Statistics of ion distribution in 1D and 2D mixed crystals

A statistical distribution of ions M in lattices or partial lattices _∞[A_{1 - x}M_xL_m] of mixed crystals being assumed, the probability functions P_J^(V)(x) for single coordination polyhedra ML_n (j = 1), pairs (j = 2) and larger clusters of ML_n groups (j > 2) in 1D (v = 2) and v-connected 2D systems (v = 3, 4, 6) are calculated. On raising the connecting number v, increasing cluster probabilities are distinctly shifted to lower concentrations x of the foreign ion M. For a relation between the numerical values P(x) and experimental results, the mixed-crystal series (BaF)₂[Zn_{1 - x}Cu_xF₄] of a four-connected 2D structure is given as an illustrative example.