Structure Invariants, Seminvariants and Origin Definition

  • H. Krabbendam
Part of the NATO ASI Series book series (NSSB, volume 274)


In a crystal a system of basic vectors a, b, c is chosen that complies with the crystal symmetry. These basic vectors represent the translational symmetry of the crystal; this means that in the three-dimensional lattice, generated by the basic vectors, the lattice points represent positions that are translational equivalent, i.e. points that have identical surroundings in the same orientation. In order to define a crystal structure completely it is sufficient to specify the atomic positions within one translational unit, the unit cell, of the lattice. However, the lattice, and thus the origin of the unit cell, can be moved freely, in a parallel fashion, within the crystal; in order to specify the atomic coordinates unambiguously, the origin of the unit cell must be defined in a unique way. Though the position of the origin can be chosen anywhere, some positions are preferred, dependent on the symmetry of the crystal. An example is space group P1, in which one of the eight different positions of the centres of symmetry is usually chosen as the origin position (Fig. 1.1). The choice of origin is not trivial, as the eight positions are not equivalent (do not have the same surroundings).


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    C. Giacovazzo, “Direct Methods in Crystallography”, Academic Press, London, New York, Toronto, Sydney, San Francisco (1980).Google Scholar
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    D. Rogers, in “Theory and Practice of Direct Methods in Crystallography”, eds. M.F.C. Ladd & R.A. Palmer, Plenum Press, New York and London (1980).Google Scholar
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    H.A. Hauptman, “Crystal Structure Determination. The Role of the Cosine Seminvariants”, Plenum Press, New York and London (1972).CrossRefGoogle Scholar
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    J. Karle, in International Tables for X-ray Crystallography”, eds. J.A. Ibers and W.C. Hamilton, The Kynoch Press, Birmingham (1974).Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • H. Krabbendam
    • 1
  1. 1.Laboratorium voor Kristal- en StructuurchemieBijvoet Centrum voor Biomoleculair OnderzoekRijksuniversiteit UtrechtThe Netherlands

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