International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A1, ch. 1.2, pp. 1822
Section 1.2.7. Application to domain structures^{a}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany 
In this section, the grouptheoretical aspects of domain (twin) formation (domain structure, transformation twin) from a homogeneous single crystal (phase A, parent phase) to a crystalline phase B (daughter phase, deformed phase) are discussed, where the space group of phase B is a subgroup of the space group of phase A, . This happens, e.g., in a displacive or order–disorder phase transition. In most cases phase B, the domain structure, is inhomogeneous, consisting of homogeneous regions which are called domains, defined below.
Only the basic grouptheoretical relations are considered here. A deeper discussion of domain structures and their properties needs methods using representation theory, thermodynamic points of view (Landau theory), lattice dynamics and tensor properties of crystals. Such treatments are beyond the scope of this section. A detailed discussion of them is given by Tolédano et al. (2003) and by Janovec & Přívratská (2003).
In order to make the grouptheoretical treatment possible, the parentclamping approximation, abbreviated PCA, is introduced, by which the lattice parameters of phase A are not allowed to change at and after the transition to phase B, cf. Janovec & Přívratská (2003). Under the assumption of the PCA, two essential conditions hold:

The domain walls, i.e. the boundaries which separate different domains, will not be treated here because their symmetries are layer groups which are twoperiodic groups in threedimensional space and not space groups with threedimensional periodicity. Layer groups are described in International Tables for Crystallography, Vol. E (2002).
Under these assumptions the domains formed may exhibit different chiralities and polarities of their structures and different spatial orientations of their symmetry elements, but 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 .
In the discussion of domain structures, the following basic concepts are established: domain, domain state, symmetry state, orientation state. These concepts are defined and then applied in different examples of phase transitions in which the grouptheoretical procedures and their results are explained.
The second step, the physically realistic situation at a temperature with the removal of the PCA, is only partly considered in this section. The relaxation of the PCA does not change the relations in a nonferroelastic phase because all crystal regions suffer the same affine deformation. On the other hand, in ferroelastic phases the different spontaneous strains complicate the relations.
In a (continuous) phase transition with the symmetry reduction of the space group to a subgroup , a splitting of the parent phase A into many crystals of the type B is observed. The number of such crystals of B is not limited; they differ in their locations in space, in their orientations, in their shapes and in their space groups which, however, all belong to the same spacegroup type. In order to describe what happens in such a transition, a few notions are useful. If not explicitly stated, the validity of the PCA is assumed.
Definition 1.2.7.2.1. A connected homogeneous part of a domain structure or of a twinned crystal with structure type B is called a domain. Each domain is a single crystal. The part of the space that is occupied by a domain is the region of that domain.
If the domains of phase B have been formed from a single crystal of phase A, then relations between the domains exist which are determined by group theory. In particular, the domains belong to a finite (small) number of domain states which have well defined relations to the original crystal A and its space group . In order to describe the relation of B to A, 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 which contains translations. Here, this (infinite) periodic object is called a crystal pattern, cf. Section 1.2.2.1.
Definition 1.2.7.2.2. Two domains belong to the same domain state if their crystal patterns are identical, i.e. if they occupy different regions of space that are part of the same crystal pattern. In other words: a domain state is a crystal pattern.
The number of domain states which are observed after a phase transition is limited and determined by the space groups and . The number of domains which 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.
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, in particular under an external electric field or under external stress. Such a procedure is known as `detwinning'. The corresponding domain state is a singledomain state. For a phase transition of the type considered, there are always several singledomain states which have the same a priori probability of appearing after a phase transition. In reality not all of them will be observed and/or their relative frequencies and sizes will be rather different.
Singledomain states are introduced in theoretical considerations in order to avoid the complications which may be caused by the coexistence of domains with different spontaneous strain in ferroelastic crystals of the structure B if the PCA cannot be assumed. In polydomain structures, the domains would distort or rotate each other a little and thus disturb the simple relations described now. These disturbances do not occur in nonferroelastic transitions, so for them the simple relations also hold in polydomain structures without the PCA.
Lemma 1.2.7.2.3. The number Z of possible domain states after a phase transition under the PCA is equal to the index i of in , . Let be the coset decomposition of relative to , where are the coset representatives, and is the space group of the domain state . The other domain states are obtained from by . For the space group of the domain state the following holds: is obtained by conjugation of the space group of with the same element : .
If in a group–subgroup relation with index the subgroups belong to more than one conjugacy class, then each conjugacy class corresponds to a separate phase transition , etc. These different phase transitions lead to different lowsymmetry structures , have different transition temperatures and different probabilities of happening.
There are more elements of the group than just that map the domain state onto the domain state . The elements of the space group map the domain state of onto itself: , . Therefore, not just the element but all elements of the coset map the domain state of onto the domain state : . This can be expressed in the form:
There is a onetoone correspondence between the cosets of the decomposition and the possible domain states which may be observed after the transition.
Similarly, all elements map the space group onto by conjugation: .
Note that due to these formulae the i different domain states do not necessarily belong to i different space groups and that different domain states may belong to the same space group, cf. Lemma 1.2.7.2.6 and Example 1.2.7.2.4.
The terms just defined shall be explained in a few examples. By Example 1.2.7.2.4 a translationengleiche transition is displayed; i.e. is a translationengleiche subgroup of . Then the relation between and is essentially reflected by the relation between the point groups and .
Example 1.2.7.2.4.
Perovskite BaTiO_{3} exhibits a ferroelastic and ferroelectric phase transition from cubic to tetragonal, phase A with space group , No. 221, and phase with a translationengleiche subgroup , No. 99. 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). These two domain states belong to the same space group of the type , i.e. the domains of each pair belong to the same symmetry state according to the following definition:
Definition 1.2.7.2.5. Two domains belong to the same symmetry state if their space groups are identical.
Note that here as in many other places of this section one has to distinguish strictly between `space group' as a specimen, e.g. in `space group of a crystal', and `spacegroup type', which is one of the 230 classes frequently called simply but inexactly `the 230 space groups', see the last paragraph of Section 1.2.5.3.
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 limited and is smaller than or equal to the number of domain states. Moreover, the number of symmetry states is determined by the space groups and .
Let be the normalizer of the space group in the space group . Then with the indices and with . By Lemma 1.2.7.2.3, the number of domain states is determined. For the number of symmetry states the following lemma holds:
Lemma 1.2.7.2.6. The number of symmetry states for the transition A B with space groups and is . To each symmetry state there belong domain states, i.e. , cf. Janovec & Přívratská (2003).
For the perovskite transition of Example 1.2.7.2.4, 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 . Because of the index 3 of in , there are three conjugate subgroups of the type with their fourfold axes directed along the z, x and y directions of the cubic space group . The group is a subgroup of index 2 of . Therefore, two domain states (with opposite polar axes) belong to each of the subgroups of the spacegroup type .
In reality, i.e. without the PCA, the additional degree of freedom in tetragonal space groups leads to a metrical inequivalence of the directions of the fourfold axis and the axes perpendicular to it, due to the spontaneous strain. If this `true physical situation' is considered, the 90° angles between the fourfold axes of the domains of different pairs will change slightly because of the ferroelastic tetragonal deformation with subsequent small rotations of the domains. The antiparallel nature of each of the three domain pairs, however, is preserved because they exhibit the same symmetry state, i.e. belong to the same space group.
The space groups and in the phase transition in perovskites, cf. Example 1.2.7.2.4, are translationengleiche. Therefore, , and the coset decomposition of the space groups corresponds to the coset decomposition of their point groups, cf. Section 1.2.6.2. Indeed, in the literature the perovskite transition is nearly always treated using point groups. Other than the microscopic description by space groups, the description by point groups is called the macroscopic or continuum description. Because there are no translations involved, the continuum approach does not require the PCA at all; the point group is for any translations of phase B a subgroup of the point group . However, the spontaneous strains will appear in this ferroelastic transition and will cause the complications mentioned at the end of Section 1.2.7.1 and in the paragraph preceding Lemma 1.2.7.2.3. For nonferroelastic translationengleiche transitions, the application of point groups and of space groups yields equivalent results.
Example 1.2.7.2.4. presents an opportunity to mention another feature of this phase transition which, however, will not be discussed further. The group is not a maximal subgroup of but with of index 3 in and of index 2 in . Such `intermediate' domain states between A and B, like the domain state with the space group , are called secondary domain states for thermodynamic reasons. They do not appear in the transition but is the symmetry of the spontaneous strain in the domain state with space group . Accidentally, in the present example holds. Secondary domain states are treated by Janovec & Přívratská (2003).
In Example 1.2.7.2.4, a phase transition was discussed which involves only translationengleiche group–subgroup relations and, hence, only orientational relations between the domains. The following two examples treat klassengleiche transitions, i.e. is a klassengleiche subgroup of , and translational domain structures, also called translation twins, may appear.
Translational domain structures consist of domains which are parallel, i.e. have the same orientation of their structures (and thus of their lattices) but differ in their location because of the loss of translations of the parent phase in the phase transition. The origins of the larger unit cells of the phase B with subgroup may coincide with any of the origins of the smaller unit cells of the parent structure A with space group . Again the number of such domain states is equal to the index of in , ; the number of symmetry states is .
Example 1.2.7.3.1.
Let , No. 225, with lattice parameter a and , No. 221, with the same lattice parameter a. The relation is 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 Flattice statistically; in the ordered compound the Au atoms occupy the positions of a cubic Plattice 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 I centring translations of are not translations of and thus . There are four domain states, each one with its own distinct space group and symmetry state and consequently its own conventional origin relative to the origin of the disordered crystal A with the space group . The origin shifts of relative to the origin of are and .
These shifts do not show up in the macroscopic properties of the domains. Indeed, one is normally neither interested in those translations of which are lost in the transition to the subgroup , nor in the position of the conventional origin of relative to that of but only in the orientation of the domain states . If so, the observed relations are not governed by the space groups and but by and Hermann's group , , cf. Lemma 1.2.8.1.2. The group is uniquely determined as the space group with the translations of and the pointgroup operations of . The group can thus be characterized as that translationengleiche subgroup of which is at the same time a klassengleiche supergroup of . This group plays a role in the practical treatment of domains. It was applied to domain structures first by Janovec (1976).
In the current literature, the following considerations are mostly restricted to the point groups of the phases involved. In the following, the use of Hermann's group is discussed in parallel with the normal use of the point groups. The (admittedly rather abstract) discussion may thus be unfamiliar to the reader. Nevertheless, it is offered here because it opens up the possibility of treating phase transitions on a microscopic or atomistic level, whereas the pointgroup approach can only deal with the continuum or macroscopic aspect. The microscopic approach is necessary in particular when discussing domain boundaries, which will not be done here.
Definition 1.2.7.3.2. Two domain states and with space groups and and point groups and have the same orientation state if their orientation is identical, i.e. if the linear part of the operation of Lemma 1.2.7.2.3 is the identity operation. This means that is a translation and implies that the point groups of and are the same: . Thus the space groups and are subgroups of the same space group .
Lemma 1.2.7.3.3. The number of orientation states in the transition A B with space groups is , i.e. the index of in , where is Hermann's group in the sequence . These orientation states belong to space groups. The number of domain states which belong to the same orientation state is , i.e. the index of in .
In Example 1.2.7.3.1 of AuCu_{3}, because is a klassengleiche subgroup of . Therefore, and all four domain states belong to the same orientation state. This is obvious visually, because, as stated above, all four domain states are parallel and only shifted against each other.
Lemma 1.2.7.3.4. Because of the isomorphism between the factor groups and , the results of the application of the groups and are the same as the results of the application of the groups and . The latter application is called the `continuum approach to phase transitions' which is nearly always applied in practice.
Lemma 1.2.7.3.3 is the microscopic formulation of the (macroscopic) continuum treatment of phase transitions and forms the bridge from the (macroscopic) continuum to the (microscopic) atomistic approach to phase transitions.
Example 1.2.7.3.5.
There is an order–disorder transition of the alloy βbrass, CuZn. In the disordered state the Cu and Zn atoms occupy statistically 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, such that a space group , No. 221, is formed, see also Example 1.3.3.1 . For the space groups the relation of index 2 holds with the same cubic lattice parameter a. In this case, , see IT A, Table 15.2.1.4 . As the index , there are two domain states with their crystal structures shifted relative to each other by . Thus, both domain states belong to the same orientation state. This also follows from . Because and thus , there is one symmetry state, and both domain states belong to the same space group.
Up to now, the examples have been concerned either with translationengleiche or with klassengleiche transitions only. In this section, the domain structure of a general transition will be considered, i.e. a transition where is a general subgroup of . General subgroups are 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 1.2.7.4.2, the chain has two steps. The results obtained under the PCA and without it are different and, therefore, will be discussed in some detail. Example 1.2.7.4.2 further shows how the subgroup data of this volume can be used for the analysis of continuous or quasicontinuous phase transitions.
We start with a lemma for general subgroups which contains the results of Lemmata 1.2.7.3.3 and 1.2.7.3.4.
Lemma 1.2.7.4.1. For general subgroups, owing to the existence of the group of Hermann, it always holds that . Here is the index of the point groups of and and is the index of the translation subgroups of and .
Example 1.2.7.4.2.
βGadolinium molybdate, Gd_{2}(MoO_{4})_{3}, is ferroelectric and ferroelastic. The hightemperature phase A has space group , No. 113, and basis vectors a, b and c. At K, a phase transition to a lowtemperature phase B occurs with spacegroup type , No. 32, basis vectors , and . The group , is of the type with the lattice parameters of . The index of in is . A factor of 2 stems from the reduction and leads to two orientation states. The other factor of 2 is caused by the loss of half of the translations, because .
In the continuum description, we consider the point groups and . There is only one subgroup of of the type . Thus, the two orientation states belong to the same point group. The orientation state is obtained from by the (lost) operation of .
This is the macroscopic or continuum treatment; it is the most common treatment of domains in phase transitions. In reality, i.e. lifting the PCA, due to the orthorhombic symmetry of phase B the domains will be slightly distorted and rotated, and thus the symmetry planes of the two domain states are no longer parallel.
The full microscopic or atomistic treatment has to consider the crystal structures of phases A and B. Under the PCA, the length of and is , the content of the unit cell of is twice that of A. Because the index there are four domain states to of . The domain state is obtained from by the (lost) operation of . The same holds for the pair and . Thus, & are rotated by 90° around a centre in the plane with respect to the pair & , and the axes are antiparallel for & relative to those of & . The orientation state of the pair & is different from that of & . The two pairs & and & are shifted relative to each other by a (lost) translation of , e.g. by in the basis of , corresponding to in the basis of .
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 . Therefore, under the PCA all four domain states belong to one symmetry state, i.e. to one space group .
Analysing the group–subgroup relations between and with the tables of this volume, one finds only one chain . For only one maximal subgroup of type is listed, for which again only one maximal subgroup of type is found, in agreement with the previous paragraph.
In reality, i.e. relaxing the PCA, the observations are made at temperatures where the lattice parameters deviate from those of phase A and the basis no longer has tetragonal symmetry, but orthorhombic symmetry, . The previous single space group now splits into two different space groups of type with orthorhombic metrics at , one belonging to the pair & , the other to & . The (, ) bases of these pairs are oriented perpendicular to each other and the axes of their domains are antiparallel. The loss of the centring translation of does not produce a new space group.
The number, two, of these space groups if the PCA is not valid can also be calculated in the usual way with the help of the normalizer. The Euclidean normalizer of with is . This is an orthorhombic group with continuous translations along the direction. with is not really a subgroup of because the translations of and thus of and are not strictly translations of . The first three groups have orthorhombic lattices and the last a tetragonal one. However, by relaxing the PCA only gradually, the difference between the orthorhombic groups and the corresponding groups with tetragonal lattices is arbitrarily small. Therefore, one considers the sequence , i.e. as a group–subgroup chain, forms the intersection as if the groups would have common translations, and finds with approximately the lattice parameters of . The index , such that there are two space groups of type which are approximately subgroups of . To each of these space groups belong two domain states of phase B, see above.
This example shows that without the PCA, in order to cope with real observations, the terms `subgroup', `intersection of groups' etc. must not be used sensu stricto but have to be relaxed. The orthorhombic translations in this example are not group elements of but are slightly modified from the original translations of . All group–subgroup relations in crystal chemistry, e.g. diamond (C)–sphalerite (ZnS), as well as many phase transitions, as in this example, require such a `softened' approach.
It turns out that the transition of Gd_{2}(MoO_{4})_{3} can be considered both under the PCA (allowing exact grouptheoretical arguments) and under physically realistic arguments (which require certain relaxations of the grouptheoretical methods). The results are different but the realistic approach can be developed by means of an increasing deviation from the PCA, starting from idealized but unrealistic considerations.
References
International Tables for Crystallography (2005). Vol. A, Spacegroup symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. (Abbreviated IT A.)Janovec, V. (1976). A symmetry approach to domain structures. Ferroelectrics, 12, 43–53.
Janovec, V. & Přívratská, J. (2003). Domain structures. International tables for crystallography, Vol. D, edited by A. Authier, Physical properties of crystals, ch. 3.4. Dordrecht: Kluwer Academic Publishers.
Tolédano, J.C., Janovec, V., Kopský, V., Scott, J.F. & Boček, P. (2003). Structural phase transitions. International tables for crystallography, Vol. D, Physical properties of crystals, edited by A. Authier, ch. 3.1. Dordrecht: Kluwer Academic Publishers.