Quantum theory of atom-surface scattering: Debye-Waller factor

Abstract

The Debye-Waller factor, introduced historically for X-rays, was used later for electrons, neutrons, and atoms as well. In this process of extension, however, the assumptions on which the Debye-Waller theory rested became more and more questionable until in the case of atoms (whose scattering from surfaces is both strong and slow) serious modifications are necessary. In the present article four models are discussed in order. In Model 1 a fast atom impinges on a surface whose atoms all vibrate deviating from their equilibrium positions by the same vector displacement ϱ. In Model 2 again the impinging atom is fast, but the atoms in the surface vibrate incoherently rather than coherently. It is shown that both Models 1 and 2 yield the conventional Debye-Waller result in the infinite crystal atom mass limit (for Model 2 Einstein oscillators have also to be assumed) and it is also shown how corrections to this result can be built. Turning then to slow impinging atoms, in Model 3 a slow atom impinges on a hard crystal surface, interacting with the rapidly varying potential of the vibrating solid. Model 3 is discussed in detail and it is shown that the Debye-Waller exponent can be written in terms of a time integral of the product of two correlations: the force correlation and the displacement correlation. The result is a dramatic increase of diffraction of relatively heavy atoms (with respect to the conventional theory). Finally, in Model 4 the impinging atom is again slow but the crystal is soft rather than hard. This case is more difficult to treat but a preliminary analysis again indicates a dramatic increase of diffraction since the soft solid adjusts itself to the instantaneous atom position leading to elastic scattering. The experimental implications of the present theory, especially for neon scattering from surfaces, are discussed.