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The Journal of Chemical Physics

The temperature dependence of diffracted beam intensities in atom–surface scattering

Bret Jackson, Horia I. Metiu

Published Aug 15, 1985• 44 citations
Publisher

Abstract

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.

DOI
10.1063/1.449333

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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.
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.
Surface diffusion of small adsorbates is analyzed in terms of the so-called intermediate scattering function and dynamic structure factor, observables in experiments using the well-known quasielastic Helium atom scattering and Helium spin echo techniques. The linear theory applied is an extension of the neutron scattering due to van Hove and considers the time evolution of the position of the adsorbates in the surface. This approach allows us to use a stochastic trajectory description following the classical, quantum and Bohmian frameworks. Three different regimes of motion are clearly identified in the diffusion process: ballistic, Brownian and intermediate which are well characterized, for the first two regimes, through the mean square displacements and Einstein relation for the diffusion constant. The Langevin formalism is used by considering Ohmic friction, moderate surface temperatures and small coverages. In the Bohmian framework, analyzed here, the starting point is the so-called Schrödinger-Langevin equation which is a nonlinear, logarithmic differential equation. By assuming a Gaussian function for the probability density, the corresponding quantum stochastic trajectories are given by a dressing scheme consisting of a classical stochastic trajectory followed by the center of the Gaussian wave packet, and issued from solving the Langevin equation (particle property), plus the time evolution of its width governed by the damped Pinney differential equation (wave property). The Bohmian velocity autocorrelation function is the same as the classical one when the initial spread rate is assumed to be zero. If not, in the diffusion regime, the Brownian-Bohmian motion shows a weak anomalous diffusion.
The authors use analytic models valid for neutron scattering and numerical calculations to examine a method proposed by Drolshagen and Heller for computing atom diffraction from surfaces. Our main concern is the manner in which the use of narrow packets causing a partial coherence of the incident beam, might affect the diffracted intensity. We find that the demands of representing beam coherence correctly conflict with the requirements for the validity of the propagation scheme. We suggest some improvements that should increase the reliability of the method without affecting greatly its remarkable computational and conceptual advantages.
Analytic methods based on matter-wave diffraction are a cornerstone in condensed-matter research, providing access to static and dynamic materials properties down to the atomic level. In these experiments, the shape of the diffraction pattern is largely determined by the lattice at equilibrium whereas vibrationally-induced distortions are treated perturbatively. Here, we show that the perturbative approach does not hold for helium diffracted at kiloelectronvolt energy through freestanding single-layer graphene. In this case, we enter a new regime of strong coupling where the projectile strongly interacts with the electron density of several lattice atoms simultaneously, leading to phase shifts of several radians. In consequence, lattice distortions introduce a significant phase spread that cannot be described by the typically employed Debye-Waller factor. We show that the weak-coupling regime is retained for atomic hydrogen diffraction. The experimental results are supported by simulations, providing a regime-independent approach to describe the influence of phonons on atom diffraction phenomena.
The Debye-Waller factor may be close to one, hence ineffective for decoherence, if it is caused by high-frequency vibrations. This appears as a manifestation of the Zeno effect, where the quantum properties of a system do not deteriorate (or do so very slowly) if the system is subjected to frequently repeated interactions. In the case of atom-surface scattering, the decoherence time is shown to be long for high frequencies, such that the oscillation period is short in comparison to the duration of a collision. For a periodic perturbation the restoration of coherence, in the case where many oscillation periods take place during a prescribed time, is shown.