The von Mises distribution of the phase of a structure invariant

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.