A General Theory of X-ray Intensity Statistics for Twins by Merohedry

A general theory of X-ray intensity statistics for twins by merohedry is developed based on mathematical analogy of this problem to a one-dimensional random walk. Probability distribution functions are derived (a) for intensities from a single twinned specimen, and (b) for the differences in intensities from two separate twinned specimens. Theoretical values of a discrepancy index between observed intensities from a twin, and calculated intensities from both correct and incorrect structural models are evaluated. A new method for determining volume fractions of crystals in a twin is also proposed, based on the distribution of differences in intensities between twin-related reflections. These results have implications for the treatment of isomorphous replacement data collected from twins by hemihedry.