The Breadth and Shape of Instrumental Line Profiles in High-Resolution Powder Diffraction

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1 913 J. Appl. Cryst. (1991). 24, The Breadth and Shape of Instrumental Line Profiles in High-Resolution Powder Diffraction By J. I. LANGFORD School of Physics and Space Research, University of Birmingham, Birmingham B15 2TT, England R. J. CERNIK SERC Daresbury Laboratory, Warrington WA4 4AD, England AND O. LOU~R Laboratoire de Cristallochimie, Universit# de Rennes I, Rennes CEDEX, France (Received 15 February 1991; accepted 10 April 1991) Abstract The instrumental resolution function for the Daresbury 9.1 high-resolution powder diffractometer has been measured as a function of wavelength, slit width and specimen geometry. It has been demonstrated that the effects of size and strain from the sample of annealed barium fluoride used as a reference material were negligible, and that it is a suitable standard for studying instrumental broadening. The results of considering the angular dependence of the Lorentzian and Gaussian components of the integral breadths of instrumental line profiles are in excellent agreement with those predicted by the receiving-slit width and the known wavelength spread in the incident beam. The measured instrumental resolution function was found to be dependent on the slit dimensions at low and intermediate angles, and to be dominated by the effects of angular dispersion at higher angles. The line shapes are predominantly Gaussian at low angles and for a flat sample tend to a pure Lorentzian shape at 180 ~. The line profiles for a capillary sample are similar at low and intermediate angles and tend to an intermediate Gaussian/ Lorentzian form at high angles. 1. Introduction A knowledge of instrumental line shapes and breadths is a basic requirement in applications of powder diffraction which depends on a precise representation of the form of line profiles. Errors in the data used to correct for instrumental broadening can profoundly affect the reliability of information on microstructural properties derived from diffraction experiments. High-resolution instruments, such as the 9.1 High-Resolution Powder Diffractometer (HRPD) at the SERC Daresbury Laboratory, have extended the range over which domain size and /91/ lattice deformation can be measured reliably, provided that the instrumental contribution is known accurately. This information is also required in ab initio structure solution from powder data and the refinement of structures by the Rietveld method. Here the angular dependence of the breadth and shape of the instrument function indicates if sample broadening is significant. If it is not, then the instrumental data can be used to model line breadths and shapes throughout the pattern. The general principles involved in instrument characterization are demonstrated below by considering data from the 9.1 HRPD for flat and capillary samples HRPD 2. Experimental considerations Station 9.1 at the Daresbury Laboratory is situated approximately 15 m from the tangent point of a three-pole 5 T superconducting wiggler magnet. A schematic layout of the station is shown in Fig. 1. The unfocused polychromatic beam enters the hutch and is reduced in size by a water-cooled fixedaperture slit. The beam is then further reduced to the desired size for the monochromator in current use by Tungsten Water-cooled carbide heal sink Scintdtahon laws ~,-, counter "~w;ndow \, \, ;ra~ per War...!,A \ II '\ ~,/~ d ~ B... profile-defining slits Be P~'~x"~Slrt ~T R \ ~" W... in'~ "7 Vertical Hori... I Kapton~... /~-'~ I'--u NII~ '~ / \ "/foil --i-~o.y / ~. zz-.~_~/,,. ;,' fj ' ~ "g ~ 1 I /,s,,, " Vertical Horizontal S1(111) ] I I,i s e Fix. water-laws cooled ~ jaws ~l-cut '>water-cooled H~t g FL[I I I P... aperture Centre opening monochromator adlustment I I shts for slit I_~ assembly '\Monitor scintillation counter 91 Monochromator vessel in He gas 91 Hutch Beamline or evacuated to 1 3Pa and diffractometer Fig. 1. Schematic layout of station International Union of Crystallography

2 914 INSTRUMENTAL LINE PROFILES Table 1. Slit widths, U, V, W and FWHM,,i, (BaF2, ,h,) b h Receiving slit Entrance slit U V Sample Width (mm) Angular aperture (o) Height (mm) Width (mm) {[ (20)] 2 x 10 ~} (a) Flat (b) Flat ( ) Flat (d) Flat (e) Flat 0-2 0" (]) Capillary W FWHMmm [ (2o) " " " a set of water-cooled movable horizontal and vertical jaws. The monochromator used for this study was a water-cooled Si(lll) channel-cut-crystal (Cernik & Hart, 1989) mounted in the orientation shown. The beam-profile-defining slits consist of a set of two fixed horizontal and vertical apertures on removable carriages facilitating easy removal and accurate setting of slit widths. The slit assembly is movable in the vertical direction. The beam decay is monitored by observing the scattering from a kapton foil held at 45 to the incident beam. The monochromator beam emerges from the evacuated front end and monochromator vessel and falls on the specimen (fiat plate or capillary) which is mounted on the 0 axis of a heavy-duty two-circle diffractometer. The 0 and 28 circles are independently driven, thereby allowing either 20 or 0:20 scans. For the experiments described here a double slit assembly and scintillation counter were mounted on the 20 or detector axis Line broadening reference materials Parameters for assessing instrumental line profiles are obtained from data for a standard reference material, for which intrinsic broadening must be negligible compared with that due to the instrument geometry and the wavelength spread in the incident beam. Of possible materials studied so far, annealed BaF: (Merck Suprapur), with a particle size < 38~m, is ideal for characterizing conventional divergentbeam X-ray diffractometers (Lou5r & Langford, 1988), though LaB6 is also used (Fawcett et al., 1988). Recent measurements made with the tripleaxis diffractometer at the Brookhaven SRS have indicated that a small amount of intrinsic broadening may be present for this sample (Cox, 1989). However, the effect of this on data obtained from the 9.1 HRPD is negligible, as can be demonstrated by considering the (sins)/k dependence of line breadths. If the material exhibits broadening due to sample inperfections in general there will be an orderindependent contribution arising from the finite size of diffracting domains ('size' effects) and an orderdependent part due to deformation of the lattice ('strain' broadening) Experimental conditions Data were obtained with the 9.1 HRPD operating in the angle-dispersive mode at three wavelengths in the range A, selected by the monochromator. Both fiat and capillary samples were used. The fiat sample was rotated about an axis perpendicular to its surface, to improve counting statistics. Thus, for an illuminated area of mm, the effective area is about 80 mm: and there are -104 particles per mm:. The 0.3 mm Lindeman-glass capillary was rotated about its long axis for the same reason. Dimensions of the slits defining the incident beam and the aperture of the receiving slit are listed in Table Analysis of line breadths From the variation of the full-width-at-halfmaximum intensity (FWHM) with diffraction angle (Fig. 2), it is evident that the breadth of lines increases with diffraction angle. In order to ascertain if the sample contributes to this angular dependence, some analytical function must be ascribed to each o :~ O0l,- u,_ ~_.~_~ e + (b) lb 3b io 7'o 9'o 1;o rio ~o 28 degrees Fig. 2. Instrument resolution function (IRF) for 9.1 HRPD. (a) Flat sample, conditions (a) (Table 1), A, A, i-6008 A. (b) Capillary sample, conditions (f) (Table 1), /~. a)

3 J. I. LANGFORD, R. J. CERNIK AND D. LOUER 915 a (A) hkt 0"7039 I i i ,10, ,6 Table 2. BaF2 line-profile parameters (flat sample) Peak position [~(20)] FWHM [~(20)] /3 [ (20)] q~ 11"230 0"024 0"028 0"884 0"163 21" "031 0"875 0"153 28"594 0"030 0'034 0"864 0"202 39" " "284 50"224 0"034 0"043 0"806 0" " "776 0"470 79"880 0"052 0"068 0"763 0" "771 0"072 0"099 0"728 0" "385 0"084 0"122 0"691 0'696 m 5"64 10"52 6"11 3"57 2"64 2"08 I "86! "60 I "25 R.~(%) 9"9 12"3 9"5 7"5 6"6 6"1 11"3 9"7 10"0 GOF! "864 0'031 0"036 0"860 0" "033 0'040 0" '647 0"035 0"044 0"797 0"395 68"299 0' "529 85"377 0"047 0'065 0"731 0"613 92"920 0"051 0"070 0"729 0" "148 0"058 0"082 0"709 0" "139 0"071 0"100 0"705 0" "403 0"078 0"111 0" " "42 1 "84 1 "63 1 "57 1 " "41 7"3 3"1 7" "3 3"7 9"0 4"0 6' i I l " "037 0'831 0"297 42"803 0"034 0"043 0"788 0"421 50'679 0'037 0" '452 68"478 0'044 0"057 0"761 0' " "744 0" '731 0' "528 0"065 0"092 0"706 0" '892 0" '700 0" ' "i " "08 1"87 1"70 1 "64 1 '45 i '42 1'33 7"6 5"4 3"0 3"2 3"2 3"2 5"3 5"3 3" peak in the diffraction pattern. The Voigt function, a convolution of a Lorentzian and a Gaussian, is a good approximation to the shape of instrumental line profiles for the 9.1 H RPD (Table 2). The Voigtian can be characterized by a form factor (q~), defined as the ratio of the FWHM (2Wv) to the integral breadth, the width of a rectangle having the same height and area as the peak (fly) (Langford, 1978). The parameters 2Wv, fly and ~0 for the flat BaF2 sample [with conditions (a) in Table 1] are listed in Table 2 and it can be seen that ~ has values between the Gaussian limit [2(ln2/Tr) ''2 = ] and the Lorentzian limit (2/rr = 0"6366) for all reflections and each of the three wavelengths used. There is no simple (exact) relationship for combining the breadths of convoluted Voigt functions, but, as is well known, the breadths of convoluted Lorentzian functions are additive and for Gaussians the squares of the breadths are additive. The angular dependence of flv can thus be obtained by considering the breadths/3/- and/3c of the Lorentzian and Gaussian components. /3/. and /3a can be obtained from /3v and ~, by means of the approximate expressions given by Langford, Delhez, de Keijser & Mittemeijer (1988) [equations (9) and (10)]. The orderindependent and order-dependent contributions to /3/. and /3G can be separated by means of Williamson-Hall plots (Langford, LouEr, Sonneveld & Visser, 1986): and fllcoso = A + Bsin0 (1) fl6cos20 = C + Dsin20. (2) From (1) and Fig. 3, A = (1)~(20) and B (2) (20). The line profile due to size effects has an inverse-square variation of intensity in the tails (e.g. Wilson, 1962) and thus tends to be Lorentzian in character, but the zero intercept in (1) indicates that there is no order-independent ('size') contribution to ill. Strain profiles, on the other hand, normally tend to be Gaussian, but if the slope is m_~ ~- ++~+ 0 03, "~ j~...'+ o'i 0'2 3 oy-o's 0'6. o"~- o's, o9-~o S,n O Fig. 3. Williamson-Hall plot for BaF2, Lorentzian components, A as in Fig. 2.

4 916 INSTRUMENTAL LINE PROFILES interpreted as a Lorentzian strain component, then B -- 2"5<eZ) '/z, (3) where <e 2) is the r.m.s, strain. The slope could thus be equivalent to a Lorentzian strain of For the Gaussian components [(2) and Fig. 4], C =0.73 (2) [ (20)]2 and D = (4) x 10-3 [ (20)] z. The order-dependent part of/3~ thus decreases with sin0, whereas the broadening due to microstrains must increase. The negative slope of Fig. 4 indicates that microstrains cannot be a dominant source of broadening. From these results it can be inferred that the influence of domain/crystallite size and microstrains is negligble and that the annealed BaF2 sample is suitable for characterizing the instrumental contribution to data obtained from the 9.1 HRPD. 3. Resolution: fine breadths 3. I. Instrument resolution function A useful 'general-purpose' curve for characterizing powder diffractometers is the instrument resolution function (IRF), the variation of the FWHM of instrumental line profiles with diffraction angle (or energy). It is customary to model the FWHM variation by means of some analytical function. The form of this need not necessarily have any physical interpretation and a quadratic in tan0 is normally used for the angle-dispersive case, or (FWHM) 2 = Utan20 + Vtan0 + W. (4) The IRF can be used for making a periodic check that alignment of the instrument is maintained, for comparing the performance with that of other diffractometers and for deciding if the resolution is adequate for a particular application. It also provides a basis for ascertaining if a powder pattern contains measurable sample broadening. This is achieved by comparing a similar plot for the sample of interest with IRF. If intrinsic broadening is significant the FWHM will clearly be greater than the IRF at all angles and in general the scatter about a % bs o;s o~s o~s o~s o'ss o~ss o.~s o'ss o~s Sin? 0 Fig. 4. Williamson-Hall plot for BaFz, Gaussian components,,~ as in Fig. 2. least-squares curve will be greater than that expected from counting statistics, due to the shape of diffracting domains, anisotropy of the elastic properties of the.material or the presence of structural 'mistakes'. In studies of microcrystalline properties by lineprofile analysis, the IRF, together with a similar curve for the integral breadth, can be used to correct line widths for instrumental effects. The curve can also be used to model the variation of line width in structure refinement by the Rietveld method. The IRF for conventional powder diffractometers with an incident-beam monochromator has been discussed by Lou6r & Langford (1988), and Hastings, Thomlinson & Cox (1984) have considered in detail the resolution of a triple-axis system which uses synchrotron radiation in conjunction with an analysing crystal. For the 9.1 HRPD a typical IRF for a flat sample is given in Fig. 2, curve (A). The slit widths were set for high resolution, consistent with an adequate level of intensity [(a) in Table 1]. Under these conditions the FWHM is dominated by the aperture of the receiving slit (0.026 ) at low angles ( 3.2). The IRF has a minimum of (Ad/d-3 x 10-3) at about 30 (20) and has twice this value at 100 (Ad/d ). The IRF is, to a first approximation, independent of,~ in the wavelength range A although there is a slight increase in breadth at low angles as A increases. For a capillary sample, with the same slit configuration and wavelengths, the IRF has a shallow minimum of at about 30 (20) [Fig. 2, curve (B)] and here the FWHM at low angles largely depends on the diameter of the sample. The values of U, V and W are listed in Table _ ul. 0 0~ (a) (c) (e) ~ 1~ s~ o 7~ 9~ 1;o 1~o ~o 2e degrees Fig. 5. Effect of changing slit widths on IRF, flat sample, (a) to (e) as in Table 1.

5 J. I. LANGFORD, R. J. CERNIK AND D. LOUER 917 It is sometimes desirable to increase the count rate, at the expense of resolution, by increasing the illuminated area of the sample. This can be achieved by increasing the width h of the slit controlling the axial length of the illuminated area, or by increasing the width b of the equatorial slit. The effect on the IRF of changing the slits which define the incident beam is shown in Fig. 5 for the flat sample; the slit conditions are summarized in Table 1. For curves (a) to (c), b was increased from 0.20 to 0.65 ram, with h = 5 mm. Curve (d) is as (c), but with h = 10 mm. From these curves it can be seen that the resolution at low angles is strongly dependent on the equatorial width of the illuminated area, but is not affected appreciably by the axial length. At high angles these changes in slit configuration have negligible effect on the IRF. Curves (a) to (d) were obtained with a 0.3 mm receiving slit (0.026 aperture). By using a 0.2 mm receiving slit [0.018 aperture], b = 0.10 and h = 5 mm, even higher resolution is possible [curve (e)], but for most purposes the intensity is then unacceptably low. For all the slit configurations used the influence of slit widths on the resolution of the instrument is inappreciable at angles greater than about 100 (20) Instrumental line breadths It has been shown ( 2.4) that diffraction data from the BaF2 sample adequately represent the instrument function over a wide range of scattering angles and it is of interest to ascertain which parameters have a significant effect on resolution. An alternative interpretation of B in (1) is that it arises from a spread in the wavelength selected by the monochromator or B = [2AA(~D/A]cotO, (5) where Aa(B,)/a = AL is the contribution to the fractional wavelength spread, in terms of the integral breadth, from the Lorentzian part of/3v. Thus AL = B/2 = 4.4 (2)x l0-4. If/3a is assumed to include a fractional wavelength contribution zl,~(~c)/,~ = aa, this can be separated from an angle-independent part by means of /32 = E + Ftan20. (6) The plot of/3~ versus tan20 (Fig. 6) is reasonably linear, with E ~/2 = (6) (20) and F 1'2 = (4). The width of the receiving slit was 0.3 mm and it was at a distance of 650 mm from the diffractometer axis, corresponding to an aperture of The intercept of Fig. 6 can thus be attributed to the aperture of the receiving slit. The slope is equivalent to Aa = 1-4 (1) At and Ao can be combined to give the Voigtian integral breadth of the wavelength dispersion, Av, by means of [Langford (1978), equation (25)1 Av = Aaexp(- k2)/[1 - erf(k)], (7) where k = AtJ~r~/2Ac, from which Av = 5.0 (2) This is fractional spread of,~ in terms of the integral breadth, whereas the FWHM is normally considered. The FWHM can be obtained from q~, which, from Ahtee, Unonias, Nurmela & Suortti [1984, equation (7)], is (5). Thus, if A,~ is the FWHM of the wavelength distribution, A,~/,~ = 3.4 (1) This spread in wavelength is largely due to the Darwin width of the Si monochromator, equivalent to A,~/a Other factors which contribute to A,~/A are the divergence of the beam in the vertical plane (0.1 mrad) and the effect of the source height (0.15 mm at 15 m from the monochromator) Voigt function 4. Line-profile shape The variation of the Voigt form factor (~) for the flat sample is given in Fig l,- co? x ~ + ~ 08- ca. 12- o._ ~~ " 020 0'60 1' '80 ~20 2~0 3'00 3't,O 380' tan ~ e Fig. 6./3d versus tan0, fiat sample Lorenfzion limit 2 e degrees Fig. 7. Voigt model: q~ versus 20 for flat sample, symbols as in Fig. 2. \

6 918 INSTRUMENTAL LINE PROFILES The form of the curve is similar for all three wavelengths; q~ tends to the Gaussian limit as 20 approaches zero and to the Lorentzian limit for 20 = 180. At intermediate angles the variation is roughly linear. The line shape is essentially the same for the longer wavelengths, but is slightly more Gaussian for A = 0.70 A. This suggests that the contributions from the source height and beam divergence to AA/A, which vary as cot0m, are predominently Gaussian. For the capillary sample, A = A, q~ again approaches the Gaussian limit at low angles, but evidently tends to an intermediate value as 20 increases (Fig. 10). Provided that ~o lies within the Voigt range ( 2.4), the ease with which Voigt functions can be deconvoluted makes it desirable to use this function in lineprofile analysis. However, the pseudo-voigt, which is a reasonable approximation to a Voigtian, and the Pearson VII are widely used in pattern decomposition and structure refinement [see Langford et al. (1986) for definitions of these functions]. It is therefore of interest to consider how their form varies when they are used to model the instrumental line profiles Pseudo- Voigt function The form of the pseudo-voigt function is characterized by the mixing or Lorentzion fraction 77. For the three wavelengths used, r/tends to 0 (Lorentzian limit) at 0 (20) and increases linearly with angle for 20 > 50 (Fig. 8). Thus, over this range,,7 =,4 + B(20), (8) where A and B are listed in Table 3. The slopes of these lines are identical, to within experimental error, but for the intercept A for a = 0-70 A is less than for Table 3. Coefficients A, B in (8) and r/,8o.,~ (A) A B["(20)] ' n,,o 0" (3) (3) 0.97 (7) "22 (3) (3) 1.01 (7) "21 (3) (3) 0.99 (7) the longer wavelengths by The profiles for the shorter wavelength are thus more Gaussian by this amount, confirming the trend which is apparent in the variation of ~o. For all A, r/ extrapolates to 1-0 (pure Lorentzian) at 20 = 180. From Fig. 10, the variation of r/ for the capillary sample again indicates that the line profiles are intermediate between a Lorentzian and a Gaussian as 20 increases Pearson VII function The index m [Langford et al. (1986), equation (3)] is a measure of the form of a Pearson VII function. For the flat sample m tends to infinity as 20 approaches zero (pure Gaussian), decreases steadily as 20 increases and again extrapolates at 1-0 (pure Lorentzian) at 180 (Fig. 9). The behaviour of m for different wavelengths also indicates that the profiles are slightly more Gaussian for,l = 0-70 A. The variation of m can be described by a polynomial of the form m = a _ 4(2 O) I- ao + am (2 8), and for the two longer wavelengths a , a o = 2"52 and al = (9) = ~ 07 lo ///" ///// // E 3- "o c g o6. -- " 05, = cs t,. > g_ o2 ol- O0 // // degrees Fig. 8. Pseudo-Voigt: r/ versus 20 for flat sample, symbols as in Fig s~ 10~ 1~o z~0 20 degrees Fig. 9. Pearson VII: m versus 20 for flat sample, symbols as in Fig. 2.

7 J. I. LANGFORD, R. J. CERNIK AND D. LOUER 919 The variation of m for the capillary sample follows the same trend as for the Voigt and pseudo-voigt models (Fig. 10). A similar behaviour of ~o, ~/and m with increasing scattering angle has been obtained for a conventional divergent-beam diffractometer with an incident-beam monochromator (Lou6r & Langford, 1988). Concluding remarks Reliable instrument characterization is an essential requirement in many applications of powder diffraction and line-profile breadths and shapes due to a diffractometer are most readily obtained by means of a standard reference material for which sample effects are negligible. From the procedure outlined for assessing possible standards, it is demonstrated that annealed BaF2 is suitable for studying the characteristics of high-resolution instruments, such as the 9.1 HRPD. A useful overview of the performance of a diffractometer is given by the instrument resolution function, a plot of FWHM versus 20. With 9.1 configured for high resolution and a reasonably intense incident beam, the IRF has a minimum of for a flat sample and ~ for a 0.3 mm capillary. If the slit apertures defining the beam are increased to obtain a large illuminated area, and hence recorded count rate, the resolution at low and intermediate angles is reduced as shown in Fig. 5, but there is little effect at high angles. A more detailed analysis of line breadths indicates that there is an angle-independent contribution, largely due to G It \ Y \ 07. * m -2,'~ // L If ////// tlo 06 / O degrees Fig. 10. Shape parameters remus 20 for capillary sample, q~, rl and m. the aperture of the receiving slit, and a contribution which increases with angle, mainly due to dispersion of the Darwin width of the Si monochromator. The Voigt function is a good approximation to the form of instrumental line profiles, though the pseudo-voigt and Pearson VII functions are also used. From the behaviour of all three functions, it is evident that the line profiles become progressively less Gaussian as 20 approaches 180% The reason for this is not clearly understood at present; although the rocking curve for the Si monochromator has slowly decaying tails, convolution of this with profiles due to the source height and divergence predicts a near Gaussian curve. The same Lorentzian behaviour has been observed for the 8.3 diffractometer (Cernik, Murray, Pattison & Fitch, 1990) and instrumental line profiles for the Brookhaven SRS (Cox, 1989) also have a dominant Lorentzian component at high angles. Further work is being undertaken to study the nature of the various contributions to the instrumental line profiles in greater detail. It is hoped that this will provide a quantitative explanation from the powder data for the near-lorentzian spectral distribution. The authors would like to thank the SERC for the provision of research facilities and for supporting the research through a senior visiting fellowship (DL). We are also grateful to Mr R Pflaumer for his assistance in collecting and analysing the data. References AHTEE, M., UNONIUS, L., NURMELA, M. & SUORTTI, P. (1984). J. Appl. Cryst. 17, CERNIK, R. J. & HART, M. (1989). Nucl. lnstrum. Methods, A281, CERNIK, R. J., MURRAY, P. K., PATTISON, P. & FITCH, A. N. (1990). J. Appl. Cryst. 23, Cox, D. E. (1989). Private communication. FAWCETT, T. G., CROWDER, C. E., BROWNELL, S. J., ZHANG, Y., HUBBARD, C., SCHREINER, W., HAMILL, G. P., HUANG, T. C., SABINO, E., LANGFORD, J. I., HAMIL- TON, R. & LOUI~R, D. (1988). Powder D(ff. 3, HASTINGS, J. B., THOMLINSON, W. & Cox, O. E. (1984). J. Appl. Cryst. 17, LANGEORD, J. I. (1978). J. Appl. Cryst. II, LANGEORD, J. I., DELHEZ, R., DE KEIJSER, TH. H. & MITTEMEIJER, E. J. (1988). Aust. J. Phys. 41, LANGFORD, J. I., LOUF.R, D., SONNEVELD, E. J. & VISSER, J. W. (1986). Powder D/ff. l, LOU~R, D. & LANGFORD, J. I. (1988). J. Appl. Cryst. 21, WILSON, A. J. C. (1962). X-ray Optics. London: Methuen.