Труды российских конференций

Any crystal structure may be represented by a weighted digraph, the vertex set of which represents atoms and the edge set of which represents chemical bonds. We may write the tetrahedrally coordinated cations and their associated anions as {T_2nΘ_m}. For {T_2nΘ_m} to be a chain or ribbon, 5n<m≤6n, and we may write m as 5n + N, where N is an integer. Within the {T_2nΘ_m) unit, we may recognize three types of anion vertices: (1) bridging anions, Θ^br, that are bonded to two T cations; (2) apical anions, Θ^ap, that are involved in linkage to other cations out of the plane of the bridging anions; and (3) linking anions, Θ^l, that link to non-T cations in the plane of the bridging anions.

Consider first the case where N = 1: we may write the general form of the tetrahedral ribbon as {T_2nΘ_5n+1} and {T_2nΘ^br_3n-1Θ^ap_2nΘ^l₂} where we specify the linking characteristics of the anions. In layered structures, the apical anions of the {T_2nΘ^br_3n-1Θ^ap_2nΘ^l₂} ribbon also constitute the anions of the layer of octahedra, one on each side of the octahedrally coordinated cations, M. Each T-Θ^br-T linkage spans an octahedron, and hence there are 3n-1 octahedrally coordinated cations between opposing {T_2nΘ^br_3n-1Θ^ap_2nΘ^l₂} ribbons. In the pyribole series, the {T_2nΘ^br_3n-1Θ^ap_2nΘ^l₂} ribbon is unbranched and contains (n-1) loops. Each loop must correspond to one additional anion (denoted φ) that is bonded to the M cations and not bonded to a T cation. We may thus write the general structure-generating function, S_n, for the pyribole frameworks as follows: S_n = [M_3n-1 φ_2(n-1) {T_2nΘ^br_3n-1Θ^ap_2nΘ^l₂}₂] = [M_3n-1φ_2(n-1){T_2nΘ_5n+1}₂]; note that the anion notations Θ^br, Θ^ap and Θ^l carry information on the structural linkage that the formula M_3n-1φ_2(n-1){T_2nΘ_5n+1}₂ does not. Interstitial sites are occupied by XI cations so as to (1) achieve electroneutrality, and (2) satisfy the local-sum rule of bond-valence theory.

This process results in the biopyribole (T-O-T) structures. For n = 1, S₁ = XI[M₂ φ₀{T₂Θ₆}₂], the pyroxenes (e.g., diopside: 2(Ca{MgSi₂O₆}). For n = 2, S₂ = XI[M₅φ₂{T₄Θ₁₁}₂], the amphiboles (e.g., tremolite: Ca₂{Mg₅Si₈O₂₂(OH)₂}. For n = 3, S₃ = XI[M₈φ₄{T₆Θ₁₆}₂], the triple-chain pyriboles (e.g., clinojimthompsonite: Mg₂{Mg₈Si₁₂O₃₂(OH)₄}. For n = 4, S₄ = XI[M₁₁φ₆{T₈Θ₂₁}₂], e.g., XI[Mg₁₁{Si₁₆O₄₂}(OH)₆]; no structure of this formula has yet been found. For n = ∞, S_∞ = XI [M_(3n-1)/nφ_2(n-1)/n{T₂Θ_(5n+1)/n}₂ = XI[M_(3-1/n)φ_2-2/n{T₂Θ_5+1/n}₂] = XI[M₃φ₂{T₂Θ₅}₂] (e.g., phlogopite, K[Mg₃{AlSi₃O₁₀}(OH)₂]).

Next, consider next the case where N = 2: the general form of the tetrahedral ribbon is {T_2nΘ_5n+2}. These correspond to the silicate moities of the H-layers in a polysomatic series of H-O-H silicates in which the ribbons are linked laterally by [5]- or [6]-coordinated cations that we will designate as D and which have the coordination (DΘ^l₄Θ^apΩ^ap), where Ω^ap may or may not be present depending on the coordination of the D cation. The handshaking lemma for weighted digraphs requires that the rank of the subset of linking anions, R^l, is the same in both the T ribbon and the linking D polyhedron: 4. The rank of the subset of apical anions, R^ap, is equal to the rank of the subset of T atoms, 2n, and the rank of the subset of bridging anions, R^br, is equal to m-R^l-2n = 5n+2-4-2n. Thus we have the general formula for an H layer: [DΘ^apΩ^ap{T_2nΘ^br_3n-2Θ^ap_2nΘ^l₄}]. The H-layer links to the O-layer via its apical anions, and hence the number of anions contributed by an H-layer to the O-sheet is the number of apical anions from the tetrahedra, 2n, plus the number of apical anions from the D polyhedron: 1. There need to be an additional n anions, designated Ψ, to complete the close-packing of each anion layer in the O-sheet. Thus the total number of anions in the O-sheet is 2(2n+1+n) = 2(3n+1). The general formula of the O-sheet is [MΘ₂]_n and hence there are 2(3n+1)/2 = (3n+1) M cations in the O-sheet, and we can write the general formula of an H-O-H sheet as [M_3n+1(DΘΩΨ_n{T_2nΘ_5n+2})₂]. These H-O-H sheets can link directly through the Ω^ap anions of the (DΘ^l₄Θ^ap Ω^ap) octahedra, giving the general formula XI[M_3n+1Θ₂Ψ_2nΩ(D₂{T_2nΘ_5n+2}₂)]. For n = 1, S₁ = XI[{M₄Θ₂Ψ₂Ω(D₂{T₂Θ₇}₂)], the group-1 TS-block structures (e.g., seidozerite: Na₂[Na₂MnTiO₂F₂(Zr₂{Si₂O₇}₂)]. For n = 2, S₂ = XI[M₇Θ₂Ψ₄Ω₂(D₂{T₄Θ₁₂}₂)], the astrophyllite-group structures (e.g., K₂Na[Fe²⁺₇O₂(OH)₄F(Ti₂{Si₄O₁₂}₂)]. For n = 3, S₃ = XI[M₁₀Θ₂Ψ₆(D₂{T₆Θ₁₇}₂)]Ω, e.g., nafertisite: ideally Na₂[Fe²⁺₁₀O₂(OH)₆(Ti₂{Si₁₂O₃₄})](H₂O). For n = 4, S₄ = XI[M₁₃O₂Φ₈Ω(M₂{Si₈O₂₂}₂)], e.g., Na₂[Fe²⁺₁₃O₂(OH)₈Ω(Ti₂{Si₁₆O₄₄})]; no structure of this formula has yet been found.

In deriving the formulae in these series, the bond topologies may also be constructed. Moreover, the approach is quite general and may be used for other families of layered structures.