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Physical review. B, Solid state

Quantum scattering from a sinusoidal hard wall: Atomic diffraction from solid surfaces

Richard I. Masel, Robert P. Merrill, William Hughes Miller

Published Dec 15, 1975• 104 citations
Publisher

Abstract

An exact quantum formalism for atom scattering from a sinusoidal hard-wall surface is presented. The Lippmann-Schwinger equation is solved for a scattering kernel consistent with the hard-wall boundary conditions on the Schr\"odinger equation. It results in an infinite-dimensional matrix equation for the Fourier coefficients of the scattering kernel which can be solved in a finite-dimensional limit to convergence. The results show either rainbow or specular patterns depending on the surface roughness and incident $k$ vector, as predicted by semiclassical and coupled-channel calculations. Bragg-like structure is present with the periodicity of the amplitude of the sinusoidal hard wall and the effects of multiple scattering are evidenced at large surface amplitudes.

DOI
10.1103/physrevb.12.5545

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An exact quantum formalism for atom scattering from a sinusoidal hard-wall surface is presented. The Lippmann-Schwinger equation is solved for a scattering kernel consistent with the hard-wall boundary conditions on the Schr"odinger equation. It results in an infinite-dimensional matrix equation for the Fourier coefficients of the scattering kernel which can be solved in a finite-dimensional limit to convergence. The results show either rainbow or specular patterns depending on the surface roughness and incident kkk vector, as predicted by semiclassical and coupled-channel calculations. Bragg-like structure is present with the periodicity of the amplitude of the sinusoidal hard wall and the effects of multiple scattering are evidenced at large surface amplitudes.
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The scattering of a monoenergetic beam of atoms by a hard hemisphere on a hard flat surface is formulated quantum mechanically. Differential and integral cross sections are calculated as a function of incident beam energy and incident angle and compared with the familiar results of scattering from a free hard sphere. Two types of interference processes are identified. In addition to the previously discussed off-specular reflection symmetry interferences, Rayleigh oscillations are found in the integral cross sections. They result from the interference of specular scattering from the hemisphere and from the flat surface. Their phases are found to be independent of the angle of incidence when plotted versus the normal component of the incident wave vector.
The semiclassical perturbation theory formalism of Hubbard and Miller [J. Chem. Phys. 78, 1801 (1983)] for atom surface scattering is used to explore the possibility of observation of heavy atom diffractive scattering. In the limit of vanishing ℏ the semiclassical theory is shown to reduce to the classical perturbation theory. The quantum diffraction pattern is sensitive to the characteristics of the beam of incoming particles. Necessary conditions for observation of quantum diffraction are derived for the angular width of the incoming beam. An analytic expression for the angular distribution as a function of the angular and momentum variance of the incoming beam is obtained. We show both analytically and through some numerical results that increasing the angular width of the incident beam leads to decoherence of the quantum diffraction peaks and one approaches the classical limit. However, the incoherence of the beam in the parallel direction does not destroy the diffraction pattern. We consider the specific example of Ar atoms scattered from a rigid LiF(100) surface.
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Motivated by recent experiments that demonstrated a strongly enhanced reflection efficiency of atoms on microfabricated, gratinglike surface structures, we develop a detailed description of the scattering of atomic matter waves on such structures. We start with a simple model that describes the ridged surface as an equivalent absorbing medium. This model correctly predicts the experimentally observed scaling and magnitude of the reflection efficiency. We then calculate the accurate solutions for the Fresnel diffraction of waves at grazing incidence at a periodic set of idealized, parallel edges. Both numerical and analytical solutions are presented. The specular reflectivity and its phase, as well as the efficiencies of higher diffraction orders are calculated. We compare our results to experimental data in a wide range of parameters. We extend the model including the finite size of the grating ridges and the van der Waals attraction and suggest a more general estimate of the reflection coefficient.