Transport Equations of Amplitudes in the Eikonal Approximation of Dynamical Diffraction Equations M. K. Balyan Yerevan State University, Yerevan, Armenia Received June 20, 2012 Abstract⎯An approach of the eikonal approximation of the dynamical diffraction equations of X-rays in deformed crystals, based on the second-order differential equations for the transmitted and diffracted waves, is presented. By analogy with usual optics, this approach allows one not only to obtain the eikonal equation and to study the behavior of the amplitude in zero-order approximation, which usually is performed in the eikonal dynamical diffraction theory, but also to establish for all orders of the amplitude asymptotic expansion the corresponding transport equations and to present their solutions as integrals over the amplitude propagation trajectory. Summarizing the transport equations, an equation for the total amplitude, analogous with the parabolic diffraction equation in optics, is obtained. DOI: 10.3103/S1068337213010088 Keywords: X-ray waves, dynamical diffraction, eikonal approximation, transport equation 1. INTRODUCTION The eikonal approximation of dynamical diffraction equations of X-ray waves [1–6] gives a possibility to determine the amplitudes and eikonal and to construct a solution for slowly varying fields. Based on the system of dynamical diffraction equations [5] one can derive the eikonal equation and transport equation for amplitudes. However, transport equations for amplitudes of high orders of the asymptotic expansion are presented in the abstract form and only the equations for the zero-order approximation of the amplitude is discussed explicitly. In the present paper, the eikonal approximation is considered by passing from the system of dynamical diffraction equations to the second-order equations in partial derivatives for each amplitude separately. This allows us to present the transport equations and their solutions for all terms of the asymptotic expansion of amplitudes in a compact form, which gives a possibility to summarize the whole sequence of successive approximations and to write down the equation for amplitudes as a whole. 2. BASIC FORMULAS OF THE EIKONAL APPROXIMATION Here, in contrast to the common approach, we give a derivation of the eikonal approximation equations, passing from the system of dynamical diffraction equations to separate second-order equations for amplitudes. In two-wave diffraction regime, when two dynamically coupled waves connected with the reciprocal lattice sites 0 and h are present in a crystal, a sufficiently slowly varying wave field in the crystal is represented in the form ( ) 0 0 2cos 0 . h ik z i i i i h E Ee Ee e e e χ − Φ θ = + Kr Kr hu (1) Here 0 E and h E are slowly varying amplitudes, 0 K and 0 h = + K K h are, respectively, wave vectors of transmitted and diffracted waves satisfying the exact Bragg condition ( ) 2 2 2 2 0 2 , h k = = = πλ K K λ is the radiation wavelength in vacuum, u is the vector of displacement of atoms for their equilibrium positions in a perfect crystal, Φ is eikonal, 0 χ is the zero Fourier-component of the crystal polarizability, θ is the Bragg angle, the z-coordinate is directed along the reflecting planes, and the x-coordinate is directed ISSN 1068–3372, Journal of Contemporary Physics (Armenian Academy of Sciences), 2013, Vol. 48, No. 1, pp. 46–50. © Allerton Press, Inc., 2013. Original Russian Text © M.K. Balyan, 2013, published in Izvestiya NAN Armenii, Fizika, 2013, Vol. 48, No. 1, pp. 68–74. 46