A foundation for the statistical dynamical theory of diffraction

Abstract

The statistical dynamical theory [Kato (1980). Acta Cryst. A36, 763–769] is reformulated on a sounder basis. The starting wave equation is free from the so-called Takagi–Taupin (T–T) approximation. Functional calculus, an operational technique and the concept of the Green function are used as mathematical tools. Integro-differential equations are derived for the coherent (averaged) wave field and the energy flow vector of the incoherent intensity field. The formulae are exact except for assuming a model in which the fluctuation of the lattice phase is a set of Gaussian random variables defined in three-dimensional space. The general framework of the previous theory is justified within the T–T approximation. In general, however, new terms must be added and some terms have to be revised by introducing a Green function matrix. The theory may be used as a starting point when any approximate theory is developed for practical purposes.