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The Journal of Chemical Physics

Quantum trajectories in atom–surface scattering with single adsorbates: The role of quantum vortices

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

Published Apr 19, 2004• 53 citations
Publisher

Abstract

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.

DOI
10.1063/1.1683136

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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.
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 elastic resonant scattering of He atoms off the Cu(117) surface is fully described with the formalism of quantum trajectories provided by Bohmian mechanics. Within this theory of quantum motion, the concept of trapping is widely studied and discussed. Classically, atoms undergo impulsive collisions with the surface, and then the trapped motion takes place covering at least two consecutive unit cells. However, from a Bohmian viewpoint, atom trajectories can smoothly adjust to the equipotential energy surface profile in a sort of sliding motion; thus the trapping process could eventually occur within one single unit cell. In particular, both threshold and selective adsorption resonances are explained by means of this quantum trapping considering different space and time scales. Furthermore, a mapping between each region of the (initial) incoming plane wave and the different parts of the diffraction and resonance patterns can be easily established, an important issue only provided by a quantum trajectory formalism.
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.
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.
Vortices are known to play a key role in many important processes in physics and chemistry. Here, we study vortices in connection with the quantum trajectories that can be defined in the framework provided by the de Broglie–Bohm formalism of quantum mechanics. In a previous work, it was shown that the presence of a single moving vortex is enough to induce chaos in these trajectories. Here, this situation is explored in more detail by discussing the relationship between Lyapunov exponents and the parameters characterizing the vortex dynamics. We also consider the issue when more than one vortex exists. In this case, the interaction among them can annihilate or create pairs of vortices with opposite vorticity. This phenomenon is analyzed from a dynamical point of view, showing how the size of the regular regions in phase space grows, as vortices disappear.