A joint probability distribution of seven structure factors inP\overline{1}

Abstract

For a fixed crystal structure in P{\bar 1}, it is assumed that h, k, l, m are random variables (vectors) uniformly and independently distributed over the subset of reciprocal space defined by h + k + l + m = 0. Then the seven structure factors Eh, Ek, El, Em, Eh + k, Ek + l, El + h, as functions of the primitive random variables h, k, l, 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 four-phase structure invariants ϕh + ϕk + ϕl + ϕm.