The evaluation of the elastic scattering differential cross section of electron scattering from a planar surface of a vibrating lattice is reduced to the solution of a set of coupled algebraic equations for the associated scattering amplitude. This reduction is valid both for overlapping potentials (thus removing the restriction of previous analyses to muffin-tin potentials) and for the nonspherical potentials associated with ion cores at solid surfaces. The algebraic equations are solved using a double-diffraction analysis of the inelastic-collision model. Surface scatterers are taken to be geometrically equivalent but electronically and vibronically inequivalent to those in the bulk. A Debye model is used to describe the phonon spectrum of the solid. Numerical results are presented for a hypothetical fcc metal with the lattice parameters of aluminum. Thermal expansion alters the energies of peaks in the elastic intensity profiles ($I-V$ curves), whereas the thermal vibration of the ion cores alters the intensities of the peaks. The temperature dependence of the peak heights can be described by the kinematic model in which it is attributed to the Debye-Waller factor associated with an "effective" Debye temperature. However, the multiple scattering of the electron from the lattice causes these "effective" Debye temperatures to be related to the parameters of the model (e.g., bulk and surface electron—ion-core scattering phase shifts, the inelastic-collision mean free path, bulk- and surface-model Debye temperatures) in a complicated fashion. Although the trends evident in the dependence of the effective Debye temperature on the model parameters can be rendered plausible, it appears almost impossible to extract from a kinematical model reliable quantitative information about the average thermal displacements of the surface and bulk ion cores.

• Received 19 June 1970

DOI:https://doi.org/10.1103/PhysRevB.2.4783