Anisotropy in thermal motion and in secondary extinction

Abstract

Within certain good approximations the probability distribution function (p.d.f.) used to describe mosaic-block orientation in secondary-extinction models is exactly analogous to the p.d.f. for atomic thermal motion in the harmonic approximation. Use is made of this relationship to explain carefully, with the aid of several diagrams, certain distinctions and relationships common to both p.d.f.'s - which if not properly understood can lead (and have led) to some important confusions. For example, if the three-dimensional p.d.f. is Gaussian, surfaces of constant probability density are ellipsoidal (e.g. the thermal-vibration ellipsoid); but the scattering process 'sees' this p.d.f. as a one-dimensional projection, the half-width of which lies on a fourth-order surface (shaped, for example, like a peanut shell). For extinction it is shown explicitly that the form of this projected one-dimensional function is independent of experimental conditions (e.g. collimation), and that an earlier form [Coppens & Hamilton (1970), Acta Cryst. A26, 417-425], still commonly used and tested, is always incorrect. Apart from the intentional restriction of the detailed analysis of secondary extinction to type I extinction (in which mosaic-block orientation is the dominant effect), the approximations adopted are shown to have a wide range of validity. The (unusual) conditions under which the approximations may be sufficiently invalid to produce detectable effects are examined qualitatively in relation to the possibility of experimental investigations.