Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-04T20:09:05.020Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 are involved 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: {T2nΘbraΘapbΘlc} 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 {T2nΘ(5n+1)} = {T2nΘbr(3n–1)Θap2nΘl2}. Each T–Θbr–T linkage spans an octahedron, and hence there are (3n – 1) octahedrally coordinated cations between opposing {T2nΘbr(3n–1)Θap2nΘl2} 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)Θap2nΘl2}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-ribbons are linked laterally by [5]- or [6]-coordinated high-valence cations, D, which have the coordination (Dφ41φapφt), where ft may or may not be present depending on the coordination number, [6] or [5], of the D cation. The general formula for an H-sheet is [Dφap{T2nΘbr(3n–2)Θap2nΘl4}φt0–1], where φt (written after the T-sheet) occurs on the outside of the H-sheet and may 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)}φt0–1)2]. These H–O–H blocks may link directly or indirectly through the φt anions of the (DΘl4φapφt) octahedra, giving S(2;n) = Xi[M(3n+1)Ψ2n(D2φap2{T2nΘbr(3n–2)Θap2nΘl4}2)φt0–2]. 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)f2ap(N–1){T2nT(3n–N)brT2napT2N1}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
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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Belov, N.V. (1963) Crystal Chemistry of Silicates with Large Cations. Akademia Nauk SSSR, Moscow.Google Scholar
Belov, N.V. (1976) Essays on Structural Mineralogy. Nedra, Moscow, [in Russian].Google Scholar
Bragg, W.L. (1930) The structure of silicates. Zeitschrift für Kristallographie, 74, 237–305.Google Scholar
Brown, I.D. (2002) The Chemical Bond in Inorganic Chemistry. The Bond Valence Model. Oxford University Press, Oxford, UK.Google Scholar
Cámara, F., Sokolova, E., Abdu, Y. and Hawthorne, F.C. (2010) The crystal structures of niobophyllite, kupletskite-(Cs) and Sn-rich astrophyllite: revisions to the crystal chemistry of the astrophyllite-group minerals. The Canadian Mineralogist, 48, 1–16.CrossRefGoogle Scholar
Cámara, F., Sokolova, E., Hawthorne, F.C., Rowe, R., Grice, J.D. and Tait, K.T. (2011) Veblenite, IMA 2010–2050.Google Scholar
CNMNC Newsletter No. 7, February 2011, page 29; Mineralogical Magazine, 75, 27–31.Google Scholar
Chisholm, J.E. (1973) Planar defects in fibrous amphiboles. Journal of Materials Science, 8, 475–483.CrossRefGoogle Scholar
Chisholm, J.E. (1981) Pyribole structure types. Mineralogical Magazine, 44, 205–216.CrossRefGoogle Scholar
Christianssen, C.C., Johnsen, O. and Makovicky, E. (2003) Crystal chemistry of the rosenbuschite group. The Canadian Mineralogist, 41, 1203–1224.Google Scholar
Drits, V.A., Goncharov, Y.I., Aleksandrova, V.A., Khadzhi, V.E. and Dmitrik, A.L. (1974) New type of strip silicate. Soviet Physics Crystallography, 19, 737–741.Google Scholar
Egorov-Tismenko, Yu.K. and Sokolova, E.V. (1987) Comparative crystal chemistry of a group of titanium silicate analogues of mica. Pp. 96–106 in: Comparative Crystal Chemistry. Moscow State University, Moscow, [in Russian].Google Scholar
Ferraris, G. (1997) Polysomatism as a tool for correlating properties and structure. Pp. 275–295 in: Modular Aspects of Minerals (S. Merlino, editor). European Mineralogical Union Notes in Mineralogy, 1. Eötvö s University Press, Budapest. Hungary.Google Scholar
Ferraris, G. and Gula, A. (2005) Polysomatic aspects of microporous minerals - heterophyllosilicates, polysepides and rhodesite-related structures. Pp. 69–104 in: Micro- and Mesoporous Mineral Phases (G. Ferraris and S. Merlino, editors). Reviews in Mineralogy and Geochemistry, 57. Mineralogical Society of America, Washington DC and the Geochemical Society, St Louis, Missouri, USA.Google Scholar
Ferraris, G., Ivaldi, G., Khomyakov, A.P., Sobolova, S.V., Belluso, E. and Pavese, A. (1996) Nafertisite, a layer titanosilicate member of a polysomatic series including mica. European Journal of Mineralogy, 8, 241–249.CrossRefGoogle Scholar
Goldschmidt, V.M. (1926) Laws of crystal chemistry. Naturwissenschaften, 14, 477–485.CrossRefGoogle Scholar
Goldschmidt, V.M. (1927) Construction of crystals. Zeitschrift für Technische Physik, 8, 251–264.Google Scholar
Hawthorne, F.C. (1981) Crystal chemistry of the amphiboles. Pp. 1–102 in: Amphiboles and Other Hydrous Pyriboles: Mineralogy (D.R. Veblen, editor). Reviews in Mineralogy 9A. Mineralogical Society of America, Washington DC.CrossRefGoogle Scholar
Hawthorne, F.C. (1983) The crystal chemistry of the amphiboles. The Canadian Mineralogist, 21, 173–480.Google Scholar
Hawthorne, F.C. (1985) Towards a structural classification of minerals: the VIMIVT2jn minerals. American Mineralogist, 70, 455–473.Google Scholar
Hawthorne, F.C. (1990) Structural hierarchy in [6]M[4] Tjn minerals. Zeitschrift für Kristallographie, 192, 1–52.CrossRefGoogle Scholar
Hawthorne, F.C. (1994) Structural aspects of oxide and oxysalt crystals. Acta Crystallographica, B50, 481–510.CrossRefGoogle Scholar
Hawthorne, F.C. (1997) Structural aspects of oxide and oxysalt minerals. Pp. 373–429 in: Modular Aspects of Minerals (S. Merlino, editor). European Mineralogical Union Notes in Mineralogy 1. Eötvö s University Press, Budapest, Hungary.Google Scholar
Hawthorne, F.C. and Oberti, R. (2007) Amphiboles: Crystal chemistry. Pp. 1–54 in: Amphiboles: Crystal Chemistry, Occurrence and Health Issues (F.C. Hawthorne, R. Oberti, G. Della Ventura and A. Mottana, editors) Reviews in Mineralogy and Geochemistry 67. Mineralogical Society of America, Washington DC and the Geochemical Society, St Louis, Missouri, USA.CrossRefGoogle Scholar
Krivovichev, S.V., Armbruster, T., Yakovenchuk, V.N., Pakhomovsky, Ya.A. and Men’shikov, Yu.P. (2003) Crystal structures of lamprophyllite-2M and lamprophyllite- 2O from the Lovozero alkaline massif, Kola peninsula, Russia. European Journal of Mineralogy, 15, 711–718.CrossRefGoogle Scholar
Landé, A. (1920) The size of atoms. Zeitschrift für Technische Physik, 2, 87–89.CrossRefGoogle Scholar
Law, A.D. and Whittaker, E.J.W. (1980) Rotated and extended model structures in amphiboles and pyroxenes. Mineralogical Magazine, 43, 565–574.CrossRefGoogle Scholar
Lewis, G.N. (1923) Valence and the Structure of Atoms and Molecules. American Chemical Society, Monograph Series, New York.Google Scholar
Lima-de-Faria, J., Hellner, E., Liebau, F., Makovicky, E. and Parthé, E. (1990) Nomenclature of inorganic structure types. Report of the International Union of Crystallography Commission on Crystallographic Nomenclature: subcommittee on the nomenclature of inorganic structure types. Acta Crystallographica, A46, 1–11.Google Scholar
Makovicky, E. (1997) Modularity - different type and approaches. Pp. 315–343 in: Modular Aspects of Minerals (S. Merlino, editor). European Mineralogical Union Notes in Mineralogy, 1. Eötvö s University Press, Budapest, Hungary.Google Scholar
Pauling, L. (1929) The principles determining the structures of complex ionic crystals. Journal of the American Chemical Society, 51, 1010–1026.CrossRefGoogle Scholar
Pautov, L.A., Agakhanov, A.A. and Bekenova, G.K. (2006) Sokolovaite CsLi2AlSi4O10F2 - a new mineral species of the mica group. New Data on Minerals, 41, 5–13.Google Scholar
Petersen, O.V., Johnsen, O., Christiansen, C.C., Robinson, G.W. and Niedermayr, G. (1999) Nafertisite - Na3Fe10Ti2(O,OH,F)43 - from the Nanna pegmatite, Narsaarsuup Qaava, south Greenland. Neues Jahrbuch für Mineralogie, Monatshefte, 1999, 303–310.Google Scholar
Piilonen, P.C., McDonald, A.M. and Lalonde, A.E. (2003) Insights into astrophyllite-group minerals. II. Crystal chemistry. The Canadian Mineralogist, 41, 27–54.CrossRefGoogle Scholar
Pyatenko, Yu.A., Voronkov, A.A. and Pudovkina, Z.V. (1976): Mineralogical Crystal Chemistry of Titanium. Nauka, Moscow, [in Russian].Google Scholar
Shannon, R.D. (1976) Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallographica, A32, 751–767.CrossRefGoogle Scholar
Sokolova, E.V. (1998) Polysomatic series seidozeri t e-n acaphite. Zap i s k i Vs esoyuzno go Mineralogicheskogo Obshchestva, 127, 111–114.[in Russian].Google Scholar
Sokolova, E. (2006) From structure topology to chemical composition. I. Structural hierarchy and stereochemistry in titanium disilicate minerals. The Canadian Mineralogist, 44, 1273–1330.CrossRefGoogle Scholar
Sokolova, E. (2010) Predictive crystal-chemical relations in Ti-silicates based on the TS block. Geology of Ore Deposits, 52, 410–427.CrossRefGoogle Scholar
Sokolova, E. and Cámara, F. (2008) Re-investigation of the crystal structure of magnesium astrophyllite. European Journal of Mineralogy, 20, 253–260.CrossRefGoogle Scholar
Sokolova, E. and Hawthorne, F.C. (2004) The crystal chemistry of epistolite. The Canadian Mineralogist, 42, 797–806.CrossRefGoogle Scholar
Sokolova, E. and Hawthorne, F.C. (2008) From structure topology to chemical composition. V. Titanium silicates: the crystal chemistry of nacareniobsite- (Ce). The Canadian Mineralogist, 46, 1333–1342.CrossRefGoogle Scholar
Sokolova, E., Cámara, F., Hawthorne, F.C. and Abdu, Y. (2009) From structure topology to chemical composition. VII. Titanium silicates: the crystal structure and crystal chemistry of jinshajiangite. European Journal of Mineralogy, 21, 871–883.CrossRefGoogle Scholar
Thomson, J.B. Jr (1978) Biopyriboles and polysomatic series. American Mineralogist, 63, 239–249.Google Scholar
Veblen, D.R. (1991) Polysomatism and polysomatic series: a review and applications. American Mineralogist, 76, 801–826.Google Scholar
Veblen, D.R. and Burnham, C.W. (1978) New biopyriboles from Chester, Vermont: II. The crystal chemistry of jimthompsonite, clinojimthompsonite, and chesterite, and the amphibole-mica reaction. American Mineralogist, 63, 1053–1073.Google Scholar
Wadsley, A.D. (1957) Crystal chemistry of nonstoichiometric pentavalent vanadium oxides: crystal structure of Li1+xV3O8. Acta Crystallographica, 10, 239–249.CrossRefGoogle Scholar
Warren, B.E. (1929) The structure of tremolite H2Ca2Mg5(SiO3 ) 8. Neues Jahrbuch fü r Mineralogie, Monatshefte, 72, 42–57.Google Scholar
Wasastjerna, J.A. (1923) On the radii of ions. Societas Scientiarum Fennica. Commentationes Physico- Mathematicae, 38, 1–25.Google Scholar
Wilson, R. (1979) Introduction to Graph Theory. Longman, London.Google Scholar