Analysis of the crystal structure of alloys of the [(Ge, Sn, Pb)(Te, Se)]m[(Bi, Sb)2(Te, Se)3]n (m, n = 0, 1, 2,...) family within the theory of closely packed spheres

The processes of the formation of complex crystal structures–layered crystals of [(Ge, Sn, Pb)(Te, Se)]m[(Bi, Sb)2(Te,Se)3]n (m, n = 0, 1, 2,…) ternary alloys with different symmetries (space groups R3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline 3 $$\end{document}m, P3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline 3 $$\end{document}m1, and P21/m)—are investigated within the theory of closely packed spheres (CPS theory). The structures of layered binary alloys of the Bi2Te3 (R3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline 3 $$\end{document}m) and PbTe (Fm3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline 3 $$\end{document}m and R3m) types are found to be related and formed based on the s3 triplet with cubic symmetry (Fm3¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline 3 $$\end{document}m), which is constructed by chalcogen (Se or Te) atoms.