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

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 TÀOÀT and HÀOÀH structures. We may write tetrahedrally coordinated cations and their associated anions as {T 2n Y m }. For {T 2n Y m } to be a chain or ribbon, 5n < m 4 6n, and we may write m as 5n + N, where N is an integer. Within the {T 2n Y (5n+N) } unit, we may recognize three types of anion vertices: (1) bridging anions, Y br , that are bonded to two T cations; (2) apical anions, Y ap , that are involved in linkage to other cations out of the plane of the bridging anions; and (3) linking anions, Y l , 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: {T 2n Y br a Y ap b Y l 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 the interaction 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 {T 2n Y (5n+1) } = {T 2n Y br (3nÀ1) Y ap 2n Y l 2 }. Each TÀY br ÀT linkage spans an octahedron, and hence there are (3n À 1) octahedrally coordinated cations between opposing {T 2n Y br (3nÀ1) Y ap 2n Y l 2 } ribbons. There are an additional (n À 1) vertices, C, required to complete the coordination of the M cations on one side of the O-sheet, and we may write the structuregenerating function for biopyriboles as follows:5n+1) } 2 ]. Where N = 2, the general form of the T-ribbon is {T 2n Y (5n+2) }, a component of the H-sheet in the polysomatic HÀOÀH minerals in which the T-ribbons are linked laterally by [5]-or [6]-coordinated high-valence cations, D, which have the coordination (DY l 4 F ap F t ), where F t may or may not be present depending on the coordination number, [6] or [5], of the D cation. The general formula for an H, where F t (written after the T-sheet) occurs on the outside of the H-sheet and may be involved in linkage between adjacent HÀOÀH blocks. The H-sheet links via its apical anions to the O-sheet, giving the general formula of an HÀOÀH block as [M (3n+1) (DF ap C n {T 2n Y (5n+2) }F t 0À1 ) 2 ]. These HÀOÀH blocks may link directly or indirectly through the F t anions of the (DY l 4 F a p F t ) octahedra, giving S ( 2 ; n ) =. Combining the expressions for the structuregenerating functions gives a single function for TÀOÀT and HÀOÀH structures:* 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 s...

A structure hierarchy for silicate minerals: chain, ribbon, and tube silicates

“…We may divide these structures into two types: (1) those that involve different stacking sequences in the a-direction (C2/m, Pnma and Pnmn), and (2) those that are derivatives of (1) and involve differences in coordination (P2 1 /m) and/or topochemistry (P2/a or C1) . Complete details of the amphibole structure and chemistry are given in and Oberti et al (2007), and Hawthorne (2012b) and mica supergroups) and HOH (astrophyllite, nafertisite, veblenite and mica supergroups) minerals via (algebraic) generating functions.…”

A structure hierarchy for silicate minerals: sheet silicates

“…Let it be understood that each of these questions below is addressed to the sheet-silicate minerals: Why is the 6 3 net so dominant a motif?Why is the net 3.12 2 not represented in mineral structures?All 3-connected nets have the stoichiometry [T 2 O 5 ]; what factors dictate which net is used for a specific mineral?Why do lower-valent tetrahedrally coordinated cations prefer 4-connected tetrahedra?Note that u 6 and d 6 rings of tetrahedra tend to occur in minerals with sheets of edge-sharing octahedrally coordinated (usually divalent) cations. Hawthorne (2012 b ) showed that where two parts of an atomic arrangement join, there must be a one-to-one mapping of the apical anions of the sheet of tetrahedra onto the anions of the interstitial complex. This may occur where a 6 3 sheet of tetrahedra links to an O-sheet of octahedra, accounting for the occurrence of u 6 and d 6 rings of tetrahedra in minerals with sheets of edge-sharing octahedra.…”

“…Mixed rings of tetrahedra are associated with interstitial cations of coordinations > [6]. The one-to-one mapping argument of Hawthorne (2012 b ) must hold, and it will be of considerable interest to relate the u–d character of the rings of silicate tetrahedra to the arrangements of the linking interstitial species.Why are many double-layer silicates based on parent nets that do not occur in single-layer structures?What is the relation between minerals with the same connectivity but different dimensions of polymerisation?…”

“…Note that u 6 and d 6 rings of tetrahedra tend to occur in minerals with sheets of edge-sharing octahedrally coordinated (usually divalent) cations. Hawthorne (2012 b ) showed that where two parts of an atomic arrangement join, there must be a one-to-one mapping of the apical anions of the sheet of tetrahedra onto the anions of the interstitial complex. This may occur where a 6 3 sheet of tetrahedra links to an O-sheet of octahedra, accounting for the occurrence of u 6 and d 6 rings of tetrahedra in minerals with sheets of edge-sharing octahedra.…”

“…Mixed rings of tetrahedra are associated with interstitial cations of coordinations > [6]. The one-to-one mapping argument of Hawthorne (2012 b ) must hold, and it will be of considerable interest to relate the u–d character of the rings of silicate tetrahedra to the arrangements of the linking interstitial species.…”