International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 910-911

Section 9.8.1.4.1. Bravais classes of vector modules

T. Janssen,a A. Janner,a A. Looijenga-Vosb and P. M. de Wolffc‡

a Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and cMeermanstraat 126, 2614 AM, Delft, The Netherlands

9.8.1.4.1. Bravais classes of vector modules

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For a modulated crystal structure with a one-dimensional modulation, the positions of the diffraction spots are given by vectors [{\bf H}=\textstyle\sum\limits^3_{i=1}\, h_i{\bf a}^*_i+m{\bf q}. \eqno (9.8.1.15)]This set of vectors is a vector module M*. The vectors [{\bf a}^*_1, {\bf a}^*_2,{\bf a}^*_3] form a basis of the reciprocal lattice Λ* of the basic structure and q is the modulation wavevector. The choice of the basis of Λ* has the usual freedom, the wavevector q is only determined up to a sign and up to a reciprocal-lattice vector of the basic structure.

A vector module M* has point-group symmetry K, which is the subgroup of all elements R of O(3) leaving it invariant.

In the case of an incommensurate one-dimensional modulation, M* is generated by the lattice Λ* of main reflections and the modulation wavevector q. It then follows that K is characterized by the following properties:

  • (1) It leaves Λ* invariant. (Only in this way are main reflections transformed into main reflections and satellites into satellites.)

  • (2) Any element R of K then transforms q into ±q (modulo reciprocal-lattice vectors of Λ*).

An element R of K then transforms the basic vectors [{\bf a}^*_1], [{\bf a}^*_2], [{\bf a}^*_3], q into ones of the form (9.8.1.15)[link]. If one denotes, as in (9.8.1.2)[link], q by [{\bf a}^*_4], this implies [R{\bf a}^*_i=\textstyle\sum\limits^4_{j=1}\,\Gamma^*(R)_{ji}{\bf a}^*_j,\quad i=1,\ldots,4, \eqno (9.8.1.16)]with Γ*(R) a 4 × 4 matrix with integral entries. In the case of an incommensurate modulated crystal structure, only two vectors with the same length as q are q and −q. As Λ* is left invariant, it follows that for a one-dimensionally modulated structure Γ*(R) has the form [\Gamma^*(R)= \left(\matrix{ \Gamma^*_E(R)&\Gamma^*_M(R) \cr 0&\varepsilon(R)}\right), \quad\hbox{where }\kern3pt\varepsilon(R)=\pm1. \eqno (9.8.1.17)]This matrix represents the orthogonal transformation R when referred to the basis vectors [{\bf a}^*_i] (i = 1, 2, 3, 4) of the vector module M*. As in the case of lattices, two vector modules of modulated crystals are equivalent if they have bases (i.e. a basis for the reciprocal lattice Λ* of the basic structure together with a modulation wavevector q) such that the set of matrices Γ*(K) representing their symmetry is the same for both vector modules. Equivalent vector modules form a Bravais class.

Again, as in the case of three-dimensional lattices, it is sometimes convenient to consider a vector module that includes as subset the one spanned by all diffraction spots as in (9.8.1.15)[link]. Within such a larger vector module, the actual diffraction peaks then obey centring conditions. For a vector module associated with a modulated structure, centring may involve main reflections (the basic structure then has a centred lattice), or satellites, or both. For example, if in a structure with primitive orthorhombic basic structure the modulation wavevector is given by [\alpha {\bf a}^*_1+{1\over2}{\bf a}^*_2], one may describe the diffraction spots by means of the non-primitive lattice basis [{\bf a}^*_1], [{1\over2}{\bf a}^*_2], [{\bf a}^*_3] and by the modulation wavevector [\alpha{\bf a}^*_1].

Crystallographic point groups are denoted generally by the same letter K.








































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