research papers
Experimentally obtained and computersimulated Xray noncoplanar 18beam pinhole topographs for a silicon crystal
^{a}NanoEngineering Research Center, Institute of Engineering Innovation, Graduate School of Engineering, The University of Tokyo, 21116 Yayoi, Bunkyoku, Tokyo 1138656, Japan, and ^{b}Japan Synchrotron Radiation Research Institute, SPring8, 111 Kouto, Mikazukicho, Sayogun, Hyogo 6795198, Japan
^{*}Correspondence email: okitsu@soyak.t.utokyo.ac.jp
This article is dedicated to Professor K. Kohra, who passed away on 29 January 2019.
Noncoplanar 18beam Xray pinhole topographs for a silicon crystal were computer simulated by fast Fourier transforming the Xray rocking amplitudes that were obtained by solving the nbeam (n = 18) Ewald–Laue (EL&FFT method). They were in good agreement with the experimentally obtained images captured using synchrotron Xrays. From this result and further consideration based on it, it has been clarified that the Xray diffraction intensities when n Xray waves are simultaneously strong in the crystal can be computed for any n by using the EL&FFT method.
Keywords: Xray diffraction; dynamical theory; multiple reflection; nbeam reflection; phase problem; protein crystallography.
1. Introduction
The present authors have reported coplanar eightbeam pinhole topographs experimentally obtained and computer simulated by fast Fourier transforming (FFT) the rocking amplitudes calculated based on the nbeam Ewald–Laue (EL) theory. This technique (EL&FFT simulation) was reported by Kohn & Khikhlukha (2016) and Kohn (2017). In Okitsu et al. (2019), it was shown that the EL&FFT simulation can also be performed for a case where the Xrays do not exit from a single plane (hereafter this paper is denoted as O et al. 2019). Furthermore, the feasibility of calculating the Xray 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 EL&FFT simulation could be performed even for a case where (noncoplanar case), the intensities of Xray diffraction spots from a lysozyme (protein) crystal as shown in Fig. 1(b) could be calculated. Here a large number (over 200) of reflected Xray beams are simultaneously strong.
2. Experimental
Fig. 2 shows the experimental arrangement. The horizontally polarized synchrotron Xrays at BL09XU of SPring8 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 × 25 µm. The goniometer system on which a []oriented floatingzone (FZ) silicon crystal was mounted was adjusted such that the 000 forwarddiffracted (FD) and 440, 484, 088, and transmittedreflected (TR) Xrays are simultaneously strong; this was achieved by monitoring the 000 FD, 440 and 484 TR Xrays 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.
In addition to the hexagonal sixbeam 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.
3. Computer simulation
The length of the wavevector K (= 1/λ, where λ is the wavelength in vacuum) was calculated to be 1.7702394 Å^{−1} for a photon energy of 22.0 keV. The position of the Laue point La whose distance from reciprocallattice nodes 000, 440, 484, 088, and was an identical value K, was calculated on a computer. From Fig. 3(a), other reciprocallattice nodes were likely to exist in the vicinity of the surface of the that is, their distance from La was approximately , i.e. is the sufficient condition for a reciprocallattice node with indices hkl to exist on the surface of the Here, a is the lattice constant of the silicon crystal, and . Because was calculated to be 18.21, the distances of reciprocallattice nodes with indices hkl from La were calculated in the range of . Then, in addition to the six reciprocallattice nodes, others with were observed, as summarized in Table 1. Here, i is the ordinal number of the reciprocallattice node in the first column of Table 1. Then, all topograph patterns surrounding 000 FD, 440, 484, 088, and 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 reciprocallattice nodes () 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 reciprocallattice nodes are present. For these 18 FD or TR Xray beams with indices h_{i}, k_{i}, l_{i} , the Bragg reflection angle , , , and were calculated and are summarized in Table 1. is the angle spanned by and where H_{i} is the ithnumbered reciprocallattice node in Fig. 4. . is the inclination angle of from .
Fig. 5 is a drawing around the Laue point La. Here, let another Laue point be defined in the vicinity of La as shown in Fig. 5 such that . Because Q is the circumcentre of the normal hexagon whose vertices are H_{i} () as shown in Fig. 4, is evidently for and is an identical vector in the direction of (= for . Here, let be defined such that as shown in Fig. 5. k_{i}  K on the lefthand side of equation (4) in O et al. 2019 can be described as follows:
Because , and = , where and are the twodimensional 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 18beam case, was defined to be for and to be for . was defined to be for .
Laue's fundamental equation of the ; Authier, 2005) restricts the amplitude and wavevector of the Bloch wave as follows:
(von Laue, 1931Here , where λ is the wavelength of the Xrays in vacuum, and is the component vector of perpendicular to . By applying the approximation , 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 is a 2 norder column vector and is a 2 n ×2 n matrix whose element in the pth row (p = 2 i + l + 1) and qth column (q = 2 j + m + 1) is given by
Here, is the Kronecker delta. Moreover, for the present 18beam case, the procedure described by equations (10)–(16) in O et al. 2019 can be used to solve the eigenvalue problem of equations (9) and (10). The values of , and listed in Tables 1 and 2 were used.

Furthermore, for the FFT to compute the EL&FFT topographs, the description using equations (17)–(20) in O et al. 2019 can also be applied to the present 18beam case. The FFT in equation (20) in O et al. 2019 was carried out with L = 50 mm and N = 4096.
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 E52680v3) 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 × 36 matrix was several times as timeconsuming as the coplanar eightbeam case solving the eigenvalue problem described with two 16 × 16 matrices described in O et al. 2019.
4. Results
Fig. 6(c) shows the EL&FFT simulated result with a photon energy of 21.98440 keV. In this figure, Xray diffraction intensities due to the outer 12 reciprocallattice 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 Xrays 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 EL&FFT simulated images.
5. Discussion
Fig. 8 shows an image of 088 TR Xrays obtained by the EL&FFT simulation omitting the presence of the outer 12 reciprocallattice 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 reciprocallattice nodes affected the features of the inner six diffraction patterns.
Incidentally, referring to Fig. 5, let another Laue point be defined at a position on Pl_{0} such that it is not far from La and . Further, let be defined such that on Pl_{i} whose distance from H_{i} is K (). By replacing , and in equations (9) and (10) with , and , respectively, the following equation is obtained:
Here, is a 2 norder column vector and is a 2 n ×2 n matrix whose element in the pth row (p = 2 i + l + 1) and qth column (q = 2 j + m + 1) is given by
This way of defining , , , and and equations (11) and (12) are more general than equations (9) and (10). 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 Xrays can be calculated with the EL&FFT method when a pinhole Xray 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 Xrays reflected from the crystal completely bathed in the incident Xrays can be calculated by incoherently superposing the pinhole topograph intensities with the incident position twodimensionally scanned over the incident side of the crystal.
6. Summary
In the present noncoplanar 18beam 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 nbeam case for was computer simulated using the EL&FFT method and was reasonably consistent with the experimentally obtained result. The constraint that has been originally placed such that n reciprocallattice nodes are on a circle in the In the case of protein crystals as shown in Fig. 1(b), the situation where a large number of reciprocallattice nodes are simultaneously present in the vicinity of the surface of the cannot be circumvented.
However, the constraint on n has been removed completely from the nbeam EL&FFT method to calculate the Xray diffraction intensities. N is the number of reciprocallattice nodes present in the vicinity of the surface of the 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 Xray diffraction spots as shown in Fig. 1(b) under the assumption that the crystal is perfect.
Acknowledgements
The SGI ICE XA supercomputer system `Sekirei', which consists of Intel Xeon E52680v3 processors, of the Institute for Solid State Physics of the University of Tokyo was used for the computer simulation. The experiment was performed at BL09XU of SPring8 under the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2009B1384). The authors are indebted to Dr T. Oguchi and Dr G. Ishiwata for their technical support in the present experiments and also to Professor Emeritus S. Kikuta for his encouragement and effective discussions with respect to the present work.
Funding information
The theoretical part and computer simulation of the present work were supported by the Nanotechnology Platform Project (No. 12024046) of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
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