Modern equations of diffractometry. Diffraction geometry

Abstract

The various geometries of area-detector diffractometers and cameras are best described using a coordinate-free abstract operator notation. Modem methods of geometry, including especially the combined application of vectors and covectors, are used; they confer the simultaneous advantages of simplifying, virtualizing and unifying the analysis, which becomes applicable to all methods and machines. A second, and most valuable, prize arising from this approach, itself a major theme of this paper, is the complete avoidance of computationally expensive and analytically inconvenient trigonometric functions in area diffractometry. The very few occasions when they are unavoidable have already been discussed fully in a previous paper on goniometry. Basic diffraction geometry is presented first, giving all the equations necessary to identify diffraction spots and to calculate a useful generalization of the Lorentz factor. These are a formalized and extended version of those presented to the EEC Cooperative Workshop on Position-Sensitive Detector Software held at LURE in Paris in 1986.Then, various previously unpublished formulae describing beam divergence, dispersion and polarization, crystal mosaicity and angular widths of diffraction spots are presented. Finally, three specific calculations appropriate to the use of an area diffractometer are given, including a calculation of window sizes, a model of the backstop shadow and a method of surveying a diffraction pattern for assessment and prealignment.