Discontinuous modulation functions and their application for analysis of modulated structures with the computing system JANA2006

Václav Petříček 1 , Václav Eigner 1 , Michal Dušek 1  and Antonín Čejchan 1
  • 1 Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, 182 21 Praha 8, Czech Republic
Václav Petříček, Václav Eigner, Michal Dušek and Antonín Čejchan


Discontinuous modulation functions called crenel and saw-tooth have been developed for description and refinement of strongly modulated crystal structures with abrupt changes of modulation parameters. Although used for refinement of many modulated structures and mentioned in books on aperiodic crystals, technical details of such refinements have never been published and remained hidden in the source code of the refinement program Jana2006. In this article we explain how to recognize discontinuous modulations in a Fourier map and how to refine structures where crenel or saw-tooth functions are combined with additional secondary modulation. Three sets of functions suitable for such combination are presented: the traditional ortho-harmonics, and newly developed sets of Legendre polynomials and x-harmonics. Tiny differences between refinements based on particular function sets are demonstrated using simulated as well as existing modulated structures.

    • Supplementary material
  • [1]

    S. van Smaalen, Incommensurate Crystallography, Oxford University Press, Oxford, 2007.

  • [2]

    T. Janssen, G. Chapuis, M. de Boissieu, Aperiodic Crystals, Oxford University Press, Oxford, 2007.

  • [3]

    P. M. de Wollf, T. Janssen, A. Janner, The superspaee groups for incommensurate crystal structures with a one-dimensional modulation. Acta Crystallogr. 1981, A37, 625.

  • [4]

    V. Petříček, P. Becker, P. Coppens, Structure analysis of displacively modulated molecular crystals. Acta Crystallogr. 1985, A41, 478.

  • [5]

    V. Petříček, Y. Gao, P. Lee, P. Coppens, The incommensurate modulation in the Bi2Sr2-xCaxCuO6 superconductor and its relation to the modulation in Bi2Sr2–xCaxCu2O8. Phys. Rev. B 1990, 42, 387.

  • [6]

    V. Petříček, A. van der Lee, M. Evain, On the use of crenel functions for occupationally modulated structures. Acta Crystallogr. 1995, A51, 529.

  • [7]

    M. Evain, F. Boucher, O. Gourdon, V. Petříček, M. Dušek, P. Bezdíčka, Incommensurate versus commensurate description of the AxBX3 hexagonal perovskite-type structure. Chem. Matter. 1998, 14, 3068.

  • [8]

    J. M. Peréz-Mato, M. Zakhour-Nakhl, F. Weill, J. Darriet, Structure of composites A1+x(A′xB1–x)O3 related to the 2H hexagonal perovskite: relation between composition and modulation. J. Mater. Chem. 1999, 9, 2795.

    • Crossref
    • Export Citation
  • [9]

    H. Leligny, D. Grebille, O. Pérez, A. C. Masset, Herviue, B. Raveau, A five-dimensional structural investigation of the misfit layer compound [Bi0.87SrO2]2[CoO2]1.82. Acta Crystallogr. 2000, B56, 173.

  • [10]

    P. M. De Wolff, The pseudo-symmetry of modulated crystal structures. Acta Crystallogr. 1974, A30, 777.

  • [11]

    L. Palatinus, G. Chapuis, SUPERFLIP – a computer program for the solution of crystal structures by charge flipping in arbitrary dimensions. J. Appl. Cryst. 2007, 41, 786.

  • [12]

    J. M. Peréz-Mato, G. Madariaga, M. J. Tello, Diffraction symmetry of incommensurate structures. J. Phys. C 1986, 19, 2613.

    • Crossref
    • Export Citation
  • [13]

    V. Petříček, P. Coppens, P. Becker, Structure analysis of displacively modulated molecular crystals. Acta Crystallogr. 1985, A41, 478.

  • [14]

    A. Yamamoto, Structure factor of modulated crystal structures. Acta Crystallogr. 1982, A38, 87.

  • [15]

    M. Dušek, V. Petříček, L. Palatinus, Advances in solution of modulated structures reflected by Jana system. J. Phys. C 2010, 226, 012014.

  • [16]

    M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, Dover publications, inc., New York, 1972.

  • [17]

    V. Petříček, M. Dušek, J. Černák, Modulated one-dimensional structure of [Cd(NH3)3Ni(CN)4]. Acta Cryst. B 2005, 61, 280.

    • Crossref
    • Export Citation
  • [18]

    V. Petříček, M. Dušek, L. Palatinus, Crystallographic computing system JANA2006 – general features. Z. Kristallogr. 2014, 229, 345.

  • [19]

    S. van Smaalen, L. Palatinus, Schneider: the maximum-entropy method in superspace. Acta Cryst. A 2003, 61, 459.

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Zeitschrift für Kristallographie – Crystalline Materials offers a place for researchers to present results of their crystallographic studies. The journal includes theoretical as well as experimental research. It publishes Original Papers, Letters and Review Articles in manifold areas of crystallography.