Basic (microscopic) domain states and their partition into translation subsets

Janovec, V.; Přívratská

doi:10.1107/97809553602060000645

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 3.4, pp. 462-470

Section 3.4.2.5. Basic (microscopic) domain states and their partition into translation subsets

V. Janovec^a ^* and J. Přívratská^b

^a Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic, and ^bDepartment of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
Correspondence e-mail: janovec@fzu.cz

3.4.2.5. Basic (microscopic) domain states and their partition into translation subsets

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The examination of principal domain states performed in the continuum approach can be easily generalized to a microscopic description. Let us denote the space-group symmetry of the parent (high-symmetry) phase by $[\cal G]$ and the space group of the ferroic (low-symmetry) phase by $[{\cal F}_1]$ , which is a proper subgroup of $[{\cal G}]$ , $[{\cal F}_1\subset {\cal G} ]$ . Further we denote by $[{\sf S}_1]$ a basic (microscopic) low-symmetry structure described by positions of atoms in the unit cell. The stabilizer $[{\cal I}_{\cal G}]$ ( $[{\sf S}_1]$ ) of the basic structure $[{\sf S}_1]$ in a single-domain orientation is equal to the space group $[{\cal F}_1]$ of the ferroic (low-symmetry) phase, $[{\cal I}_{\cal G}({\sf S}_1)={\cal F}_1. \eqno(3.4.2.41) ]$

By applying a lost symmetry operation $[{\sf g}_j]$ on $[{\sf S}_1 ]$ , one gets a crystallographically equivalent low-symmetry basic structure $[{\sf S}_j]$ , $[{\sf g}_j{\sf S}_1 = {\sf S}_j \not= {\sf S}_1, \quad {\sf g}_j\in {\cal G}, \quad {\sf g}_j \not\in {\cal F}_1. \eqno(3.4.2.42) ]$ We may recall that $[{\sf g}_j]$ is a space-group symmetry operation consisting of a rotation (point-group operation) [g_j] and a non-primitive translation $[{\bf u}(g_j)]$ , $[{\sf g}_j=\{g_j|{\bf u}(g_j)\}]$ (see Section 1.2.3 ). The symbol $[\{g_j|{\bf u}(g_j)\}]$ is called a Seitz space-group symbol (Bradley & Cracknell, 1972). The product (composition law) of two Seitz symbols is $[\{g_1|{\bf u}(g_1)\}\{g_2|{\bf u}(g_2)\}=\{g_1g_2|g_1{\bf u}(g_2)+ {\bf u}(g_1)\}. \eqno(3.4.2.43) ]$

All crystallographically equivalent low-symmetry basic structures form a $[{\cal G}]$ -orbit and can be calculated from the first basic structure $[{\sf S}_1]$ in the following way: $[{\cal G}{\sf S}_1=\{{\sf S}_1, {\sf S}_2,\ldots, {\sf S}_j\ldots, {\sf S}_N\} = \{{\sf e}{\sf S}_1, {\sf g}_2{\sf S}_1,\ldots, {\sf g}_j{\sf S}_1\ldots, {\sf g}_N{\sf S}_1\}, \eqno(3.4.2.44) ]$ where $[{\sf g}_1=]$ $[{\sf e},]$ $[{\sf g}_2,\ldots, {\sf g}_j,\ldots,{\sf g}_N ]$ are the representatives of the left cosets $[{\sf g}_j{\cal F}_{1} ]$ of the decomposition of $[{\cal G}]$ , $[{\cal G}={\cal F}_{1} \cup {\sf g}_2{\cal F}_{1} \cup\ldots\cup {\sf g}_j{\cal F}_{1} \cup\ldots\cup {\sf g}_N{\cal F}_{1}. \eqno(3.4.2.45) ]$ These crystallographically equivalent low-symmetry 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 $[{\cal G}]$ and $[{\cal F}_{1}]$ 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 space-group symmetry from a parent space group $[{\cal G} ]$ , with translation subgroup $[{\cal T}]$ and point group G, to a low-symmetry space group $[{\cal F}_1]$ , with translation subgroup $[{\cal U}_1]$ and point group [F_1] . There exists a unique intermediate group $[{\cal M}_1]$ , called the Hermann group, which has translation subgroup $[{\cal T}]$ and point group [M_1=F_1] (see e.g. Hahn & Wondratschek, 1994; Wadhawan, 2000; Wondratschek & Aroyo, 2001): $[\eqalignno{{\cal F}_{1} \,&{\buildrel {c}\over {\subseteq}}\, {\cal M}_1 \,{\buildrel {t} \over {\subseteq}}\, \cal{G}, &(3.4.2.46)\cr F_1 &= M_1 \subseteq G, &(3.4.2.47)\cr {\cal U}_1&\subseteq {\cal T} = {\cal T},&(3.4.2.48)}%fd3.4.2.48 ]$ where $[{\buildrel {c}\over {\subset}}]$ denotes an equiclass subgroup (a descent at which only the translational subgroup is reduced but the point group is preserved) and $[{\buildrel {t}\over {\subset}}]$ signifies a equitranslational subgroup (only the point group descends but the translational subgroup does not change). Group $[{\cal M}_1]$ is a maximal subgroup of $[{\cal G}]$ that preserves all macroscopic properties of the basic domain state $[{\sf S}_1]$ with symmetry $[{\cal F}_1]$ .

At this point we have to make an important note. Any space-group symmetry descent $[{\cal G}\subset {\cal F}_1]$ requires that the lengths of the basis vectors of the translation group $[{\cal U}_1]$ of the ferroic space group $[{\cal F}_1]$ are commensurate with basic vectors of the translational group $[{\cal T}]$ of the parent space group $[{\cal G}]$ . 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 first-order 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 high-symmetry approximation (Zikmund, 1984) or parent clamping approximation (PCA) (Janovec et al., 1989; Wadhawan, 2000), requires that the lengths of the basis vectors $[{\bf a}^{\kern1pt f}]$ $[{\bf b}^{\kern1pt f}]$ $[{\bf c}^{\kern1pt f} ]$ of the translation group $[{\cal U}_1]$ of the ferroic space group $[{\cal F}_1]$ are either exactly the same as, or are integer multiples of, the basic vectors $[{\bf a}^{\kern1pt p}]$ $[{\bf b}^{\kern1pt p}]$ $[{\bf c}^{\kern1pt p}]$ of the translational group $[{\cal T}]$ of the parent space group $[{\cal G}]$ . Then the relation between the primitive basis vectors $[{\bf a}^{\kern1pt f}]$ $[{\bf b}^{\kern1pt f} ]$ $[{\bf c}^{\kern1pt f}]$ of $[{\cal U}_1]$ and the primitive basis vectors $[{\bf a}^{\kern1pt p}]$ $[{\bf b}^{\kern1pt p}]$ $[{\bf c}^{\kern1pt p}]$ of $[{\cal T}]$ can be expressed as $[\pmatrix{{\bf a}^{\kern1pt f},\!\! & {\bf b}^{\kern1pt f},\!\! & {\bf c}^{\kern1pt f}}= \pmatrix{{\bf a}^{\kern1pt p},\!\! &{\bf b}^{\kern1pt p},\!\! & {\bf c}^{\kern1pt p}} \left(\matrix{m_{11} &m_{12} &m_{13} \cr m_{21} &m_{22} &m_{23} \cr m_{13} &m_{23} &m_{33}}\right), \eqno(3.4.2.49) ]$ where $[m_{ij}]$ , [i, j=1,2,3] , 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 $[\{{\sf S}_1,{\sf S}_2,\ldots,{\sf S}_{d_{t}}\} ]$ be the set of all basic translational domain states that can be generated from $[{\sf S}_1]$ by lost translations. The stabilizer (in $[{\cal G} ]$ ) of this set is the Hermann group, $[{\cal I}_{\cal G}\{{\sf S}_1,{\sf S}_2,\ldots,{\sf S}_{d_t}\} = {\cal M}_1, \eqno(3.4.2.50) ]$ 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 $[{\cal G}]$ into left cosets of $[{\cal M}_1]$ : $[\eqalignno{{\cal G}&=\{g_1|{\bf v}(g_1)\}{\cal M}_1 \cup \{g_2|{\bf v}(g_2)\}{\cal M}_1 \cup\ldots\cup \{g_j|{\bf v}(g_j)\}{\cal M}_1 &\cr &\quad\cup\ldots\cup \{g_{n}|{\bf v}(g_{n})\}{\cal M}_1. &(3.4.2.51)} ]$ Representatives $[{\sf g}_j=\{g_j|{\bf u}(g_j)\}]$ are space-group operations, where [g_j] is a point-group operation and $[{\bf u}(g_j) ]$ is a non-primitive translation (see Section 1.2.3 ).

We note that the Hermann group $[{\cal M}_1]$ can be found in the software GI $[\star]$ KoBo-1 as the equitranslational subgroup of $[{\cal G}]$ with the point-group descent $[G \subset F_1 ]$ for any space group $[{\cal G}]$ and any point group [F_1] of the ferroic phase.

The decomposition of the point group G into left cosets of the point group [F_1] is given by equation (3.4.2.10): $[G=g_1F_{1} \cup g_2F_{1} \cup\ldots\cup g_jF_{1}\cup\ldots\cup g_{n}F_{1}. \eqno(3.4.2.52) ]$ Since the space groups $[{\cal M}_1]$ and $[{\cal F}_1]$ have identical point groups, [M_1=F_1] , the decomposition (3.4.2.51) is identical with a decomposition of G into left cosets of [M_1] ; one can, therefore, choose for the representatives in (3.4.2.10) the point-group parts of the representatives $[\{g_j|{\bf u}(g_j)\}]$ 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: $[n =[{\cal G}:{\cal M}_1]=[G:F_1]=|G|:|F_1|, \eqno(3.4.2.53) ]$ where [|G|] and [|F_1|] are the number of operations of G and [F_1] , respectively.

The first `representative' basic domain state $[{\sf S}_j]$ of each subset can be obtained from the first basic domain state $[{\sf S}_1]$ : $[{\sf S}_j =\{g_j|{\bf v}(g_j)\}{\sf S}_1, \quad j=1,2,\ldots,n, \eqno(3.4.2.54) ]$ where $[\{g_j|{\bf v}(g_j)\}]$ are representatives of left cosets of $[{\cal M}_1]$ in the decomposition (3.4.2.51).

Now we determine basic domain states belonging to the first subset (first principal domain state). Equiclass groups $[{\cal M}_1]$ and $[{\cal F}_1 ]$ have the same point-group operations and differ only in translations. The decomposition of $[{\cal M}_1]$ into left cosets of $[{\cal F}_1 ]$ can therefore be written in the form $[{\cal M}_1=\{e|{\bf t}_1\}{\cal F}_1\cup \{e|{\bf t}_2\}{\cal F}_1 \cup\ldots\cup \{e|{\bf t}_k\}{\cal F}_1 \cup\ldots\cup \{e|{\bf t}_{d_{t}}\}{\cal F}_1, \eqno(3.4.2.55) ]$ where e is the identity point-group operation and $[{\cal T}_k ]$ , $[k=1,2\ldots, d_t]$ , are lost translations that can be identified with the representatives in the decomposition of $[{\cal T}]$ into left cosets of $[{\cal U}_1]$ : $[{\cal T}={\bf t}_{1}{\cal U}_1+{\bf t}_{2}{\cal U}_1+\ldots+ {\bf t}_k{\cal U}_1 +\ldots+ {\bf t}_{d_t}{\cal U}_1. \eqno(3.4.2.56) ]$ The number [d_t] of basic domain states belonging to one principal domain state will be called a translational degeneracy. For the translations $[{\bf t}_1,{\bf t}_2,\ldots,{\bf t}_k,\ldots,{\bf t}_{d_t}]$ , one can choose vectors that lead from the origin of a `superlattice' primitive unit cell of $[{\cal U}_1]$ to lattice points of $[{\cal T}]$ located within or on the side faces of this `superlattice' primitive unit cell. The number [d_t] of such lattice points is equal to the ratio $[v_{\cal F}:v_{\cal G} ]$ , where $[v_{\cal F}]$ and $[v_{\cal G}]$ are the volumes of the primitive unit cells of the low-symmetry and parent phases, respectively.

The number [d_t] can be also expressed as the determinant det $[(m_{ij})]$ of the $[(3\times 3)]$ matrix of the coefficients $[m_{ij}]$ that in equation (3.4.2.49) relate the primitive basis vectors $[{\bf a}^{\kern1pt f},{\bf b}^{\kern1pt f},{\bf c}^{\kern1pt f}]$ of $[{\cal U}_1]$ to the primitive basis vectors $[{\bf a}^{\kern1pt p},{\bf b}^{\kern1pt p},{\bf c}^{\kern1pt p} ]$ of $[{\cal T}]$ (Van Tendeloo & Amelinckx, 1974; see also Example 2.5 in Section 3.2.3.3 ). Finally, the number [d_t] equals the ratio $[Z_{\cal F}:Z_{\cal G}]$ , where $[Z_{\cal F}]$ and $[Z_{\cal G}]$ 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 [d_f] three expressions: $[d_t=[{\cal M}_1: {\cal F}_1]=[{\cal T}:{\cal U}]=v_{\cal F}:v_{\cal G}= {\rm det}(m_{ij})=Z_{\cal F}:Z_{\cal G}. \eqno(3.4.2.57) ]$ The basic domain states belonging to the first subset of translational domain states are $[{\sf S}_j=\{e|{\bf t}_k\}{\sf S}_1, \quad k=1,2,\ldots,d_t, \eqno(3.4.2.58) ]$ where $[\{e|{\bf t}_k\}]$ 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 [ab] , where the first integer $[a=1,2,\ldots,n]$ specifies the principal domain state (translational subset) and the integer $[b=1,2,\ldots,d_t]$ 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 $[{\sf S}_1={\sf S}_{11}]$ (see Proposition 3.2.3.30 in Section 3.2.3.3 ): $[{\sf S}_{jk}=\{g_j|{\bf v}(g_j)\}\{e|{\bf t}_k\}{\sf S}_{11}, \quad j=1,2,\ldots,n, \quad k=1,2,\ldots,d_t. \eqno(3.4.2.59) ]$ In a shorthand version, the letter $[{\sf S}]$ can be omitted and the symbol can be written in the form [a_b] , 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 [d_t] (number of translational domain states within one translational subset) is equal to the total number N of all basic domain states: $[\eqalignno{N&=nd_t=(|G|:|F_1|)(v_{\cal F}:v_{\cal G})= (|G|:|F_1|){\rm det}(m_{ij})&\cr&=(|G|:|F_1|)(Z_{\cal F}:Z_{\cal G}). &(3.4.2.60)\cr} ]$

Example 3.4.2.6. Basic domain states in gadolinium molybdate (GMO). Gadolinium molybdate [Gd₂(MoO₄)₃] undergoes a non-equitranslational ferroic phase transition with parent space group $[{\cal G}=P\bar42_1m]$ $[(D_{2d}^3)]$ and with ferroic space group $[{\cal F}_1=Pba2]$ $[(C_{2v}^8)]$ (see Section 3.1.2 ). From equation (3.4.2.53) we get n = $[|\bar42m|:|mm2| =]$ [8:4 = 2] , i.e. there are two subsets of translational domain states corresponding to two principal domain states. In the software GI $[\star]$ KoBo-1 one finds for the space group $[P\bar42_1m]$ and the point group [mm2] the corresponding equitranslational subgroup $[{\cal M}_1=Cmm2 ]$ $[(C_{2v}^{11})]$ with vectors of the conventional orthorhombic unit cell (in the parent clamping approximation) $[{\bf a}^{o}={\bf a}^{t}-{\bf b}^{t} ]$ , $[{\bf b}^{o}={\bf a}^{t}+{\bf b}^{t}]$ , $[{\bf c}^{o}={\bf c}^{t} ]$ , where $[{\bf a}^{t},]$ $[{\bf b}^{t},]$ $[{\bf c}^{t}]$ is the basis of the tetragonal space group $[P\bar42_1m]$ . Hence, according to equation (3.4.2.49), $[\left(\matrix{{\bf a}^o,\!\! &{\bf b}^o,\!\! &{\bf c}^o}\right) = \left(\matrix{{\bf a}^t,\!\! &{\bf b}^t,\!\! &{\bf c}^t}\right) \left(\matrix{1 &1 &0 \cr -1 &1 &0 \cr 0 &0 &1}\right). \eqno(3.4.2.61) ]$ The determinant of the transformation matrix equals two, therefore, according to equation (3.4.2.57), each principal domain state can contain [d_t=2] translational domain states that are related by lost translation $[{\bf a}^{t}]$ or $[{\bf b}^{t}]$ . 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, Hg₂Cl₂, 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 body-centred parent phase with the conventional tetragonal basis a^t, b^t, c^t and with space group $[{\cal G}=I4/mmm]$ . The structure of this phase projected onto the [z=0] 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 Hg₂Cl₂ molecules at the levels [z=0] and [z=c/2] , respectively.

Figure 3.4.2.5 | top | pdf |

Four basic single-domain states $[{{\sf S}_1}\equiv 1_1]$ , $[{{\sf S}_2}\equiv 1_2]$ , $[{{\sf S}_3}\equiv 2_1]$ , $[{{\sf S}_4}\equiv 2_2 ]$ of the ferroic phase of a calomel (Hg₂Cl₂) crystal. Full $[\bullet]$ and empty $[\circ]$ circles represent centres of gravity of Hg₂Cl₂ molecules at the levels [z=0] and [z=c/2] , respectively, projected onto the [z=0] 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 point-group symmetry $[F_1=m_{xy}m_{x\bar y}2_z ]$ , hence there are n = $[|\bar42m|:|m_{xy}m_{x\bar y}2_z|]$ = 2 ferroelastic principal domain states . The conventional orthorhombic basis is $[{\bf a}^{o}={\bf a}^{t}-{\bf b}^{t}, ]$ $[{\bf b}^{o}={\bf a}^{t}+{\bf b}^{t},]$ $[{\bf c}^{ o}={\bf c}^{t} ]$ (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 [d_t=2] , i.e. each ferroelastic domain state can contain two basic domain states.

The structure $[{\sf S}_1]$ of the ferroic phase in the parent clamping approximation is depicted in the left-hand 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 frozen-in displacements of a transverse acoustic soft mode with the k vector along the [110] direction in the first domain state $[{\sf S}_1]$ , hence all molecules in the (110) plane passing through the origin O are shifted along the $[[1\bar10]]$ direction, whereas those in the neighbouring parallel planes are shifted along the antiparallel direction $[[\bar110]]$ (the indices are related to the tetragonal coordinate system). The symmetry of $[{\sf S}_1]$ is described by the space group $[{\cal F}_1=Amam]$ $[(D^{17}_{2h})]$ ; this symbol is related to the conventional orthorhombic basis and the origin of this group is shifted by $[{\bf a}^{t}/2]$ or $[{\bf b}]$ with respect to the origin 0 of the group $[{\cal G}=I4/mmm ]$ .

Three more basic domain states $[{\sf S}_2]$ , $[{\sf S}_3]$ and $[{\sf S}_4]$ can be obtained, according to equation (3.4.2.44), from $[{\sf S}_1]$ by applying representatives of the left cosets in the resolution of $[{\cal G}]$ [see equation (3.4.2.42)], for which one can find the expression $[{\cal G}=\{1|000\}{\cal F}_1 \cup \{1|100\}{\cal F}_1 \cup \{4_z|000\}{\cal F}_1 \cup \{4_{z}{^3}|000\}{\cal F}_1.\eqno(3.4.2.62) ]$

All basic domain states $[{{\sf S}_1},]$ $[{{\sf S}_2},]$ $[{{\sf S}_3}]$ and $[{{\sf S}_4}]$ are depicted in Fig. 3.4.2.5. Domain states $[{{\sf S}_1}]$ and $[{{\sf S}_2}]$ , and similarly $[{{\sf S}_3}]$ and $[{{\sf S}_4}]$ , are related by lost translation $[{\bf a}^{t}]$ or $[{\bf b}^{t}]$ . Thus the four basic domain states $[{{\sf S}_1},]$ $[{{\sf S}_2},]$ $[{{\sf S}_3} ]$ and $[{{\sf S}_4}]$ can be partitioned into two translational subsets $[\{{{\sf S}_1},{{\sf S}_2}\}]$ and $[\{{{\sf S}_3},{{\sf S}_4}\}]$ . Basic domain states forming one subset have the same value of the secondary macroscopic order parameter $[\lambda]$ , which is in this case the difference $[{\varepsilon}_{11}-{\varepsilon}_{22}]$ of the components of a symmetric second-rank tensor $[\varepsilon]$ , 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: $[{{\sf S}_1}\equiv 1_1,]$ $[{{\sf S}_2}\equiv 1_2,]$ $[{{\sf S}_3}\equiv 2_1, ]$ $[{{\sf S}_1}\equiv 2_2]$ , 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 $[{\cal G}]$ ) of basic domain states can be calculated from a space-group version of equation (3.4.2.13): $[\eqalign{{\cal F}_2&=\{1|100\}{\cal F}_2\{1|100\}^{-1}={\cal F}_1\semi\cr \ {\cal F}_3&= \{4_z|000\}{\cal F}_2\{4_z|000\}^{-1}=Bbmm,} ]$ with the same conventional basis, and $[{\cal F}_4=\{1|100\}{\cal F}_3\{1|100\}^{-1}]$ $[={\cal F}_3]$ , where the origin of these groups is shifted by $[{\bf a}^{t}/2 ]$ or $[{\bf b}]$ with respect to the origin of the group $[{\cal G}=I4/mmm]$ .

In general, a space-group-symmetry descent $[{\cal G}\supset {\cal F}_1 ]$ can be performed in two steps:

(1) An equitranslational symmetry descent $[{\cal G}\,{\buildrel {t}\over {\supseteq}}\, {\cal M}_1 ]$ , where $[{\cal M}_1]$ is the equitranslational subgroup of $[{\cal G}]$ (Hermann group), which is unequivocally specified by space group $[{\cal G}]$ and by the point group of the space group $[{\cal F}_1]$ . The Hermann group $[{\cal M}_1]$ can be found in the software GI $[\star]$ KoBo-1 or, in some cases, in IT A (2005) under the entry `Maximal non-isomorphic subgroups, type I'.

(2) An equiclass symmetry descent $[{\cal M}_1\,{\buildrel{c}\over{\supseteq} }\,{\cal F}_1 ]$ , which can be of three kinds [for more details see IT A (2005), Section 2.2.15 ]:

(i) Space groups $[{\cal M}_1]$ and $[{\cal F}_1 ]$ have the same conventional unit cell. These descents occur only in space groups $[{\cal M}_1]$ with centred conventional unit cells and the lost translations are some or all centring translations of the unit cell of $[{\cal M}_1]$ . In many cases, the descent $[{\cal M}_1\,{\buildrel{c}\over{\supseteq} }\,{\cal F}_1 ]$ can be found in the main tables of IT A (2005), under the entry `Maximal non-isomorphic subgroups, type IIa'. Gadolinium molybdate belongs to this category.
(ii) The conventional unit cell of $[{\cal M}_1 ]$ is larger than that of $[{\cal F}_1]$ . Some vectors of the conventional unit cell of $[{\cal U}_1]$ are multiples of that of $[{\cal T}]$ . In many cases, the descent $[{\cal M}_1\,{\buildrel{c}\over{\supseteq} }\,{\cal F}_1 ]$ can be found in the main tables of IT A (2005), under the entry `Maximal non-isomorphic subgroups, type IIb'.
(iii) Space group $[{\cal F}_1]$ is an isomorphic subgroup of $[{\cal M}_1]$ , i.e. both groups are of the same space-group type (with the same Hermann–Mauguin symbol) or of the enantiomorphic space-group type. Each space group has an infinite number of isomorphic subgroups. Maximal isomorphic subgroups of lowest index are tabulated in IT A (2005), under the entry `Maximal non-isomorphic subgroups, type IIc'.

References

International Tables for Crystallography (2005). Vol. A, Space-group symmetry, 5th edition, edited by Th. Hahn. Heidelberg: Springer.Google Scholar

Barkley, J. R. & Jeitschko, W. (1973). Antiphase boundaries and their interactions with domain walls in ferroelastic–ferroelectric Gd₂(MoO₄)₃. J. Appl. Phys. 44, 938–944.Google Scholar

Bradley, C. J. & Cracknell, A. P. (1972). The mathematical theory of symmetry in solids. Oxford: Clarendon Press.Google Scholar

Hahn, Th. & Wondratschek, H. (1994). Symmetry of crystals. Introduction to International Tables for Crystallography Vol. A. Sofia: Heron Press.Google Scholar

Janovec, V. (1976). Symmetry approach to domain structures. Ferroelectrics, 12, 43–53.Google Scholar

Janovec, V., Schranz, W., Warhanek, H. & Zikmund, Z. (1989). Symmetry analysis of domain structure in KSCN crystals. Ferroelectrics, 98, 171–189.Google Scholar

Van Tendeloo, G. & Amelinckx, S. (1974). Group-theoretical considerations concerning domain formation in ordered alloys. Acta Cryst. A30, 431–440.Google Scholar

Wadhawan, V. K. (2000). Introduction to ferroic materials. The Netherlands: Gordon and Breach.Google Scholar

Wondratschek, H. & Aroyo, M. I. (2001). The application of Hermann's group $[{\cal M}]$ in group–subgroup relations between space groups. Acta Cryst. A57, 311–320.Google Scholar

Wondratschek, H. & Jeitschko, W. (1976). Twin domains and antiphase domains. Acta Cryst. A32, 664–666.Google Scholar

Zikmund, Z. (1984). Symmetry of domain pairs and domain walls. Czech. J. Phys. B, 34, 932–949.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.4, pp. 462-470