Mathematical methods in the solution of the Hamilton-Darwin and the Takagi-Taupin equations

Abstract

The diffraction of neutrons by a single crystal is intrinsically a multiple scattering problem. For an ideally imperfect mosaic crystal the Hamilton-Darwin transfer equations describe the coupling of the incident and diffracted beams; whereas, for a perfect crystal one must use the dynamical theory of diffraction, which can be recast in the form of two coupled partial differential equations commonly referred to as the Takagi-Taupin equations. From a mathematical point of view these two problems are equivalent, although the physical manifestations of the solutions are quite different. For the occasion of Professor Shull's seventieth birthday celebration, we bring together in this paper some of the mathematical techniques which we have found useful in elucidating the subtleties of the Bragg diffraction of neutron by crystals.