Coordination formulation of tetrahedrally close packed structures: An addendum to the observations of Yarmolyuk and Kripyakevich
Ya. P. Yarmolyuk and P. I. Kripyakevich ( Kristallographiya19, 539 (1974)) showed that all tetrahedrally close packed (t.c.p.) structures have coordination formulae P pQ qR rX x → (PX 2) i(Q 2R 2X 3) j (R 3X) 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. B42, 3 (1986)).