Quintets: the probabilistic theory of the structure invariant ϕ_h + ϕ_k + ϕ_l + ϕ_m + ϕ_n

It is assumed that a crystal structure in P1 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 variables (vectors) h, k, l, m, n are assumed to be uniformly and independently distributed in the regions of reciprocal space 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 + n}| = 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, 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 leads to estimates for cos φ in terms of the 15 magnitudes (1) and (2).