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Bond topology and structure-generating functions: graph-theoretic prediction of chemical composition and structure in polysomatic T–O–T (biopyribole) and H–O–H structures

Published online by Cambridge University Press:  05 July 2018

F. C. Hawthorne
F. C. Hawthorne*
Affiliation:
Department of Geological Sciences, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
*
*E-mail: frank_hawthorne@umanitoba.ca
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Abstract

Aspects of the bond topology and chemical composition of a mineral may be incorporated into a general formula by writing the local topological details of each cation and anion, along with their chemical identity, as a general expression called a structure-generating function. Here,this procedure is described for polysomatic T–O–T and H–O–H structures. We may write tetrahedrally coordinated cations and their associated anions as {T2nΘm}. For {T2nΘm} to be a chain or ribbon, 5n < m ≤6n, and we may write m as 5n + N, where N is an integer. Within the {T2nΘ(5n+N)} unit, we may recognize three types of anion vertices: (1) bridging anions, Θbr, that are bonded to two T cations; (2) apical anions, Θap, that areinvolved in linkage to other cations out of the plane of the bridging anions; and (3) linking anions, Θl, that link to non-T cations in the plane of the bridging anions. We may incorporate the connectivity of the cations in our algebraic representation of the chain as follows:{T2nΘbr aΘap bΘl c} where a + b + c = 5n + N. The apical anions of the T- or H-sheets provide some anions of the layer of octahedra. We may use the handshaking di-lemma of graph theory to examine theinteraction between the two types of layers, and write a Structure-Generating Function, S(N;n), that gives both the stoichiometry and aspects of the bond topology of the structures.

Where N = 1, the T-sheet consists of ribbons of the form {T2nΘ(5n+1)}= {T2nΘbr (3n–1)Θap 2nΘl 2}. Each T–Θbr–T linkage spans an octahedron, and hence there are (3n – 1) octahedrally coordinated cations between opposing {T2nΘbr (3n–1)Θap 2nΘl 2}ribbons. There are an additional (n–1) vertices, Ψ, required to complete the coordination of the M cations on one side of the O-sheet, and we may write the structure-generating function for biopyriboles as follows: S(1;n) = Xi[M(3n–1)Ψ2(n–1){T2nΘbr (3n–1)Θap 2nΘl 2}2]= [M(3n–1)Ψ2(n–1){T2nΘ(5n+1)}2]. Where N = 2, the general form of the T-ribbon is {T2nΘ(5n+2)}, a component of the H-sheet in the polysomatic H–O–H minerals in which the T-ribbonsare linked laterally by [5]- or [6]-coordinated high-valence cations, D, which have the coordination (Dφ4 1φapφt), where ft may or may not be present depending on the coordination number, [6] or [5], of the D cation. Thegeneral formula for an H-sheet is [Dφap{T2nΘbr (3n–2)Θap 2nΘl 4}φt0–1], where φt (written after the T-sheet) occurs on the outside of the H-sheet andmay be involved in linkage between adjacent H–O–H blocks. The H-sheet links via its apical anions to the O-sheet, giving the general formula of an H–O–H block as [M(3n+1)(DφapΨn{T2nΘ(5n+2)}φt 0–1)2].These H–O–H blocks may link directly or indirectly through the φt anions of the (DΘl 4φapφt) octahedra, giving S(2;n) = Xi[M(3n+1)Ψ2n(D2φap 2{T2nΘbr (3n–2)Θap 2nΘl 4}2)φt 0–2].Combining the expressions for the structure-generating functions gives a single function for T–O–T and H–O–H structures:

S(N;n) = Xi[M(3n+2N–3)?2(n+N–2)(D2(N–1)f2 ap (N–1){T2nT(3n–N) brT2n apT2N 1}2)f0–2(N–1) t]

This expression also generates mixed-ribbon polysomatic structures. Thus S(1;2+3) gives the chemical composition and structure of the mixed-chain pyribole chesterite, and S(2;1+4) gives the chemical composition and structure of the mixed-chain H–O–H mineral,veblenite.

Keywords

bond topologystructure-generating functionbiopyriboleT–O–T structuresH–O–H structurespolysomatic seriesstructure prediction

Information

Type
Research Article
Information
Mineralogical Magazine , Volume 76 , Issue 5 , October 2012 , pp. 1053 - 1080
Copyright
Copyright © The Mineralogical Society of Great Britain and Ireland 2016

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