Multiphonon Scattering of Low-Energy Electrons

The multiphonon scattering in low-energy electron diffraction experiments has been observed and analyzed. It is shown that the elastically scattered intensity integrated over a Brillouin zone is given by $\int \stackrel{}{\mathrm{zone}}{I}_{\mathrm{total}}\left(\mathrm{S}\right)d^{2}S\propto e^{-2M}\left(1+2M+\frac{{\mathrm{}\left(2M\right)\mathrm{}}^2}{2!}+\cdots \right),$ where $2M$ is the exponent in the Debye-Waller factor. The first two terms are the zero- and one-phonon terms, respectively; the remaining terms are the multiphonon scattering. The scattering from the (111) face of Ni has been measured. The observed diffuse intensity may be separated into two components: a uniform background which is a function of energy and temperature, and a remaining intensity which has all the properties expected of the one-phonon scattering. The uniform background is then shown to be the multiphonon scattering by comparing the results with the above relation using measured values of $2M$. The total elastic current integrated over back angles has been measured as a function of temperature and is essentially constant, as expected from above. It is thus possible to account for the phonon contribution in experiments to investigate other sources of diffuse scattering such as surface imperfections and amorphous layers of adsorbed foreign atoms.