t-Matrix Approach in Low-Energy-Electron Diffraction

A $t$-matrix perturbation approach in the microscopic treatment of low-energy-electron-diffraction theory is developed. The total scattering matrix of a crystal is expanded in terms of the individual ion-core $t$ matrix. Propagation of the incident electron inside the crystal is expressed in terms of a one-particle Green's-function propagator containing the proper self-energy of an interacting electron gas. The intraplanar structural propagator of the crystal is evaluated in real space while the interplanar structural propagator of the crystal is transformed into a $\overrightarrow{\mathrm{k}}$-space representation. It is shown that using such a $\overrightarrow{\mathrm{k}}$-space representation of the interplanar structural propagator, a general scheme for summing scattering contributions from all layers of the crystal can be formulated. Temperature correction to the elastic cross section is also included. The diagonal nature of the $t$ matrix leading to separable sums of partial waves for interplanar scattering terms, making the $t$-matrix method fast and practical even for the case of complicated structures and many partial-wave phase shifts, is discussed.