The von Mises distribution of the phase of a structure invariant

Abstract

The main terms in the exponential forms of the phase probability distributions of structure invariants are von Mises distributions P(Φ = [2πI0(k)]-1 exp [k cos (Φ - ϕ)] (a). It is shown that in this formula k and ϕ are given by k exp iϕ = 2 〈F〉 |F|/σ2 (b), where F, a complex variable, is identified with a structure invariant, 〈F〉 is its mean value andσ2 is its variance. From (a) and (b) are obtained, by simple algebraic calculations of 〈F〉 and σ2, the formulae for phase probability distributions of structure invariants, the derivation of which up to now required lengthy calculations via joint probability distributions of many structure factors. A new application is the calculation of the phase probability distribution of a quartet employing both the magnitudes of the cross terms and a priori structural information.