Quintets in P: probabilistic theory of the five-phase structure invariant in the space group P

It is assumed that a crystal structure in P is fixed and that the 15 non-negative numbers R₁, R₂, R₃, R₄, R₅; R₁₂, R₁₃, R₁₄, R₁₅, R₂₃, R₂₄, R₂₅, R₃₄, R₃₅, R₄₅ are also specified. The random vector (h, k, l, m, n) is assumed to be uniformly distributed over the subset of the fivefold Cartesian product W × W × W × W × W of reciprocal space W defined by |E_h| = R₁, |E_k| = R₂, |E_l| = R₃, |E_m| = R₄, |E_n| = R₅; (1) |E_{h + k}| = R₁₂, |E_{h + l}| = R₁₃, |E_{h + m}| = R₁₄, |E_{h + n}| = R₁₅, |E_{k + l}| = R₂₃ |E_{k + m}| = R₂₄, |E_{k + n}| = R₂₅, |E_{l + m}| = R₃₄, |E_{l + m}| = R₃₅, |E_{m + n}| = R₄₅; (2) and h + k + l + m + n = 0. (3) Then the structure invariant (φ = φ_h + φ_k + φ_l + φ_m + φ_n, (4) as a function of the primitive random variables h, k, l, m, n, is itself a random variable, and its conditional probability distribution, given (1) and (2), is derived. The distribution yields reliable estimates for large numbers of quintets φ in terms of the 15 magnitudes (1) and (2).