Dynamical Method And Quantum Trajectory Method To Study Of Electron Diffraction And Imaging In Crystals

Electron microscopy is one of the most important tools for material characteriza-tion and analysis,and is relevant to the advancement of frontier science and technology and the improvement of daily living standards.Quantitative analysis and interpretation of experimental images obtained by electron microscopy,on the other hand,is the basis for material characterization and analysis.Therefore,the in-depth study of the physi-cal processes of electron-solid interactions and the development of mature theoretical models are the bridges between humans and the microscopic world.Coherent and incoherent scattering of electrons occurs in solids,where the inter-actions in amorphous materials are mostly incoherent scattering,which is usually sim-ulated by classical trajectory Monte Carlo(CTMC)method;while in crystals,coher-ent scattering of electrons is more important and cannot be treated by CTMC method.Therefore,other theoretical models need to be developed to consider the coherent scat-tering of electrons in crystals.In this paper,we have developed some computational methods based on dynamical method and quantum trajectory theory to simulate and study a variety of diffraction processes of electrons in crystals.Chapter 1 introduces the development history,principles and applications of elec-tron microscopes.We briefly outline the background of the birth of electron micro-scopes and the means of resolution enhancement,and introduce the structure,uses and differences of common types of electron microscopes.While electron microscopes rely on the complex signals generated by the interaction of electron beams with materials for analysis and characterization of materials,accordingly,we introduce the principles and uses of various electron energy spectra,electron microscopic imaging,and elec-tron diffraction patterns.As for simulation,this chapter introduces the CTMC method for electrons scattering in amorphous materials,the quantum mechanical method for electrons scattering in crystals,and the quantum trajectory method,and outlines the achievements of our group in developing these methods.Chapter 2 introduces the theoretical basis covered in this thesis,which mainly in-cludes electron diffraction theory and quantum trajectory theory.In electron diffraction theory we introduce the basics of crystallography,and the kinematical and dynamical theories of electron diffraction in crystals.The kinematical theory is based on Bragg's law,which is simple and intuitive and suitable for qualitative analysis,but it is only applicable to thin samples,where the intensity of the diffracted beam is small and the intensity variation of the zero-order beam can be neglected,so the kinematical theory has great limitations.Therefore,the dynamical theory is needed in the quantitative simulation of multibeam diffraction.The dynamical theory used in this paper is based on Bethe's eigenvalue equation and takes into account the effects of various inelas-tic scattering processes such as thermal diffusive scattering,plasmon excitation,and single-electron excitation,which can effectively simulate the diffraction of electrons in crystals as well as the absorption of wave functions.The quantum trajectory theory is a probabilistic flow description of quantum mechanics,which can take into account the particle and wave nature in quantum mechanics and investigate the quantum system both intuitively and precisely.The quantum trajectory theory is often used in the theoretical study of a variety of quantum systems,and at the same time,with the advancement of experimental techniques,quantum trajectory has become observable in experiments and is now a very important theoretical interpretation of quantum mechanics.In addition,this chapter introduces the principles and theoretical form of special relativity,which requires relativistic corrections to the mass and wavelength of electrons when studying the interaction of electrons with crystals.Chapter 3 investigates electron diffraction in transmission electron microscopy by dynamical methods,including the diffraction patterns for parallel beams and convergent beams.For the parallel electron beam,we give diffraction dot patterns in several crystal directions and compare them with the kinematical model.We also studied the variation of the intensity of the zero-order beam and the diffraction beam with the thickness of the sample,which cannot be considered by the kinematical model,and gave the variation curves for multiple crystal directions.In addition,we studied the effect of the incident angle of the electron beam on the diffraction pattern and gave a theoretical method to find the diffraction conditions such as two-beam and three-beam.As for the convergent beam electron diffraction pattern,we analyzed the effects of various parameters such as convergence half-angle,sample thickness,and acceleration voltage.The simulated results are in perfect agreement with the experimental results and can show more infor-mation of the experimental images compared with the simulated results given by other software.Moreover,by matching the images,we can reverse the experimental infor-mation from the experimental results with insufficient parameters,which can be used for the analysis of crystal structures.In Chapter 4,we develop a new quantum trajectory calculation method based on the dynamics of Bloch waves.The previous quantum trajectory calculation mainly uses the split operator method,multi-slice method and other spatial grid methods,which are computationally intensive and prone to encounter trajectory errors due to phase prob-lems.In particular,the spatial grid method cannot calculate the case of divergent tra-jectories.In contrast,our Bloch wave quantum trajectory method does not need to divide the grid,and the velocity field of the full space can be calculated directly by the superposition coefficients of the wave function.We also develop the tracking al-gorithm to calculate the quantum trajectories,which greatly reduces the computational effort,increases the computational speed,and can calculate the divergent trajectories.We applied the Bloch-wave quantum trajectory method to study the channelling effect in crystals,quantitatively investigating the effects of material type,thickness,and inci-dent electron energy,and providing an intuitive interpretation of the extinction distance in crystals.Compared with the results of wave function,quantum trajectory and wave function give the same distribution,but wave function can only give the spatial distribu-tion of probability density without evolutionary sequential information;while quantum trajectory not only gives the distribution density of electrons,but also can show the se-quential relationship from incidence to diffraction to emission,which is more intuitive.The details that are difficult to find in the wave function can be easily found in the im-age of quantum trajectory,and the quantum trajectory has better representation than the wave function especially in the three-dimensional case,which can clearly reflect the system state at each place in the three-dimensional space.In addition,we also study the effect of inelastic scattering on the channel effect by the quantum trajectory method.In Chapter 5,the electron backscatter diffraction pattern is studied by the dynam-ical method and the quantum trajectory method,respectively.And a faster momentum expectation method is developed based on the idea of quantum trajectory theory.We studied the effect of electron energy,sample thickness,crystal orientation and other factors on the electron backscatter diffraction pattern by simulation.The simulation results are in good agreement with the experimental results.Meanwhile,the quantum trajectory method can explain the formation process of electron backscatter diffraction pattern from the perspective of single electron,and more intuitively describe the diffrac-tion of the wave function of multibeam interference in the crystal.In addition,we have developed a method to construct the diffraction spheres for electron backscatter,i.e.,by calculating the diffraction pattern of the crystal in all directions,and then stitching them together according to the crystal direction to form a sphere,so that the diffraction pattern of the crystal in all directions can be represented in one picture,and the rela-tionship between the crystal direction and the Kikuchi band as well as the higher-order Laue zone rings can be described more comprehensively.The diffraction spheres are useful for building a more intuitive and efficient database or simulation software.In Chapter 6,we develop a new quantum trajectory Monte Carlo(QTMC)calcu-lation method,which is more accurate in the calculation of trajectories than the original QTMC method and has no limitation on the depth of calculation.At the same time,a N-nearest neighbor model is introduced to take into account all atoms interacting with electrons,making it possible to simulate arbitrary crystals with this method instead of limiting it to elemental crystals.We analyze the effect of various excitation processes on the results of atomic resolution secondary electron imaging(ARSEI)using the QTMC method,and thus construct a new imaging mechanism.Other existing theoretical mod-els assume that the atomic resolution of secondary electron imaging originates from the secondary electrons directly excited from the inner-shell and that the cascade processes do not contribute to the atomic resolution.We found through quantitative calculations that the high-energy secondary electrons excited by the inner-shell do not contribute directly to the experimental image,and the real atomic resolution actually comes from the large number of low-energy secondary electrons generated by these high-energy secondary electrons after successive cascade excitations.In addition,based on the new imaging mechanism,we also found that the atomic resolution of secondary electrons can distinguish the same elements at different depths near the sample surface,i.e.,it has ultra-high surface three-dimensional resolution,which can be widely used for material surface analysis as well as two-dimensional material characterization.Chapter 7 is a summary and outlook of the whole thesis.