Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Zeitschrift für Kristallographie - Crystalline Materials

Editor-in-Chief: Pöttgen, Rainer

Ed. by Antipov, Evgeny / Boldyreva, Elena V. / Friese, Karen / Huppertz, Hubert / Jahn, Sandro / Tiekink, E. R. T.

IMPACT FACTOR 2018: 1.090
5-year IMPACT FACTOR: 2.159

CiteScore 2018: 1.47

SCImago Journal Rank (SJR) 2018: 0.892
Source Normalized Impact per Paper (SNIP) 2018: 0.722

See all formats and pricing
More options …
Volume 234, Issue 5


Coordination sequences and layer-by-layer growth of periodic structures

Anton Shutov / Andrey Maleev
Published Online: 2018-12-20 | DOI: https://doi.org/10.1515/zkri-2018-2144


A new approach to the problem of coordination sequences of periodic structures is proposed. It is based on the concept of layer-by-layer growth and on the study of geodesics in periodic graphs. We represent coordination numbers as sums of so called sector coordination numbers arising from the growth polygon of the graph. In each sector we obtain a canonical form of the geodesic chains and reduce the calculation of the sector coordination numbers to solution of the linear Diophantine equations. The approach is illustrated by the example of the 2-homogeneous kra graph. We obtain three alternative descriptions of the coordination sequences: explicit formulas, generating functions and recurrent relations.

Keywords: coordination sequences; 2-homogeneous graphs; layer-by-layer growth; periodic structures


  • [1]

    B. K. Vainshtein, V. M. Fridkin, V. L. Indenbom, Modern Crystallography 2. Structure of Crystals. Springer-Verlag, Berlin, Heidelberg, 2000.Google Scholar

  • [2]

    G. O. Brunner, F. Laves, Zum problem der koordinationszahl. Wiss. 7. Techn. Univers. Dresden 1971, 20, 387.Google Scholar

  • [3]

    W. Fischer, Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. Z. Kristallogr. 1973, 138, 129.CrossrefGoogle Scholar

  • [4]

    W. M. Meier, H. J. Moeck, The topology of three-dimensional 4-connected nets: Classification of zeolite framework types using coordination sequences. J. Solid State Chem. 1979, 27, 349.CrossrefGoogle Scholar

  • [5]

    G. O. Brunner, The properties of coordination sequences and conclusions regarding the lowest possible density of zeolites. J. Solid State Chem. 1979, 29, 41.CrossrefGoogle Scholar

  • [6]

    R. W. Grosse-Kunstleve, G. O. Brunner, N. J. A. Sloane, Algebraic description of coordination sequences and exact topological densities for zeolites. Acta Crystallogr 1996, A52, 879.Google Scholar

  • [7]

    W. M. Meier, D. H. Olson, in Atlas of Zeolite Structure Types, Butterworth-Heinemann, London, 3rd edn. 1992.Google Scholar

  • [8]

    G. O. Brunner, “Quantitative zeolite topology” can help to recognize erroneous structures and to plan syntheses. Zeolites 1993, 13, 88.CrossrefGoogle Scholar

  • [9]

    M. O’Keeffe, N-Dimensional diamond, sodalite and rare sphere packings. Acta Crystallogr. 1991, A47, 748.Google Scholar

  • [10]

    M. O’Keeffe, Dense and rare four-connected nets. Z. Kristallogr. 1991, 196, 21.CrossrefGoogle Scholar

  • [11]

    V. A. Blatov, A. P. Shevchenko, D. M. Proserpio, Applied Topological analysis of crystal structures with the program package ToposPro. Cryst. Growth Des. 2014, 14, 3576.Web of ScienceCrossrefGoogle Scholar

  • [12]

    C. P. Herrero, Coordination sequences of zeolites revisited:asymptotic behaviour for large distances. J. Chem. Soc. Faraday Trans. 1994, 90, 2597.CrossrefGoogle Scholar

  • [13]

    J.-G. Eon, Topological density of nets: a direct calculation. Acta Crystallogr 2004, A60, 7.Google Scholar

  • [14]

    M. Baake, U. Grimm, Coordination sequences for root lattices and related graphs. Z. Kristallogr. 1997, 212, 253.Google Scholar

  • [15]

    J. H. Conway, N. J. A. Sloane, Low dimensional lattices VII: coordination sequences. P. Roy. Soc. A-Math. Phy. 1997, 453, 2369.CrossrefGoogle Scholar

  • [16]

    J.-G. Eon, Algebraic determination of generating functions for coordination sequences in crystal structures. Acta Crystallogr 2002, A58, 47.Google Scholar

  • [17]

    A. V. Shutov, The number of words of a given length in the planar crystallographic groups. J. Math. Sci. 2005, 129, 3922.CrossrefGoogle Scholar

  • [18]

    C. Goodman-Strauss, N. J. A. Sloane, A coloring book approach to finding coordination sequences. Acta Crystallogr 2019, A75, (to be published).Web of ScienceGoogle Scholar

  • [19]

    V. G. Rau, V. G. Zhuravlev, T. F. Rau, A. V. Maleev, Morphogenesis of crystal structures in the discrete modeling of packings. Crystallogr. Rep. 2002, 47, 727.CrossrefGoogle Scholar

  • [20]

    V. G. Zhuravlev, Self-similar growth of periodic partitions and graphs. St. Petersburg Math. J. 2002, 13, 201.Google Scholar

  • [21]

    J.-G. Eon, From symmetry-labeled quotient graphs of crystal nets to coordination sequences. Struct. Chem. 2012, 23, 987.CrossrefWeb of ScienceGoogle Scholar

  • [22]

    J. L. Ramirez Alfonsin, The Diophantine Frobenius Problem. Oxford University Press, Oxford, 2005.Google Scholar

  • [23]

    Reticular Chemistry Structure Resource (RCSR), http://rcsr.net.

  • [24]

    The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, https://oeis.org.

  • [25]

    A. V. Maleev, V. G. Zhuravlev, A. V. Shutov, V. G. Rau, Software package for studying coordination shells in layer-by-layer growth of the connectivity graphs. Rospatent. Certificate no. 2013619399, 2013.Google Scholar

  • [26]

    S. K. Lando, Lectures on Generating Functions. American Mathematical Society, Providence, 2003.Google Scholar

About the article

Received: 2018-10-31

Accepted: 2018-11-23

Published Online: 2018-12-20

Published in Print: 2019-05-27

Citation Information: Zeitschrift für Kristallographie - Crystalline Materials, Volume 234, Issue 5, Pages 291–299, ISSN (Online) 2196-7105, ISSN (Print) 2194-4946, DOI: https://doi.org/10.1515/zkri-2018-2144.

Export Citation

©2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in