Research Article: Experimentally obtained and computer-simulated X-ray asymmetric eight-beam pinhole topographs for a silicon crystal

Date Published: May 01, 2019

Publisher: International Union of Crystallography

Author(s): Kouhei Okitsu, Yasuhiko Imai, Yoshitaka Yoda, Yoshinori Ueji.


Experimentally obtained eight-beam pinhole topographs for a silicon crystal were compared with computer simulations based on the n-beam Takagi–Taupin equation and Ewald–Laue theory.

Partial Text

We previously reported a derivation of the n-beam Takagi–Taupin (T-T) equation and an algorithm to integrate it (Okitsu, 2003 ▸; Okitsu et al., 2006 ▸). We verified these by comparing computer-simulated and experimentally obtained topographs using a six-beam case (Okitsu et al., 2003 ▸, 2006 ▸, 2011 ▸) and three-, four-, five-, six-, eight- and 12-beam cases (Okitsu et al., 2012 ▸). Hereafter, Okitsu et al. (2006 ▸), Okitsu et al. (2011 ▸) and Okitsu et al. (2012 ▸) are denoted by O et al. 2006, O et al. 2011 and OIY 2012, respectively.

The optics used in the present work, shown in Fig. 1 ▸(a), were fundamentally the same as those in Fig. 2 ▸, which is a reproduction of Fig. 7 of O et al. 2006, showing the experimental setup used when taking the six-beam pinhole topographs. However, the goniometer axes were adjusted such that the 000 forward-diffracted (FD) and 004, 026, 066, 084, 080, and transmitted-reflected (TR) X-rays were simultaneously strong, and the vector product of the 000 FD and 066 TR beam directions was horizontal. The polarization state of the incident synchrotron X-rays with a photon energy of 18.245 keV at BL09XU of SPring-8 was controlled by using a rotating four-quadrant phase retarder system. Its schematic and photograph are shown in Figs. 3 ▸(a) and 3 ▸(b), respectively [these are reproductions of Figs. 3(a) and 3(b) of OIY 2012]. Its usage was described in O et al. 2006. An X-ray beam whose dimensions were limited to 25 × 25 µm was incident on a position on the entrance surface of a floating zone (FZ) silicon crystal with a thickness of 9.6 mm. The incident point was 16.5 mm from the corner of the crystal block, as shown in Fig. 1 ▸(a). The orientation of the crystal is also shown in Fig. 1 ▸. An imaging plate (IP) with a pixel size of 50 × 50 µm was placed 47.5 mm behind the crystal, such that its surface was approximately perpendicular to the [100] direction of the crystal.

Fig. 5 ▸ [Ex] (x ∈ {h, v}) shows the experimentally obtained pinhole topograph images recorded on the IP for the incidence of the horizontally polarized (x = h) and vertically polarized (x = v) X-rays. Fig. 5 ▸ [Sx(T-T)] and [Sx(E-L)] show the T-T and E-L&FFT simulated images corresponding to Fig. 5 ▸ [Ex]. In Figs. 6 ▸ and 7 ▸, enlargements of the 000 FD and 066 TR images, respectively, are shown. Fig. 6 ▸ [Eh] and [Ev] are reproductions of Fig. 11 [S(a)] and [S(b)] in OIY 2012. Fig. 6 ▸ [Sh(T-T)] and [Sv(T-T)] are obtained by solving the n-beam T-T equation layer by layer with a thickness of (19.015/4000) mm, whereas the number of layers was 3600 for Fig. 11 [S(a)] and [S(b)] in OIY 2012. In Figs. 6 ▸ and 7 ▸, there is good agreement between the experimentally obtained and computer-simulated topographs (both the T-T and E-L&FFT simulated ones).

In Fig. 6 ▸ [Sh(T-T)], a sharp line similar to a knife edge (KEL) is observed. However, a KEL is not seen in Fig. 6 ▸ [Eh] or [Sh(E-L)]. The width of the KEL is extremely narrow. In the case of the T-T simulation, a boundary condition of the incident X-rays whose amplitude is the delta function was assumed. Then the incident X-rays have a plane-wave component whose initial point of the wavevector was far from the Laue point. However, the incident X-rays used in the experiment have a finite angular width. In addition, in the E-L&FFT simulation, the integration range is finite. This is probably the reason for the absence of the KEL in Fig. 6 ▸ [Eh] and [Sh(E-L)].

Experimentally obtained and computer-simulated (both T-T simulated and E-L&FFT simulated) asymmetric eight-beam pinhole topographs, which were in good agreement, were reported. This is for a case where the exit surface was not a single plane. It was verified that the X-ray wavefield could be computed not only based on the n-beam T-T equation but also on the n-beam E-L theory.




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