Extinction within the limit of validity of the Darwin transfer equations. I. General formalism for primary and secondary extinction and their applications to spherical crystals

Abstract

A theory of extinction is derived which is valid within the limit of the Darwin intensity transfer equations. An expression describing the effect of n-fold rescattering within an ideal crystallite is derived, which differs from the equation given by Zachariasen because independent coordinates x1 and x2 based on an external coordinate system have been used, rather than the coordinates t1 and t2 which are only mutually independent if the crystal is a parallelepiped with faces parallel to the incident and diffracted beams. Furthermore, the derivation of the earlier expressions is based on a generally unjustifiable reversal of the direction of the diffracted ray (interchange of t2 and t'2). An exact expression is derived for the diffraction cross section σ(ε1) in the perfect crystallite, which contains a factor sin 2θ neglected in the earlier work. As a result, the previously used classification of crystals into type I and type II becomes less well defined because at very small Bragg angles particle size always becomes the dominant effect. It is shown that the extinction factor yp (p = primary), for a perfect spherical crystallite, calculated with the present theory, is in good agreement with calculations based on the dynamical theory. Furthermore, the limiting behavior of the expressions at 2θ = 0 and π justifies some of the mathematical approximations made. For a mosaic crystal the extinction coefficient y is written as yp . ys (s = secondary), yp is evaluated numerically from the expressions derived. An analytical expression for yp is obtained by least-squares fit to the numerical values. A similar procedure is followed for ys, in the case of a Gaussian, Lorentzian and Fresnellian distributions of the crystallites and a spherical mosaic crystal. Analysis of the results shows that the Zachariasen expression can be used for small extinction (y > 0.8), provided the θ dependent factor is properly introduced for particle-size-affected extinction. Allowance for polarization effects in the X-ray case is discussed. Absorption effects cannot be treated separately from extinction for all but small values of 1 -y. Coefficients of the analytical extinction expressions are given for absorbing spherical crystals with μR values ≤ 4. Application of the expressions and extension to non-spherical geometries will be treated in following publications.