Sextets in P1: The joint probability distribution of thirty-one structure factors

It is assumed that a crystal structure in P1 is fixed and that the random variables (vectors) h, k, l, m, n, p are uniformly and independently distributed over the subset of reciprocal space defined by h + k +l + m + n + p = 0. Then the 31 structure factors E_h, E_k, . . . ., E_p; E_{h + k}, E_{h + l}, . . ., E_{n + p}; E_{h + k + l}, E_{h + k + m}, . . ., E_{h + n + p}, as functions of the primitive random variables h, k, l, m, n, p, are themselves random variables, and their joint probability distribution is found. This distribution plays the central role in the theory and estimation of the six-phase structure invariant φ_h + φ_k + φ_l + φ_m + φ_n + φ_p