Coherent nuclear resonant scattering of X-rays: Time and space picture

The problem of coherent resonant scattering of X-rays by an ensemble of nuclei is solved directly in time and space. In a first step the problem with a single coherently scattered beam is considered – nuclear forward scattering. The wave equation describing the propagation of the radiation through the nuclear ensemble is derived. It is a first order integro-differential equation. Its kernel is a double time function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$K(t,\tilde t)$$ \end{document} which represents the coherent single scattering response of the nuclear system at time t to excitation at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde t$$ \end{document}. The kernel is defined by the character of the interactions the nuclei experience with the environment and by the character of their motion. A general procedure of solution of the wave equation is introduced which is independent of the type of kernel. In a second step the wave equation is generalized to the case of many coherently scattered beams, which is, e.g., the case of nuclear Bragg diffraction. Kernels of the wave equations are derived for some particular cases: collective motion of nuclei in space, thermal lattice vibrations, time-independent hyperfine interactions, and time-dependent hyperfine interactions due to external magnetic-field switching.