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Physical review. B, Condensed matter

Temperature effects in diffractive atom-surface scattering

Vittorio Celli, A. A. Maradudin

Published Jan 15, 1985• 31 citations
Publisher

Abstract

A theory is presented of the diffractive scattering of a light atom by a semi-infinite crystal in which the atoms are executing thermal vibrations about their equilibrium positions. The intensities of the diffracted beams can be expressed in terms of the average of the scattered wave function over the canonical ensemble defined by the vibrational Hamiltonian of the crystal (this is not true for the inelastic component of the scattered intensity). With the aid of projection operators we obtain the integral equation satisfied by the averaged atomic wave function. The effective potential entering this equation is nonlocal in space and time, complex, and temperature dependent. The change in intensity of each diffracted beam due to scattering out of the beam by the thermal vibrations of the atoms constituting the crystal is related to this effective potential.

DOI
10.1103/physrevb.31.825

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A theory is presented of the diffractive scattering of a light atom by a semi-infinite crystal in which the atoms are executing thermal vibrations about their equilibrium positions. The intensities of the diffracted beams can be expressed in terms of the average of the scattered wave function over the canonical ensemble defined by the vibrational Hamiltonian of the crystal (this is not true for the inelastic component of the scattered intensity). With the aid of projection operators we obtain the integral equation satisfied by the averaged atomic wave function. The effective potential entering this equation is nonlocal in space and time, complex, and temperature dependent. The change in intensity of each diffracted beam due to scattering out of the beam by the thermal vibrations of the atoms constituting the crystal is related to this effective potential.
We develop a new method for the calculation of the atom scattering analog of the Debye–Waller factor. Unlike x-ray and neutron scattering the properties of an atom scattered by a solid surface cannot be computed by perturbation theory; therefore the simple Debye–Waller theory cannot be applied. Nevertheless the Debye–Waller phenomenon has a close analog: the elastic intensity is depressed due to the uncorrelated part of the thermal motion of the lattice atoms. To compute this effect we develop a time dependent scattering theory in which the quantum properties of the scattered atom are described by propagating coherently an ensemble of wave packets and lattice motion is simulated by a classical Langevin equation. Applications are made to He and Ne scattering from a surface whose lattice dynamics mimics that of Pt(111) but whose corrugation was slightly increased to enrich the diffraction structure.
We develop a new method for the calculation of the atom scattering analog of the Debye-Waller factor. Unlike x-ray and neutron scattering, the properties of an atom scattered by a solid surface cannot be computer by perturbation theory, therefore the simple Debye-Waller theory cannot be applied. Nevertheless the Debye-Waller phenomenon has a close analog: the elastic intensity is depressed due to the uncorrelated part of the thermal motion of the lattice atoms. To compute this effect we develop a time dependent scattering theory in which the quantum properties of the scattered atom are described by propagating coherently an ensemble of wave packets and lattice motion is simulated by a classical Langevin equation. Applications are made to He and Ne scattering from a surface whose lattice dynamics mimics that of Pt(111) but whose corrugation was slightly increased to enrich the diffraction structure.
Diffraction of fast helium atoms at alkali-halide surfaces under grazing angles of incidence shows intriguing diffraction patterns. The persistence of quantum coherence is remarkably strong, even though high surface temperatures and high (keV) kinetic energies of the incident atoms would strongly suggest the dominance of dissipative and decohering processes. The main source of decoherence is the excitation or absorption of surface vibrations upon impact. The momentum transfer between the surface and the incident helium atom depends on the amplitude of the thermal vibrations of the surface atoms and the energy of the incident particle. We present an ab initio simulation of the quantum diffraction of fast helium beams at a LiF (100) surface in the (110) direction, and compare with recent experimental diffraction data.
The diffraction of a coherent, stationary, beam of atoms from a standing wave of light is considered. Particular attention is paid to long interaction times for which the motion of the atoms inside the light field must be taken into account. This is the regime of dynamical diffraction, and is treated within the framework of a set of differential difference equations due to Raman and Nath. The popular Raman-Nath approximation, in which the kinetic energy imparted to the atoms by the light wave is ignored, is no longer valid and the full equations must be solved. When the interaction between the atoms and the light is very weak then few beams are produced. To describe spontaneous emission from the atom one can invoke a complex potential as a model for dissipation. If the atom beam is obliquely incident one finds anomalously high transmission close to the Bragg angles which is the physical manifestation of degeneracies in non-Hermitian matrices. The classical limit is approached when the atom-light interaction is strong, producing many beams. The classical dynamics (geometrical rays) reveal that the farfield intensity pattern is dominated by caustics. These proliferate with increasing interaction distance. Two methods are presented for analytically treating the semiclassical limit of the wave mechanics. The first begins with W.K.B. solutions to the continuised Raman-Nath equations. Mapping onto parabolic cylinder functions removes the spurious divergences of the W.K.B. solutions, giving eigenvectors which are uniformly valid for all scattering angles and may be summed to give the wavefunction. The second method transforms the original W.K.B. eigensum, using the Poisson summation formula, giving a new sum, each term of which has classical significance. This method shows that wave theory decorates the classical caustics with an Airy function intensity profile, as predicted by catastrophe theory.
When cold atoms approach a periodically magnetized surface, they are scattered by the effective repulsive potential $U(x,y)={\ensuremath{\mu}}_{B}{m}_{F}gB(x,y)$. If the period aaa of the magnetized surface is larger than the de Broglie wavelength $\ensuremath{\lambda}$ of the atoms, the atoms are diffracted. We have developed a method for calculating the location and intensity of the diffraction peaks using formal scattering theory. We solve the exact two-dimensional Schr\"odinger equation with the finite element method and calculate the scattering amplitude $f(\ensuremath{\phi})$ using the integral formula; a plot of $\ensuremath{\mid}f(\ensuremath{\phi}){\ensuremath{\mid}}^{2}$ provides a visualization of the diffraction pattern. By varying the experimental parameters within a realistic range, we predict the optimal conditions for observing atom diffraction.
Diffraction of atoms by a particular absorbing `crystal of light', that has been studied experimentally, is described by the complex potential . The diffracted beam intensities can be calculated exactly, as a function of (dimensionless) potential strength , angle of incidence , and crystal thickness . Only the beams have nonzero intensity; this `lop-sidedness' is a dramatic violation of Friedel's law. The n th beam is strong at Bragg angles , and the peaks get sharper with increasing ; they correspond to degeneracies of the (non-Hermitian) governing matrix. For normal incidence , the are periodic in , and as n increases they approach a self-similar function of .