Some applications of the theory of electron diffraction contrast - Saldin, Dilano Kerzaman (1975)
﻿ The theory of the diffraction of electrons from crystals originated in the classical paper of Bethe (1928) very shortly after the enunciation of the principles of quantum mechanics. The use of the electron microscope as a practical tool for studying crystal structures (and their defects) led to the development of the theory in forms suitable for the explanation of contrast effects on micrographs. The fundamental problem is the calculation of the electron wave-function emerging from the exit force of the crystal. Broadly speaking, the two types of electron diffraction theory are the quantum mechanical and the wave optical. Chapter 1, which begins with a review of the development of diffraction theories since the early years of this century, is mainly concerned with the quantum mechanical approach. The elegant formulation of Yoshioka (1957) in which the phenomenon of Absorption is shown to be capable of explanation by the use of a complex potential is outlined, and the usefulness of the dispersion surface construction is demonstrated. The conditions under which the Bloch wave and the Darwin (1914) formulations or equivalent are discussed, as is the validity of various approximations normally used in the theory as applied to deformed crystals. The wave-optical approach is discussed in Chapter 2 with particular reference to Sturkey's (1962) theory. It is shown that the discrepancy noted by Fukuhara (1966) between the results of Sturkey's and Bethe's theories may be explained by the fact that Sturkey's theory is concerned with positron diffraction. Numerical calculations are given to support this explanation and a corrected form of Sturkey's theory is given. The theory of the weak-beam technique (Cockayne, Ray and Whelan 1969) is discussed in Chapter 3. It is shown that under such conditions the dynamical theory expressed in the Darwin formalism asymptotically approaches the form of a kinematical theory modified to take account of the effects of absorption. Under such conditions it is shown that only two Bloch waves contribute appreciably to the image contrast of defects and that as a result, a theory based on the kinematical scattering of Bloch waves gives results identical with those of kinematic theory applied to diffracted wave amplitudes. For particular experimental situations, such as contrast from extended dislocation nodes in silicon, numerical calculations, which confirm excellent agreement between the results of dynamical theory and kinematical theory are described. Applications of the kinematical Bloch wave scattering theory to the predictions of the main features of contrast from small defects, such as dislocation loops, are reviewed in Chapter 4. The computer program used for calculating the image contrast from defects is described in Chapter 5. Also given is a critical discussion of numerical methods of solving a set of coupled ordinary differential equations, such as the Howie-Whelan equations (1961), for complicated strain-fields. A method of generating computer simulated electron microscope images is discussed with careful attention to the simulation of the photographic processes involved. Chapter 6 begins with a review of linear elasticity theory and its application to the theory of dislocations. It is shown how integration around a suitably chosen Burgers' circuit yields the formulae derived by Yoffe (1960) for the displacement field, and how these formulae may be used to compute the displacement field due to a regular polygonal dislocation loop of an arbitrary number of sides. This field is expressed in terms of its components in the directions of the cube axes of the crystal in a form convenient for its use in a dynamical theory computer program. An introduction to the theory of point defects and their mechanisms of clustering during annealing and after particle irradiation is given in Chapter 7. The reasons for the observation of dislocation loops and stacking-fault tetrahedron are discussed. The electron microscope contrast features of such defects as well as others, such as voids and spherical inclusions,under both kinematical and dynamical imaging conditions are considered. The various methods used so far for the evaluation of the displacement fields due to dislocation loops and previous computer simulation work on the images due to such defects are reviewed. Finally a description of the present investigations of image contrast carried out using the polygonal loop model of Chapter 6 is described. The computer simulated images and their corresponding experimental images are displayed,and the effects of varying parameters-such as the loop size, depth in the foil, operating Bragg reflection etc. are studied. Such computer simulation work is shown to be helpful in the identification not only of Frank loops, which are common in f.c.c. metals, but also of perfect loops (with shear components of Burgers vector) which are common in b.c.c. metals. Chapter 8 begins with a consideration of the various mechanisms of formation of a stacking-fault tetrahedron in an f.c.c metal and contains a discussion of the nature of the displacement field around it, including the nature of its stacking-faults. A method of constructing its displacement field from four triangular edge loops on {111} planes is described. The constituent triangular loops may be constructed by the methods outlined in Chapter 6. Finally the results of computations of the image contrast due to stacking-fault tetrahedra are shown and discussed. It is shown, for instance, that interstitial and vacancy-type tetrahedra may be distinguished by essentially the same methods as those used for dislocation loops. The main conclusions of this thesis and remaining problems are discussed in Chapter 9.