International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 486490
Section 3.4.2.1. Principal and basic domain states^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic 
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. Corresponding domain states in the macroscopic description 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').
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, dissymmetrization, symmetry lowering or reduction.
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.
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. Another expression is nonoriented symmetry.) 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 usually 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 in onetoone correspondence 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 from degenerate 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.
References
Aizu, K. (1969). Possible species of `ferroelastic' crystals and of simultaneously ferroelectric and ferroelastic crystals. J. Phys. Soc. Jpn, 27, 387–396.Chen, X. J., Liu, J. S., Zhu, J. S. & Wang, Y. N. (2000). Group theoretical analysis of the domain structure of SrBi_{2}Ta_{2}O_{9} ferroelectric ceramic. J Phys. Condens. Matter, 12, 3745–3749.
Van Tendeloo, G. & Amelinckx, S. (1974). Grouptheoretical considerations concerning domain formation in ordered alloys. Acta Cryst. A30, 431–440.