Estimation of the Mosaic Spread and the Mosaic-Block Size of Zinc Single-Crystal Spheres by Simulations of ~, Scans with UMWEG90

Transcription

1 510 J. Appl. Cryst. (1994). 27, Estimation of the Mosaic Spread and the Mosaic-Block Size of Zinc Single-Crystal Spheres by Simulations of ~, Scans with UMWEG90 BY ELSABETH ROSSMANTH, GUNAD ADWDJAJA, JOACHM ECK, GABRELA KUMPAT AND GERD ULRCH Mineralogisch-Petrographisches lnstitut der Universitfit Hamburg, D Hamburg, Grindelallee 48, Germany (Received 25 March 1993; accepted 24 November 1993) Abstract n a previous paper [Rossmanith (1992). Acta Cryst. A48, ], new formulae for the Lorentz factor and peak width for single and multiple diffraction were given. These formulae are used in the program UMWEG90 for the calculation of if-scan simulations. n this paper, simulations are compared with five different i scans of two different zinc single-crystal spheres measured with Mo and Ag K~ radiations, respectively. t is shown that, by fitting the calculated to the measured i scans, consistent and physically significant parameters for the mosaic structure parameters - mosaic spread and mosaic-block size - and for the divergence parameters of the X-ray beam are obtained. Moreover, it is shown that the parameters obtained from multiple-diffraction experiments are consistent with those resulting from single-diffraction experiments performed with synchrotron radiation. ntroduction During a i scan (Renninger, 1937), the crystal is rotated about the normal to the reflecting plane, whose Bragg intensity is measured (Fig. 1). n the scan technique, for each step in i, a 0-20 scan is performed. ntensity due to multiple diffraction is recorded in the counter if at least three reciprocallattice points lie simultaneously on the Ewald sphere (Fig. 2). These are the zero point, O, of the lattice, the point B belonging to the primary reflection hprim and the point O' belonging to the operative reflection hop. The X-ray beam, incident parallel to s o, is diffracted in the sl direction as well as in the s2 direction. The reflected beam in the s2 direction acts as the incident beam for the cooperative reflection hcoop, which in turn reflects part of this beam in the s 1 direction. n the case of a forbidden or, at least, very weak primary reflection hprim, the intensity observed in the sl direction is caused only by this double reflection, i.e. by Umweganregung. ntegrating the intensity measured in the scan technique with respect to 0 and plotting it against i results in the case of a forbidden or, at least, very weak primary reflection in the so-called Umweganregung pattern (Figs. 3a, 4a, 5a, 6a and 7a). The nonzero intensity between Umweganregung events in these experimental patterns corresponds to the background intensity of the measurements.. s, oax,s ' ~ " 20 Fig. 1. The 0-20-@ scan (1994) nternational Union of Crystallography Printed in Great Britain - all rights reserved Fig. 2. The geometry of multiple diffraction in reciprocal space. Journal of Applied Crystallography SSN ((~ 1994

2 E. ROSSMANTH, G. ADWDJAJA, J. ECK, G. KUMPAT AND G. ULRCH 51 1 The Umweganreyung pattern is calculated by UMWEG90 (Rossmanith, 1992) using l(oi) = scale factor x ~.f(~l.z)l(oi)op l(0i),,p plzlolq, F 2 2 = o~f~oo~[(1/o,) + (1/o9] x ((1 GL) exp [--1(0 i op) /0"1,2] +GL{ +[2(Oi-O,,p)/AO~.2]2}-~), op (la) (lb) where (0~) is the total intensity of the pattern for a particular step i in 0, J'(~.z) depends on the intensity ratio of the K~ and K:~ 2 radiations, l(0i)op is the intensity contribution of a particular Umweganregunq event, with the profile maximum at the azimuthal angle 0op, P~ 2 is the polarization factor, L 0 and L, are the Lorentz factors for the 0 and 0 rotations, respectively, and Fop and F~oop are the structure factors of the operative and cooperative reflections. The intensity profiles are approximated by asymmetric pseudo-voigt distributions, o-1. 2 and A01.2 are the standard deviations and the full widths at halfmaxima (FWHMs) used for the calculation of the left and right halves of the theoretical profile, respectively, and G L is a measure of the percentage of the Gaussian and Lorentzian contributions to the distribution. Lo and L o, as well as a~.2 and AO,2, are functions of the divergence parameters 6, and 6p (defined in Fig. 8 of Rossmanith, 1992) and of the wavelength spread A2/2 of the incident beam, as well as of the mosaic spread q and mosaic-block radius r of the single-crystal sample.* A new concept for the calculation of L o, Lo, a and A0 is given by Rossmanith (1992). The concept used in UMWEG90 for the 0-scan simulation, especially for the calculation of the FWHMs and the Lorentz factors, differs appreciably from the concepts used in previous work as published, for example, by Caticha-Ellis (1969), Soejima, Okazaki & Matsumoto (1985) and Rossmanith (1986). t is obvious from (lb) that in the actual version of the program UMWEG90 extinction and absorption effects are neglected. This and the fact that the actual version of UMWEG90 does not allow for the anisotropic mosaic parameters q and r results in discrepancies between measured and calculated intensities, especially for the strong peaks in the Umweganregun9 patterns. For this reason, no least-squares fitting routine has yet been added to the program so no statistics (agreement index, standard deviations, correlations) have yet been output by UMWEG90. Experimental Two zinc single-crystal spheres (Zn3 and Zn8), with diameters given in Table 1, were used for the * t is shown by Rossmanith (1994) that r' = [in 2/(4n)]l/2r has to be used instead of r in the expressions defined by Rossmanith (1992, 1993a,b). Table 1. Parameters used for calculation of the Umweyanregun9 patterns Characteristics of the incident X-ray beams Radiation A2/), 6p ( ) Mo K~ Mo Ka{ Ag K~ Ag K~ Characteristics of the samples Sample Diameter (gm) q () Zn Zn ,6~ C) GL r (gm) measurements. (Space group P63/mmc; structure h.c.p.; forbidden reflections hhl, 1 2n; 'almost' forbidden reflections h - / = 3n, l : 2n; lattice constants a = 2.666, c= A: anisotropic temperature parameters fill = fl22 = , fl33 = , fl12 = fl,1/2, fl13 = fl23 = ) The measurements were performed with a CAD-4 diffractometer (Enraf-Nonius) using Mo K~ and Ag K:~ radiations. The wavelength spreads A2/2 (Table 1) used for the calculation of L, a and A0 are equal to the half-widths of the corresponding characteristic lines (Compton & Allison, 1935). The 002 reflection of a graphite single crystal was used for monochromatization of the primary X-ray beam. The number of steps in all the 0-20 scans was 96. The scan width was ~ for Zn3 and ~ for Zn8. The 10 min measuring time for each 0-20 scan per 0 step was chosen to guarantee that the standard deviations a(l~) of the individual integrated intensities (Oi) were less than --~ 1% of these intensities, even in the low-intensity region. The minimum and maximum measured integrated intensities 1(0;) of each Umweganregun9 pattern are given in the captions of the corresponding figures. Comparison between measured and calculated Umweganregung patterns Because of the lack of a least-squares routine in UMWEG90, the simulations given in Figs. 3, 4, 5, 6 and 7 had to be chosen from theoretical 0 scans calculated with varying parameters 6~, 6p, r/, r and the scale factor. (A2/;~ is known for X-ray tubes, see Table 1.) Figs. 3, 4, 5, 6 and 7 display the Umweffanregung patterns with the best agreement between measured and calculated intensities (0i) that are not affected by extinction. [The magnitude of the extinction correction for the individual reflections is known from Rossmanith, (1977).] The parameters used for the calculation of these patterns are also given in Table 1. The quality of the agreement was ascertained by visual inspection, i.e. by superimposing transparent copies of the measured scans

3 512 ESTMATON OF MOSAC SPREAD AND MOSAC-BLOCK SZE and the respective simulations. This qualitative fitting procedure was facilitated by the fact that it is not just the widths of the peak profiles that are determined by the parameter set used for calculation. t is obvious from (la) and (lb) that the heights of the peaks are also dependent on these parameters. The influence of the individual parameters strongly depends on the locus at which the particular reciprocal point passes the Ewald sphere. Each peak profile in the Umweganregun9 pattern is therefore individually sensitive to parameter changes. Although only a qualitative and therefore preliminary estimation of the parameters is possible using this method, it is remarkable that all the scans given in Figs 3, 4, 5, 6 and 7 are obtained using the consistent and physically significant parameter values given in Table 1. [Further examples will be published by Rossmanith & Bengel (1994).] The patterns in each of the Figs. 3(c), 4(c), 5(c), 6(c) and 7(c) represent the Umweganregun9 peak location plot (~-2 diagram) for the 2 region KOtl-K~t2, corresponding to the measured and calculated 1 l i i t /, t t t t ~ ! (c) KOt2K~l - ' -" "''-'~'"-" ',\ ' ~,/' ~ ' " ""J ~b [degreesl Fig. 3. The 1il reflection of Zn8 (Mo Kct, AqJ =0.05 ). Experimental Umweoanregung pattern. Calculated Umweganregun 9 pattern. (c) Umweganregun 9 peak-location plot. l(qji),,i. = counts/10 min, (~'i)max = counts/ patterns and in Figs Only the 2 regions above (A2x~,2) and below (A2r~,,) the respective lines in the patterns (c) are involved in the experiment. Because of the short wavelengths used in the experiment, the density of multiple-diffraction events [lines in the patterns (c)] is large. However, from the patterns of Figs. 3-7, it is obvious that most of the corresponding multiple-diffraction peaks are too small (in comparison with the background) to be observed in the Umweganregung pattern. The calculated patterns predict this fact correctly. Because of the known orientation matrix, the ~O values defined by the CAD-4 geometry can be calculated with UMWEG90 on an absolute scale. The quality of the agreement between the experimental and calculated ff values is apparent in Figs Because of the different orientation matrices of the two zinc samples, the 301 Umweganregung patterns presented in Figs. 5 and 6 are not identical. The results for sample Zn8 The measurements of the 17 pattern of the forbidden 111 reflection (Fig. 3a), the 15 ~, scan of the 'almost' forbidden 303 reflection (Fig. 4a) and the 55 pattern of the 'almost' forbidden 301 reflection (Fig. 5a) were all performed with an Mo Ka tube and a step width in ff equal to 0.05% The patterns given in Figs. 3, 4 and 5 were calculated with the same set of parameters A2/2, 6s, 6p, r/and r (Table 1), in accordance with the fact that the same X-ray tube and crystal were used for the corresponding measurements. The 111 Umweganregung pattern - Fig. 3 General views of the scans are given in the inserts of Figs. 3 and. The intensity is given in arbitrary units. The two largest peaks of the pattern are mainly due to the four-beam cases 111/200, 100/011 (ff = 58.8 ) and 1]-0/021,201/]-10 (~ = 65.5 ), respectively. Except for the peak at ~, = 58.8, the overall agreement between measured profiles and those calculated using the parameters given in Table 1 is satisfactory. n the main parts of Figs. 3 and, the scans are given on an enlarged intensity scale. n this presentation, the surprisingly good agreement between experiment and theory for the weak multiplediffraction events becomes more obvious. n particular, the complicated structure of the intensity profile at ~ = is well predicted by UMWEG90, indicating that the formulae deduced in the framework of the kinematical theory are correct. Considering the simple model underlying the calculations - neglection of extinction and absorption, isotropic mosaic spread, approximation of the mosaic blocks by spheres - the disagreement observed for the peak

4 ' ' E. ROSSMANTH, G. ADWDJAJA, J. ECK, G. KUMPAT AND G. ULRCH 513 at ~O = 58.8, which is due to strong reflections, is not surprising. Especially because of the neglect of extinction and absorption, this peak is overestimated by UMWEG90. The 303 Umweyanregung pattern - Fig. 4 The intensity of each peak of the pattern is mainly due to one three-beam case. The largest peak is remarkable because of its large angle fl = 148 [see formula (17) and Fig. 8 of Rossmanith (1992)]. n Fig. 4(d), the geometry of this multiple-diffraction event is given in reciprocal space in correct proportion. The operative reflection 100 passes the Ewald sphere in the vicinity of the zero point of the reciprocal lattice. The contribution of the wavelength spread A2/2 to the width of this peak is therefore small. The width of the corresponding calculated peak is mainly due to the enlargement of the reciprocal-lattice points to 'spheres', i.e. to the particle size effect. The excellent agreement between the two corresponding profiles in Figs. 4 and confirms the concept for peak-width calculation introduced by Rossmanith (1992). (c) The 301 Umweyanregun9 pattern - Fig. 5 The main contribution to each peak in Fig. 5 is due to one three-beam case. The largest peak in Fig. 5 is caused by the strongest possible operative reflection. Once more, the extinction effect explains the overestimation of this peak by UMWEG90. The results for sample Zn3 For Zn3, two scans were measured: the 55 Umweganregung pattern of the 'almost' forbidden 301 reflection with Mo K~ radiation and a step width in ~p equal to 0.05 (Fig. 6a); and the 30 ~ Umweganregung pattern of the forbidden 003 reflection with Ag K~ radiation and a step width in ~ equal to 0.02 (Fig. 7a). For the 301 scan, the characteristics of the incident beam are those of the three previous scans. Only the 100 / 203 K~ ~b axis ~_~ B:\ / ~ e ~ " ~ Ka J i i a J i l i i,... 1oo s,,...a......,x......, (c) t r aject or y,~x,... ~ 0 of 100 m [degrees] (d) Fig. 4. The 303 reflection of Zn8 (Mo Kct, A@ = 0.05").,, (c) as in Fig. 3. /(]/i)mi n = 9584 counts/10 min, /(]/i)ma x = counts/ (d) The geometry of the multiple diffraction event at ~k = 27.2T' (K~tt) and ~9 = ~ (K~z) in reciprocal space: elevation and plan of the trajectory of the operative lattice point 100.

5 514 ESTMATON OF MOSAC SPREAD AND MOSAC-BLOCK SZE characteristics of the sample may be different. n the case of the 003 reflection, 6, r/and r had to be adjusted. The mosaic-spread parameter for Zn3, 1/= 0.03 ~ of Table 1, is larger than that for ZnS. f it is assumed that the mosaicity is more pronounced at the surfaces of the crystal spheres, the larger overall mosaic spread for Zn3 may be explained by its smaller diameter, i.e. its larger surface-to-volume ratio. t is also physically significant that the same values for the divergence were obtained for the Ag K~ beam and the Mo K~ beam. The 301 Umweyanre,qun9 pattern - Fig. 6 Besides the two four-beam cases 3 and 4 in Fig. 6, all the peaks of the scan are mainly built up by one three-beam case. Peaks 1 and 2, which are due to the strongest operative reflections, are again overestimated in the simulation because of extinction effects. [ i i L i i l i L o 5 i i ] i. i The 003 Umwe,qanre,qun9 pattern - Fig. 7 Because of the symmetry in the hexagonal-closepacked crystal parallel to the [001] direction, all the peaks in the scan are due to four-beam cases. Nevertheless, the agreement between the experiment and the simulation calculated in the framework of the kinematical theory is surprisingly good. The overestimation of the two left-most profiles in Fig. 7, which are due to the strongest possible operative reflection, can again be explained by the neglect of extinction by UMWEG90. i (c) Kc~ ,,... j\... 'i... ldegrees] Fig. 5. The 301 reflection ofzn8 (Mo K:t, AqJ = 0.05").,, (c) as in Fig. 3. l(~ki)mi, = 9052 counts/10 min, l(~'i)max = counts/ -50 i i T F i J L i L L ( >,,' t 1 / i F i i S A (c) K~2.k..... JVLJ/a,k_ b, ((9 K~2 Kal ' [degreesl Fig. 6. The 301 reflection ofzn3 (Mo Ks, Aft = 0.05':).,,(c) as in Fig. 3. l(//i)mi n = 6971 counts/10 min,/(,ki)ma x = counts/ Kcd 4' [degrees] -~ Fig. 7. The 003 reflection of Zn3 (Ag Ks, Aft = 0.02~).,, (c) as in Fig. 3. l(]/i)mi n = 9000 counts/10 min,/(,bi)ma x = counts/

6 E. ROSSMANTH, G. ADWDJAJA, J. ECK, G. KUMPAT AND G. ULRCH 515 Comparison with results obtained from Bragg reflections measured with synchrotron radiation n Rossmanith (1993a), the new concept for the calculation of the half-width of intensity profiles, proposed in Rossmanith (1992), was applied to Bragg reflections measured with the triple-crystal diffractometer at a synchrotron radiation source. n Rossmanith (1993b), a simple approximation for the expression of the FWHM defined in Rossmanith (1993a) is given. t was shown in Rossmanith (1993a) that, for the experimental equipment used at the beam line D3 at HASYLAB (DESY, Hamburg, Germany), the divergence 6 and wavelength spread A2/2 recorded by the sample can be calculated using formulae (4), (5a) and (5b) of Rossmanith (1993a). nserting the radius of the Zn3 sample and 2 = /~ into these formulae, we obtain 6Zn3 = and (A2/2)z,3 = With the formulae (6a)-(6c) of Rossmanith (1993a) or with the simplified version (6b) of Rossmanith (1993b), the FWHM can be calculated as a function of the Bragg angle 0. n Fig. 8, experimental FWHMs of Bragg intensity profiles of various reflections of the sample Zn3 (symbols) are compared with theoretical ones (curves 1 to 3). The measurement of the Bragg reflections was carried out in the routine og-step-scan mode on the Stoe five-circle diffractometer at HASYLAB with 2 = A. Owing to the small divergence of the synchrotron radiation, the FWHMs mainly depend on the mosaic spread q, the mosaic-block radius r and the wavelength spread A2/2. The contribution to the FWHM caused by the particle size effect A 0r can be calculated approximately ~ OO6 ~- 004 "r u ~ ~ " ~ ~ ~ ~)~) 0 O [degrees] Fig. 8. Comparison of calculated FWHMs with the experimental FWHMs of Zn3 obtained with ). = ~. ~ Experimental FWHMs measured with synchrotron radiation. The size of the symbols corresponds to the uncertainties of the measurement. 001 reflections with l= 2, 4, 6, 8, 10. A2/2 = , 6= Curve 1:q=0, r=50pm. Curve 2:r/=0.03, r = 50 pm. Curve 3: q = 0.03% r = 0.23 ~m. by expression (3c) of Rossmanith (1992), taking into account expression (3h) of Rossmanith (1994) (see footnote in the ntroduction). By insertion of the radius of the Zn3 sphere, r = 50 pm, the vanishingly small contributions AOr = for the Bragg angle 0 = 45 and AOr = for 0 = 6 are obtained. For the mosaic-block size r = 0.23 pm, on the other hand, the particle-size effect results in AOr = for 0 = 45"' and AOr = for 0 = 6L Curve 1 in Fig. 8 is calculated by insertion of A2/2= ,,6=0.0019, q=0 and r=50pm into (6a)-(6c) of Rossmanith (1993a). [or (6b) of Rossmanith (1993b)]. t represents, therefore, the FWHM contribution due to the wavelength spread and divergence, which were both calculated from purely geometrical considerations. The deviation between this curve and the experimental FWHMs must therefore be caused by the Zn3 sample. nsertion of the mosaic parameters q and r obtained from the Umweyanreyun9 patterns results in curves 2 and 3. Curve 2 is calculated with the parameters A2/2= , 6=0.0019, r/=0.03 ~" and r= 50 pm, i.e. with the particle-size effect neglected. Curve 3 is calculated as before but the parameter r is replaced by 0.23 pm, i.e. by the value used for the calculation of the 0-scan simulations shown in Figs. 5, 6 and 7. The distance between curves 2 and 3 therefore represents the particle-size effect caused by ideally perfect mosaic blocks with r = 2300/~. Because of the large anisotropy of the mosaic structure in the zinc samples, known from conventional structure analyses, i.e. from extinction corrections (Rossmanith, 1977), the FWHMs at a particular Bragg angle 0 in Fig. 8 scatter within a band of,~ 0.02 On the other hand, the mosaic-structure parameters should be the same for reflections whose scattering vectors are parallel. As an example, the 001 FWHMs are marked by black symbols in Fig. 8 (l = 2, 4, 6, 8, 10). Curve 3 agrees very well with the FWHMs of the 001 reflections. t can therefore be concluded from Fig. 8 that the parameters used for the calculation of curve 3 characterize the mosaic structure in the c direction of the lattice and that the anisotropic mosaic spread 1/ in Zn3 takes values between 0.01 and 0.03 ~'. The isotropic r/ obtained from the C-scan simulation corresponds to the upper limit of this q band, indicating that the magnitude of the values obtained for r/ and r by the C-scan fits measured with X-ray tubes are consistent with the parameters obtained from the fit of the Bragg reflection FWHMs measured with synchrotron radiation. Concluding remarks The new concept for the calculation of the Lorentz factors and half-widths of multiple-diffraction Bragg

7 516 ESTMATON OF MOSAC SPREAD AND MOSAC-BLOCK SZE intensity profiles has been tested on five different scans measured with radiation of two different wavelengths and with two different zinc samples. The results are compared with those of single diffraction. Consistent and physically significant parameters for the characteristics of the incident beam as well as for the mosaic-structure parameters are obtained. Disagreements between experiments and calculations can be explained by the simplicity of the underlying model - the assumption of isotropic mosaic structure and the neglect of extinction and absorption. t is therefore concluded that the concept for the calculation of the Lorentz factor and peak width that was deduced within the framework of kinematical theory can be successfully applied to ideally mosaic crystals. An even better agreement between experiment and simulation can be expected when a forthcoming version of UMWEG90 that takes into account extinction and absorption is used. References CATCHA-ELLS, S. (1969). Acta Cryst. A25, COMPTON, A. H. 8~ ALLSON, S. K. (1935). X-rays in Theory and Experiment. Princeton, New Jersey: van Nostrand. RENNNGER, M. (1937). Z. Phys. 106, ROSSMANTH E. (1977). Acta Cryst. A33, ROSSMANTH E. (1986). Acta Cryst. A42, ROSSMANTH E. (1992). Acta Cryst. A48, ROSSMANTH E. (1993a). Acta Cryst. A49, ROSSMANTH E. (1993b). J. Appl. Cryst. 26, ROSSMANTH E. (1994). Acta Cryst. AS0, ROSSMANTH E. ~ BENGEL, K. (1994). Acta Cryst. Submitted. SOEJMA, Y., OKAZAK, A., MATSUMOTO, T. (1985). Acta Cryst. A41,