Dynamical theory of x-ray diffraction at Bragg angles nearπ2

Abstract

The dynamical theory of x-ray diffraction normally makes use of some approximations that, however, cease to be valid when ${\ensuremath{\theta}}_{B}\ensuremath{\simeq}\frac{\ensuremath{\pi}}{2}$. In this paper we analyze why this happens and establish the theory applicable to this case, obtaining new appropriate expressions the analysis of which allowed us to distinguish three different regimes of diffraction in the neighborhood of $\frac{\ensuremath{\pi}}{2}$ (one related to the usual Bragg diffraction, one a transition regime, and a third related to the normal soft-x-ray propagation). The reflectivity of a semi-infinite crystal is then calculated, and extremely large linewidths are found for the rocking curve as compared to the common cases for perfect crystals; the absorption effect on the profiles and the relatively small effect of the orientation of the crystal surface, also studied here, turn out to be quite interesting and may have important practical consequences. As we expected, an extreme sensibility to minute variations of the lattice parameters is found. The peculiar peak shapes and the large linewidths could be of use in high-precision measurements of lattice parameters. Our treatment also provides the theoretical basis for the design of resonant cavities for x rays and other such interferometric devices.