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. 3.1, pp. 370381

In the Landau theory, presented in the preceding Section 3.1.2, symmetry considerations and thermodynamics are closely interwoven. These two aspects can be, at least to some extent, disentangled and some basic symmetry conditions formulated and utilized without explicitly invoking thermodynamics. Statements which follow directly from symmetry are exact but usually do not yield numerical results. These can be obtained by a subsequent thermodynamic or statistical treatment.
The central point of this section is Table 3.1.3.1, which contains results of symmetry analysis for a large class of equitranslational phase transitions and presents data on changes of property tensors at most ferroic phase transitions. Notions and statements relevant to these two applications are explained in Sections 3.1.3.1 and 3.1.3.2, respectively. Table 3.1.3.1 with a detailed explanation is displayed in Section 3.1.3.3. Examples illustrating possible uses of the table are given in Section 3.1.3.4.
A basic role is played in symmetry considerations by the relation between the space group of the high symmetry parent or prototype phase, the space group of the lowsymmetry ferroic phase and the order parameter : The lowsymmetry group consists of all operations of the highsymmetry group that leave the order parameter invariant. By the term order parameter we mean the primary order parameter, i.e. that set of degrees of freedom whose coefficient of the quadratic invariant changes sign at the phasetransition temperature (see Sections 3.1.2.2.4 and 3.1.2.4.2).
What matters in these considerations is not the physical nature of but the transformation properties of , which are expressed by the representation of . The order parameter with components can be treated as a vector in a dimensional carrier space of the representation , and the lowsymmetry group comprises all operations of that do not change this vector. If is a real onedimensional representation, then the lowsymmetry group consists of those operations for which the matrices [or characters ] of the representation equal one, . This condition is satisfied by one half of all operations of (index of in is two) and thus the real onedimensional representation determines the ferroic group unambiguously.
A real multidimensional representation can induce several lowsymmetry groups. A general vector of the carrier space of is invariant under all operations of a group , called the kernel of representation , which is a normal subgroup of comprising all operations for which the matrix is the unit matrix. Besides that, special vectors of – specified by relations restricting values of orderparameter components (e.g. some components of equal zero, some components are equal etc.) – may be invariant under larger groups than the kernel . These groups are called epikernels of (Ascher & Kobayashi, 1977). The kernel and epikernels of represent potential symmetries of the ferroic phases associated with the representation . Thermodynamic considerations can decide which of these phases is stable at a given temperature and external fields.
Another fundamental result of the Landau theory is that components of the order parameter of all continuous (secondorder) and some discontinuous (firstorder) phase transitions transform according to an irreducible representation of the space group of the highsymmetry phase (see Sections 3.1.2.4.2 and 3.1.2.3). Since the components of the order parameter are real numbers, this condition requires irreducibility over the field of real numbers (socalled physical irreducibility or Rirreducibility). This means that the matrices of Rirreducible representations (abbreviated Rireps) can contain only real numbers. (Physically irreducible matrix representations are denoted by instead of the symbol used in general considerations.)
As explained in Section 1.2.3 and illustrated by the example of gadolinium molybdate in Section 3.1.2.5, an irreducible representation of a space group is specified by a vector of the first Brillouin zone, and by an irreducible representation of the little group of k, denoted . It turns out that the vector k determines the change of the translational symmetry at the phase transition (see e.g Tolédano & Tolédano, 1987; Izyumov & Syromiatnikov, 1990; Tolédano & Dmitriev, 1996). Thus, unless one restricts the choice of the vector , one would have an infinite number of phase transitions with different changes of the translational symmetry.
In this section, we restrict ourselves to representations with zero vector (this situation is conveniently denoted as the point). Then there is no change of translational symmetry at the transition. In this case, the group is called an equitranslational or translationengleiche (t) subgroup of , and this change of symmetry will be called an equitranslational symmetry descent . An equitranslational phase transition is a transition with an equitranslational symmetry descent .
Any ferroic spacegroupsymmetry descent uniquely defines the corresponding symmetry descent , where G and F are the point groups of the space groups and , respectively. Conversely, the equitranslational subgroup of a given space group is uniquely determined by the pointgroup symmetry descent , where G and F are point groups of space groups and , respectively. In other words, a pointgroup symmetry descent defines the set of all equitranslational spacegroup symmetry descents , where runs through all space groups with the point group G. All equitranslational spacegroup symmetry descents are available in the software GIKoBo1, where more details about the equitranslational subgroups can also be found.
Irreducible and reducible representations of the parent point group G are related in a similar way to irreducible representations with vector for all space groups with the point group G by a simple process called engendering (Jansen & Boon, 1967). The translation subgroup of is a normal subgroup and the point group G is isomorphic to a factor group . This means that to every element there correspond all elements of the space group with the same linear constituent g, the same nonprimitive translation and any vector of the translation group (see Section 1.2.3.1 ). If a representation of the point group G is given by matrices , then the corresponding engendered representation of a space group with vector assigns the same matrix to all elements of .
From this it further follows that a representation of a point group G describes transformation properties of the primary order parameter for all equitranslational phase transitions with pointsymmetry descent . This result is utilized in the presentation of Table 3.1.3.1.
The primary order parameter expresses the `difference' between the lowsymmetry and highsymmetry structures and can be, in a microscopic description, identified with spontaneous displacements of atoms (frozen in soft mode) or with an increase of order of molecular arrangement. To find a microscopic interpretation of order parameters, it is necessary to perform mode analysis (see e.g. Rousseau et al., 1981; Aroyo & PerezMato, 1998), which takes into account the microscopic structure of the parent phase.
Physical properties of crystals in a continuum description are described by physical property tensors (see Section 1.1.1.2 ), for short property tensors [equivalent expressions are matter tensors (Nowick, 1995; Wadhawan, 2000) or material tensors (Shuvalov, 1988)]. Property tensors are usually expressed in a Cartesian (rectangular) coordinate system [in Russian textbooks called a crystallophysical system of coordinates (Sirotin & Shaskolskaya, 1982; Shuvalov, 1988)] which is related to the crystallographic coordinate system (IT A , 2005) by convention (see IEEE Standard on Piezoelectricity, 1987; Sirotin & Shaskolskaya, 1982; Shuvalov, 1988). In what follows, Cartesian coordinates will mean coordinates in the crystallophysical system and tensor components will mean components in this coordinate system.
As explained in Section 1.1.4 , the number of independent components of property tensors depends on the pointgroup symmetry of the crystal: the higher this symmetry is, the smaller this number is. Lowering of pointgroup symmetry at ferroic phase transitions is, therefore, always accompanied by an increased number of independent components of some property tensors. This effect manifests itself by the appearance of morphic (Strukov & Levanyuk, 1998) or spontaneous tensor components, which are zero in the parent phase and nonzero in the ferroic phase, and/or by symmetrybreaking increments of nonzero components in the ferroic phase that break relations between these tensor components which hold in the parent phase. Thus, for example, the strain tensor has two independent components in a tetragonal phase and four independent components in a monoclinic phase. In a tetragonaltomonoclinic phase transition there is one morphic component and one relation is broken by the symmetrybreaking increment .
Changes of property tensors at a ferroic phase transition can be described in an alternative manner in which no symmetrybreaking increments but only morphic terms appear. As we have seen, the transformation properties of the primary order parameter are described by a dimensional Rirreducible matrix representation of the group G. One can form linear combinations of Cartesian tensor components that transform according to the same representation . These linear combinations will be called components of a principal tensor parameter of the ferroic phase transition with a symmetry descent . Equivalent designations are covariant tensor components (Kopský, 1979a) or symmetry coordinates (Nowick, 1995) of representation of group G. Unlike the primary order parameter of a ferroic phase transition, a principal tensor parameter is not uniquely defined since one can always form further principal tensor parameters from Cartesian components of higherrank tensors. However, only the principal tensor parameters formed from components of one, or even several, property tensors up to rank four are physically significant.
A principal tensor parameter introduced in this way has the same basic properties as the primary order parameter: it is zero in the parent phase and nonzero in the ferroic phase, and transforms according to the same Rirep . However, these two quantities have different physical nature: the primary order parameter of an equitranslational phase transition is a homogeneous microscopic distortion of the parent phase, whereas the principal tensor parameter describes the macroscopic manifestation of this microscopic distortion. Equitranslational phase transitions thus possess the unique property that their primary order parameter can be represented by principal tensor parameters which can be identified and measured by macroscopic techniques.
If the primary order parameter transforms as a vector, the corresponding principal tensor parameter is a dielectric polarization (spontaneous polarization) and the equitranslational phase transition is called a proper ferroelectric phase transition. Similarly, if the primary order parameter transforms as components of a symmetric secondrank tensor, the corresponding principal tensor parameter is a spontaneous strain (or spontaneous deformation) and the equitranslational phase transition is called a proper ferroelastic phase transition.
A conspicuous feature of equitranslational phase transitions is a steep anomaly (theoretically an infinite singularity for continuous transitions) of the generalized susceptibility associated with the primary order parameter, especially the dielectric susceptibility near a proper ferroelectric transition (see Section 3.1.2.2.5) and the elastic compliance near a proper ferroelastic transition (see e.g. Tolédano & Tolédano, 1987; Tolédano & Dmitriev, 1996; Strukov & Levanyuk, 1998).
Any symmetry property of a ferroic phase transition has its pendant in domain structure. Thus it appears that any two ferroic single domain states differ in the values of the principal tensor parameters, i.e. principal tensor parameters ensure tensor distinction of any two ferroic domain states. If, in particular, the principal order parameter is polarization, then any two ferroic domain states differ in the direction of spontaneous polarization. Such a ferroic phase is called a full ferroelectric phase (Aizu, 1970). In this case, the number of ferroic domain states equals the number of ferroelectric domain states. Similarly, if any two ferroic domain states exhibit different spontaneous strain, then the ferroic phase is a full ferroelastic phase. An equivalent condition is an equal number of ferroic and ferroelastic domain states (see Sections 3.4.2.1 and 3.4.2.2 ).
The principal tensor parameters do not cover all changes of property tensors at the phase transition. Let be a dimensional matrix Rirep of G with an epikernel (or kernel) L which is an intermediate group between F and G, in other words, L is a supergroup of F and a subgroup of G, This means that a vector of the dimensional carrier space of is invariant under operations of L. The vector specifies a secondary order parameter of the transition, i.e. is a morphic quantity, the appearance of which lowers the symmetry from G to L (for more details on secondary order parameters see Tolédano & Tolédano, 1987; Tolédano & Dmitriev, 1996). Intermediate groups (3.1.3.1) can be conveniently traced in lattices of subgroups displayed in Figs. 3.1.3.1 and 3.1.3.2.

Lattice of subgroups of the group . Conjugate subgroups are depicted as a pile of cards. In the software GIKoBo1, one can pull out individual conjugate subgroups by clicking on the pile. 

Lattice of subgroups of the group . Conjugate subgroups are depicted as a pile of cards. In the software GIKoBo1, one can pull out individual conjugate subgroups by clicking on the pile. 
One can form linear combinations of Cartesian tensor components that transform according to . These combinations are components of a secondary tensor parameter which represents a macroscopic appearance of the secondary order parameter .
If a secondary tensor parameter is a spontaneous polarization and no primary order parameter with this property exists, the phase transition is called an improper ferroelectric phase transition (Dvořák, 1974; Levanyuk & Sannikov, 1974). Similarly, an improper ferroelastic phase transition is specified by existence of a secondary tensor parameter that transforms as components of the symmetric secondrank tensor (spontaneous strain) and by absence of a primary order parameter with this property. Unlike proper ferroelectric and proper ferroelastic phase transitions, which are confined to equitranslational phase transitions, the improper ferroelectric and improper ferroelastic phase transitions appear most often in nonequitranslational phase transitions. Classic examples are an improper ferroelectric phase transition in gadolinium molybdate (see Section 3.1.2.5.2) and an improper ferroelastic phase transition in strontium titanate (see Section 3.1.5.2.3). Examples of equitranslational improper ferroelectric and ferroelastic symmetry descents can be found in Table 3.1.3.2.
Secondary tensor parameters and corresponding susceptibilities exhibit less pronounced changes near the transition than those associated with the primary order parameter (see e.g. Tolédano & Tolédano, 1987; Tolédano & Dmitriev, 1996; Strukov & Levanyuk, 1998).
In tensor distinction of domains, the secondary tensor parameters play a secondary role in a sense that some but not all ferroic domain states exhibit different values of the secondary tensor parameters. This property forms a basis for the concept of partial ferroic phases (Aizu, 1970): A ferroic phase is a partial ferroelectric (ferroelastic) one if some but not all domain states differ in spontaneous polarization (spontaneous strain). A nonferroelectric phase denotes a ferroic phase which is either nonpolar or which possesses a unique polar direction available already in the parent phase. A nonferroelastic phase exhibits no spontaneous strain.
The first systematic symmetry analysis of Landautype phase transitions was performed by Indenbom (1960), who found all equitranslational phase transitions that can be accomplished continuously. A table of all crystallographic point groups G along with all their physically irreducible representations, corresponding ferroic point groups F and related data has been compiled by Janovec et al. (1975). These data are presented in an improved form in Table 3.1.3.1 together with corresponding principal tensor parameters and numbers of ferroic, ferroelectric and ferroelastic domain states. This table facilitates solving of the following typical problems:

We note that Table 3.1.3.1 covers only those pointgroup symmetry descents that are `driven' by Rireps of G. All possible pointgroup symmetry descents are listed in Table 3.4.2.7 . Principal and secondary tensor parameters of symmetry descents associated with reducible representations are combinations of tensor parameters appearing in Table 3.1.3.1 (for a detailed explanation, see the manual of the software GIKoBo1 and Kopský, 2000). Necessary data for treating these cases are available in the software GIKoBo1 and Kopský (2001).

Example 3.1.3.4.1. Phase transition in triglycine sulfate (TGS). Assume that the space groups of both parent (highsymmetry) and ferroic (lowsymmetry) phases are known: , . The same number of formula units in the primitive unit cell in both phases suggests that the transition is an equitranslational one. This conclusion can be checked in the lattice of equitranslational subgroups of the software GIKoBo1. There we find for the lowsymmetry space group the symbol , where the vector in parentheses expresses the shift of the origin with respect to the conventional origin given in IT A (2005).
In Table 3.1.3.1, one finds that the corresponding pointgroupsymmetry descent is associated with irreducible representation . The corresponding principal tensor parameters of lowest rank are the pseudoscalar (enantiomorphism or chirality) and the vector of spontaneous polarization with one nonzero morphic component – the transition is a proper ferroelectric one. The nonferroelastic () full ferroelectric phase has two ferroelectric domain states (). Other principal tensor parameters (morphic tensor components that transform according to ) are available in the software GIKoBo1: , , , ; , , , , , , , . Property tensors with these components are listed in Table 3.1.3.3. As shown in Section 3.4.2 , all these components change sign when one passes from one domain state to the other. Since there is no intermediate group between G and F, there are no secondary tensor parameters.
Example 3.1.3.4.2. Phase transitions in barium titanate (BaTiO_{3}). We shall illustrate the solution of the inverse Landau problem and the need to correlate the crystallographic system with the Cartesian crystallophysical coordinate system. The spacegroup type of the parent phase is , and those of the three ferroic phases are , , , all with one formula unit in the primitive unit cell.
This information is not complete. To perform mode analysis, we must specify these space groups by saying that the lattice symbol P in the first case and the lattice symbol R in the third case are given with reference to the cubic crystallographic basis (), while lattice symbol C in the second case is given with reference to crystallographic basis . If we now identify vectors of the cubic crystallographic basis with vectors of the Cartesian basis by , , , where , , are three orthonormal vectors, we can see that the corresponding point groups are , , .
Notice that without specification of crystallographic bases one could interpret the point group of the space group as . Bases are therefore always specified in lattices of equitranslational subgroups of the space groups that are available in the software GIKoBo1, where we can check that all three symmetry descents are equitranslational.
In Table 3.1.3.1, we find that these three ferroic subgroups are epikernels of the Rirep with the following principal tensor components: , , , respectively. Other principal tensor parameters can be found in the main tables of the software GIKoBo1. The knowledge of the representation allows one to perform softmode analysis (see e.g. Rousseau et al., 1981).
For the tetragonal ferroelectric phase with , we find in Fig. 3.1.3.1 an intermediate group . In Table 3.1.3.1, we check that this is an epikernel of the Rirep with secondary tensor parameter . This phase is a full (proper) ferroelectric and partial ferroelastic one.
More details about symmetry aspects of structural phase transitions can be found in monographs by Izyumov & Syromiatnikov (1990), Kociński (1983, 1990), Landau & Lifshitz (1969), Lyubarskii (1960), Tolédano & Dmitriev (1996) and Tolédano & Tolédano (1987). Group–subgroup relations of space groups are treated extensively in IT A1 (2004).
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