research papers 1. Introduction 2. Experimental E. Rossmanith, a * A Hupe, a R. Kurtz, a H. Schmidt a and H.-G. Krane b

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1 Journal of Applied Crystallography ISSN Received 14 September 2000 Accepted 17 January 2001 Kinematical two-dimensional multiple-diffraction intensity profiles. Application to x±w scans of silicon and diamond obtained with synchrotron radiation E. Rossmanith, a * A Hupe, a R. Kurtz, a H. Schmidt a and H.-G. Krane b a Mineralogisch-Petrographisches Institut der UniversitaÈt Hamburg, D Hamburg, Grindelallee 48, Germany, and b Mineralogisch-Petrologisches Institut, UniversitaÈt Bonn, Poppelsdorfer Schloû, D Bonn, Germany. Correspondence rossmanith@mineralogie.uni-hamburg.de # 2001 International Union of Crystallography Printed in Great Britain ± all rights reserved In a previous paper by Rossmanith [J. Appl. Cryst. (2000), 33, 1405±1414], expressions for the calculation of multiple-diffraction patterns observed in!± scans of Bragg re ections were derived within the framework of the kinematical theory, taking into account the divergence and wavelength spread of the incident beam, as well as the mosaic structure of the crystal sample. Agreement with Cu K experiments was demonstrated. In this paper, it is shown that the theoretical expressions are also suitable for synchrotron radiation experiments. 1. Introduction In a paper by Rossmanith (2000c), cited as Ro-00c hereinafter, expressions for the calculation of two-dimensional multiplediffraction intensity pro les were presented, taking into account the pro le-broadening effects caused by the magnitude and shape of the perfect crystallites (mosaic blocks) and the mosaic spread of the sample, as well as the divergence and wavelength spread of the incident X-ray beam. The theoretical results obtained within the framework of the kinematical approach were compared with measurements obtained in a `real experiment', i.e. performing a routine 190!± scan with Cu K radiation for the 222 multiple-diffraction pattern of a mosaic diamond crystal. In the case of the Cu K experiment, the widths of most of the intensity pro les are dominated by the divergence of the incident beam, of about 0.12, which masks the comparatively small contributions of the wavelength spread and the crystallite-size effect. Consequently, the intensity pro les of Bragg re ections of perfect and mosaic crystals with moderate Lorentz factors will have similar widths and shapes. On the other hand, crystal-monochromatized synchrotron radiation (the properties of which will be discussed in x2.1) differs appreciably from the Cu K radiation used in Ro-00c. Because of the small divergence of about 0.002, it can be expected that the contribution of the wavelength spread will dominate in the case of perfect crystals, whereas in the case of a mosaic crystal, the much larger contributions arising from the mosaic block size and the mosaic spread will have the determining in uence on the width and shape of the intensity distribution. Consequently, it is expected that!± scans measured with synchrotron radiation should differ appreciably between mosaic and perfect samples. Agreement between theoretical and experimental!± patterns of a perfect as well as a mosaic crystal of the same substance would therefore con rm the validity of the kinematical derivation, i.e. it would be an indication for the justi- cation of the approximations made in the corresponding program 3d-UMWEG, a modi ed version of UMWEG99 (Rossmanith, 1999, cited as Ro-99 hereinafter). Because of the lack of a perfect crystal of diamond, in this paper equivalent!± scan ranges of a perfect silicon crystal and a mosaic diamond crystal will be compared instead, making use of the additional advantage of the synchrotron radiation, namely its tunability. The choice of an appropriate ± range for silicon and diamond will be discussed in x2.2. The surface and contour plots of the multiple-diffraction patterns presented in x3 are vivid examples of the applicability of the expressions given in Ro-00c and used in the corresponding program 3d-UMWEG.In x3, attention is focussed on the in uence of the divergence and wavelength spread of the incident synchrotron radiation beam. The effect of primary and secondary extinction and absorption on the width and height of the intensity pro les will be discussed elsewhere. J. Appl. Cryst. (2001). 34, 157±165 E. Rossmanith et al. Multiple-diffraction intensity profiles Experimental 2.1. Divergence and wavelength spread recorded by the silicon and diamond samples The arrangement of the triple-crystal system used at the synchrotron radiation source at beamline D3 at HASYLAB (DESY, Hamburg, Germany), Fig. 1, has been discussed fully by Rossmanith (1993), cited as Ro-93 hereinafter. It was shown therein that, neglecting in a rst approximation the effects on the intensity distribution caused by the two monoelectronic reprint

2 by the second monochromator crystal with negative M p and smaller wavelength ( M p < M p ˆ0) will correspond to a positive angle, S p, between the particular ray and the central ray at the sample, and vice versa, i.e. S p = M p. The wavelengths + in the expressions (14) and (15) given in Ro-00c and used in 3d- UMWEG are therefore related to S p according to ˆ S p ˆ 2d 111 sin 111 S p : 3 Figure 1 The arrangement of the triple-crystal system used at the synchrotron radiation source at HASYLAB. chromator crystals (I and II in Fig. 1), the divergence recorded by a small sample bathed in the incident beam depends solely on geometrical factors and is independent of the wavelength: ˆ 2 arctan s=2 r =L; where s = 0.11 cm is the vertical dimension of the synchrotron radiation source, r is the radius of the spherical perfect crystal and L = 3731 cm is the distance between the source and the sample. Following the reasoning given in Ro-93, each ray diffracted by the (111) planes of the two perfect silicon monochromator crystals parallel to the re ection plane will have its characteristic wavelength, M p, depending on the angle M p between the particular ray and the central ray: 1 M p ˆ 2d 111 sin 111 M p ; 2 where d 111 and 111 are respectively the interplanar spacing and the Bragg angle of the (111) plane of the silicon monochromator crystals. The same natural cloudy diamond sample and perfect silicon sample as in the previous papers (Ro-00c; Rossmanith, 2000b, cited as Ro-00b hereinafter) were used for the experiment. The divergence,, and the corresponding wavelength range,, calculated for these samples according to expressions (1) and (2) are given in Table 1. The measurements were performed with the samples arranged antiparallel (position III in Fig. 1) with respect to the second monochromator crystal. Consequently, rays diffracted In a routine experiment, the intensity distribution function of the part of the beam impinging on the sample will not be known. Assuming an equal probability for all of the rays of this part of the beam, the normalized distribution function corresponds to a rectangle with the area = 1 and with a full width at half-maximum (FWHM) corresponding to the divergence de ned in equation (1). Taking into account the modi cation of the intensity distribution caused by the diffraction by the monochromator crystals, in 3d-UMWEG this normalized rectangular distribution is replaced by a normalized Lorentzian or Gaussian distribution function with the same FWHM. The additional divergence and wavelength spread caused by the Darwin width of the monochromator crystals can also be taken into account in 3d-UMWEG. But, because of CPU-time considerations, the Darwin width of the monochromator crystals will be neglected in the calculations presented in this paper Choice of an appropriate w±k range In Fig. 2, the ± diagram (peak-location plot) for the `almost forbidden' 222 re ection of diamond and silicon is shown. Each line in this plot corresponds to an Umweganregung event, i.e. an operative lattice point, which crosses the Ewald sphere twice during the 360 rotation about the axis (out! in, in! out). The angle of rotation between these two Table 1 Parameters used as input for 3d-UMWEG. Diamond Silicon R (mm) r (mm) ( ) ( ) (A Ê ) (A Ê ) a (A Ê ) (8) (17) ii (A Ê 2 ) f 0 () f 00 () F Figure 2 The ± diagram (peak-location plot) for the `almost forbidden' 222 re ection of diamond and silicon. Only strong Umweganregung events are shown. Ordinate (A Ê ) on the left side of the diagram corresponds to silicon, ordinate (A Ê ) on the right side of the diagram corresponds to diamond. 158 E. Rossmanith et al. Multiple-diffraction intensity profiles J. Appl. Cryst. (2001). 34, 157±165

3 crossing points, 2 (Cole et al., 1962; Post, 1975; Rossmanith, 1992) decreases with increasing wavelength. The maximum of each line in Fig. 2 corresponds to 2 =0,i.e. to the case in which the operative lattice point does not penetrate the sphere of re ection, but touches it at one point only. These maxima are of particular interest from the point of view of this paper, because it can be expected that in their vicinity the effect of the wavelength range on the multiple-diffraction pattern will be most pronounced. Because of the equivalent structure of diamond and silicon, their peak-location plots differ only in their ordinate, which is given in Fig. 2 on the left side of the diagram for silicon and on the right side of the diagram for diamond. The wavelength corresponding to the diamond crystal is related to that of silicon according to C = Si ˆ a C =a Si ; 4 where a C and a Si are the lattice parameters of diamond and silicon, respectively. The experimental lattice parameters a, given in Table 1, are obtained from the orientation matrix of the Huber diffractometer. In Fig. 2, only the strongest Umweganregung events are shown. It is obvious from the gure that there is a repetition of the ± pattern after each 60. Because of the additional strong lines in the vicinities of the maxima at =30 and = 150, corresponding to the operative re ections 515 and 155, respectively, these maxima were chosen for the experiment. The geometry of these Umweganregung events in reciprocal space is represented in Fig. 3, showing the reciprocal-lattice vectors and the radius of the Ewald sphere of the central ray in the correct proportion, but exaggerating the divergence and wavelength range of the incident beam and the magnitude of the `reciprocal-lattice spheres' (see Rossmanith, 2000a, cited as Ro-00a hereinafter). 3. Comparison of experimental 222 multiple-diffraction patterns of silicon and diamond with theoretical patterns The multiple-diffraction patterns of the two samples, diamond and silicon, were measured using the routine!± scanning technique with a Huber four-circle diffractometer. The experimental parameters used for input to 3d-UMWEG are given in Table 1. Mainly because of the different wavelength used for this experiment, the mosaic parameters of diamond given in Table 1, which were estimated according to the method described in Ro-93, differ slightly from those given in previous work. This point will be discussed in more detail in a forthcoming paper concerning the effect of secondary extinction on multiple-diffraction intensities. In the case of the perfect spherical silicon crystal, the sample radius R is equal to the mosaic-block radius r and the mosaic spread is zero. The same temperature parameters, ii (Table 1) and ij =0, as in the previous papers, Ro-00a, Ro-00b and Ro-00c, were used as input for 3d-UMWEG. The anomalous-dispersion corrections for the atomic form factor, f 0 and f 00, of silicon, were evaluated with the program ABSORB (Brennan & Cowan, 1992) The multiple-diffraction pattern of the perfect silicon crystal The ± diagram of the experimental 10 scan of the `almost forbidden' 222 re ection of silicon calculated for = A Ê, with = A Ê (see Table 1), is given in Fig. 4. Very weak Umweganregung and Aufhellung events are omitted. The lines with stronger intensities are marked by the indices of the corresponding operative re ections. The zero point of the axis corresponds to the reference direction [110] of the silicon crystal. As a result of the wavelength range, 2, in Fig. 4 the angle 2 of the line corresponding to the 515 operative re ection decreases from about 4 to zero. In Fig. 5, the surface plot of the experimental 0.081±10!± scan measured at = A Ê is shown. For each of the 100 Figure 3 The geometry of the Umweganregung events near the maxima denoted in Fig. 2. Shaded spheres correspond to the `reciprocal-lattice spheres' of the primary re ection 222 and the operative re ection 515 (silicon) or 155 (diamond), respectively. Figure 4 The ± diagram (peak-location plot) for the `almost forbidden' 222 re ection of silicon corresponding to the wavelength range and range of the experiment. Very weak Umweganregung and Aufhellung events are omitted. The stronger lines are marked by the indices of the corresponding operative re ections. The zero point for corresponds to the [110] direction. J. Appl. Cryst. (2001). 34, 157±165 E. Rossmanith et al. Multiple-diffraction intensity profiles 159

4 Figure 5 Surface plot of the experimental!± scan of the `almost forbidden' 222 re ection of silicon. The zero point for corresponds to that of the Huber diffractometer. The measurement was performed with 81 steps in! with a step width of! = and 100 steps per degree in at = A Ê. steps per degree in,an! scan was performed with 81! steps and a step width of! = The zero point for in Fig. 5 corresponds to that of the Huber diffractometer, which is displaced from the reference direction [110] of the crystal by (see Fig. 5 in Ro-99 for the calculation of the displacement). In Fig. 6, the theoretical contour plot (Fig. 6b) of the 0.02± 10!± scan calculated by 3d-UMWEG with 200 steps in! and steps in is compared with the corresponding part of the experimental!± scan (Fig. 6a). For reasons of CPUtime minimization, for this gure, only the! region corresponding to the 21 most interesting experimental! steps in the peak maximum of the primary re ection was included in the calculation. The theoretical pattern (Fig. 6b) corresponds to the `real experiment', i.e. the expressions (14)±(16) given in Ro-00c and the parameters of Table 1 were used for calculation, taking into account the relationship (3) between the wavelength + and the divergence p. The intensity distribution with respect to the divergence of the incident beam was assumed to be a Lorentzian distribution with an intensity maximum at the central ray with = A Ê and FWHM = Using a PC with a 500 MHz Intel Pentium III processor, the calculations were performed in about 40 h, neglecting (for CPU-time limitation) the divergence normal to the re ecting plane, n. It is obvious from Fig. 6(a), as well as Fig. 6(b), that, as a result of the wavelength range of the incident X-ray beam, the Umweganregung event corresponding to the 515 operative re ection is visible in the whole range, of more than 4 with two intensity maxima at = and , corresponding to the wavelength of the central ray. Remarkable similarity between the ± diagram (Fig. 4) and the contour plots (Fig. 6) is noticeable. The intensity distribution of the!± scan of the perfect silicon crystal sphere is obviously mainly determined by the intensity distribution of the divergence and the corresponding wavelength range of the incident X-ray beam [expressions (1) and (3)]. This becomes even more obvious from Figs. 7(a) and 7(b), where the theoretical!± intensity distributions of the 515 Umweganregung event calculated with the Lorentzian distribution and with the Figure 6 Contour plot of the!± scan of the `almost forbidden' 222 re ection of silicon. The zero point for corresponds to that of the Huber diffractometer. Only the middle 21 steps per in! are considered. (a) Measurement. (b) Calculation with 3d-UMWEG. Figure 7 Surface plot of the 515 Umweganregung event calculated with 3d- UMWEG. Only the middle 41 steps per in! are considered. (a) Lorentzian distribution of the intensity distribution of the incident beam. (b) Gaussian distribution of the intensity distribution of the incident beam. 160 E. Rossmanith et al. Multiple-diffraction intensity profiles J. Appl. Cryst. (2001). 34, 157±165

5 Gaussian distribution, respectively, are compared. The! range in Figs. 7(a) and7(b) corresponds to the middle 41! steps (the rst 20 and last 20 steps being ignored) of the measurement. There is no doubt that the Lorentzian distribution (Fig. 7a) results in a better t with the experiment than the Gaussian distribution (Fig. 7b). Figs. 8 and 9, which show a part of Fig. 5 (! range as in Fig. 5; range 222±228 ), provide the evidence for this fact. The theoretical surface plots of the 0.081±6!± scans (Figs. 8b and 9b) were calculated as before (Fig. 6b), but with the number of steps reduced to 81 steps in!, with 100 steps per degree in, according to the experimental conditions of the measured scans given in Figs. 8(a) and 9(a). The saw-toothed appearance of the theoretical Umweganregung events is caused by the insuf cient number of calculated steps in! as well as. Fig. 9 represents a 180 rotation of Fig. 8 about the z axis (intensity axis), i.e. it represents a view from behind of the plots of Fig. 8. The agreement between theory and experiment in the case of the perfect silicon crystal is amazing. For = 222 and 228, the peak corresponding to the 515 Umweganregung event starts and ends at the same! step deep in the background of the primary 222 re ection, in both cases, theory (Fig. 8b) as well as experiment (Fig. 8a). It crosses the peak of the `almost forbidden' primary 222 re ection with increasing intensity, Figure rotation of Fig. 8 about the z axis (intensity axis). (a) Measurement. (b) Calculation with 3d-UMWEG using the Lorentzian distribution. Figure 8 Part of Fig. 5:! range 0.08 ; range 222±228.(a) Measurement. (b) Calculation with 3d-UMWEG using the Lorentzian distribution. both peaks having their maximum at the same! step. Looking from behind the 222 peak (Fig. 9), the decreasing intensity of the 515 Umweganregung event can be observed. The minimum intensity of the 515 Umweganregung event at = corresponds to 2 =0. The behaviour of the intensity distributions of the 515 Umweganregung event and the `almost forbidden' primary 222 re ection shown in Figs. 8 and 9 can easily be understood by considering Fig. 3. For each step in! and, the contributions of all incident rays have to be considered and added. The magnitudes of the contributions of the individual rays will depend on the intensity distribution of the incident beam as well as the particular distances, prim and op, between the reciprocal-lattice points and the sphere of re ection under consideration in the direction of the re ected beams [see expressions (14) and (15) in Ro-00c]. The three reciprocallattice vectors h prim, h op and h coop corresponding to the primary, operative and cooperative re ections, respectively, form a rigid triangle. Rotating the crystal from small to large! values (! is de ned as the angle between the 222 plane and the incident beam), the reciprocal-lattice points 222 and 515 cross (from outside to inside) consecutively the Ewald spheres with decreasing radius. Consequently, for small!, the very strong J. Appl. Cryst. (2001). 34, 157±165 E. Rossmanith et al. Multiple-diffraction intensity profiles 161

6 515 re ection crosses the largest Ewald sphere (with the smallest wavelength) with a large value of the angle 2 (see Fig. 4). The contribution of the ray corresponding to this Ewald sphere to the total!± step intensity recorded in the counter will therefore dominate. The reciprocal-lattice point of the very strong cooperative re ection 313, which is identical to the reciprocal-lattice point of the primary 222 re ection, is outside of this sphere of re ection. Because of this fact and because of the small probability of the corresponding X-ray, the Umweganregung intensity is small and the intensity of the `almost forbidden' primary 222 re ection is negligible. The intensity of the Umweganregung event as well as that of the primary re ection is largest for the most intense central ray, for which the two reciprocal-lattice points lie simultaneously on the corresponding sphere of re ection. Increasing further the value of! will result in a decrease of 2 as well as the intensities. The intensity of the Umweganregung event corresponding to the Ewald sphere with 2 = 0 is strongly dependent on the intensity distribution of the incident X-ray beam. It is therefore evident from the comparison of the theoretical and experimental surface plots that the selection of the Lorentzian distribution function as well as the method of estimating the divergence-to-wavelength relation [expressions (1) and (3)] results in a very good t between theory and experiment in the case of the perfect silicon crystal sphere. The very sharp peaks of additional Umweganregung events in close vicinity to the 515 maxima in Figs. 8(b) and 9(b) correspond to the eight lines (see Fig. 4) which cross one another at ' and ' 226. The wings of the strongest of these lines are also visible in Figs. 8(a) and 9(a), but because of the insuf cient experimental step widths, their sharp peak maxima obviously are not measured properly. The interference effect between the primary re ection and the Umweganregung event manifests itself in an enlargement of the 222 intensity between the two 515/313 multiplediffraction peaks and a reduction of the intensity on the left side and on the right side of the peak pair (Figs. 5, 8 and 9). Although less pronounced, the divergence±wavelength range relation of the incident beam also has an effect on all the other multiple-diffraction events shown in the experimental contour plot of the total scan (Fig. 6a). The inclination of the four additional strong Umweganregung events in Fig. 6(a), which is similar to the tilt of the corresponding lines of the operative re ections 151, 115, 511 and 151 in the ± diagram (Fig. 4), is well predicted by the theoretical contour plot calculated with 3d-UMWEG (Fig. 6b). The experimental surface and contour plots of the part, 228±229 (! range as in Fig. 5) are compared with 3d-UMWEG results in Figs. 10 and 11. The appearance of the wings of the peaks as well as the pronounced interference effect between the primary re ection and the Umweganregung event (Fig. 10a) are well predicted by the kinematical approximation (Figs. 10b). The difference between the experimental and theoretical peak tops is very probably caused by the insuf cient experimental step widths of the measurement as well as by the fact that the divergence normal to the re ection plane, n, is neglected in the calculations. Figure 10 Part of Fig. 5 rotated 180 about the z axis (intensity axis):! range 0.08 ; range 228±229.(a) Measurement. (b) Calculation with 3d-UMWEG using the Lorentzian distribution. Figure 11 Contour plot corresponding to Fig. 10. (a) Measurement. (b) Calculation with 3d-UMWEG using the Lorentzian distribution The multiple-diffraction pattern of the mosaic diamond crystal The ± diagram of the experimental 17 scan of the `almost forbidden' 222 re ection of diamond, calculated for = A Ê, with = A Ê (Table 1), is presented in Fig. 12. Because of the small mosaic blocks, in the case of the mosaic diamond crystal the wavelength range recorded by a particular coherent block is similar to the wavelength range of the silicon experiment. As in Fig. 4, very weak Umweganregung and Aufhellung events are omitted and 162 E. Rossmanith et al. Multiple-diffraction intensity profiles J. Appl. Cryst. (2001). 34, 157±165

7 Figure 12 The ± diagram (peak-location plot) for the `almost forbidden' 222 re ection of diamond corresponding to the wavelength range and range of the experiment. Very weak Umweganregung and Aufhellung events are omitted. The stronger lines are marked by the indices of the corresponding operative re ections. The zero point for corresponds to the [110] direction. only the strongest lines are marked by the indices of the corresponding operative re ections. The zero point of the axis corresponds to the reference direction [110] of the diamond as well as the silicon crystal. In Fig. 13, the theoretical contour plot (Fig. 13b) of the 0.81± 17!± scan calculated with 3d-UMWEG with 81! steps, a step width! of 0.01 and 100 steps per degree in, is compared with the corresponding part of the experimental!± scan (Fig. 13a). As in the case of the Cu K experiment on diamond (Ro-00c), the `real experiment' expressions (14)± (16) given in Ro-00c were used for the calculation, additionally taking into account the relationship (3) between the wavelength + and the divergence p. In the case of the mosaic diamond, a better t with the experiment was obtained, assuming a Gaussian distribution for the probability P(, n, p, n, p ), with a FWHM = (the sum of the parameters for the divergence and mosaic spread given in Table 1), simultaneously taking into account, in a very rst approximation, the divergence of the incident beam and the mosaic spread of the sample. The calculations were performed in about 10 h, neglecting (for CPU-time limitation) the divergence normal to the re ecting plane, n. The zero point for in Fig. 13 corresponds to that of the Huber diffractometer, which is displaced from the reference direction [110] of the crystal by The surface plot of the experimental 0.81±17!± scan measured at = A Ê is shown in Fig. 14(a). For each of the 100 steps per degree in,an! scan was performed with 81! steps and a step width of! = It should be noted that, because of the mosaicity of the crystal, the! scan width of the diamond experiment is ten times larger than the! scan width of the silicon experiment. Consequently, the intensity pro les of the `almost forbidden' 222 re ection as well as the multiplediffraction events of the diamond sample are mainly dominated by the radius of the mosaic blocks, r, and the mosaic spread,, of the sample. Figure 13 Contour plot of the!± scan of the `almost forbidden' 222 re ection of diamond. The zero point for corresponds to that of the Huber diffractometer. Only the middle 41 steps per 0.01 in! are considered. (a) Measurement. (b) Calculation with 3d-UMWEG using the Gaussian distribution. Figure 14 Surface plot of the experimental!± scan of the `almost forbidden' 222 re ection of diamond. The zero point for corresponds to that of the Huber diffractometer. The measurement was performed with 81 steps in! with a step width of! = 0.01 and 100 steps per degree in at = A Ê.(a) Measurement. (b) Calculation with 3d-UMWEG using the Gaussian distribution. J. Appl. Cryst. (2001). 34, 157±165 E. Rossmanith et al. Multiple-diffraction intensity profiles 163

8 Figure 15 Full line: the measured! scan of silicon with = 220. Dashed-dotted line: the measured! scan of diamond with = Discussion The diamond contour plot (Fig. 13) differs appreciably from that of the silicon crystal (Fig. 6). As in the case of silicon, the Umweganregung event, corresponding in the case of diamond to the 155 operative re ection (Fig. 12), is visible in a range of more than 4 (Figs. 13 and 14), but there is neither a similarity to the ± diagram (Fig. 12) nor a similarity to the corresponding part of the contour plot of silicon (Fig. 6). This fact can easily be understood by comparing the FWHM (0.05) of the diamond 222! pro le (dashed-dotted line in Fig. 15), which is mainly caused by the mosaic structure of the sample, with that of the silicon 222! pro le (FWHM ' 0.005; full line in Fig. 15), which represents the contribution of the divergence and wavelength range of the incident beam to the width of the intensity pro le. The 21! steps of the silicon 222 peak in Fig. 6 correspond to two! steps in the maximum of the diamond 222 peak in Figs. 13 and 14. The two peaks in Fig. 15 differ not only in their FWHM values, but also in their peak shapes. Whereas the theoretical! pro le calculated with the Lorentzian distribution ts the 222 peak of the perfect silicon crystal very well on an absolute scale (Fig. 16a), disagreement between theory and experiment is observed in the case of the 222 peak of the mosaic diamond crystal (Fig. 16b). The diamond crystal is obviously not ideally imperfect, but consists of a few ideally imperfect blocks, i.e. the mosaic-structure of the sample is neither isotropic nor homogenous. This is clearly visible in Fig. 17, in which an enlargement of the measured Umweganregung peak at ' is shown. In view of this fact, the t between the kinematical approach (Fig. 14b), which is based on an isotropic and homogenous mosaic structure model, and the synchrotron measurement (Fig. 14a) is satisfactory. 5. Conclusions Figure 16 (a) The! scan of the silicon 222 re ection at = 220. Dotted pro les: scans measured at three successive steps (step width 0.01 ). Full line: calculated with 3d-UMWEG. (b) The! scan of the diamond 222 re ection at = 225. Dotted pro le: scans measured at three successive steps (step width 0.01 ). Full line: calculated with 3d-UMWEG. Figure 17 Part of Fig. 15(a) (! range 0.81 ; range 227.4±228.4 ) showing the inhomogenous mosaic structure of the diamond crystal. Con rming the expectation discussed in x1, it was shown in this paper that, in the case of synchrotron radiation experiments, corresponding multiple-diffraction patterns of perfect and mosaic crystals are signi cantly different. Furthermore, it was shown that satisfactory agreement between the kinematical and experimental contour and surface plots of!± scans can be obtained with the kinematical approach discussed in Ro-00c for perfect silicon as well as for a mosaic diamond 164 E. Rossmanith et al. Multiple-diffraction intensity profiles J. Appl. Cryst. (2001). 34, 157±165

9 crystal, although in the case of the mosaic diamond, crude approximations were used for the modelling of the probability P(, n, p, n, p ). In the case of mosaic crystals, additional improvement of the t between theory and measurement can be expected, replacing the approximation for the calculation of the probability P(, n, p, n, p ), used in this paper, by a more sophisticated modelling of the distribution functions involved. This point will be discussed in more detail in a forthcoming paper considering the effect of secondary extinction on multiple-diffraction intensities. References Brennan, S. & Cowan, P. L. (1992). Rev. Sci. Instrum. 63, 850±853. Cole, H., Chambers, F. W. & Dunn, H. M. (1962). Acta Cryst. 15, 138± 144. Post, B. (1975). J. Appl. Cryst. 8, 452±456. Rossmanith, E. (1992). Acta Cryst. A48, 596±610. Rossmanith, E. (1993). Acta Cryst. A49, 80±91. Rossmanith, E. (1999). J. Appl. Cryst. 32, 355±361. Rossmanith, E. (2000a). J. Appl. Cryst. 33, 323±329. Rossmanith, E. (2000b). J. Appl. Cryst. 33, 921±927. Rossmanith, E. (2000c). J. Appl. Cryst. 33, 1405±1414. J. Appl. Cryst. (2001). 34, 157±165 E. Rossmanith et al. Multiple-diffraction intensity profiles 165