Generating functions for stoichiometry and structure of single- and double-layer sheet-silicates

摘要

Two-dimensional nets may be used to generate the stoichiometry and structure of single-layer and double-layer sheet-silicate minerals. Many sheet-silicate minerals are based on the 3-connected plane nets 6(3), 4.8(2), (4.6.8)(2)(6.8(2))(1) and (5(2).8)(1)(5.8(2))(1), and some more complicated nets, e.g. (5.6.7)(4)(5.7(2))(1)(6(2).7)(1), (4.12(2))(2) (4(2).12)(1), (5(2).8)(1)(5.6(2))(1)(5.6.8)(2)(6(2).8)(1), have one or two representative structures. Many complicated sheet-silicate minerals are based on sheets of 2-, 3- and 4-connected tetrahedra that may be developed from 3- and 4-connected plane nets by a series of oikodomeic operations on 3-or 4-connected nets that change the topology of the parent net. There are three classes of oikodomeic operations: (1) insertion of 2- and 3-connected vertices into 3- and 4-connected plane nets; (2,3) replication of single-layer sheets by topological mirror or two-fold-rotation operators, and condensation of the resulting two single-layer sheets to form double-layer sheets. The topological aspects of these sheet structures may be described by functions that express stoichiometry in terms of tetrahedron connectivities (formula-generating functions) and functions that associate these formula-generating functions with specific two-dimensional nets. Using these functions, we may generate formulae and structural arrangements of single-layer and double-layer silicate structures with specific local and long-range topological features.