International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.10, p. 243
Section 1.10.1.1. Introduction
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Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands |
Many materials are known which show a well ordered state without lattice translation symmetry, often in a restricted temperature or composition range. This can be seen in the diffraction pattern from the appearance of sharp spots that cannot be labelled in the usual way with three integer indices. The widths of the peaks are comparable with those of perfect lattice periodic crystals, and this is a sign that the coherence length is comparable as well.
A typical example is K2SeO4, which has a normal lattice periodic structure above 128 K with space group Pcmn, but below this temperature shows satellites at positions , where is an irrational number, which in addition depends on temperature. These satellites cannot be labelled with integer indices with respect to the reciprocal basis , , of the structure above the transition temperature. Therefore, the corresponding structure cannot be lattice periodic.
The diffraction pattern of K2SeO4 arises because the original lattice periodic basic structure is deformed below 128 K. The atoms are displaced from their positions in the basic structure such that the displacement itself is again periodic, but with a period that is incommensurate with respect to the lattice of the basic structure.
Such a modulated structure is just a special case of a more general type of structure. These structures are characterized by the fact that the diffraction pattern has sharp Bragg peaks at positions that are linear combinations of a finite number of basic vectors: Structures that have this property are called quasiperiodic. The minimal number n of basis vectors such that all are integers is called the rank of the structure. If the rank is three and the vectors do not all fall on a line or in a plane, the structure is just lattice periodic. Lattice periodic structures form special cases of quasiperiodic structures. The collection of vectors forms the Fourier module of the structure. For rank three, this is just the reciprocal lattice of the lattice periodic structure.
The definition given above results in some important practical difficulties. In the first place, it is not possible to show experimentally that a wavevector has irrational components instead of rational ones, because an irrational number can be approximated by a rational number arbitrarily well. Very often the wavevector of the satellite changes with temperature. It has been reported that in some compounds the variation shows plateaux, but even when the change seems to be continuous and smooth one can not be sure about the irrationality. On the other hand, if the wavevector jumps from one rational position to another, the structure would always be lattice periodic, but the unit cell of this structure would vary wildly with temperature. This means that, if one wishes to describe the incommensurate phases in a unified fashion, it is more convenient to treat the wavevector as generically irrational. This experimental situation is by no means dramatic. It is similar to the way in which one can never be sure that the angles between the basis vectors of an orthorhombic lattice are really 90°, although this is a concept that no-one has problems understanding.
A second problem stems from the fact that the wavevectors of the Fourier module are dense. For example, in the case of K2SeO4 the linear combinations of and cover the c axis uniformly. To pick out a basis here could be problematic, but the intensity of the spots is usually such that choosing a basis is not a problem. In fact, one only observes peaks with an intensity above a certain threshold, and these form a discrete set. At most, the occurrence of scale symmetry may make the choice less obvious.