Coordination formulation of tetrahedrally close packed structures: An addendum to the observations of Yarmolyuk and Kripyakevich

Ya. P. Yarmolyuk and P. I. Kripyakevich (Kristallographiya 19, 539 (1974)) showed that all tetrahedrally close packed (t.c.p.) structures have coordination formulae PpQqRrXx → (PX2)i(Q2R2X3)j (R3X)k, where P, Q, R, and X represent coordination numbers (CN) 16, 15, 14, and 12 polyhedra respectively: p, q, r, and x indicate the numbers of such polyhedra in the unit cells of t.c.p. structures and i, j, and k are positive integers. We propose and demonstrate a limitation to the above formulation: if i ≥ 1 and k ≥ 1, then j ≥ 1 (or if both p > 0 and r > 0, then q > 0). We give reasons for this and discuss the Aufbauprinzip of t.c.p. structures and the results of C. B. Shoemaker and D. P. Shoemaker (Acta Crystallogr. B 42, 3 (1986)).