Generalized Laue dynamical theory for x-ray reflectivity at low and high incidence angles on ideal crystals of finite size

We present a generalized dynamical theory within the Laue formalism valid for x-ray Bragg and Laue diffraction on ideal crystals of finite thickness (crystal slab, film). In our model only the following two approximations are made: to consider a two-beam case and to neglect quadratic terms of the dielectric susceptibility. In fact, 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 of finite thickness in a dynamical way in the whole angular range from 0 to {π}/2. Our improved theory can be applied for describing: (i) symmetric and asymmetric reflections both close to the Bragg angle and at the far tails of the Bragg peaks; (ii) Bragg and Laue diffractions of x rays at very small incidence 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; (iv) diffractions with Bragg angles close to {π}/2.