A method is developed for calculating the scattering of a beam of x rays striking the plane surface of a crystal at an angle below the critical angle for total external reflection. The low penetration under this condition offers the possibility of determining the special structure of the surface layers, as has been pointed out by Marra, Eisenberger, and Cho. A distorted-wave approach is developed whereby the crystal is first taken to be a homogeneous dielectric slab for the purpose of calculating the distorted wave. The distorted wave is considered to illuminate the actual crystal, from which a scattering pattern can then be calculated. The effects of absorption can be taken into account; in some cases absorption may offer the possibility of observing near-surface structures at angles of incidence larger than the critical angle. Such illumination may also be useful for performing fluorescence analysis of the near-surface layers to determine their impurity content. Synchrotron sources offer new opportunities for measurements of these kinds. Thermal neutrons may also be used in place of x rays and offer unique opportunities for studying surface magnetization and, through inelastic scattering, surface phonons and, conceivably, surface magnons.

• Received 1 July 1982

DOI:https://doi.org/10.1103/PhysRevB.26.4146