International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 493495
Section 3.4.2.3. Property tensors associated with ferroic domain states^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic 
In the preceding section we derived relations for domain states without considering their specific physical properties. Basic formulae for the number of principal and secondary domain states [see equations (3.4.2.11) and (3.4.2.17), respectively] and the transformation properties of these domain states [equations (3.4.2.12) and (3.4.2.21), respectively] follow immediately from the symmetry groups G, of the parent and ferroic phases, respectively. Now we shall examine which components of property tensors specify principal and secondary domain states and how these tensor components change in different domain states.
A property tensor is specified by its components. The number of independent tensor components of a certain tensor depends on the pointgroup symmetry G of the crystal (see Chapter 1.1 ). The number of nonzero Cartesian (rectangular) components depends on the orientation of the crystal in the reference Cartesian coordinate system and is equal to, or greater than, the number of independent tensor components; this number is independent of orientation. Then there are linear relations between Cartesian tensor components. The difference is minimal for a `standard' orientation, in which symmetry axes of the crystal are, if possible, parallel to the axes of the reference coordinate system [for more on this choice, see Nye (1985) Appendix B, Sirotin & Shaskolskaya (1982), Shuvalov (1988) and IEEE Standards on Piezoelectricity (1987)]. Even in this standard orientation, only for point groups of triclinic, monoclinic and orthorhombic crystal systems is the number of nonzero Cartesian components of each property tensor equal to the number of independent tensor components, i.e. all Cartesian tensor components are independent. For all other point groups , i.e. there are always relations between some Cartesian tensor components. One can verify this statement for the strain tensor in Table 3.4.2.2.
The relations between Cartesian tensor components can be removed when one uses covariant tensor components. [Kopský (1979); see also the manual of the the software GIKoBo1 and Kopský (2001). An analogous decomposition of Cartesian tensors into irreducible parts has been performed by Jerphagnon et al. (1978).] Covariant tensor components are linear combinations of Cartesian tensor components that transform according to irreducible matrix representations of the group G of the crystal (i.e. they form a basis of irreducible representations of G; see Chapter 1.2 ). The number of covariant tensor components equals the number of independent components of the tensor .
The advantage of expressing property tensors by covariant tensor components becomes obvious when one considers a change of a property tensor at a ferroic phase transition. A symmetry descent is accompanied by the preservation of, or an increase of, the number of independent Cartesian tensor components. The latter possibility can manifest itself either by the appearance of morphic Cartesian tensor components in the lowsymmetry phase or by such changes of nonzero Cartesian components that break some relations between tensor components in the highsymmetry phase. This is seen in our illustrative example of the strain tensor u. In the highsymmetry phase with , the strain tensor has two independent components and three nonzero components: . In the lowsymmetry phase with , there are three independent and three nonzero components: , i.e. the equation does not hold in the parent phase. This change cannot be expressed by a single Cartesian morphic component.
Since there are no relations between covariant tensor components, any change of tensor components at a symmetry descent can be expressed by morphic covariant tensor components, which are zero in the parent phase and nonzero in the ferroic phase. In our example, the covariant tensor component of the spontaneous strain is , which is a morphic component since for the symmetry but for symmetry .
Tensorial covariants are defined in an exact way in the manual of the software GIKoBo1 and in Kopský (2001). Here we give only a brief account of this notion. Consider a crystal with symmetry G and a property tensor with independent tensor components. Let be a dimensional physically irreducible matrix representation of G. The covariant of consists of the following covariant tensor components: , where a = and numbers different tuples formed from components of . These covariant tensor components are linear combinations of Cartesian components of that transform as socalled typical variables of the matrix representation , i.e. the transformation properties under operations of covariant tensor components are expressed by matrices .
The relation between two presentations of the tensor is provided by conversion equations, which express Cartesian tensor components as linear combinations of covariant tensor components and vice versa [for details see the manual and Appendix E of the software GIKoBo1 and Kopský (2001)].
Tensorial covariants for all nonequivalent physically irreducible matrix representations of crystallographic point groups and all important property tensors up to rank four are listed in the software GIKoBo1 and in Kopský (2001). Thus, for example, in Table D of the software GIKoBo1, or in Kopský (2001) p. 5, one finds for the twodimensional irreducible representation E of group 422 the following tensorial covariants: , , , , , .
Let us denote by a tensorial covariant of in the first singledomain state . A crucial role in the analysis is played by the stabilizer of these covariants, i.e. all operations of the parent group G that leave invariant. There are three possible cases:
Now we shall indicate how one can find particular property tensors that fulfil conditions (3.4.2.39) or (3.4.2.40). The solution of this grouptheoretical task consists of three steps:
Phase transitions associated with reducible representations are treated in detail only in the software GIKoBo1 and in Kopský (2001). Fortunately, these phase transitions occur rarely in nature.
A rich variety of observed structural phase transitions can be found in Tomaszewski (1992). This database lists 3446 phase transitions in 2242 crystalline materials.
Example 3.4.2.4. Morphic tensor components associated with symmetry descent
The use of covariant tensor components has two practical advantages:
Firstly, the change of tensor components at a ferroic phase transition is completely described by the appearance of new nonzero covariant tensor components. If needed, Cartesian tensor components corresponding to covariant components can be calculated by means of conversion equations, which express Cartesian tensor components as linear combinations of covariant tensor components [for details on tensor covariants and conversion equations see the manual and Appendix E of the software GIKoBo1 and Kopský (2001)].
Secondly, calculation of property tensors in various domain states is substantially simplified: transformations of Cartesian tensor components, which are rather involved for higherrank tensors, are replaced by a simpler transformation of covariant tensor components by matrices of the matrix representation of , or of [see again the software GIKoBo1 and Kopský (2001)]. The determination of the tensor properties of all domain states is discussed in full in the book by Kopský (1982).
The relations between morphic properties, tensor parameters, order parameters and names of domain states are summarized in Table 3.4.2.4. Macroscopic principal domain states can be distinguished by various property tensors that transform either according to the same representation (tensors T and U) or different representations and (tensors T and S). In the microscopic description, a basic domain state may sometimes be shared by two physically different order parameters: a primary order parameter (the order parameter, components of which form a quadratic invariant with a temperaturedependent coefficient in the free energy) and a pseudoproper order parameter that transforms according to the same representation as the primary order parameter but has a temperature coefficient that is almost independent of temperature. This is, however, rather rare (see, e.g., Tolédano & Dmitriev, 1996).

References
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