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The Journal of Chemical Physics

Quantum-mechanical scattering of an atom from a rigid hemisphere on a flat surface

B. H. Choi, Kwong Tin Tang, Jan Peter Toennies

Published Dec 8, 1997• 17 citations
Publisher

Abstract

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.

DOI
10.1063/1.475240

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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.
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.
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.
We studied the low-energy limit of atomsurface scattering and the phenomenon of quantum reflection by using hydrogen atoms and the surface of liquid helium. Measuring the specular reflectivity of a liquid-helium-covered hemispheric mirror for a beam of hydrogen atoms we obtained information on elastic and inelastic scattering phenomena. At 100 mK, reflectivities larger than 90% were achieved; this can be used to demonstrate atomic beam focusing. Approximately three-fourths of the atoms escaping from a small orifice in a cell were reflected back into the cell. The dominant inelastic processes at work here are surface absorption and direct inelastic scattering over small angles under excitation of surface ripplons (quantized capillary waves). Helium-covered surfaces maybe useful for atomic-beam manipulation of atomic hydrogen.
Quantum reflection of thermal He atoms from various surfaces (glass slide, GaAs wafer, flat, and structured Cr) at grazing conditions is studied within the elastic close-coupling formalism. Comparison with the experimental results of Zhao et al. [Phys. Rev. Lett. 105, 133203 (2010)] is quite reasonable but the conclusions of the present theoretical analysis are different from those discussed in the experimental work. The universal linear behavior observed in the dependence of the reflection probability on the incident wave-vector component perpendicular to the surface is only valid at small values of the component whereas, at larger values, deviation from the linearity is evident, approaching a quadratic dependence at higher values. The surface roughness seems to play no important role in this scattering. Moreover, the claim that one observes a transition from quantum to classical reflection seems to be imprecise.
We present a theoretical analysis of the influence of surface roughness on the atomic reflection from an evanescent wave mirror. In our calculations we assume that light scattered at the rough dielectric surface interferes with the evanescent wave, creating a rough potential that scatters the atoms. We calculate the probability of diffuse reflection and the momentum distribution of the scattered atoms with a statistical model for the rough surface. The atomic reflection is diffuse rather than specular if the surface roughness is comparable to the wavelength of the incident atoms. We discuss the spatial coherence of the reflected matter waves. We indicate how to generalize our treatment to the case of multilevel atoms.
We solve the two-dimensional Schrödinger equation for particles with momentum px=ℏk scattering off of a hard circular cylinder using the finite element method; we compare our results with the exact analytic solution. The quantity of interest to experimentalists is the differential cross section σ(ϕ)=|fk(ϕ)|2, which represents the final angular distribution of only the scattered particles. Here, we are also interested in the interference between the incident and scattered wave, which can be seen in the probability density for the total wave function, ρ(x,y)=|ψk(x,y)|2. We also apply the finite element method to the problem of particles scattering off of a hard rectangular cylinder, for which there is no analytic solution.