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Dianzi xianwei xuebao

Study on electron diffraction using bohmian quantum trajectory method

Ding Ze-jun

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Diffraction is one of the main characteristics of the undulatory motion of electron.It is widely used to study the structure of materials.In this work the dynamics of electron diffraction processes were investigated using bohmian quantum trajectory method,which gives an intuitive interpretation of the physical phenomena and a prediction of quantum mechanics.Crystal potential of electron scattering from two dimensional model crystal was constructed by analytical pseudo potential.Quantum trajectories were calculated by a numerical solution of the time-dependent Schrdinger equation.

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Diffraction is one of the main characteristics of the undulatory motion of electron.It is widely used to study the structure of materials.In this work the dynamics of electron diffraction processes were investigated using bohmian quantum trajectory method,which gives an intuitive interpretation of the physical phenomena and a prediction of quantum mechanics.Crystal potential of electron scattering from two dimensional model crystal was constructed by analytical pseudo potential.Quantum trajectories were calculated by a numerical solution of the time-dependent Schrdinger equation.
Based on Bohmian quantum trajectory theory, an investigation of electron scattering in a thin crystal has been performed. A time-independent interaction potential was employed to describe electron scattering from model crystal. Quantum trajectories were calculated by a numerical solution of the time-dependent Schrödinger equation. The probability density functions and quantum trajectories representing the electron diffraction in real space were obtained from the calculation. The quantum trajectories provide an intuitive dynamics of the interaction process during the electron diffraction.
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 Bohmian trajectory method is employed to study electron diffraction in crystalline materials.It provides a fresh understanding of the process of electron diffraction, including traveling channels of electrons and formation of diffraction patterns.By combining it with the Bloch wave method, the electron trajectories can be calculated more efficiently than the traditional wave-packet propagation algorithm.Meanwhile, we propose a momentum expectation approach which is a good approximation method with even higher computational efficiency.Both methods result in intuitive and accurate electron trajectories for the simulation of the electron backscatter diffraction (EBSD) pattern.Excellent agreement has been obtained between the simulated trajectory distributions and the experimental EBSD pattern from Mo (001) at 20 kV, where the Kikuchi patterns and higher order Laue zone rings are characterized.
We present in this work the calculation of Bohmian quantum trajectories representing the wave function propagation in a crystal for a focused electron probe in a scanning transmission electron microscope (STEM). The wave function and quantum trajectories are obtained from the calculation of time-dependent Schrödinger equation by fast Fourier transformation multislice algorithm. In our work, the Bohmian quantum trajectories of a scanning probe penetrating a Cu crystal are studied as an example of this calculation scheme. The results help us to better understand the electron diffraction process in a microscopic imaging from a trajectory-based point of view. This Bohmian quantum trajectory method can be used to extend the application of classical Monte Carlo method from the study of electron interaction with amorphous solid to crystalline structure.
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 trajectory-based formalism of Bohmian mechanics constitutes an alternative (but equivalent at a predictive level) approach to the standard formulation of quantum mechanics. Here we show the advantages of this formalism in providing both an accurate description and a novel interpretation when it is applied to different phenomena of interest in elastic particle scattering, such as diffraction by “hard” and “soft” slits, and quantum vortical dynamics due to the presence of single adsorbates. These problems illustrate fairly well how quantum trajectories are able to satisfactorily reproduce the main features of real scattering experiments as well as to provide a causal insight of the underlying dynamics