Improved dynamical theory for X-ray reflectivity of ideal crystals of finite size at low and high incidence angles

In this work we present an improved dynamical theory within the Laue formalism for the two-beam case, which is valid for X-ray Bragg and Lane diffraction on ideal crystals of finite thickness. In our calculations no approximations, which are generally used in the conventional dynamical theory, are made. We limit ourselves to the two-beam case and neglect only the quadratic terms of the dielectric susceptibility. We take into account: i) the asymptotic sphericity of the dispersion surface and all the four solutions of the secular equation; ii) the difference between electric and displacement fields; iii) the boundary conditions of continuity of the tangential components of the electric and magnetic fields at the two crystal-vacuum interfaces. With the equations derived it is possible to describe the interaction of the X-ray beam with the crystal in a dynamical way in the whole angular range from 0 to x/2. Our improved theory can be applied for describing: i) symmetric and asymmetric reflections both near the Bragg condition and at the far tails of the Bragg peaks; ii) Bragg diffraction of X-rays at very small glancing angles of the order of the critical angle; iii) Bragg and Laue diffractions in which the diffracted beam travels almost parallel to the crystal surface.