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. 3.4, pp. 449505
doi: 10.1107/97809553602060000645 Chapter 3.4. Domain structures^{a}Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic This chapter is devoted to the crystallographic aspects of static ferroic domain structures. The exposition is based on well defined concepts and rigorous relations that follow from the symmetry lowering at the ferroic phase transition. Necessary mathematical tools are explained in Section 3.2.3 and important points are illustrated with simple examples. Synoptic tables provide useful readytouse data accessible even without knowledge of deeper theory. Three main concepts needed in a rigorous analysis (both in a continuum and a microscopic description) of any domain structure are thoroughly discussed. (1) Domain states (orientation states or structural variants) representing inner structures of domains are classified according to their characteristic properties (ferroelastic, ferroelectric etc.) and their hierarchy (primary, secondary, principal, basic etc.). A synoptic table is given with all possible symmetry lowerings at ferroic transitions and contains the numbers of ferroic, ferroelectric and ferroelastic domain states, Aizu's classification and the representation characterizing the principal domain states. (2) Relations between domain states (twin laws) determine domain distinction, switching of domain states in external fields and properties of interfaces (domain walls) between coexisting domains. Tables give for each possible transition all independent twin laws and for each twin law the number of equal and distinct tensor components of material tensors up to rank 4 in two coexisting domains of a domain twin. (3) The basic properties of domain twins and domain walls are determined by their symmetry, which is expressed by crystallographic layer groups. The fundamental significance of this description is explained and illustrated. Synoptic tables give for each twin law the possible orientations of compatible domain walls and their symmetries. 
It was demonstrated in Section 3.1.2 that a characteristic feature of structural phase transitions connected with a lowering of crystal symmetry is an anomalous behaviour near the transition, namely unusually large values of certain physical properties that vary strongly with temperature. In this chapter, we shall deal with another fundamental feature of structural phase transitions: the formation of a nonhomogeneous, textured lowsymmetry phase called a domain structure.
When a crystal homogeneous in the parent (prototypic) phase undergoes a phase transition into a ferroic phase with lower pointgroup symmetry, then this ferroic phase is almost always formed as a nonhomogeneous structure consisting of homogeneous regions called domains and contact regions between domains called domain walls. All domains have the same or the enantiomorphous crystal structure of the ferroic phase, but this structure has in different domains a different orientation, and sometimes also a different position in space. When a domain structure is observed by a measuring instrument, different domains can exhibit different tensor properties, different diffraction patterns and can differ in other physical properties. The domain structure can be visualized optically (see Fig. 3.4.1.1) or by other experimental techniques. Powerful highresolution electron microscopy (HREM) techniques have made it possible to visualize atomic arrangements in domain structures (see Fig. 3.4.1.2). The appearance of a domain structure, detected by any reliable technique, provides the simplest unambiguous experimental proof of a structural phase transition.
Under the influence of external fields (mechanical stress, electric or magnetic fields, or combinations thereof), the domain structure can change; usually some domains grow while others decrease in size or eventually vanish. This process is called domain switching. After removing or decreasing the field a domain structure might not change considerably, i.e. the form of a domain pattern depends upon the field history: the domain structure exhibits hysteresis (see Fig. 3.4.1.3). In large enough fields, switching results in a reduction of the number of domains. Such a procedure is called detwinning. In rare cases, the crystal may consist of one domain only. Then we speak of a singledomain crystal.
There are two basic types of domain structures:
Another important kind of domain structure is a ferroelectric domain structure, in which domains differ in the direction of the spontaneous polarization. Such a domain structure is formed at ferroelectric phase transitions that are characterized by the appearance of a new polar direction in the ferroic phase. Ferroelectric domains can usually be switched by external electric fields. Two ferroelectric domains with different directions of spontaneous polarization can have different spontaneous strain [e.g. in dihydrogen phosphate (KDP) crystals, two ferroelectric domains with opposite directions of the spontaneous polarization have different spontaneous shear strain], or two ferroelectric domains with antiparallel spontaneous polarization can possess the same strain [e.g. in triglycine sulfate (TGS) crystals].
The physical properties of polydomain crystals are significantly influenced by their domain structure. The values of important material property tensor components, e.g. permittivity, piezoelectric and elastic constants, may be enhanced or diminished by the presence of a domain structure. Owing to switching and detwinning phenomena, polydomain materials exhibit hysteresis of material properties. These features have important practical implications, e.g. the production of anisotropic ceramic materials or ferroelectric memories.
The domain structure resulting from a structural phase transition belongs to a special type of twinning referred to as transformation twinning (see Section 3.3.7.2 ). Despite this, the current terminology used in domainstructure studies is different. The main terms were coined during the first investigations of ferroelectric materials, where striking similarities with the behaviour of ferromagnetic materials led researchers to introduce terms analogous to those used in studies of ferromagnetic domain structures that had been examined well at that time.
Bicrystallography (see Section 3.2.2 ) provides another possible frame for discussing domain structures. Bicrystallography and domain structure analysis have developed independently and almost simultaneously but different language has again precluded deeper confrontation. Nevertheless, there are common features in the methodology of both approaches, in particular, the principle of symmetry compensation (see Section 3.2.2 ), which plays a fundamental role in both theories.
In Chapter 3.1 , it is shown that the anomalous behaviour near phase transitions can be explained in the framework of the Landau theory. In this theory, the formation of the domain structures follows from the existence of several equivalent solutions for the order parameter. This result is a direct consequence of a symmetry reduction at a ferroic phase transition. It is this dissymmetrization which is the genuine origin of the domain structure formation and which determines the basic static features of all domain structures.
This chapter is devoted to the crystallographic aspects of static domain structures, especially to the symmetry analysis of these structures. The main aim is to explain basic concepts, derive relations that govern the formation of domain structures and provide tables with useful readytouse data on domain structures of ferroic phases. The exposition uses algebraic tools that are explained in Section 3.2.3 , but the important points are illustrated with simple examples comprehensible even without mathematical details. The synoptic tables in Sections 3.4.2 and 3.4.3 present the main results of the analysis for all possible ferroic domain structures. More detailed information on certain points can be found in the software GIKoBo1.
All these results are definite – their validity does not depend on any particular model or approximation – and form thus a firm basis for further more detailed quantitative treatments. `For the most part, the only exact statements which can be made about a solid state system are those which arise as a direct consequence of symmetry alone.' (Knox & Gold, 1967.)
The exposition starts with domain states, continues with pairs of domain states and domain distinction, and terminates with domain twins and walls. This is also the sequence of steps in domainstructure analysis, which proceeds from the simplest to more complicated objects.
In Section 3.4.2, we explain the concept of domain states (also called variants or orientational states), define different types of domain states (principal, ferroelastic, ferroelectric, basic), find simple formulae for their number, and disclose their hierarchy and relation with symmetry lowering and with order parameters of the transition. Particular results for all possible ferroic phase transitions can be found in synoptic Table 3.4.2.7, which lists all possible crystallographically nonequivalent pointgroup symmetry descents that may appear at a ferroic phase transition. For each descent, all independent twinning groups (characterizing the relation between two domain states) are given together with the number of principal, ferroelastic and ferroelectric domain states and other data needed in further analysis.
Section 3.4.3 deals with pairs of domain states and with the relationship between two domain states in a pair. This relationship, in mineralogy called a `twin law', determines the distinction between domain states, specifies switching processes between two domain states and forms a starting point for discussing domain walls and twins. We show different ways of expressing the relation between two domain states of a domain pair, derive a classification of domain pairs, find nonequivalent domain pairs and determine which tensor properties are different and which are the same in two domain states of a domain pair.
The presentation of nonequivalent domain pairs is divided into two parts. Synoptic Table 3.4.3.4 lists all representative nonequivalent nonferroelastic domain pairs, and for each pair gives the twinning groups, and the number of tensor components that are different and that are the same in two domain states. These numbers are given for all important property tensors up to rank four. We also show how these data can be used to determine switching forces between two nonferroelastic domain states.
Then we explain specific features of ferroelastic domain pairs: compatible (permissible) domain walls and disorientation of domain states in ferroelastic domain twins. A list of all nonequivalent ferroelastic domain pairs is presented in two tables. Synoptic Table 3.4.3.6 contains all nonequivalent ferroelastic domain pairs with compatible (coherent) domain walls. This table gives the orientation of compatible walls and their symmetry properties. Table 3.4.3.7 lists all nonequivalent ferroelastic domain pairs with no compatible ferroelastic domain walls.
Column K_{1j} in Table 3.4.2.7 specifies all representative nonequivalent domain pairs that can appear in each particular phase transition; in combination with Tables 3.4.3.4 and 3.4.3.6, it allows one to determine the main features of any ferroic domain structure.
Section 3.4.4 is devoted to domain twins and domain walls. We demonstrate that the symmetry of domain twins and domain walls is described by layer groups, give a classification of domain twins and walls based on their symmetry, and present possible layer groups of nonferroelastic and ferroelastic domain twins and walls. Then we discuss the properties of finitethickness domain walls. In an example, we illustrate the symmetry analysis of microscopic domain walls and present conclusions that can be drawn from this analysis about the microscopic structure of domain walls.
The exposition is given in the continuum description with crystallographic point groups and property tensors. In this approach, all possible cases are often treatable and where possible are covered in synoptic tables or – in a more detailed form – in the software GIKoBo1. Although the grouptheoretical tools are almost readily transferable to the microscopic description (using the space groups and atomic positions), the treatment of an inexhaustible variety of microscopic situations can only be illustrated by particular examples.
Our attempt to work with well defined notions calls for introducing several new, and generalizing some accepted, concepts. Also an extended notation for the symmetry operations and groups has turned out to be indispensable. Since there is no generally accepted terminology on domain structures yet, we often have to choose a term from several existing moreorless equivalent variants.
The specialized scope of this chapter does not cover several important aspects of domain structures. More information can be found in the following references. There are only two monographs on domain structures (both in Russian): Fesenko et al. (1990) and Sidorkin (2002). The main concepts of domain structures of ferroic materials are explained in the book by Wadhawan (2000) and in a review by Schranz (1995). Ferroelastic domain structures are reviewed in Boulesteix (1984) and Wadhawan (1991), and are treated in detail by Salje (1990, 1991, 2000a,b). Different aspects of ferroelectric domain structures are covered in books or reviews on ferroelectric crystals: Känzig (1957), Jona & Shirane (1962), Fatuzzo & Merz (1967), Mitsui et al. (1976), Lines & Glass (1977), Smolenskii et al. (1984), Zheludev (1988) and Strukov & Levanyuk (1998). Applications of ferroelectrics are described in the books by Xu (1991) and Uchino (2000). Principles and technical aspects of ferroelectric memories are reviewed by Scott (1998, 2000).
As for all crystalline materials, domain structures can be approached in two ways: In the microscopic description, a crystal is treated as a regular arrangement of atoms. Domains differ in tiny differences of atomic positions which can be determined only indirectly, e.g. by diffraction techniques. In what follows, we shall pay main attention to the continuum description, in which a crystal is treated as an anisotropic continuum. Then the crystal properties are described by property tensors (see Section 1.1.1 ) and the crystal symmetry is expressed by crystallographic point groups. In this approach, domains exhibit different tensor properties that enable one to visualize domains by optical or other methods.
The domain structure observed in a microscope appears to be a patchwork of homogeneous regions – domains – that have various colours and shapes (see Fig. 3.4.1.1). Indeed, the usual description considers a domain structure as a collection of domains and contact regions of domains called domain walls. Strictly speaking, by a domain one understands a connected part of the crystal, called the domain region, which is filled with a homogeneous lowsymmetry crystal structure. Domain walls can be associated with the boundaries of domain regions. The interior homogeneous bulk structure within a domain region will be called a domain state. Equivalent terms are variant or structural variant (Van Tendeloo & Amelinckx, 1974). We shall use different adjectives to specify domain states. In the microscopic description, domain states associated with the primary order parameter will be referred to as primary (microscopic, basic) domain states. In the macroscopic description, the primary domain states will be called principal domain states, which correspond to Aizu's orientation states. (An exact definition of principal domain states is given below.)
Further useful division of domain states is possible (though not generally accepted): Domain states that are specified by a constant value of the spontaneous strain are called ferroelastic domain states; similarly, ferroelectric domain states exhibit constant spontaneous polarization etc. Domain states that differ in some tensor properties are called ferroic or tensorial domain states etc. If no specification is given, the statements will apply to any of these domain states.
A domain is specified by a domain state and by domain region : . Different domains may possess the same domain state but always differ in the domain region that specifies their shape and position in space.
The term `domain' has also often been used for a domain state. Clear distinction of these two notions is essential in further considerations and is illustrated in Fig. 3.4.2.1. A ferroelectric domain structure (Fig. 3.4.2.1a) consists of six ferroelectric domains , , , but contains only two domain states , characterized by opposite directions of the spontaneous polarization depicted in Fig. 3.4.2.1(d). Neighbouring domains have different domain states but nonneighbouring domains may possess the same domain state. Thus domains with odd serial number have the domain state (spontaneous polarization `down'), whereas domains with even number have domain state (spontaneous polarization `up').

Hierarchy in domainstructure analysis. (a) Domain structure consisting of domains and domain walls and ; (b) domain twin and reversed twin (with reversed order of domain states); (c) domain pair consisting of two domain states and ; (d) domain states and . 
A great diversity of observed domain structures are connected mainly with various dimensions and shapes of domain regions, whose shapes depend sensitively on many factors (kinetics of the phase transition, local stresses, defects etc.). It is, therefore, usually very difficult to interpret in detail a particular observed domain pattern. Domain states of domains are, on the other hand, governed by simple laws, as we shall now demonstrate.
We shall consider a ferroic phase transition with a symmetry lowering from a parent (prototypic, highsymmetry) phase with symmetry described by a point group G to a ferroic phase with the pointgroup symmetry , which is a subgroup of G. We shall denote this dissymmetrization by a group–subgroup symbol (or in Section 3.1.3 ) and call it a symmetry descent or dissymmetrization. Aizu (1970a) calls these symmetry descents species and uses the letter F instead of the symbol .
As an illustrative example, we choose a phase transition with parent symmetry and ferroic symmetry (see Fig. 3.4.2.2). Strontium bismuth tantalate (SBT) crystals, for instance, exhibit a phase transition with this symmetry descent (Chen et al., 2000). Symmetry elements in the symbols of G and are supplied with subscripts specifying the orientation of the symmetry elements with respect to the reference coordinate system. The necessity of this extended notation is exemplified by the fact that the group has six subgroups with the same `nonoriented' symbol : , , , , , . Lower indices thus specify these subgroups unequivocally and the example illustrates an important rule of domainstructure analysis: All symmetry operations, groups and tensor components must be related to a common reference coordinate system and their orientation in space must be clearly specified.

Exploded view of singledomain states , , and (solid rectangles with arrows of spontaneous polarization) formed at a phase transition from a parent phase with symmetry to a ferroic phase with symmetry . The parent phase is represented by a dashed square in the centre with the symmetry elements of the parent group shown. 
The physical properties of crystals in the continuum description are expressed by property tensors. As explained in Section 1.1.4 , the crystal symmetry reduces the number of independent components of these tensors. Consequently, for each property tensor the number of independent components in the lowsymmetry ferroic phase is the same or higher than in the highsymmetry parent phase. Those tensor components or their linear combinations that are zero in the highsymmetry phase and nonzero in the lowsymmetry phase are called morphic tensor components or tensor parameters and the quantities that appear only in the lowsymmetry phase are called spontaneous quantities (see Section 3.1.3.2 ). The morphic tensor components and spontaneous quantities thus reveal the difference between the high and lowsymmetry phases. In our example, the symmetry allows a nonzero spontaneous polarization , which must be zero in the highsymmetry phase with .
We shall now demonstrate in our example that the symmetry lowering at the phase transition leads to the existence of several equivalent variants (domain states) of the lowsymmetry phase. In Fig. 3.4.2.2, the parent highsymmetry phase is represented in the middle by a dashed square that is a projection of a square prism with symmetry . A possible variant of the lowsymmetry phase can be represented by an oblong prism with a vector representing the spontaneous polarization. In Fig. 3.4.2.2, the projection of this oblong prism is drawn as a rectangle which is shifted out of the centre for better recognition. We denote by a homogeneous lowsymmetry phase with spontaneous polarization and with symmetry F_{1} = . Let us, mentally, increase the temperature to above the transition temperature and then apply to the highsymmetry phase an operation , which is a symmetry operation of this highsymmetry phase but not of the lowsymmetry phase. Then decrease the temperature to below the transition temperature. The appearance of another variant of the lowsymmetry phase with spontaneous polarization obviously has the same probability of appearing as had the variant . Thus the two variants of the lowsymmetry phase and can appear with the same probability if they are related by a symmetry operation suppressed (lost) at the transition, i.e. an operation that was a symmetry operation of the highsymmetry phase but is not a symmetry operation of the lowsymmetry phase . In the same way, the lost symmetry operations and generate from two other variants, and , with spontaneous polarizations and , respectively. Variants of the lowsymmetry phase that are related by an operation of the highsymmetry group G are called crystallographically equivalent (in G) variants. Thus we conclude that crystallographically equivalent (in G) variants of the lowsymmetry phase have the same chance of appearing.
We shall now make similar considerations for a general ferroic phase transition with a symmetry descent . By the state S of a crystal we shall understand, in the continuum description, the set of all its properties expressed by property (matter) tensors in the reference Cartesian crystallophysical coordinate system of the parent phase (see Example 3.2.3.9 in Section 3.2.3.3.1 ). A state defined in this way may change not only with temperature and external fields but also with the orientation of the crystal in space.
We denote by a state of a homogeneous ferroic phase. If we apply to a symmetry operation of the group G, then the ferroic phase in a new orientation will have the state , which may be identical with or different. Using the concept of group action (explained in detail in Section 3.2.3.3.1 ) we express this operation by a simple relation:
Let us first turn our attention to operations that do not change the state : The set of all operations of G that leave invariant form a group called a stabilizer (or isotropy group) of a state in the group G. This stabilizer, denoted by , can be expressed explicitly in the following way: where the righthand part of the equation should be read as `a set of all operations of G that do not change the state ' (see Section 3.2.3.3.2 ).
Here we have to explain the difference between the concept of a stabilizer of an object and the symmetry of that object. By the symmetry group F of an object one understands the set of all operations (isometries) that leave this object invariant. The symmetry group F of an object is considered to be an inherent property that does not depend on the orientation and position of the object in space. (The term eigensymmetry is used in Chapter 3.3 for symmetry groups defined in this way.) In this case, the symmetry elements of F are `attached' to the object.
A stabilizer describes the symmetry properties of an object in another way, in which the object and the group of isometries are decoupled. One is given a group G, the symmetry elements of which have a defined orientation in a fixed reference system. The object can have any orientation in this reference system. Those operations of G that map the object in a given orientation onto itself form the stabilizer of in the group G. In this case, the stabilizer depends on the orientation of the object in space and is expressed by an `oriented' group symbol with subscripts defining the orientation of the symmetry elements of . Only for certain `prominent' orientations will the stabilizer acquire a symmetry group of the same crystal class (crystallographic point group) as the eigensymmetry of the object.
We shall define a singledomain orientation as a prominent orientation of the crystal in which the stabilizer of its state is equal to the symmetry group which is, after removing subscripts specifying the orientation, identical with the eigensymmetry of the ferroic phase: This equation thus declares that the crystal in the state has a prominent singledomain orientation.
The concept of the stabilizer allows us to identify the `eigensymmetry' of a domain state (or an object in general) with the crystallographic class (nonoriented point group) of the stabilizer of this state in the group of all rotations O(3), .
Since we shall further deal mainly with states of the ferroic phase in singledomain orientations, we shall use the term `state' for a `state of the crystal in a singledomain orientation', unless mentioned otherwise. Then the stabilizer will often be replaced by the group , although all statements have been derived and hold for stabilizers.
The difference between symmetry groups of a crystal and stabilizers will become more obvious in the treatment of secondary domain states in Section 3.4.2.2 and in discussing disoriented ferroelastic domain states (see Section 3.4.3.6.3).
As we have seen in our illustrative example, the suppressed operations generate from the first state other states. Let be such a suppressed operation, i.e. but . Since all operations that retain are collected in , the operation must transform into another state , and we say that the state is crystallographically equivalent (in G) with the state , .
We define principal domain states as crystallographically equivalent (in G) variants of the lowsymmetry phase in singledomain orientations that can appear with the same probability in the ferroic phase. They represent possible macroscopic bulk structures of (1) ferroic singledomain crystals, (2) ferroic domains in nonferroelastic domain structures (see Section 3.4.3.5), or (3) ferroic domains in any ferroic domain structure, if all spontaneous strains are suppressed [this is the socalled parent clamping approximation (PCA), see Section 3.4.2.5]. In what follows, any statement formulated for principal domain states or for singledomain states applies to any of these three situations. Principal domain states are identical with orientation states (Aizu, 1969) or orientation variants (Van Tendeloo & Amelinckx, 1974). The adjective `principal' distinguishes these domain states from primary (microscopic, basic – see Section 3.4.2.5) domain states and secondary domain states, defined in Section 3.4.2.2, and implies that any two of these domain states differ in principal tensor parameters (these are linear combinations of morphic tensor components that transform as the primary order parameter of an equitranslational phase transition with a pointgroup symmetry descent , see Sections 3.1.3.2 and 3.4.2.3). A simple criterion for a principal domain state is that its stabilizer in G is equal to the symmetry of the ferroic phase [see equation (3.4.2.4)].
When one applies to a principal domain state all operations of the group G, one gets all principal domain states that are crystallographically equivalent with . The set of all these states is denoted and is called an Gorbit of (see also Section 3.2.3.3.3 ), In our example, the Gorbit is .
Note that any operation g from the parent group G leaves the orbit invariant since its action results only in a permutation of all principal domain states. This change does not alter the orbit, since the orbit is a set in which the sequence (order) of objects is irrelevant. Therefore, the orbit is invariant under the action of the parent group G, .
A ferroic phase transition is thus a paradigmatic example of the law of symmetry compensation (see Section 3.2.2 ): The dissymmetrization of a highsymmetry parent phase into a lowsymmetry ferroic phase produces variants of the lowsymmetry ferroic phase (singledomain states). Any two singledomain states are related by some suppressed operations of the parent symmetry that are missing in the ferroic symmetry and the set of all singledomain states (Gorbit of domain states) recovers the symmetry of the parent phase. If the domain structure contains all domain states with equal partial volumes then the average symmetry of this polydomain structure is, in the first approximation, identical to the symmetry of the parent phase.
Now we find a simple formula for the number n of principal domain states in the orbit and a recipe for an efficient generation of all principal domain states in this orbit.
The fact that all operations of the group leave invariant can be expressed in an abbreviated form in the following way [see equation (3.2.3.70 )]: We shall use this relation to derive all operations that transform into : The second part of equation (3.4.2.8) shows that all lost operations that transform into are contained in the left coset (for left cosets see Section 3.2.3.2.3 ).
It is shown in group theory that two left cosets have no operation in common. Therefore, another left coset generates another principal domain state that is different from principal domain states and . Equation (3.4.2.8) defines, therefore, a onetoone relation between principal domain states of the orbit and left cosets of [see equation (3.2.3.69 )], From this relation follow two conclusions:

This result can be illustrated in our example. Table 3.4.2.1 presents in the first column the four left cosets of the group . The corresponding principal domain states , and the values of spontaneous polarization in these principal domain states are given in the second and the third columns, respectively. It is easy to verify in Fig. 3.4.2.2 that all operations of each left coset transform the first principal domain state into one principal domain state ,

The left coset decompositions of all crystallographic point groups and their subgroup symmetry are available in the software GIKoBo1, path: Subgroups\View\Twinning Group.
Let us turn briefly to the symmetries of the principal domain states. From Fig. 3.4.2.2 we deduce that two domain states and in our illustrative example have the same symmetry, , whereas two others and have another symmetry, . We see that symmetry does not specify the principal domain state in a unique way, although a principal domain state has a unique symmetry .
It turns out that if transforms into , then the symmetry group of is conjugate by to the symmetry group of [see Section 3.2.3.3 , Proposition 3.2.3.13 and equation (3.2.3.55 )]: One can easily check that in our example each operation of the second left coset of (second row in Table 3.4.2.1) transforms into itself, whereas operations from the third and fourth left cosets yield . We shall return to this issue again at the end of Section 3.4.2.2.3.
In this section we demonstrate that any morphic (spontaneous) property appears in the lowsymmetry phase in several equivalent variants and find what determines their number and basic properties.
As we saw in Fig. 3.4.2.2, the spontaneous polarization – a principal tensor parameter of the phase transition – can appear in four different directions that define four principal domain states. Another morphic property is a spontaneous strain describing the change of unitcell shape; it is depicted in Fig. 3.4.2.2 as a transformation of a square into a rectangle. This change can be expressed by a difference between two strain components , which is a morphic tensor parameter since it is zero in the parent phase and nonzero in the ferroic phase. The quantity is a secondary order parameter of the transition (for secondary order parameters see Section 3.1.3.2 ).
From Fig. 3.4.2.2, we see that two domain states and have the same spontaneous strain, whereas and exhibit another spontaneous strain . Thus we can infer that a property `to have the same value of spontaneous strain' divides the four principal domain states , , and into two classes: and with the same spontaneous strain and and with the same spontaneous strain . Spontaneous strain appears in two `variants': and .
We can define a ferroelastic domain state as a state of the crystal with a certain value of spontaneous strain , irrespective of the value of the principal order parameter. Values and thus specify two ferroelastic domain states and , respectively. The spontaneous strain in this example is a secondary order parameter and the ferroelastic domain states can therefore be called secondary domain states.
An algebraic version of the above consideration can be deduced from Table 3.4.2.1, where to each principal domain state (given in the second column) there corresponds a left coset of (presented in the first column). Thus to the partition of principal domain states into two subsets there corresponds, according to relation (3.4.2.9), a partition of left cosets where we use the fact that the union of the first two left cosets of is equal to the group . This group is the stabilizer of the first ferroelastic domain state , . Two left cosets of correspond to two ferroelastic domain states, and , respectively. Therefore, the number of ferroelastic domain states is equal to the number of left cosets of in , i.e. to the index of in , : : , and the number of principal domain states in one ferroelastic domain state is equal to the index of in , i.e. : : .
A generalization of these considerations, performed in Section 3.2.3.3.5 (see especially Proposition 3.2.3.30 and Examples 3.2.3.10 and 3.2.3.33 ), yields the following main results.
Assume that is a secondary order parameter of a transition with symmetry descent . Then the stabilizer of this parameter is an intermediate group, Lattices of subgroups in Figs. 3.1.3.1 and 3.1.3.2 are helpful in checking this condition.
The set of n principal domain states (the orbit ) splits into subsets
Each of these subsets consists of principal domain states,The number is called a degeneracy of secondary domain states.
The product of numbers and is equal to the number n of principal domain states [see equation (3.2.3.26 )]:
Principal domain states from each subset have the same value of the secondary order parameter and any two principal domain states from different subsets have different values of . A state of the crystal with a given value of the secondary order parameter will be called a secondary domain state . Equivalent terms are degenerate or compound domain state.
In a limiting case , the parameter is identical with the principal tensor parameter and there is no degeneracy, .
Secondary domain states are in a onetoone correspondence with left cosets of in the decompositiontherefore
Principal domain states of the first secondary domain state can be determined from the first principal domain state : where is the representative of the kth left coset of of the decomposition
The partition of principal domain states according to a secondary order parameter offers a convenient labelling of principal domain states by two indices , where the first index j denotes the sequential number of the secondary domain state and the second index k gives the sequential number of the principal domain state within the jth secondary domain state [see equation (3.2.3.79 )]:where and are representatives of the decompositions (3.4.2.20) and (3.4.2.23), respectively.
The secondary order parameter can be identified with a principal order parameter of a phase transition with symmetry descent (see Section 3.4.2.3). The concept of secondary domain states enables one to define domain states that are characterized by a certain spontaneous property. We present the three most significant cases of such ferroic domain states.
The distinction ferroelastic–nonferroelastic is a basic division in domain structures. Ferroelastic transitions are ferroic transitions involving a spontaneous distortion of the crystal lattice that entails a change of shape of the crystallographic or conventional unit cell (Wadhawan, 2000). Such a transformation is accompanied by a change in the number of independent nonzero components of a symmetric secondrank tensor that describes spontaneous strain.
In discussing ferroelastic and nonferroelastic domain structures, the concepts of crystal family and holohedry of a point group are useful (IT A , 2005). Crystallographic point groups (and space groups as well) can be divided into seven crystal systems and six crystal families (see Table 3.4.2.2). A symmetry descent within a crystal family does not entail a qualitative change of the spontaneous strain – the number of independent nonzero tensor components of the strain tensor u remains unchanged.

We shall denote the crystal family of a group M by the symbol FamM. Then a simple criterion for a ferroic phase transition with symmetry descent to be a nonferroelastic phase transition is
A necessary and sufficient condition for a ferroelastic phase transition is
A ferroelastic domain state is defined as a state with a homogeneous spontaneous strain . [We drop the suffix `s' or `(s)' if the serial number of the domain state is given as the superscript . The definition of spontaneous strain is given in Section 3.4.3.6.1.] Different ferroelastic domain states differ in spontaneous strain. The symmetry of a ferroelastic domain state R_{i} is specified by the stabilizer of the spontaneous strain of the principal domain state [see (3.4.2.16)]. This stabilizer, which we shall denote by , can be expressed as an intersection of the parent group G and the holohedry of group , which we shall denote Hol (see Table 3.4.2.2):This equation indicates that the ferroelastic domain state R_{i} has a prominent singledomain orientation. Further on, the term `ferroelastic domain state' will mean a `ferroelastic domain state in singledomain orientation'.
The number of ferroelastic domain states is given byIn our example, . In Table 3.4.2.7, last column, the number of ferroelastic domain states is given for all possible ferroic phase transitions.
The number of principal domain states compatible with one ferroelastic domain state (degeneracy of ferroelastic domain states) is given byIn our example, , i.e. two nonferroelastic principal domain states are compatible with each of the two ferroelastic domain states (cf. Fig. 3.4.2.2).
The product of and is equal to the number n of all principal domain states [see equation (3.4.2.19)],The number of principal domain states in one ferroelastic domain state can be calculated for all ferroic phase transitions from the ratio of numbers n and that are given in Table 3.4.2.7.
According to Aizu (1969), we can recognize three possible cases:

A similar classification for ferroelectric domain states is given below. Both classifications are summarized in Table 3.4.2.3.

Example 3.4.2.1. Domain states in leucite. Leucite (KAlSi_{2}O_{6}) (see e.g. Hatch et al., 1990) undergoes at about 938 K a ferroelastic phase transition from cubic symmetry to tetragonal symmetry . This phase can appear in singledomain states, which we denote , , . The symmetry group of the first domain state is . This group equals the stabilizer of the spontaneous strain of since Hol( (see Table 3.4.2.2), hence this phase is a full ferroelastic one.
At about 903 K, another phase transition reduces the symmetry to . Let us suppose that this transition has taken place in a domain state with symmetry ; then the roomtemperature ferroic phase has symmetry . The phase transition is a nonferroelastic one [ ] with nonferroelastic domain states, which we denote and . Similar considerations performed with initial domain states R_{2} and R_{3} generate another two couples of principal domain states , and , , respectively. Thus the roomtemperature phase is a partially ferroelastic phase with three degenerate ferroelastic domain states, each of which can contain two principal domain states. Both ferroelastic domains and nonferroelastic domains within each ferroelastic domain have been observed [see Fig. 3.3.10.13 in Chapter 3.3 , Palmer et al. (1988) and Putnis (1992)].
Ferroelectric domain states are defined as states with a homogeneous spontaneous polarization; different ferroelectric domain states differ in the direction of the spontaneous polarization. Ferroelectric domain states are specified by the stabilizer of the spontaneous polarization in the first principal domain state [see equation (3.4.2.16)]:The stabilizer is one of ten polar groups: 1, 2, 3, 4, 6, m, , , , . Since must be a polar group too, it is simple to find the stabilizer fulfilling relation (3.4.2.31).
The number of ferroelectric domain states is given byIf the polar group does not exist, we put . The number of ferroelectric domain states is given for all ferroic phase transitions in the eighth column of Table 3.4.2.7.
The number of principal domain states compatible with one ferroelectric domain state (degeneracy of ferroelectric domain states) is given by
The product of and is equal to the number n of all principal domain states [see equation (3.4.2.19)],The degeneracy of ferroelectric domain states can be calculated for all ferroic phase transitions from the ratio of the numbers n and that are given in Table 3.4.2.7.
According to Aizu (1969, 1970a), we can again recognize three possible cases (see also Table 3.4.2.3):

The classification of full, partial and nonferroelectrics and ferroelastics is given for all Aizu's species in Aizu (1970a).
This classification for all symmetry descents is readily available from the numbers n, , in Table 3.4.2.7. One can conclude that partial ferroelectrics are rather rare.
Example 3.4.2.3. Domain structure in tetragonal perovskites. Some perovskites (e.g. barium titanate, BaTiO_{3}) undergo a phase transition from the cubic parent phase with to a tetragonal ferroelectric phase with symmetry . The stabilizer Hol . There are 3 ferroelastic domain states each compatible with 2 principal ferroelectric domain states that are related e.g. by inversion , i.e. spontaneous polarization is antiparallel in two principal domain states within one ferroelastic domain state.
A similar situation, i.e. two nonferroelastic domain states with antiparallel spontaneous polarization compatible with one ferroelastic domain state, occurs in perovskites in the trigonal ferroic phase with symmetry and in the orthorhombic ferroic phase with symmetry .
Many other examples are discussed by Newnham (1974, 1975), Newnham & Cross (1974a,b), and Newnham & Skinner (1976).
In our illustrative example (see Fig. 3.4.2.2), we have seen that two domain states and have the same symmetry group (stabilizer) . In general, the condition `to have the same stabilizer (symmetry group)' divides the set of n principal domain states into equivalence classes. As shown in Section 3.2.3.3 , the role of an intermediate group is played in this case by the normalizer of the symmetry group of the first domain state . The number of domain states with the same symmetry group is given by [see Example 3.2.3.34 in Section 3.2.3.3.5 and equation (3.2.3.95 )], The number of subgroups that are conjugate under G to can be calculated from the formula [see equation (3.2.3.96 )]The product of and is equal to the number n of ferroic domain states,
The normalizer enables one not only to determine which domain states have the symmetry but also to calculate all subgroups that are conjugate under G to (see Examples 3.2.3.22 , 3.2.3.29 and 3.2.3.34 in Section 3.2.3.3 ).
Normalizers and the number of principal domain states with the same symmetry are given in Table 3.4.2.7 for all symmetry descents . The number of subgroups conjugate to is given by .
All these results obtained for pointgroup symmetry descents can be easily generalized to microscopic domain states and spacegroup symmetry descents (see Section 3.4.2.5).
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; for irreducible representations 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 compared in Table 3.4.2.4, from which it is seen that what matters in distinguishing different domain states is the stabilizer of the spontaneous (morphic) property, where physically different parameters may possess a common stabilizer. The latter thermodynamic division, based on conditions of the stability, is finer than the former division, which is based on symmetry only. This difference manifests itself, for example, in the fact that two physically different tensor parameters, such as the principal order parameter and a `similar' order parameter , transform according to different representations and but have the same stabilizer (such symmetry descents are listed in Table 3.1.3.2 ) and possess common domain states. This `degeneracy' of domain states can be even more pronounced in the microscopic description, where the same stabilizer and therefore a common basic domain state can be shared by three 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), a pseudoproper order parameter that transforms according to the same representation as the primary order parameter but has a temperature coefficient that is not as strongly temperaturedependent as the primary order parameter, and a `similar' order parameter with a representation different from .

The considerations of this and all following sections can be applied to any phase transition with pointgroup symmetry descent . All such nonmagnetic crystallographically nonequivalent symmetry descents are listed in Table 3.4.2.7 together with some other data associated with symmetry reduction at a ferroic phase transition. These symmetry descents can also be traced in lattices of point groups, which are displayed in Figs. 3.1.3.1 and 3.1.3.2 .
The symmetry descents listed in Table 3.4.2.7 are analogous to Aizu's `species' (Aizu, 1970a), in which the symbol F stands for the symbol in our symmetry descent, and the orientation of symmetry elements of the group with respect to G is specified by letters p, s, ps, pp etc.
As we have already stated, any systematic analysis of domain structures requires an unambiguous specification of the orientation and location of symmetry elements in space. Moreover, in a continuum approach, the description of crystal properties is performed in a rectangular (Cartesian) coordinate system, which differs in hexagonal and trigonal crystals from the crystallographic coordinate system common in crystallography. Last but not least, a readytouse and userfriendly presentation calls for symbols that are explicit and concise.
To meet these requirements, we use in this chapter, in Section 3.1.3 and in the software GIKoBo1 a symbolism in which the orientations of crystallographic elements and operations are expressed by means of suffixes related to a reference Cartesian coordinate system. The relation of this reference Cartesian coordinate system – called a crystallophysical coordinate system – to the usual crystallographic coordinate system is a matter of convention. We adhere to the generally accepted rules [see Nye (1985) Appendix B, Sirotin & Shaskolskaya (1982), Shuvalov (1988), and IEEE Standards on Piezoelectricity, 1987].
We list all symbols of crystallographic symmetry operations and a comparison of these symbols with other notations in Tables 3.4.2.5 and 3.4.2.6 and in Figs. 3.4.2.3 and 3.4.2.4.



Oriented symmetry operations of the cubic group and of its subgroups. The Cartesian (rectangular) coordinate system is identical with the crystallographic and crystallophysical coordinate systems. Correlation with other notations is given in Table 3.4.2.5. 

Oriented symmetry operations of the hexagonal group and of its hexagonal and trigonal subgroups. The coordinate system corresponds to the Cartesian crystallophysical coordinate system, the axes of the crystallographic coordinate system are parallel to the twofold rotation axes and to the sixfold rotation axis . Correlation with other notations is given in Table 3.4.2.6. 
Now we can present the synoptic Table 3.4.2.7.

Example 3.4.2.5. Orthorhombic phase of perovskite crystals. The parent phase has symmetry and the symmetry of the ferroic orthorhombic phase is . In Table 3.4.2.7, we find that , i.e. the phase is fully ferroelectric. Then we can associate with each principal domain state a spontaneous polarization. In column there are four twinning groups. As explained in Section 3.4.3, these groups represent four `twin laws' that can be characterized by the angle between the spontaneous polarization in singledomain state and , . If we choose along the direction [110] ( does not specify unambiguously this direction, since !), then the angles between and , representing the `twin law' for these four twinning groups , , , , are, respectively, 60, 120, 90 and 180°.
The examination of principal domain states performed in the continuum approach can be easily generalized to a microscopic description. Let us denote the spacegroup symmetry of the parent (highsymmetry) phase by and the space group of the ferroic (lowsymmetry) phase by , which is a proper subgroup of , . Further we denote by a basic (microscopic) lowsymmetry structure described by positions of atoms in the unit cell. The stabilizer () of the basic structure in a singledomain orientation is equal to the space group of the ferroic (lowsymmetry) phase,
By applying a lost symmetry operation on , one gets a crystallographically equivalent lowsymmetry basic structure , We may recall that is a spacegroup symmetry operation consisting of a rotation (pointgroup operation) and a nonprimitive translation , (see Section 1.2.3 ). The symbol is called a Seitz spacegroup symbol (Bradley & Cracknell, 1972). The product (composition law) of two Seitz symbols is
All crystallographically equivalent lowsymmetry basic structures form a orbit and can be calculated from the first basic structure in the following way: where are the representatives of the left cosets of the decomposition of , These crystallographically equivalent lowsymmetry structures are called basic (elementary) domain states.
The number N of basic domain states is equal to the number of left cosets in the decomposition (3.4.2.45). As we shall see in next section, this number is finite [see equation (3.4.2.60)], though the groups and consist of an infinite number of operations.
In a microscopic description, a basic (elementary) domain state is described by positions of atoms in the unit cell. Basic domain states that are related by translations suppressed at the phase transition are called translational or antiphase domain states. These domain states have the same macroscopic properties. The attribute `to have the same macroscopic properties' divides all basic domain states into classes of translational domain states.
In a microscopic description, a ferroic phase transition is accompanied by a lowering of spacegroup symmetry from a parent space group , with translation subgroup and point group G, to a lowsymmetry space group , with translation subgroup and point group . There exists a unique intermediate group , called the Hermann group, which has translation subgroup and point group (see e.g. Hahn & Wondratschek, 1994; Wadhawan, 2000; Wondratschek & Aroyo, 2001): where denotes an equiclass subgroup (a descent at which only the translational subgroup is reduced but the point group is preserved) and signifies a equitranslational subgroup (only the point group descends but the translational subgroup does not change). Group is a maximal subgroup of that preserves all macroscopic properties of the basic domain state with symmetry .
At this point we have to make an important note. Any spacegroup symmetry descent requires that the lengths of the basis vectors of the translation group of the ferroic space group are commensurate with basic vectors of the translational group of the parent space group . It is usually tacitly assumed that this condition is fulfilled, although in real phase transitions this is never the case. Lattice parameters depend on temperature and are, therefore, different in parent and ferroic phases. At ferroelastic phase transitions the spontaneous strain changes the lengths of the basis vectors in different ways and at firstorder phase transitions the lattice parameters change abruptly.
To assure the validity of translational symmetry descents, we have to suppress all distortions of the crystal lattice. This condition, called the highsymmetry approximation (Zikmund, 1984) or parent clamping approximation (PCA) (Janovec et al., 1989; Wadhawan, 2000), requires that the lengths of the basis vectors of the translation group of the ferroic space group are either exactly the same as, or are integer multiples of, the basic vectors of the translational group of the parent space group . Then the relation between the primitive basis vectors of and the primitive basis vectors of can be expressed as where , , are integers.
Throughout this part, the parent clamping approximation is assumed to be fulfilled.
Now we can return to the partition of the set of basic domain states into translational subsets. Let be the set of all basic translational domain states that can be generated from by lost translations. The stabilizer (in ) of this set is the Hermann group, which plays the role of the intermediate group. The number of translational subsets and the relation between these subsets is determined by the decomposition of into left cosets of : Representatives are spacegroup operations, where is a pointgroup operation and is a nonprimitive translation (see Section 1.2.3 ).
We note that the Hermann group can be found in the software GIKoBo1 as the equitranslational subgroup of with the pointgroup descent for any space group and any point group of the ferroic phase.
The decomposition of the point group G into left cosets of the point group is given by equation (3.4.2.10): Since the space groups and have identical point groups, , the decomposition (3.4.2.51) is identical with a decomposition of G into left cosets of ; one can, therefore, choose for the representatives in (3.4.2.10) the pointgroup parts of the representatives in decomposition (3.4.2.51). Both decompositions comprise the same number of left cosets, i.e. corresponding indices are equal; therefore, the number of subsets, comprising only translational basic domain states, is equal to the number n of principal domain states: where and are the number of operations of G and , respectively.
The first `representative' basic domain state of each subset can be obtained from the first basic domain state : where are representatives of left cosets of in the decomposition (3.4.2.51).
Now we determine basic domain states belonging to the first subset (first principal domain state). Equiclass groups and have the same pointgroup operations and differ only in translations. The decomposition of into left cosets of can therefore be written in the formwhere e is the identity pointgroup operation and , , are lost translations that can be identified with the representatives in the decomposition of into left cosets of : The number of basic domain states belonging to one principal domain state will be called a translational degeneracy. For the translations , one can choose vectors that lead from the origin of a `superlattice' primitive unit cell of to lattice points of located within or on the side faces of this `superlattice' primitive unit cell. The number of such lattice points is equal to the ratio , where and are the volumes of the primitive unit cells of the lowsymmetry and parent phases, respectively.
The number can be also expressed as the determinant det of the matrix of the coefficients that in equation (3.4.2.49) relate the primitive basis vectors of to the primitive basis vectors of (Van Tendeloo & Amelinckx, 1974; see also Example 2.5 in Section 3.2.3.3 ). Finally, the number equals the ratio , where and are the numbers of chemical formula units in the primitive unit cell of the ferroic and parent phases, respectively. Thus we get for the translational degeneracy three expressions: The basic domain states belonging to the first subset of translational domain states are where is a representative from the decomposition (3.4.2.55).
The partitioning we have just described provides a useful labelling of basic domain states: Any basic domain state can be given a label , where the first integer specifies the principal domain state (translational subset) and the integer designates the the domain state within a subset. With this convention the kth basic domain state in the jth subset can be obtained from the first basic domain state (see Proposition 3.2.3.30 in Section 3.2.3.3 ): In a shorthand version, the letter can be omitted and the symbol can be written in the form , where the `large' number a signifies the principal domain state and the subscript b (translational index) specifies a basic domain state compatible with the principal domain state a.
The number n of translational subsets (which can be associated with principal domain states) times the translational degeneracy (number of translational domain states within one translational subset) is equal to the total number N of all basic domain states:
Example 3.4.2.6. Basic domain states in gadolinium molybdate (GMO). Gadolinium molybdate [Gd_{2}(MoO_{4})_{3}] undergoes a nonequitranslational ferroic phase transition with parent space group and with ferroic space group (see Section 3.1.2 ). From equation (3.4.2.53) we get n = , i.e. there are two subsets of translational domain states corresponding to two principal domain states. In the software GIKoBo1 one finds for the space group and the point group the corresponding equitranslational subgroup with vectors of the conventional orthorhombic unit cell (in the parent clamping approximation) , , , where is the basis of the tetragonal space group . Hence, according to equation (3.4.2.49), The determinant of the transformation matrix equals two, therefore, according to equation (3.4.2.57), each principal domain state can contain translational domain states that are related by lost translation or . In all, there are four basic domain states (for more details see Barkley & Jeitschko, 1973; Janovec, 1976; Wondratschek & Jeitschko, 1976).
Example 3.4.2.7. Basic domain states in calomel crystals. Crystals of calomel, Hg_{2}Cl_{2}, consist of almost linear Cl—Hg—Hg—Cl molecules aligned parallel to the c axis. The centres of gravity of these molecules form in the parent phase a tetragonal bodycentred parent phase with the conventional tetragonal basis a^{t}, b^{t}, c^{t} and with space group . The structure of this phase projected onto the plane is depicted in the middle of Fig. 3.4.2.5 as a solid square with four full circles and one empty circle representing the centres of gravity of the Hg_{2}Cl_{2} molecules at the levels and , respectively.

Four basic singledomain states , , , of the ferroic phase of a calomel (Hg_{2}Cl_{2}) crystal. Full and empty circles represent centres of gravity of Hg_{2}Cl_{2} molecules at the levels and , respectively, projected onto the plane. The parent tetragonal phase is depicted in the centre of the figure with a full square representing the primitive unit cell. Arrows are exaggerated spontaneous shifts of molecules in the ferroic phase. Dotted squares depict conventional unit cells of the orthorhombic basic domain states in the parent clamping approximation. If the parent clamping approximation is lifted, these unit cells would be represented by rectangles elongated parallel to the arrows. 
The ferroic phase has pointgroup symmetry , hence there are n = = 2 ferroelastic principal domain states. The conventional orthorhombic basis is (see upper left corner of Fig. 3.4.2.5). This is the same situation as in the previous example, therefore, according to equations (3.4.2.57) and (3.4.2.61), the translational degeneracy , i.e. each ferroelastic domain state can contain two basic domain states.
The structure of the ferroic phase in the parent clamping approximation is depicted in the lefthand part of Fig. 3.4.2.5 with a dotted orthorhombic conventional unit cell. The arrows represent exaggerated spontaneous shifts of the molecules. These shifts are frozenin displacements of a transverse acoustic soft mode with the k vector along the [110] direction in the first domain state , hence all molecules in the (110) plane passing through the origin O are shifted along the direction, whereas those in the neighbouring parallel planes are shifted along the antiparallel direction (the indices are related to the tetragonal coordinate system). The symmetry of is described by the space group ; this symbol is related to the conventional orthorhombic basis and the origin of this group is shifted by or with respect to the origin 0 of the group .
Three more basic domain states , and can be obtained, according to equation (3.4.2.44), from by applying representatives of the left cosets in the resolution of [see equation (3.4.2.42)], for which one can find the expression
All basic domain states and are depicted in Fig. 3.4.2.5. Domain states and , and similarly and , are related by lost translation or . Thus the four basic domain states and can be partitioned into two translational subsets and . Basic domain states forming one subset have the same value of the secondary macroscopic order parameter , which is in this case the difference of the components of a symmetric secondrank tensor , e.g. the permittivity or the spontaneous strain (which is zero in the parent clamping approximation).
This partition provides a useful labelling of basic domain states: , where the first number signifies the ferroic (orientational) domain state and the subscript (translational index) specifies the basic domain state with the same ferroic domain state.
Symmetry groups (stabilizers in ) of basic domain states can be calculated from a spacegroup version of equation (3.4.2.13): with the same conventional basis, and , where the origin of these groups is shifted by or with respect to the origin of the group .
In general, a spacegroupsymmetry descent can be performed in two steps:

Different domains observed by a single apparatus can exhibit different properties even though their crystal structures are either the same or enantiomorphic and differ only in spatial orientation. Domains are usually distinguished by their bulk properties, i.e. according to their domain states. Then the problem of domain distinction is reduced to the distinction of domain states. To solve this task, we have to describe in a convenient way the distinction of any two of all possible domain states. For this purpose, we use the concept of domain pair.
Domain pairs allow one to express the geometrical relationship between two domain states (the `twin law'), determine the distinction of two domain states and define switching fields that may induce a change of one state into the other. Domain pairs also present the first step in examining domain twins and domain walls.
In this section, we define domain pairs, ascribe to them symmetry groups and socalled twinning groups, and give a classification of domain pairs. Then we divide domain pairs into equivalence classes (Gorbits of domain pairs) – which comprise domain pairs with the same inherent properties but with different orientations and/or locations in space – and examine the relation between Gorbits and twinning groups.
A qualitative difference between the coexistence of two domain states provides a basic division into nonferroelastic and ferroelastic domain pairs. The synoptic Table 3.4.3.4 lists representatives of all Gorbits of nonferroelastic domain pairs, contains information about the distinction of nonferroelastic domain states by means of diffraction techniques and specifies whether or not important property tensors can distinguish between domain states of a nonferroelastic domain pair. These data also determine the external fields needed to switch the first domain state into the second domain state of a domain pair. Synoptic Table 3.4.3.6 contains representative ferroelastic domain pairs of Gorbits of domain pairs for which there exist compatible (permissible) domain walls and gives for each representative pair the orientation of the two compatible domain walls, the expression for the disorientation angle (obliquity) and other data. Table 3.4.3.7 lists representatives of all classes of ferroelastic domain pairs for which no compatible domain walls exist. Since Table 3.4.2.7 contains for each symmetry descent all twinning groups that specify different Gorbits of domain pairs which can appear in the ferroic phase, one can get from this table and from Tables 3.4.3.4, 3.4.3.6 and 3.4.3.7 the significant features of the domain structure of any ferroic phase.
A pair of two domain states, in short a domain pair, consists of two domain states, say and , that are considered irrespective of their possible coexistence (Janovec, 1972). Geometrically, domain pairs can be visualized as two interpenetrating structures of and . Algebraically, two domain states and can be treated in two ways: as an ordered or an unordered pair (see Section 3.2.3.1.2 ).
An ordered domain pair, denoted (), consists of the first domain state and the second domain state. Occasionally, it is convenient to consider a trivial ordered domain pair () composed of two identical domain states .
An ordered domain pair is a construct that in bicrystallography is called a dichromatic complex (see Section 3.3.3 ; Pond & Vlachavas, 1983; Sutton & Balluffi, 1995; Wadhawan, 2000).
An ordered domain pair () is defined by specifying and or by giving and a switching operation that transforms into , For a given and , the switching operation is not uniquely defined since each operation from the left coset [where is the stabilizer (symmetry group) of ] transforms into , .
An ordered domain pair with a reversed order of domain states is called a transposed domain pair and is denoted . A nontrivial ordered domain pair is different from the transposed ordered domain pair,
If is a switching operation of an ordered domain pair , then the inverse operation of is a switching operation of the transposed domain pair :
An unordered domain pair, denoted by , is defined as an unordered set consisting of two domain states and . In this case, the sequence of domains states in a domain pair is irrelevant, therefore
In what follows, we shall omit the specification `ordered' or `unordered' if it is evident from the context, or if it is not significant.
A domain pair can be transformed by an operation into another domain pair, These two domain pairs will be called crystallographically equivalent (in G) domain pairs and will be denoted .
If the transformed domain pair is a transposed domain pair , then the operation g will be called a transposing operation, We see that a transposing operation exchanges domain states and : Thus, comparing equations (3.4.3.1) and (3.4.3.7), we see that a transposing operation is a switching operation that transforms into , and, in addition, switches into . Then a product of two transposing operations is an operation that changes neither nor .
What we call in this chapter a transposing operation is usually denoted as a twin operation (see Section 3.3.5 and e.g. Holser, 1958a; Curien & Donnay, 1959; Koch, 2004). We are reserving the term `twin operation' for operations that exchange domain states of a simple domain twin in which two ferroelastic domain states coexist along a domain wall. Then, as we shall see, the transposing operations are identical with the twin operations in nonferroelastic domains (see Section 3.4.3.5) but may differ in ferroelastic domain twins, where only some transposing operations of a singledomain pair survive as twin operations of the corresponding ferroelastic twin with a nonzero disorientation angle (see Section 3.4.3.6.3).
Transposing operations are marked in this chapter by a star, (with five points), which should be distinguished from an asterisk, (with six points), used to denote operations or symmetry elements in reciprocal space. The same designation is used in the software GIKoBo1 and in the tables in Kopský (2001). A prime, ′, is often used to designate transposing (twin) operations (see Section 3.3.5 ; Curien & Le Corre, 1958; Curien & Donnay, 1959). We have reserved the prime for operations involving time inversion, as is customary in magnetism (see Chapter 1.5 ). This choice allows one to analyse domain structures in magnetic and magnetoelectric materials (see e.g. Přívratská & Janovec, 1997).
In connection with this, we invoke the notion of a twin law. Since this term is not yet common in the context of domain structures, we briefly explain its meaning.
In crystallography, a twin is characterized by a twin law defined in the following way (see Section 3.3.2 ; Koch, 2004; Cahn, 1954):

An analogous definition of a domain twin law can be formulated for domain twins by replacing the term `twin components' by `domains', say and , where , and , are, respectively, the domain state and the domain region of the domains and , respectively (see Section 3.4.2.1). The term `transposing operation' corresponds to transposing operation of domain pair as we have defined it above if two domains with domain states and coexist along a domain wall of the domain twin.
Domain twin laws can be conveniently expressed by crystallographic groups. This specification is simpler for nonferroelastic twins, where a twin law can be expressed by a dichromatic space group (see Section 3.4.3.5), whereas for ferroelastic twins with a compatible domain wall dichromatic layer groups are adequate (see Section 3.4.3.6.3).
Restriction (ii), formulated by Georges Friedel (1926) and explained in detail by Cahn (1954), expresses a necessity to exclude from considerations crystal aggregates (intergrowths) with approximate or accidental `nearly exact' crystal components resembling twins (Friedel's macles d'imagination) and thus to restrict the definition to `true twins' that fulfil condition (i) exactly and are characteristic for a given material. If we confine our considerations to domain structures that are formed from a homogeneous parent phase, this requirement is fulfilled for all aggregates consisting of two or more domains. Then the definition of a `domain twin law' is expressed only by condition (i). Condition (ii) is important for growth twins.
We should note that the definition of a twin law given above involves only domain states and does not explicitly contain specification of the contact region between twin components or neighbouring domains. The concept of domain state is, therefore, relevant for discussing the twin laws. Moreover, there is no requirement on the coexistence of interpenetrating structures in a domain pair. One can even, therefore, consider cases where no real coexistence of both structures is possible. Nevertheless, we note that the characterization of twin laws used in mineralogy often includes specification of the contact region (e.g. twin plane or diffuse region in penetrating twins).
Ordered domain pairs and , formed from domain states of our illustrative example (see Fig. 3.4.2.2), are displayed in Fig. 3.4.3.1(a) and (b), respectively, as two superposed rectangles with arrows representing spontaneous polarization. In ordered domain pairs, the first and the second domain state are distinguished by shading [the first domain state is grey (`black') and the second clear (`white')] and/or by using dashed and dotted lines for the first and second domain state, respectively.

Transposable domain pairs. Singledomain states are those from Fig. 3.4.2.2. (a) Completely transposable nonferroelastic domain pair. (b) Partially transposable ferroelastic domain pair. 
In Fig. 3.4.3.2, the ordered domain pair and the transposed domain pair are depicted in a similar way for another example with symmetry descent = .

Nontransposable domain pairs. (a) The parent phase with symmetry is represented by a dotted hexagon and the three ferroelastic singledomain states with symmetry are depicted as drastically squeezed hexagons. (b) Domain pair and transposed domain pair . There exists no operation from the group that would exchange domain states and , i.e. that would transform one domain pair into a transposed domain pair. 
Let us now examine the symmetry of domain pairs. The symmetry group of an ordered domain pair consists of all operations that leave invariant both and , i.e. comprises all operations that are common to stabilizers (symmetry groups) and of domain states and , respectively, where the symbol denotes the intersection of groups and . The group is in Section 3.3.4 denoted by and is called an intersection group.
From equation (3.4.3.8), it immediately follows that the symmetry of the transposed domain pair is the same as the symmetry of the initial domain pair :
Symmetry operations of an unordered domain pair include, besides operations of that do not change either or , all transposing operations, since for an unordered domain pair a transposed domain pair is identical with the initial domain pair [see equation (3.4.3.4)]. If is a transposing operation of , then all operations from the left coset are transposing operations of that domain pair as well. Thus the symmetry group of an unordered domain pair