduced intensity of the secondary structure of these reflections indicates that the secondary structure is not a result of energetic, angular, or spatial structure in the beam, as these would persist in measurements made at all angles. Measurements of other rocking-curves indicate that the secondary structure is not due to the response function of the scintillator. The reproducibility of the measurement over a range of count rates and the smoothness of the two-peak signature indicates that this structure is not a saturation effect. The shaping observed in Figs 7.3 and 7.4 is consistent with a defect in the analyser crystal, quite probably due to a strained layer of material on the etched surface of the germanium analyser crystal.

168 140 Chapter 7. DETERMINING THE PHOTON ENERGIES Figure 7.4: Rocking-curves of reflections from the (333) and (444) planes showing the presence of secondary peaks of reduced intensity and separation compared to those observed in Fig φ φ 3 x-ray beam germanium crystal 2φ diffraction profiles with crystal artefacts indicated φ 177 Figure 7.5: Illustration of the influence of imperfections in the analyser crystal on the diffraction profile. The grey and white regions tag the contributions of the crystal in the diffraction profile. Upper: measurements at small positive φ angles. Lower: measurements at φ 180. The tagged grey region appears adjacent to the φ = 0 location in the upper diagram but away from the φ = 180 location in the lower diagram. This behaviour is in direct contrast to a similar arguments in which the beam has intensity structure. Angular and energetic structure in the beam are ruled out because the shaping in the diffraction profile does not appear for peaks measured at diffraction angles above φ 5.

169 7.5. IMPLICATIONS OF PEAK SHAPING 141 The rocking-curves depicted in Fig. 7.3 were measured in complementary symmetrical geometries, with the germanium crystal inclined at angles of about 3 and about 177 to the x-ray beam. Figure 7.5 illustrates the effect of a defective region of the analyser crystal on the observed rocking-curve. This illustration shows that the symmetry of the structure in the rocking-curves measured in these complementary geometries is consistent with a defect in the analyser crystal. A portion of the x-ray beam could exit through the end face rather than through the top face of the analyser crystal. However, such a transmission effect would lead to structure appearing symmetrically about the φ = 0 axis, which is not consistent with the observations. When φ 3, the footprint of the beam on the analyser crystal is 1 mm sin 3 20 mm (for a 1 mm 2 beam), and covers most of the length of the 25 mm analyser crystal. In this orientation, defects along the entire length of the analyser crystal are illuminated and contribute to the rocking curve. At higher diffraction angles less of the crystal is exposed to the beam, and the likelihood of probing such defects is reduced. This behaviour is consistent with the reduced intensity of the secondary structure of the rocking-curves measured at higher diffraction angles. 7.5 Implications of peak shaping Detector saturation Detector saturation has a significant effect on the measured rocking-curve around the location of the maximum of the peak intensity. We therefore require a technique for determining the location of the peak which is either insensitive to the peak distortion resulting from saturation or in which one can quantify the likely error resulting from the saturated regions. The background measurements leading up to the peak are obviously not saturated. Measurements made up the leading-edge of the rocking-curve are similarly unaffected until the detector response becomes non-linear in the saturation region. Saturated measurements are of course compromised. Once the measured intensity

170 142 Chapter 7. DETERMINING THE PHOTON ENERGIES drops below the saturation level it is possible that the measured intensities again accurately reflect the diffracted intensities. However, it is also possible that the detector may undergo a non-linear recovery phase following saturation, which may compromise all immediately following measurements Crystal imperfections The analyser crystal imperfections have resulted in a secondary structure in the rocking-curves. This structure is always located on the high-angle side of the desired diffraction feature. An analysis which uses only the low-angle features of the rockingcurve or which uses the entire rocking-curve but which includes some measure of the uncertainty arising from the asymmetry of the rocking-curve can therefore provide a robust analysis of these data. 7.6 Determination of peak positions In both cases where the measured rocking-curve has been adversely affected (by detector saturation or imperfections in the analyser crystal) the leading-edge of the diffraction profile is expected to be unaffected. The observed peak shaping precludes the use of a straightforward (Gaussian or Lorentzian) fitting regime such as that developed in chapter 10. Avoiding such an approach, we analyse the measured rocking-curves using two alternate and independent techniques Technique 1: fitting the leading-edge It is well known that the centroid locations of diffraction peaks are shifted from their ideal Bragg locations, and that these shifts depend on the diffraction geometry and the Miller indices of the diffracting planes. This shift results primarily from refractive index and depth-penetration effects. Furthermore, the width of a diffraction peak also depends on the Miller indices of the reflection. In light of these facts it is appropriate to discuss the validity of performing an energy determination based on

171 7.6. DETERMINATION OF PEAK POSITIONS 143 Figure 7.6: Predicted rocking-curves for the reflection of 25-keV x-rays from the (111) (solid black line), (333) (solid grey line), and (444) (dashed black line) planes of a germanium crystal. The abscissa is the angular difference from the Bragg angle for each of the reflections. the location of the leading-edge of the measured rocking-curves. Figure 7.6 shows the predicted diffraction profiles resulting from the reflection of 25-keV x-rays from the (111), (333), and (444) planes of a perfect and infinite single crystal of germanium. These calculated curves do not include experimental broadening. The abscissa is the angular difference from the ideal Bragg locations of the three reflections. The shift of the centroid of the profiles from their Bragg locations and their associated reflection widths change as the order of the reflection increases. We observe that the leading left edge of the diffraction profiles remains in good correspondence with the calculated Bragg angle for all three reflections. Therefore, if one aims to use a simple Bragg model to describe the peak locations, the left edge location of the peaks corresponds more closely to the Bragg angle than the centroid location. Experimental peak broadening has a significant impact on this observation. The unbroadened profiles shown in Fig. 7.6 have widths of between and , and are shifted from the Bragg location by around one-half of their width. However, our measured profiles have been broadened to a width of around In this case it is appropriate to question the validity of using the left edge. Independent peak broadening causes add in quadrature to the observed peak

172 144 Chapter 7. DETERMINING THE PHOTON ENERGIES width and do not alter the centroid location of the peak. Therefore the centroids of the broadened peaks will be shifted from their Bragg locations by the amounts indicated in Fig The width of the observed peak is approximately w 2 + b 2, where w is the intrinsic reflection width (indicated in Fig. 7.6) and b is the experimental broadening width. The left-edge location will occur at an angle θ LEL determined by the shift (approximately w ) and the half-width of the observed peak, giving 2 θ LEL θ B = w w2 + b 2. (7.3) We observe that w b, and so this becomes θ LEL θ B w 2 b 2 (1 + w2 2b 2 ) (7.4) = w 2 w2 4b b 2. (7.5) Therefore the left-edge of the broadened profile will occur at an angle which is shifted by about w 2 w2 4b b 2 the constant b 2 from the Bragg angle. As we shall see in section 7.7, component of this shift can be removed, and is of no concern to us here. The relative shifts for reflections of various widths are therefore w 2 w2 4b, which is not a constant as it was for the case of an unbroadened profile. correction term w2 4b The residual is small when the experimental broadening width is greater than the intrinsic diffraction width, and thus the left-edge location is more stable and reliable than a centroid analysis. Accordingly, we conclude that we are justified in using the leading or left edge locations of the rocking-curves with a Bragg-law analysis to determine the photon energies. A peak profile is usually described by a set of basis parameters which determine the peak centroid location c, the peak width w, and the peak height h. However, as we wish to determine the location of the leading-edge of the rocking-curves it is convenient to choose alternate basis parameters. The new parameters are: the leftedge position l, the right-edge position r, and the peak height, h. This parameter

173 7.6. DETERMINATION OF PEAK POSITIONS 145 set is related to the usual set by l = c w/2, (7.6) r = c + w/2, (7.7) h = h. (7.8) We use this parametrisation of the diffraction profile to fit the measured rockingcurves and to extract the locations of the left edges of the rocking-curves. The rocking-curves were fitted with a Lorentzian-slit profile. This profile allows the shape of the leading-edge to change dramatically and thus to fit a variety of leading-edge profiles. In order to avoid shifts to the fitted leading-edge position resulting from peak shaping, the Lorentzian-slit profile was fitted to measurements recorded on the leading-edge of the measured rocking-curve up to 50% of the maximum peak height. We therefore avoid probing the saturated measurements occurring around the peak maximum and the non-linear region on the trailing edge of the peak. Measurements made following the fifty-percentile location are excluded from the fit. Five parameters were fitted to the rocking-curves: a constant background, the left-edge position, the height, the right-edge position, and a parameter describing the proportion of Lorentzian and slit character of the profile which we term the slit fraction. The peak height was restricted to vary between 90% and 110% of the peak intensity after background subtraction and the slit fraction was restricted to vary between 20% and 90% of the entire peak width. The recorded counts were assumed to follow a Poisson distribution, and accordingly uncertainties of I are used, where I is the count recorded for each orientation of the analyser crystal. Figure 7.7 presents rocking-curves measured with a fine scan interval. The leading-edges of these rocking-curves have been fitted to determine their locations. The light grey curve indicates the Lorentzian-slit fitted to measurements recorded up to 50% of the maximum (measured) intensity. The location of the leading-edge is determined consistently by this procedure. Figure 7.8 presents rocking-curves measured with coarse scan interval. The

174 146 Chapter 7. DETERMINING THE PHOTON ENERGIES Figure 7.7: Results of fitting to determine the location of the leading-edge of two rocking-curves measured with a fine interval. The pluses indicate the measured count rates. The fitted profile is shown in grey with the vertical line denoting the location of the leading-edge. Points measured following the fifty-percentile position are excluded from the fit but are shown here for reference. Figure 7.8: Results of fitting to determine the location of the leading-edge of two rocking-curves measured with a coarse interval. The pluses indicate the measured count rates. The fitted profile is shown in grey with the vertical line denoting the location of the leading-edge which corresponds to the location of the fifty-percentile of the fitted profile peak. Points measured following this fifty-percentile position are excluded from the fit but are shown here for reference.

175 7.6. DETERMINATION OF PEAK POSITIONS 147 Figure 7.9: χ 2 r resulting from fitting the location of the leading edge of the rockingcurve profiles with a Lorentzian-slit profile, presented separately for the coarsely and finely-measured profiles. These results show that the coarsely-measured profiles often have a significantly better χ 2 r (χ 2 r 0.3 2) than the finely-measured profiles (χ 2 r 6 60). This is due to the fact that the coarse profile may often sample the leading-edge at only one or two angles, and thus the detailed form of the leading-edge is not probed by these measurements. leading-edges of these rocking-curves have been fitted as before. The leading-edge location is well determined for the rocking-curve on the left of this figure. In contrast, for peak on the right of the figure the data is not measured at sufficient angular resolution for the leading edge to be accurately determined. Consider the measures of the quality of our fitting of the leading-edge of the rocking-curves with a Lorentzian-slit profile: the reduced chi-squared χ 2 r of the fit, the σ sd error estimates on the leading-edge position, and the σ sd χ 2 r. Figure 7.9 presents the χ 2 r of the fit to (a) the coarsely-scanned peaks, with measurement densities of 2 5 points per fwhm, and (b) the finely-scanned peaks, with 6 20 measurements recorded over the fwhm. The coarse measurements show χ 2 r 0.3 2, indicating possibly that the leading-edge model may be adequate for the description of these data. The fine measurements show a significantly worse χ 2 r In this case the lower χ 2 r associated with the coarse measurements does not indicate better conformity with our model, but rather that the coarse measurements do not probe sufficiently the shape of the leading-edge. The fine scans show a very large χ 2 r, indicating indeed that the model is in some respects deficient. In this case the measurement-interval dependence of the χ 2 r (i.e., fine versus coarse scans) informs us of a failure of the model. The differing

176 148 Chapter 7. DETERMINING THE PHOTON ENERGIES Figure 7.10: One-σ error estimates resulting from fitting the location of the leading edge of the rocking-curve profiles with a Lorentzian-slit profile, presented separately for the coarsely and finely-measured profiles. As expected, the location of the leading-edge is less well determined for the coarsely-measured profiles than for the finely-measured profiles. distributions of the χ 2 r measures for the two scan frequencies (fine and coarse) imply that we must either treat these populations separately and/or reject one of them. Figure 7.10 presents the one standard deviation error estimates for the leadingedge location, σ LEL. The results are again presented separately for the (a) coarse and (b) fine-scanned peaks. This figure shows that the leading-edges of the coarselyscanned profiles are determined to a lower accuracy than the finely-scanned profiles. This lower accuracy is consistent with the reduced measurement density of the coarse scans. If the model, the input data, and their uncertainties are reliable, then we should achieve χ 2 r 1. This result can then be used to validate the fitting function and the uncertainties. However, this is not applicable for the determination of the locations of these sometimes saturated or misshapen profiles. In our case the appropriate uncertainty estimates are obtained from the one-sigma uncertainties determined from the fit (σ) multiplied by the square-root of the reduced chi-squared ( χ 2 r). Figure 7.11 presents σ χ 2 r for the fitted profiles, showing that the uncertainty determined for the finely-scanned profiles is relatively stable at millidegrees, with only a few exceptions. The uncertainty determined for the coarsely-scanned profiles is less stable, with variations between millidegrees. The reported accuracy of these measurements is not considered to be reasonable given the sig-

177 7.6. DETERMINATION OF PEAK POSITIONS 149 nificantly coarser angular grid upon which the measurements were made. The 1 3 measurements along the leading-edge do not enable uncertainties to be determined robustly. Consequently we reject all of the coarsely-measured profiles from our analysis of the photon energies Technique 2: centre of mass In view of the potential model-dependence of the energy determined using the leading-edge technique, we have also applied a robust and complementary technique using of the centre-of-mass to determine the peak location. The centre-of-mass and the leading-edge analyses should determine peak locations at significantly different angular positions, differing by around half of the profile width w. 2 The centre-of-mass of each rocking-curve was determined after background subtraction using the background values defined by the leading-edge fitting. The centre of mass (φ CM ) is defined by φ CM = i φ ii i, (7.9) i I i Figure 7.11: σ χ 2 r measure of uncertainty resulting from fitting the location of the leading edge of the rocking-curve profiles with a Lorentzian-slit profile, presented separately for the coarsely and finely-measured profiles. The leading edge location determined using the finely-measured profiles is approximately twice as accurate as that obtained using the coarsely-measured profiles. The finely-measured profiles are stable, ranging only over a factor of four for reflections variously affected by peak shaping. The coarsely-measured profiles show a much greater variation in the reported accuracy of the edge location, with a significant number of the reported values falling into the lower end of the range associated with the finely-scanned measurements.

178 150 Chapter 7. DETERMINING THE PHOTON ENERGIES Figure 7.12: Determination of the centres of mass of two finely-measured rockingcurves. The pluses represent the measured data. The location of the centre of mass is denoted by the black vertical line. The Gaussian curve shown in grey is fitted to determine the asymmetry of the peak and thus to provide an estimate for the uncertainty of the centre of mass. where I i is the count rate recorded with the φ axis of the diffractometer at angle φ i. It is possible to determine an expression for the uncertainty of the location of the centre of mass based on statistical considerations. However, it is less straightforward to determine an uncertainty with allowance for peak shaping which may give rise to a significant error of the centre of mass locations. We use a simple model to quantify the shaping of the measured rocking-curves. The asymmetry of the rocking-curves is estimated by fitting the rocking-curves with a Gaussian peak of fixed centroid (located at the centre of mass), but with the height and width fitted as free parameters. The χ 2 r determined from these fits was used to provide a measure of the quality of the rocking-curve profiles and thus the reliability of the centres of mass. Uncertainty estimates obtained in this manner were scaled on a relative basis. Figure 7.12 shows examples of determining the centres of mass and their relative uncertainties. The vertical line indicates the angular location of the centre of mass. The grey line shows the fitted Gaussian curve used to estimate the peak symmetry. Figure 7.13 presents the relative uncertainties χ 2 r determined for each of the peaks, presented separately for the coarsely and the finely-scanned profiles. The uncertainty attributed to the finely scanned profiles is less uniform than that determined for the coarsely-scanned profiles. This is due to the robustness of the

179 7.7. DETERMINING THE PHOTON ENERGIES 151 Figure 7.13: χ 2 r uncertainties determined by fitting the rocking curves with a Gaussian beam profile, presented separately for the coarsely and finely-measured profiles. The fitted Gaussian profile has its centroid fixed at the location of the centre-of-mass of the profile. The Gaussian fit probes the symmetry of the peak and quantifies the likely uncertainty in the determined centre-of-mass location. These plots show that the relative uncertainties associated with the coarsely-measured profiles are significantly lower than those associated with the finely-measured profiles, indicating a sensitivity of the uncertainty estimation technique to the sampling frequency. Accordingly, we cannot apply the technique equally to the measurements measured at very different sampling frequencies, and we discard the coarsely-measured profiles. fitted Gaussian as an indicator of the asymmetry of the measured profiles. The coarsely-measured profiles again reflect a lower relative uncertainty which is due to their undersampling of the rocking-curves and which does not reflect the relative accuracy of the determination. We thus (again) discard measurements made using the coarse scans as their locations and uncertainties are not reliable. The distribution of the fine-scan uncertainties is quite dissimilar to that depicted in Fig. 7.11, indicating the complementarity of the techniques. The two techniques are quite independent, one probing only the leading-edge of the measured profiles and the other the centre of mass. The application of these complementary techniques will therefore provide a robust determination of both the photon energies and their associated uncertainties despite the influence of any profile structure. 7.7 Determining the photon energies The angular location of the centre of mass or of the leading-edge of a Bragg reflection is related to the photon energy E, the Miller indices of the reflecting planes (hkl),

180 152 Chapter 7. DETERMINING THE PHOTON ENERGIES and the lattice parameter of the reflecting crystal (a 0,Ge = Å [165) by hc E = 2a 0,Ge sin φ h2 + k 2 + l 2 = 2d Ge sin φ. (7.10) Misalignment of the zero-angle of the φ axis of the diffractometer and a constant profile shift both result in a constant shift to the measured angle φ φ + δ φ, where δ φ is the (small) effective alignment offset. Introducing this shift into Eq. (7.10) yields hc E = 2d Ge sin (φ + δ φ ) (7.11) = 2d Ge (cos δ φ sin φ + sin δ φ cos φ) (7.12) 2d Ge sin φ + δ φ 2d Ge cos φ, (7.13) or d Ge sin φ = hc 2E δ φ d Ge cos φ. (7.14) A plot of d Ge sin φ versus d Ge cos φ produces a straight line with gradient proportional to the misalignment δ φ and with the d Ge cos φ = 0 intercept equal to hc 2E. We can avoid extrapolating to d Ge cos φ = 0 by performing measurements on both sides of the φ = 0 location. Such measurements have been made at several energies. The photon energies were obtained by performing a linear fit to the peak locations determined by the two techniques. Figure 7.14 presents the determined peak positions on the d Ge sin φ versus d Ge cos φ axes. On the left side are the data for the leading-edge locations. The data are presented with error-bars; however, in most cases these uncertainties are too small to be viewed on this scale. The data fall on a straight line which is extrapolated to the d Ge cos φ = 0 axis to determine the energy. The determined energy, the 1-σ uncertainty of the fit, the χ 2 r, and the uncertainty in the determined energy (evaluated using σ E χ 2 r ) are all presented on the figure, indicating that the photon energy determined by the leading-edge technique is ± kev. On the right side of Fig we present the results of the similar treatment of

181 7.7. DETERMINING THE PHOTON ENERGIES 153 Figure 7.14: Results of a linear extrapolation to determine a single energy. Left: using the leading-edge locations; right: using the centres of mass. The results of the coarse scans have been excluded due to the inconsistency of their uncertainty estimates. The (relative) uncertainties shown on the centre-of-mass determination have been rescaled so as to be displayed on the figure. The gradient of the line on the two graphs is very different, indicating a different effective δ φ, consistent with the different definition of the peak locations. The energies determined are ± kev and ± kev. Despite the very different approach taken by these techniques, the energy is consistent to within 0.2 ev with uncertainties of order ev. the centre-of-mass locations. The error-bars have been scaled to show their relative sizes. The results of the energy determination using the centre of mass of the profiles is ± kev, in excellent agreement with the leading-edge value. The large differences in the locations of the peaks determined using the centre of mass and the leading-edge are absorbed into the variable δ φ, leaving the determined energy unchanged. Figure 7.15 presents the value of the fitted offset δ φ determined using the two analyses. This figure shows a strong correspondence of the determined offset parameters. This observation is confirmed in Fig. 7.16, where we present the difference between the offset parameters determined using the two approaches, i.e., δ φ,leading-edge δ φ,centre-of-mass. The systematic difference between the offsets determined using the two techniques is consistent with the different definitions of the peak location. The difference of about is in good correspondence with the half-width of the measured profiles depicted in Figs 7.7 and The energies obtained using the two techniques are in good agreement and are presented in Table 7.1. We have used σ χ 2 r to estimate the uncertainty in the determined energies because this allows us to determine robust uncertainties even

182 154 Chapter 7. DETERMINING THE PHOTON ENERGIES Figure 7.15: Fitted offset parameters determined from the leading-edge locations (left) and the centre of mass locations (right). Due to the different definitions of the peak location the offset parameters are different, however, the correspondence between the variation of the angular offset parameters thus determined is good. Figure 7.16: The difference between the fitted offset parameters shown in Fig is approximately constant at around 5 millidegrees. This angle corresponds well to the difference between the location of the leading-edge and the centre of mass which is equal to half of the peak width.

183 7.8. INTERPOLATING THE PHOTON ENERGIES 155 Figure 7.17: Comparison of the energy determined using the leading edge and the centre-of-mass locations. The determined energies agree to better than about 3 ev. Indicated on the plot is the χ 2 r of the difference between the determined energies which indicates that there is a σ difference between the results obtained by the two complementary techniques. though the centre-of-mass uncertainties are only relatively scaled. The difference between the energies determined using the two definitions of peak location is generally within their stated uncertainties, indicating no preference for the results of either technique. We thus proceed using the weighted average of the energies determined using the two techniques. The uncertainty in this weighted mean was estimated as the maximum of the weighted error [Eq. (A.2) and the weighted mean standard error [Eq. (A.3). 7.8 Interpolating the photon energies We have determined photon energies from measurements of a series of rocking curves reflected from a germanium crystal. Such a series has been recorded at regular 1 2 kev energy intervals throughout the experiment. We determine the photon energy for the measurements made at the intermediate energies by interpolating the directly determined energies against the monochromator angle. The monochromator angle was recorded by an angular feedback transducer which was located on the rotation stage which drives the primary crystal of the monochromator. This value has been logged in the electronic data file immediately prior to every measurement recorded throughout this experiment.

184 156 Chapter 7. DETERMINING THE PHOTON ENERGIES Table 7.1: Comparison of the energies determined using the leading-edge and the centre of mass techniques. The columns list the nominal photon energy, the energy determined using the leading-edge location (with uncertainty), the energy determined using the centre of mass (with uncertainty), the difference between the energies determined using the two techniques, the expected difference, based on the quadrature sum of the uncertainties associated with the two techniques, and the significance of the difference, evaluated as the difference divided by the expected difference. The energies determined using the two techniques are generally in agreement. The average of the significances is about 0.55, indicating that there is no significant systematic difference between the results obtained using the two techniques. The average of the absolute value of the significances is around 0.8, indicating that the results obtained using the two techniques are usually within one standard deviation. nominal leading-edge centre of mass energy expected signifenergy energy energy difference difference icance (kev) (kev) (kev) (ev) (ev) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

185 7.8. INTERPOLATING THE PHOTON ENERGIES 157 The photon energy E can be related to the silicon monochromator angle θ by the Bragg equation (for a cubic lattice) E = hc h 2 + k 2 + l 2 2a 0,Si sin θ = hc 11 2a 0,Si sin θ, (7.15) where h, k, and l are the Miller indices of the reflecting planes of the monochromator. In this experiment monochromation was effected by double-reflection of the x-ray beam from the (311) planes of two crystals of silicon arranged in a [+, (parallel) configuration. Operationally, the angle of the second crystal is offset slightly from the position parallel to the first in order to reject harmonic components from the beam. The zero of the monochromator crystal angular scale is not perfectly aligned with respect to the beam trajectory. This alignment error leads to a constant offset of the monochromator angle θ. Heat load, arising from the illumination by the intense polychromatic x-ray beam, results in an expansion of the monochromator crystal and its lattice parameter a 0,Si. We determine these parameters by fitting a modified form of the Bragg equation, E = hc 2d(1 + δ d ) sin (θ + δ θ ) (7.16) to the measured energies. The heat-load parameter has been cast as a small number, δ d 0. We have explored the inclusion of a linear stretch term on the angular scale of the monochromator but this is not supported by the measurements. Equation (7.16) does not provide a satisfactory fit to the measured energies across the entire range of monochromator angles due to discontinuities in experimental conditions. Such discontinuities include changing the harmonic order of the undulator and a cycling of the monochromator angle, leading possibly to hysteresis in the monochromator offset angle δ θ. These experimental discontinuities preclude the interpolation of the measured energies across the entire range using a single calibration curve. We have investigated the effect of these discontinuities by splitting the data into

186 158 Chapter 7. DETERMINING THE PHOTON ENERGIES Figure 7.18: Results of the interpolation of the photon energies. The difference between the determined energies and the nominal energy is plotted against the nominal synchrotron energy. The determined energies diverge from the nominal values as the photon energy increases. The break in the interpolation at 25 kev coincides with the change from the third to the fifth order of the undulator spectrum. separate pools delimited by these events. Our tests showed a significant improvement in the quality of the interpolation when the data was divided into two pools only, corresponding to the change of the undulator harmonic order at 25 kev, and which has resulted in a significantly different heat load on the monochromator crystal. The values of the fitted parameters and their uncertainties returned from fitting the two pools were physically reasonable and stable. Further partitioning of the data to account for monochromator hysteresis provided no significant improvement in the quality of the fit. Figure 7.18 presents the result of fitting the modified Bragg equation [Eq. (7.16), represented by the black line, to the measured energies, presented with their errorbars. The small break in the fit at around 25 kev is due to the change in the undulator harmonic. The two regions are nearly continuous. The grey line around the fitted Bragg equation is the uncertainty in the fitted photon energies, determined by evaluation of the covariant error matrix returned from the fitting procedure. The uncertainty ranges from about 3 ev at 41.5 kev to less than 1 ev at 18 kev, and is much better than the level of accuracy required for the measurement of the mass attenuation coefficients, as discussed in section 7.1. Figure 7.19 compares the

187 7.9. CONCLUSIONS 159 Figure 7.19: Uncertainties associated with the determined photon energies, plotted against the nominal synchrotron energy. The curved segments indicate the uncertainty of the energy interpolation. The break at about 25 kev is a result of our use of two separate interpolation regions. The dashed line indicates the 0.01% uncertainty level in the energy. A major goal of this analysis was to determine the photon energies to below this level of uncertainty so that the energy determination would not limit the accuracy of the attenuation measurement in regions away from the absorption edge and the XAFS. uncertainty in the determined photon energies against our requirements, showing that the precision of this determination is in excess of our earlier stated requirements. 7.9 Conclusions The measured rocking-curves of reflections from a germanium single crystal appear inferior to those we have previously recorded using a powder sample. This is likely to be due to structure in our germanium crystal and to the presence of saturation in some of the measured reflections. The locations of the measured rocking-curves were determined by the leadingedge and centre-of-mass techniques. These techniques probe very different features of the rocking-curves, and so provide orthogonal interpretations of the measured rocking-curves. The photon energies were determined separately using the results of these techniques. The consistency of these determined energies demonstrates the validity of each technique while also allowing the robust quantification of the limitations of each approach. Accordingly, we have determined the energy as the weighted

188 160 Chapter 7. DETERMINING THE PHOTON ENERGIES mean of the energies determined by the two techniques, and have evaluated the uncertainties from the consistency of the two techniques and their directly determined uncertainties. Our use of these very different techniques has resulted in a robust determination despite the presence of the strong peak-shaping which has affected the measured rocking-curves. The photon energies have been determined to better than 0.01% everywhere. The measurement accuracy is by a factor of better than the requirements described at the beginning of this chapter.

189 Chapter 8 Determining the mass attenuation coefficient, the photoelectric absorption coefficient, and the imaginary component of the atomic form-factor of molybdenum We determine the mass attenuation coefficients from the measured attenuations and examine these for effects of systematic errors. Equations predicting divergences resulting from systematic errors are derived. A single parameter describes the magnitude of each systematic error, such as the fraction of harmonic components in the x-ray beam. A fitting routine is used to determine the presence of the systematic error and, if present, a correction to the observed values. We treat effects relating to foil thickness variations (requiring -0.4 to +0.8% corrections), the finite x-ray bandwidth ( % corrections), and the dark current value ( % correction). The mass attenuation coefficients are determined to accuracies of %. 8.1 Principles of the treatment of systematic contributions to attenuation measurements Systematic contributions to the observed attenuation are often exhibited by a smooth and continuous divergence of values obtained across a range of experimental conditions. Such divergences are probed using the XERT and provide signatures by which systematic effects can be identified. We use a Levenberg-Marquardt least-squares fitting program to minimise the 161

190 162 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM discrepancies between the measured mass attenuation coefficients. The χ 2 statistic χ 2 = points ( ) 2 data function. (8.1) σ data is often used to quantify the agreement between measured data and a function. In our case we aim to optimise the self consistency of the measured data. Accordingly, we evaluate the χ 2 by quantifying the discrepancies from the weighted mean of the data using χ 2 = E i F j ([ µ EiFj [ µ E i σ( [ µ ) E i F j ) 2. (8.2) Equation (8.2) sums measurements obtained at all energies E i using all measured foils F j. The term in parentheses is the difference between the mass attenuation coefficient obtained using the foil F j at energy E i, [ µ and the weighted mean E i F j of all values obtained at energy E i, [ µ, divided by the measurement uncertainty E i σ( [ µ ). This χ 2 measures the least-squares inconsistency of the mass attenuation E i F j coefficients and is insensitive to the functional form of the measured values. The term thickness effect has been used to refer to any systematic error that can be probed by making measurements using foils of different thickness [138. Common recommendations for the correction of such effects include the extrapolation of measurements made using a variety of foil thicknesses to the zero-thickness limit ([ , [137), or the use of absorbers with [ µ [t 1, in an attempt to realise an effective zero-thickness limit [166, 167. However, such approaches are patently incorrect for thickness effects that do not disappear in the zero thickness limit. Our approach differs from these techniques in two important ways. Firstly, the form of the extrapolation function is determined by the cause of the systematic shift. This refinement enables proper treatment of all effects, regardless of the functional form of the effect in the zero thickness limit. Secondly, we examine the functionality of systematic discrepancies over an extremely wide range of the measurement parameter-space. The use of this extensive data set enables a thorough investigation of the functional form of the systematic discrepancy, makes the technique robust in the presence of noise, and allows accurate corrections to be made even where the systematic shifts are below the noise level.

191 8.2. DETERMINING THE LOCAL INTEGRATED COLUMN DENSITIES Determining the local integrated column densities Figure 8.1 presents the mass attenuation coefficient [ µ determined by dividing the calculated attenuations by the average integrated column densities of each of the foils. The measurements appear in excellent agreement with one another, with the exception of a few measurements between 25 and 30 kev. We investigate the detailed deviations and discrepancies in the measurements in Fig. 8.2 where we present the percentage difference between the measured values and the weighted mean at each energy. The percentage difference is percentage difference = 100 ([ µ F,E [ µ [ µ E E ), (8.3) where the subscripts refer to measurements made using foil F at x-ray energy E, and [ µ represents the weighted mean of all measurements at energy E. Systematic E differences of up to 2% between the mass attenuation coefficients measured with different foils are due to the non-uniformity of the integrated column density, and Figure 8.1: The mass attenuation coefficients [ µ calculated by dividing the measured attenuations by the average integrated column densities. The values obtained using the various foil thicknesses are plotted using different coloured symbols and appear to be in excellent agreement, but cannot be resolved on this scale.

192 164 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.2: The percentage difference between the measured mass attenuation coefficients and the weighted mean value at each energy. The colours represent foils of nominal thickness: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm; light blue 150 µm; maroon 200 µm; orange 250 µm. The symbol indicates the value determined from the full-foil mapping procedure. Six different symbols represent measurements using three different aperture combinations using the two downstream ion chambers: closed symbols (,, ) and line symbols (, +, ) indicate measurements recorded using the first and second downstream ion chambers with the small, medium, and large aperture combinations, respectively. Point symbols ( ) occurring between and kev represent measurements affected by numerical saturation. in particular to the difference between the average integrated column density and the value realised at the measurement location. The beam passes through the same point of the foil for all measurements using that foil, so the integrated column density is common for all measurements made with a given foil. We therefore determine the local integrated column density by scaling the measurements according to [ µ [ µ = [ µ [t [t T, (8.4) where [t T is the local integrated column density for a particular foil. The fitting routine determines [t T by minimising the χ 2 difference between the mass attenuation coefficients. The full-foil absolute value is included in the evaluation of χ 2

193 8.2. DETERMINING THE LOCAL INTEGRATED COLUMN DENSITIES 165 but (of course) not varied. Minimisation of χ 2 optimises the weighted agreement between the values determined using each of the foils and the absolute value at the full-foil mapping energy and the weighted agreement of the determined values at other energies. The removal and replacement of a foil in the sample stage can result in the presentation of a slightly different part of the foil to the beam. Our use of accurately machined components and high-precision motorised translation stages places tight constraints on this replacement error. We estimate a maximum shift of around 200 µm. However, even this small shift has the effect that the integrated column density of the local region of the foil may be different for multiple replacements. Therefore, nine integrated column densities relating to seven foils have been varied to minimise the discrepancies between 5028 measurements at 526 energies. Figure 8.3 presents the percentage difference of each measurement from the weighted mean at each energy after determining the local integrated column densities. The unresolved large-diamond markers represent the full-foil mapping values determined in chapter 6. The 2% differences present in Fig. 8.2 have been eliminated by determining the local integrated column densities, and most measurements are now consistent to better than 0.2%. This procedure can be assessed by comparing the fitted integrated column densities against their average values. The difference between the foil averages and the fitted values is less than 0.8% for all of the foils. The full-foil mapping has revealed nonuniformities of order 0.1% and 0.5% for the 250 µm and 50 µm foils, respectively, and so the adjustments to the average integrated column densities are consistent with the thickness variations quantified by use of the full-foil mapping. The significance of the difference between the average and fitted integrated column densities ([t [t fit )/ [t, (8.5) where [t is the quadrature sum of the fitting and measurement uncertainties, has a mean of 1.2 with a standard deviation of 1.6. This is consistent with random deviations from the average values.

194 166 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.3: Percentage difference of the mass attenuation coefficients from the weighted mean at each energy after determining the local integrated column density of each foil. Several divergences are now apparent, and correlate with increasing foil absorption. Symbols: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm; light blue 150 µm; maroon 200 µm; orange 250 µm. The symbol indicates the value determined from the full-foil mapping procedure. The six different symbols represent measurements using three different aperture combinations using the two downstream ion chambers: closed symbols (,, ) and line symbols (, +, ) indicate measurements recorded using the first and second downstream ion chambers with the small, medium, and large aperture combinations, respectively. Point symbols ( ) occurring between and kev represent measurements affected by numerical saturation. The improved consistency of the measured values is indicated by the lower reduced-chi-squared χ 2 r (χ 2 per degree of freedom) of the data set. Without determining the local integrated column density (using [t), the measurements obtained using different foils differed by up to 2% due to local thickness variations, with a χ 2 r of 112. After determining the local integrated column density this χ 2 r is reduced to 3.95, reflecting the improved consistency across all of the parameter space. Figure 8.3 shows a number of prominent divergences from the zero line, including two inconsistencies at 30 and 35 kev which are readily distinguished from systematic trends by virtue of their transience, and a prominent complex of deviations occurring around the absorption edge at 20 kev. Four further systematic divergences are observed, where the measured value obtained using one of the foils diverges systematically below the zero line as the x-ray energy decreases. These

195 8.2. DETERMINING THE LOCAL INTEGRATED COLUMN DENSITIES 167 Figure 8.4: Significance of deviations of the mass attenuation coefficients from the weighted mean after scaling [Eq. (8.6). Significant outliers are seen in the near-edge region, implying the existence of uncorrected systematics such as the bandwidth effect. Symbols: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm; light blue 150 µm; maroon 200 µm; orange 250 µm. The symbol indicates the value determined from the full-foil mapping procedure. The six different symbols represent measurements using three different aperture combinations using the two downstream ion chambers: closed symbols (,, ) and line symbols (, +, ) indicate measurements recorded using the first and second downstream ion chambers with the small, medium, and large aperture combinations, respectively. Point symbols ( ) occurring between and kev represent measurements affected by numerical saturation. divergences fall to around 4% below the zero line at 25 kev (light blue, 150 µm foil), 2 6% below at 20 kev (red, 100 µm foil), 2% below at 15 kev (orange, 250 µm foil) and 0.8% below at 13.5 kev (light blue, 150 µm foil). These divergences correlate with rising foil attenuation, and are onset when the foil attenuation increases above 4 5, as can be seen by comparison with Fig The statistical significance of the discrepancies can be appreciated by examining the individual differences from the weighted mean divided by the measurement uncertainties, which we term the significance significance = [ µ Ei [ µ Fj σ( [ µ ) E i F j E i. (8.6)

196 168 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM The significance describes the contribution of each measurement to χ 2 [cf. Eq. (8.2). Figure 8.4 shows the low significance of the four systematic divergences of Fig. 8.3, reflecting the low statistical precision associated with those measurements. Those divergences represent real systematic deviations in the results, but fall within experimental uncertainty and do not exert significant influence on the weighted mean of the measured values. In the region below 20 kev the values obtained using the 100 µm foil (red marker) are up to 4σ higher than the weighted mean. In the same energy range we see the 50 µm foil values (green marker) are up to 4σ lower than the weighted mean. However, the tensions cannot be resolved by redetermining the integrated column densities of these foils as there are opposing discrepancies elsewhere. Figure 8.4 is particularly useful for the close examination of the remaining discrepancies in the fitted result. The next section deals with the statistically-significant complex of discrepancies observed about the absorption edge. 8.3 The effect of x-ray bandwidth The synchrotron undulator produces a highly-structured spectrum of x-ray energies. From this spectrum the monochromator selects a small bandwidth of energies whose attenuation we measure. The fact that one is in practice always dealing with a finite bandwidth rather than a single x-ray energy has a number of consequences which we explore in this section. The technique has been used to determine the bandwidth of the x-ray beam at beamline 1-ID [23. The work includes a thorough discussion of the effects of the x-ray bandwidth on various absorption experiments and has been included in Appendix D of this thesis. A distribution of energies in the x-ray beam implies that we measure the combined attenuation at these energies weighted by the intensity of each x-ray energy component. Since each energy component has a different attenuation coefficient, the original distribution of energies in the x-ray beam the beam energy profile will change as the beam is attenuated, with the less attenuated components increasing their relative intensity over the more attenuated components. This change in the beam energy profile results in a decrease in the differential attenuation of the beam

197 8.3. THE EFFECT OF X-RAY BANDWIDTH 169 Figure 8.5: Measured values of the mass attenuation coefficient in the near-edge region. At each energy, measurements have been made with a a number of thicknesses of foil, represented by three different coloured markers: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm. The consistency of the experimental results in the lower edge region is too good for the three independent measurements at each energy to be clearly resolved on this scale. Above the absorption edge the 100 µm foil (red marker) has an attenuation of about 8, and these measurements are seriously affected by a number of systematic effects. The gradient of the weighted mean of the measurements is indicated on a relative scale using a dotted line. as it passes through an absorber, and is manifest as a systematic decrease of the measured mass attenuation coefficient as the absorber thickness is increased. The effect of the bandwidth is greatest on the absorption edge where the beam energy profile is distorted strongly by the rapidly changing mass attenuation coefficient. Figure 8.5 shows the measured mass attenuation coefficients within a small range of energies about the absorption edge, after determining the local integrated column density (section 8.2). The results appear consistent and the near-edge structure seems well determined. The result could clearly be used quantitatively, for example to report XANES. However, the precision of each measurement is sufficiently good for the discrepancies in the data to be further interpreted. Figure 8.6 presents the percentage difference between the mass attenuation coefficients and the weighted mean value at each energy for the measurements made along the absorption edge. This figure shows a consistent trend in the differences, peaking at kev. The thicker foils return consistently lower values than the thin foils. The deviation is proportional to the gradient of the mass attenuation

198 170 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.6: Percentage differences between the mass attenuation coefficients [ µ and the weighted mean value at each energy in the neighbourhood of the absorption edge. The dotted line indicates structure in the gradient of the mass attenuation coefficient (presented as a percentage of its value, appropriate for the ordinate of this plot), on a relative scale. The effect of the bandwidth of the x-ray beam is clearly visible in the form of a prominent deviation which correlates with the gradient of the mass attenuation coefficient. The deviation also correlates with the thickness of the foil used to make the measurement, with the thinner foils returning higher values for the mass attenuation coefficient. This gradient and thickness-correlated deviation, peaking at around kev, is the signature of the effect of the bandwidth. Symbols: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm. coefficient. The correlation of the deviations presented in Fig. 8.6 with the gradient of the mass attenuation coefficient uniquely identifies the deviations as caused by the finite bandwidth of the beam. If deviations correlate with the gradient, the gradient must be probed: i.e., there must be a finite spread of energies in the beam and hence a finite bandwidth. To correct the mass attenuation coefficients for the effect of the bandwidth, we include the finite bandwidth of the x-ray beam in the calculation of the mass attenuation coefficients derived from measured beam intensities. We assume that the detector efficiencies are constant across the relatively narrow bandwidth of the beam. We begin with the Beer-Lambert absorption law: { exp [ } µ [t = I, (8.7) me 0 I 0

199 8.3. THE EFFECT OF X-RAY BANDWIDTH 171 where [ µ represents the mass attenuation coefficient of the foil material as me 0 measured (subscript m) with a beam whose central energy is E 0, I the attenuated beam intensity, I 0 the unattenuated beam intensity, and [t the integrated column density of the attenuating foil. Explicit inclusion of the bandwidth gives: { exp [ } µ [t me 0 = 0 I de 0 I 0 de, (8.8) where I 0 now represents the intensity of the incident beam at each energy E distributed around the central energy E 0, i.e., the incident beam energy profile. The transmitted intensity I results from the attenuation of each of the energy components in the incident beam by an amount determined by the true mass attenuation coefficient at each energy E, [ µ. Thus te { exp [ } µ [t me 0 = { I 0 0 exp [ } µ te [t de. (8.9) I 0 0 de We define the normalised incident beam energy profile as and write Eq. (8.9) more simply as { exp [ } µ [t me 0 Ĩ 0 = = I 0 0 I 0 de, (8.10) 0 { Ĩ 0 exp [ µ te [t } de. (8.11) To invert Eq. (8.11), we isolate the correction c E0 to the measured mass attenuation coefficient at each measured energy, E 0 : c E0 = [ µ te0 [ µ = [ µ + te0 1 [t ln (8.12) me 0 [ { Ĩ 0 exp [ } µ te [t de, (8.13) 0 obtained from Eq. (8.12) by use of Eq. (8.11). Equation (8.13) refers to true mass attenuation coefficients [ µ, which are unknown at this stage. We linearise te

200 172 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM [ µ by expanding in a Taylor series about energy E 0, keeping only terms linear in te (E E 0 ) [ µ te [ µ + (E E 0 ) d[ µ (8.14) te0 so that c E0 [ [ µ + te0 1 [t ln 0 [ = [ µ + te0 1 [t ln [ = 1 [t ln Ĩ 0 exp 0 { exp { Ĩ 0 exp [ µ { d[ µ te 0 de } (E E 0)[t de [ µ [t d[ µ te 0 te0 de } [t Ĩ 0 exp { d[ µ } te 0 te 0 0 de (E E 0)[t de } (E E 0)[t de. (8.15) te 0 de The validity of this linearisation relies on the second-order correction being small: d 2[ µ te 0 (E E 0 ) 2 d 2 E 2 d[ µ te 0 de (E E 0) (8.16) for the range of energies across the bandwidth. The absorption edge is some 20 ev wide and, as will be seen, the bandwidth is of order 1 ev. The mass attenuation coefficient has a strong nonlinearity with respect to energy across the absorption edge, but on the relevant electron-volt scale the deviation from nonlinearity is quite small. Equation (8.15) tells us that the correction due to the finite bandwidth of the beam is due to the gradient of the true mass attenuation coefficients. Since the difference between the measured and true values is proportional to the gradient, the difference between their gradients varies with the curvature. The true gradient is therefore the same as the measured gradient, to the same order of approximation as Eq. (8.15), and therefore [ c E0 1 [t ln Ĩ 0 exp 0 { d[ µ me 0 de } (E E 0)[t de. (8.17) Here the gradient of the measured values of the mass attenuation coefficient d [ µ me 0 de

201 8.3. THE EFFECT OF X-RAY BANDWIDTH 173 has replaced the gradient of the true values d [ µ te 0 de. Inaccuracies due to the use of measured gradients have been checked by testing the correction algorithm in a two-pass manner: a first-pass approximate correction was used to calculate the gradient for determination of the second-pass correction. The results of the twopass calculations were consistent with the single-pass results, and validate the use of Eq. (8.17). Equation (8.17) describes a first-order dependence of the effect of the x-ray bandwidth on the gradient of the mass attenuation coefficient. This first-order dependence is readily observed by comparing the deviations with the (similarly-scaled but laterally-shifted) mass attenuation coefficient gradient presented in Fig Given a model beam energy profile Ĩ0, the correction in Eq. (8.17) can be evaluated by employing appropriate numerical methods. We have evaluated the integral using a finite sampling of the integrand over a finite energy range, parametrised using r W, the profile range in units of the full-width at half-maximum (fwhm), and s F W, the sampling frequency in units of the number of sample points within each fwhm, using a Gaussian profile [ j c E0 = 1 { [t ln Ĩ 0 (E 0 + k E) exp k= j d [ µ m,e 0 de } (k E)[t (8.18) j = r W (s F W 1), E = fwhm 2 s F W 1, s F W odd integers. (8.19) We force s F W to be odd to ensure that the dominant, central contribution from the beam energy profile is evaluated. The sampling frequency s F W and range r W were varied and found to produce robust and well-converged results. The derivative in the summation is evaluated as the numerical derivative of the weighted mean of the measured mass attenuation coefficients using the 3-point Lagrangian interpolation algorithm of Hildebrand [168. Results of the minimisation of the deviations are presented in Fig Comparison with Fig. 8.6 shows that the systematic deviations have been significantly reduced. The value of the average fwhm bandwidth of the x-ray beam that minimises

202 174 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.7: Percentage differences between the mass attenuation coefficients [ µ and the weighted mean values after correcting for the effect of the finite bandwidth of the x-ray beam. The prominent, gradient-correlated difference presented in Fig. 8.6 has disappeared. the discrepancies is 1.57 ± 0.03 ev, assuming the Gaussian distribution of energies. The uncertainty attributed to the bandwidth determination comes from the fitted one standard deviation error estimate multiplied by χ 2 r, thus assuming that the functional approach is valid and that χ 2 r is due to inadequate estimation of input uncertainties. Although measurements taken in the XAFS region may not satisfy the conditions of the linearised approximation, Eq. (8.15) is used to apply a first-order correction. The likely error of this approach is estimated from the effect of the bandwidth in regions where the linearised approach may fail, i.e., at the extrema of the attenuation in the XAFS. We observe that the effect of the bandwidth at the attenuation extrema is less than twice that predicted by the linearised model due to the two-sided nature of the extremum when the gradient is taken to be the maximum gradient within the energy span of the beam. Taking this energy span to be equal to twice the fwhm bandwidth of the beam, we estimate the upper bound of the bandwidth effect at the first attenuation maximum to be twice that calculated within the range from ( ) kev to ( ) kev. At these energies the applied linearised correction is less than around 0.03% (from Fig. 8.9), and so an upper

203 8.3. THE EFFECT OF X-RAY BANDWIDTH 175 bound of 0.06% is established. The error arising from the use of the linearised approximation for the measurements made around the more weakly curved extrema is significantly less than this upper bound, and is generally well below the uncertainty of the measurements in the XAFS region. The determined value for the average bandwidth of the beam is consistent with the nominal monochromaticity of the x-ray beam at beamline 1-ID ( E E the bandwidth of E = E E = 10 4), or E = kev = 2 ev [149. This nominal bandwidth is quoted by many synchrotron radiation facilities employing silicon doublereflection monochromators, and is a rough estimate of the bandpass of this type of monochromator. We characterise the monochromaticity of synchrotron beams at such facilities better than the previous literature. Such a measurement is otherwise very difficult to accomplish. The χ 2 r associated with the reduced data set presented in Figs 8.6 and 8.7 has decreased from 7.8 to 4.5, clearly confirming the presence and successful correction of this effect. Table 8.1 presents the χ 2 r before and after the correction for the effect of the beam bandwidth, evaluated from the entire data set. This table shows that the χ 2 r has decreased by about 10%, in spite of the fact that less than 10% of the measurements were adjusted by more than 0.04%, again providing clear indication of the effect of the x-ray bandwidth. This large reduction in the χ 2 r is due to improved agreement of the values on the absorption edge and to improved agreement away from the edge which has occurred in response to the removal of the systematic discrepancy on the absorption edge. In the above, a single average bandwidth is fitted for the entire data-set. While we have used a consistent method for setting our x-ray energy which is likely to result in a stable bandwidth, the x-ray bandwidth does vary. The energy variation Model χ 2 r Input data 112 Determining the local integrated column densities 3.95 Effect of the x-ray bandwidth 3.63 Table 8.1: Reduction to the χ 2 r resulting from the correction of the bandwidth effect. The bandwidth correction is now made in conjunction with the determination of the local integrated column densities.

204 176 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.8: Sketch of discretely sampled Gaussian (left) and Lorentzian (right) beam energy profiles used to evaluate the effect of the bandwidth. The continuous profile is indicated by the continuous line. The five vertical bars per fwhm represent the discrete energies used to evaluate the effect of the bandwidth. The Lorentzian tails contain a far greater proportion of the intensity than the Gaussian tails. The profiles have been normalised [Eq. (8.10). of the x-ray bandwidth contributes to residual discrepancies of the corrected values on the absorption edge, where the effect of the bandwidth is greatest, compared with those well away from the edge, where the effect of the bandwidth is insignificant. The weak structure in the residual discrepancies in Fig. 8.7 may be the result of such variations of the beam bandwidth. The finite sampling of the summation can be interpreted as an adjustment to the beam energy profile. Figure 8.8 illustrates the effective beam energy profiles resulting from the use of a Gaussian and Lorentzian beam energy profiles with r W = 5 fwhm and s F W = 5 points per fwhm. Table 8.2 shows the fitted values of the bandwidth when the summed function is evaluated using a variety of choices of r W and s F W, using Gaussian and Lorentzian beam energy profiles. This table indicates that s F W = 3 samples per fwhm evaluated over an energy range of r W = 3 fwhm around the central energy is sufficient for the effect of the bandwidth to converge under the assumption that the beam energy profile is Gaussian in distribution. The convergence of the minimising E E to a well defined value for these energy ranges indicates that the correction due to finite bandwidth is consistent with a Gaussian distribution of energies in the beam. The convergence to a well-defined E E with a variety of sampling frequencies validates our use of the finite sampling approximation. The corrected values of the mass

205 8.3. THE EFFECT OF X-RAY BANDWIDTH 177 Profile Gaussian Lorentzian s F W fwhm bandwidth E E 105 (points r W (energy range in units of fwhm) per fwhm) Table 8.2: The fwhm bandwidth, E E 105, when the systematic due to the finite x-ray bandwidth is corrected using Gaussian and Lorentzian profiles with a variety of parameter choices. When the Gaussian profile is used the bandwidth converges for r W 3, s F W 3. However, when the Lorentzian profile is used the bandwidth does not converge for any values of r W and s F W. attenuation coefficients have been evaluated using a Gaussian beam energy profile over an energy range of r W = 5 fwhm with a sampling frequency of s F W = 5 points per fwhm. When a Lorentzian beam energy profile is assumed, the determined E E does not converge for any parameter choices. The trend to narrower bandwidth as the energy range increases indicates that the Lorentzian tail intensities provide major corrections to the determined mass attenuation coefficients. The lack of convergence indicates that the beam energy profile is not Lorentzian in the tail regions. Non- Lorentzian tails are common on otherwise Lorentzian profiles, and such Lorentzian profiles are often observed to be truncated or convolved by instrumental and other processes, resulting in Lorentzian-slit or Voigt profiles. The convergence behaviour of corrections determined using Gaussian and Lorentzian profiles demonstrates the extreme limits of the possible profile form. We change variables (E E = E E 0 ) and retain only energy-dependent terms to observe the asymptotic extremes of the correction. The (linearised) attenuation term becomes e E and, for a Gaussian beam profile, Ĩ0 e E 2, so the correction behaves like c ln [ E 0 e E 2 e E de, (8.20)

206 178 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM which is well converged even when the range of values used to evaluate the integral is narrow. In the case of a Lorentzian beam profile Ĩ0 E 2, the correction behaves like [ e E c ln E 0 E 2 de, (8.21) which does not converge on the negative axis. The correction to the mass attenuation coefficients determined using the Lorentzian profile grows as the range broadens. Consequently, the fitted bandwidth decreases to compensate for the over-correction. This explains the observed behaviour of the derived bandwidths in Table 8.2. Figure 8.9 presents the percentage correction to the mass attenuation coefficients for the measurements made along the absorption edge. The bandwidth has resulted in a systematic effect rising to about 0.25%, 0.45% and 0.9% for the thin, intermediate, and thick foils respectively. The structure shown in Fig. 8.9 is of great interest in XANES investigations: we have shown that such investigations must correct for the bandwidth effect along the absorption edge if they wish to avoid the false attribution of such structure to bond- Figure 8.9: The percentage effect of the x-ray bandwidth on the measured mass attenuation coefficients: a 1.5 ev bandwidth of a 20 kev beam can affect the measured attenuation coefficient by one percent. The effect is significant for all foil thicknesses, and contains structure of great interest in XANES investigations. Symbols: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm.

207 8.4. THE EFFECT OF FLUORESCENCE RADIATION 179 ing rearrangement or to orientation effects, as in [166, corrected in [137. Further implications of the bandwidth effect for a variety of absorption-based techniques are discussed by us in [ The effect of fluorescence radiation Photoelectric absorption involves the absorption of an x-ray and the promotion of a bound electron to a higher (possibly continuum) energy level. At some later time the vacancy left by the promoted electron is filled, possibly resulting in the emission of a fluorescent x-ray. The detection of such fluorescence radiation by the ion chambers used to measure the incident and transmitted beam intensities can result in a systematic shift of the measured mass attenuation coefficients. Immediately above the K-shell absorption edge, the attenuation of molybdenum is dominated by the resonant absorption of the x-ray by the 1s electron, where the atom is ionised and the electron is dissociated from the target atom. The ionised atom may decay to the ground state via a number of decay sequences. The most common decay sequences involve the 2p 1 1s, 2p s and 3p 1, 3p 3 1s transitions 2 2 accompanied by the emission of Kα 2, Kα 1, Kβ 1, and Kβ 3 radiations of approximate energy , , , and kev respectively [169. The fluorescent photon energy is below the absorption-edge energy, and so they are attenuated by up to an order of magnitude less strongly than the incident beam photons. When the incident photon energy is near to but above the absorption edge, the fluorescent photons may have a significantly greater chance of penetrating the absorber than the incident beam 1. It is therefore possible that the count-rate recorded by an ion chamber located downstream of the absorber may be dominated by fluorescent photon events. To model the fluorescence we assume that the fluorescent photons are all of one 1 For this reason x-ray apertures and collimators are often layered with materials of successively decreasing atomic number Z. Fluorescence radiation escaping the highly absorbing, high Z material is absorbed by the (otherwise more penetrable) low Z layers. Each successive absorption leads to a reduction in photon numbers due to geometrical factors and fluorescence yield, and the unwanted x-ray intensity is reduced to a low level. See, for example, [68.

208 180 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM energy, corresponding to the high-yield Kα 1 radiation. We do not treat multipleinteraction processes except for the absorption of the fluorescent photon. This is a single interaction approximation. We assume initially that the detection efficiency for the fluorescent photons is the same as that for the beam photons, although this value may differ by a factor of two. All parameters influencing the detected fluorescence signal are known, so the effect can be calculated from the experimental conditions. However, to account for assumptions affecting the magnitude of the modelled shift, we scale the predicted systematic shift. The scale factor combines the efficiency of the detection of the fluorescent photons and the fluorescence yield in our modelling of the shift. The fluorescence is modelled by calculating the number of fluorescent photons produced at each location within the absorber, the subsequent attenuation of these photons as they exit the foil, and the probability that they will enter the detector. The number of photons removed from the incident beam is given by the differential form of the Beer-Lambert law di dx = [ µ Ix x = [ µ I0 exp { [ } µ x. (8.22) The number of fluorescent photons reaching the downstream detector is I f,down = t 0 A { 4πr exp 2 [ µ where the subscript f denotes the fluorescent photons. f (t x) } di x, (8.23) A 4πr 2 describes the probability that the fluorescent photon will reach the detector, with A the aperture area located at a distance r from the absorber. exp { [ µ (t x)} describes the attenuation f of the fluorescent photons as they escape the absorber, where we assume that the aperture is narrow so cos θ 1. di x represents the number of photons produced at

209 8.4. THE EFFECT OF FLUORESCENCE RADIATION 181 depth x through the foil. Using Eq. (8.22), this becomes I f,down = A 4πr 2 = A 4πr 2 t 0 { exp [ } µ f [t [ µ I0 [ µ f [ µ ( { exp exp [ µ { [ µ } [ x µ f I0 exp { [ } µ x dx } [t exp (8.24) { [ }) µ f [t. (8.25) Similarly, the number of fluorescent photons reaching the upstream detector is I f,up = A 4πr 2 [ µ I0 [ µ f + [ µ ( { 1 exp ([ µ f + [ }) ) µ [t. (8.26) The effect of the fluorescence radiation can be determined using Eqs (8.25) and (8.26). Of course, a correction is only made when the incident x-ray energy is above the absorption edge energy. The effect of the fluorescence radiation is presented in Fig The effect is a strong function of foil thickness in the neighbourhood of the absorption edge. The model indicates that a 0.3% effect may be observed for measurements using a 100 µm foil near the absorption edge. However, the measurements of the 100 µm foil are Figure 8.10: Modelling of the effect of fluorescence on the measured mass attenuation coefficients, assuming a 100% fluorescence yield. The effect has been calculated for the three foil thicknesses and three aperture sizes used most extensively for measurements in the region of the absorption edge, where the effect is most pronounced. The apertures were in the shape of a 9 mm diameter circle, a 9 mm by 1.8 mm rectangle and a 1.8 mm diameter circle, spanning a factor of 25 in the solid-angle subtended at the sample. For the geometry of this experiment the discrepancy reaches 0.3% for the 100 µm foil immediately above the absorption edge.

210 182 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM of low statistical precision generally no better than 1% near to the absorption edge (see Fig. 5.16) and cannot be used to determine the presence of such a 0.3% discrepancy. Thus, we cannot observe the effect of the fluorescent photons on the measured mass attenuation coefficients by examining the discrepancies between the measured values. The effect of the fluorescent photons on the high-precision measurements (i.e., where [ µ [t 5) is less than 0.01%. We have minimised the effect of the fluorescence on the measured values by use of our fitting routine, and have confirmed that this resulted in no significant reduction to χ 2 r. A more sensitive test for the effect of all secondary photons including Rayleigh and Compton scattered photons on the mass attenuation coefficients is obtained by examining the measurements made across the entire energy range for discrepancies between the values obtained using the three aperture combinations. significance of the difference between the values measured using each of the apertures is defined as significance = [ µ larger [ µ smaller σ 2 larger + σ2 smaller The, (8.27) and shows no deviation from zero, indicating the aperture independence of the measured mass attenuation coefficients, entirely consistent with our modelling. We have recently reported the observation and correction of the effect of fluorescence photons on a measurement of the mass attenuation coefficient of silver [69, 122. In that experiment fluorescence radiation was responsible for a 2% systematic effect on the mass attenuation coefficient measured using one of the foils. The methods of calculation were developed from the investigation reported. The different magnitude of that signature was due to different experimental conditions and geometry: the silver measurements used larger apertures located closer to the absorber.

211 8.5. THE EFFECT OF ROUGHNESS The effect of roughness The determination of the local integrated column densities presented in section 8.2 corrects the observed mass attenuation coefficients for the effect of thickness variations across the sample. Here we consider nonuniformity of the integrated column density within the range of the beam footprint. The most obvious contribution to the variation of the integrated column density of the absorber under the beam footprint results from the presence of nonuniformity and microstructure on the surface(s) of the absorber, sometimes referred to as roughness. However, in transmission, nonuniformity of the distribution of material within the absorber volume may also contribute to variations of the integrated column density. Consider the attenuation across the beam area. Measurement with a beam of macroscopic dimension determines the intensity-weighted average of the attenuation at each point through the surface. Assuming uniform incident intensity, we write the measured mass attenuation coefficient [ µ in terms of the true mass attenuation m coefficient [ µ as t { exp [ } µ m [t = 1 A area { exp [ µ t [t local } da, (8.28) where [t local is the integrated column density at each point under the illuminated beam area A. Factoring out the average integrated column density over the beam footprint, [t, gives { exp [ } µ m [t = { exp [ } µ t [t A area { exp [ } µ ([t local [t) da. (8.29) t The argument of the exponential appearing in the integral is small, allowing expansion in a Taylor Series: = { exp [ } µ t [t A area { 1 [ µ ([t local [t) + t [ µ 2 t ([t local [t) 2 2! } +... da. (8.30)

212 184 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM The first term in the Taylor series tells us that the measured attenuation is approximately equal to the true attenuation in the limit of low roughness. The second term is identically zero since the integral of the average is equal to the integral of the local values, A [t da = A [t local da. Accordingly, { exp [ } µ m [t { exp [ µ t [t } { 1 + [ µ 2 t 2! [ } 1 ([t local [t) 2 da. A area (8.31) Identifying the integral in the square brackets as the mean-square deviations of the integrated column density, σ[t 2, we write this as { exp [ } µ m [t { exp [ µ t [t } { 1 + [ µ 2 t σ2 [t 2! }. (8.32) We isolate the true mass attenuation coefficient by replacing the true mass attenuation coefficient in the correction term with the measured value, { exp [ } µ m [t valid when the correction term { exp [ µ t [t } { 1 + [ µ 2 m σ2 [t ( [ µ ) 2 t σ2 [t /2! is small. Rearranging gives [ µ t = [ µ m + 1 { [ µ [t ln m σ2 [t 2! 2! }, (8.33) }. (8.34) Figure 8.11 presents a simulation of the effect of roughness on the measured attenuations, calculated using Eq. (8.34). This figure shows the effect of a fixed degree of roughness for the three foil thicknesses that we have used for measurements in the region of the absorption edge, where the effect of the roughness is most pronounced. Roughness in the integrated column density can result in a significant systematic shift of the measured attenuations. For a constant roughness the systematic shift increases as the foil thickness tends to zero, in direct contrast to the zero-thickness extrapolation recommended for the correction of thickness effects discussed in section 8.1.

213 8.6. THE EFFECT OF HARMONIC COMPONENTS 185 Figure 8.11: Modelling of the effect of sample roughness on the determined mass attenuation coefficients. Each plot shows the effect of a constant roughness σ t on measurements made using the three thicknesses of foil that we have used. These plots show that the effect of foil roughness is not likely to be detectable (i.e., over the statistical precision of the measurements: cf. Fig. 5.16) unless our thinnest 25-µm foil has a σ t roughness greater than about 0.3 µm. AFM measurements such as that presented in Fig show that the surface roughness is less than 0.2 µm for the thinner foils, and that the surface roughness is well below 1 µm for all foil thicknesses. Assuming that this surface roughness dominates the σ [t, the simulated curves presented in Fig suggest that the effect of roughness will not be discernible in our measurements. A single σ [t roughness was fitted to the measured values to determine a correction for this effect. The χ 2 r was unchanged by this additional round of fitting, and roughness has not had a significant impact on the measurements. This is entirely consistent with modelling (Fig. 8.11). 8.6 The effect of harmonic components We examine the effect of harmonic energy components in the x-ray beam on the determination of the mass attenuation coefficients, and show that the effect of harmonic components in this experiment is significantly less than another effect which we treat in a subsequent section. When harmonic components of energy E n are present in the x-ray beam, the total transmission probability I I 0 will be equal to the sum of the transmission probabilities [ for each of the energy components exp { [ µ [t } weighted by the relative E n intensity f n = I 0,En I 0 of each component in the incident beam.

214 186 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM The energy dependence of the ion chamber efficiency has a significant effect on the effective harmonic component. For example, if the detectors are blind to the harmonic x-rays then the presence of these x-rays in the beam will be of no consequence. As we aim to determine the effect of the harmonic components on the measured attenuation, we refer to this effective harmonic component. When there is only one harmonic component of energy E n in a beam of fundamental energy E 1, the measured intensity ratio is given by [119 ( I I 0 ) { = exp [ } µ m [t { = (1 f n ) exp [ µ } [t E 1 { + f n exp [ µ } [t. E n (8.35) We describe the effect of each systematic error on the attenuation measurements by using a model function with a small number of fitted parameters. Our use of a small number of parameters to determine a correction for all measurements places a significant constraint on the applied correction and ensures the robustness of the corrected result. For example, by using a single bandwidth for all measurements we determine the best fit to the divergences within the measured data, and prevent fitting to spurious signals or noise. The effective fraction of harmonic components is highly dependent on the operation of the undulator and the monochromator (particularly with detuning), and on the operation of the ion chamber detectors. These strong dependences prevent description of the effect of the harmonic components across a wide range of energies. For example, while the detection efficiency can be described in the model function, the electron yield per detected x-ray (leading to the recorded count-rate, and required to compare the harmonic and fundamental x-ray fluences across a wide range of energies) is not so easily treated. We quantify the effective harmonic component using the daisy-wheel technique developed in [119, 120, 121, and briefly summarised here. We have measured the attenuation of a large number of molybdenum and aluminium foils whose thicknesses were chosen so that their attenuations ranged from 0.05 to The foils, mounted on the perimeter of the daisy-wheels shown in Fig. 4.2, were placed in the path of the beam by suitable rotation of the daisy wheel.

215 8.6. THE EFFECT OF HARMONIC COMPONENTS 187 Figure 8.12: The results of daisy-wheel measurements, showing a clear deviation from linearity of the measured attenuation as a function of foil thickness. Measurements for aluminium ( ) and molybdenum ( ) foils are plotted. On the left we present the result of fitting the measured values assuming a significant harmonic component, following Eq. (8.35). This model fails to describe the plateau in the measured attenuation occurring at [ µ [t 8.5. On the right we present the results of fitting the measured values assuming a dark current offset, following Eq. (8.36). The dark current offset is clearly significant to explain the data. The attenuations of the daisy wheel foils, measured using a beam of energy 39 kev, are shown in Fig The triangular markers indicate measurements using the molybdenum foils and the crosses those using aluminium. The measurements made using the molybdenum foils initially follow a linear trend as the foil thickness increases, the gradient defined by the mass attenuation coefficient of molybdenum for x-rays of the fundamental energy. about 8, no longer increasing with the foil thickness. The measured value becomes constant at The solid lines drawn on the left plot indicate the result of fitting a single, thirdorder (n = 3) harmonic component using the function defined in Eq. (8.35). performing this fit, we have allowed the value of the mass attenuation coefficient at the fundamental energy [ µ and the effective harmonic fraction f 3 to vary, but E 1 have used the FFAST tabulated value for the mass attenuation coefficient at the harmonic energy [ µ. The general shape of the fitted curve is in agreement with E 3 the measured values, although the measurement taken using the thickest foil does not agree with the fitted curve. In particular, the gradient of the second portion of the fitted curve is steeper than that described by the measured values. The plot on the right side of this figure presents the same measured values, but In

216 188 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM now fitted using { exp [ µ m [t } = I dc δ dc I 0 dc (8.36) where δ dc describes an error in the value of the dark current used for the downstream ion chamber. This function describes the effect on the measured attenuations of an incorrect value for the dark current, and is derived in the next section. The observed divergence from linearity is explained more satisfactorily by the effect of the dark current than by the harmonic component model. The plateau in the attenuations indicates that the thick-foil measurements are dominated by a dark-current offset rather than by a beam harmonic component, which would result in a qualitatively different signature. We have performed such fits for over 250 such daisy-wheel measurements. In Fig we present the χ 2 r measures of the quality of these fits determined using the harmonic component and dark current error models. The quality of the fit for the measurements made using the aluminium foils is unchanged because those measurements do not probe the highly absorptive and nonlinear attenuation region. The quality of the fit for the molybdenum foils is dramatically improved by allowance for a dark current offset, often reporting improvements to χ 2 r by factors of Figure 8.13: The χ 2 r statistic resulting from fitting for (left) a fraction of harmonic components in the beam and (right) for an error in the determined dark current. Measurements for the aluminium ( ) and molybdenum ( ) foils are plotted. The χ 2 r for the aluminium measurements do not change between the two models as they do not probe the high-attenuation region where the difference between the models is exhibited. The χ 2 r for the molybdenum measurements are improved by a factor of , indicating the clear preference for the dark current error model.

217 8.7. THE EFFECT OF A DARK-CURRENT ERROR over the harmonic component model. The relevance of the dark current model would not have been discernible had we not recorded measurements using foils whose nominal attenuation of the fundamental beam was well over 20. For example, the measurement using the cm molybdenum foil requires the dark current model (Fig. 8.12). The attenuation of this foil is 25 < [ µ [t < 175 over the range of energies used in this measurement. Even at an attenuation [ µ [t = 25, this foil should transmit only one photon per second for a beam flux of photons/s. This foil has an attenuation of 1.4 for the thirdorder harmonic photons. Accordingly, our ability to clearly resolve the effects of the harmonic components from those of a dark current offset is due to these extremely high-attenuation measurements and the recording of the daisy-wheel measurements over an extended range of energies. 8.7 The effect of using an incorrect value of the dark current In this section we examine the effect of using an incorrect value of the dark current for the determination of the mass attenuation coefficients. Dark currents were measured by allowing the detectors to count with the beam shuttered by inserting a thick filter into the beam. A positive count-rate was recorded due to the presence of a leakage current in the ion chambers which was negated in part by the adjustment of an offset current in the current amplifier. The x-ray beam is assumed to be completely shuttered while taking a dark current measurement, and the linearity of the ion chamber response, established at high count rates, is assumed to continue down into the low count-rate regime. Figure 8.14 illustrates several dark current offsets and their possible impact on the determined mass attenuation coefficients. The solid black line on this plot describes a perfectly linear relationship between the photon flux passing through the ion chamber and the reported count rate. Along this line we show two + markers, representing intensities I recorded with a sample in the path of the beam and I 0 with no sample in the path of the beam. Four dark current measurements are indicated by diamond markers. The well-

218 190 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM determined dark current describes a measurement made at zero real photon flux which lies on the linear ion chamber response function. Three incorrect values of the dark current are indicated on the plot. One of these indicates a measurement made with a nonzero photon flux, resulting possibly from leakage through the beam shutter. Two other dark current measurements record an incorrect value of the dark current due to nonlinearities in the low count-rate region of the detector response curve. The horizontal dotted line on Fig indicates our use of the dark current for the determination of the mass attenuation coefficient. The vertical arrows extending upwards from the horizontal dotted line indicate the count rate resulting from the photon flux, determined by calculation of (I dc) and (I 0 dc). The transmission probability is evaluated from the ratio I dc I 0. An incorrect value of the dark current dc will affect the measured transmission probability. The effect of the use of an incorrect dark current is greatest when the measured intensity approaches the dark current Figure 8.14: Illustration of a variety of causes for the recording of an incorrect value of the dark current and the impact of the use of this incorrect value on the determined mass attenuation coefficient. The dark current is the offset count-rate that must be subtracted from the measured intensities, and sets the level of the horizontal dotted line in this figure. The attenuation is determined from the transmission probability I dc I 0, i.e., the ratio of the lengths of the vertical arrows. The use of an incorrect value dc of the dark current will alter the value of the determined transmission probability. This effect will be most significant when the attenuated intensity I is low.

219 8.7. THE EFFECT OF A DARK-CURRENT ERROR 191 value. Accordingly, this effect is most pronounced for a dark current error occurring in the downstream ion chamber, and increases with the attenuation of the sample. A first-order expression allowing for a dark current offset the downstream ion chamber begins with the Beer-Lambert law { exp [ µ i [t } = I dc i I 0 dc i, (8.37) where the subscript i indicates the ideal values, and include the effect of the dark current error by writing (dc m = dc i + δ dc ) when the measured dark currents are δ dc higher than their ideal value. Equation (8.37) becomes { exp [ } µ i [t = I dc m + δ dc I 0 dc m + δ dc I dc m + δ dc = (I 0 dc m )(1 + δ dc I 0 dc m ) [ [ I dcm δ dc δ dc + 1 I 0 dc m I 0 dc m I 0 dc m = I dc [ [ m δ dc + 1 I dc m I 0 dc m I 0 dc m I 0 dc m { exp [ } [ [ µ m [t δ { dc + 1 exp I 0 dc m so that: [ µ i = 1 ( { [t ln exp [ } µ m [t [ δ dc + I 0 dc m [ { 1 exp [( δdc + O [ µ [ µ I 0 dc m } [t, m ) 2 m [t } ). (8.38) Figure 8.15 illustrates the effect of an incorrect dark current value on the measured attenuations. The effect is most profound for the thickest foil, although small shifts are observable for thinner foils near the absorption edge. The systematic shifts are similar to the residual systematic discrepancies observed in Fig δ dc I 0 dc m The dark current errors of 10, 20, and 40 cps illustrated in Fig correspond to , , and , respectively, with I 0 set to cps. If the dark current is offset by 40 cps, then the modelled divergences match the detailed structure observed in Fig In particular, the modelled divergences rise to 6% for the 100 µm foil at the absorption edge (red marker in Fig. 8.3), to 1 2%

220 192 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.15: Simulation of the effect of an incorrect dark current value on the mass attenuation coefficients [Eq. (8.38). Calculated deviations are in broad agreement with residual variations observed in Fig for the 250 µm foil at 15 kev (orange), and to 1 2% for the 150 µm foil at 25 kev (light blue). Fig. 8.3 shows that variations are not constant for all measurements made using a given foil at each energy. Over the course of the experiment the detector gain levels have been varied by a factor of approximately 1000 to maximise statistics for all measurements. Accordingly, for each group of measurements taken with common gain settings, that is, for each ion chamber epoch, we require an independent dark current correction. To determine the dark current correction within a given ion chamber epoch, there must be a clear signature of divergences in each ion chamber epoch. We have combined small groupings to allow the dark current corrections to be determined robustly. The signature of discrepancies over the range kev and kev shows no evidence of a discontinuity in the dark current correction, so we combine these regions and stabilise the determined dark current correction compared to an evaluation in the smaller sub-regions. The measurements recorded using the first and second downstream ion chambers are fitted using different dark current values. Model χ 2 r Input data 112 Determining the local integrated column densities 3.95 Effect of the x-ray bandwidth 3.63 Dark current correction 3.44 Table 8.3: Improvement to the χ 2 r resulting from the dark current correction.

221 8.7. THE EFFECT OF A DARK-CURRENT ERROR 193 Figure 8.16: The percentage discrepancies of measurements from the weighted mean, after the dark current correction. The four largest divergences in Fig. 8.3 are now centered about the zero line. The reported uncertainty is that of the weighted mean added in quadrature to one quarter of the present correction. Symbols: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm; light blue 150 µm; maroon 200 µm; orange 250 µm. The symbol indicates the value determined from the full-foil mapping procedure. The six different symbols represent measurements using three different aperture combinations using the two downstream ion chambers: closed symbols (,, ) and line symbols (, +, ) indicate measurements recorded using the first and second downstream ion chambers with the small, medium, and large aperture combinations, respectively. Point symbols ( ) occurring between and kev represent measurements affected by numerical saturation. Figure 8.16 presents the percentage discrepancies of the measured mass attenuation coefficients after fitting for the dark current values. The agreement between the measurements across a wide range of energies is qualitatively improved. The four large systematic divergences appearing in Fig. 8.3 are now centered about the zero line, indicating that the correction is of the correct form and magnitude. The residual discrepancies suggest a significant point-to-point variation in the required dark current value. The magnitude of the residual discrepancies is about half that of the discrepancies prior to correction, so we estimate the maximum error for any particular measurement to be one-quarter of the correction. Figure 8.17 presents the change to the weighted mean arising from the dark current correction. The error bars shown in this figure indicate the uncertainty in

222 194 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.17: The percentage change of the determined mass attenuation coefficients resulting from the correction to the dark current values. The error bars indicate the uncertainty in the weighted mean. The change is everywhere less than about 0.1%, and is less than the uncertainty in the weighted mean everywhere except in the energy range kev, where the change ranges from σ. the weighted mean value at each energy, before addition of the further uncertainty equal to one quarter of the size of the dark current correction. The change to the weighted mean is less 1-σ for all measurements except those between 21 and 30 kev. Any further systematic effects are likely to have even less impact on the weighted mean values and are therefore of little consequence, justifying the claimed precision and accuracy of the measurements at % (Fig. 8.17). 8.8 Incorporation of rapid measurements In this experiment the measurement density (the number of measurements per kev) is commensurate with structure in the mass attenuation coefficient. Measurement intervals are ev away from the absorption edge and ev near to the absorption edge and in the region of the EXAFS. Measurements made at 0.5 ev intervals were recorded using a rapid measurement mode, for which slowly-varying parameters were not measured and for which different beamline optimisation procedures were adopted. In the rapid measurement mode the dark currents and the blank

223 8.8. INCORPORATION OF RAPID MEASUREMENTS 195 Figure 8.18: Incorporation of rapid or single-foil measurements. The dark error-bars indicate the weighted mean of regular measurements, made using a variety of foil thicknesses and aperture combinations. The dot markers indicate measurements made using the 50-µm foil. The discontinuity between regular and the rapid values is due to the different sampling of the measurement parameter space for these values. We incorporate the rapid values by shifting them by an amount equal to their difference from the weighted mean at nearby regular measurement energies. The grey error-bars indicate the incorporated rapid measurements. These are now smoothly consistent with the regular values. The improved consistency of the resulting data set is crucial for any subsequent XAFS analysis. count-rates were not recorded. Interpolated values of dark current and normalised incident intensity I d,0 dc d I u,0 dc u are therefore used. The photon energy was selected by rotation of the double reflection monochromator, and the undulator was not retuned at each energy. The detuning angle was not adjusted during the rapid measurements. The strongest influences on the measured values of the normalised incident intensity are the energy dependences of the x-ray production, detection, and air-path absorption. Therefore, values of I d,0 dc d I u,0 dc u are interpolated on the energy axis. As the energy scale of the interpolation was typically 5 ev, we interpolate these values using a linear function. Over this energy scale the uncertainty due to the linear assumption is negligible and dominated by individual measurement uncertainty. The rapid measurements present an issue for the smooth determination of the mass attenuation coefficients. This issue arises due to the different sampling of the measurement parameter space at the rapid and regular measurement energies: the

224 196 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM rapid measurements record the attenuation of one foil only, the regular measurements record the attenuation of several foil thicknesses. The weighted mean of the rapid measurements is therefore equal to the value determined for the measured foil: for the regular measurements, the weighted mean is determined from several values. The rapid results are not smoothly continuous with nearby weighted mean values. Figure 8.18 presents a detail of the mass attenuation coefficients measured in the region of the XAFS, where rapid measurements have been made. The four bold error bars indicate values of the mass attenuation coefficient determined using the standard range of foil thicknesses and aperture combinations. Measurements made using the 50-µm foil are indicated with a dot marker, including three measurements (employing the various aperture combinations) at the standard measurement energy, and a string of single measurements (employing only one aperture combination) at the rapid measurement energies. The 50-µm-foil measurements have contributed to the evaluation of the weighted mean at the standard measurement energies. The 50-µm-foil measurements lie systematically above the weighted mean at each of the standard measurement energies shown in Fig Ergo, the rapid measurements employing the 50-µm foil also lie systematically above the trend of the weighted mean. Therefore we shift the 50-µmfoil rapid measurement values by an amount equal to the average of the differences observed at nearby each end of a rapid measurement series, offset = 1 2 {( [ µ single foil, low energy [ ) µ weighted mean, + low energy ( [ µ single foil, high energy [ )} µ weighted mean,. high energy (8.39) Equation (8.39) determines the offset from the average difference at the ends of each series. In fact we have used the weighted average difference (i.e., including the uncertainties of the single-foil and weighted-mean measurements) to determine a robust value for the offset. The quick measurements are incorporated into the body of measurements by shifting them parallel to the [ µ axis. The incorporated quick measurements are represented in Fig by the light grey error-bars. consistent with the trend of the weighted mean values. These values are smoothly

225 8.9. CONSISTENCY OF THE MEASURED VALUES 197 The consistency gained by this procedure is particularly important for any subsequent XAFS analysis, which requires evaluation of a Fourier transform of the measured values. Without this interpolation, the small (±1σ) discontinuities occurring at regular 5 ev intervals throughout the measurements would result in a spurious signal in the transformed XAFS spectrum. 8.9 Consistency of the measured values The weighted mean for a given energy typically involves between 18 and 30 independent measurements using a variety of foil thicknesses and aperture combinations. The uncertainty in the mass attenuation coefficient is evaluated from the standard deviation of the values contributing to the weighted mean, σ s.d. σ s.d. = σ[ µ Ei (8.40) = ( F j ( [ µ Ei [ µ ) 2 Fj E i σ[ 2 µ E i F j / 1 σ[ 2 F µ j E i F j ) 1/2. (8.41) Figure 8.19 presents the deviations from the weighted mean for the values falling within a small range of the weighted mean. Also indicated on this figure are two lines representing the upper and lower bounds of the 1-σ s.d. uncertainty of the determined mass attenuation coefficients. These lines show that the uncertainty is very low, generally falling between 0.03 and 0.1%. This figure shows the high degree of consistency of the measurements made at each energy, but does not directly assess the consistency of measurements across a range of energies. Measurements made at neighbouring energies are only coupled by the dark current interpolation, the bandwidth correction, and the energy determination. The cross-talk between neighbouring values resulting from these is very small. Given the smooth energy dependence of the mass attenuation coefficient, the continuity of the measured values can be used to assess the quality of the determined values relative to their uncertainties. The continuity of the measured values provides a particularly robust test of the measurement due to the independence of values determined at

226 198 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.19: Percentage difference of the measured values from the weighted mean value within a small range of the zero line. Symbols: yellow 25 µm; dark blue 25 µm; green 50 µm; red 100 µm; light blue 150 µm; maroon 200 µm; orange 250 µm. The symbol indicates the value determined from the full-foil mapping procedure. The six different symbols represent measurements using three different aperture combinations using the two downstream ion chambers: closed symbols (,, ) and line symbols (, +, ) indicate measurements recorded using the first and second downstream ion chambers with the small, medium, and large aperture combinations, respectively. Point symbols ( ) occurring between and kev represent measurements affected by numerical saturation. neighbouring energies. The continuity of the trend of the measured mass attenuation coefficients can be evaluated using Figs 8.21 and The determined values are plotted in comparison with the FFAST tabulation to display clearly the trends of values and their uncertainties across various energy ranges. The measured values have been examined for discontinuities at those energies where sample changes were made, where ion chamber gain settings were adjusted, and where we changed from the fifth-order undulator harmonic to the third-order harmonic. In all cases the values were continuous within uncertainty and the gradients were smooth, indicating the accuracy of the determined uncertainties.

227 8.9. CONSISTENCY OF THE MEASURED VALUES 199 Figure 8.20: Figures (a), (c), and (d) present the percentage difference from the FFAST tabulation [33, 34, 35: figure (b) presents the measurements made along the absorption edge. The trends of the measured values are within uncertainty, indicating the consistency of the measured values and their determined uncertainties. Figure (a) shows that the measurements made below the absorption edge are continuous within their uncertainties of %. Along the absorption edge, figure (b) shows that the measurements are smooth, consistent with the smooth gradient of Fig Figure (c) shows good continuity except for a few measurements in the energy range from 23.4 to 25 kev. Here the operation of the undulator changed from fifth-order to third-order harmonic generation. This change may have de-stabilised the apparatus, which may be responsible for the scatter observed over a transitional energy region. The trend in the values above and below the transition region is continuous to within the determined uncertainties. Figure (d) shows an extremely high degree of continuity in the high-energy region, occurring where the measurement uncertainty drops to below 0.03%.

228 200 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Figure 8.21: Percentage difference from the FFAST tabulation [33, 34, 35 of the mass attenuation coefficients measured in the region of the XAFS. Measured values represented by their error bars. The local consistency of the results is evinced by the smoothness of these data The photoelectric absorption coefficient [ µ & the imaginary component of the pe atomic form-factor f 2 Equation (2.36) described the mass attenuation coefficient as a sum of photoelectric absorption [ µ pe, Compton scattering [ µ C, and Rayleigh scattering [ µ according R to [ µ [ µ pe + [ µ C + [ µ. (8.42) R Use of Eq. (8.42) is appropriate when Rayleigh and Compton scattering are the only significant other contributions to the total attenuation. The breakdown of the independent particle and isolated atom approximations in the regions of the EXAFS and XANES does not affect the physical interpretation of Eq. (8.42). However, results of calculations reliant on these assumptions will not be expected to be accurate in these regions. The photoelectric absorption coefficient [ µ pe has been determined from the mass attenuation coefficient by subtracting the average of the Rayleigh plus Compton

229 8.10. [ µ pe & f contribution as tabulated in XCOM [96, 97 and FFAST [33, 34, 35. We estimate the uncertainty in the subtracted Rayleigh plus Compton contributions, RC, to be half of the difference between these tabulated values. Table 8.4 presents the values of the mass attenuation coefficients measured at 526 energies between 13.5 and 41.5 kev. A tabulation of the measured values is also available electronically (see [25). The first column of this table presents the measured photon energy (in kev), with the uncertainty in the last significant figures presented in parentheses. The second column gives the mass attenuation coefficient [ µ (in cm 2 /g) with its uncertainty. In the third column we present the percentage uncertainty in the mass attenuation coefficient. The values in the second and third columns have been determined from the weighted mean of the measurements made with a variety of apertures and foil thicknesses, and using the values determined from the counts recorded in both of the downstream ion chambers. The fourth column gives the imaginary component of the atomic form-factor f 2, evaluated from f 2 = EuA[ µ pe, (8.43) 2hcr e where E is the photon energy in ev, u is the atomic mass unit, A the relative atomic mass of molybdenum, h is the Planck constant, c the speed of light, and r e the classical electron radius. The uncertainty in f 2 is evaluated from σ f2 = EuA ( ) σ 2[ µ RC, (8.44) 2hcr e which includes an uncertainty contribution of half of the difference between the tabulated values of the Rayleigh plus Compton contribution. If these measured data are used for the investigation of alternate atomic environments then, in the region of the XAFS and XANES, values of f 2 may be subject to a further uncertainty of the order of the XAFS amplitude since the oscillations are dominated by solid-state rather than atomic processes.

230 202 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM Table 8.4: Mass attenuation coefficients [ µ and the imaginary component of the atomic form-factor f 2 of molybdenum as a function of x-ray energy, with one standard deviation uncertainties in the least significant digits indicated in parentheses. We present also the percentage uncertainty in the mass attenuation coefficients, σ [ µ / [ µ. Uncertainty in f2 includes the measurement uncertainty and the difference between major tabulations of the total Rayleigh plus Compton scattering cross-sections. Values of f 2 in the energy range of kev are likely to be affected by solid-state and atomic effects. A further uncertainty, of the same order as the XAFS amplitude, may apply to these values when alternate atomic environments are investigated. Energy (kev) [ µ (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (31) (25) 0.067% (17) (31) (35) 0.098% (18) (30) (24) 0.070% (14) (30) (23) 0.072% (11) (29) (18) 0.060% (79) (29) (30) 0.11% (11) (28) (16) 0.060% (70) (28) (18) 0.071% (11) (27) (26) 0.11% (17) (27) (15) 0.065% (20) (27) (24) 0.11% (25) (27) (21) 0.10% (28) (27) (21) 0.11% (30) (28) (17) 0.091% (31) (28) (11) 0.064% (31) (29) (19) 0.11% (33) (30) (18) 0.11% (33) (31) (16) 0.11% (31) (32) (95) 0.065% (27) (34) (95) 0.068% (22) (35) (92) 0.069% (18) (35) (12) 0.086% (17) (35) (69) 0.052% (17) (36) (95) 0.071% (16) (36) (14) 0.10% (16) (36) (79) 0.060% (15) (36) (61) 0.046% (15) (36) (10) 0.078% (15) (36) (11) 0.085% (14) (36) (97) 0.074% (14) (36) (43) 0.033% (12) (36) (31) 0.024% (12) continued...

231 8.10. [ µ pe & f from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (37) (41) 0.032% (11) (37) (41) 0.032% (11) (37) (12) 0.095% (12) (37) (35) 0.027% (99) (37) (46) 0.035% (95) (37) (42) 0.033% (90) (37) (47) 0.036% (86) (37) (94) 0.072% (88) (38) (29) 0.020% (63) (38) (70) 0.049% (69) (38) (45) 0.031% (65) (38) (15) 0.11% (93) (38) (18) 0.12% (10) (38) (46) 0.31% (22) (38) (83) 0.056% (72) (38) (37) 0.025% (63) (38) (21) 0.14% (11) (38) (52) 0.35% (25) (38) (28) 0.19% (14) (38) (21) 0.14% (11) (38) (16) 0.10% (94) (38) (18) 0.12% (10) (38) (14) 0.088% (86) (38) (18) 0.11% (10) (38) (22) 0.14% (12) (38) (26) 0.17% (13) (38) (14) 0.087% (87) (38) (11) 0.070% (78) (38) (92) 0.057% (72) (38) (71) 0.043% (67) (38) (19) 0.12% (11) (38) (20) 0.12% (11) (38) (12) 0.069% (79) (38) (27) 0.16% (14) (38) (12) 0.069% (81) (38) (42) 0.23% (20) (38) (29) 0.14% (14) (38) (30) 0.14% (15) (38) (39) 0.17% (18) (38) (32) 0.13% (16) (38) (40) 0.15% (19) (38) (54) 0.19% (25) continued...

232 204 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM... from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (38) (59) 0.20% (28) (38) (93) 0.28% (43) (38) (68) 0.19% (31) (38) (59) 0.16% (27) (38) (49) 0.13% (23) (38) (43) 0.11% (20) (38) (67) 0.16% (31) (38) (40) 0.092% (19) (38) (56) 0.12% (26) (38) (60) 0.13% (28) (38) (41) 0.085% (20) (38) (32) 0.064% (16) (38) (23) 0.045% (12) (38) (44) 0.081% (21) (38) (29) 0.052% (14) (38) (26) 0.046% (13) (38) (55) 0.094% (26) (38) (35) 0.058% (17) (38) (51) 0.082% (24) (38) (27) 0.041% (13) (38) (90) 0.13% (41) (38) (88) 0.13% (40) (38) (58) 0.081% (27) (38) (47) 0.062% (22) (38) (45) 0.057% (21) (38) 81.16(15) 0.18% (67) (38) 83.88(10) 0.12% (47) (38) (70) 0.081% (32) (38) (61) 0.070% (28) (38) (52) 0.059% (24) (38) (44) 0.049% (21) (38) (40) 0.044% (19) (38) (34) 0.038% (17) (38) (28) 0.031% (14) (38) (25) 0.028% (13) (38) (42) 0.047% (20) (38) (39) 0.044% (19) (38) (40) 0.045% (19) (38) (33) 0.037% (16) (38) (24) 0.028% (12) (38) (17) 0.019% (90) (38) (51) 0.060% (24) continued...

233 8.10. [ µ pe & f from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (38) (46) 0.054% (22) (38) (36) 0.042% (17) (38) (33) 0.039% (16) (38) (32) 0.037% (15) (38) (30) 0.035% (15) (38) (28) 0.032% (14) (38) (27) 0.031% (13) (38) (25) 0.029% (12) (38) (23) 0.026% (11) (38) (20) 0.024% (10) (38) (45) 0.051% (21) (38) (64) 0.073% (29) (38) (60) 0.070% (28) (38) (56) 0.065% (26) (38) (53) 0.061% (25) (38) (49) 0.056% (23) (38) (44) 0.051% (21) (38) (41) 0.046% (19) (38) (35) 0.040% (16) (38) (27) 0.031% (13) (38) (29) 0.033% (14) (38) (37) 0.042% (18) (38) (36) 0.041% (17) (38) (35) 0.039% (17) (38) (34) 0.037% (16) (38) (32) 0.035% (15) (38) (29) 0.032% (14) (38) (29) 0.032% (14) (38) (28) 0.030% (13) (38) (24) 0.026% (12) (38) (33) 0.035% (16) (38) (41) 0.044% (19) (38) (40) 0.043% (19) (38) (38) 0.040% (18) (38) (37) 0.039% (17) (38) (33) 0.035% (16) (38) (32) 0.033% (15) (38) (30) 0.031% (14) (38) (27) 0.029% (13) (38) (31) 0.032% (15) (38) (28) 0.029% (13) (38) (93) 0.099% (43) continued...

234 206 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM... from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (38) (90) 0.096% (42) (38) (87) 0.093% (40) (38) (82) 0.089% (38) (38) (82) 0.089% (38) (38) (76) 0.084% (35) (38) (74) 0.082% (34) (38) (69) 0.077% (32) (38) (69) 0.077% (32) (38) (42) 0.048% (20) (38) (38) 0.045% (18) (38) (60) 0.071% (28) (38) (59) 0.069% (27) (38) (59) 0.069% (27) (38) (59) 0.070% (27) (38) (60) 0.070% (28) (38) (60) 0.071% (28) (38) (61) 0.071% (28) (38) (61) 0.071% (28) (38) (63) 0.073% (29) (38) (52) 0.060% (24) (38) (30) 0.034% (14) (38) (70) 0.080% (32) (38) (77) 0.088% (35) (38) (98) 0.11% (45) (38) 87.51(12) 0.13% (53) (38) 87.42(10) 0.12% (47) (38) (68) 0.078% (31) (38) (66) 0.076% (30) (38) (68) 0.078% (31) (38) (65) 0.075% (30) (38) (29) 0.033% (14) (38) (40) 0.047% (19) (38) (72) 0.085% (33) (38) (70) 0.081% (32) (38) (69) 0.081% (32) (38) (67) 0.078% (31) (38) (66) 0.077% (31) (38) (64) 0.075% (29) (38) (62) 0.072% (29) (38) (61) 0.071% (28) (38) (60) 0.069% (28) (38) (46) 0.053% (21) continued...

235 8.10. [ µ pe & f from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (39) (69) 0.076% (32) (39) (81) 0.090% (37) (39) (81) 0.090% (37) (39) (81) 0.089% (37) (39) (79) 0.087% (36) (39) (76) 0.083% (35) (39) (76) 0.083% (35) (39) (73) 0.081% (34) (39) (75) 0.082% (34) (39) (74) 0.082% (34) (39) (47) 0.052% (22) (39) (45) 0.053% (21) (39) (65) 0.076% (30) (39) (65) 0.075% (30) (39) (64) 0.075% (30) (39) (63) 0.074% (29) (39) (63) 0.074% (29) (39) (65) 0.076% (30) (39) (64) 0.075% (29) (39) (62) 0.073% (29) (39) (62) 0.073% (29) (39) (68) 0.080% (31) (39) (62) 0.072% (29) (39) (67) 0.078% (31) (39) (66) 0.077% (31) (39) (67) 0.079% (31) (39) (67) 0.078% (31) (39) (67) 0.078% (31) (39) (67) 0.078% (31) (39) (67) 0.079% (31) (39) (66) 0.077% (30) (39) (66) 0.077% (30) (39) (65) 0.075% (30) (39) (52) 0.060% (24) (39) (68) 0.078% (31) (39) (66) 0.077% (30) (39) (66) 0.077% (31) (39) (65) 0.075% (30) (39) (65) 0.075% (30) (39) (64) 0.074% (30) (39) (64) 0.074% (30) (39) (65) 0.074% (30) continued...

236 208 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM... from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (39) (64) 0.074% (29) (39) (67) 0.077% (31) (39) (83) 0.094% (38) (39) (77) 0.087% (35) (39) (75) 0.085% (35) (39) (76) 0.085% (35) (39) (76) 0.086% (35) (39) (75) 0.084% (34) (39) (74) 0.083% (34) (39) (74) 0.083% (34) (39) (73) 0.082% (34) (39) (73) 0.082% (34) (39) (67) 0.076% (31) (39) (49) 0.056% (23) (39) (63) 0.072% (29) (39) (62) 0.072% (29) (39) (63) 0.072% (29) (39) (64) 0.073% (29) (39) (66) 0.076% (30) (39) (67) 0.078% (31) (39) (68) 0.079% (31) (39) (70) 0.081% (32) (39) (69) 0.080% (32) (39) (72) 0.085% (33) (39) (51) 0.061% (23) (39) (63) 0.075% (29) (39) (61) 0.073% (28) (39) (61) 0.073% (28) (39) (60) 0.072% (28) (39) (62) 0.075% (29) (39) (60) 0.073% (28) (39) (61) 0.074% (28) (39) (60) 0.072% (28) (39) (61) 0.074% (28) (39) (53) 0.063% (24) (39) (40) 0.049% (19) (39) (67) 0.081% (31) (39) (67) 0.081% (31) (39) (66) 0.079% (30) (39) (64) 0.077% (29) (39) (64) 0.078% (30) (39) (62) 0.074% (28) continued...

237 8.10. [ µ pe & f from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (39) (62) 0.075% (29) (39) (59) 0.071% (27) (39) (60) 0.073% (28) (39) (40) 0.048% (18) (39) (64) 0.077% (29) (39) (45) 0.053% (21) (39) (40) 0.047% (18) (39) (65) 0.076% (30) (39) (62) 0.073% (29) (39) (62) 0.073% (29) (39) (61) 0.071% (28) (39) (59) 0.070% (27) (39) (59) 0.069% (27) (39) (58) 0.068% (27) (39) (57) 0.067% (26) (39) (56) 0.067% (26) (39) (40) 0.047% (18) (39) (18) 0.021% (84) (39) 86.77(19) 0.22% (87) (39) 86.92(19) 0.22% (87) (39) 87.14(18) 0.21% (85) (39) 87.43(19) 0.22% (87) (39) 87.69(19) 0.22% (89) (39) 87.80(20) 0.23% (92) (39) 87.98(19) 0.22% (89) (39) 88.16(20) 0.23% (92) (39) 88.38(20) 0.23% (93) (39) (22) 0.025% (10) (39) (14) 0.016% (65) (39) 88.93(20) 0.23% (92) (39) 88.83(20) 0.23% (93) (39) 88.71(20) 0.22% (91) (39) 88.52(20) 0.23% (92) (39) 88.33(20) 0.22% (90) (39) 88.20(19) 0.21% (86) (39) 87.95(19) 0.21% (86) (39) 87.81(19) 0.21% (86) (39) 87.57(18) 0.21% (85) (39) (32) 0.037% (15) (39) (25) 0.029% (11) (39) 83.76(15) 0.18% (68) (39) 83.40(14) 0.17% (66) continued...

238 210 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM... from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (39) 83.05(14) 0.17% (66) (39) 82.77(14) 0.17% (63) (39) 82.46(14) 0.16% (62) (39) 82.22(13) 0.16% (60) (39) 81.94(13) 0.16% (60) (39) 81.67(13) 0.15% (58) (39) 81.35(12) 0.15% (57) (39) (14) 0.017% (64) (39) (33) 0.041% (15) (39) 79.45(11) 0.14% (52) (39) 79.52(10) 0.13% (48) (39) 79.82(10) 0.13% (46) (39) 80.34(10) 0.13% (48) (39) (70) 0.086% (32) (40) (23) 0.028% (11) (40) 81.59(11) 0.14% (51) (40) 81.90(11) 0.14% (51) (40) 82.30(12) 0.15% (55) (40) 82.73(12) 0.14% (55) (40) (49) 0.059% (23) (40) (69) 0.081% (32) (40) 85.51(14) 0.16% (64) (40) 85.59(14) 0.16% (64) (40) 85.46(14) 0.16% (63) (40) 85.45(14) 0.16% (63) (40) (31) 0.036% (15) (40) (36) 0.043% (17) (40) 84.67(13) 0.16% (61) (40) 84.40(12) 0.14% (54) (40) 84.06(12) 0.14% (55) (40) 83.61(12) 0.14% (55) (40) (41) 0.050% (19) (40) (30) 0.037% (14) (40) (52) 0.064% (24) (40) (42) 0.052% (19) (40) (40) 0.050% (19) (40) (38) 0.048% (18) (40) (36) 0.046% (17) (40) (30) 0.037% (14) (40) (39) 0.049% (19) (40) (39) 0.048% (18) (40) (41) 0.050% (19) continued...

239 8.10. [ µ pe & f from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (40) (41) 0.050% (19) (40) (35) 0.042% (16) (40) (34) 0.041% (16) (40) (44) 0.053% (21) (40) (51) 0.061% (24) (40) (61) 0.073% (28) (40) (65) 0.078% (30) (40) (37) 0.045% (18) (40) (49) 0.060% (23) (40) (41) 0.050% (20) (40) (40) 0.049% (19) (40) (40) 0.048% (19) (40) (38) 0.047% (18) (40) (37) 0.045% (18) (41) (32) 0.039% (16) (41) (44) 0.054% (21) (41) (38) 0.047% (18) (41) (34) 0.043% (17) (41) (45) 0.056% (21) (41) (27) 0.034% (14) (41) (53) 0.066% (25) (41) (68) 0.084% (32) (41) (44) 0.055% (21) (41) (40) 0.050% (19) (41) (40) 0.050% (19) (41) (32) 0.040% (16) (41) (31) 0.038% (15) (41) (40) 0.050% (19) (41) (66) 0.082% (31) (41) (43) 0.053% (21) (41) (43) 0.053% (21) (41) (33) 0.042% (17) (41) (22) 0.027% (12) (41) (44) 0.054% (21) (41) (42) 0.052% (21) (41) (42) 0.052% (21) (41) (43) 0.053% (21) (41) (43) 0.054% (21) (41) (38) 0.048% (19) (41) (59) 0.074% (29) (41) (43) 0.054% (21) (41) (46) 0.058% (23) continued...

240 212 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM... from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (42) (73) 0.092% (35) (42) (44) 0.056% (22) (42) (29) 0.037% (16) (42) (41) 0.052% (21) (42) (42) 0.054% (22) (42) (43) 0.054% (22) (42) (43) 0.055% (22) (42) (30) 0.038% (16) (42) (29) 0.036% (16) (42) (45) 0.057% (23) (42) (42) 0.053% (22) (42) (44) 0.055% (23) (42) (47) 0.058% (24) (42) (33) 0.041% (18) (42) (23) 0.030% (15) (42) 77.81(11) 0.14% (53) (42) 77.75(11) 0.14% (53) (42) 77.78(10) 0.13% (49) (42) (98) 0.13% (47) (42) (34) 0.044% (19) (43) (35) 0.045% (20) (43) 77.59(11) 0.14% (53) (43) 77.67(11) 0.14% (53) (43) 77.70(11) 0.14% (52) (43) 77.72(11) 0.14% (53) (43) (27) 0.035% (17) (43) (32) 0.042% (19) (43) 77.62(11) 0.14% (53) (43) 77.62(11) 0.14% (52) (43) 77.60(10) 0.14% (51) (43) 77.59(11) 0.14% (51) (43) (25) 0.033% (17) (43) (31) 0.041% (19) (43) 77.01(14) 0.18% (65) (43) 76.80(13) 0.17% (62) (43) 76.66(13) 0.17% (62) (43) 76.55(13) 0.17% (63) (43) (24) 0.032% (17) (44) (37) 0.048% (22) (44) 75.73(12) 0.15% (57) (44) 75.70(12) 0.16% (60) (44) 75.76(12) 0.16% (59) continued...

241 8.10. [ µ pe & f from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (44) 75.79(13) 0.17% (63) (44) (35) 0.046% (22) (44) (15) 0.020% (16) (44) 76.15(14) 0.18% (66) (44) 76.01(13) 0.17% (63) (44) 75.65(12) 0.16% (59) (44) (31) 0.041% (21) (47) (24) 0.033% (23) (47) (49) 0.069% (31) (47) (41) 0.057% (29) (48) (38) 0.054% (28) (48) (37) 0.052% (28) (48) (34) 0.048% (27) (49) (42) 0.060% (31) (49) (43) 0.062% (32) (49) (41) 0.060% (31) (49) (39) 0.056% (30) (49) (35) 0.050% (30) (50) (40) 0.058% (31) (51) (22) 0.032% (28) (51) (39) 0.057% (32) (53) (32) 0.048% (33) (54) (30) 0.046% (33) (56) (26) 0.042% (34) (57) (19) 0.030% (33) (58) (28) 0.046% (35) (60) (25) 0.041% (35) (62) (19) 0.033% (35) (64) (27) 0.047% (36) (66) (32) 0.057% (37) (68) (29) 0.053% (37) (70) (45) 0.084% (41) (72) (39) 0.074% (38) (75) (20) 0.038% (33) (76) (43) 0.085% (39) (78) (25) 0.050% (33) (81) (24) 0.049% (32) (10) (24) 0.050% (32) (83) (21) 0.043% (31) (99) (46) 0.096% (38) (99) (29) 0.062% (32) (98) (21) 0.045% (29) continued...

242 214 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM... from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ f 2 (e/atom) (97) (54) 0.12% (40) (94) (38) 0.088% (32) (90) (73) 0.18% (47) (87) (59) 0.15% (39) (84) (30) 0.080% (22) (81) (32) 0.089% (21) (79) (32) 0.096% (21) (77) (25) 0.078% (16) (76) (13) 0.042% (86) (76) 28.06(11) 0.40% (77) (78) (16) 0.061% (13) (80) (13) 0.052% (12) (84) (14) 0.058% (13) (94) (14) 0.064% (13) (10) (84) 0.039% (99) (11) (93) 0.044% (10) (11) (63) 0.032% (92) (12) (17) 0.090% (15) (13) (68) 0.037% (81) (14) (68) 0.038% (77) (15) (54) 0.031% (66) (16) (44) 0.026% (54) (17) (39) 0.025% (46) (18) (42) 0.027% (43) (20) (43) 0.029% (41) (21) (67) 0.047% (61) (22) (57) 0.042% (52) (23) (40) 0.030% (36) (25) (28) 0.022% (27) (26) (32) 0.026% (30) (28) (22) 0.018% (21) (29) (29) 0.025% (28)

243 8.11. THE EXPERIMENTAL ERROR BUDGET The experimental error budget We have sought the optimal measurement configuration but have deliberately extended these measurements beyond the optimal regimes to determine the effect of systematic errors on the measurement. An exhaustive search for systematic errors has been undertaken, and has resulted in the correction of three systematic effects. Table 8.5 presents a summary of the key contributions to the uncertainties in the reported values. A number of quantified uncertainties and estimates of the maximum effect of residual contributions are detailed. Statistical uncertainties have been kept to a low level by employing a highbrilliance third-generation synchrotron radiation source, by using high-quality serialflow argon-gas ion chambers, by repeating each measurement a number of times to directly determine the reproducibility of the measurement, and by critically examining the recorded count rates for inconsistencies which may in general have any one of a number of causes. Each detail has contributed to the extreme accuracy and precision of the determined values. Table 8.5: Contributions to uncertainties in Table 8.4, with source specified. Further established limits for the systematic uncertainty are provided. Estimated magnitude Contributions & comments [ Away from the absorption edge µ 0.028% accuracy, limited by the full-foil mapping technique (chapter 6) % precision, limited by counting statistics and foil replacement errors < 0.03% dark current error: one quarter of correction (section 8.7) [ Near the absorption edge ( kev) µ % x-ray bandwidth (section 8.3) } total % sample roughness (section 8.5) accuracy < 0.01% harmonic components (section 8.6) near edge % secondary photons (section 8.4) % E % accuracy of monochromator dispersion function interpolation (chapter 7) f % inconsistency of subtracted scattering components (section 8.10)

244 216 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM The accuracy of the determined values rests heavily on the full-foil mapping technique. We have determined the attenuation at a single energy, about 41.5 kev, to 0.028%. The accuracy of this result was limited by our ability to determine the area of the absorbing foil Summary and Conclusions In chapters 4 through 8 we have detailed our determination of the mass attenuation coefficients of molybdenum to accuracies of between 0.02% and 0.15%. Using the principles of the XERT, measurements have been made over an extended range of the measurement parameter space. We have carefully filtered the measured data-set to reject inconsistencies from the body of the data, and have determined robust uncertainties. The x-ray photon energies have been determined to high accuracy. Chapter 6 presented a full-foil mapping technique and its application to determine an extremely accurate value of the mass attenuation coefficient at a single energy. The technique overcomes the largest and most-persistent source of uncertainty affecting recent measurements. Experimental values obtained under a variety of conditions have been examined for deviations and discrepancies to investigate the effects of systematic errors on the measurement. Systematic errors affecting the measured values have been investigated using a maximum-likelihood technique. In particular, we have detected a discrepancy resulting from the difference between the average and local foil integrated column densities, and have used the technique to determine the local integrated column density for each of the foils. We have detected a systematic effect due to the finite bandwidth of the x-ray beam which, if undetected, would have resulted in a systematic error of up to 1% for measurements made along the absorption edge. We have applied a correction for the bandwidth effect and have determined the bandwidth of the synchrotron x-ray beam to be 1.57 ± 0.03 ev at about 20 kev. The measured values have been investigated for systematic errors resulting from sample scattering and fluorescence and from the influence of harmonic components which can be present in an x-ray beam. If present at all, these effects were determined

245 8.12. SUMMARY AND CONCLUSIONS 217 Figure 8.22: Percentage difference from the FFAST tabulated values [33, 34, 35 of: previous experimental work [106, 107; the XCOM tabulated values [96, 97, and; this work [25. to be well below the level of the measurement uncertainties. A small correction (generally less than about 1-σ s.e. ) was required to correct a dark current error which had a small effect on a portion of the measurements. Figure 8.22 presents the percentage difference from the FFAST tabulated values [33, 34, 35 of the XCOM tabulated values [96, 97, previous measured values (taken from [106, 107), and our measured values. The large difference between our measured values and the FFAST tabulation at the absorption edge is due to the XAFS structure which is not modelled by either tabulation. Our measured values are smooth and continuous, indicating the validity of the evaluated uncertainties. The determined values are accurate to between 0.02 and 0.15%, which is over one order of magnitude more accurate than previous measurement. Our data-set exhibits a significantly greater degree of self-consistency than the other data-sets shown here. The results of atomic form-factor calculations can be assessed by comparing the calculated photoelectric absorption coefficients with our measured values. Fig-

246 218 Chapter 8. DETERMINING [ [ µ, µ, & f pe 2 OF MOLYBDENUM ure 8.23 presents the percentage difference between a variety of commonly-used tabulations of [ µ and our values. Our values form the zero (reference) line, with pe the measurement uncertainties presented as error-bars about this zero line. uncertainty in the subtracted Rayleigh plus Compton cross-sections is presented as a shaded region around the zero line. Except in the region immediately below the absorption edge the uncertainty in the subtracted Rayleigh plus Compton crosssections is generally less than our experimental error-bar. The Figure 8.23: Percentage discrepancy between various tabulated values of [ µ pe and this work. The photoelectric component has been determined by subtracting the average of the calculated Rayleigh plus Compton scattering cross-sections of FFAST and XCOM from our measured values. The results of this work appear along the zero line, with error bars reflecting the experimental uncertainties. The narrow grey region around the zero line represents half of the difference between the Rayleigh plus Compton scattering cross-sections tabulated in XCOM and FFAST, and reflects the likely error in the absorption coefficient evaluated using these different models. Tabulated values are taken from FFAST [33, 34, 35, XCOM [96, 97, CXRO [94, 95, and Brennan and Cowan [101, 105.

247 8.12. SUMMARY AND CONCLUSIONS 219 The XCOM calculation exhibits a large difference from the measured values over an extended range of energies above the absorption edge. There is some evidence of an oscillatory behavior in the XCOM values, possibly extending beyond the measured energy range. Oscillatory behavior in the calculated values has been observed elsewhere [33, 34 and may be the result of an incompletely converged calculation. The FFAST tabulation is in best agreement with the measurements. In the below-edge region differences between the various calculations and our results are remarkably similar in form, even though they differ by 4% 5% in the absolute level of the photoelectric absorption coefficient. The similarity of these differences may imply a common limitation of the calculations in this region. At the point immediately below the absorption edge the FFAST calculation is in best agreement with our photoelectric absorption coefficient. There is significant structure in the differences between our measured values and the tabulated values. However, it is impossible to determine the general validity of observations based on measurements of a single element. We therefore defer further comparison of our measurements with the tabulated values until chapter 11.

248

249 Part III Tin and Silver 221

250

251 Chapter 9 Tin Preliminary results for measurements of the mass attenuation coefficients of tin between 29 and 60 kev are presented. The energy range of the measurements covers the K-shell absorption edge of tin at around 29.2 kev, and extends over a wide range of energies above the edge. The measurements were made at BESSRC-CAT beamline 12-BM (now incorporated into XOR) of the APS facility in June, Experimental details The x-ray beam was produced by a bending magnet device at the 12-BM XOR beamline [170 of the Advanced Photon Source facility at the Argonne National Laboratory. The broad and continuous bending magnet spectrum expected from this source, calculated by use of the XOP simulation program [171, is shown in Fig At high energies the photon flux decreases rapidly with increasing energy. We expect a factor of four reduction in the x-ray flux over the energy range from 30 to 60 kev. To produce a monochromatised x-ray beam with energy in the range from 29 to 60 kev we have reflected the x-ray beam from a pair of silicon crystals located in the first optical enclosure of the 12-BM facility. The (444) planes of these crystals were oriented to select the desired photon energy. However, a crystal so oriented diffracts a harmonic series of energies, passing all allowed multiples of the first-order energy. The (4n 2) multiples of the first order energy are not reflected as they are forbidden for this crystal. Further monochromation was effected by reflecting the x-ray beam from the (333) planes of a channel-cut silicon crystal located in the experimental hutch, m downstream of the first optical enclosure. The use of reflection orders with no 223

252 224 Chapter 9. TIN Figure 9.1: The x-ray spectrum expected from the 12-BM beamline, calculated using the XOP [171 package. common factors ensures the rejection of all higher order energy components up to the 4 3 = 12 th -order multiple of the first-order energy. The lowest unwanted energy component passed by this combination of crystals is the third-order multiple of the desired beam energy. Due to the rapid drop-off of the bending magnet spectrum with energy, the intensity of this high energy component is likely to be extremely low. The effective fraction of higher-order components is determined by the sensitivity of the ion chambers to the harmonic and fundamental photon energies. On the left side of Fig. 9.2 we present calculated efficiencies for 180-mm long ion chambers employing a variety of gases. This figure shows that krypton, xenon, and argon are all suitable over the energy range of this experiment: the sub-1% efficiencies associated with the use of nitrogen would result in a significant statistical imprecision (cf. Fig. 3.4). Over this range of energies krypton has a suitable efficiency. Xenon may be a little too efficient at the (xenon) absorption edge at about 34.5 kev. We have employed argon-filled ion chambers. We note that the decreasing efficiency of argon leads to a significant decrease (by a factor of about six) in the effective photon intensity at 60 kev compared to that observed at 30 kev. The statistical level of the measurement may begin to decrease as the photon energy increases, particularly when the efficiency drops below 2%. On the right side of Fig. 9.2 we

253 9.1. EXPERIMENTAL DETAILS 225 Figure 9.2: Left: simulation of the efficiency of a 180-mm long ion chamber with a variety of gases over a range of photon energies. This plot shows that the efficiency of an argon gas ion chamber drops by a factor of 6 over the energy range of this experiment, and warns us that the statistical level of the measurement may begin to decrease as the photon energy increases above about 50 kev, when the efficiency drops below about 2%. Right: the relative efficiency of detection of the first and third-order multiples of the desired beam energy, showing that the effective intensity of the harmonic components is reduced by an order of magnitude by the use of an argon gas ion chamber. Both of these figures indicate that krypton is an ideal gas to use over these energies: however, krypton was not available for use in this experiment. present the relative efficiency for this ion chamber for the detection of the first and third-order multiples of the desired beam energy. The effective harmonic rejection using argon is by a factor of between 4 16, with the most significant gains observed in the low-energy region, where harmonic contamination may be appreciable. Figure 9.3 presents a schematic of the experimental configuration. After monochromation the x-rays passed through a pair of orthogonal adjustable slits which defined the beam cross-section to be approximately 4 mm in the horizontal direction by 1 mm in the vertical direction. These beam dimensions were chosen to optimise the intensity of the x-ray beam used for the measurement. The avail- silicon 4,4,4 monochromator with angle encoder, located in first optical enclosure beamdefining slits daisy wheel Bicron NaI detector from bending magnet silicon 3,3,3 monochromator, located in hutch 'upstream' argon-gas ion-chamber sample stage with two translational and two rotational degrees of freedom daisy wheel 'downstream' argon-gas ion-chambers germanium single-crystal Figure 9.3: Schematic of the experimental layout.

254 226 Chapter 9. TIN able x-ray intensity was significantly lower than that reported in chapter 4 due to our use of a bending-magnet source with a four-reflection monochromating system. The increased beam dimensions prevented us from making measurements using the smallest of the daisy-wheel apertures. Other details are similar to those described in chapter Attenuation The attenuations of the tin foils were measured and analysed in a manner similar to those reported in chapter 5. Accordingly, we report here the major departures from that work. In this experiment the operation and timing of the beam shutter mechanism were not a cause of data loss. There was no clipping of the x-ray beam by the apertures. Accordingly we have had no need to filter the measured count rates for these effects. Figure 9.4 shows the attenuations [ µ [t of the tin foils, determined from the counts recorded by the first downstream ion chamber. The attenuation is observed to rise by a factor of six at the absorption edge, at about 29.2 kev. Following this edge, there is a steady decrease in the attenuation until a discontinuity is observed at about 40 kev. This discontinuity has resulted from the reflection of the third-order multiple of the fundamental beam energy (i.e., the 3 4 multiple of the desired beam energy), as will be demonstrated in section 9.2. At energies higher than 42 kev, the Figure 9.4: Attenuations of tin absorbers determined using the counts recorded by the first downstream ion chamber.

255 9.2. ATTENUATION 227 mass attenuation coefficient again decreases in a smooth fashion. The increasing uncertainties associated with the measurements made at higher energies are due to the rapid decrease of the photon flux and of the ion chamber efficiencies. Figure 9.5 shows the calculated uncertainties of the attenuations measured using the first downstream ion chamber. In the energy range from 29.1 to 40 kev we see that the uncertainties are generally below 0.2% for the measurements recorded using the optimum foil thickness. The uncertainties in the measured attenuations increase in an almost exponential fashion for all measurements above about 40 kev. This increase is due to the rapid decrease in the source intensity combined with the declining detector efficiencies with photon energy. This decline was of course anticipated; a key question was where the statistical limit would dominate. A faulty gas line between the first and second downstream ion chambers lead to a decoupling of the gas pressure in the second downstream ion chamber compared to the other ion chambers. This leak significantly degraded the counting correlations between these ion chambers. Figure 9.6 presents the attenuation determined using the count rates recorded by the second downstream ion chamber, showing measurement uncertainties over ten times greater than those obtained using the first downstream ion chamber. There is no comparative statistical value in the attenua- Figure 9.5: Uncertainties of the attenuations determined from counts recorded by the first downstream ion chamber. The uncertainty for the optimum foil thickness is of order 0.2% for measurements below 35 kev. Above this energy the declining x-ray flux and detector efficiencies leads to higher uncertainty levels.

256 228 Chapter 9. TIN Figure 9.6: Attenuations of tin absorbers determined using the counts recorded by the second downstream ion chamber. Due to the use of a faulty gas line the measurements are of low statistical precision compared to those recorded using the first downstream ion chamber. tions determined from these measurements, so we discard them and proceed using the count rates recorded by the first downstream ion chamber only. A discontinuity in the measured attenuations The discontinuity in the mass attenuation coefficients determined between 40 and 42 kev is due to the spurious reflection of 3 -energy x-rays from some planes within 4 the downstream (333) monochromator, as we show in this section. The energy was changed by setting the upstream crystal so that the x-rays reflected from the (444) planes were of the desired energy. When this is done, x-rays of all allowed harmonic energies are reflected into the beam. The unwanted harmonic energies were to be removed by reflecting this partially-monochromated beam from the (333) planes of a second downstream crystal. The downstream crystal was tuned to optimise the reflected x-ray intensity by scanning it through a small range of angles about the predicted Bragg angle for the (333) planes. An intensity peak was then identified from this scan, and the crystal was set at the angle corresponding to the peak intensity. Figure 9.7 presents the tuning curves recorded between 39.2 and 42.8 kev. The ordinate presents recorded count-rate, with all plots sharing the same scale. The

257 9.2. ATTENUATION 229 Figure 9.7: Tuning curves for the downstream crystal over the energy range from kev. The abscissa is common, showing the intensity (on a logarithmic scale) recorded using an ion chamber located downstream from the second monochromator. The ordinate shows the angular location of the downstream monochromator. The stability of the intensity peak located in the centre of the scanned region is due to our scanning over a constant but narrow range about the predicted Bragg angle. A spurious peak can be seen entering the scan from the high-angle side at 40.0 kev. The spurious peak tracks across the scan for the tuning curves measured at energies between 40.0 kev and 42.0 kev, where it disappears off the low-angle end of the scan range. The central, desired peak remains stationary throughout this process. abscissa is the angular location of the downstream monochromator. All tuning curves show a peak in the intensity occurring at the centre of the range of the scan. This peak is comprised of x-rays of the desired beam energy: its reproducible presence in the centre of the scan range is due to the accurate prediction of its location by use of the Bragg law. In the tuning curve taken at 40.0 kev we observe a spurious peak on the highangle side of plot. Although the spurious peak is not fully within the scan range, its intensity is greater than that of the central peak and thus (according to our tuning algorithm) this peak was interpreted to be the desired beam energy, and its intensity was optimised for the attenuation measurement. In the subsequent tuning curves the spurious peak is seen to move rapidly across the scan region, and relative to the position of the central peak. Although partly within the scan range at 42.4 kev, the intensity of the spurious peak is below the central peak intensity, so the central peak

258 230 Chapter 9. TIN was optimised. The appearance of the spurious reflection coincides exactly with the discontinuities in the measured attenuations shown in Fig The values of the mass attenuation coefficient determined at energies between 40.0 and 42.0 kev are similar to those measured at energies between 30.0 and 32.0 kev, suggesting that the energy of the x-rays comprising the spurious reflection may be 3 4 of the desired beam energy. With our use of fourth-order and third-order reflections, such an occurrence is not unlikely, requiring only an accidental reflection of the 3 -energy x-rays by a set of planes within the (333) monochromator. Recall 4 that these photons are not removed by the upstream monochromator. The 3 -energy hypothesis is confirmed by the results of an energy determination 4 undertaken at 40.8 kev. In this determination we could only locate the (444) reflection from the germanium analyser crystal after searching within 0.5 of the nominal Bragg locations. The absence of all other reflections from their predicted locations and the sole presence of the (444) reflection is entirely consistent with the 3 4 -energy hypothesis: the measured reflection of photons of energy E is actually the (333) reflection of photons of energy 3 4 E. Further evidence for this hypothesis is presented in Fig The abscissa shows Figure 9.8: Difference between the nominal energy determined from the upstream (444) monochromator (using the Bragg equation, assuming no angular offsets) and that of the downstream (333) monochromator.

259 9.3. FULL-FOIL MAPPING 231 the upstream monochromator energy, the value of the x-ray energy calculated from the angle of the upstream (444) monochromator using the Bragg equation. This value is approximate, as it is calculated irrespective of any angular offsets of the monochromator or any nonlinearities which may affect the rotation axis motion. The ordinate presents the difference between the similarly-calculated downstream (333) monochromator energy and the upstream monochromator energy. The downstream and upstream monochromator energies are smoothly related for the measurements made outside of the anomalous region. The small variations are due to small instabilities in the experimental components. The measurements made between 40 and 42 kev show a significant and consistent deviation away from the trend of all other energies. The measurements within this region establish a consistent trend of their own, which is related to the transit of the spurious peak across the angular range of the downstream monochromator scan. If the 3 -energy photons are three-quarters of the energy of the wanted photons, 4 then these measurements can be used to determine extra values of the mass attenuation coefficient between 30 and 31.5 kev. Furthermore, comparison of these values provides a strong test of the technique. We defer further examination of the 3 -energy measurements until all other mass attenuation coefficients and photon 4 energies have been determined. 9.3 Full-foil mapping Absorbers have been mapped with the x-ray beam according to the full-foil mapping procedure at and 42.8 kev. Measurements have been made at about = 364 locations across the surface of the foil, taking 2 3 hours to accomplish. Measurements were made at 2 mm intervals in the horizontal x direction in order to over-sample the attenuation profile as measured using the 4 mm-wide beam. Figure 9.9 shows the attenuation profile of the sample-plus-holder measured at kev using sample sn 24. The measurements are smoothly continuous across the surface of the sample-plus-holder. The kev attenuation profile has been fitted with a function which models

260 232 Chapter 9. TIN Figure 9.9: The attenuation profile measured using x-rays of energy kev. The smoothness of the attenuation profile indicates the good statistical level associated with these measurements. The spikes around the foil edges arise as expected from the interaction of the x-ray beam with the mounting screws.

261 9.3. FULL-FOIL MAPPING 233 the attenuation profile of the foil-plus-holder at every measured location to enable subtraction of the holder component. The function described in section predicts the attenuation profile for a foil-plus-holder, as measured using a 1 mm 1 mm beam. That function was modified to predict the attenuation profile measured using a N mm 1 mm beam (N {I, 1}) by evaluating the logarithmic average of N neighbouring values of the predicted 1 mm 1 mm attenuation according to ([ µ [t ) N 1,x,y = ln ( 1 N N n=1 { exp [ µ [t } 1 1,x+n N+1 2,y ), (9.1) where the subscripts refers to the dimensions of the beam and the x and y ordinates of the prediction, respectively. The logarithmic averaging in Eq. (9.1) represents the real averaging of the attenuation over the measured area that occurs when the beam is of uniform intensity. However, if the intensity of the beam is highly non-uniform, Eq. (9.1) may not describe the measured profile to high accuracy. We have used a four-reflection monochromation system with a 4-mm-wide aperture to define the beam footprint. However, as the 4 mm aperture was located 210 mm downstream of the in-hutch crystal, it subtends an angle of about 1 to the crystal. With this degree of angular acceptance there may be a significant decrease of the beam intensity at angles away from the optic axis. The effect of a small beam intensity nonuniformity was discussed in chapter 6: however, the degree of the intensity nonuniformity is more significant in the present case. Measurements using a beam of uniform intensity probe every point of the sample-plus-holder equally, and so the measured intensity ratio is the logarithmic average of that at each point under the beam footprint. When the beam intensity is nonuniform, this logarithmic average is further weighted by the beam intensity at each point. Such beam intensity nonuniformities do not affect the measured intensity ratio when the attenuation profile does not vary. That is, beam intensity nonuniformities are only probed where the sample-plus-holder attenuation varies significantly over the beam footprint. For this reason we have excluded measurements from around the edge of the foil in the determination of the mass attenuation coefficient of molybdenum (chapter 6).

262 234 Chapter 9. TIN beam width [ foil edges included µ σ [ µ χ 2 r [ foil edges excluded µ σ [ µ χ 2 r (mm) (cm 2 /g) (cm 2 /g) (cm 2 /g) (cm 2 /g) Table 9.1: Values of [ µ, with 1-σ uncertainties, and χ 2 r determined by fitting the attenuation profile using various (integer-valued) beam widths. The data on the left have been obtained using all measurements in the attenuation profile except those affected by the screws, and provide a sensitive determination of the effective beam width. The χ 2 r determined from each of these fits indicates that the effective beam width is somewhere between 2 and 4 mm. Data on the right have been obtained by excluding the measurements made partially over the edge of the foil, and provide the best estimate of the mass attenuation coefficient of the foil. The mass attenuation coefficients determined using beam widths of 3 and 4 mm are consistent to within their uncertainties. Accordingly, we use the weighted mean of the 3 and 4 mm beamwidth values, with an additional component added to the uncertainty to reflect the small discrepancy between these two models. We test for effects relating to the uniformity of the beam intensity by fitting the attenuation profile using beams of various (integer multiple) dimensions between 1 mm and 7 mm. The fitting has been performed initially with the measurements made over the foil edge included to provide a strong constraint on the beam dimensions, and later with the foil-edge measurements excluded to determine the mass attenuation coefficient. Table 9.1 presents the determined values of the mass attenuation coefficient (with uncertainty) and the χ 2 r resulting from the fitting procedure. For the fits performed with the foil edges included, the χ 2 r reaches its minimum value for a beam footprint width of 3 mm, indicating an effective value between 2 and 4 mm. The value of the mass attenuation coefficient determined from the 2 4-mm beam width fits is stable at about 44.0 cm 2 /g. The uncertainty in the mass attenuation coefficient is relatively stable despite the divergence of χ 2 r. The uncertainties for these values do not reflect the model uncertainty, which we determine by examining the distribution

263 9.3. FULL-FOIL MAPPING 235 Figure 9.10: The residuals of the fit of the 4-mm-wide x-ray beam to the attenuation profile measured at kev. The absence of structure reminiscent of the shape of the holder indicates that the holder attenuation profile has been correctly modelled. The regions of middle-grey on the left and right-hand sides are the points that have been excluded from the fit due to the interaction of the beam with the foil edge. The measurements were made column-by-column in this figure, and so the horizontal structure in the residuals implies real structure in the attenuation profile. of the residuals. For the fits performed with the foil edges excluded, the χ 2 r shows a broad minimum across the entire range of trialled beam widths. This is exactly as expected, as the lack of features on the attenuation profile of the sample-plus-holder does not enable the fitting routine to probe the nonuniformity of the beam intensity. The distribution of the residuals for the fits with the beam width between 3 and 5 mm show no circular pattern, indicating that the holder has been modelled to within the statistical accuracy of the measurements. Figure 9.10 shows the distribution of the residuals for the fit using the 4 mm beam width. The values of the mass attenuation coefficient determined using the 2, 3, and 4-mm-wide beams are consistent within the fitting and measurement uncertainties. All other aspects of the measurement and fitting are well-behaved. We therefore use the weighted mean of the mass attenuation coefficient determined using the 3 and 4 mm beam widths, and add (in quadrature) an additional uncertainty component equal to half the difference between the two, i.e., σ [ µ = cm 2 /g, giving [ µ = ± cm 2 /g.

264 236 Chapter 9. TIN Figure 9.11: The attenuation profile measured using x-rays of energy 42.8 kev. The roughness of the attenuation profile indicates the lower statistical level associated with these measurements. The noise in the attenuation profile is of the same order as the magnitude of the holder contribution, which has prevented robust fitting of this attenuation profile to remove the holder component. Figure 9.11 presents the attenuation profile measured at 42.8 kev using sample sn 201. This higher-energy measurement is of much weaker statistical merit. The holder component of the attenuation profile is comparable to the statistical noise of the measurement, and so cannot be readily identified from this attenuation profile. Accordingly,we have been unable to fit the holder component of this attenuation profile. 9.4 Determination of the photon energies We have measured the angular locations of a number of (hhh) reflections from a germanium single-crystal using the same movement routines and intensity optimisation procedures as for the measurements reported in chapter 7. Similar problems with detector saturation and analyser crystal defects are observed as in that chap-

265 9.5. SCALING THE LOCAL INTEGRATED COLUMN DENSITIES 237 Figure 9.12: The determined photon energies. ter. Accordingly we use similar analysis for determining the photon energies for this experiment and present only the main results. The logic of this was detailed in chapter 7. Figure 9.12 presents the determined photon energies. These energies were determined from the locations of the leading-edge and the centre-of-mass of the measured reflections. The directly-determined energy at 41.2 kev has been compromised by the presence of the spurious peak discussed in section 9.2. However, because of the overlap of the spurious and the central reflections at this energy, we have been able to measure a number of low-intensity reflections (see Fig. 9.7). The deviations of the directly determined energies from the interpolated curve suggest that it may be appropriate to perform the interpolation over a number of ranges. However, without this refinement, the χ 2 r is about 3.1 and the estimated energy uncertainty is of the order of 1 4 ev, indicating that the photon energies can be determined to the required level of accuracy without any further development of this analysis. 9.5 Scaling the local integrated column densities Figure 9.13 presents the mass attenuation coefficients determined by division of the measured attenuations by the average integrated column density of the appropriate

266 238 Chapter 9. TIN Figure 9.13: The mass attenuation coefficients determined by use of the average integrated column density. foil. These measurements show good consistency and statistical quality. As discussed in chapter 8, the average value of the integrated column density may not accurately reflect the integrated column density of the foil at the location through which the x-ray beam actually passed. It is therefore necessary to determine local values for the integrated column density by scaling the mass attenuation coefficients to minimise the differences from one another and from the absolute value determined by use of the full-foil mapping technique (Eq. (8.4)). Figure 9.14 presents the percentage deviation of the values from their weighted mean at each energy, after determining the local foil integrated column densities. Our uncertainty in the weighted mean is % at energies below 40 kev and generally below about 1% for the measurements made at higher energies. The local integrated column densities have been rescaled by 0.6%, 0.35%, 0.25%, 0.15%, 0.02%, -0.4%, and -1.1% compared to their nominal, average values. These values are in good agreement with estimates of the thickness uniformity of the foils. Figure 9.15 shows the percentage difference between our values after determination of the local integrated column densities and the FFAST tabulated values. The point-to-point variation of this difference is consistent with the claimed uncertainties, with the exception of a 1% inconsistency observed at about 37 kev. It is likely that some transient event (such as a beam injection) has affected this particular

267 9.6. RESOLUTION OF THE 3 -ENERGY POINTS Figure 9.14: Left: percentage deviation of the mass attenuation coefficients from their weighted mean at each energy, after determining the local integrated column densities. There is a large complex of deviations around the absorption-edge at about 29.2 kev. The gradual increase of the deviations as the photon energy increases above about 40 kev is consistent with the lower precision expected for these measurements. Right: values within a small range of the zero line only. The line indicates the 1-σ s.e. variation of the weighted mean, and indicates a measurement precision of about % below 40 kev, rising to about 1% at higher energies. measurement. Figure 9.16 shows values of the attenuation within a small energy range of the absorption edge. In this energy region we have made measurements at an energy interval equal to the 1s hole-width of approximately 8 ev [18. The XAFS is plainly visible, although the measurements would have been improved by being made at a finer energy interval. 9.6 Resolution of the 3 4-energy points The discontinuity between 40 and 42 kev is due to the presence of 3 -energy photons 4 in the beam. Both third and fourth-order photons were reflected by the upstream crystal and, due to the low-energy bias of the bending-magnet spectrum, we expect the beam entering the hutch to comprise more third than fourth-order photons. We have calibrated the photon energies to the angle of the upstream monochromator. Accordingly, we multiply the calibrated energy by 3 4 to determine the photon energy for those measurements made using the third-order beam. Our determination of the energies in this way assumes that any differential shifts, arising from energy-dependent diffraction effects (such as refractive index and depth pene-

268 240 Chapter 9. TIN Figure 9.15: Percentage deviation from the FFAST tabulated values of the mass attenuation coefficients after determining the local integrated column densities. tration) are not significant over these energies. Figure 9.17 shows the determined values of the mass attenuation coefficient for the incorporated and regular measurements. Values are indicated by their uncertainties in [ µ. The large diamond marker indicates the value determined using the full-foil mapping procedure. The incorporated measurements are indicated by arrows whose labels are their nominal energies. These measurements are entirely consistent with the trend of the regular measurements, except for the value at the Figure 9.16: Determined values of the mass attenuation coefficient of tin within a narrow range of the absorption-edge at 29.2 kev. The XAFS is clearly visible on this plot.

269 9.6. RESOLUTION OF THE 3 -ENERGY POINTS Figure 9.17: Incorporation of the 3 -energy measurements. The incorporated values 4 agree with the regular measurements to within their % uncertainties, verifying the 3 -energy hypothesis and the values and the uncertainties determined here. The 4 measurement made at nominal energy 41.2 kev is inconsistent with the trend of the other values due to the presence of the 41.2 kev photons in the beam (cf. Fig. 9.7). nominal energy of 41.2 kev. Examination of Fig. 9.7 shows that the beam used for this measurement is composed of a mixture of third and fourth-order photons, and this is the cause of the observed discrepancy. The other incorporated measurements are consistent with the trend of the regular measurements to within their uncertainties of between 0.07% and 0.6%. The level of agreement shown here confirms the 3 -energy hypothesis and demonstrates the 4 reproducibility of our measurements within their uncertainties.

270 242 Chapter 9. TIN 9.7 Conclusions, current status, and further work The current chapter reports a first-pass calculation to determine the mass attenuation coefficients of tin between 29 and 60 kev. The full-foil mapping technique has been employed and used to determine a value of [ µ = (36) cm 2 /g for the mass attenuation coefficient, accurate to 0.081%. This absolute and accurate value has been used to determine the integrated column density, along the path traversed by the beam, for the other absorbers. This has determined a consistent set of values with negligible outliers. The photon energies have been determined to accuracies of 1 4 ev over the energy range of the measurement, which is sufficiently accurate so that they do not limit the measurement away from the absorption edge and the XAFS. A spurious reflection from one of our double-reflection monochromators has resulted in some of the measurements being made at 3 4 of the nominal beam energy. These measurements provide a dramatic and unexpected test of our technique. The determined values were consistent within their uncertainty estimates, which were below 0.1% in some cases. This high degree of consistency validates our methodology, calculational approach, and uncertainty estimation procedures. Figure 9.18 presents the differences between our provisional values for the mass attenuation coefficients of tin and the FFAST tabulated values [33, 34, 35. The differences of the XCOM [96, 97, 98 values are also shown, as are various measured values compiled in [106, 107, 108. Values arising from individual investigations are plotted with a unique symbol to establish the trend of the measurements. Our measurements exhibit a high degree of consistency in comparison to previous experimental work, varying smoothly with respect to the tabulated values. identification of any inconsistencies in our measurements is facilitated by our use of synchrotron sources to make measurements over a fine energy grid. Investigations employing characteristic or radioactive sources are often limited to a spares set of discrete energies and are unable to evaluate even the self-consistency of their measured values. The

271 9.7. CONCLUSIONS, CURRENT STATUS, AND FURTHER WORK 243 Table 9.2 presents the provisional values for the mass attenuation coefficients of tin. The first column presents the photon energy, with the uncertainty in the final two digits given in parentheses. The second column presents the mass attenuation coefficient [ µ, with the uncertainty in parentheses. The mass attenuation coefficient is determined as the weighted mean after determining the local integrated column density for each of the foils, and the uncertainty is determined as the 1-σ s.e. spread in these values. The third column gives the percentage uncertainty of the determined mass attenuation coefficients, and indicates that measurement accuracies are generally below 0.2% for measurements made at photon energies below about 40 kev. The measurement accuracy is 0.081%, and is limited by the measurement of the area of the foils and the removal of the holder attenuation by means of the fitting procedure. The precision of the measurements, determined from the discrepancies between measurements using various thicknesses of foil, is between 0.06% and 1%. Figure 9.18: Percentage difference from FFAST tabulated values for [ µ of: our values; the XCOM tabulated values, and; various previous experimental values, tabulated in [106, 107, 108. Our values, obtained on a fine energy grid using a synchrotron source, exhibit a significantly greater degree of consistency compared to all other measurements.

272 244 Chapter 9. TIN Figure 9.19: Significance of deviations from the weighted mean. Measurements at higher energies show no systematic trends in this figure, indicating that the spread of the values (Fig. 9.14) is indeed due to lower statistical precision. The outliers observed in the neighbourhood of the absorption edge at about 29.2 kev are expected due to the increased energy density of the measurements and the use of foils with attenuations as high as [ µ [t 8. Significance is defined in Eq. (8.6). Figure 9.19 presents the significance of the differences from the weighted mean values [defined in Eq. (8.6). There are no systematic discrepancies within the measurements above 40 kev, and individual measurements are generally within their estimated uncertainties of the weighted mean value (i.e., the discrepancies are generally between ±1 standard deviation). Between 30 and 40 kev the measurements exhibit a slight increase in the discrepancies, with little apparent systematic trend. At the absorption edge and in the region of the XAFS there are no significant outliers. Below the absorption edge there are a few outliers, but the small number of these (of order 10) is indicative of transient rather than systematic discrepancies. We have not completed the investigation of the measured values for the presence of systematic effects. However, the absence of systematic deviations in Fig indicates that such systematic effects are only slight, if of any significance at all. In the following few paragraphs we discuss several particular systematic effects and their likely impact on this measurement. The x-ray bandwidth can have an appreciable effect on the measured values in regions of high attenuation gradient (i.e., on the absorption edge and in the XAFS). Measurements were made at energy intervals down to 8 ev, and we have only one

273 9.7. CONCLUSIONS, CURRENT STATUS, AND FURTHER WORK 245 or two measurements on the absorption edge (see Table 9.2 at about 29.2 kev). Due to the subtle, gradient-correlated nature of the bandwidth effect, these few measurements are likely to provide insufficient information from which to isolate a value for the bandwidth or to determine any effect on the measured mass attenuation coefficients. We have used two double-reflection monochromators in a (+ +) or dispersive geometry (see Fig. 9.3). This configuration results in a beam of reduced bandwidth [160, 172. Therefore, we expect that any effect of the bandwidth will in any case be greatly reduced in comparison to the molybdenum measurement. We expect that x-ray harmonic energy components will have negligible effect on the measured values due to the method of monochromation employed in this investigation, the intrinsic bending-magnet spectrum (see Fig. 9.1), and the harmonic rejection resulting from the ion chamber efficiencies (see Fig. 9.2). This expectation is confirmed by the absence of systematic signatures in Figs 9.14 and The shiny appearance of the tin absorbers indicated that they were significantly smoother than their molybdenum counterparts. This was confirmed by measuring the tin absorbers at several locations across their surface using an AFM. The thickness of the tin absorbers ranged from 25 µm to 1.5 mm. We therefore expect any effects of roughness to be below that for molybdenum, for which roughness effects were not detected. The experimental geometry for the tin measurement was very similar to that employed for molybdenum, including the relative positions of the sample stage, the daisy wheels, and the ion chambers. The effect of fluorescence and other secondary radiation is therefore likely to be undetectable in the tin measurements (as for the molybdenum measurements) due to the high degree of beam collimation. We shall soon complete the null tests for these systematic effects. Following this we will determine the coefficient of photoelectric absorption by subtracting tabulated Rayleigh and Compton components, and will then determine f 2. These data will enable detailed comparison with predicted form-factors, and will provide valuable and accurate form-factors for tin over this energy range.

274 246 Chapter 9. TIN Table 9.2: Mass attenuation coefficients [ µ of tin as a function of x-ray energy, with one standard deviation uncertainties in the least significant digits indicated in parentheses. We present also the percentage uncertainty in the mass attenuation / [ µ. The five energies marked with a are measured using the coefficients, σ [ µ energy x-rays. The measurement made at the corrected nominal energy of kev has been omitted as the x-ray beam was a mixture of the third and fourth-order harmonic components. Energy (kev) [ µ (cm 2 /g) σ [ [ µ / µ (92) 7.797(61) 0.78% (92) (97) 0.12% (92) 7.854(70) 0.89% (92) 7.855(55) 0.70% (92) 7.873(20) 0.25% (92) 7.913(58) 0.74% (92) 8.015(49) 0.61% (92) 8.164(27) 0.34% (92) 8.285(21) 0.26% (92) (96) 0.10% (92) (93) 0.094% (92) (15) 0.13% (92) (17) 0.11% (92) 38.37(12) 0.31% (92) (65) 0.13% (92) (62) 0.14% (92) (54) 0.12% (92) (50) 0.11% (92) (63) 0.14% (92) (50) 0.11% (92) (86) 0.18% (92) (61) 0.13% (92) (67) 0.15% (92) (52) 0.11% (92) (87) 0.18% (92) (58) 0.12% (92) (75) 0.16% (92) (30) 0.064% (92) (41) 0.088% (92) (77) 0.16% (92) (33) 0.070% (92) (37) 0.079% (92) (71) 0.15% (92) 46.63(10) 0.22% (92) (87) 0.19% continued...

275 9.7. CONCLUSIONS, CURRENT STATUS, AND FURTHER WORK from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ (92) 46.01(10) 0.22% (92) (40) 0.087% (92) (36) 0.079% (92) (28) 0.061% (92) (41) 0.090% (92) (35) 0.077% (92) (48) 0.11% (92) (27) 0.059% (92) (50) 0.11% (92) (31) 0.070% (92) (36) 0.081% (92) (35) 0.079% (92) (54) 0.12% (92) (39) 0.087% (92) (40) 0.090% (92) (36) 0.081% (98) 42.88(26) 0.62% (91) (45) 0.11% (99) (87) 0.21% (90) (46) 0.11% (10) (70) 0.17% (90) (52) 0.13% (11) (28) 0.074% (90) (72) 0.19% (11) (53) 0.14% (89) (74) 0.20% (89) (62) 0.18% (88) (25) 0.074% (88) (62) 0.19% (88) (39) 0.12% (88) (43) 0.14% (87) (54) 0.18% (87) (37) 0.13% (87) (64) 0.23% (87) (47) 0.17% (88) (28) 0.11% (88) (58) 0.22% (88) (31) 0.12% (89) (38) 0.15% (89) 23.41(12) 0.51% (90) (38) 0.17% (91) (31) 0.14% continued...

276 248 Chapter 9. TIN... from previous page [ µ Energy (kev) (cm 2 /g) σ [ [ µ / µ (92) (51) 0.24% (93) (37) 0.18% (95) (41) 0.20% (96) (50) 0.25% (11) (39) 0.23% (11) (44) 0.27% (12) (66) 0.42% (12) (68) 0.45% (13) (48) 0.33% (13) (84) 0.59% (14) 13.63(11) 0.83% (14) (43) 0.32% (15) (48) 0.37% (16) (53) 0.43% (16) (34) 0.29% (17) (59) 0.51% (18) (53) 0.47% (18) (55) 0.51% (19) (57) 0.54% (20) (92) 0.91% (21) 9.84(14) 1.4% (22) 9.45(14) 1.5% (23) 9.218(73) 0.79% (24) 8.901(57) 0.65% (25) 8.586(87) 1.0% (26) 8.25(10) 1.2% (27) 7.968(84) 1.1% (28) 7.761(75) 0.97% (29) 7.509(71) 0.94% (30) 7.244(67) 0.92% (31) 6.99(20) 2.9% (33) 6.804(66) 0.97% (34) 6.548(97) 1.5%

277 Chapter 10 Silver: determining the photon energies In the course of my candidature I have assisted with a determination of the mass attenuation coefficients of silver over the energy range from 15 to 50 kev. This chapter reports my contribution to this measurement: the determination of the x-ray energies. The measurement was performed at sector 1-BM (now XOR) of the APS in 1999, and the results have recently been reported [ Experimental details for 1-BM XOR The x-ray beam was produced by use of the bending magnet device at the 1-BM XOR beamline [173 of the Advanced Photon Source facility at the Argonne National Laboratory. This beam was monochromatised by reflection from the (400) planes of a pair of silicon crystals located in the first optical enclosure. The second crystal of the pair was detuned slightly to suppress the passage of harmonic components into the beam. A schematic of the experimental configuration is shown in Fig We discuss here only those details of direct relevance to the energy determination. For a full discussion see [69. incoming beam daisy wheel daisy wheel detector monochromator 1st ion chamber 2nd ion chamber translational/rotational sample stage Si640b powder sample Figure 10.1: Schematic of the experimental layout. 249

278 250 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES After monochromation the x-rays passed through a pair of orthogonal adjustable slits which defined the beam cross-section to be approximately 2 2 mm 2. Located downstream from the apparatus used for the measurement of attenuation was a six-circle Huber diffractometer. On the central φ axis of this diffractometer (see Fig. 7.1) was a thin capillary of the silicon standard reference material SRM 640b [174, whose mean lattice parameter is known to very high accuracy. A sodiumiodide scintillation counter located on the δ arm of the diffractometer recorded the angular location of a number of reflections from the powder specimen. Such energy measurements were carried out on 46 occasions over the course of the measurement Determination of peak positions Figure 10.2 shows a selection of peak profiles measured using x-rays of nominal energy 35.2 kev. The profiles of a series of reflections have been recorded, at most energies extending on both sides of the δ = 0 location. The angular positions of the measured diffraction profiles have been determined by performing least-squares fits using both Gaussian and Lorentzian profiles. The indicators of the quality of the fits (values of σ s.e., χ 2 r, and σ s.e. χ 2 r ) were relatively insensitive to the choice of profile due to the sparse angular density of the measurements. The analysis reported in this chapter has been performed using the results of both profiles. In the end the energies determined using Gaussian profiles showed a higher degree of self-consistency and accordingly these values have been used. We do not report the details of the parallel Lorentzian calculation in the interests of brevity. The uncertainties in the fitted profile locations have been estimated using σ s.e. χ 2 r returned by the fitting routine, thereby accounting for any profile dependence of the determined centroid location. The high-angle measurement [Fig. 10.2(d) exhibits a reduced signal and an increased noise component compared to the other profiles. This is characteristic of all profiles that we have measured at high angles, and is due to a combination of the decrease of the form-factor, the Lorentz-polarisation factor [68, which acts to reduce the intensity of high-angle reflections, and to increased incoherent scatter at higher angles, which leads to the increased background signal. These effects have reduced

279 10.2. DETERMINATION OF PEAK POSITIONS 251 Figure 10.2: Measured profiles of the x-ray reflections of nominal energy 35.2 kev from the silicon (SRM 640b) powder specimen. The error bars represent the (Poisson) counting uncertainty for each of the measured counts. The black and grey curves indicate the fitted Gaussian and Lorentzian profiles, respectively. A constant background was fitted in each case. the signal-to-noise ratio to such a degree that it has been impractical to measure reflections at angles above about 20 at the higher energies and above about 56 at the lower energies in this experiment. A casual examination of Figs 10.2(a) and (b), showing our measurements of the complementary (111) reflections at 6.38 and 6.45, reveals that the angular locations of the measured reflections have been offset from their ideal locations as a result of a diffractometer misalignment. In the next section we investigate the likely cause of this offset and its consequences for our determination of the x-ray energies.

280 252 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES 10.3 Determining the photon energies: effects of diffractometer misalignment We use the same technique for determining the photon energy as that described in section 7.7, which we recap briefly for convenience. The main result appearing in chapter 7 is given by Eq. (7.14): when the measured angular positions are affected by a systematic angular offset, the x-ray energy can be determined by linear extrapolation of a plot of d cos (δ/2) versus d sin (δ/2) to d cos (δ/2) = 0, and is proportional to the inverse of d sin (δ/2) at the extrapolation point. We shall refer to a plot of d cos (δ/2) versus d sin (δ/2) as an energy determination plot. Figure 10.3 shows the energy determination plot for the measurements made using x-rays of nominal energy 35.2 kev. We have fitted a single linear trend to the entire data-set, shown here by the grey line. The fitted single linear trend is in poor agreement with the data-set. The measurements recorded in the δ > 0 and δ < 0 regions appear to follow separate linear trends. Accordingly, separate linear trends are fitted to the measurements made in the δ > 0 and δ < 0 regions, shown here Figure 10.3: The energy determination plot of d cos (δ/2) versus d sin (δ/2) for x-rays of nominal energy 35.2 kev. Linear trends have been fitted to the entire data-set and separately to the data measured in the δ > 0 and δ < 0 regions. The trends indicate that the data may be consistent with a linear trend in the δ > 0 and the δ < 0 regions but that the entire data-set is not consistent with a single linear trend. This small nonlinearity is observed through all of the energy determination plots.

281 10.3. DETERMINING THE PHOTON ENERGIES 253 using black lines. These separate trends show much improved agreement with the measured data. The δ > 0 and δ < 0 measurements follow separate linear trends for measurements made at all energies: we do not present them here for reasons of space. To explain the structure presented in Fig. 10.3, the effects of diffractometer misalignment on the measured peak locations have been investigated. In order for a detector mounted upon the δ arm of the diffractometer to accurately record the angle between the incident and diffracted beams the powder diffraction specimen should be situated at the centre of rotation of the δ axis and the zero of the δ axis should be aligned with the undeflected beam. We describe in the remainder of this section four diffractometer misalignments and their effects on the determination of the photon energies by use of the energy determination plot. When the zero of the δ axis is not properly aligned with the undeflected beam a constant offset in δ results. This effect has been discussed in detail in chapter 7 but we present a different discussion here in order to provide a basis from which to discuss further diffractometer offsets. Figure 10.4 shows the effect of such a constant offset in the δ axis on (a) the measured angles and (b) the distribution of the points on the energy determination plot. The solid black line on these plots indicates the Figure 10.4: Effect of a constant angular offset of the δ scale of the diffractometer axis. The black line indicates the form of the offset at all angular locations. It is clear that the energy can be determined using the interpolation method. The + markers indicate the expected locations of our measurements. These data do not fall near to the interpolation location at d cos (δ/2) = 0. In this and subsequent simulations we model the angular locations of 25 kev x-rays reflected from a silicon sample. Powder specimen absorption effects are not included in our modelling.

282 254 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES effect of the angular offset for all allowed reflections of a beam of 25 kev x-rays from a silicon powder sample. The limited angular range of our measurements has reduced our ability to observe such diffractometer offsets. Calculated reflections from planes with indices ranging up to (731) (the highest reflection index measured in this experiment) are indicated on the plot using plus (+) markers. All profiles have been measured by scanning the δ arm of the diffractometer from low values of δ to high values of δ. By consistently performing the measurement in this way the first-order effects of mechanical hysteresis are removed. In order to avoid the second-order effects of hysteresis one could measure the profiles beginning at the lowest δ and move monotonically through to higher values of δ. The energy determination at 35.2 kev involved the measurement of peaks in the following sequence: 6.45, 10.5, 12.3, 14.9, 16.2, 18.3, 19.4, 21.1, -6.38, -10.5, -12.3, -14.8, -16.2, -18.2, -19.3, From this sequence it can be seen that the peaks measured at angles δ > 0 each follow a large positive motion of the δ arm. Conversely, the peaks measured at angles δ < 0 each follow a large negative motion of the δ arm. Accordingly, it is reasonable to expect that a different second-order hysteresis might apply to the peaks measured on either side of the δ = 0 location. The effect of second-order hysteresis can be modelled using a bimodal offset, where different offsets are associated with the measurements made in the δ > 0 and δ < 0 regions. Figure 10.5 presents the effect of a constant angular offset of 600 µrad, with a second-order, bimodal component of 100 µrad. This bimodal-δ Figure 10.5: Effect of a bimodal offset δ = 600 µrad, bimodal δ = 100 µrad in the angular scale of the diffractometer axis.

283 10.3. DETERMINING THE PHOTON ENERGIES 255 offset is identical to a 700 µrad offset in the δ < 0 region and a 500 µrad offset in the δ > 0 region. The two offsets of the angular scale of the δ axis of the diffractometer result in a linear distribution of points on the energy determination plot. We now discuss the effects of an offset of the powder capillary specimen with respect to the centre-of-rotation of the δ arm, and show that such an offset results in a nonlinear error in the recorded diffraction angles. Figure 10.6 illustrates the situation where the powder capillary is offset by a distance z downstream of the centre of rotation of the δ axis of the diffractometer. Such an offset gives rise to an angular shift z of the measured angular locations from their ideal Bragg values δ B equal to { } z z = arcsin R sin δ B, (10.1) where R is the radius of the detector on the δ arm, which was set to be about 1 m. incident x-ray beam diffractometer centre ideal profile location z R δ z actual capillary location measured profile location Figure 10.6: The effect of a misalignment of the powder capillary with respect to the centre of rotation of the δ axis. The black lines emanating from the centre of the diffractometer represent reflections from a well-aligned powder capillary. The dashed lines emanate from the actual capillary location, displaced a distance z from the diffractometer centre. The grey line represents the measured angle of the reflection. The z > 0 offset shown here has the effect of shifting all of the peak locations towards δ = 0. This systematic shift of peak locations results in an apparent shift of the photon energy to higher values.

284 256 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES Figure 10.7: Effect of a misalignment of the powder capillary by a distance z = 600 µm downstream of the centre of rotation of the δ axis. The smooth black curves indicate the form of the offset at all angular locations. All measurements are shifted closer to the δ = 0 location. The measured locations (+ markers) in the δ > 0 and the δ < 0 regions have been fitted using straight lines to determine the x-ray energy: it is clear from this plot that both values are incorrect by a negligible amount (0.06%). The actual diffractometer z-offset is certainly less than the 600-µm offset simulated here, as this is beyond the tolerances of the diffractometer alignment. All symbols as for Fig The black curves in Fig show the loci of the measured angles and the points on the energy determination plot resulting from an offset of z = 600 µm. The plus markers show the locations of the reflections falling within our measured angular range, approximately 56 < δ < 56. The reflections that we measure are offset by a constant amount on the energy determination plot. The effect of the z offset is almost degenerate with an increase in the x-ray energy when measurements are made over a limited range of the measurement parameter space. In particular, the effect of the 600-µm z offset shown here is almost indistinguishable from a (negligible) = 0.06% change in the x-ray energy for our set of measurement angles. We do not expect the z offset to be greater than about 600 µm as such an offset would have been detected whilst aligning the diffractometer. Figure 10.8 illustrates the effect of an offset of the powder capillary with respect to the centre of rotation of the δ axis in the vertical or y direction. Such an offset gives rise to a shift y of the measured angular locations equal to { } y y = arcsin R cos δ B. (10.2)

285 10.3. DETERMINING THE PHOTON ENERGIES 257 actual capillary location y y δ R measured profile location ideal profile location incident x-ray beam diffractometer centre Figure 10.8: Schematic of the effect of a powder offset in the vertical or y direction. The y > 0 offset shown here shifts all of the peak locations in the direction of the offset. When only a small range of forward angles are measured the effect of the y offset is similar to the effect of a misalignment of the δ axis of the diffractometer. Figure 10.9 shows the effect of a y = 600 µm offset on (a) the angular locations of the diffraction maxima and (b) their locations on the energy determination plot. The black curves indicate the loci of the offset for all angles. The plus markers indicate the locations of reflections falling within our limited angular range. The angular shifts y for these data are similar to those resulting from a constant diffractometer offset, presented in Fig Accordingly, the energy determination plot also bears some resemblance to the corresponding plot calculated for a constant angular offset [Fig. 10.4(b). Linear segments have been fitted to the (simulated) values falling in the δ > 0 and δ < 0 regions, and are indicated here by the grey lines. The energies determined using these linear segments are shifted slightly from their ideal values. As the energies are shifted in opposite directions, the beam energy can be determined accurately from the average of the values resulting from the δ > 0 and δ < 0 regions. The powder capillary is long and thin, and so outboard or x-direction shifts are not likely. Further offsets of the measured peak locations can result from the misalignment of the plane of the δ-arm detector motion and the incident beam [68. However, such effects will be insignificant for our geometry as we have used a detector whose linear dimensions are larger than the likely extent of any such offset.

286 258 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES Figure 10.9: Effect on the energy determination plot of a misalignment of the powder capillary by a distance y = 600 µm above the centre of rotation of the δ axis. The smooth black curves indicate the form of the offset at all angular locations. All forward angles are measured at higher (more positive) angles, and all backward angles at lower angles. All symbols as for Fig The energy determined using high-index reflections is not affected. However, such reflections are not measured here due to their low signal-to-noise ratios. The measured locations (+ markers) in the δ > 0 and the δ < 0 regions have been linearly extrapolated to determine the x-ray energy: it is clear from this plot that, while both values are incorrect, the mean of the positive and negative angles produces the correct result. We have discussed four different types of diffractometer misalignment and their effects on the angular locations of the measured reflections and on the shape of the energy determination plot. These offsets produce partially degenerate signatures, especially when the reflections fall within a limited angular range. Here we discuss the implications of such offsets for our determination of the photon energies. The constant angular offset and the bimodal δ offset do not have any effect on the energy determined by fitting separate linear segments to the measurements made in the δ > 0 and δ < 0 regions. The y and z powder capillary offsets do alter the value of the energy determined by use of the separate δ > 0 and δ < 0 pools. In the case of the y offset, the correct value for the energy can be determined from the mean of the two pools. Our simulation suggests that the energies determined using the two pools should differ by less than 0.03%. The z offset is almost degenerate with a change in the photon energy. Our modelling shows that measurements made over a limited angular range cannot be used to determine the energy correctly in the case of a z offset. However, for z offsets below about 0.5 mm the maximum likely inaccuracy of the diffractometer

287 10.3. DETERMINING THE PHOTON ENERGIES 259 alignment the effect on the determined photon energy is negligible at less than about 0.06%. We find no clear evidence for the presence of y and z offsets despite extensive investigation. We do not use our quantitative descriptions for these offsets as this would involve the use of a highly-correlated parameter space. Instead we use a minimal non-degenerate (orthogonal) parameter set and determine the energy by fitting a linear trend to each of the δ > 0 and δ < 0 pools. We determine a single value for the energy by coupling the two pools at δ = 0. The fitted function is (m δ + m bimodal δ )[d cos (δ/2) + c δ < 0, [d sin (δ/2) = (10.3) (m δ m bimodal δ )[d cos (δ/2) + c δ > 0, where m δ is the gradient component arising from the constant angular offset and m bimodal δ is the bimodal component that changes sign in the δ > 0 and δ < 0 regions. Figure shows some examples of the application of this fitting procedure. The linear fit to the δ < 0 and δ > 0 regions is consistent with the data, and a single value for the energy is determined by the procedure. The uncertainties appropriate to this fitting procedure are shown on the plot as a pair of grey lines Figure 10.10: Determining the energies from the measured reflections. Two linear segments are fitted to the measurements in the δ < 0 and the δ > 0 regions respectively. These segments are united by the fixed d sin (δ/2) intercept. The light grey envelope about the fitted line indicates the uncertainty at all points along the fit determined from the covariant error matrix multiplied by the χ 2 r returned by the fitting routine.

288 260 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES Figure 10.11: Determined values of the gradient of the energy determination plot at each measured energy. This plot shows four regions of smoothly varying gradient punctuated by sudden change. These sudden changes are consistent with operator initiated changes to the diffractometer alignment. forming an envelope around the fitted values. Figure presents the values of the gradients m δ determined at each measured energy as a function of the nominal beam energy. This plot shows four regions of smoothly varying offset punctuated by periods of rapid change. These periods of rapid change correspond to operator-initiated changes to the diffractometer alignment, which suggests that the gradient can be interpreted as the constant angular offset of the δ axis Recovering lost encoder readings The angular position of the monochromator was determined by the use of an angular encoder and this value was reported using a digital display. At the beginning of this experiment (at high energies), the monochromator encoder value was recorded manually in a log book. However, on some occasions the monochromator encoder value was not recorded. In response to this, the computer controlled routines were modified and later monochromator encoder values were automatically recorded in the computer data file.

289 10.4. RECOVERING LOST ENCODER READINGS 261 Figure 10.12: Nominal x-ray energy presented as a function of the recorded monochromator encoder angles. A number of encoder readings have not been recorded at angles below 7, and most obviously in the angular region between 5.2 and 6.4. Excluding calculation and motion errors, and errors relating to the extraction of the nominal energy from the data file, these values are clearly functionally related. Figure shows the nominal photon energy as a function of the recorded values of the monochromator encoder. Encoder values have not been recorded in the region between 5.2 and 6.4. In this section we concern ourselves with the recovery of these missing encoder values. The monochromator angle should be in exact correspondence with the nominal energy, as the former is calculated from the latter by the beamline software. However, slight deviations from an exact relationship exist due to round-off errors (finite numerical precision) and motor limitations (sub-step-size positioning errors). Accordingly, we determine the missing encoder values by fitting the modified Bragg equation [Eq. (7.16) to the recorded values. Figures 10.13(a) (c) present the residuals of the fit of the modified Bragg equation to the nominal energy and the recorded encoder values. In Fig (a) we see that the fit has two significant outliers between 8.6 and 8.8. These outliers have resulted from the use of an incorrect value of nominal energy for these measurements. Usually the nominal energy is only used as an identifier for viewing the measurements on an approximate energy scale, so that the use of an incorrect value is inconsequential. The dashed curves indicate the σ s.e. χ 2 r fitting uncertainty de-

290 262 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES Figure 10.13: Interpolating the encoder readings. Figures (a) (c) present the residuals determined after fitting the modified Bragg equation to the monochromator encoder angles and the nominal energy. Figures (a) and (b) show a generally good trend around the zero line, indicating a good fit, but with outliers (marked with a diamond symbol) which have resulted from our extraction of an incorrect value of the nominal energy from the data file. The use of an incorrect value of the nominal energy is generally inconsequential, as the nominal energy is only used to provide an approximate energy scale for the figures. However, in this case we use the nominal energy to recover the absent monochromator values, and so we must discard these outliers. Figure (c) shows the residuals of the fit after all outliers have been discarded. The uncertainty of the interpolation is typically less than 1.5 ev, which is well below the tolerances of this determination. Figure (d) shows the recovered monochromator angles. termined from the covariant error matrix returned by the fitting routine. We aim to achieve an energy determination accurate to about 0.1% (determined from the precision of the attenuation measurement), and therefore require the encoder values to be recovered with an accuracy of better than ev over the kev energy range of the measurement. Figure 10.13(b) shows the residuals determined from the fit after discarding two significant outliers. This plot shows a further ev outlier at an angle of about 4.7. Figure 10.13(c) shows the residuals determined from the fit after discarding this remaining outlier. The uncertainty of the interpolated encoder values

291 10.5. INTERPOLATING THE PHOTON ENERGIES 263 is less than about 1 ev, which is significantly better than our goal. The absence of significant remaining outliers indicates that we have used a consistent data-set for the determination. Figure 10.13(d) shows the full set of monochromator encoder angles with the recovered values indicated by markers Interpolating the photon energies The photon energies have been determined at regular energy intervals throughout the experiment. Here we use these energy measurements to calibrate the monochromator encoder readings and determine the x-ray energies. We again use our modified Bragg equation [Eq. (7.16) to allow the zero of the monochromator angle and the lattice parameter to vary slightly from their nominal values, thus describing a possible monochromator misalignment and an expansion of the monochromator crystal occurring, for example, in response to the heat load of the synchrotron beam. The monochromator was moved to produce photons of energy well above 50 kev and then gradually stepped down during the attenuation measurements to avoid the effects of backlash hysteresis [13. However, at about kev a cycling of the monochromator occurred, resulting in a discontinuity of the angular offset parameter. Figure shows discrepancies between directly measured energies (crosses, with σ s.e. uncertainties indicated) and interpolated energies (zero line) in the two ranges. The calibration uncertainty, represented as the envelope, is the standard deviation determined by evaluating the covariant error matrix returned by the (Levenberg-Marquardt least-squares) fitting routine at each energy. The uncertainties E of the determined energies range between 1.6 and 4.3 ev for the lower energy measurements, corresponding to between 0.01% and 0.02%; and between 12.6 and 45.7 ev for the higher energy region, corresponding to between 0.04% and 0.09%. The uncertainties in the lower energy region are smaller than those in the higher energy region. The improved stability in the lower energy region is a result of several online optimisations that were carried out while data was taken in the higher energy region. The accuracy of the measured values is significantly better than that resulting from our maximum estimated z offset error of 0.06%.

292 264 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES Figure 10.14: Calibrating the monochromator encoder values to determine the x-ray energies. The calibration has been performed over two separate energy ranges to treat the effects of a cycling of the monochromator angle which has led to a small monochromator hysteresis. The uncertainties are of order 0.02% and 0.1% in the low and high-energy regions, respectively.

293 10.6. DISCUSSION Discussion Some limitations of this data set and this technique for measuring the x-ray energies have been observed. Some particular insights and their consequences for later experiments are noted here. Figure 10.2 presented a selection of reflection profiles measured using x-rays of nominal energy 35.2 kev. Peak count-rates of about about 300 cps are observed for the measurements of the most intense (111) reflections. The higher-angle, higherorder reflection at about 21.1 records count rates little more than a factor of two above background. These count rates and signal-to-noise ratios are typical for the profiles measured in this experiment. The low signal-to-noise ratios associated with the high-index reflections are particularly unfortunate for the analysis because those reflections may otherwise have provided useful information for the determination of the photon energies. In particular, if measured to high statistical precision these reflections may have enabled us to discriminate between several possible diffractometer offsets. There are three main reasons for the low signal-to-noise ratio for these measurements, which can be summarised as: (a) low signal; (b) high noise and; (c) optimisation technique. Further gains can be made by (d) increasing the angular resolution of the measurements and by (e) optimising the energy distribution of the direct determination of the energy. These are discussed in turn below. (A) Our use of a powder diffraction technique to determine the x-ray energy has required us to employ a small (300 µm) capillary. We have used the silicon powder standard SRM 640b as its mean lattice parameter is known to extremely high accuracy. However, small capillaries of low-z powders (including silicon) are not optimal for this experiment because of the small elastic cross-section at high energy, leading to the weak signals observed in Fig The use of a higher-z material will increase the diffracted intensity, possibly by a factor of 2 to 4. A further option is to use a single-crystal specimen, which can increase the diffracted intensity by several orders of magnitude. (B) Inelastic scattering is significant for low-z materials and increases with photon energy. The inelastic background, constituting noise in this experiment, is

294 266 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES particularly severe as the scattering angle increases towards δ = 90. Such photons have resulted in the significant background count rate shown in Fig. 10.2(d), and have restricted the angular range over which diffraction peaks could be measured. The effects of inelastic scatter can be reduced by using a higher-z material. Use of a single-crystal would further reduce the scattering contribution from defects and crystal boundaries present in the powder specimen. (C) The intensity of the x-rays diffracted into the scintillator was reduced to a level acceptable to the scintillator by locating a single thickness of absorber in the path of the beam. This absorber was fixed onto the δ arm of the diffractometer, and so all reflections have been measured using the same absorber. However, as the reflected intensity varies significantly as a function of energy and reflection index, a number of the reflections have been measured at a low count rate. Improved methods for adjusting the beam intensity have been developed for later experiments. The daisy-wheels have been used to select the optimum thickness of absorber for regulating the diffracted intensity. (D) The reflection profiles shown in Fig have been measured with significantly lower angular resolution than those analysed in chapters 7 and 9. This low angular resolution has limited the precision of the determined reflection locations. The selection of a fitting profile (Gaussian, Lorentzian) has therefore had negligible influence on the fitted positions. The precision can therefore be improved by measuring the profiles on a finer angular grid. In subsequent experiments we have made measurements using a finer angular grid. (E) A series of diffraction profiles has been measured to determine the x-ray energy on 46 separate occasions. Figure presents the measured angular locations of the reflections as a function of the nominal x-ray energy. On about 14 occasions, a series of six or more reflections has been measured in the positive angular range (δ > 0 ), with three or more reflections measured in the negative angular range. The wide angular range of the reflection profiles measured at these energies provides robust and precise information for the calibration of the monochromator encoder values. Accordingly, such measurements should be interspersed regularly throughout the energy range of the experiment with an increase in their measurement den-

295 10.6. DISCUSSION 267 Figure 10.15: Angular locations of the measurements, as a function of energy, showing that a four-peak energy determination was performed frequently. Future experiments using monolithic analyser crystals are able to measure the locations of reflections over a larger angular range and to provide a higher-quality determination of the x-ray energy less frequently throughout the experiment. sity near to the absorption edge, where a higher accuracy energy determination is required due to the high energy-gradient of the mass attenuation coefficient. On about 20 occasions only two reflections in the positive angular region have been measured, with only two reflections in the negative angular region. In the presence of the observed bimodal angular offset, this is the minimum number required to determine the energy and its uncertainty. On a further 10 or so occasions only two reflections have been measured. These measurements have not been used to determine the x-ray energy in the presence of the bimodal angular offset. Most of these limited measurements have been made at energies between 27 and 33 kev. The optimisation of the distribution of the energy determination measurements is a difficult and non-trivial problem which is created by limited beamtime and access, and is common for synchrotron investigations of all types. Many researchers have accepted the nominal energy and encoder accuracy as the definitive result for their data. We have shown this to be a dangerous practice which can lead to very large experimental error. In light of the bimodal angular offset, we no longer make measurements recording only a few reflections, preferring instead to perform measurements at fewer energies, but requiring that these measurements be of a high

296 268 Chapter 10. SILVER: DETERMINING THE PHOTON ENERGIES standard by recording many reflections. In addition to this modification, we also make more high-quality measurements at the extremes of the experimental energy range so as to establish the trend of determined energies quickly and accurately Conclusion The x-ray photon energies were determined by fitting the centroid positions, taking into account the effects of a (bimodal) misalignment of the zero angle of the diffractometer arm and further possible misalignments of the powder capillary with respect to the centre of the diffractometer axis. The x-ray photon energies have been determined to better than 0.1% at all energies and have been used to report values of the mass attenuation coefficients of silver [69. Figure shows the measured mass attenuation coefficients of silver as a function of photon energy. A detailed report of the determination of the mass attenuation coefficients is reported in [69. In that work, measurements made using various absorber thicknesses and apertures of various diameter have been used to detect the effect of fluorescent photons on the measured mass attenuation coefficients. The effect of the fluorescent photons was significant (in comparison to the Figure 10.16: Measured values of the mass attenuation coefficient of silver. The discussion presented here is only concerned with the determination of the x-axis of this plot. The FFAST tabulated values are also shown. From [69.

297 10.7. CONCLUSION 269 Figure 10.17: Percentage difference from the FFAST tabulated values [33, 34, 35 of previous experimental work [106, 107, XCOM tabulated values [96, 97, and this work [69. molybdenum and tin measurements reported in this thesis) because of the increased range of solid angles probed by the apertures. We have modelled the effect of this fluorescent contribution and have obtained very good agreement with measurement [122. This modelling has been used to apply a correction of up to 0.3% to the measured values. Structure in the thickness of the sample has been quantified by combining micrometer measurements with an x-ray scan made over the central region of the foil. These have been used to determine values of the mass attenuation coefficient with uncertainties between %. The determined values were found to be consistent with measurement uncertainty. Final measurement accuracies of the mass attenuation coefficients of silver range between 0.27 and 0.4%. Figure presents the percentage difference from the FFAST tabulated values [33, 34, 35 of the XCOM tabulated values [96, 97, previous measured values (compiled in [106, 107), and the results of this analysis [69. As with the corresponding plots for molybdenum (Fig. 8.22) and tin (Fig. 9.18), our data-set exhibits improved consistency across the entire measured energy range.

298

299 Chapter 11 Conclusions In chapter 2 we discussed the theoretical basis for the calculation of the atomic formfactor and showed that the various tabulations exhibited significant differences. We observed there the difficulty of assessing the merits of the different tabulations in the absence of reliable measurements. In chapter 3 we observed that earlier measurements show discrepancies indicative of the effects of undiagnosed systematic errors. In chapters 4 8 and 9 we have determined the mass attenuation coefficients of molybdenum and tin, and in chapter 10 we have determined the photon energies for measurements of silver. In this section we use our measured values to examine the relative merits of the various tabulations Molybdenum Figure 11.1 presents the percentage discrepancy between a variety of commonly-used tabulations of [ µ and our measured mass attenuation coefficients of molybdenum. pe On this plot our experimental values form the zero (reference) line, with our measurement uncertainties presented as error-bars about this line. The uncertainty in the subtracted Rayleigh plus Compton cross-sections is presented as a shaded region around the zero line, and is generally less than our experimental error-bars, except in the region immediately below the absorption edge. Also shown are the results of tabulations of the photo-electric absorption coefficients due to FFAST [33, 34, 35, XCOM [96, 97, 98, CXRO [94, 95, and Brennan and Cowan [101, 105. We concentrate this and following discussions on the FFAST and XCOM tabulations as these refer specifically to the full range of x-ray energies covered by our measurements. A striking feature of this plot is the large inconsistency of the XCOM tabulation in comparison to our measurements and to the other tabulations. The pattern of the XCOM differences indicates an oscillation of the calculated values in the above-edge 271

300 272 Chapter 11. CONCLUSIONS Figure 11.1: Percentage discrepancy between various tabulated values of [ µ pe for molybdenum and this work. We have determined the photo-electric absorption coefficient by subtracting the average of the calculated Rayleigh plus Compton scattering cross-sections of FFAST and XCOM from our measured values. Our measurements appear along the zero line, with error bars reflecting the experimental uncertainties. The narrow grey region around the zero line represents half of the difference between the Rayleigh plus Compton scattering cross-sections tabulated in XCOM and FFAST, and reflects the likely error in the absorption coefficient evaluated using these different models. Tabulated values are taken from FFAST [33, 34, 35, XCOM [96, 97, CXRO [94, 95, and Brennan and Cowan [101, 105.

301 11.1. MOLYBDENUM 273 region, and which possibly extends beyond the energy range of our measurements. Such oscillations have been observed elsewhere [34, and have been identified as the result of an incompletely converged calculation. Those appearing here in the XCOM values are likely to have resulted from a similar lack of convergence. Figure 2.1 reproduced the NIST-calculated comparison of the FFAST and XCOM tabulations. Those figures showed a significant difference between the tabulations in the above-edge region for molybdenum (Z = 42) which is also evident in Fig Our measurements resolve that difference firmly in favour of the FFAST tabulation. Immediately above the absorption edge all tabulations are lower than our measured values. As the x-ray energy increases from the absorption edge the tabulated values of FFAST, CXRO, and Brennan and Cowan each increase rapidly before reaching a stable level at about 4 7 kev or 20 30% above the molybdenum absorption-edge energy. In the below-edge region the various tabulations are approximately parallel to one another, indicating constant differences of about 3 5% between the tabulations. The trend of the measured values certainly does not agree with the trend of the tabulations in this energy region. This signature warrants a discussion of the interpretation of this plot. We have presented our measured values along the zero line so as to compare them equally against all tabulations. A limitation or failure common to all tabulations would be shown on this plot as structure common to all of the tabulations. However, a systematic error in the measured values could produce an identical signature. For example, the small noise-like structure appearing in the tabulations between 23 and 25 kev is a result of a variation of the measured values, and does not provide information about the tabulations. The different trend of the measurements relative to the tabulations is significant in the below-edge region. This divergence has not resulted from a systematic error in the measured values. Therefore this divergence may indicate a common limitation of the tabulations in the below-edge region. We discuss this point further after considering the measurements of tin and silver.

302 274 Chapter 11. CONCLUSIONS 11.2 Tin Figure 11.2 presents the differences between the FFAST and XCOM tabulated values and the preliminary results of our measurement of the mass attenuation coefficient of tin. The measured values form the zero (reference) line, with the uncertainties presented as error-bars about this line. Our measured values are preliminary, as they require null testing for a range of systematic effects. It is highly unlikely that these tests will result in a change to the structure depicted in Fig The discontinuities appearing in the tabulated values at 37 kev and at a few energies above 47 kev are artifacts resulting from discontinuities in the measured values with large σ s.e. uncertainties. They do not affect any structural conclusions. The XCOM values exhibit an oscillation about our measured values, again indicating incomplete convergence of the calculation. Further examination of Fig. 2.1 suggests that such ( 2%) oscillations are observed above the K edge of six elements out of those with Z > 20. Although we have measured two such elements Figure 11.2: Percentage discrepancy between the FFAST tabulated values and the preliminary results of measurements of the mass attenuation coefficients of tin. The measured values appear along the zero line, with error bars reflecting the experimental uncertainties. Although the absolute values of the attenuation coefficients may differ in the final analysis, the shape of the divergences presented here will be unaltered.

303 11.3. SILVER 275 in succession, we cannot assume that all XCOM tabulated values suffer from such convergence issues. The pattern of the FFAST differences is very similar to that presented in Fig for molybdenum. As before, the FFAST values increase rapidly from several percent below the measured values at the K-edge, and reach a stable level of difference at 6 8 kev or 20 30% above the K-edge energy Silver Figure 11.3 presents the percentage differences of the tabulated values from our measurements of the mass attenuation coefficient of silver [69. In this figure the measured data forms the zero line, with the error bars reflecting the experimental uncertainties. The tabulated values plotted here are again those due to FFAST and XCOM. Immediately above the K-shell absorption edge at about 25.5 kev the FFAST and XCOM tabulations are 4 6% below our measured values. For the FFAST tabulation, this difference decreases rapidly, reaching zero at about 5 kev or 20% above Figure 11.3: Percentage discrepancy between the FFAST tabulated values and the measurements of the mass attenuation coefficients of silver, reported in [69. The measured values appear along the zero line, with error bars reflecting the experimental uncertainties.

304 276 Chapter 11. CONCLUSIONS the absorption-edge energy. The XCOM values do not appear to converge to our measured values within the range of the measurements shown here. The XCOM values do not exhibit the oscillatory structure observed for molybdenum and tin. In the below-edge region the XCOM and FFAST tabulated values show similar trends although their values differ by about 2%. The trends of these differences are similar to those observed earlier for molybdenum, and may establish a pattern in this region. We shall examine this idea in more detail in the next section Informing form-factor calculations The differences presented in the previous sections have revealed patterns which we examine here. Figure 11.4 shows the differences between the FFAST tabulation from our measured values by again presenting the percentage difference from the measured values, but this time on an energy axis which is normalised to the K-shell absorption-edge energies. This plot clearly shows the consistent structure of the FFAST tabulated values in the above-edge and below-edge regions. Increased attenuation levels in the above-edge region have been observed previously. For example, Ref. [175 explicitly included an attenuation enhancement in the above-edge region when fitting measured mass attenuation coefficients to tabulated values. Creagh has observed that the attenuation enhancement in the above-edge region may be due to some new scattering mechanism [36, 37, 176, only tentatively speculating as to what this mechanism may be. Figure 11.4 clearly confirms this above-edge attenuation enhancement relative to the FFAST tabulation for our measurements of molybdenum, tin, and silver. The consistency of the below-edge divergence establishes a clear pattern of differences in this region. Such below-edge divergences are not expected to occur in the calculation as all approximations are assumed to be valid in this energy range. Figure 11.4 also shows that there may be a very small % oscillation in the FFAST values in the above-edge region. This oscillation may be restricted to the above-edge region or, alternately, it may be related to the resonance-like discrepancy at the absorption edge.

305 11.4. INFORMING FORM-FACTOR CALCULATIONS 277 Figure 11.5 shows the differences between the XCOM tabulation and our measurements of molybdenum, tin, and silver. The pattern of differences in the belowedge region is in qualitative agreement with the below-edge discrepancy observed for the FFAST tabulation. The instability of the XCOM calculation in the above-edge region resulting from poor convergence prevents the clear identification of similar trends in that region. The FFAST tabulation is in best agreement with our measurements. Following a 3 5% divergence of the FFAST values in the region of the absorption edge, the FFAST values are generally within 2% of our measured values. The consistent repetition of the differences between the FFAST tabulation and our measurements for the different elements suggests a common cause for the differences which may be identified and subsequently corrected. The pattern of the discrepancies appearing here is statistically significant and suggests that these measurements are indeed capable of informing form-factor calculations. Figure 11.4: Comparison of the difference between the FFAST tabulated values and our measurements for molybdenum, tin, and silver. We have scaled the energies to align the K-shell absorption edges. Our measurements form the zero line but are not presented here, for clarity. Uncertainties are not re-presented here for the same reason. The difference between the FFAST tabulations and our measurements is remarkably stable for the three elements. The stability of the FFAST tabulation inspires confidence in the calculation and enables the calculation to be critically examined to determine possible causes of the discrepancies.

306 278 Chapter 11. CONCLUSIONS Figure 11.5: Difference between the XCOM tabulation and our measurements of the mass attenuation coefficients of molybdenum, tin, and silver. We have scaled the energies to align the K-shell absorption edges. Our measurements form the zero line, but are not presented here for clarity. Uncertainties are not re-presented here for the same reason. The difference of the XCOM tabulations from our measurements agrees with our identified below-edge discrepancy. The XCOM tabulation is significantly less stable in the above-edge region, and this prevents the clear identification of the above-edge discrepancy which is so readily identified from the FFAST values. So far we have separated our discussions of the differences in the below and aboveedge regions. However, the discrepancies in these regions exhibit a symmetry which suggests a causal relationship. In particular, the percentage discrepancies from the FFAST values (Fig. 11.4) have the form of a derivative of a Lorentzian resonance curve. We discuss causes for such discrepancies arising from the implementation of the calculation and from physical processes not described within the form-factor formalism. We reproduce our earlier expression for the form-factor [Eq. (2.26) for convenience f = e { V e e i α. re dv ω e ( dg ) dω e 1 ( ω e ) 2 ω i κ e ω } dω. (11.1) Equation (11.1) describes the form factor as a sum over atomic electrons of the Fourier transform of the charge density multiplied by the resonant response of the electrons. The resonant contribution from each electron extends over wavelengths from the threshold energy ω e to infinity. The integrand is a product of a Lorentzian,

307 11.4. INFORMING FORM-FACTOR CALCULATIONS 279 whose width and height are determined by the parameters ω e and κ e, and the density of states. The parameters ω e, κ e, and ( ) dg determine the energy location and dω step height of the absorption edge. Early calculations of atomic form-factors used calculated values, but later calculations have tended towards the use of experimental values [84 to improve qualitative agreement with experiment. Use of an incorrect value of ω e or κ e would result in a pattern of discrepancies in the form of a Lorentzian derivative. However, as the integral in Eq. (11.1) only applies when the photon energy exceeds the threshold energy, such an error should not produce matching discrepancies below the threshold energy. Lorentzian tails can result in significant contributions to the evaluation of integrals. Such integrals are evaluated numerically by truncating integration at some point where the residual contributions are considered insignificant. Although the FFAST calculation has sought to apply consistent convergence criteria, it is possible that these integrals are truncated, giving a pattern of discrepancies as observed in Fig However, again, such an effect should not be observed on the low-energy side of the threshold location. The observed pattern of discrepancies could result from errors in the wavefunction, expressed in Eq. (11.1) by the charge density e, or in the density of states, ). Equation (11.1) is a sum over all electrons of the Fourier transform of the ( dg dω e charge density FT[ e multiplied by an infinite integral of the product of a step function, the density of states, and a Lorentzian, i.e., e f = e { FT[ e H(ω e ) ( ) dg dω e 1 1 ( ω e ) 2 ω i κ e ω } dω, (11.2) where H(ω e ) is the Heaviside-Lorentz step function 0 ω < ω e, H(ω e ) = 1 ω ω e. (11.3) Differentiation can be achieved by deconvolution with a step function. Accordingly, a parameter error or an incorrect integration range could give an error in f 1 which

308 280 Chapter 11. CONCLUSIONS is similar to a convolution term, and which may either differentiate or smooth the calculated structure. discrepancies observed in Fig Either of these processes could give rise to the pattern of The atomic energy levels and transition rules may be in error. These combine to determine the density of states term ( ) dg in Eq. (11.2). However, an error in dω the density of states term is expected to modify the calculated attenuation curve over an energy range which is equal to that of the lowest-energy available orbital, of order ten ev. Again, this mechanism does not produce the structure on the kev energy scale on both sides of the threshold energy observed in Fig The observed discrepancies could be due to physical processes not described within the form-factor formalism. e One example of such a process is hole-width broadening, where the energy width of the vacancy broadens the absorption spectrum by convolution with a Lorentzian of half-width equal to the hole width. The hole widths for molybdenum, silver and tin are 4, 6, and 8 ev, respectively [18. This width seems far too narrow to cause discrepancies on the kev-scale, and cannot be responsible for a similar effect below the threshold energy. However, it is difficult to estimate the likely effect of Lorentzian broadening processes X-ray absorption fine structure Structure in the attenuation coefficient arises from the interaction of an ejected photo-electron with the surrounding atoms. As this interaction is only strong for reasonably low photo-electron energies, the extent of the effect is usually limited to within 1 kev of the absorption edge. The field of XAFS analysis is primarily concerned with the interpretation of this structure to extract information about the local atomic environment. Attenuation measurements on a relative scale may not be sufficient to determine the structure of interest. Implicit in this analysis is the assumption that the desired structure is an oscillation of the mass attenuation coefficient about a mean level which would be observed if measurements were made using an isolated atom. The near-edge enhancements have not received attention from either the XAFS or the atomic communities.

309 11.6. CONCLUSION 281 The role of XAFS and XANES in the near-edge enhancement can be investigated using the finite difference methods of Joly [21. In particular, the accuracy of these methods can be tested by comparison with our measurements in the near-edge region, a sample of which is shown in Fig Recent studies indicate that these calculations may not yet be fully converged [70. The results of our absolute measurements, which provide robust uncertainty estimates, may be used to test the accuracy of XAFS analysis packages. There has been some concern that the number of free parameters used in XAFS analysis may adversely affect the validity of the recovered information [177. These questions can be answered by comparing the results of XAFS analyses using various parameter sets with our absolute measurements Conclusion We have developed the full-foil mapping technique which overcomes a significant limitation to the accuracy of the measurement of mass attenuation coefficients. This technique has been applied at one x-ray energy to determine the value of the mass Figure 11.6: Measured values of the mass attenuation coefficients of molybdenum in the region of the XAFS. These absolute values have properly-quantified uncertainties and can be used to test ab initio XAFS calculations and the robustness of XAFS analysis packages.

310 282 Chapter 11. CONCLUSIONS attenuation coefficient which is accurate to better than 0.02%, which is over one order of magnitude more accurate than any previous measurement. The full-foil technique has been reported (Ref. [24), and that article is included in Appendix D. We have examined measurements of the attenuation made over a wide range of the measurement parameter-space for deviations and discrepancies which may indicate the presence of systematic errors. We have developed a technique for identifying and correcting such errors, and have avoided the use of incorrect foil integrated column densities, have applied corrections for the effect of the bandwidth of the x-ray beam, and for a residual discrepancy due to a dark-current offset. These corrections have enabled us to determine the mass attenuation coefficient of molybdenum over the x-ray energy range from 13.5 kev to 41.5 kev to an accuracy of between 0.02% and 0.15%. Three articles have so far resulted from this investigation, reporting our determination of the bandwidth of the x-ray beam (Ref. [23), the full-foil mapping technique, and the measured values of the mass attenuation coefficients of molybdenum (Ref. [25). These are included in Appendix D. The preliminary results of a measurement of the mass attenuation coefficient of tin over the energy range from 29 kev to 60 kev are presented. It seems likely that this measurement will result in a determination accurate to between 0.1% and 1%. The results of an energy calibration for measurements of the mass attenuation coefficients of silver were presented. These measurements have been reported in Ref. [69. The thesis concludes with a detailed discussion of the significance of these measurements. We have focussed on the particular significance for the calculation of atomic form-factors, and have briefly discussed developments in the fields of XAFS analysis and ab initio structure prediction calculations. We have identified a significant pattern of discrepancies between our measured values and the FFAST tabulation which suggests that these measurements may indeed be capable of providing valuable information for form-factor calculations. There is a strong motivation to develop an improved theoretical understanding and framework to address the observed structural discrepancies between these experiments and available theory.

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323 BIBLIOGRAPHY 295 [166 S M Heald and E A Stern, Anisotropic absorption in layered compounds, Phys. Rev. B 16(12), (1977). [167 A I Frenkel, E A Stern, M Qian, and M Newville, Multiple-scattering x- ray-absorption fine-structure analysis and thermal expansion of alkali halides, Phys. Rev. B 48(17), (1993). [168 F B Hildebrand, Introduction to Numerical Analysis (Mc Graw Hill, New York, 1956). [169 A C Thompson and D Vaughan, editors, X-ray data booklet (Lawrence Berkeley National Laboratory, Berkeley, California, 2001), LBNL/PUB-490 Rev.2. [170 M A Beno, M Engbretson, G Jennings, G S Knapp, J Linton, C Kurtz, U Rütt, and P A Montano, BESSRC-CAT bending magnet beamline at the advanced photon source, Nucl. Instrum. and Meth. in Phys. Res. A: Accelerators, Spectrometers, Detectors and Associated Equipment 467-8, (2001). [171 R J Dejus and M Sanchez del Río, XOP: A graphical user interface for spectral calculations and x-ray optics utilities, Rev. Sci. Instrum. 67(9), (1996). [172 S Kraft, J Stümpel, P Becker, and U Kuetgens, High-resolution x-ray absorption spectroscopy with absolute energy calibration for the determination of absorption edge energies, Rev. Sci. Instrum. 67(3), (1996). [173 J C Lang, G Srajer, J Wang, and P L Lee, Performance of the advanced photon source 1-BM beamline optics, Rev. Sci. Instrum. 70(12), (1999). [174 W Parrish, A J C Wilson, and J I Langford, International tables for x-ray crystallography, Volume C: mathematical, physical and chemical tables (Kluwer academic publishers, Dordrecht, 1999), chap [175 P Dreier, P Rabe, W Malzfeldt, and W Niemann, Anomalous x-ray scattering factors calculated from experimental absorption spectra, J. Phys. C 17(17), (1984). [176 D C Creagh, The resolution of discrepancies in tables of photon attenuation coefficients, Nucl. Instrum. and Meth. in Phys. Res. A: Accelerators, Spectrometers, Detectors and Associated Equipment 255, 1 16 (1987). [177 L F Smale, C T Chantler, E C Cosgriff, M D de Jonge, Z Barnea, and C Q Tran, Failure of XAFS interpretation for ab initio investigations a new way forward, in 16th congress of the Australian Institute of Physics, Canberra, Australia, [178 J Topping, Errors of observation and their treatment (Chapman and Hall, London, 1972).

324 296 BIBLIOGRAPHY [179 M Takeda, H Ina, and S Kobayashi, Fourier-transform method of fringepattern analysis for computer based topography and interferometry, J. Opt. Soc. Am. 72(1), (1982). [180 F J Cuevas, J H Sossa-Azuela, and M Servin, A parametric method applied to phase recovery from a fringe-pattern based on a genetic algorithm, Opt. Commun. 203, (2002). [181 P H Chan, P J Bryanston-Cross, and S C Parker, Fringe-pattern analysis using a spatial phase-stepping method with automatic phase unwrapping, Meas. Sci. & Technol. 6, (1995). [182 J M Huntley and H O Saldner, Shape measurement by temporal phase unwrapping: comparison of unwrapping algorithims, Meas. Sci. & Technol. 8, (1997). [183 T Kreis, Digital holographic interference-phase measurement using the Fourier-transform method, J. Opt. Soc. Am. A 3(6), (1986). [184 S De Nicola and P Ferraro, Fourier-transform method of fringe analysis for moiré interferometry, J. Opt. A: Pure Appl. Opt. 2, (2000). [185 S De Nicola and P Ferraro, Fourier-transform calibration method for phase retrieval of carrier-coded fringe pattern, Opt. Commun. 151, (1998). [186 J A Quiroga, M Servin, and F Cuevas, Modulo 2π fringe orientation angle estimation by phase unwrapping with a regularized phase tracking algorithm, J. Opt. Soc. Am. A 19(8), (2002).

325 Appendix A The weighted mean The weighted mean of a pool of n values x i, i = 1, 2,... n with one-sigma uncertainties σ i is defined [178 x w = i i x i σ 2 i 1 σ 2 i. (A.1) Estimation of the uncertainty of the weighted mean depends on the consistency of the data in comparison to the individual uncertainties. When individual values are consistent, the sample variance is given by σ 2 s.e. = 1 (n 1) i 1. (A.2) σi 2 When the pool is inconsistent, a measure of the inconsistency is included by multiplying the variance by the population χ 2, σ 2 w = χ2 i 1 σ 2 i = ( xi x w ) 2 σ i i (n 1) i 1. (A.3) σi 2 Equation (A.3) needs to be used with caution. When the pool is small, accidental agreements can determine unreasonably small values for χ 2. When this occurs, the χ 2, whose purpose is to reflect the pool inconsistency, instead describes the consistency. In such cases Eq. (A.2) should be used. Accidental underestimation of the uncertainty is avoided by using the maximum of Eqs (A.2) and (A.3), which is appropriate for consistent (χ 2 = 1) or inconsistent (χ 2 > 1) pools. Alternately and equivalently, one may use Eq. (A.3) with the proviso χ

326

327 Appendix B Interferometric thickness measurement Section 6.4 described a full-foil mapping technique which can determine extremely accurate values of [ µ. The accuracy of that technique is in part due to the quantification of variations of the integrated column density across the foil. Some previous measurements have mapped the thickness profile using a combination of a limited-area x-ray mapping and micrometer measurements across the foil surface [13, 14, 117, 145. We have also attempted to determine a relatively-scaled but high-resolution profile of the absorber thicknesses by interferometric techniques. In this appendix we report the results of our investigation, and in particular document a novel technique for decoding low signal-to-noise, high fringe-density interferograms. B.1 Recording the Interferogram Interferometric surface profiles have been obtained using thin-film measurement techniques and a pair of optical flats. Each foil was placed between two high-quality quartz optical flats and compacted by applying a pressure of between tons / sq. inch. Pressing the foils reduced the air gap between the foil and the optical flats and flattened any burrs or ridges around the perimeter of the foil which may have resulted from their preparation. Immediately following removal from the press, the optical flat - foil - optical flat sandwich was held together by surface tension. To prevent the sandwich from detaching over time, two low spring force clamps were attached to opposing sides of the sandwich. The optical flats were flat to within 0.5 fringe across their (2 ) diameter when illuminated using 523 nm light. Interference fringes from the air gap between the surface of the foil and the optical flat could be observed by illuminating the optical flat - foil with monochromatic light. We initially attempted to use a helium-neon laser for this purpose. However, the extremely high temporal coherence of this laser source resulted in the observation 299

328 300 Appendix B. INTERFEROMETRIC THICKNESS MEASUREMENT Sodium vapour lamp TOP VIEW Optical flat foil optical flat, on rotating stage. 100 mm 100 mm CCD camera 600 mm Figure B.1: Schematic of apparatus and geometry used to record interferograms. of interference fringes from optical interfaces separated by large distances, such as the two surfaces of the optical flat. While these auxiliary fringes can be reduced by painstaking optimisation of the optical system, a simpler and sufficient technique involves the use of an illuminating source with reduced temporal coherence. In particular, we required a source with wavetrain lengths of the order of the air gap, but much shorter than the thickness of the optical flats. A sodium vapour lamp, with bichromatic emission at λ D1 = nm and λ D2 = nm, was chosen for this purpose. The bichromaticity of this source is inconsequential for this pilot study. Figure B.1 shows the configuration used to record interferograms. As indicated, the sodium vapour lamp and a CCD camera were located close to but on either side of the perpendicular bisector from the face of the optical flat and foil. Both surfaces of each foil was photographed with minimal disturbance to the air gap by rotating the stage upon which the sandwich was located. Figure B.2 shows an interferogram obtained in this manner, with interference fringes clearly visible. The image intensity is nonuniform, decreasing away from the foil centre. The image exhibits significant contamination, which may be the result of dust on the optical flats or on the foil. The signal-to-noise ratio of the interference fringes is clearly quite low. The image resolution of about pixels is barely sufficient to resolve fringes where their density is greatest.

329 B.2. INTERFEROMETRIC INFORMATION 301 Figure B.2: Air-gap interferogram taken using one of the molybdenum foils. The signal-to-noise ratio is well below 10%, with the noise due to foil roughness and dust in the optical system. B.2 Interferometric information The fringe distribution depends on the air gap between the optical flat and the foil, the curvature of the illuminating wavefront, the location of the camera and the illuminating lamp, and the wavelength of the illuminating light source. Although interferograms recorded using planar wavefronts and under normal incidence and reflection are more easily interpreted, the adopted geometry is sufficient for this pilot study. In the following we concentrate on the recovery of information from a general interferogram and therefore ignore details of geometrical and wavefront curvature effects. Subject to these simplifications, our interferometric problem can be stated briefly as the determination of the phase excursion φ(x, y) given an interferogram I cos 2 φ(x, y). This description of the interferometric problem neglects a variety of scale factors and offsets which are inherent to the recording of interferometric images. For our foils it is assumed that φ(x, y) varies smoothly and continuously over the foil surface.

330 302 Appendix B. INTERFEROMETRIC THICKNESS MEASUREMENT B.3 Decoding an interferogram A number of techniques of varying complexity and sophistication have been developed to solve the interferometric problem. The selection of the appropriate technique depends primarily on the quality of the interferogram. The parameters which best describe the quality of an interferogram are the signal-to-noise ratio and the image resolution relative to the fringe spacing. A high-quality interferogram has a high signal-to-noise ratio and image resolution well in excess of the minimum fringe spacing, so that there are many pixels per interference fringe. The phase excursion modulo 2π, φ 2π, can be obtained from cos 1 I after normalising the interferogram. When the interferogram is of high quality, φ can be determined from φ 2π by applying continuity conditions [179. This technique has been used to determine phase excursions of less than one fringe per image by using a tilted reference wavefront to produce a carrier signal and then demodulating this by use of the Fourier shift theorem [179. When the signal-to-noise ratio approaches unity it is not possible to use pointto-point continuity to unwrap φ 2π. In such cases it is necessary to use information over an extended region to unwrap the phase excursion. For example, Cuevas et al. have fitted a polynomial phase excursion to an interferogram [180. In their technique, a priori phase gradient and boundary restrictions are used to limit the available solution space of the fit. Many further techniques have been developed [181, 182, 183, 184, 185, 186, however we do not discuss these in this appendix. In contrast to the phase approach described above, we have developed a technique which examines only the density and orientation of the interference fringes. These quantities are suitable for the interpretation of low S/N interferograms, and are only defined with reference to a region of the interferogram: they are not defined at a point. The density and orientation of the interference contours are equivalent to the gradient of the phase excursion, φ. In our technique φ is determined from the interferogram. Once obtained, this numerical gradient can be used to determine the phase excursion.

331 B.4. THE FINITE-AREA FOURIER TRANSFORM TECHNIQUE 303 B.4 The finite-area Fourier transform technique High levels of noise destroy the pixel-to-pixel continuity of the interferogram. Figure B.3 shows a simulated interferogram with constant fringe spacing and S/N of 0.2. The fringes are clearly visible. By considering finite areas of the interferogram, it is possible to quantify the fringe spacing and direction. Our technique for determining φ uses the Fourier transform of selected regions of the interferogram. The Fourier transform of a constant fringing pattern produces a delta-function spike in reciprocal space whose location is determined by the direction and frequency of the fringes. The two-dimensional discrete Fourier transform of a function f(x, y) is defined by F (u, v) = 1 N 1 [ N 1 1 { 2πiux } { 2πivy } f(x, y) exp exp. (B.1) N N N N y=0 x=0 For a general harmonic f(x, y) with spatial wavelength λ = λ 2 x + λ 2 y pixels / fringe { x f(x, y) = exp 2πi( + y ) } { 2πix } { 2πiy } = exp exp, (B.2) λ x λ y λ x λ y Figure B.3: Simulated interferogram with constant fringe spacing. The noise level is Gaussian distributed with variance equal to 5 times that of the signal.

332 304 Appendix B. INTERFEROMETRIC THICKNESS MEASUREMENT so that F (u, v) = 1 N 1 { ( 1 exp 2πix u ) } 1 N λ x N x=0 N 1 N y=0 { ( 1 exp 2πiy v ) }. (B.3) λ y N The orthogonality of the Fourier basis functions implies that F (u, v) is non-zero when 1 λ x = u N and 1 λ y = v N. (B.4) Accordingly, there is a peak in the Fourier transform or reciprocal space at u = N λ x, v = N λ y. The fringe spacing and direction can therefore be determined by locating the peaks in reciprocal space. Figure B.4 shows the absolute value of the Fourier transform of Fig. B.3 (the absolute value is required because the Fourier transform is complex). The two peaks represent the positive and negative-travelling waves. We have used Eq. (B.4) to determine λ x and λ y, and from these the fringe spacing and direction. The results of this determination are shown on Fig. B.3 as a small black line whose direction and length indicate one wavelength of the fringes. Figure B.4: Absolute value of the Fourier transform of the simulated interferogram (Fig. B.3), showing two spikes whose positions determine the fringe density and direction. The small black line at the centre of Fig. B.3 is calculated from the location of these peaks, and indicates the gradient of the phase excursion φ (equivalent to the fringe spacing and direction) determined from the Fourier transform.

333 B.5. APPLICATION OF THE FOURIER TECHNIQUE 305 When the sampled region of the interferogram has a constant fringe gradient the Fourier transform contains a delta-function which reflects the spacing and direction of those fringes. Where the fringe gradient is not constant over the sampled region, the reciprocal-space peak spreads out, but does not change location. The centre of mass of the reciprocal-space peak determines the mean value of φ over the transformed region. Noise in the image produces a noise-like signal in reciprocal space, whose amplitude is generally less than the peak resulting from the interference fringes. Accordingly, the technique is robust in the presence of noise. The interferogram has been multiplied by a two-dimensional Hanning window prior to taking the Fourier transform to avoid high-frequency edge-truncation ringing effects in the Fourier spectrum. The Hanning window has the effect of broadening all features in the Fourier spectrum (by convolution with the Fourier transform of the Hanning window), but does not affect the peak location. The small oscillations in Fig. B.4 are due to the noise spectrum in the original (simulated) interferogram convolved with the image of the Hanning window. B.5 Application of the Fourier technique We have developed an algorithm to evaluate the Fourier transform of a small portion of the interferogram, and to determine the fringe direction and gradient from the location of peaks in the Fourier spectrum. This process is repeated with the region centered on every point within the interferogram. Figure B.5 presents again the interferogram presented in Fig. B.2, recorded using a molybdenum foil. The dark frame indicated in Fig. B.5 indicates the boundary of the points that have been treated in this manner. Points located outside of this frame cannot be treated in this manner as they do not have sufficient surrounding points. Below this interferogram we present the determined fringe angle at each point of the interferogram. The angles fall between 90 and +90, and are represented by a grey scale where black = -90, white = +90. The two stationary points within the

334 306 Appendix B. INTERFEROMETRIC THICKNESS MEASUREMENT image are clearly located in the angular plot. These stationary points correspond to a saddle point (right) and an extremum (left). In general the fringe angle is well determined by the technique. However, there are many regions where the angle is discontinuous. It is likely that these discontinuities are due to the influence of the dirt in the image, as these features produce a high amplitude peak in the Fourier spectrum. Figure B.5: Angular recovery from an interferogram obtained from a molybdenum foil. The top figure re-presents the interferogram shown in Fig. B.2, and indicates a frame within which angular information has been extracted. A region of pixels is used to determine φ for each point in the interferogram. Points located outside of the frame are not processed. The lower figure shows the recovered angular information: the shade indicates the fringe direction. Black: -90, grey: 0, white: 90. The fringe direction is in good correspondence with the interferogram, and the stationary points are well defined. Defects in the angular information have resulted from the dust-like contamination in the interferogram.

335 B.6. DISCUSSION AND CONCLUSION 307 B.6 Discussion and conclusion The Fourier mouse operates by taking a Fourier transform of a small region of the interferogram. The Fourier transform contains peaks whose magnitudes are determined by the amplitude of the interferometric fringes. The location of the peaks can be used to determine the direction and frequency of the fringes. We have outlined preliminary work towards the development of a technique to decode interferograms with low S/N ratios. The technique avoids the detrimental influence of noise and high fringe gradients by examining finite regions of the interferogram. A number of steps remain to be overcome before the technique can be widely applicable. Some investigations have been made using simulated (but noisy) interferograms, and these show greatly reduced incidence of inconsistencies in the determined φ. Once φ can be determined accurately, it is necessary to develop a method for determining φ from φ. It is apparent that a number of topological theorems will prove very useful for this purpose. The developed method appears to be robust in the presence of noise, but is not robust in the presence of dirt and dust. These problems have limited the application of the technique for the current purpose. Furthermore, an interferometric solution was no longer necessary once the full foil mapping technique was developed. However, the increased lateral resolution offered by interferometric techniques remains attractive.

336

337 Appendix C Experimental design modifications C.1 Foil holder The sample holder shown in Figs 6.4 and 6.7 was designed so as to minimise sample motion and to allow measurements to be made with the sample inclined over a wide range of angles with respect to the incident x-ray beam. In view of the full-foil mapping technique developed in this thesis, it is apparent that this robust sample holder design is not optimal for the application of this new technique. Accordingly, we present here a modified design for the sample holders. The improved holder design is shown in Fig. C.1. The sample is held in the holder by applying Crystalbond (R), a water-soluble, heat-softening wax, along both edges of the holder. The use of wax is a drawback of the new design as it prevents the sample from being mounted and remounted easily. This drawback has been overcome in later developments. The holder design shown in Fig. C.1 overlaps a significantly smaller fraction of the total sample area when compared to the previous design. Using the new holder design, approximately 10% of the sample area is overlapped by the holder, representing a dramatic reduction from the earlier design where 60 70% of the sample area was covered by the holder. Holders with this new design have been used for measurements at experiments performed in May, November, and December The results of these measurements have not yet been analysed sufficiently for the value of this development to be fully assessed. 309

338 310 Appendix C. EXPERIMENTAL DESIGN MODIFICATIONS FRONT VIEW 35 mm 26 mm 22 mm SIDE VIEW 10 mm mm 80 mm 60 mm MATERIAL : Sheet Perspex, unmachined, hand polished to thickness tolerance as discussed CRITICAL TOLERANCES: * thickness & width of holder (both to fit into bases) * height of holder * channel width * vertical symmetry 10 mm mm 3 mm mm END VIEW 1 mm mm 2 mm mm Figure C.1: New sample holder design used in a number of recent experiments. C.2 Modified design for daisy-wheels: wedges We report a modification to our procedure for determining the harmonic content of the beam and the linearity of the detection system. Measurements of the attenuation of the foils mounted on the daisy-wheels can be used to determine the effective fraction of harmonic components in the beam, to check the quality of the determined dark currents, and to investigate the linearity of the ion chambers [119. Furthermore, apertures mounted on the perimeter of the daisy-wheel have been used to determine the influence of secondary photons on the measured attenuations. In order that the multiple-foil measurement be properly orthogonal to the scattering measurements, the measured foils should be equidistant from the upstream and downstream ion chambers so that the amount of secondary photons reaching the two detectors is approximately equal and stable. This restriction would imply that the multiple foils be located in the plane of the absorbing samples.

339 C.2. MODIFIED DESIGN FOR DAISY-WHEELS: WEDGES 311 The three-aperture measurement is properly optimised by locating the apertures at a distance from the samples so that they subtend a significant but small range of solid angle at the sample location. This range of solid angles can be well defined by locating the ion-chambers directly behind the apertures. Our use of the daisy wheels for the multiple-foil and the multiple-aperture measurements results in a conflict of the optimisation of the daisy-wheel location. Furthermore, the collocation of the multiple foils and the multiple apertures necessarily excludes the simultaneous use of the multiple foils and the multiple apertures. We have observed that the relative inaccuracy of the knowledge of the local thickness of the multiple-foils places an additional constraint on the accuracy of the multiple-foil measurement. An earlier multiple-foil measurement by this group used multiples of a single thickness of aluminium foil, and was able to determine the relative thicknesses of each of the compound foils to high accuracy [119. We have developed an alternative to the daisy-wheels which in principle overcomes these limitations and objections. We employ a small wedge-shaped absorber which is mounted onto the sample stage. Measurements recorded along the length of the wedge probe different thicknesses of absorber, the thickness increasing linearly as the wedge is inserted into the path of the beam. Due to the linearity of the wedge, the relative thickness at each measurement location is well determined. The location of the wedge on the sample stage allows the measurements to be made in conjunction with any selection of apertures. The use of a wedge-shaped absorber presents an absorber with a linear thickness variation to the x-ray beam. The attenuation through a wedge-shaped absorber can be calculated exactly from [117 I { = exp I 0 [ µ [t }sinh ( [ µ t ) [ µ t, (C.1) where t is the change of the thickness of the absorber over the region illuminated by the beam. As the linearity of the thickness variation is well known from the geometry of the wedge, its effect can be corrected to determine a flat-foil attenuation at each location along the wedge.

340 312 Appendix C. EXPERIMENTAL DESIGN MODIFICATIONS Figure C.2: Modelled attenuation measurements taken along the length of two wedge-shaped absorbers. The modelled measurements are represented by their error bars. The lines are the result of fitting the modelled values with a function describing the attenuation of a harmonically-contaminated beam passing through the wedge-shaped absorber with a possible dark-current offset. The fitted values agree within uncertainty with the modelled values. We have modelled the performance of the wedges to inform the selection of the wedge materials and their geometrical parameters. The modelling includes calculation of the attenuation of a 1 mm 1 mm x-ray beam with a fractional harmonic content passing through a wedge-shaped absorber with measurements recorded every 1 mm along the length of the wedge. A random uncertainty of σ = 0.05 mm has been added to the measurement location to determine whether such a measurement inaccuracy is likely to invalidate the measurement. An error in the determined dark-current is modelled. We show in Fig. C.2 the results of this simulation, and the extraction of the dark-current error and the beam harmonic content from the simulated measurements. These simulated results have been used to optimise the wedge design. By way of these simulated results we have found that this design can be expected to detect harmonic components comprising as little as 0.05% of the incident beam intensity. Figures C.3 and C.4 present the optimised design for the wedges, constructed from 7075 aircraft grade aluminium and 316 stainless steel respectively, and the wedge holder, use to mount the wedges onto the sample stage. These have been

341 C.2. MODIFIED DESIGN FOR DAISY-WHEELS: WEDGES ALUMINIUM WEDGE 316 STAINLESS STEEL WEDGE SIDE VIEW FRONT VIEW SIDE VIEW FRONT VIEW * scale drawing 2 * scale drawing M2-thread M2-thread aluminium wedge. dimensions: front view: 8.0 +/- 0.5 mm /- 0.5 mm side view: thin end 0.5 +/- 0.1 mm thick end /- 0.5 mm angle /- 0.4 degrees construct 3*required width (ie ~24 mm wide) wedge surfaces to be ground & then contact martin (x 45575) who will lap them down to required thickness and smoothness after which he will return them for narrowing back to 8 mm (on milling machine) stainless steel wedge. dimensions: front view: 8.0 +/- 0.5 mm /- 0.5 mm side view: thin end 0.0 +/- 0.1 mm thick end 2.0 +/- 0.1 mm angle / degrees wedge to zero by gluing job to similar steel support prior to final cut. construct 3*required width (ie ~24 mm wide) wedge surfaces to be ground & then contact martin (x 45575) who will lap them down to required thickness and smoothness after which he will return them for narrowing back to 8 mm (on milling machine) and de-mounting from support material Figure C.3: Design of the 7075 aluminium wedge and the 316 (stainless) steel wedge.

342 314 Appendix C. EXPERIMENTAL DESIGN MODIFICATIONS M2 throughholes to mount 45.0 wedges mm ledge with sharp corner to locate wedges FRONT SIDE material: 7075 aluminium Countersunk through-holes to match holder base 2* scale drawing 20.0 TOP Figure C.4: Design of a wedge holder to enable the wedge to be mounted on the sample stage. built and employed in experiments performed in May and November of 2004 at beamline 11-ID of the APS. Figure C.5 presents preliminary results arising from measurements along the length of the wedges using an x-ray beam of energy 28 kev. These measurements indicate that the steel wedge has not been manufactured to required tolerances, and that its thickness does not increase linearly along its length. This has resulted from difficulties associated with the manufacture of such a thin wedge; the steel wedge tapers from 2 mm thick down to zero thickness. The wedge was constructed by gluing the steel to a large steel support whose faces were tapered at the angle required for the wedge. The wedge and support were ground parallel to the base of the support until the wedge was observed to taper to zero thickness. However, difficulties were encountered with the glue which came unstuck a number of times during the grinding process. Future attempts to construct the wedge may involve the use of wire-cutting apparatus which would reduce the stresses applied to the wedge and would therefore result in the production of a cleaner and more uniform wedge.

343 C.2. MODIFIED DESIGN FOR DAISY-WHEELS: WEDGES 315 Figure C.5: Attenuation measurements made along the length of the two wedges using a beam of 28 kev x-rays. The measurements represented by the small plus markers (+) were made using the aluminium wedge and show excellent linearity across the entire range of attenuation, from 0.5 < [ µ [t < 6. The measurements represented by the filled circles ( ) show an interesting deviation from linearity. Comparison of the attenuations measured below [ µ [t = 6 for the two wedges, where the aluminium wedge records excellent linearity, shows that the wedge itself is not linear. This non-linearity is associated with difficulties encountered during the manufacture of the thin stainless steel wedge.

344

345 Appendix D Resulting publications This appendix presents a number of publications resulting directly from the work presented in this thesis and for which I have contributed a majority of the content. 317

346

347 PHYSICAL REVIEW A 71, Measurement of the x-ray mass attenuation coefficient and determination of the imaginary component of the atomic form factor of molybdenum over the keV energy range Martin D. de Jonge, Chanh Q. Tran, Christopher T. Chantler, Zwi Barnea, and Bipin B. Dhal School of Physics, University of Melbourne, Melbourne, Victoria 3010, Australia David J. Cookson* Chem-Mat-CARS, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA Wah-Keat Lee and Ali Mashayekhi XOR 1-ID, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA Received 19 July 2004; published 4 March 2005 We use the x-ray extended-range technique XERT Chantler et al., Phys. Rev. A 64, to measure the mass attenuation coefficients of molybdenum in the x-ray energy range of kev to % accuracy. Measurements made over an extended range of the measurement parameter space are critically examined to identify, quantify, and correct where necessary a number of experimental systematic errors. These results represent the most extensive experimental data set for molybdenum and include absolute mass attenuation coefficients in the regions of the x-ray absorption fine structure XAFS and x-ray-absorption near-edge structure XANES. The imaginary component of the atomic form-factor f 2 is derived from the photoelectric absorption after subtracting calculated Rayleigh and Compton scattering cross sections from the total attenuation. Comparison of the result with tabulations of calculated photoelectric absorption coefficients indicates that differences of 1 15 % persist between the calculated and observed values. DOI: /PhysRevA PACS number s : Cy, Ht, Fb, Ci I. INTRODUCTION The attenuation of x rays by materials provides a wide variety of information about the fundamental properties of matter in the atomic, molecular, and solid states. In particular, relative and absolute measurements of the mass attenuation coefficient are used to test theoretical predictions of photoelectric absorption using bound-state electron wave functions 1,2, to investigate the dynamics of atomic processes, including shake-up, shake-off, and Auger transitions 3 6, and to provide information on the density of electronic states 7, molecular bonding, and other solid-state properties 8. The diversity of these studies is evidence of the wide variety of processes that influence the attenuation of x rays. In order to develop a deeper understanding of the interactions between x rays and matter it is necessary to make accurate measurements, so that each attendant process may be studied and compared with theoretical models. While relative measurements are adequate for some applications, absolute attenuation measurements provide additional, crucial, and demanding tests of theoretical predictions. For example, while finite-difference calculations 9 have recently had significant success in predicting extended x-ray absorption fine structure EXAFS on a relative scale, they are in relatively poor agreement with the results of absolute measurements 10. Measurement inaccuracy and discrepancies between theoretical calculations seriously impede the understanding of x-ray interactions with matter. *Also at ANSTO, Private Mail Bag 1, Menai, New South Wales 2234, Australia. X-ray atomic form factors are calculated by using atomic theory, quantum mechanics, and quantum electrodynamics to describe the scattering of x rays using calculated atomic wave functions. Major differences in the calculated values of form factors result from the various theoretical frameworks that are employed for calculating these atomic wave functions, each of which treats exchange, correlation, and overlap effects in a different manner. Further differences stem from the diverse application of approximate methods employed to describe multielectron atomic wave functions. We present in Fig. 1 a comparison between the results of two commonly used tabulations of mass attenuation coefficients for molybdenum. These results have been derived directly from form-factor calculations with small Rayleigh and Compton scattering cross sections added. The ordinate of this plot is the percentage difference from the FFAST tabulation This figure shows the large differences that can occur when alternate methodologies are applied, with the differences rising to around 17% in the region above the absorption edge at around 20 kev. The differences are stable at around 3 4 % in the energy region below the absorption edge and there is reasonable agreement at the higher energies shown in this figure. The presence of regions of moderate agreement and large differences suggests that the approach and the implementation of the calculation may have significant and varying consequences for the predicted values in different energy regions. These models and their implementations can be tested by comparing tabulated and measured values. In Fig. 1 we have included the results of a number of measurements of the mass attenuation coefficient of molybdenum 14,15. We im /2005/71 3 / /$ The American Physical Society

348 320 de JONGE et al. FIG. 1. Discrepancies between theoretical predictions and experimental measurements of the mass attenuation coefficient of molybdenum presented as a percentage difference from the FFAST tabulation The XCOM tabulation is from 16,17. The various previously measured values have been sourced from the compilation of Hubbell et al. 14,15. Multiple values arising from individual experiments are marked with the same symbol, establishing the trend of each set of measurements. The % variation between the measured values whose typical claimed uncertainties are around 2% indicates the presence of unquantified systematic errors affecting these measurements. mediately see from this plot that the measurements do not commend either tabulation. These reported measurements typically claim accuracies of %, sufficient to decide between the theoretical values. However, despite these claimed accuracies, the different sets of measurements differ by up to 20%. In order to discriminate between the different tabulated values, measurements are required to be both accurate and precise to better than about 1% below the absorption edge, about 4% immediately above the absorption edge, and possibly 0.2% far above the absorption edge. The discrepancies between the theories, between different experiments, and between theory and experiment have prompted the International Union of Crystallography, representing one of the world s largest group of users of formfactor data, to undertake a systematic investigation of formfactor-based calculations of mass attenuation coefficients and their measurement 18,19. A primary conclusion of their survey of measurement techniques was that a variety of poorly understood and unquantified sources of systematic error may be adversely affecting the measurements. The x-ray extended-range technique XERT 1,2 employs measurements made over an extended range of the measurement parameter space to probe systematic errors affecting the measurement. The specific extended ranges of the measurement parameter space investigated were the attenuation / t of the absorbers, the x-ray energy, the angular acceptance of the detectors, the angle of the absorbing sample relative to the incident x-ray beam, and the variations in integrated column density of the absorbing foil. These parameter-space explorations sought the optimal measurement configuration but were deliberately extended outside the optimal regimes to determine the effect of systematic errors on the measurement. PHYSICAL REVIEW A 71, In this article we report measurements of the mass attenuation coefficients of molybdenum. The results of an extensive investigation of systematic errors affecting the measurement are presented. The mass attenuation coefficients are determined to an accuracy of 0.028% away from the K absorption edge and 0.1% in the vicinity of the K absorption edge. The precision of the measurements is % at over 500 energies between 13.5 and 41.5 kev. This article is divided into eight sections. In Sec. II we describe the attenuating samples and the experimental setup. Section III describes the detailed interpretation of the measurements leading to the determined mass attenuation coefficients. In Sec. IV we report the method by which we determine the energy of the x rays. We provide a tabulation of the results in Sec. V, and quantify contributions to the accuracy and the precision of the results. In Sec. VI we compare our results with a variety of tabulations of the photoelectric absorption coefficients and find that the currently available tabulations differ significantly from our measured values. Section VII is a summary of our conclusions. We have relegated to the Appendix further details of the interpretation of the measurements leading to the mass attenuation coefficients. II. EXPERIMENTAL DETAILS A. Samples The molybdenum foil samples were of various thicknesses between 25 and 250 m and were all approximately mm 2 in area as supplied by ESPI 20. The quoted purity of all foils was 99.98%. A typical assay provided by the manufacturer listed the impurities as iron 52 ppm, potassium 40 ppm, chromium 32 ppm, nickel 25 ppm, and copper 16 ppm 21. The effect of these impurities on the measured mass attenuation coefficient was estimated by use of the tabulated values of their mass attenuation and found to be less than 0.01% for all measurements in the range of energies between 13.5 and 41.5 kev. Each foil was weighed to determine its mass m using a microgram-accuracy Mettler microbalance which was buoyancy compensated for a mass of density =8.4 g/cm 3. The residual effect of the buoyancy of the molybdenum samples nominal density =10.2 g/cm 3 is to alter the apparent mass by around %, and this effect was not corrected as it is far below the measurement uncertainty. Each foil had its projected facial area A measured with a Mitutogo PJ300 traveling-stage shadow-projection optical comparator. The mass and area of each foil was used to determine its average integrated column density t from t =m/a. We have measured the surface roughness of a number of the foils used in this measurement using an atomic force microscope AFM. The AFM measurements determine rms roughnesses t of nm over scan areas typically of the order of m 2. The effect of these measured roughnesses on the measured attenuation was evaluated from 22,23 = t t ln 1+ / 2, 1 2! resulting in a correction of less than 0.004% for the foils used in this experiment. The effect of thickness variations

349 MEASUREMENT OF THE X-RAY MASS ATTENUATION 321 PHYSICAL REVIEW A 71, FIG. 2. Schematic of the experimental layout. over longer length scales will be investigated in a later section. B. Experimental components The x-ray beam was produced by an undulator insertion device at the 1-ID XOR beamline of the Advanced Photon Source facility at the Argonne National Laboratory 24. The 3,1,1 planes of a silicon double-reflection monochromator were used to select a narrow range of energies from the undulator spectrum. The x-ray energy range covered by this investigation, from 13.5 to 41.5 kev, includes the K-shell absorption edge of molybdenum at around 20 kev, and extends over a wide range of energies above and below the edge. The energy range was limited primarily by the operational characteristics of the synchrotron beamline facility. The energy spacing of the measurements was varied in accordance with the structure in the mass attenuation coefficient of molybdenum: it was kept down to 0.5 ev within 100 ev of the absorption edge, and was increased to 500 ev at energies far from the absorption edge. The fifth-order component of the undulator spectrum was selected to provide x rays with energies between 41.5 and 25 kev and the third-order component for x-ray energies below 25 kev. To reduce the passage of harmonic components into the beam the second crystal in the monochromator was detuned slightly from its position parallel to the first crystal such that the beam intensity decreased to between 35% and 55% of its peak, undetuned value 25,26. After monochromation the x-ray beam traveled approximately 30 m down an evacuated pipe into the experimental hutch see Fig. 2. On entry to the hutch the x rays passed first through a beryllium window and then through a pair of orthogonal adjustable slits which defined the beam cross section to be approximately 1 1 mm 2. The x-ray beam then passed through the first of three 95-mm-long, argon gas ion chambers. The ion chambers were of identical construction, and argon gas flowed through the detectors in series at a rate of around 1 l/ min. Two downstream ion chambers were employed to improve the counting statistics, to investigate the ion chamber and electronic nonlinearities, and to provide a cross-check of the measured attenuated beam intensity. The molybdenum samples were clamped between two Perspex holders which could slot neatly into a stainless steel base to provided wobble-free location of the sample. Five samples at a time were mounted on the stage, shown in Fig. 2, which was located midway between the upstream and the first of the downstream ion chambers. The sample thicknesses were chosen such that at each energy they typically spanned a range of attenuation / t The stage could be rotated about two axes and translated in two directions orthogonal to the beam. The samples could thus be placed and replaced in the path of the beam to high precision by the use of a computer-controlled motorized driving system. The estimated reproducibility of the translation was of order 10 m and the rotational reproducibility was of order 0.1. Counter normalization was determined by recording the count rates in the detectors with the samples translated out of the path of the beam. The attenuated and unattenuated intensities were measured in rapid succession at each energy by an automated movement routine. Daisy wheels 27 were located between the sample stage and the ion chambers. These had on their perimeters three apertures subtending solid angles of 8.7, 33, and 150 sr at the sample which were used to admit different amounts of secondary photons into the ion chambers. In addition to these apertures, 30 attenuating foils were mounted on the perimeter of the daisy wheels and these too could be placed in the path of the beam by suitable rotation of the daisy wheel. The thicknesses of these foils were chosen to span approximately three orders of magnitude in the x-ray attenuation / t. III. DETERMINING THE MASS ATTENUATION COEFFICIENT A. Intensity measurements Counts were recorded simultaneously in the upstream u and downstream d ion chambers with a sample s interposed into the x-ray beam recording intensities I d,s,i u,s, without a sample in the x-ray beam I d,b,i u,b, b for blank, and with the x-ray beam shutter closed I d,d,i u,d, d for dark. Each measurement of 0.1 s counting time was repeated ten times to yield a measure of the reproducibility of the measurement and to enable proper treatment of correlations in the counting chain 28,

350 322 de JONGE et al. Dark current measurements of the apparent count rate recorded in the absence of the x-ray beam were made regularly throughout the experiment to account for amplifier offsets. The trend of the dark current count rates was linearly interpolated within regions where the ion-chamber electronics settings were unchanged to account for any variation of this offset due, for instance, to electronic drifts. The error attributed to the dark-current count rates was one standard deviation of the results about the trend of the measured values. Dark-current counts were typically of the order of 17± 1 over a 0.1 s counting interval. The upstream ion chamber was used to monitor the beam intensity and to normalize the downstream readings, thus enabling the separation of the synchrotron beam intensity fluctuations from other noise components. The normalized count rates for the blank and sample measurements were obtained from the ratio of the counts recorded simultaneously in the upstream and downstream ion chambers after subtraction of their dark currents. The counts recorded by each of the downstream ion chambers were processed separately at this and every successive stage of the calculation. The normalized intensities were determined from the mean of the ratios of ten successive measurements, I x = I d,x I d,d 2 I u,x I u,d, and their uncertainties were determined from Ix = var I d,x I d,d I u,x I u,d + + I d,x I d,d Iu,d I u,x I u,d I d,x I d,d I u,x I u,d Iu,d 2 1/2 I u,x I u,d 2 I u,x I u,d, 3 where the subscript x denotes the use of blank b or sample s measurements to determine the unattenuated and attenuated normalized intensities I b and I s and their uncertainties Ib and Is, respectively. Id,d and Iu,d are the uncertainties in the dark currents determined in the downstream and upstream detectors, respectively. As discussed elsewhere 28,29, the variance of the measurements is appropriate for the high correlation coefficient R of 0.99 between the measurements recorded with the upstream and downstream ion chambers. The additional terms in Eq. 3 are the contributions to the uncertainty in the normalized intensities arising from the corresponding uncertainties in the measured dark currents. The attenuation / t is evaluated for measurements with each sample at each energy using t = ln I s I b =ln I d,b I d,d I u,b I u,d ln I d,s I d,d I u,s I u,d, with the uncertainty in the attenuation determined by / t = 2 I s I s + 2 1/2 I b I b. 5 4 PHYSICAL REVIEW A 71, FIG. 3. Measured attenuations ln I s /I b = / t. The markers represent results obtained using foils of the following nominal thicknesses:, 25 m;, 50 m;, 100 m;, 150 m;, 200 m;, 250 m. A subset of the foils was measured at each energy. The absorbers span a wide range of attenuations at each measured energy, allowing attenuation-dependent systematic errors to be detected. We present in Fig. 3 the attenuations calculated by use of Eqs The results of the calculations using the counts recorded in the two downstream ion chambers and those with apertures of various diameters placed between the absorber and the ion chambers are plotted on this figure but cannot be resolved except where the statistical precision of the measurement is poor, particularly when the foil attenuation rises above about 5 6. In Fig. 4 we show the percentage uncertainties for the attenuations presented in Fig. 3. This figure shows that, as expected, a higher level of uncertainty is associated with measurements where the foil attenuation differs markedly from the optimal Nordfors range of 2 / t The uncertainties presented here are in broad agreement with the statistical limit of the precision given by the Nordfors criterion. The discontinuities in the uncertainties are due to replacement of one sample with another and with adjustments made to the ion-chamber electronics settings at 41, 35, 30, 25, 21.8, and 20.8 kev. These adjustments change the noise level associated with the ion chambers, but the continuity of the measured attenuations Fig. 3 shows that the normalization procedure prevents these adjustments from having any significant impact on the measured attenuations. B. A full-foil absolute measurement of the mass attenuation coefficient In this section we summarize our use of a full-foil x-ray mapping technique to determine the mass attenuation coefficient of the thickest foil at the highest available energy to high accuracy 31. We determine an attenuation profile of the foil plus holder / t xy F+H by performing a raster measurement of the attenuation at x,y locations across the entire foil mounted in the holder. We determine the relatively small holder contribution to the attenuation profile by use of a fitting routine. The determined holder contribution was less

351 MEASUREMENT OF THE X-RAY MASS ATTENUATION FIG. 4. Percentage uncertainties in the measured attenuations following Eq. 5. Note that despite a consistent approach to the data acquisition, some measurements have statistical uncertainties exceeding 1%, due to the foil thickness. However, in each case other foils of more optimum thickness measured at the same energies have uncertainties of %, so that the final results are not compromised by the poor data. The results of measurements obtained using the two downstream ionization chambers with variously sized apertures placed between the absorber and the ionization chamber, that cannot be resolved in Fig. 3, are clearly resolved here. Foil markers as for Fig. 3. The points marked by the large diamonds at 41.5 kev unresolved are discussed in Sec. III B. than 5% of the foil attenuation. Subtraction of the fitted holder profile from the total measured profile then produces an attenuation profile of the foil, / t xy F. The average of the attenuation profile measured at a number of x,y locations across the entire foil is related to the average integrated column density t of the foil by t xy = = t m A, 6 so that t xy F = = A m t xy F, 7 yielding a precise and absolute value of the mass attenuation coefficient. By this method we determine / to be ± cm 2 /g and ± cm 2 /g, using the counts recorded in the first and second of the downstream ion chambers. These results are consistent within their associated measurement and fitting uncertainties. The absolute results determined by this technique are represented in the figures by the large diamond markers. Further details of the technique are discussed in 31. We did not use the full-foil x-ray mapping technique to determine the mass attenuation coefficient at each energy as this would have been too time consuming. C. Scaling other mass attenuation coefficients If the other measured attenuations are divided by the average integrated column density t =m/a of the relevant foil, this results in systematic differences of up to 2% between the mass attenuation coefficients measured at the same energy with different foils. Our work with full-foil measurements indicates that a difference of this magnitude could easily result from foil nonuniformities which this averaging procedure neglects 10,23,31. We have the beam passing through the same point of the foil for all measurements for which that foil was used, so that the integrated column density of the foil is common for all measurements made with a given foil. The measurements were therefore scaled by varying the integrated column density of each of the foils according to = / t t T, 8 where t T is the trial value of the integrated column density of the foil. We used a fitting routine to vary t T until the difference between the mass attenuation coefficients was minimized. The minimized parameter is 2 = E i 323 PHYSICAL REVIEW A 71, F j / EiFj / Ei / Ei F j 2, 9 which is defined by analogy with the 2 measure of deviations. The summations in Eq. 9 cover measurements obtained at all energies E i using all measured foils F j. The term in the parentheses is the difference between the scaled value obtained using the foil F j at energy E i, / Ei F, and j the weighted mean of the scaled values obtained at energy E i, / Ei, divided by the measurement uncertainty / Ei F j. The full-foil absolute value is included in the evaluation of 2 but, of course, not varied. Minimization of 2 optimizes the weighted agreement between the results determined using each of the foils and the absolute value at the full-foil mapping energy and also the weighted agreement of the relative measurements. The scaling is based on an iterative least-squares minimization of the difference between the results. The physical removal and replacement of a foil in the sample stage generally results in the presentation of a slightly different part of the foil to the beam. We estimate that this replacement shifts the location of the beam footprint on the foil by less than around 300 m. This shift has the effect that the integrated column density of the local region of the foil may be significantly different for the two placements. In the scaling procedure nine integrated column densities relating to seven foils have therefore been varied to minimize the discrepancies between 5161 measurements at 527 energies. The mass attenuation coefficients are evaluated as the weighted mean of the measurements obtained with all foil and aperture combinations at each energy. We examine the consistency of the scaled values in Fig. 5 where we present the percentage difference of the measurements for each foil from the weighted mean at each energy. The unresolved large diamond markers represent the results of the absolute measurements described in Sec. III B. In this figure we can see a number of prominent divergences from the zero line

352 324 de JONGE et al. PHYSICAL REVIEW A 71, FIG. 5. Percentage difference of the measured values from the weighted mean at each energy after scaling. A number of divergences have become apparent. These divergences correlate with increasing thickness of the foil used to make the measurement. The markers represent results obtained using foils of the following nominal thicknesses:, 25 m;, 50 m;, 100 m;, 150 m;, 200 m;, 250 m;, full-foil measurement. These include two inconsistencies, at 30 and 35 kev, which are readily distinguished from systematic trends by virtue of their transience, and a prominent complex of deviations occurring around the absorption edge at 20 kev. Four further systematic divergences can be seen, where the measured value obtained using one of the foils diverges systematically below the zero line. These divergences fall to around 4% below the zero line at 25 kev marker, 2 4 % below at 20 kev marker, 2% below at 15 kev marker, and 0.8% below at 13.5 kev marker. Comparison with Fig. 3 shows that these divergences correlate with rising foil attenuations. The onset of the divergence typically occurs when the foil attenuation increases above 4 5. We have assessed the accuracy of the scaling procedure by comparing the fitted integrated column densities against their measured average values determined in Sec. II A. Assuming that the variation of the integrated column density across the foil is random, the fitted and average integrated column densities should, on average, be in agreement. On average the fitted integrated column densities are 1.2 standard deviations below the measured average values, where the standard deviation is evaluated as the quadrature sum of the fitting and measurement uncertainties. As the spread in the differences between the fitted and measured integrated column densities is 1.6 standard deviations, these results are consistent within the observed variation. A quantitative measure of the improvement in the consistency of the measured values is also gained from the reduction of the reduced r 2 2 per degree of freedom. Without scaling using t the measurements obtained using different foils differed by up to 2% due to local thickness variations yielding a large r 2 of 114. After scaling this r 2 is dramatically reduced to 3.94, reflecting the high degree of consistency of the scaled values across all of the parameter space but especially across the extended energy range. The statistical significance of the discrepancies presented in Fig. 5 can be appreciated by comparing the magnitudes of FIG. 6. Significance of deviations from weighted mean, after scaling. Significance is defined in Eq. 10. Symbols as for Fig. 3. Significant outliers are seen in the near-edge region, implying the existence of uncorrected systematics such as the bandwidth effect see Appendix A 1. the discrepancies with the measurement uncertainties. In Fig. 6 we present the significance of the deviations from the weighted mean, defined as / Ei F j / Ei significance =. 10 / Ei F j As can be seen from a comparison with Eq. 9, the significance describes contributions to r 2. In Fig. 6 we can see that the significance of the four regions of large divergence shown in Fig. 5 is generally very low, reflecting the low statistical precision associated with these measurements. Thus, while the divergences represent real systematic deviations in the results, they fall within the experimental uncertainty and do not exert significant influence on the weighted mean of the measurements. The persistence of the prominent complex of discrepancies about the absorption edge in Fig. 5 indicates their statistical significance. D. Other systematic effects Numerous other key systematics must be dealt with in this paper, in order to achieve the accuracies claimed below. In several cases these required techniques and approaches not elsewhere discussed. We therefore present these in the Appendix in order to avoid distracting attention from the major conclusions, but allowing the reader to follow these details as they see fit. IV. CALIBRATION OF X-RAY PHOTON ENERGIES The photon energy was directly determined by diffraction from a germanium crystal mounted on a Huber four-circle diffractometer as depicted in Fig. 2. With the attenuating samples out of the x-ray beam, foils mounted on the daisy wheel were introduced into the beam path to decrease the intensity of the x-ray beam used to measure the rocking curves

353 MEASUREMENT OF THE X-RAY MASS ATTENUATION 325 PHYSICAL REVIEW A 71, E = hc h 2 + k 2 + l 2 2a 0 1+ a0 sin +, 11 FIG. 7. Results of the energy calibration process. A series of hhh peaks was measured at a number of energies. These were used to determine the x-ray energy, represented here by the error bars. These energies were fitted to the monochromator crystal angle by Eq. 11, and this fitted function was used to interpolate the x-ray energy from the monochromator angle for all measurements. The results of this fitting process and the interpolation are shown here by the line of best fit heavy, black and the uncertainty light, gray, estimated from the covariant error matrix returned by the fitting program. Rocking curves were recorded with the x-ray intensity reflected into a stationary sodium iodide scintillation detector whose face was centered on the predicted Bragg angle of the reflection. The detector used was wide open, with no further angular selection applied to the reflected beam. The germanium crystal was rotated through a small range of angles about the Bragg angle to record the rocking curve of the reflection. Between three and thirteen such rocking curves were recorded at each directly measured energy, diffracted by lattice planes of the form hhh with h ranging from 1 up to 17. The angular locations of these rocking curves were determined by fitting with a Lorentzian and also by determining their centers of mass. Two independent techniques for determining the angular locations were employed to avoid the effects of the saturation of the detector used to measure the diffracted intensities. The largest single source of systematic error in the energy determined in this manner is due to the misalignment of the zero-angle position of the germanium crystal. We have corrected for this source of error using an adaptation of a standard technique 32. Extrapolation of a plot of a 0 sin hkl / h 2 +k 2 +l 2 versus a 0 cos hkl / h 2 +k 2 +l 2 to the limit cos hkl =0 allows one to determine the energy of the beam from the sin hkl intercept, as well as the magnitude of the zero-angle misalignment of the germanium crystal. The lattice parameter of germanium was taken to be a 0 = Å 33. The determined energies, depicted as points with error bars in Fig. 7, were used to calibrate the x-ray energy across the entire measurement range. This was achieved by fitting a modified Bragg function which related the monochromator angle to the directly determined energies. The fitting function used was which follows directly from Bragg s law, with a small adjustment to the monochromator lattice parameter via the parameter a0, allowing for an expansion of the crystal due to the x-ray heat load; and an offset angle of the monochromator crystal, which allows for mechanical slack in the crystal rotation stage and errors in the crystal alignment. We have used values for hc and the lattice parameter for silicon taken from Ref. 34. Diffraction was from the hkl = 3,1,1 planes of the silicon monochromator. The monochromator angles were fitted separately over two energy ranges corresponding to the change from the fifth to the third undulator harmonic at about 25 kev, possibly resulting in a change in the value of the lattice parameter a0 due to the different heat load. In Fig. 7 we show the results of this process, where the abscissa is the nominal synchrotron x-ray energy and the ordinate is the difference between the calibrated and nominal energies. The error bars represent the directly determined energies and the solid lines are the best fits to these energies, determined using Eq. 11. The gray lines above and below the fitted energies are the error estimates evaluated from the covariant error matrix returned from the fitting procedure. By this procedure the x-ray energies have been determined to a precision of between % and 0.007% across the entire measurement range. The directly determined energies are generally consistent with the smoothly interpolated fit with a few points indicating a possible additional small variation of the beam energy not correlated with the monochromator angle. Accordingly we use the smoothly interpolated values. The accuracy of the energy determination can be assessed by comparing the absorption edge energy against its most accurate literature value. The first point of inflection of the mass attenuation coefficient on the absorption edge occurs at ±0.0002± kev, where the first uncertainty reflects our ability to locate the position of the point of inflection and the second is our uncertainty in determining the energy. Comparison with the result of Ref. 35, ± kev, indicates a discrepancy of 6 ev or 0.03%. The most likely cause of this discrepancy is a difference in the interpretation of the absorption-edge location, chemical or thermal effects on the edge location, or further errors in the energy determination. We consider an upper limit on the accuracy of our determined energies to be half of the difference between these absorption-edge locations at about 0.015%. V. TABULATION OF THE RESULTS In Table I we present the values of the mass attenuation coefficients measured at 101 energies between 13.5 and 41.5 kev. A further 425 measurements made at energies between and kev are not detailed here due to space limitations. The complete tabulation of measured values is available electronically 42. In the first column we present the calibrated photon energy in kev with the uncertainty in the last significant figures presented in parentheses

354 326 de JONGE et al. PHYSICAL REVIEW A 71, TABLE I. Mass attenuation coefficients / and the imaginary component of the atomic form factor f 2 as a function of x-ray energy, with one standard deviation uncertainties in the least significant digits indicated in parentheses. We present also the percentage uncertainty in the mass attenuation coefficients / / /. Uncertainty in f 2 includes the measurement uncertainty and the difference between major tabulations of the total Rayleigh plus Compton scattering cross sections. Values of f 2 in the energy range of kev are likely to be affected by solid-state and atomic effects. A further uncertainty, of the same order as the XAFS amplitude, may apply to these values when alternate atomic environments are investigated. The complete tabulation of measured values is available electronically 42. Energy kev / cm 2 /g / / / f 2 e/atom Energy kev / cm 2 /g / / / f 2 e/atom % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

355 MEASUREMENT OF THE X-RAY MASS ATTENUATION 327 PHYSICAL REVIEW A 71, TABLE I. Continued. Energy kev / cm 2 /g / / / f 2 e/atom Energy kev / cm 2 /g / / / f 2 e/atom % % % % % % % % % % % % % % % In the second column we present the mass attenuation coefficient / in cm 2 /g with its uncertainty. In the third column we present as an aid to the reader the percentage uncertainty in the mass attenuation coefficient. The values in the second and third columns have been determined from the weighted mean of the measurements made with a variety of apertures and foil thicknesses, and using the values determined from the counts recorded in both of the downstream ion chambers. The weighted mean typically involves between 18 and 30 determinations. The uncertainty in the mass attenuation coefficient was evaluated from sd defined in Eq. A3. The measured mass attenuation coefficients are plotted as a function of energy in Figs. 8 and 9. In the fourth column of Table I we present the imaginary component of the atomic form factor f 2, evaluated from f 2 = EuA / pe 2hcr e, 12 where E is the photon energy in ev, u is the atomic mass unit, A is the relative atomic mass of molybdenum, h is the Planck constant, c is the speed of light, r e is the classical electron radius, and / pe is the photoelectric component of the attenuation. / pe has been evaluated by subtracting the average of the Rayleigh plus Compton contributions as tabulated in XCOM 16,17 and FFAST In parentheses following the reported values is the uncertainty in f 2, evaluated from f2 = EuA 2 2hcr / e + 2 RC 1/2, 13 which includes an uncertainty contribution of half of the difference RC between the tabulated values of the Rayleigh plus Compton contributions. The use of the photoelectric component of the attenuation determined in this manner is appropriate when Rayleigh and Compton scattering are the only significant other contributions to the total attenuation. This is certainly the case in the energy range covered by this experiment except near the absorption edge and in the region of the XAFS. In these regions the influence of solid-state and bonding effects is difficult to calculate. It may well be that values of f 2 in the energy range from 19.9 to 20.9 kev should be subject to a further uncertainty, of the same order as the XAFS amplitude, when alternate atomic environments are investigated. Estimates of the individual error contributions to the reported values are presented in Table II. FIG. 8. Energy dependence of the measured mass attenuation coefficients. FIG. 9. Detail of the measured mass attenuation coefficients in the region of the XAFS. Marker size corresponds approximately to the measurement uncertainty

356 328 de JONGE et al. PHYSICAL REVIEW A 71, TABLE II. Error contributions to the values reported in Table I, with source specified. Further established limits for the systematic uncertainty are quoted here. VI. COMPARISON WITH TABULATED VALUES OF THE PHOTOELECTRIC ABSORPTION COEFFICIENT / pe The mass attenuation coefficient can be written as a sum of photoelectric absorption / pe, Rayleigh scattering / R, and Compton scattering / C according to + C pe R We do not include further attenuating processes in this summation as they are negligible in the energy region of this experiment. The results of atomic form-factor calculations can be assessed by comparing the calculated photoelectric absorption coefficients with our measured values. Note that we have not directly measured the Rayleigh and Compton crosssections, but instead estimate the Rayleigh plus Compton cross section to be equal to the average of the values reported by the FFAST and XCOM tabulations. We estimate the uncertainty in the Rayleigh plus Compton cross section to be half of the difference between these tabulations. We have subtracted these scattering components from the measured values to determine photoelectric absorption coefficients. In Fig. 10 we present the percentage discrepancy between a variety of commonly used tabulations of / pe and our results. Our experimental results form the zero reference line, with the measurement uncertainties presented as error bars about this zero line. The uncertainty in the subtracted Rayleigh plus Compton cross-sections is presented as a shaded region around the zero line. Except in the region immediately below the absorption edge the uncertainty in the subtracted Rayleigh plus Compton cross-sections is generally less than our experimental error-bars. Figure 10 shows that the XCOM calculation exhibits a large difference from the measured values over an extended range of energies above the absorption edge. There is some evidence of an oscillatory behavior in the XCOM values, possibly extending beyond the measured energy range. Oscillatory behavior in the calculated values has been observed elsewhere 11,12 and may be the result of an incompletely converged calculation. The FFAST tabulation is in best agreement with the measurements. The difference between the various calculations and our results in the below-edge region are remarkably similar in form, even though they differ by 4 5 % in the absolute level of the photoelectric absorption coefficient. The similarity of these differences may imply a common limitation of the calculations in this region. At the point immediately below the absorption edge the FFAST calculation is in best agreement with our photoelectric absorption coefficient. Previous highly accurate measurements for copper 1 and silver 39 have reported a similar difference between the measured values and the FFAST tabulation in the region immediately above the absorption edge, extending out to approximately 25% of the absorption-edge energy. Over this region the measured values typically decrease from being 3 5 % higher than the FFAST values to around the level of the FFAST values. This difference is again observed in the results for molybdenum, as can be seen in Fig. 10. The presence of this effect in three elements indicates either a systematic problem with the FFAST formalism or the presence of an unrecognized contribution to the measured attenuation in this region. A further systematic difference between the measured values and the FFAST tabulation, similar to that observed in silver 39, is observed below the absorption edge. In the case of silver the FFAST tabulation is approximately 2.5% below the measured values at energies well below the absorption edge. This difference begins to decrease at about 5 kev below the absorption edge, converging to the measured value immediately below the absorption edge. Such a pattern of systematic differences is also observed for molybdenum and may provide further insight into the limitations of the FFAST formalism. VII. CONCLUSION We have determined the mass attenuation coefficients of molybdenum on an absolute scale. The measurements are

357 MEASUREMENT OF THE X-RAY MASS ATTENUATION 329 PHYSICAL REVIEW A 71, results of this work. The FFAST tabulation is in best agreement with our set of measurements, with discrepancies of about 1% far above the absorption edge and up to 4% near and below the absorption edge. Systematic differences between the FFAST calculation and the results of a number of recent experiments are confirmed for molybdenum. The systematic nature of these differences indicates that the FFAST calculation needs to be refined in certain regions, in particular immediately below and above the K-shell absorption edge. These discrepancies may indicate new physics, particularly in the above-edge region. Absolute measurements in the near-edge region will be of interest in solid-state and bonding studies, and in particular for those wishing to compute XAFS and XANES on an absolute scale. ACKNOWLEDGMENTS We wish to acknowledge the assistance of our collaborators at the Advanced Photon Source including David Paterson and the staff of SRI-CAT and BESSRC-CAT. This work was supported by the Australian Synchrotron Research Program, which is funded by the Commonwealth of Australia under the Major National Research Facilities Program, and by a number of grants of the Australian Research Council. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Basic Energy Sciences, Office of Energy Research, under Contract No. W Eng-38. FIG. 10. Percentage discrepancy between various tabulated values of / pe and this work. We have determined / pe by subtracting the average of the calculated Rayleigh plus Compton scattering cross sections of FFAST and XCOM from our measured values. The results of this work appear along the zero line, with error bars reflecting the experimental uncertainties. The narrow gray region around the zero line represents half of the difference between the Rayleigh plus Compton scattering cross sections tabulated in XCOM and FFAST, and reflects the likely error in the absorption coefficient evaluated using these different models. Tabulated values are taken from FFAST 11 13, XCOM 16,17, Henke et al. 36, and Brennan and Cowan 37,38. placed on an absolute scale by comparison with the results of a full-foil mapping procedure which has been used to determine the mass attenuation coefficient at a single energy. Measurements have been made over an extended range of the measurement parameter space. The values obtained from this extended investigation have been examined for the effect of systematic errors on the measurement. We have corrected a systematic error in the measured values arising from the effect of the bandwidth of the x-ray beam used to make the measurement whose effect is particularly pronounced along the rise of the absorption edge. A small residual deviation in the measurements is consistent with an incorrectly determined dark current, and corrected as such. The measurements are compared with a variety of predictions of the photoelectric absorption coefficients. Some of the available tabulations are in very poor agreement with the APPENDIX: DETERMINING THE MASS ATTENUATION COEFFICIENT: OTHER SYSTEMATIC EFFECTS 1. The x-ray bandwidth We have shown elsewhere that the discrepancies between measurements on the absorption edge are correlated with the gradient of the mass attenuation coefficient, and that this correlation is due to the energy bandwidth of the x-ray beam 40. The bandwidth effect arises from the energy dependence of the mass attenuation coefficient, so that the different spectral components of the x-ray beam are attenuated to varying degrees by the absorber. As the beam penetrates the absorber this differential attenuation modifies the beam energy profile such that the intensities of the lesser attenuated components gradually increase relative to those of the more attenuated components. This modification of the beam energy profile leads to the systematic decrease of the measured mass attenuation coefficient with increasing thickness of the attenuating foil. In 40 a particularly sensitive subset of measurements around the absorption edge was used to determine the bandwidth of the synchrotron beam to be 1.57±0.03 ev at 20 kev. Away from absorption edges the mass attenuation coefficient varies sufficiently slowly for the bandwidth effect to be insignificant. However, on the absorption edge and in the region of the XAFS, where the mass attenuation coefficient changes rapidly as a function of energy, the effect of the bandwidth is significant. This effect has obviously contributed to the discrepancies at around 20 kev in Figs. 5 and

358 330 de JONGE et al. PHYSICAL REVIEW A 71, FIG. 11. Mass attenuation coefficients in the neighborhood of the absorption edge and the XAFS. The values on the absorption edge can be linearized in order to correct for the effect of the bandwidth of the x-ray beam. Points are marked by their se error bars. When the change in the mass attenuation coefficient over the scale of the bandwidth of the x-ray beam is approximately linear, its effect can be corrected by following 40 = + 1 t ln j k= j exp d / E 0 k E t, de Ĩ 0 E 0 + k E j = r W s FWHM 1, E = W FWHM 2 s FWHM 1, A1a A1b FIG. 12. Correction to the mass attenuation coefficients measured in the neighborhood of the absorption edge and in the region of the XAFS, evaluated using the linearized approximation for the bandwidth. The correction in the XAFS region is everywhere less than 0.03%, indicating that the correction resulting from this approximation is at the level of the experimental uncertainties. Symbols as for Fig. 3. where Ĩ 0 is the incident beam energy profile, representing the distribution of energies within the beam about the central energy E 0. d / E0 /de is the energy derivative of the weighted mean of the mass attenuation coefficients at the central energy. The summation is evaluated over a range of energies corresponding to r W times the full width at half maximum FWHM of the bandwidth and is discretely sampled s FWHM times per FWHM bandwidth. Here we use r W =10 and s FWHM =9 40. In Fig. 11 we present the results of measurements made along the absorption edge and within the first few XAFS oscillations. Along the rise of the absorption edge the gradient of the mass attenuation coefficient reaches its maximum value and the bandwidth effect is greatest. Furthermore, along this rise the mass attenuation coefficient is approximately linear on the scale of the bandwidth of the beam 1.57 ev 40. However, the mass attenuation coefficient is not linear on the scale of the bandwidth of the beam at around kev, where the mass attenuation coefficient reaches its first and most strongly curved maximum. We have included a correction for the bandwidth effect Eq. A1 in the fitting routine. We now fit the FWHM bandwidth of the beam W FWHM with appropriate allowance for the energy dependence of the bandwidth, and scale the nine integrated column densities as before to minimize the 2 differences between the measurements at each energy. In Fig. 12 we present the results of this fitting in the form of the correction applied to the mass attenuation coefficients measured on the absorption edge. From this figure we see that the correction for the bandwidth effect is correlated with the gradient of the mass attenuation coefficient and increases with the thickness of the foil used to make the measurement. The baseline value for each of the foils differs slightly from zero due to a further small adjustment to the scale of the integrated column densities. The linearization adopted in 40 is valid when the variation of the mass attenuation coefficient on the scale of the bandwidth of the beam is approximately linear. Alternate Fourier deconvolution techniques for correcting the bandwidth effect encounter other difficulties preventing their simple application, as discussed in 40. We have also compared the linearized and the Fourier-deconvolution techniques in the region of the XAFS and have found that the linearized approach is quite adequate for addressing the effect of the bandwidth on these measurements. We estimate the effect of the bandwidth in regions where the linearized approach may fail, i.e., at the extrema of the XAFS. We observe that the effect of the bandwidth at these extrema is less than twice that predicted by the linearized model due to the two-sided nature of the extremum when the gradient is taken to be the maximum gradient within the energy span of the beam. We take this energy span to be equal to twice the FWHM bandwidth of the beam, thereby including 95% of the bandwidth. Accordingly, we estimate the upper bound for the bandwidth effect at the first maximum to be twice that calculated at ± kev. At these energies the applied linearized correction is less than around 0.03% from Fig. 12, and so an upper bound of 0.06% is established. The error arising from the use of the linearized approximation is significantly less than this upper bound for the measurements made around the more weakly curved extrema in the XAFS region. The error arising from

359 MEASUREMENT OF THE X-RAY MASS ATTENUATION 331 PHYSICAL REVIEW A 71, FIG. 13. The percentage discrepancies of individual measurements from the weighted mean after correction for the effect of the bandwidth. Only those measurements falling within a small range of the zero line are shown. The line indicates the sd uncertainty in the weighted mean determined from Eq. A3. the use of the linearized approximation is generally well below the measurement uncertainty in the XAFS region. Allowance for the bandwidth effect results in a reduction of the r 2 from r 2 =3.94 to This reduction is quite significant considering that only a small fraction of the data are corrected by more than 0.03%. 2. Residual discrepancies and their treatment The percentage discrepancies between the measurements after correction for the bandwidth effect are presented in Fig. 13. Here we show only the results within a small range of the zero line since the discrepancies are very similar to those presented in Fig. 5. Included on this plot is a line marking the one-standard-deviation uncertainty in the weighted mean, sd, determined as the weighted uncertainty multiplied by the 2 of the population by use of = F j sd = / Ei / Ei F j / Ei 2 2 / Ei F j F j 1 2 / Ei F j 1/2, A2 A3 where the summation is over all measurements F j at each energy, and / Ei is the weighted mean of the measurements at each energy. This sd properly quantifies our uncertainty in the weighted mean for measurements that are not necessarily consistent within their individual uncertainties. The sd uncertainty is typically below about 0.07%. At a number of energies between 25 and 30 kev the uncertainty associated with the measurement rises due to instabilities in the apparatus and adjustments of the settings made during the course of the experiment. The complex of discrepancies occurring near the absorption edge is still present in this plot and has resulted in a slight increase in the sd uncertainty in that region. FIG. 14. Percentage difference between the measured mass attenuation coefficients and the FFAST tabulation. By comparing the measured values with a smooth and near-lying result we are able to examine closely the trend of the measured values. This plot demonstrates that the trend of the measured values is continuous to within their estimated uncertainties. The minimization of the differences between measurements made at each energy does not alter the relationship between measurements made at different energies. In order to examine closely the smoothness of the measurements we calculate the percentage difference between the measured values and a near-lying and smooth function. In Fig. 14 we have used for this purpose the interpolated results of the FFAST calculation. The measured values fall on a continuous and smooth curve to within the determined uncertainty. The correspondence between the measurement variation and the uncertainties confirms the procedure used to estimate the uncertainties. We have tested the measurement for further systematic errors. These tests have employed a number of approaches including i statistical analysis t test of differences between subsets of the measurement population; ii comparison of the predictions of the correction equation for example, Eq. A1 against the observed discrepancies Figs. 5 and 13 ; iii fitting the results to test for the presence of trial systematic errors; and iv interpretation of on-line diagnostic measurements. In the following subsections we will briefly describe these investigations and their findings. a. Secondary photons There is no significant difference between the results obtained with different aperture sizes. This is in agreement with modeling of scattering processes and aperture sizes. The apertures used probe one-tenth of the range of solid angles reported in 41. We therefore predict the effect to be significantly smaller than that observed in 41, and estimate a maximum correction of %, which is below the sensitivity of our measurements. b. Beam harmonic content Daisy wheels were used to make measurements using a large number of molybdenum and aluminum absorbers with

360 332 de JONGE et al. attenuations typically covering the range 0.05 / t 30. These attenuations have been analyzed following Ref. 27, yielding a maximum effective beam harmonic content of less than about a 10 4 fraction of harmonic photons at all energies measured in this experiment. The beam harmonic content was also tested by attempting to fit the mean beam harmonic fraction to the results of the attenuation measurements. Here we used a correction equation similar to that in 27, but refined to include the energy dependence of the ion-chamber efficiencies and the air absorption. This fitting determined an average harmonic component that was consistent with zero, with the sensitivity of the diagnostic at around 1 in 10 4 photons. These diagnostics indicate that the detuning of the monochromator successfully suppressed the harmonic components and their effect on the result was insignificant. c. The effect of roughness The effect of roughness in the integrated column density 23,31 was checked by use of the fitting procedure. In this case the fitted parameters were the rms deviations of the integrated column density t for each of the foils. The effect was found to be consistent with zero, confirming the absence of a systematic error due to sample roughness and voids within the sample volume. PHYSICAL REVIEW A 71, FIG. 15. The percentage discrepancies of measurements from the weighted mean, after correction for the effect of the residual discrepancy. The four large systematic divergences seen in Fig. 5 are now centered about the zero line. The reported uncertainty is the uncertainty of the weighted mean added in quadrature to onequarter of the present correction. d. Linearity of the detection system In order for an attenuation measurement to be accurate, the detectors and counting chain must be linear over the range of the measured intensities. It is difficult to guarantee the linearity of the ion chambers and the detection chain to the level of precision indicated by Fig. 13, of order 0.07%, without performing explicit tests. A number of effects can lead to nonlinearities in the detector and counting chain. It is beyond the scope of this article to discuss all physical mechanisms responsible for detector nonlinearity. Instead we consider the possibility of a nonlinear response in the current amplifiers and the counting chain used in this experiment. One consequence of the energy dependence of the ionchamber detection efficiencies is that we have had to manually adjust the electronic gain settings on the current amplifiers. When these gain settings were altered, usually by a factor of 2 or 5, any effect of nonlinearities in the current amplification and counter scaling might be observed as a discontinuity in the measured attenuations. However, such discontinuities are not present in the measured data presented in Fig. 14, and thus such effects are insignificant in this experiment. Nonlinearities resulting from the molecular dynamics within the detector volume can be investigated by examining discrepancies between the results of measurements using very different incident beam intensities. In this experiment the incident x-ray intensity decayed to around half of its initial value over the course of 12 h, at which time electrons were injected into the synchrotron ring to return the beam intensity to its initial value. We have used the pre- and postinjection period as an opportunity to test the effect of intensity variations on the measurements of the mass attenuation coefficient. We found no significant discontinuities in the results measured immediately before and after the beam injections and conclude that the response of the counting chain is linear to the level of the claimed uncertainties in the relevant high-count region. e. Residual discrepancies A residual systematic deviation of the measurements made with the thickest foil is clearly present in Fig. 5. The low precision of the measurements contributing to the dominant signature, results in the low significance of this residual, as seen from Fig. 6. These residual deviations can be due to a residual harmonic component in the synchrotron beam, a poorly determined dark current, or sample roughness. The extremely low r 2 attached to this residual signature gives rise to a similar pattern of discrepancy for each source of error. We have corrected the systematic deviations present in Fig. 5 by treating the residual deviations as if they were the result of an incorrectly determined dark current in the downstream ion chambers. This dark-current error is corrected according to = + 1 dc, t I d,s A4 where the fitted parameter dc describes the offset to the measured dark-current value in the downstream detectors. In this fitting a single value of dc is used for each period in which the ion-chamber electronics settings remained unaltered. In Fig. 15 we present the percentage discrepancies of the measured mass attenuation coefficients after fitting for this signature. The agreement between the measurements across a wide range of energies is qualitatively improved, indicating that this correction is of the correct form and magnitude. The change to the weighted mean arising from this correction is

361 MEASUREMENT OF THE X-RAY MASS ATTENUATION less than the original uncertainty for all measurements except those between kev. To account for the model uncertainty and the small remnant signature of the deviations we have added a further uncertainty to these measurements equal to one-quarter of the magnitude of the correction. 333 PHYSICAL REVIEW A 71, The rejection of corrupted measurements In a highly automated large experimental run it is almost inevitable to end up with some procedural errors. In response to a number of experimental variables such as the changing detector efficiencies and sample attenuations, and the energy dependence of the undulator spectrum, the apparatus and the experimental settings were adjusted and optimized a number of times during the five days in which measurements were made. The results have been examined closely to determine situations in which the measurements exhibited systematic deviations and low accuracy. Where such deviations are present in the data we have investigated the measurements to determine the cause of their inconsistency. Where possible we have isolated the affected data and discarded them from the measurement. In this section we will briefly detail inconsistencies observed in the data and their treatment. The time taken for the beamline shutter to open and close resulted in the recording of a number of 0.1 s measurements before the shutter had completed its motion. The incomplete motion of the shutter, which was located upstream of the monitor counter and was evidenced by the monitor count, typically affected only the first two measurements in each series of ten. Around 2300 out of measurements were affected by the incomplete shutter motion and were rejected from the data set. Measurements recorded immediately after the daisy wheel was rotated to the position of the smallest aperture were affected by a small vibration in the daisy wheel, which led to some clipping of the x-ray beam. This resulted in a large variation in the measured downstream intensity which, unlike the shutter problem, was uncorrelated with the monitor count. The high correlation coefficient R 0.99 between the measurements recorded with the upstream and downstream ion chambers enabled us to easily identify and reject the affected measurements. In the energy range between and 20.9 kev measurements were made with a higher than appropriate counter amplification setting in the downstream ion chambers. The effect of this was to cause the amplifier output voltage on some occasions to exceed the scaler s input range, leading to what we shall refer to as counter saturation. This counter saturation is akin to reaching the full-scale deflection of a measuring instrument, and does not necessarily imply a loss of linearity of nonsaturated measurements. As it turned out, the counters were just barely saturated and only some of the measurements were affected. Counter saturation in downstream ion chambers affected only the high-intensity measurements, i.e., the unattenuated normalized intensities I b, made with the sample removed from the beam. The removal of the saturated measurements from the data set was again facilitated by the high correlation of the counts recorded in the upstream and downstream ion chambers. The rejection of some measurements from each series of ten results in a greater uncertainty for I b ; however, there is no significant decrease in the overall precision of the counter-saturated measurements as the precision of I b turned out not to limit the experimental precision. The measurements made at energies between 20.9 and kev were also affected by counter saturation. However, here the measurements recorded in both of the downstream ion chambers were fully saturated and could not be recovered. A significant proportion of the results in the region of the XAFS have been determined from measurements using only a single foil. As a result, these results do not necessarily share the same baseline as the results derived from the weighted mean of many measurements. Correction of the measurement discontinuities in the XAFS region is particularly appropriate for subsequent XAFS partial-wave analysis. We treated these single-foil measurements as being on a relative scale and placed them on an absolute scale by comparing them with neighboring weighted-mean values. These data have been rescaled by an amount that is less than the uncertainty in the measurements, so any analysis correctly propagating input uncertainties will be unaffected by this procedure. Measurements recorded with a single foil whose attenuation was / t 8 in the XAFS region have been omitted because the high attenuation of this foil resulted in a low precision of these values of about 1 3 %. This has resulted in the gaps in the measurement seen in Fig C. T. Chantler et al., Phys. Rev. A 64, C. Q. Tran, C. T. Chantler, Z. Barnea, D. Paterson, and D. J. Cookson, Phys. Rev. A 67, M. S. Freedman and F. T. Porter, Phys. Rev. A 6, D. W. Lindle et al., Phys. Rev. A 38, P. Weightman, E. D. Roberts, and C. E. Johnson, J. Phys. C 8, M. O. Krause and J. H. Oliver, J. Phys. Chem. Ref. Data 8, Y. Joly, D. Cabaret, H. Renevier, and C. R. Natoli, Phys. Rev. Lett. 82, D. Sayers, E. Stern, and F. Lytle, Phys. Rev. Lett. 27, Y. Joly, Phys. Rev. B 63, C. T. Chantler, C. Q. Tran, D. Paterson, D. J. Cookson, and Z. Barnea, Phys. Lett. A 286, C. T. Chantler, J. Phys. Chem. Ref. Data 29, C. T. Chantler, J. Phys. Chem. Ref. Data 24, C. T. Chantler et al., 14 J. H. Hubbell, Bibliography of Photon Total Cross Section (At

362 334 de JONGE et al. tenuation Coefficient) Measurements 10 ev to 13.5 GeV, , NISTIR 5437 National Institute of Standards and Technology, Gaithersburg, MD, J. H. Hubbell, J. S. Coursey, J. Hwang, and D. S. Zucker, Bibliography of Photon Total Cross Section (Attenuation Coefficient) Measurements version 2.3 National Institute of Standards and Technology, Gaithersburg, MD, 2003, available at 16 M. J. Berger and J. H. Hubbell, XCOM: Photon Cross Sections on a Personal Computer, NBSIR National Bureau of Standards, Gaithersburg, MD, M. J. Berger, J. H. Hubbell, S. M. Seltzer, J. S. Coursey, and D. S. Zucker, XCOM: Photon Cross Section Database version 1.2 National Institute of Standards and Technology, Gaithersburg, MD, 1999, available at 18 D. C. Creagh and J. H. Hubbell, Acta Crystallogr., Sect. A: Found. Crystallogr. 43, D. C. Creagh and J. H. Hubbell, Acta Crystallogr., Sect. A: Found. Crystallogr. 46, J. Goulon, C. Goulon-Ginet, R. Cortes, and J. M. Dubois, J. Phys. France 43, C. Q. Tran, Z. Barnea, C. T. Chantler, and M. D. de Jonge, Rev. Sci. Instrum. 75, bd display pkg.display beamline?i beamline id 1-ID 25 J. H. Beaumont and M. Hart, J. Phys. E 7, U. Bonse, G. Materlik, and W. J. Schröder, J. Appl. Crystallogr. 9, C. Q. Tran et al., X-Ray Spectrom. 32, C. T. Chantler, C. Q. Tran, D. Paterson, Z. Barnea, and D. J. Cookson, X-Ray Spectrom. 29, C. T. Chantler, C. Q. Tran, D. Paterson, D. J. Cookson, and Z. PHYSICAL REVIEW A 71, Barnea, X-Ray Spectrom. 29, B. Nordfors, Ark. Fys. 18, M. D. de Jonge, Z. Barnea, C. T. Chantler, and C. Q. Tran, Meas. Sci. Technol. 15, B. D. Cullity, Elements of X-Ray Diffraction Addison-Wesley, Reading, MA, R. D. Deslattes, E. M. Kessler, W. C. Sauder, and A. Henins, Ann. Phys. N.Y. 129, P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 72, S. Kraft, J. Stümpel, P. Becker, and U. Kuetgens, Rev. Sci. Instrum. 67, B. L. Henke, E. M. Gullikson, and J. C. Davis, At. Data Nucl. Data Tables 54, S. Brennan and P. L. Cowan, Rev. Sci. Instrum. 63, S. Brennan and P. L. Cowan, C. Q. Tran et al., J. Phys. B 38, M. D. de Jonge, Z. Barnea, C. Q. Tran, and C. T. Chantler, Phys. Rev. A 69, C. Q. Tran, M. D. de Jonge, Z. Barnea, and C. T. Chantler, J. Phys. B 37, See EPAPS Document No. E-PLRAAN for a complete tabulation. The electronic tabulation includes further 425 measurements between kev and kev, made at energy intervals down to 0.5 ev. These further measurements include detailed x-ray-absorption near-edge structure XANES and EXAFS. A direct link to this document may be found in the online article s HTML reference section. The document may also be reached via the EPAPS homepage or from ftp.aip.org in the directory/epaps/. See the EPAPS homepage for more information

363 INSTITUTE OFPHYSICS PUBLISHING Meas. Sci. Technol. 15 (2004) MEASUREMENTSCIENCE AND TECHNOLOGY PII: S (04) Full-foil x-ray mapping of integrated column density applied to the absolute determination of mass attenuation coefficients M D de Jonge, Z Barnea, C Q Tran and C T Chantler School of Physics, University of Melbourne, Australia chantler@physics.unimelb.edu.au Received 25 March 2004 Published 23 July 2004 Online at stacks.iop.org/mst/15/1811 doi: / /15/9/019 Abstract Recent measurements of mass attenuation coefficients have identified the determination of the thickness of the absorbing specimen as the major limitation to the accuracy of the measurement. We present a technique for determining the mass attenuation coefficient with high accuracy. The technique uses the integral of the density along a column extending through the thickness of the absorber, which we term the integrated column density. Attenuation measurements mapped across the entire absorber are used to determine a relative map of the integrated column density. These relative measurements are then placed on an absolute scale by comparison with the average integrated column density and are used to determine the mass attenuation coefficient. This approach correctly treats variations in the integrated column density across the foil. We illustrate the technique with an absolute measurement of the x-ray mass attenuation coefficient of molybdenum using a synchrotron beam of energy kev ± kev. We obtain [ µ = cm 2 g 1 ± cm 2 g 1, accurate to 0.028% over one order of magnitude more accurate than any previous work. The full-foil technique used to determine the mass attenuation coefficient is used to determine an integrated column density profile of a sample to a precision of around 0.05% of the thickness of the absorber. We demonstrate the sensitivity of the technique by observing a periodic thickness variation of order 0.1 µm occurring over a 5 mm length scale on a nominally 50 µm thick molybdenum foil. Keywords: integrated column density, full-foil mapping, x-ray, mass attenuation coefficient Introduction The attenuation of x-rays by materials provides a wide variety of information about the fundamental properties of matter at the atomic, molecular and solid-state levels. In particular, relative and absolute measurements of the mass attenuation coefficient are used to test theoretical predictions of photoelectric absorption using bound-state electron wavefunctions [1, 2, to investigate the dynamics of atomic processes, including shake-up, shake-off and Auger transitions [3 6, and to provide information on the density of electronic states [7, molecular bonding and other solidstate properties [8. Furthermore, a number of surface-science investigations have been undertaken to probe the qualities, properties and interactions at and between the surfaces of materials [9. The diversity of these studies is evidence of /04/ $ IOP Publishing Ltd Printed in the UK 1811

364 336 M D de Jonge et al the wide variety of processes that influence the attenuation of x-rays. In order to develop a better understanding of these processes, it is necessary to make accurate measurements, allowing each process to be isolated, studied and compared with theoretical models. While relative measurements are adequate for some comparisons with theory, absolute attenuation measurements provide an additional, crucial and demanding test of theoretical predictions. For example, while the finite-difference calculations of Joly [10 have had significant recent success in predicting EXAFS on a relative scale, they are in relatively poor agreement with the results of absolute measurements [11. The lack of highly accurate measurements and current limitations faced by theoretical prediction provide serious barriers to the understanding of x-ray interactions with matter. Many independent measurements of x-ray attenuation coefficients have been published. These measurements exhibit considerable discrepancies [12, 13 which, in the 1980s, led the International Union of Crystallography (IUCr) to devote a multi-laboratory project to the investigation of their causes [14, 15. An important conclusion of that project was that the discrepancies were the result of an inadequate understanding of a wide range of random and systematic sources of uncertainty. In a number of recent reports [16 22 it has been observed that, at accuracies between 0.5% and 2%, the dominant and limiting source of error in the measurement of mass attenuation coefficients is the inaccuracy in the determination of the thickness of the absorber along the path traversed by the x-ray beam. In this paper, we develop a technique for determining the mass attenuation coefficient on an absolute scale which overcomes the limitations imposed by knowledge of the thickness of the absorber. In this technique raster measurements of the beam attenuation over the entire absorber are combined with an accurate determination of the mass of the measured area of the absorber, i.e., the average integrated column density, to yield the absolute value of the mass attenuation coefficient. We demonstrate the technique with an absolute measurement of the x-ray mass attenuation coefficient [ of molybdenum using a synchrotron beam, yielding µ = cm 2 g 1 ± cm 2 g 1. This result, one to two orders of magnitude more accurate than any such previous work, is limited by the accuracy of our measurement of the area of the foil sample. 2. Full-foil mapping The Beer Lambert equation describes the attenuation of x-rays of a given energy passing through an absorber by ( ) [ I µ ln = [t xy, (1) I 0 xy where I and I 0 represent the attenuated and unattenuated beam intensities respectively, [ µ the mass attenuation coefficient of the absorbing material at a given energy, and [t xy the integrated column density along the path taken by the x-ray beam through the location (x, y) on the absorber. The integrated column density represents the path integral of the density through the absorber according to [t xy = xyz dz = txy 0 xyz dz, (2) where xyz represents the three-dimensional variation of the density within the absorber and t xy the thickness of the absorber through the point (x, y). The integrated column density provides the best macroscopic measure of the total amount of absorbing material in the path of the beam. We use the notation [t to represent the integrated column density in order to maintain connection with the traditional quantities density and thickness t, but employ the square brackets to indicate that [t (and likewise [ µ ) are directly measured quantities and not combinations of µ, and t. Traditionally, the local value of the integrated column density has been determined as the product of the density and the thickness. However, since the local density of a sample is difficult to determine, such techniques have in the past proceeded by assuming a specimen to be homogeneous, and thus that the bulk density accurately reflects the local density of the sample. Subject to this assumption, the problem of determining the integrated column density was reduced to one of determining the local thickness of the specimen. This problem had been studied in detail, and the local thickness had been determined by a variety of techniques using micrometry [1, 2, 21, 23, 24, profilometry [1, optical microscopy [25, step-profilometry [26 and x-ray scanning techniques [1, 2, 23. We have also previously attempted to address this problem by the use of interferometric measurements on the surface of foil absorbers to determine the relative thickness variations on each surface of the absorber. Measurements of sample thickness have the advantage that they probe the variation of the thickness across the surface of the foil. However, each of the techniques mentioned above is subject to a range of fundamental limitations affecting their precision and accuracy which are difficult to overcome [1, 2, 27, and which represent a major limitation on the precision and accuracy of the determination of the mass attenuation coefficient. More recent measurements have used the areal density, which we term the integrated column density, of the absorber for the determination of the mass attenuation coefficient [1, 2, 16, 18 20, 22, However, these measurements have generally been limited to accuracies of 0.5% 2% due to structure in the thickness, which has limited the determination of the local integrated column density of the absorbing specimen along the column traversed by the beam. The mass attenuation coefficient of a foil absorber can be determined more accurately by using attenuation measurements made across the entire surface of the absorber. We write the average of the attenuation measurements made at a number of (x, y) locations on a foil (from equation (1)) as ( ) [ I µ ln = [t xy. (3) I 0 xy As the mass attenuation coefficient [ µ is a constant for all measurements at a given energy, we recast this as ( ) [ I µ ln = [t xy. (4) I 0 xy When the entire surface of the absorber is probed by the (x, y) x-ray mapping, [t xy can be identified with the average integrated column density of the specimen [t. Thus ( ) [ I µ ln = [t, (5) I 0 xy 1812

365 Full-foil x-ray mapping of integrated column density mm 356 mm 368 mm 203 mm 'upstream' Ar gas ion-chamber daisy wheel with foils & 3 scattering apertures sample with two translational and two rotational degrees of freedom 'downstream' Ar gas ion-chamber Figure 1. Schematic of the experimental set-up. and by mapping the attenuation across the entire sample we can determine the mass attenuation coefficient without directly determining the local integrated column density at any point on the absorber. As the local value of the integrated column density does not appear in equation (5), variations of this quantity across the surface of the absorber, which have limited other non-local measurements, do not limit this technique. The average integrated column density of the specimen can be measured to high accuracy using well-established techniques, for example by using an optical comparator to determine area and an accurate microgram balance to measure mass. 3. Absolute determination of the mass attenuation coefficient of molybdenum 3.1. The attenuation profile The measurements reported here are part of an experiment designed to measure the mass attenuation coefficients [ µ of molybdenum at x-ray energies between 13.5 kev and 41.5 kev. The experiment was performed at beamline 1-ID of the Advanced Photon Source synchrotron facility at Argonne National Laboratory. The x-ray beam was produced by an undulator insertion device, where the fifth order of the undulator spectrum was tuned to the requested experimental energy. This spectrum was monochromated by reflection from the (3,1,1) planes of a silicon double-reflection monochromator. The second crystal of this monochromator was detuned slightly from the parallel position to suppress the passage of the undulator harmonic components into the experimental beam [31, 32. A schematic of the experimental arrangement is shown in figure 1. Argon gas ionization-chamber detectors were placed at a distance of approximately 572 mm upstream and downstream of the absorbing specimen. Daisy-wheels were placed between the absorbing specimen and the ionization chambers. Three scattering apertures located on the perimeter of each daisy-wheel were in the shape of a 2 mm diameter circular hole and 2 6mm 2 and 6 9mm 2 rectangular holes. These scattering apertures were used to limit the amount of scattered radiation reaching the upstream and downstream ionization chambers. The upstream ionization chamber was used to normalize the measured intensities so as to isolate the beam intensity fluctuations from the other noise components. The normalized attenuated and unattenuated counts were determined from I = I down I dc,down I up I dc,up I 0 = I down,0 I dc,down I up,0 I dc,up, (6) where 0 refers to measurements made with no absorber in the path of the beam, down and up refer to the measurements recorded in the downstream and upstream ionization chambers, respectively, and dc refers to measurements made with the shutter closed so as to fully block the x-ray beam. With the air-path and sample thickness reported here, an undetected air-pressure fluctuation of 1% would, if not corrected, result in a change in the mass attenuation coefficient of around 0.01%. The normalized unattenuated intensity, which determines the detector efficiencies and the air-path attenuation, was measured on a number of occasions before and after the full-foil mapping. The weighted mean and variance of these values were used to account for the first-order variation in the air-path attenuation. Further experimental details relating to the counter normalization and air-path attenuation corrections are similar to those reported elsewhere [1, 2. The nominally mm 3 molybdenum foil used for this measurement was mounted in a holder, which was in turn mounted on a combination of translatable and rotatable stages, so that the sample could be accurately positioned in the path of the beam. The sample holder was machined from two sheets of mm 3 Perspex. These were constructed by drilling a hole of approximately 13 mm diameter through the Perspex. This hole was then bevelled, meeting the full thickness of the holder at a diameter of approximately 24 mm. Through-holes and threads for eight screws were drilled and tapped around the perimeter of the holder. The sample foil was placed between two of these holders and the screws tightened so that sample motion was prevented with minimal stress applied to the sample. This design minimized any motion of the sample within the holder so that subsequent attenuation measurements could be made through the same location on the absorber. In figure 2 we present the result of the two-dimensional x-ray scan which has been processed to return a value for ln ( ) I I 0 xy = ([ ) µ [txy at each measured (subscript S+H,mea mea ) (x, y) location on the sample mounted in the holder (subscript S + H ), i.e., an attenuation profile of the sampleplus-holder. The measurements were made with a 1 1mm 2 x-ray beam at 1 mm intervals in the x and y directions indicated in the figure. The attenuation profile clearly exhibits a number of features which have resulted from the attenuation of the x-ray beam by the sample and the holder. In the central region we see values resulting from measurements where the beam has passed through the molybdenum sample only. Surrounding these points is a conical ramp in the measured attenuation resulting from the increasing thickness 1813

366 338 M D de Jonge et al Figure 2. Attenuation profile of the sample mounted in the holder. The attenuation profile was produced from the x-ray scan, processed to determine a value of ln ( I I 0 ) xy = ([ µ [t xy) S+H,mea at every (x, y) location across the surface of the foil. The x-ray beam used to make the measurements was 1 1mm 2 and measurements were taken at 1 mm intervals across the foil. of the (bevelled) Perspex holder in the path of the beam. These measurements plateau at a value corresponding to the attenuation of the sample plus the full thickness of the holder. The skirt surrounding this plateau corresponds to measurements that have been made with the x-ray beam either fully or partially by-passing the sample. Thus, the values around the edge drop sharply from the sample-plus-holder value to that of the holder alone. The several sharp spikes in the measured attenuation occurring near the corners and mid-way between the corners of the sample are the result of the x-ray beam hitting the screws which have been used to mount the sample in the holder. Measurements at each (x, y) location on the absorber were repeated ten times in rapid succession to yield a direct measure of precision and reproducibility and to optimize the treatment of correlations in the counting chain [33, 34. In figure 3 we present the directly-quantified uncertainties in the measured data σ ([ ) µ [t, evaluated as the standard error of the S+H,mea results obtained from the ten repeated measurements. This figure shows that the uncertainty is relatively constant for all the measurements at about (units of ln ( I I 0 )) Removal of the holder attenuation In order to use equation (5) to determine the mass attenuation coefficient on an absolute scale, we need to remove the effect of the holder attenuation on the measured attenuation profile. In our case the simple and uniform geometrical shape of the holder and the strong signature that it presents in the attenuation profile allow the holder component of the Figure 3. Uncertainties σ([ µ [t) S+H,mea in the measured attenuation at every point in the x-ray scan, determined from the standard error of ten repeated measurements. The directly-determined uncertainty is relatively constant at around (units of ln ( I I 0 )). attenuation profile to be modelled and then subtracted from the total measured attenuation profile. The sample can be removed from the holder and the holder attenuation profile measured in isolation [27. This alternate approach is useful but the holder and the sample-plusholder attenuation profiles must be exactly registered prior to subtraction. This registration is of similar complexity as the approach adopted here, and we prefer to leave the sample undisturbed in the holder. We have constructed a program to fit the total attenuation profile using a standard Levenberg Marquardt least-squares fitting routine. The fitting function takes as input a number of parameters describing the geometrical properties of the holder and of the sample and calculates the resulting attenuation profile at any given (x, y) location on the sample-plus-holder. The sample attenuation profile is recovered by subtracting the fitted holder component from the total measured attenuation profile according to ([ µ [t xy ) S,rec = ([ ) µ [t xy S+H,mea ([ ) µ [t xy, (7) H,fit where the subscripts rec, mea and fit refer to the recovered, measured and fitted attenuation profiles, respectively. For example, a set of 14 parameters is required to fit the total attenuation profile shown in figure 2 in the following manner: the maximum [ µ [t of the sample and of the holder (two parameters), the location of the centre of the circles defining the bevel in the holder (two parameters), their radii (two parameters) and the locations of the corners of the sample (eight parameters). 1814

367 The fitting routine requires the prediction of the attenuation of the sample and holder combination at every sampled (x, y) point. The measured values in figure 2 result from the interaction of an x-ray beam of finite cross-sectional area A with the sample and holder, and are thus predicted from ([ µ = ln [t xy ) S+H,mod { beam 0,xyexp[ I ([ µ [t xy ) beam I 0,xy da S,mod ([ µ [t xy ) H,mod where the subscript mod refers to the modelled attenuation profiles. For a beam of uniform intensity I 0,xy = I 0 this reduces to ([ µ = ln [t xy ) S+H,mod { [ ([ ) 1 µ exp [t xy A beam ([ ) } µ [t xy da [ ([ µ = ln exp H,mod [t xy ) S,mod S,mod da } ([ ) µ [t xy H,mod. (8) In practice the average of the exponential of the attenuation over the illuminated region only needs to be calculated at those locations where the modelled attenuation varies significantly over the beam footprint. Thus, averaging has been undertaken only for those measurements made on the bevel of the holder and around the edge of the foil sample. The result of fitting the combined sample and holder attenuation profile is shown in figure 4. We note very good agreement with the general form of the measured attenuation profile of figure 2. A more detailed investigation of the quality of this fit can be undertaken by examining the distribution of the residuals, defined as ([ µ [t )S+H,mea residual = ([ ) µ [t S+H,fit σ ([ ) µ, (9) [t S+H,mea and presented in figure 5. The grey scale in this figure suppresses the residual magnitudes but displays their distribution. Any structure in the residual pattern indicates an inadequate description of the modelled sample or holder. The pattern of residuals shows no significant structure and, in particular, shows no structure reminiscent of the shape of the holder. This indicates that the holder component of the attenuation profile has been successfully modelled. It also shows a good normal distribution of values, with all levels of the grey scale well-represented. To ensure that the holder component of the attenuation profile is properly determined it is necessary for the measured data to be correctly modelled by the fitting program. However, measurements taken with the beam overlapping the edge of the foil are subject to significant variation, resulting either from a tiny displacement of the foil in the beam or from a small change in the intensity distribution over the beam area. These variations depend only on the properties of the beam and the, Full-foil x-ray mapping of integrated column density Figure 4. Results of fitting the attenuation profile presented in figure 2. The fitted profile has been produced by calculation whose inputs are the fitted geometrical dimensions of the sample and of the holder scaled by their fitted attenuations. The function has been evaluated at each (x, y) location by summing the attenuation of the sample and holder calculated for measurement with a 1 1mm 2 beam. Figure 5. Residuals of the fit to the attenuation profile. The random appearance of the distribution of the residuals implies the absence of any additional significant systematic or geometrical correction. The grey scale is arbitrary. Measurements around the foil edges (extreme bottom and right, second row from top, second column from left) exhibit variations resulting from small displacements of the foil in the beam. These variations are not described by the model function and provide no information for fitting the holder. To enable the holder attenuation profile to be more properly isolated the weighting of these measurements has been decreased in the fit

368 340 M D de Jonge et al foil, and thus provide no information for the fitting of the holder component of the attenuation profile. Furthermore, as these variations are not quantified in the directly determined input error estimates, they can confuse the fitting of the holder component. Thus, the uncertainty of these measurements has been increased by a factor of 100. Similarly, measurements made where the x-ray beam interacted with the screws were discarded. The frame of middle-grey on the edges of figure 5, with a residual of approximately zero, results directly from the weighting applied to these measurements. We have modelled a sample which is perfectly flat but which may have a wedge-like shape, becoming linearly thicker as one traverses the surface of the absorber. By employing such a model the fitting program is able to resolve these features in the attenuation profiles of the sample and of the holder. The need to include such geometrical features of the sample and holder in the fitting program has been determined empirically by examining the distribution of the residuals of the fit. For instance, the wedge-like shape of the foil has been included in the fitting program in response to an observed systematic left-right pattern in the residuals of the fit when the foil was modelled as a perfectly flat object with parallel surfaces. Any second-order variation in the integrated column density of the absorber (i.e., curvature) would show up in the residuals as a series of rings of alternating positive and negative deviation from zero. There is no such correlation in the residuals, so such curvature is not significant. The holder has also been allowed to have a wedge-like shape. The wedge-like features of the sample and holder are not degenerate due to the large number of sampled points where the foil and the holder are probed in isolation. The number and distribution of these points is sufficient to allow these parameters to be well resolved by the fitting routine, with low correlation. While the recovered sample attenuation profiles may differ when further higher-order contributions to the model are included, the average of these attenuations required to evaluate [ µ from equation (5) is insignificantly affected. In particular, the difference between the average sample attenuation obtained with and without the assumed wedgelike character of the foil and holder is significantly less than the uncertainty associated with each of the fitting schemes. Similarly, errors in the fit resulting from the beam size and non-uniform intensity profile have negligible effect on the recovered average sample attenuation. The reduced-chi-squared χr 2 of the fit is χr 2 1 because the model is not intended to describe the attenuation profile of the sample. We carry out the fit in order to determine the holder contribution from its strong attenuation profile signature. After fitting, the true sample attenuation profile is determined by subtracting the fitted holder attenuation profile from the total measured attenuation profile according to equation (7). In figure 6 we present the recovered sample attenuation profile after subtraction of the fitted holder contribution. The attenuation in the central region has no holder component, and thus is unchanged in the process of the recovery. This central region is completely consistent with the attenuation profile in the region where the contribution of the holder has been subtracted. The two diagnostics therefore confirm the Figure 6. The recovered absorber attenuation profile with the edge omitted. The recovered attenuation profile is the measured attenuation profile of the (sample plus holder) minus the fitted holder profile. The variations of order in the attenuation are partially explained by the statistical uncertainty in the measured attenuation. The statistical uncertainty is approximately for all measurements across the foil. The remaining variation could be explained by long-range, aperiodic deviations of order 0.3 µminthe foil thickness, but is more likely due to the statistical uncertainty requiring scaling by a factor of χ 2 r. appropriateness of the fitting model and the quality of the result. However, the aperiodic variation between neighbouring measurements in figure 6 has a standard deviation of approximately This is greater than the determined uncertainty of approximately attributed to the points by means of the ten repeated measurements at each point (figure 3). Therefore, either the structure in the attenuation depicted in figure 6 is real (corresponding to pointwise randomly-distributed thickness variations of about 0.3 µm) or the input uncertainties are underestimated. Other work [33 has noted that ten consecutive measurements of a very short period of time (0.1 s each) repeated in rapid succession may not fully probe the random variation in intensities when compared to measurements made over a longer time interval (the full-foil mapping takes about one hour). Thus the directly determined uncertainty in the attenuation may be underestimated. The absence of any artefacts of the measurement sequence in figures 5 and 6 is consistent with the claimed measurement reproducibility. The dominant component of χr 2 appears to be due to the underestimation of the input measurement uncertainties by a factor of χr 2. The χr 2 of the fit to the combined sample and holder is 3.8. Hence the scaled uncertainties are used for the remainder of the calculation. The determined uncertainty at each point in the recovered sample attenuation profile is evaluated by adding the 1816