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. 1.4, pp. 99100
Section 1.4.1. Definition, symmetry and representation surfaces^{a}Institut für Geowissenshaften, Universität Kiel, Olshausenstrasse 40, D24098 Kiel, Germany 
If the temperature T of a solid is raised by an amount ΔT, a deformation takes place that is described by the strain tensor : The quantities are the coefficients of thermal expansion. They have dimensions of and are usually given in units of . Since is a symmetrical polar tensor of second rank and T is a scalar, is a symmetrical polar tensor of second rank . According to the properties of the strain tensor (cf. Section 1.3.1.3.2 ), the `volume thermal expansion', β, is given by the (invariant) trace of the `linear' coefficients .
The magnitudes of thermal expansion in different directions, , can be visualized in the following ways:
The three possible graphical representations are shown in Fig. 1.4.1.1.
The maximum number of independent components of the tensor is six (in the triclinic system). With increasing symmetry, this number decreases as described in Chapter 1.1 . Accordingly, the directions and lengths of the principal axes of the representation surfaces are restricted as described in Chapter 1.3 (e.g. in hexagonal, trigonal and tetragonal crystals, the representation surfaces are rotational sheets and the rotation axis is parallel to the nfold axis). The essential results of these symmetry considerations, as deduced in Chapter 1.1 and relevant for thermal expansion, are compiled in Table 1.4.1.1.

The coefficients of thermal expansion depend on temperature. Therefore, the directions of the principal axes of the quadrics in triclinic and monoclinic crystals change with temperature (except the principal axis parallel to the twofold axis in monoclinic crystals).
The thermal expansion of a polycrystalline material can be approximately calculated if the tensor of the single crystal is known. Assuming that the grains are small and of comparable size, and that the orientations of the crystallites are randomly distributed, the following average of [(1.4.1.4)] can be calculated: If the polycrystal consists of different phases, a similar procedure can be performed if the contribution of each phase is considered with an appropriate weight.
It should be mentioned that the true situation is more complicated. The grain boundaries of anisotropic polycrystalline solids are subject to considerable stresses because the neighbouring grains have different amounts of expansion or contraction. These stresses may cause local plastic deformation and cracks may open up between or within the grains. These phenomena can lead to a hysteresis behaviour when the sample is heated up or cooled down. Of course, in polycrystals of a cubic crystal species, these problems do not occur.
If the polycrystalline sample exhibits a texture, the orientation distribution function (ODF) has to be considered in the averaging process. The resulting overall symmetry of a textured polycrystal is usually (see Section 1.1.4.7.4.2 ), showing the same tensor form as hexagonal crystals (Table 1.4.1.1), or mmm.