Darwin's approach to X-ray dynamical diffraction in distorted crystals

Abstract

The well known Darwin approach is generalized to study the problem of X-ray dynamical diffraction in distorted crystals with a three-dimensional deformation ®eld of arbitrary form. The recursion equations describing X-ray diffraction in the above crystals have been derived. The analytical solution of these equations has been obtained within the kinematical approximation.At present, there exist at least two different methods that are widely used in theoretical studies of X-ray dynamical diffraction from distorted crystals. One method is based on the Takagi±Taupin differential equations (Taupin, 1964; Takagi, 1969) and the other on the recursion equations for the re¯ection and transmission coef®cients of layered crystals. The recursion equations were ®rst obtained by Darwin (1914) for solving the problem of Bragg-case dynamical diffraction by perfect crystals. Then Borie (1967) demonstrated that Darwin's approach could be extended to Laue-case diffraction. In fact, the method based on recursion formulae is a convenient approach to the problem of X-ray scattering from layered crystals and multilayered materials such as superlattices, heterostructures etc. [see e.g. Belyaev & Kolpakov, 1983; Vardanyan et al., 1985; Bartels et al., 1986; Caticha, 1994;Andreev & Prudnikov, 1998]. Up to now, the recursion formulae have generally been used for the description of X-ray diffraction by layered crystals with one-dimensional variation of the deformation ®eld uz and/or the polarizability 1z along the z axis normal to the crystal surface. The aim of the present communication is to extend Darwin's approach to the case of X-ray dynamical diffraction from distorted crystals which are characterized by a three-dimensional deformation ®eld ur of arbitrary form.In the spirit of Darwin's approach, we consider ®rst the problem of X-ray scattering by a single distorted atomic plane (Fig. 1). Let the atomic plane be irradiated by a plane monochromatic wave where r e is the classical electron radius, F25 0 is the structure factor, N xy is the number of scattering centers per unit area in the XY plane, 5 0 is the incidence angle, P is the polarization factor (P 1 for ' polarization and P cos25 0 for % polarization), and r H xy , r xy are the distances depending on the position of dxdy. In the case of the distorted atomic plane (see Fig. 1), r H xy q uq, r xy r 0 À q uq, where r 0 is the position vector of the observation point A, q xY yY 0 is the two-dimensional position vector in the XY plane, and uq is the displacement ®eld. The vector r 0 lies in the XZ plane and makes the angle 5 0 with the X axis. Note that in (1) we omit the time factor expÀi3t. If the conditions uqar 0 ( 1 and &ar 0 ( 1 are simultaneously ful®lled, then the phase 9 k 0 Á r H xy k 0 r xy in (1) can be written in the following approximate form:Here, Q k s À k 0 and k s k 0 r 0 ar 0 is the wave vector of the scattered wave. For a distorted atomic plane, the phase 9 differs from that in the case of an ideal (undistorted) atomic plane by the term Q Á uq. In the...

X-Ray Diffraction in Distorted Crystals and Imperfect Multilayered Structures. A Theoretical Approach Based on Recursion Equations

“…In our previous paper [19], we calculated the reflected and transmitted wave fields for the case of scattering by the deformed atomic plane and took into account only the displacement field ur. Basing on the results of [19], we now investigate scattering by the atomic plane that is characterized by both the displacement function ur and the structure factor variation function f r. In the following, we will consider, for the sake of definiteness, the two-beam Bragg-case diffraction with the 726 I. R. Prudnikov Here E 0 n k is an angular spectrum of the incident wave field at z 0. In the small vicinity of the Bragg peak, the coefficients t s and t t in (7) can be considered constant, t s % t s j J 0 " JB and t t % t t j J 0 " JB , where " J B is the Bragg angle for the averaged crystal lattice.…”

“…If the variation of the structure factor is negligible, then Eqs. (9) and (10) become much simpler in their form [19]. Equations (9) and (10) should be supplemented with the boundary conditions…”

“…(9). Then one obtains the expression for the spectral amplitude E s 0 k k of the diffracted wave field at the entrance surface [19], (11) can be used to describe kinematical X-ray diffraction both in distorted crystals and non-ideal multilayered structures. In the next section, this formula will be applied to calculate the intensity of diffuse scattering by MSs with random deformation field.…”

“…In the next section, this formula will be applied to calculate the intensity of diffuse scattering by MSs with random deformation field. Note that an example of the numerical solution of (9), (10) for the case of dynamical diffraction from a distorted crystal is given in [19].…”

“…Up to now, the recursion formulae have mainly been used to study X-ray scattering in layered crystals and multilayered materials with the deformation field uz and the polarizability cz that only depend on the z-coordinate along the normal to the sample entrance surface. Previously [19], applying Darwin's approach, we have obtained the recursion equations that govern X-ray dynamical diffraction from distorted crystals with an arbitrary three-dimensional deformation field. The aim of the present paper is to extend Darwin's method to the case of X-ray dynamical diffraction in distorted crystals and non-ideal multilayered structures (MSs) where both the deformation field and the polarizability (i.e., the structure factor) are functions of three coordinates xY yY z of the point inside the sample.…”