Date Published: May 01, 2019
Publisher: International Union of Crystallography
Author(s): Kouhei Okitsu, Yasuhiko Imai, Yoshitaka Yoda.
Experimentally obtained non-coplanar 18-beam pinhole topographs were compared with computer simulations based on the Ewald–Laue theory.
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 (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 × 25 µm. The goniometer system on which a -oriented floating-zone (FZ) silicon crystal was mounted was adjusted such that the 000 forward-diffracted (FD) and 440, 484, 088, and 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.
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 reciprocal-lattice nodes 000, 440, 484, 088, and 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 , i.e. 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, and . Because was calculated to be 18.21, the distances of reciprocal-lattice nodes with indices hkl from La were calculated in the range of . Then, in addition to the six reciprocal-lattice nodes, others with 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, 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 reciprocal-lattice 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 reciprocal-lattice nodes are present. For these 18 FD or TR X-ray beams with indices , the Bragg reflection angle , , , and were calculated and are summarized in Table 1 ▸. is the angle spanned by and where is the ith-numbered reciprocal-lattice node in Fig. 4 ▸. . is the inclination angle of from .
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 (= 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. 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.
In the present non-coplanar 18-beam case, the 18 reciprocal-lattice nodes are on two circles, drawn in red and blue in Fig. 4 ▸. The most important aspect of the present work is that a non-coplanar n-beam case for was computer simulated using the E-L&FFT method and was reasonably consistent with the experimentally obtained result. The constraint that 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.