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New Journal of Physics

Bohmian trajectory-bloch wave approach to dynamical simulation of electron diffraction in crystal

Long Cheng, Ming Yi, Zejun Ding

Published Oct 17, 2018• 15 citations
Publisher

Abstract

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.

DOI
10.1088/1367-2630/aae8f1

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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.
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 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.
The numerical analysis of the diffraction features rendered by transmission electron microscopy typically relies either on classical approximations (Monte Carlo simulations) or quantum paraxial tomography (the multislice method and any of its variants). Although numerically advantageous (relatively simple implementations and low computational costs), they involve important approximations and thus their range of applicability is limited. To overcome such limitations, an alternative, more general approach is proposed, based on an optimal combination of wave-packet propagation with the on-the-fly computation of associated Bohmian trajectories. For the sake of clarity, but without a loss of generality, the approach is used to analyze the diffraction of an electron beam by a thin aluminum slab as a function of three different incidence (working) conditions which are of interest in electron microscopy: the probe width, the tilting angle, and the beam energy. Specifically, it is shown that, because there is a dependence on particular thresholds of the beam energy, this approach provides a clear description of the diffraction process at any energy, revealing at the same time any diversion of the beam inside the material towards directions that cannot be accounted for by other conventional methods, which is of much interest when dealing with relatively low energies and/or relatively large tilting angles.
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.
We apply the mixed quantum/classical method based on the Bohmian formulation of quantum mechanics [E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys. 113, 9369 (2000)] to the case of rotational diffractive surface scattering of a diatomic molecule. The rotation as well as the normal translational degree of freedom are treated classically while the two parallel degrees of freedom that account for the diffraction are treated quantum mechanically. The effects of treating some degrees of freedom classically are discussed in detail by comparing our novel approximate method to quantum wave packet results obtained by the multiconfiguration time-dependent Hartree method.