Concentration dependent probabilities for clusters of impurities of one kind in simple cubic, body-centered cubic, face-centered cubic, and hexagonal close-packed lattice structures are given. Polynomials which give the probabilities of singles, pairs, and different configurations of triads are tabulated for concentration values ranging from 10.0% to 0.01 %. The existence of the next-nearest-neighbor and nearest-neighbor interactions between impurities randomly distributed throughout the lattice is assumed. Results obtained by considering only nearest-neighbor interactions are graphically compared with those of the present paper, and a significant reduction in the probability of singles, pairs, and triads is claimed in the present work. The inclusion of next-nearest-neighbor interactions produces a change in the number of single, pair, or triad clusters similar to that caused by increasing the concentration. Among other conclusions, it is noted that, for impurity concentrations below 3%, changes in the cluster probabilities are influenced by the consideration of interactions with more distant neighbors, while for higher concentrations it is wise to consider, in addition, clusters of four or more.• See, for example, the general algebraic technique by P. J. Wojtowicz, Mol. Phys. 6, 157 (1963) and also

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