Conditional probability distributions of the four-phase structure invariant φh+ φk+ φl+ φminP\overline{1}
Herbert A. Hauptman, E. A. Green · Acta Crystallographica Section A · 1976
A crystal structure in P{\bar 1} is assumed to be fixed and the seven non-negative numbers R1, R2, R3, R4, R12, R23, R31 are also given. It is assumed that h, k, l, m are random variables uniformly and independently distributed over the subsets of reciprocal space defined by |Eh| = R1, |Ek| = R2, |El| = R3, |Em| = R4, (1) |Eh + k| = R12, |Ek + l| = R23, |El + h| = R31, (2) and h + k + l + m = 0. (3) Then the structure invariant ϕ = ϕh + ϕk + ϕl + ϕm (4) is a function of the primitive random variables h, k, l, m. The conditional probability distribution of ϕ, given (1) and (2), is obtained and compared with the conditional probability distribution of ϕ when only (1) is given. Some calculations are presented which show the usefulness of the distribution, given (1) and (2), in estimating the value of ϕ.