Bloch wave degeneracies in systematic high energy electron diffraction

1976 ◽  
Vol 282 (1308) ◽  
pp. 485-525 ◽  
Keyword(s):  
Band Structure ◽  
Incident Beam ◽  
High Energy ◽  
Real Space ◽  
Bloch Wave ◽  
Critical Voltage ◽  
Bloch Waves ◽  
One Dimensional ◽  
Symmetry Properties ◽  
Reflexion Coefficient

For the systematic diffraction of high energy electrons by a thin crystalline slab, we study the accidental degeneracies of the Bloch waves excited in the specimen by the incident beam. It is shown that these Bloch waves are the eigenstates of a one dimensional band structure problem, and this is solved by wave matching methods. For a symmetric potential, the symmetry properties of the Bloch waves are discussed, and it is shown how accidental degeneracies of these waves can occur when the reflexion coefficient for waves incident on one unit cell of the one dimensional periodic potential vanishes. The form of the band structure and the Bloch waves in the neighbourhood of a degeneracy are derived by expanding the Kramers function in a Taylor series. It is then shown analytically how the degeneracy affects the diffracted waves emerging from the crystalline specimen (in particular, the Kikuchi pattern). To understand these effects fully, W. K. B. approximations for the Bloch waves are used to derive the Bloch wave excitations and the absorption coefficients. However, to predict the degeneracies themselves, it is shown that a different formula for the reflexion coefficient, due to Landauer, must be used. This formula shows how the critical voltage at which the Bloch waves degenerate depends on the form of the potential, and allows quick, accurate, computations of the critical voltages to be made. Also, a new higher order degeneracy is predicted for some of the systematic potentials of cadmium, lead and gold. Finally, to infer the potential in real space from measurements of critical voltages and several other quantities, we suggest an inversion scheme based on the Landauer formula for the reflexion coefficient. To a close approximation this potential is proportional to V 2 of the crystal charge density.

Intervalley kinetic energy terms in the effective mass equation: a one-dimensional analytical study

1989 ◽  
Vol 67 (9) ◽  
pp. 896-903 ◽  
Author(s):  
Lorenzo Resca
Keyword(s):  
Kinetic Energy ◽  
Effective Mass ◽  
Analytical Study ◽  
Three Dimensional ◽  
Real Space ◽  
The Other ◽  
Alternative Formulation ◽  
One Dimensional ◽  
Symmetry Properties ◽  
Band Case

We show that a one-dimensional analytical study allows us to test and clarify the derivation, assumptions, and symmetry properties of the intervalley effective mass equation (IVEME). In particular, we show that the IVEME is consistent with a two-band case, and is in fact exact for a model that satisfies exactly all its assumptions. On the other hand, an alternative formulation in k-space that includes intervalley kinetic energy terms is consistent with a one-band case, provided that intra-valley kinetic energy terms are also calculated consistent with one band. We also show that the standard symmetry assumptions for both real space and k-space formulations are not actually exact, but are consistent with a "total symmetric" projection, or with taking spherical averages in a three-dimensional case.

Theory of bulk resonance diffraction in THEED

1993 ◽  
Vol 440 (1908) ◽  
pp. 95-115 ◽  
Keyword(s):  
Elastic Scattering ◽  
Incident Beam ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Zone Axis ◽  
Bloch Waves ◽  
Diffraction Geometry ◽  
Ewald Sphere ◽  
Lattice Images ◽  
New Type

A full dynamical theory has been developed for an off-axis diffraction geometry. A new type of resonance elastic scattering is found and discussed. This occurs when the Ewald sphere is almost tangential to one of the minus high order Laue zones, and is termed bulk resonance diffraction. It is shown that under certain diffraction conditions, i. e. bulk resonance diffraction conditions, effectively only a single distinct tightly bound Bloch wave localized around atom strings is excited within the crystal, and selection can be made of the particular bound Bloch waves by appropriately tilting the incident beam or the crystal. A new scheme for imaging individual tightly bound Bloch waves is proposed. Full dynamical calculations have been made for 1T–V Se 2 single crystals. It is demonstrated that chemical lattice images of V and Se atom strings can be obtained along the [0001] zone axis of a 1T–V Se 2 crystal for angles of incidence of 109.54 and 109.90 mrad respectively.

The interpretation of the zone axis critical voltage effect

1978 ◽  
Vol 36 (1) ◽  
pp. 194-195
Author(s):  
B. F. Buxton
Keyword(s):  
Structural Information ◽  
Incident Beam ◽  
Beam Theory ◽  
High Energy ◽  
Zone Axis ◽  
Critical Voltage ◽  
High Symmetry ◽  
Brillouin Zone Boundary ◽  
Extinction Length ◽  
Thickness Parameter

In cross-grating high energy electron diffraction, the scattering of the incident beam by the atomic string potentials is often so strong that there are many zone axis critical voltages below 1 MV. Steeds et al. (1976, 1977) have therefore explored the possibility of obtaining structural information from these critical voltages. In particular, for simple zone axes of high symmetry with only one string of atoms in each unit cell of the projected potential, they were able to characterize zone axis patterns by the critical voltage Ec and a thickness parameter ξ24 defined as the (2)-(4) extinction length at an orientation approximately midway between the zone axis and the first Brillouin zone boundary. Here a simple model atomic string potential will be used to investigate the information which can be gleaned from these parameters.The atomic string approximation (ASA) developed by Buxton and Tremewan (1977) will be used in order to avoid the large matrices encountered in the conventional many-beam theory.

Scattering of Quasi-Bloch Wave Sets by a Crystal With Defects

1990 ◽  
Vol 48 (2) ◽  
pp. 538-539
Author(s):  
N. I. Borgardt
Keyword(s):  
Electron Microscopy ◽  
High Resolution Electron Microscopy ◽  
Quantitative Information ◽  
Real Space ◽  
Bloch Wave ◽  
Qualitative Description ◽  
Real Crystal ◽  
Bloch Waves ◽  
Diffracted Waves ◽  
Diffraction Patterns

Application of electron microscopy methods (high resolution electron microscopy (HREM), convergent-beam electron diffraction (CBED)) permitting quantitative information of a defect structure to be obtained causes the necessity of developing a theory of scattering fast electrons by the real crystal.Determination of the electron wave function by the multislice or real space methods which provide a high accuracy requires numerical calculations for a crystal of a large volume and is restricted to foils with thicknesses of several tens of nanometrs. Effective application of approaches based on the solution of differential equations for amplitudes of diffracted waves or Bloch waves and being widely used for qualitative description of images with a diffraction contrast, is possible only in combination with a column approximation. This approximation, however, is not sufficient for a quantitative analysis, nor is it enough for interpretation of images and diffraction patterns obtained under CBED conditions.In the present paper, formalism of the Bloch waves is generalized by introdusing the idea of quasi-Bloch wave sets.

Bloch-wave interpretations of Gjønnes–Moodie lines in convergent-beam electron diffraction for non-symmorphic space-group crystals

2021 ◽  
Vol 77 (3) ◽  
pp. 222-231
Author(s):  
Hirofumi Matsuhata
Keyword(s):  
Electron Diffraction ◽  
Dimensional Space ◽  
High Energy ◽  
Bloch Wave ◽  
Convergent Beam Electron Diffraction ◽  
Bloch Waves ◽  
Beam Electron ◽  
Space Group ◽  
Space Groups ◽  
Convergent Beam

The contrast of Gjønnes–Moodie (GM) lines which appear in convergent-beam electron diffraction patterns for non-symmorphic space-group crystals is explained using Bloch waves. In the two-dimensional space groups p2mg and pg the Bloch waves for electron diffraction are described. In both space groups along the Δ line, Bloch waves are arranged as two different types, and it is shown that the two types of Bloch waves do not contribute to the intensity of forbidden reflections. Along the position where the forbidden reflection satisfies the Bragg condition, degeneracies of two Bloch waves are found and it is shown that the degenerated pair of Bloch waves do not contribute to the intensity. These Bloch-wave results provide a new perspective in the understanding of the contrast mechanism of GM lines previously described using scattering polynomials. They also advance the understanding of Bloch-wave behaviour in high-energy electron diffraction.

Inclusion of the weak beam effects in the bloch-wave simulation of dynamical diffraction by using the bethe potential

1992 ◽  
Vol 50 (2) ◽  
pp. 1450-1451
Author(s):  
J. M. Zuo
Keyword(s):  
Electron Diffraction ◽  
Computing Time ◽  
Incident Beam ◽  
Simulation Method ◽  
High Energy ◽  
Waller Factor ◽  
Bloch Wave ◽  
Wave Method ◽  
Weak Beam ◽  
Debye Waller Factor

The Bloch wave method of high energy transmission electron diffraction involves diagonalization of an nxn matrix, where n is the number of beams included in the simulation. This method is used widely in quantitative electron diffraction as the simulation method because of its flexibility in treating electron diffraction of arbitrary incident beam directions, while the multislice method of simulation is mostly used for low index zone axes of large unit cell crystals. The number of beams included in the simulation is limited by the computing time and the memory size of the computer. The computing time of the Bloch wave method is proportional to the n2. Thus only a limited number of beams can be included in the simulation. The truncation of beams may introduce a systematic error, if the number of beams included fails to converge. This is illustrated by fig. 1, in which the thickness pendelösung for MgO crystal along [001] zone axis is calculated with different number of beams in ZOLZ only, with no absorption, no Debye-Waller factor, atomic scattering factors for neutral spherical Mg and O atoms, and a = 4.2112 Å.

scholarly journals Electron and Positron Density Distribution of Bloch Waves

1973 ◽  
Vol 28 (1) ◽  
pp. 1-8
Author(s):  
G. Lehmpfuhl
Keyword(s):  
Density Distribution ◽  
Charge Density ◽  
Incident Beam ◽  
Bloch Wave ◽  
Zone Axis ◽  
Charge Density Distribution ◽  
Bloch Waves ◽  
Electrons And Positrons ◽  
Beam Diffraction ◽  
Small Charge

The charge density distribution in the strongest Bloch waves for a dynamical many-beam diffraction situation was calculated for electrons and positrons. Near the [110] zone axis of MgO there exist three strong Bloch waves for electrons. One Bloch wave is concentrated at the rows of Mg-atoms, a second at the rows of O-atoms and a third one between the atoms. The positron Bloch waves are mainly concentrated between the atom rows and have only small charge density at the positions of the atoms. For an incident beam parallel to the [110] axis there exists only one strong positron Bloch wave while for electrons more than three Bloch waves are strong, explaining the channeling behaviour of positrons and electrons. Strong partial waves of different electron Bloch waves can be identified in the diffraction pattern from a MgO crystal wedge.

Electron and Positron Density Distribution of Bloch Waves

1973 ◽  
Vol 28 (5) ◽  
pp. 661
Author(s):  
G. Lehmpfuhl
Keyword(s):  
Density Distribution ◽  
Charge Density ◽  
Incident Beam ◽  
Bloch Wave ◽  
Zone Axis ◽  
Charge Density Distribution ◽  
Bloch Waves ◽  
Electrons And Positrons ◽  
Beam Diffraction ◽  
Small Charge

The charge density distribution in the strongest Bloch waves for a dynamical many-beam diffraction situation was calculated for electrons and positrons. Near the [110] zone axis of MgO there exist three strong Bloch waves for electrons. One Bloch wave is concentrated at the rows of Mg-atoms, a second at the rows of O-atoms and a third one between the atoms. The positron Bloch waves are mainly concentrated between the atom rows and have only small charge density at the positions of the atoms. For an incident beam parallel to the [110] axis there exists only one strong positron Bloch wave while for electrons more than three Bloch waves are strong, explaining the channeling behaviour of positrons and electrons. Strong partial waves of different electron Bloch waves can be identified in the diffraction pattern from a MgO crystal wedge.

Sign in / Sign up

Export Citation Format

Share Document