Calculation Of Bohmian Quantum Trajectories For STEM

Electron diffraction is widely used in the study of material structure characteristics as a main behavior of the wave nature of electrons. The material analysis technique based on electron diffraction applied for the crystal structure research well. Depending on the analysis of the diffraction pattern for the structure which obtained using these analysis tools, we can get the information of the crystal structure,In order to analysis the measured data, comprehensive study of the interaction process of electronics and solid should be made. In this thesis, Bohmian quantum trajectory method has intuitive and accurately described the dynamics of electron diffraction in scanning transmission electron microscopy (STEM). The evolution of electron wave function in the crystal is obtained by the numerical solution of time-dependent schrodinger equation, and the quantum trajectory results intuitively shows the dynamic diffraction process of the electron and crystal structure.Quantum trajectory theory is proposed by de Broglie and developed by Bohm. The key element of quantum trajectory theory is the introduction of the concept of particle into the quantal description of nature. Quantum trajectory theory assumes that the wave function represented an objective field and is not only a mathematical symbol, it defines a quantum mechanism system made up of particles, the particle has a precise definition of the position, but the location is a time-dependent function, the momentum of the particle is the gradient of the phase of wave function. Quantum trajectory theory not only provides a new way to explain the quantum phenomenon, and provides a new research method for calculating the electron interaction with crystal also. It can be accurate to a specific crystal structure, this can be widely used in study of the latest atomic materials characterization.In the introduction, we introduced the interaction process of electrons with crystal, including electron diffraction and scattering problem, and scanning transmission electron microscopy theory and the simulation study. Scanning transmission imaging is different from general parallel beam transmission electron microscopy imaging, it is made up of convergent beam. We considered the interaction of the incident convergent beam with the crystal potential in our study.The Bohmian quantum trajectory theory presented in chapter2is the theoretical basis of this thesis, including the basic formula of Bohmian mechanics, the algorithm of trajectory and the applications of Quantum trajectory method. Monte Carlo method which has been widely used in the interactions research of electrons with amorphous or polycrystalline materials is based on classical trajectory concept, it ignores the coherent scattering of electrons and cannot consider the structure information of crystal. While Bohmian quantum trajectory theory can describe wave-particle duality of electron diffraction process, it has both the intuitiveness of classical mechanics and accurate foresight of quantum mechanics.Chapter3discusses the dynamical simulation of electron diffraction. It converts into the problem of solving the schrodinger equation. It’s divided into the time-dependent and time-independent calculation methods. The general algorithm of time-dependent method is splitting operator method, Multislice method, Bloch wave method, and the scattering matrix method for time-independent method. Time-dependent algorithm is mainly used for the low-energy electron evaluation problem, as to the high incident energy occasion, time-independent method is appropriate. Moreover in the simulation of high resolution STEM image, multislice method is better than Bloch wave method.We present in Chapter4the 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 Bohmian quantum trajectory theory not only provides an intuitive way to understand electron diffraction phenomena in electron microscopic imaging but more importantly offers a numerical scheme to build a quantum Monte Carlo simulation technique for study of electron interaction with a crystal and nanomaterial. 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 numerical solution of Schrodinger equation is calculated based on the FFT multislice approach. Firstly, a series of defocus conditions for electron probe are discussed. Then we investigate two representative scanning locations, at the atom column and the tunnel center between atom columns, has been displayed. Based on the wave function obtained Bohmian quantum trajectories are shown to tend to be attracted to move along atom columns during their propagation. These trajectories in the crystal in fact represent essentially electron diffraction process and show actually how an electron probe are scattered dynamically in a crystal. The results help us to better understand the electron diffraction process in a microscopic imaging from a trajectory-based point of view. In chapter5, we have developed a new theoretical method based on Bohmian quantum trajectory theory to calculate the differential cross section of electron scattering from atoms in near field region, which provides an intuitive interpretation of the dynamic interaction process during the electron interaction with matter in precision of quantum mechanics. Quantum trajectories are calculated by a numerical solution of the time-dependent Schrodinger equation using the split-operator method; a central potential is constructed with screening Coulomb potential. Depending on the definition, the differential cross section can be described as the number of trajectories scattered into unit solid angle. Bohmian quantum trajectory method enables in principle the study of electron coherent scattering within more complex material structures for applications to nanomaterial characterization.Finally we summarized the thesis in Chapter6.