Experimentally obtained and computer-simulated X-ray non-coplanar 18-beam pinhole topographs for a silicon crystal

Experimentally obtained non-coplanar 18-beam pinhole topographs were compared with computer simulations based on the Ewald–Laue theory.


Introduction
The present authors have reported coplanar eight-beam pinhole topographs experimentally obtained and computer simulated by fast Fourier transforming (FFT) the rocking amplitudes calculated based on the n-beam Ewald-Laue (E-L) theory. This technique (E-L&FFT simulation) was reported by Kohn & Khikhlukha (2016) and Kohn (2017). In Okitsu et al. (2019), it was shown that the E-L&FFT simulation can also be performed for a case where the X-rays do not exit from a single plane (hereafter this paper is denoted as O et al. 2019). Furthermore, the feasibility of calculating the X-ray intensities diffracted from a crystal that has plural facets, as shown in Fig. 9 of O et al. (2019), was discussed. In addition to this, if the E-L&FFT simulation could be performed even for a case where n 6 ¼ f3; 4; 5; 6; 8; 12g (non-coplanar case), the intensities of X-ray diffraction spots from a lysozyme (protein) crystal as shown in Fig. 1(b) could be calculated. Here a large number (over 200) of reflected X-ray beams are simultaneously strong. Fig. 2 shows the experimental arrangement. The horizontally polarized synchrotron X-rays at BL09XU of SPring-8 were monochromated to be 22.0 keV. The phase retarder system was not used in the present experiment. The beam size was limited to 25 Â 25 mm. The goniometer system on which a [111]-oriented floating-zone (FZ) silicon crystal was mounted was adjusted such that the 000 forward-diffracted (FD) and 440, 484, 088, 484 and 404 transmitted-reflected (TR) X-rays are simultaneously strong; this was achieved by monitoring the 000 FD, 440 and 484 TR X-rays with PIN photodiodes. The thickness of the crystal was 10.0 mm. An imaging plate (IP) was placed 24 mm behind the crystal such that the surface of the IP was parallel to the exit surface of the crystal.

Experimental
In addition to the hexagonal six-beam topograph images, a further 12 images surrounding them were found on the IP as shown in Fig. 3(a). The exposure time was 300 s.

Computer simulation
The length of the wavevector K (= 1/, where is the wavelength in vacuum) was calculated to be 1.7702394 Å À1 for a photon energy of 22.0 keV. The position of the Laue point La whose distance from reciprocal-lattice nodes 000, 440, 484, 088, 448 and 404 was an identical value K, was calculated on a computer. From Fig. 3(a), other reciprocal-lattice nodes were likely to exist in the vicinity of the surface of the Ewald sphere; that is, their distance from La was approximately jna Ã j 2K, i.e. jnj 2a= is the sufficient condition for a reciprocal-lattice node with indices hkl to exist on the surface of the Ewald sphere. Here, a is the lattice constant of the silicon crystal, a Ã ¼ 1=a and n 2 fh; k; lg. Because 2a= was calculated to be 18.21, the distances of reciprocal-lattice nodes with indices hkl from La were calculated in the range of À18 n 18. Then, in addition to the six reciprocal-lattice nodes, others with i 2 f6; 7; 8; . . . ; 17g were observed, as summarized in Table 1. Here, i is the ordinal number of the reciprocal-lattice node in the first column of Table 1. Then, all topograph patterns surrounding 000 FD, 440, 484, 088, 448 and 404 TR images have been indexed as shown in Fig. 3(b). For obtaining this figure, a photon energy of 21.98415 keV was assumed. It was observed that the ith reciprocal-lattice nodes (i 2 f6; 7; 8; . . . ; 17g) were on another circle (drawn as a blue circle in Fig. 4) outside the circle (drawn as a red circle whose centre is Q in Fig. 4) on which the inner six reciprocal-lattice nodes are present. For these 18 FD or TR X-ray beams with indices h i ; k i ; l i ði 2 f0; 1; 2; . . . ; 17gÞ, the Bragg reflection angle ð B i Þ, Â i , ÁK i =K, i and h i were calculated and are summarized in Table 1. Â i is the angle spanned by LaQ ! and Schematic drawing of the experimental setup. The horizontally polarized synchrotron X-rays were incident on a [111]-oriented floating-zone (FZ) silicon crystal with a thickness of 10.0 mm such that the six beams are simultaneously strong. The angle of the monochromator was adjusted such that the photon energy of the X-rays was 22.0 keV. However, the practical value of the photon energy was considered to be marginally different from this value. An IP was placed 24 mm behind the crystal.

Figure 1
Diffraction spots for (a) a sucrose (small molecular weight) crystal and (b) a hen egg-white lysozyme (protein) crystal taken on the imaging plate (IP) of a Rigaku Micro7 HFM-AXIS7 diffractometer. The distance between the crystal and the IP was 150 mm. The IP was exposed for 60 s by oscillating the crystal in the range of 0.1 for both (a) and (b).
circumcentre of the normal hexagon whose vertices are H i (i 2 f0; 1; 2; 3; 4; 5g) as shown in Fig. 4, LaLa 0 i ! is evidently 0 ! for i 2 f0; 1; 2; 3; 4; 5g and is an identical vector in the direction of LaQ ! =jLaQ ! j (= n z Þ for i 2 f6; 7; . . . ; 17g. Here, let 00 i be defined such that 00 i n z ¼ LaLa 0 i ! as shown in Fig. 5. k i À K on the left-hand side of equation (4) in O et al. 2019 can be described as follows: where ð0Þ and ð1Þ are the two-dimensional angular deviation of P 1 from La as shown in Fig. 5. Therefore, equation (2) can be modified as follows: The polarization factors C and S are defined as In the present 18-beam case, e ð0Þ i was defined to be s i Â s modðiþ3;6Þ =js i Â s modðiþ3;6Þ j for i 2 f0; 1; 2; 3; 4; 5g and to be s i Â s ½modði;12Þþ6 =js i Â s ½modði;12Þþ6 j for i 2 f6; 7; 8; . . . ; 17g. e ð1Þ i was defined to be s i Â e ð0Þ i for i 2 f0; 1; 2; . . . ; 17g. Laue's fundamental equation of the dynamical theory (von Laue, 1931;Authier, 2005)    Six reciprocal-lattice nodes are on a red circle in reciprocal space. Outside of this circle, a blue circle was observed on which 12 reciprocal-lattice nodes were present. Q is the centre of the red circle.

Figure 5
Geometry around the Laue point La. Pl 0 and Pl 3 are planes whose distance from H 0 and H 3 is K. Pl h is a plane normal to n z (downward surface normal). The Laue point La and point P 00 1 exist on Pl h . Pl i ði 2 f1; 2; 4; 5; . . . ; 17gÞ were not drawn for simplicity. La 0 i is a point whose distance from H i ði 2 f6; 7; . . . ; 17gÞ is K. P 0 1 is the initial point of the wavevector of the Bloch wave. P 0 1;k that appears in equation (14) in O et al. (2019) is the kth-numbered P 0 1 , i.e. the initial point of the wavevector of the kth-numbered Bloch wave where k 2 f1; 2; 3; . . . ; 2ng.
Here K ¼ 1=, where is the wavelength of the X-rays in vacuum, and ½D j ?k i is the component vector of D j perpendicular to k i . By applying the approximation k i þ K ' 2K, equation (6) becomes Substituting equation (4) into equation (7), the following equation can be obtained: Equation (8) is represented by using a vector and a matrix as follows: Here D is a 2n-order column vector and A 0 is a 2n Â 2n matrix whose element in the pth row ðp ¼ 2i þ l þ 1Þ and qth column ðq ¼ 2j þ m þ 1ÞA 0 p;q is given by Here, p;q is the Kronecker delta. Moreover, for the present 18-beam case, the procedure described by equations (10) (9) and (10). The values of Â i , h i Àh j and 00 i listed in Tables 1 and 2 were used. Furthermore, for the FFT to compute the E-L&FFT topographs, the description using equations (17) It required 1080 s (890 s for solving the eigenvalue problem, 20 s for FFT and 170 s for writing the topographs to the hard disk) to obtain the 18 topograph images shown in Fig. 3(b) using one node (Intel Xeon E5-2680v3) of the supercomputer system 'Sekirei' of the Institute of Solid State Physics of the University of Tokyo. The calculation to solve the eigenvalue problem for a 36 Â 36 matrix was several times as timeconsuming as the coplanar eight-beam case solving the eigenvalue problem described with two 16 Â 16 matrices  Table 1 The position of the point La whose distance from the ith-numbered reciprocal-lattice nodes H i ði 2 f0; 1; 2; 3; 4; 5gÞ is an identical length K, calculated for a photon energy of 22.0 keV. The Miller indices were 000, 440, 484, 088, 440 and 404. Â i ( ) ði 2 f0; 1; 2; . . . ; 17gÞ is the angle spanned by the directions of n z and LaH i ! . n z is a unit vector in the direction of ½111 (downward surface normal). When   Table 2 Values of 00 i for ÁE ð¼ E À E 0 Þ are À0.75, À0.50, À0.25, 0.00, +0.25, +0.50 and +0.75 eV, where E 0 = 21.98440 keV. +2.11782 Â 10 5 Fig. 6(b) 21.98390 À0.50 +1.40582 Â 10 5 Fig. 3(b) 21.98415 À0.25 +0.69386 Â 10 5 Fig. 6(c) 21.98440 0.00 0.01805 Â 10 5 Fig. 6 Fig. 6(c) shows the E-L&FFT simulated result with a photon energy of 21.98440 keV. In this figure, X-ray diffraction intensities due to the outer 12 reciprocal-lattice nodes on the blue circle in Fig. 4 are as strong as the inner six diffraction patterns that are substantially different from the experimentally obtained topograph in Fig. 3(a). However, the outer 12 topograph patterns are almost unobservable when the energy deviation from E 0 (= 21.98440 keV) is over 0.50 eV. Thus the present authors conclude that the photon energy of the synchrotron X-rays used in the present experiment was $21.98415 keV with which Fig. 3(a) was obtained. Fig. 7 shows enlargements of 088 TR and 000 FD images from Figs. 3(a) and 3(b). There is remarkable consistency between the experimentally obtained and the E-L&FFT simulated images. Fig. 8 shows an image of 088 TR X-rays obtained by the E-L&FFT simulation omitting the presence of the outer 12 reciprocal-lattice nodes. The assumed photon energy was identical to that in Fig. 7 [S(a)] (21.984150 keV). The vertical centre line in Fig. 8 was divided into two lines, whereas only one vertical line was observed in Fig. 7 [S(a)]. Further, an evident difference in the central part was observed between Fig. 7 [S(a)] and Fig. 8. It has been clarified that the presence of the outer 12 reciprocal-lattice nodes affected the features of the inner six diffraction patterns.

Discussion
Incidentally, referring to Fig. 5, let another Laue point La 00 0 be defined at a position on Pl 0 such that it is not far from La and P 1 La 00 0 . Further, let La 00 i be defined such that La 00 0 La 00 i ! ¼ 000 i n z on Pl i whose distance from H i is K (i 2 f0; 1; 2; . . . ; n À 1g). By replacing ð0Þ , ð1Þ and 00 i in equations (9) and (10) with ð0Þ0 , ð1Þ0 and 000 i , respectively, the following equation is obtained: Here, D 0 is a 2n-order column vector and A 00 is a 2n Â 2n matrix whose element in the pth row ðp ¼ 2i þ l þ 1Þ and qth column ðq ¼ 2j þ m þ 1ÞA 00 p;q is given by This way of defining La 00 i , ð0Þ0 , ð1Þ0 , 000 i and A 00 and equations (11) and (12) are more general than equations (9)  When the photon energy is 21.9843937 keV, the inner six and outer 12 reciprocal-lattice nodes (see Fig. 4) can be present simultaneously on an identical surface of the Ewald sphere. The deviations of photon energies from E 0 (= 21.98440 keV) were assumed to be À0.75, À0.50, 0.00, +0.25, +0.50 and +0.75 eV for (a), (b), (c), (d), (e) and (f), respectively.

Figure 7
½EðaÞ and ½EðbÞ are enlargements of 088 TR and 000 FD X-ray patterns of Fig. 3(a) obtained experimentally. ½SðaÞ and ½SðbÞ are enlargements of 088 TR and 000 FD X-ray patterns of Fig. 3(b) obtained by the E-L&FFT simulation.
Even when La cannot be defined as shown in Fig. 5, the eigenvalue problem represented by (11) can be solved. Then, the intensity distribution of reflected X-rays can be calculated with the E-L&FFT method when a pinhole X-ray beam is incident on an arbitrary position of the surface of the crystal. This is also the case for a crystal as shown in Fig. 9 of O et al. 2019 owing to the description given therein. The total intensities of X-rays reflected from the crystal completely bathed in the incident X-rays can be calculated by incoherently superposing the pinhole topograph intensities with the incident position two-dimensionally scanned over the incident side of the crystal.

Summary
In the present non-coplanar 18-beam case, the 18 reciprocallattice nodes are on two circles, drawn in red and blue in Fig. 4. The most important aspect of the present work is that a noncoplanar n-beam case for n 6 ¼ f3; 4; 5; 6; 8; 12g was computer simulated using the E-L&FFT method and was reasonably consistent with the experimentally obtained result. The constraint that n 2 f3; 4; 5; 6; 8; 12g has been originally placed such that n reciprocal-lattice nodes are on a circle in the reciprocal space. In the case of protein crystals as shown in Fig.  1(b), the situation where a large number of reciprocal-lattice nodes are simultaneously present in the vicinity of the surface of the Ewald sphere cannot be circumvented.
However, the constraint on n has been removed completely from the n-beam E-L&FFT method to calculate the X-ray diffraction intensities. N is the number of reciprocal-lattice nodes present in the vicinity of the surface of the Ewald sphere whose presence should be considered. Another difficulty caused by the complex shape of the crystal has also been overcome with the description in O et al. 2019. Thus, the present authors could calculate the intensities of X-ray diffraction spots as shown in Fig. 1(b) under the assumption that the crystal is perfect. E-L&FFT simulated 088 TR topograph images with a photon energy of 21.98415 keV under an assumption of the six-beam case; here, the 000 FD, 440, 484, 088, 448 and 404 TR X-rays are strong by neglecting the outer 12 beams. An evident discrepancy is observed between this figure and Fig. 7 ½SðaÞ.