Research Article: X-ray interference fringes from a weakly bent plane-parallel crystal with negative strain gradient

Date Published: November 01, 2019

Publisher: International Union of Crystallography

Author(s): Tomoe Fukamachi, Sukswat Jongsukswat, Dongying Ju, Riichirou Negishi, Keiichi Hirano, Takaaki Kawamura.


In the waves emitted from the entrance, back and lateral surfaces of a very weakly bent plane-parallel perfect crystal with negative strain gradient, X-ray interference fringes between two refracted beams with different hyperbolic trajectories were observed when the strain was very weak, of the order of 10−7.

Partial Text

When X-rays are incident on a thin plane-parallel perfect crystal in the symmetric Bragg geometry, the refracted beam in the crystal reaches the back surface as illustrated in Fig. 1 ▸(a). A part of the beam is reflected (S4) and the rest is emitted from the back surface as the transmitted beam (Pt). The refracted beam is referred to as the beam corresponding to the Poynting vector excited at a point on the dispersion surface defined in the dynamical theory of diffraction. The refracted beam is called the wavefield by Authier (2001 ▸). In the two-wave approximation, it is composed of two waves: one is propagating in the forward direction and the other in the diffracted-wave direction. In an anomalous transmission condition, the divergence of the refracted beam is much larger than that of the incident wave (Authier, 2001 ▸). If the divergence angle of the refracted beam is large enough, interference fringes can be formed between once (S1)- and twice (S2)-reflected beams from the back surface (abbreviated as IFRB) as shown in Fig. 1 ▸(a). Similar interference fringes can be observed in the transmitted beam (IFTB) from the back surface. If the crystal is small along the incident azimuth direction (x) compared with the propagation length of the X-ray, interference fringes can be observed in the emitted beams from the lateral surface both in the diffracted- (IFLSD) and the transmitted-wave directions (IFLST). Both IFLSD and IFLST have been observed by Fukamachi et al. (2004 ▸, 2005 ▸) from a thin Ge plane-parallel crystal. These fringes were formed by interference between the beams directly propagating to the lateral surface (S3) and once reflected from the back surface (S4) (Hirano et al., 2008 ▸, 2009a ▸,b ▸; Fukamachi, Hirano et al., 2011 ▸).

The sample was a plane-parallel single Si crystal. The top (entrance) and bottom (back) surfaces of the crystal were polished by a non-disturbance polishing method at Sharan Inc. The size was 50 mm long, 15 mm wide and 0.28 mm thick. One end of the sample was clamped and the other end was free along the gravity direction as shown in Fig. 3 ▸(b). The sample was bent due to gravity and the residual strain. The experiments were carried out using X-rays from synchrotron radiation at the bending-magnet beamline 15C, Photon Factory, Tsukuba, Japan. The measuring optical system is shown in Fig. 3 ▸(a). The X-rays were σ-polarized and had a very narrow band of energy achieved by using an Si(111) double-crystal monochromator. The X-ray energy was 11 100 eV, which was determined by measuring XANES (X-ray absorption near-edge structure) from a thin Ge plate near the Ge K absorption edge (11 103 eV) with an accuracy of ±0.5 eV. The distance from the source to slit 1 was 30 m and that from slit 1 to the sample was 300 mm. The vertical width of slit 1 was 0.02 mm. In Fig. 3 ▸(a), Ph, Pr, Pt, Plr and Plt are the intensities of the diffracted wave, the reflected beam except for the diffracted beam, the transmitted beam, the emitted beam from the lateral surface in the diffracted-wave direction and that in the transmitted-wave direction, respectively. As shown in Fig. 3 ▸(b), the X-rays were incident on the crystal with the azimuth perpendicular to the bending direction. The incident glancing angle was fixed at the angle where the rocking curve of Pr in Fig. 3 ▸(e) showed the peak and the anomalous transmission was maximized. The X-ray intensities were measured by the scintillation counters (SC1 and SC2) and recorded on a nuclear plate (ILFORD L4; emulsion thickness, 25 µm). The incident and reflected X-ray geometries with respect to the sample are shown for positive and negative values of β in Figs. 3 ▸(c) and 3 ▸(d), respectively. For observation of IFRB, it is important to make the value of |β| small. If l denotes the distance from the free edge to the incident plane and L that from the free edge to the fixed edge, |β| is proportional to (Jongsukswat et al., 2013 ▸). In the present experiment the distance l was between 3.25 and 4 mm, and L was 48 mm.

The trajectory of the X-ray refracted beam in a very weakly bent perfect crystal with negative strain gradient shows a hyperbolic form opening down, while it shows a hyperbolic form opening up when the strain gradient is positive. This difference in beam trajectory results in quite different X-ray interference fringes. The interference fringes for β < 0 were mainly studied in the above. In the wave emitted from the entrance surface IFRB were observed between once- and twice-reflected beams from the back surface. In the transmitted wave from the back surface, IFTB were observed between the beam directly reaching the back surface and the beam once reflected from the entrance surface. These interference fringes for β < 0 were compared with the corresponding interference fringes for β > 0. IFRB and IFTB were analysed by using the dynamical theory of diffraction for a bent crystal. The results showed very good agreement between the observed and the calculated values of the interference fringe spacing, which enabled us to evaluate the strain gradient of the sample crystal. If a thinner crystal is used when , it is possible to observe Bragg–Pendellösung fringes reported by Batterman & Hildebrandt (1968 ▸), which are formed by interference between two waves corresponding to the two branches of the dispersion surface as shown by Authier (2008 ▸). But if , it is not possible to observe them because of the mirage effect.




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