The dynamical theory of X-ray Bragg diffraction from a crystal with a uniform strain gradient. The Green–Riemann functions

Abstract

The dynamical problem of X-ray Bragg diffraction from a thick (semi-infinite) crystal deformed by a uniform strain gradient (USG) is treated on the basis of the Green-Riemann function formalism. The rigorous solution of the problem is formulated by means of the Huygens-Fresnel principle. The exact Green functions are obtained in the form of the Laplace integrals suitable in physical applications. The quasi-classical and the Born (kinematical) asymptotic expansions of the Green functions are constructed as functions of the effective USG parameter B. Special attention is paid to the analysis of the wave-field propagation in a crystal with USG. The spatial harmonics Re(q) of the diffracted Green function, when Re(qB) < 0, as is shown, propagate within the proper 'waveguides', while the ones with Re(qB) > 0 are damped exponentially in the bulk of the crystal. The Taupin problem of the Bragg dynamical diffraction of the X-ray incident plane wave from a thick crystal, the lattice spacing being a linear function of the coordinate z (along the inward normal to the entrance surface) only is solved exactly in analytical form. In the latter case the waveguide nature of the propagation of the spatial harmonics inside such a crystal, provided that Re(qB) < 0, is clearly revealed.

X-ray diffraction of bent crystals in Bragg geometry. I. Perfect-crystal modelling

“…Models following an eikonal approximation based on rays were developed and solved for the Laue geometry, and solutions of wave equations were presented for the Laue case with a spherical incident wave (Katagawa & Kato, 1974;Chukhovskii & Petrashen', 1977). Forms of solution in Bragg reflection for thick crystals in certain limits have been developed (Chukhovskii, Gabrielyan & Petrashen', 1978;Chukhovskii, Gabrielyan, Kislovskii & Prokopenko, 1987;Chukhovskii & Malgrange, 1989) but precision calculation of profiles or extension to non-ideal crystals has not been achieved. Standard implementations for curved-crystal Johann spectroscopy of synchrotron, stationary and fast-beam sources use idealized crystal elements, neglecting centroid, width and profile effects (Beyer & Liesen, 1988;Suortti & Freund, 1989).…”

X-ray topography of bent crystals

“…Here they are split into two parts, the outgoing and reflected waves, and so the process continues (Chukhovskii, Gabrielyan & Petrashen', 1978). In other words, wave-field components with sign (aofl) > 0 gain the capability of a 'waveguide' travelling along the entrance surface over large distances, restricted only by their absorption lengths.…”

“…An analysis of wave-field propagation in the bulk of a crystal can be carried out on the basis of (3.1)-(3.5), like the analysis performed by Balibar, Chukhovskii & Malgrange (1983) (see also Chukhovskii, Gabrielyan & Petrashen', 1978;Chukhovskii, 1981;Gronkowski & Malgrange, 1984) in the Laue diffraction case; it yields the following picture.…”

“…In the Bragg geometry of X-ray diffraction in a crystal with USG the section-topograph pattern is described with the Green function introduced by Chukhovskii, Gabrielyan & Petrashen' (1978). The expansion of this function as a wave packet takes the form Po~ i°° Gho--dp Ph(P),…”

“…The dynamical scattering of X-rays (DSXR) in distorted crystals with a uniform strain gradient (USG) has been investigated in a number of studies (Kato, 1964;Bonse, 1964;Penning, 1966;Hart, 1966;Blech & Meieran, 1967;Malgrange, 1969;Ando & Kato, 1970;Petrashen', 1973;Chukhovskii, 1974;Petrashen' & Chukhovskii, 1975;Chukhovskii & Petrashen', 1977;Chukhovskii, Gabrielyan & Petrashen', 1978;Kushnir, Suvorov & Mukhin, 1980;Khrupa, Kislovskii & Datzenko, 1980;Petrashen' & Yaroslavskaya, 1981;Balibar, Chukhovskii & Malgrange, 1983;Shulpina, Petrashen', Chukhovskii & Gabrielyan, 1984) and is a starting point for the quantitative theory of X-ray topographic images and the theory of DSXR in distorted crystals in general.…”