Tolerance Factor Approach for the Design of Quaternary Materials as Applied to the A₂Ln₄Cu_{2 n}Q_{7+ n} Homologous Series

The design of novel quaternary materials can be a chemically challenging endeavor. Here, we show that a new tolerance factor, we herein call the quaternary tolerance factor (QTF), akin to the Goldschmidt ratio, can be applied to a homologous series of quaternary materials to aid in the prediction of structural stability in new phases. Single crystals of nine new members of the A₂Ln₄Cu_2nQ_7+n homologous series were synthesized (A = K, Rb, Cs, Ln = Sm, Gd, Tb, Dy, Er, Yb, Lu, Q = S, Se) for n = 1, 2, and 3 using a reactive KCl flux method as the sole potassium source. This homologous series is notable for being the largest of all known chalcogenide homologies and thus serves as a case study for understanding quaternary structural relationships. We present a thorough structural analysis of these newly synthesized compounds and place them in the context of all previously reported members of the homologous series. We then report a generalizable tolerance factor capable of predicting structurally interrelated quaternary materials from an a priori sphere-packing approach for the design of new materials.