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

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 y_p (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 y_p . y_s (s = secondary), y_p is evaluated numerically from the expressions derived. An analytical expression for y_p is obtained by least-squares fit to the numerical values. A similar procedure is followed for y_s, 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.