Ab initio crystal structures and relative phase stabilities for the aleksite series, PbnBi4Te4Sn+2.

Density functional theory methods are applied to crystal structures and stabilities of phases from the aleksite homologous series, Pb_nBi₄Te₄S_n+2 (n = homologue number). The seven phases investigated correspond to n = 0 (tetradymite), 2 (aleksite-21R and -42R), 4 (saddlebackite-9H and -18H), 6 (unnamed Pb₆Bi₄Te₄S₈), 8 (unnamed Pb₈Bi₄Te₄S₁₀), 10 (hitachiite) and 12 (unnamed Pb₁₂Bi₄Te₄S₁₄). These seven phases correspond to nine single- or double-module structures, each comprising an odd number of atom layers, 5, 7, (5.9), 9, (7.11), 11, 13, 15 and 17, expressed by the formula: S(M_pX_p+1)·L(M_p+1X_p+2), where M = Pb, Bi and X = Te, S, p ≥ 2, and S and L = number of short and long modules, respectively. Relaxed structures show a and c values within 1.5% of experimental data; a and the interlayer distance d_sub decrease with increasing PbS content. Variable Pb-S bond lengths contrast with constant Pb-S bond lengths in galena. All phases are n-fold superstructures of a rhombohedral subcell with c/3 = d_sub*. Electron diffraction patterns show two brightest reflections at the centre of d_sub*, described by the modulation vector q_F = (i/N) · d_sub*, i = S + L. A second modulation vector, q = γ · c_sub*, shows a decrease in γ, from 1.8 to 1.588, across the n = 0 to n = 12 interval. The linear relationship between γ and d_sub allows the prediction of any theoretical phases beyond the studied compositional range. The upper PbS-rich limit of the series is postulated as n = 398 (Pb₃₉₈Bi₄Te₄S₄₀₀), a phase with d_sub (1.726 Å) identical to that of trigonal PbS within experimental error. The aleksite series is a prime example of mixed layer compounds built with accretional homology principles.