Development of an innovative diffraction scattering theory of X-rays and electrons in imperfect crystals

Fundamental equations describing the X-ray and electron diffraction scattering in imperfect crystals have been derived in the form of the matrix Fredholm-Volterra integral equation of the second kind. A theoretical approach has been developed using the perfect-crystal Green function formalism. In contrast, another approach utilizes the wavefield eigenfunctions related to the diagonalized matrix propagators of the conventional Takagi-Taupin and Howie-Whelan equations. Using the Liouville-Neumann-type series formalism for building up the matrix Fredholm-Volterra integral equation solutions, the general resolvent function solutions of the X-ray and electron diffraction boundary-valued Cauchy problems have been obtained. Based on the resolvent-type solutions, the aim is to reveal the features of the diffraction scattering onto the crystal lattice defects, including the mechanisms of intra- and interbranch wave scattering in the strongly deformed regions in the vicinity of crystal lattice defect cores. Using the two-stage resolvent solution of the second order, this approach has been supported by straightforward calculation of the electron bright- and dark-field contrasts of an edge dislocation in a thick foil. The results obtained for the bright- and dark-field profiles of the edge dislocation are discussed and compared with analogous ones numerically calculated by Howie & Whelan [Proc. R. Soc. A (1962), 267, 206].