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Europhysics Letters (EPL)

On the classical limit in atom-surface diffraction

Ángel S. Sanz, F. Borondo, Salvador Miret-Artés

Published Aug 1, 2001• 32 citations
Publisher

Abstract

The transition to the classical limit in atom-surface diffraction is studied using the de Broglie-Bohm causal formalism. In particular, we focus on rainbow scattering, which is a well-defined effect in classical mechanics and has a clear counterpart in quantum mechanics. In order to achieve this limit, we consider the scattering of particles with increasing masses off a Cu(110) surface. Although the classical limit seems strictly unreachable, quantum trajectories mimic the characteristics of the classical intensity distribution, and their use allows to unveil the mechanism by which the quantum rainbow condition takes place.

DOI
10.1209/epl/i2001-00415-5

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The transition to the classical limit in atom-surface diffraction is studied using the de Broglie-Bohm causal formalism. In particular, we focus on rainbow scattering, which is a well-defined effect in classical mechanics and has a clear counterpart in quantum mechanics. In order to achieve this limit, we consider the scattering of particles with increasing masses off a Cu(110) surface. Although the classical limit seems strictly unreachable, quantum trajectories mimic the characteristics of the classical intensity distribution, and their use allows to unveil the mechanism by which the quantum rainbow condition takes place.
The method of quantum trajectories proposed by de Broglie and Bohm is applied to the study of atom diffraction by surfaces. As an example, a realistic model for the scattering of He off corrugated Cu is considered. In this way, the final angular distribution of trajectories is obtained by box counting, which is in excellent agreement with the results calculated by standard SSS matrix methods of scattering theory. More interestingly, the accumulation of quantum trajectories at the different diffraction peaks is explained in terms of the corresponding quantum potential. This nonlocal potential ``guides'' the trajectories causing a transition from a distribution near the surface, which reproduces its shape, to the final diffraction pattern observed in the asymptotic region, far from the diffracting object. These two regimes are homologous to the Fresnel and Fraunhofer regions described in undulatory optics. Finally, the turning points of the quantum trajectories provide a better description of the surface electronic density than the corresponding classical ones, usually employed for this task.
Diffraction and interference of matter waves are key phenomena in quantum mechanics. Here we present some results on particle diffraction in a wide variety of situations, ranging from simple slit experiments to more complicated cases such as atom scattering by corrugated metal surfaces and metal surfaces with simple and isolated adsorbates. The principal novelty of our study is the use of the so-called Bohmian formalism of quantum trajectories. These trajectories are able to satisfactorily reproduce the main features of the experimental results and, more importantly, they provide a causal intuitive interpretation of the underlying dynamics. In particular, we will focus our attention on: (a) a revision of the concepts of near and far field in undulatory optics; (b) the transition to the classical limit, where it is found that although the quantum and classical diffraction patterns tend to be quite similar, some quantum features are maintained even when the quantum potential goes to zero; and (c) a qualitative description of the scattering of atoms by metal surfaces in the presence of a single adsorbate.
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.
In this work, a full quantum study of the scattering of He atoms off single CO molecules, adsorbed onto the Pt(111) surface, is presented within the formalism of quantum trajectories provided by Bohmian mechanics. By means of this theory, it is shown that the underlying dynamics is strongly dominated by the existence of a transient vortitial trapping with measurable effects on the whole diffraction pattern. This kind of trapping emphasizes the key role played by quantum vortices in this scattering. Moreover, an analysis of the surface rainbow effect caused by the local corrugation that the CO molecule induces on the surface, and its manifestation in the corresponding intensity pattern, is also presented and discussed.
Typically one expects that when a heavy particle collides with a surface, the scattered angular distribution will follow classical mechanics. The heavy mass usually assures that the coherence length of the incident particle in the direction of the propagation of the particle (the parallel direction) will be much shorter than the characteristic lattice length of the surface, thus leading to a classical description. Recent work on molecular interferometry has shown that extreme collimation of the beam creates a perpendicular coherence length which is sufficiently long so as to observe interference of very heavy species passing through a grating. Here we show, using quantum mechanical simulations, that the same effect will lead to quantum diffraction of heavy particles colliding with a surface. The effect is robust with respect to the incident energy, the angle of incidence, and the mass of the particle.
The quantum trajectories in the de Broglie-Bohm formulation of quantum mechanics depend on an additional quantum potential derived from the full wave solution of Schrödinger's equation. The task of supplying collectively all the correct quantum results strongly alters the characteristics of the corresponding classical trajectories, which underlie semiclassical approximations to the evolving wave function. Both classical and quantum trajectories are here considered to be conservative with no influence of an external environment, even though this is the source of eventual classicality in quantum systems, that is, decoherence. The concept of integrability, closely correspondent in classical and quantum mechanics, is not preserved by the quantum trajectories. General systems, in which classical chaotic motion participates, are much harder to treat semiclassically, but quantum trajectories can be chaotic even for integrable systems. This discrepancy between the character of classical and quantum trajectories in the de Broglie-Bohm interpretation does not clarify the singular classical-quantum transition.