Unified model of diffractive and multiphonon He atom scattering from adsorbates: Holstein renormalization of the interactions and the complete Debye-Waller factor

Abstract

We derive general expressions for the energy and lateral momentum resolved scattering spectrum describing collisions of beams of thermal energy He atoms with adsorbates on a flat substrate surface. Elastic and inelastic components of the projectile-adlayer interaction, of which the latter can be nonlinear in adsorbate displacements, are treated to all powers in the coupling constant g and with full account of the projectile recoil. The established formalism enables a combined treatment of elastic scattering (either diffuse or diffractive) and inelastic excitation of multiple phonons and overtones in the adlayer on an equivalent footing. For nonlinear vibrational coupling the distorted wave matrix elements of the interactions are Holstein-renormalized by zero point motion of adsorbates. In the case of scattering intensities calculated to lowest order in g this gives rise to a direct analog of the standard Glauber\char21{}van Hove Debye-Waller factor. The closed form solutions for the scattering spectra to all orders in g are characterized by a unified or complete Debye-Waller factor which embodies the effects discussed so far only separately in the literature: (i) attenuation of the scattered beam intensities due to zero point motion of adsorbates, and (ii) attenuation of the beam intensity in the elastic channel due to inelastic scattering from adlayer phonons and overtones. The complete Debye-Waller factor acquires a form of an exponentiated sum of Holstein-renormalized scattering intensities and acts to preserve the unitarity of the scattering spectrum in accord with the optical theorem. The developed model facilitates evaluation of the various approximate and limiting forms of the scattering spectra and the associated Debye-Waller factors characteristic of the different scattering regimes and different types of adlayer vibrational dynamics. Potential applications of the model are illustrated by estimating the effect of Holstein renormalization occurring in the unified Debye-Waller factor and scattering intensities for several prototype adlayer systems.