The Ewald–Oseen theorem in the x-ray frequency region: A microscopic analysis

We address the problem of the proper microscopic description of the propagation of x rays in condensed matter. In the optical region of the spectrum, Fresnel boundary conditions may be used; these boundary conditions can be regarded as a consequence of the Ewald–Oseen extinction theorem. This theorem is, however, generally derived in the dipole limit, where the wavelength of the light is large compared with the dimensions of the scatterers. Since x-ray wavelengths are comparable to atomic dimensions, it is necessary to re-examine the validity of the Ewald–Oseen theorem and the consequent boundary conditions for x-ray optics. In this paper, we demonstrate that the Ewald–Oseen extinction theorem may also be derived in the limit where the radiation frequency is high compared with atomic transition frequencies, a condition which holds for x-ray radiation. This result therefore justifies the use of macroscopic dielectric theory with Fresnel boundary conditions to describe the x-ray reflectivity of liquid surfaces.