Generalization of the nonstandard approach in the dynamic theory of diffraction for deformed crystals
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IFFRACTION AND SCATTERING OF IONIZING RADIATIONS
Generalization of the Nonstandard Approach in the Dynamic Theory of Diffraction for Deformed Crystals A. A. Dyshekov KabardinoBalkarian State University, Nalchik, 360004 Russia email: [email protected] Received February 22, 2013
Abstract—The nonstandard theory of Xray scattering in a deformed crystal has been generalized. The vector of atomicplane displacement is introduced into the crystal polarizability model like in the generalized Takagi dynamic theory. The solution to the wave equation is sought for using the procedure of expanding the field amplitude and vector operators in the Fourier components of polarizability χH in a series according to the multiscale method. It is shown that considering lattice strain generally calls for introducing various charac teristic spatial regions for the diffraction equation, which is in complete agreement with the main concept of the multiscale method. A particular case of a strain field depending on one scale is considered. If a relative change in strain occurs at a length on the order of the extinction length, one can obtain equations generalizing the Takagi equations to the case of arbitrary diffraction geometries. DOI: 10.1134/S1063774513070067
INTRODUCTION The Ewald–Laue theory is limited because a wave field cannot be described in crystals with lattice defects [1]. This circumstance stimulated the generalization of the dynamic theory based on the Takagi equations [2]. The latter were obtained using the model concept of the character of lattice distortions, which makes it possible to directly take into account the displacement of atomic planes in the threedimensional periodic function of crystal polarizability. This concept allows for describing lattice displacements and strain using the methods of classical theory of elasticity formulated within the continuum approximation. To satisfy the condition of the dynamic character of scattering in the generalized dynamic theory, the lattice distortion is assumed to be rather weak; correspondingly, the strain is small. The character of variation in strain is implic itly taken into account only when the wave field is cho sen in the form of a Bloch function with slowly varying amplitudes; thus, the question of applicability of this concept remains open. There is another limitation of the generalized dynamic theory, which is related to the correct state ment of boundary conditions. Mathematically, the Takagi equations form a firstorder differential system with respect to the scalar amplitudes of transmitted and diffracted waves. The procedure of determining these amplitudes at the crystal–vacuum interface does not correspond to the classical boundary conditions. This discrepancy is due to the fact that the Takagi equations are obtained disregarding the second deriv atives of the field amplitudes with respect to coordi nates. Thus, the boundary conditions impose funda
mental limitations on the applicability of the Takagi equations (for example, when analyzing extremely
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