Theoretical Considerations on Bragg-case Diffraction of X-rays at a Small Glancing Angle

Abstract

X-ray diffraction phenomenon, for which the glancing angle of the incident beam is so small that the effect of specular reflection can not be neglected, is studied theoretically. Two wave fields, one of which is usually neglected, are considered as the solution of the dispersion equation, and the boundary conditions of the waves are considered for the gradient of amplitude normal to the surface besides the amplitude. Numerical calculations are made for (220) reflection of CuKα1 from a germanium single crystal. The diffracted beam is stronger in intensity and broader in half-value width than that where specular reflection is not considered.

Comparison between experimental and theoretical rocking curves in extremely asymmetric Bragg cases of X-ray diffraction

“…The intensity of extremely asymmetric diffraction from an Si(001) crystal has been measured around three glancing angles by changing the wavelength. Furthermore, we have compared these results with theoretical calculations based on the formulation first presented by Kishino & Kohra (1971) (hereinafter referred to as the extended theory) and also on the ordinary dynamical theory for absorbing crystals (Zachariasen, 1945). The calculation of the extended theory is shown to be clearly superior for the case of extremely asymmetric diffraction.…”

“…Under these conditions, however, ordinary treatments of the dynamical theory of diffraction (Zachariasen, 1945) are not valid, since the effects of specular reflection can no longer be neglected. Consequently, several theoretical studies (Kishino & Kohra, 1971;Bedyfiska, 1973;Rustichelli, 1975;Zeilinger & Beatty, 1983;Afanas'ev & Melikyan, 1990) have been reported on diffraction phenomena in such an extremely asymmetric case. According to the work of these authors:…”

“…Although Kishino & Kohra (1971) approximated the last two factors, 2 2 kokh, on the right-hand side of (4) as k 4, we do not, since we would like to make as few approximations as possible to obtain a more exact solution. Equation (4) shows the dispersion surface, which describes the positions of the origin points of wave vectors k0 and kh in the reciprocal-lattice space, as shown in Fig.…”

“…The boundary conditions for the electric-field vectors and magnetic-field vectors must be used as proposed by Kishino & Kohra (1971 …”

“…Kishino & Kohra (1971) have predicted theoretically that the FWHM of the rocking curve should have a maximum at a small glancing angle and cannot be larger than this maximum even if the glancing angle becomes smaller. On the other hand, if we follow the ordinary treatment of dynamical theory, the FWHM diverges when the glancing angle tends to zero.…”