Theory of Atom-Surface One-Phonon Inelastic Scattering: Application to HELIUM+COPPER(111) System
Abstract
Using the ab initio close-coupling approach we formulate the atom-nonrigid surface scattering dynamics which is applicable to a general interaction potential between the projectile and the crystal. The total scattering wave function and the interaction potential matrix are derived from the translational invariance of the full Hamiltonian. An integral form of the scattering equations is obtained for general multiphonon scattering. Assuming a sum of pair potential an expression of the potential matrix of the total interation potential is obtained. The iterative coupled integral equations are used to calculate the transition matrix in which multiphonon processes can be taken into account. Using the distorted wave Born approximation an expression of the transition matrix is obtained for one -phonon transitions. The surface is taken to be a slab of 80 layers with nearest neighbor interaction for the computation of phonon dispersion relations and polarization vectors. Using a pair potential obtained by Eichenauer et al. the differential cross-section for one-phonon transitions corresponding to the lowest two dispersion curves of the He+Cu(111) system is calculated. The phonon wave vectors are chosen to be along the < 1| 10> direction with lengths ranging from 0.05A^{-1} to 0.60A ^{-1}. The projectile mainly interacts with the vibrational motion of the atoms on the surface. It is found that when the interaction involving displacements of the second or deeper layers is also taken into account the differential cross-section reduces slightly. For short phonon wave vectors the nearly linear dependence of the component of the polarization vector normal to the surface on the phonon wave vector dominates the differential cross -section. This dominance is overcome by the interaction of pair potential for phonon wave vectors with length greater than about 0.30A^{-1}. The shape of the sum of the differential cross-sections for the lowest two dispersion curves is smooth and in good agreement with the experiment by Eichenauer et al. The change of the differential cross-section due to the Debye -Waller factor is small as the temperature is changed from 60K to 300K. The differential cross-section increases with the temperature due to the Bose-Einstein distribution of the phonons and is larger for phonon annihilation than for emission because of the density of the final states of the projectile, the kinematic condition of the scattering, and the crystal properties. Finally we discuss the dependence of the potential matrix on the number of unit cells on the surface and the relation between the truncation of the interaction potential and the Debye -Waller factor. The method of treating the Debye-Waller factor and the calculation of the differential cross-section require large amount of computer time on CRAY at the San Diego Supercomputer Center.
- Publication:
-
Ph.D. Thesis
- Pub Date:
- 1989
- Bibcode:
- 1989PhDT.......147W
- Keywords:
-
- HELIUM;
- COPPER;
- Physics: Condensed Matter