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. 497505
Section 3.4.2.5. Basic (microscopic) domain states and their partition into translation subsets^{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 
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.
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:

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