A joint probability distribution of seven structure factors

Abstract

It is assumed that a crystal structure in P1 is fixed and that the random variables (vectors) h, k, l, m, subject to h + k + I + m = 0, are uniformly and independently distributed in reciprocal space. Then the seven structure factors Eh, Ek, El, Em, Eh + k, Ek + l, El + h, as functions of the primitive random variables h, k, 1, m, are themselves random variables, and their joint probability distribution is found. This distribution plays the central role in the theory and estimation of the cosine invariants cos (ϕh + ϕk + ϕ + ~l + ϕm).