Diffraction contrast of electron microscope images of crystal lattice defects - II. The development of a dynamical theory

The dynamical theory of electron diffraction is developed in a form suitable for the computation of images of crystal lattice defects such as dislocations observed by transmission electron microscopy. As shown in a previous kinematical theory, the contrast arises because the waves diffracted by atoms near the defect are changed in phase as a result of the displacements of these atoms from the perfect crystal positions. The two-beam dynamical theory of diffraction in the symmetrical Laue case is derived from simple kinematical principles by methods similar to those used by Darwin in the Bragg case. Simultaneous differential equations describing the changes of incident and diffracted wave amplitudes with depth in a crystal are obtained. In a perfect crystal these equations lead to the well-known Laue solutions of the dynamical equations of electron diffraction and in a deformed crystal they reduce to the kinematical theory when the deviation from the reflecting position is large. The effects of absorption can be included phenomenologically by use of a complex atomic scattering factor (complex lattice potential). Finally it is shown that an equivalent theory may be derived directly from wave mechanics in a way which allows the effects of absorption and several diffracted beams to be included. From the formal solution of this general theory some important symmetry relations for electron microscope images of defects can be deduced.