A new method in the probabilistic theory of the structure invariants

It is assumed that a crystal structure in P1 is fixed and that the seven non-negative numbers R₁, R₂, R₃, R₄, R₁₂, R₂₃, R₃₁ are also specified. The random variables (vectors) h, k, l, m 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, (1) |E_{h + k}| = R₁₂, |E_{k + l}| =R₂₃, |E_{l + h}| = R₃₁, (2) and h + k + l + m = 0. (3) Then the structure invariant, (φ = φ_h + φ_k + φ_l + φ_m, as a function of the primitive random variables h, k, l, m, is itself a random variable, and its conditional probability distribution, given (1) and (2), is derived and compared with the conditional distribution when only (1) is given. The distribution leads to improved estimates for cos φ in terms of the seven magnitudes (1) and (2). The results secured here suggest a generalization which is described in terms of the 'neighborhood concept'; and a 'principle of nested neighborhoods' is formulated. This terminology permits, in turn, an analogy with interpolation formulas.