engleski

Different aspects of the Debye-Waller factor in various atom-surface scattering theories

We discuss differences in the formal appearance, structure and physical origin of the Debye-Waller factor (DWF) obtained in various theoretical treatments of inelastic atom-surface collisions. These differences are illustrated on the example of two most commonly used formalisms for description of inelastic atom scattering by multiple excitation of surface phonons. In the first of these approaches, one starts from the expression for transition rates for the projectile atom which is coupled to the phonon field to all orders in lattice displacements. In first-order Born approximation, which is usually employed to express the corresponding T-matrix in a tractable form, the problem reduces to calculation of the exponentiated lattice displacement correlation function. By using a diagrammatic expansion to represent such correlation function we demonstrate that in this description the DWF occurs as a consequence of atom scattering by zero point motion of the lattice and is therefore expressed through dynamical variables which are not constrained to the energy shell. In the second approach one starts from the expression for the scattering spectrum of the projectile and linear atom-phonon field coupling, which is then treated in a nonperturbative fashion to all orders in the coupling constant. Here the dominant quantum contribution to the DWF arises from the projectile scattering through real uncorrelated multiphonon emission processes, which reduce the scattered beam intensity in the elastic channel. Using analogous diagrammatic expansion we show that the corresponding DWF is expressed in terms of the on-the-energy-shell quantities, which then leads to a normalization of the total scattering spectrum so as to comply with the optical theorem. Only in the limit of fast collision and trajectory approximation for the scattered particle motion the two approaches lead to the same result for the DWF because in this case the on-the-energy-shell features are washed out. We conclude by arguing that full correspondence between the expressions for the DWF in two discussed treatments could be achieved only by going beyond the presently employed approximations in the evaluation of the scattered beam intensities.

atom-solid interactions ; scattering ; diffraction ; surface waves

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