INELASTIC-SCATTERING OF HIGH-ENERGY ELECTRONS IN A CRYSTAL IN THERMAL-EQUILIBRIUM WITH THE ENVIRONMENT .2. SOLUTION OF THE EQUATIONS AND APPLICATIONS TO CONCRETE CASES

The equations governing the dynamical diffraction of high-energy electrons in a crystal in thermal equilibrium with the environment, derived earlier [C. Fanidis, D. Van Dyck and J. Van Landuyt, Ultramicroscopy 41 (1992) 55], have been solved. The solutions for the different wave-function components have been obtained in a closed analytical form. Complex potentials, accounting for absorption of the electrons have been derived in the solutions. For phonon scattering, analytical expressions of these potentials have been calculated, and for independent vibrations a parametrized form is presented. It is shown that even in the case of no energy transfer the thermal equilibrium will influence the image and the diffraction pattern intensities. For phonon scattering, images and diffraction patterns and the complex potentials have been calculated numerically for concrete cases. The inelastic scattering results in diffuse intensity in the diffraction pattern. The images formed from inelastically scattered electrons and the relative intensities in the corresponding diffraction patterns depend on the polarization of the excited vibration modes. From the results it is concluded that thermal equilibrium and inelastic scattering should be taken into account in the calculations.