International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 
International Tables for Crystallography (2011). Vol. A1, ch. 1.2, pp. 2023
Section 1.2.7. Application to domain structures^{a}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany 
In this section, the basic grouptheoretical aspects of this chapter are exemplified using the topic of domain structures (transformation twins). Domain structures result from a displacive or order–disorder phase transition. A homogeneous single crystal phase A (parent or prototypic phase) is transformed to a crystalline phase B (daughter phase, distorted phase). In most cases phase B is inhomogeneous, consisting of homogeneous regions which are called domains.
Definition 1.2.7.1.1. A connected homogeneous part of a domain structure or of a twinned crystal of phase B is called a domain. Each domain is a single crystal. The part of space that is occupied by a domain is the region of that domain.
The space groups of phase B are conjugate subgroups of the space group of phase A, . The number of domains is not limited; they differ in their locations in space, in their orientations, in their sizes, in their shapes and in their space groups which, however, all belong to the same spacegroup type . The boundaries between the domains, called the domain walls, are assumed to be (infinitely) thin.
A deeper discussion of domain structures or transformation twins and their properties needs a much more detailed treatment, as is given in Volume D of International Tables for Crystallography (2003) (abbreviated as IT D) Part 3, by Janovec, Hahn & Klapper (Chapter 3.2 ), by Hahn & Klapper (Chapter 3.3 ) and by Janovec & Přívratská (Chapter 3.4 ) with more than 400 references. Domains are also considered in Section 1.6.6 of this volume.
In this section, nonferroelastic phase transitions are treated without any special assumption as well as ferroelastic phase transitions under the simplifying parent clamping approximation, abbreviated PCA, introduced by Janovec et al. (1989), see also IT D, Section 3.4.2.5 . A transition is nonferroelastic if the strain tensors (metric tensors) of the lowsymmetry phase B have the same independent components as the strain tensor of the phase A.^{12} There are thus no spontaneous strain components which distort the lattices of the domains. In a ferroelastic phase transition the strain tensors of phase B have more independent components than the strain tensor of phase A. The additional strain components cause lattice strain. By the PCA the lattice parameters of phase B at the transition are adapted to those of phase A, i.e. to the lattice symmetry of phase A. Therefore, under the PCA the ferroelastic phases display the same behaviour as the nonferroelastic phases.
If in this section ferroelastic phase transitions are considered, the PCA is assumed to be applied.
Under a nonferroelastic phase transition or under the assumption of the PCA, the translations of the constituents of the phase B are translations of phase A and the space groups of B are subgroups of the space group of A, .
Under this supposition the domain structure formed may exhibit different chiralities and/or polarities of its domains with different spatial orientations of their symmetry elements. Nevertheless, each domain has the same specific energy and the lattice of each domain is part of the lattice of the parent structure A with space group .
The description of domain structures by their crystal structures is called the microscopic description, IT D, Section 3.4.2.1 . In the continuum description, the crystals are treated as anisotropic continua, IT D, Section 3.4.2.1 . The role of the space groups is then taken over by the point groups of the domains. The continuum description is used when one is essentially interested in the macroscopic physical properties of the domain structure.
Different kinds of nomenclature are used in the discussion of domain structures. The basic concepts of domain and domain state are established in IT D, Section 3.4.2.1 ; the terms symmetry state and directional state are newly introduced here in the context of domain structures. All these concepts classify the domains and will be defined in the next section and applied in different examples of phase transitions in Section 1.2.7.3.
In order to describe what happens in a phase transition of the kind considered, a few notions are useful.
If the domains of phase B have been formed from a single crystal of phase A, then they belong to a finite (small) number of domain states with space groups . The domain states have well defined relations to the original crystal of phase A and its space group . In order to describe these relations, the notion of crystal pattern is used. Any perfect (ideal) crystal is a finite block of the corresponding infinite arrangement, the symmetry of which is a space group and thus contains translations. Here, this (infinite) periodic object is called a crystal pattern, cf. Section 1.2.2.1. Each domain belongs to a crystal pattern.
Definition 1.2.7.2.1. The set of all domains which belong to the same crystal pattern forms a domain state . The domains of one domain state occupy different regions of space and are part of the same crystal pattern: to each domain state belongs a crystal pattern.
From the viewpoint of symmetry, different domains of the same domain state cannot be distinguished. The following considerations concern domain states and thus indirectly their constituents, the domains.
The number of domain states that may be observed in the distorted phase B is limited and determined by the space groups and . The number of domains that belong to the same domain state is not limited. The diversity of the domains and their shapes is due to mechanical stresses, defects, electrical charges and nucleation phenomena, which strongly influence the kinetics of the phase transition; see, for example, Fig. 3.4.1.1 in IT D.
A trivial domain structure is formed when phase B consists of one domain only, i.e. when it forms a singledomain structure. This is possible, for example, in nanocrystals (Chen et al., 1997) and in particular under an external electric field or under external stress, and is stable under zero field and zero stress (detwinning). For a phase transition of the type considered, the different domain states have the same a priori probability of appearing in the distorted phase. In reality not all of them may be observed and/or their relative frequencies and sizes may be rather different.
In order to calculate the number of possible domain states, the space group will be decomposed into cosets relative to its subgroup , which is the space group of the domain state ,
The elements of are the coset representatives. The elements of map the domain state onto itself, , . The elements of the other cosets map onto the other domain states, for example, .^{13}
It follows:
Having determined the number of domain states, we turn to their space groups, i.e. to the space groups of their crystal patterns.
From for any , written as , and or , we conclude that or .
This means: the space group leaves the domain state invariant and is a subgroup of which is conjugate to for any value of .
For the classification of the domain states according to their space groups we now introduce the term symmetry state.
Definition 1.2.7.2.3. A symmetry state is a set of all domain states (crystal patterns) the space groups of which are identical.
Domains of the same domain state always belong to the same symmetry state. Domains of different domain states may or may not belong to the same symmetry state. The number of symmetry states is determined by the normalizer of in .
Let be the normalizer of the space group in the space group . Then with the indices and with . By Lemma 1.2.7.2.2 the number of domain states is determined. For the number of symmetry states the following lemma holds:
Lemma 1.2.7.2.4. The number of symmetry states for the transition with space groups and is . To each symmetry state there belong domain states, i.e. , cf. IT D, Section 3.4.2.4.1 .
In a group–subgroup relation such as that between and , has in general lost translations as well as nontranslational symmetry operations. Thus, for the translation subgroups (the lattices) holds, and for the point groups holds. Then a uniquely determined space group exists, called Hermann's group, such that has the translations of and the point group of , IT D, Example 3.2.3.32 . The group can thus be characterized as that translationengleiche subgroup of which is simultaneously a klassengleiche supergroup of . For general subgroups , for translationengleiche subgroups and for klassengleiche subgroups holds.
For the corresponding indices one finds by coset decomposition the following lemma, see IT D, equation (3.2.3.91) .
Lemma 1.2.7.2.5. The index of in can be factorized into the pointgroup part and the lattice part : .
In the continuum description the lattices are ignored and only the relation between the point groups is considered. Because is a translationengleiche subgroup of , the index of in is the same as the index of the point group of in the point group of .
In many cases one is only interested in the orientation of the domain states of a domain structure in the space. These orientations are determined by the crystal patterns and may be taken also from the values of the components of the property tensors of the domain states. A classification of the domain states according to their orientations into classes which shall be called directional states may be achieved in the following way.
Definition 1.2.7.2.6. A directional state is a set of all domain states that are parallel to each other.
Parallel domain states (crystal patterns) have space groups with the same point group and the same lattice; they are distinguished only by the locations of the conventional origins. If and are parallel, then the coset representative is a translation and and are klassengleiche subgroups of . Both subgroups have also the same translations because a translation commutes with all translations and thus leaves invariant, see Examples 1.2.7.3.2 and 1.2.7.3.3.
To determine the number of directional states, we consider the coset decomposition of relative to Hermann's group . Each coset of the decomposition maps the domain state onto a domain state with another directional state which results in the following lemma:
Lemma 1.2.7.2.7. The number of directional states in the transition with space groups and is , i.e. the index of in .
The number of directional states does not depend on the type of description, whether microscopic or continuum, and is thus the same for both. Therefore, the microscopic description of the directional state may form a bridge between both kinds of description.
The terms just defined shall be explained in a few examples. In Example 1.2.7.3.1 a translationengleiche transition is considered; i.e. is a translationengleiche subgroup of . Because , the relation between and is reflected by the relation between the space groups and and the results of the microscopic and continuum description correspond to each other.
Example 1.2.7.3.1
Perovskite BaTiO_{3} exhibits a ferroelastic and ferroelectric phase transition from phase A with the cubic space group , No. 221, to phase with tetragonal space groups of type , No. 99. Several aspects of this transition are discussed in IT D. Let be one of the domain states of phase B. Because the index , there are six domain states, forming three pairs of domain states which point with their tetragonal c axes along the cubic x, y and z axes of . Each pair consists of two antiparallel domain states of opposite polarization (ferroelectric domains), related by, for example, the symmetry plane perpendicular to the symmetry axes 4.
The six domain states also form six directional states because (antiparallel domain states belong to different directional states).
To find the space groups of the domain states, the normalizers have to be determined. For the perovskite transition, the normalizer can be obtained from the Euclidean normalizer of in Table 15.2.1.4 of IT A, which is listed as . This Euclidean normalizer has continuous translations along the z direction (indicated by the lattice part of the HM symbol) and is thus not a space group. However, all additional translations of are not elements of the space group , and = = = is a subgroup of with index 3 and with the lattice of due to the PCA. There are thus three symmetry states, i.e. three subgroups of the type with their fourfold axes directed along the z, x and y directions of the cubic space group . Two domain states (with opposite polar axes) belong to each of the three subgroups of type .
In Example 1.2.7.3.1, a phase transition was discussed which involves only translationengleiche group–subgroup relations and, hence, only directional relations between the domain states occur. Each domain state forms its own directional state. The following two examples treat klassengleiche transitions, i.e. is a klassengleiche subgroup of . Then and there is only one directional state: a translational domain structure, also called translation twin, is formed.
The domain states of a translational domain structure differ in the origins of their space groups because of the loss of translations of the parent phase in the phase transition.
Example 1.2.7.3.2
Let , No. 225, with lattice parameter a and , No. 221, with the same lattice parameter a. The relation is klassengleich of index 4 and is found between the disordered and ordered modifications of the alloy AuCu_{3}. In the disordered state, one Au and three Cu atoms occupy the positions of a cubic F lattice at random; in the ordered compound the Au atoms occupy the positions of a cubic P lattice whereas the Cu atoms occupy the centres of all faces of this cube. According to IT A, Table 15.2.1.4 , the Euclidean normalizer of is with lattice parameter a. The additional Icentring translations of are not translations of and thus . There are four domain states, each one with its own distinct space group , and symmetry state. The shifts of the conventional origins of relative to the origin of are ; ; ; and . These shifts do not show up in the macroscopic properties of the domains but in the mismatch at the boundaries (antiphase boundaries) where different domains (antiphase domains) meet. This may be observed, for example, by highresolution transmission electron microscopy (HRTEM), IT D, Section 3.3.10.6 .
In the next example there are two domain states and both belong to the same space group, i.e. to the same symmetry state.
Example 1.2.7.3.3
There is an order–disorder transition of the alloy βbrass, CuZn. In the disordered state the Cu and Zn atoms occupy at random the positions of a cubic I lattice with space group , No. 229. In the ordered state, both kinds of atoms form a cubic primitive lattice P each, and one kind of atom occupies the centres of the cubes of the other, as in the CsCl crystal structure. Its space group is , No. 221, which is a subgroup of index [2] of with the same cubic lattice parameter a. There are two domain states with their crystal structures shifted relative to each other by . The space group is a normal subgroup of and both domain states belong to the same symmetry state.
Up to now, only examples with translationengleiche or klassengleiche transitions have been considered. Now we turn to the domain structure of a general transition, where is a general subgroup of . General subgroups are never maximal subgroups and are thus not listed in this volume, but have to be derived from the maximal subgroups of each single step of the group–subgroup chain between and . In the following Example 1.2.7.3.4, the chain has two steps.
Example 1.2.7.3.4
βGadolinium molybdate, Gd_{2}(MoO_{4})_{3}, is ferroelectric and ferroelastic. It was treated as an example from different points of view in IT D by Tolédano (Section 3.1.2.5.2 ) and by Janovec & Přívratská (Example 3.4.2.6 ). The hightemperature phase A has space group , No. 113, and basis vectors a, b and c. A phase transition to a lowtemperature phase B occurs with spacegroup type , No. 32, basis vectors , and . In addition to the reduction of the pointgroup symmetry the primitive unit cell is doubled. The PCA will be applied because the transition from the tetragonal to the orthorhombic crystal family would without the PCA allow for the lattice parameters. The index of in is such that there are four domain states. These relations are displayed in Fig. 1.2.7.1.

Group–subgroup relations between the high (HT) and low temperature (LT) forms of gadolinium molybdate (Bärnighausen tree as explained in Section 1.6.3 ). 
To calculate the number of space groups , i.e. the number of symmetry states, one determines the normalizer of in . From IT A, Table 15.2.1.3 , one finds for the Euclidean normalizer of under the PCA, which includes the condition . is a supergroup of . Thus, and is a normal subgroup. Therefore, under the PCA all four domain states belong to one space group , i.e. there is one symmetry state. Indeed, the tables of Volume A1 list only one subgroup of type under and only one subgroup under with in both cases. Hermann's group , is of spacegroup type with the point group of and with the lattice of . Because , there are two directional states which belong to the same space group. The directional state of is obtained from that of by, for example, the (lost) operation of : the basis vector a of is parallel to b of , b of is parallel to a of and c of is antiparallel to c of . The other factor of 2 is caused by the loss of the centring translations of , . Therefore, the domain state is parallel to but its origin is shifted with respect to that of by a lost translation, for example, by of , which is in the basis , of . The same holds for the domain state relative to .
References
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Janovec, V., Schranz, W., Warhanek, H. & Zikmund, Z. (1989). Symmetry analysis of domain structure in KSCN crystals. Ferroelectrics, 98, 171–189.