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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 496-497

## Section 3.4.2.4.1. Explanation of Table 3.4.2.7

V. Janoveca* and J. Přívratskáb

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, 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

#### 3.4.2.4.1. Explanation of Table 3.4.2.7

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• G: point group expressing the symmetry of the parent (prototypic) phase. Subscripts of generators in the group symbol specify their orientation in the Cartesian (rectangular) crystallophysical coordinate system of the group G (see Tables 3.4.2.5 and 3.4.2.6, and Figs. 3.4.2.3 and 3.4.2.4).

 Table 3.4.2.7| top | pdf | Group–subgroup symmetry descents
 G: point-group symmetry of parent phase; : point-group symmetry of single-domain state ; : representation of G; : normalizer of in G; : twinning groups, ; n: number of principal single-domain states; : number of principal domain states with the same symmetry; : number of ferroelectric single-domain states; : number of ferroelastic single-domain states.
Gn
1 2 2 2 1
1 B 2 2 2 2
1 2 2 2 2
2 2 2 1
2 2 2 1
2 2 0 2
1 Reducible , , 4 4 4 2
2 2 2 2
2 2 2 2
2 2 2 2
1 Reducible , , 4 4 4 4
2 2 2 2
2 2 2 2
2 2 1 2
1 Reducible , , 4 2 4 4
2 2 2 1
2 2 2 1
2 2 2 1
2 2 0 1
2 2 0 2
2 2 0 2
2 2 0 2
Reducible , , 4 4 4 2
Reducible , , 4 4 4 2
Reducible , , 4 4 4 2
Reducible , , 4 4 2 2
Reducible , , 4 4 2 2
Reducible , , 4 4 2 2
Reducible , , 4 4 0 4
1 Reducible , , , , , , 8 8 8 4
B 2 2 1 2
1 4 4 4 4
B 2 2 2 2
1 4 2 4 4
2 2 0 1
2 2 2 1
2 2 0 2
, 4 4 4 2
Reducible , , 4 4 2 2
, 4 4 0 4
1 Reducible , , , , 8 8 8 4
2 2 2 1
2 2 0 2
2 2 0 2
E , 4 2 2 2
Reducible , , 4 4 2 2
E , 4 2 2 2
1 E , , , 8 8 8 8
2 2 1 1
2 2 1 2
2 2 1 2
E , 4 2 4 4
E , 4 2 4 4
Reducible , , 4 4 2 2
1 E , , , 8 8 8 8
2 2 0 1
2 2 2 2
2 2 0 2
E , 4 2 4 4
Reducible , , 4 4 2 2
E , 4 2 4 4
1 E , , , 8 8 8 8
2 2 0 1
2 2 2 2
2 2 0 2
E , 4 2 4 4
E , 4 2 4 4
Reducible , , 4 4 2 2
1 E , , , 8 8 8 8
2 2 0 1
2 2 0 1
2 2 2 1
2 2 0 1
2 2 0 1
Reducible , , 4 4 0 1
Reducible , , 4 4 2 1
2 2 0 2
2 2 0 2
, 4 2 4 2
, 4 2 4 2
Reducible , , 4 4 2 2
Reducible , , 4 4 2 2
Reducible , , 4 4 0 2
Reducible , , 4 4 0 2
, 4 2 0 4
Reducible , , 4 4 0 4
, 4 2 0 4
Reducible , , , , 8 4 8 4
, , , 8 8 8 4
Reducible , , , , 8 4 8 4
Reducible , , , , 8 4 8 4
Reducible , , , , , , 8 8 2 4
Reducible , , , , 8 4 4 4
, , , 8 8 0 8
1 Reducible , , , , , , , , 16 16 16 8
1 E 3 3 3 3
2 2 2 1
3 3 0 3
1 , , 6 6 6 3
2 2 2 1
, E 3 1 3 3
1 E 6 6 6 6
2 2 2 1
, E 3 1 3 3
1 E , 6 6 6 6
2 2 1 1
, E 3 1 3 3
1 E , 6 6 6 6
2 2 1 1
, E 3 1 3 3
1 E , 6 6 6 6
2 2 2 1
2 2 0 1
2 2 0 1
Reducible , , 4 4 2 1
, 3 1 0 3
, , , 6 2 6 3
, , , 6 2 6 3
, 6 6 0 6
1 , , , , 12 12 12 6
2 2 2 1
2 2 0 1
2 2 0 1
Reducible , , 4 4 0 1
, 3 1 2 1
, , , 6 2 0 3
, , , 6 2 6 3
, 6 6 0 3
1 , , , , 12 12 12 6
B 2 2 1 1
3 3 1 3
1 , , 6 6 6 6
2 2 2 1
3 2 3 3
1 , , 6 6 6 6
2 2 0 1
2 2 2 1
2 2 0 1
Reducible , , 4 4 2 1
3 3 0 3
, , 6 6 6 3
, , 6 6 2 3
, , 6 6 0 6
1 Reducible , , , , , , 12 12 12 6
2 2 2 1
2 2 0 1
2 2 0 1
Reducible , , 4 4 2 1
, 3 1 0 3
, 6 6 2 6
, , , 6 2 6 6
, , , 6 2 6 6
1 , , , , 12 12 12 12
2 2 1 1
2 2 1 1
2 2 1 1
Reducible , , 4 4 1 1
, 3 1 1 3
, , , 6 2 6 6
, , , 6 2 6 6
, 6 6 1 6
1 , , , , 12 12 12 12
2 2 0 1
2 2 2 1
2 2 0 1
Reducible , , 4 4 2 1
, 3 1 3 3
6 6 6 6
, , , 6 2 6 6
, , , 6 2 3 6
1 , , , , 12 12 12 12
2 2 0 1
2 2 2 1
2 2 0 1
Reducible , , 4 4 2 1
, 3 1 3 3
, 6 6 6 6
, , , 6 2 6 6
, , , 6 2 3 6
1 , , , , 12 12 12 12
2 2 0 1
2 2 0 1
2 2 2 1
2 2 0 1
2 2 0 1
Reducible , , 4 4 0 1
Reducible , , 4 4 2 1
2 2 0 1
2 2 0 1
Reducible , , 4 4 2 1
Reducible , , 4 4 2 1
Reducible , , 4 4 0 1
Reducible , , 4 4 0 1
Reducible , , 4 4 0 1
Reducible , , , , , , 8 8 2 1
, 3 1 0 3
, , , 6 2 2 3
, , , 6 2 6 3
, , , 6 2 6 3
, , , 6 6 0 3
, 6 6 0 6
, , , 6 2 0 6
, , , 6 2 0 6
, , , , 12 12 12 6
, Reducible , , , , , , 12 4 12 6
, Reducible , , , , , , 12 4 12 6
, , , , 12 12 2 6
, Reducible , , , , , , 12 4 6 6
, Reducible , , , , , , 12 4 6 6
, , , , 12 12 0 12
1 Reducible , , , , , , , , , , 24 24 24 12
, , T 23 4 1 4 4
E 23 23 3 3 0 3
, T , 6 2 6 6
1 T , 12 12 12 12
23 2 2 0 1
, , 4 1 0 4
, , 8 2 8 4
3 3 0 3
, 6 2 6 3
, , 6 6 0 3
, , 6 2 0 6
, , , , 12 4 12 6
, Reducible , , , , 12 4 6 6
, 12 12 0 12
1 , , , , 24 24 24 12
23 432 2 2 0 1
, , 432 4 1 0 4
, , , 8 2 8 4
, E 432 3 1 0 3
, , 6 2 6 3
E 432 , 6 6 0 6
, , 6 2 0 6
, Reducible 23, , , , 12 4 6 12
, , , , , , , , , 12 2 12 12
1 , 432 , , , 24 24 24 24
23 2 2 0 1
, , 4 1 4 4
, , , , 8 2 4 4
, E 3 1 0 3
, 6 2 0 3
, , 6 2 6 6
E 6 6 0 6
, , , , , , , , 12 2 12 12
, Reducible 23, , , , 12 4 6 12
1 , , , , 24 24 24 24
2 2 0 1
432 2 2 0 1
2 2 0 1
23 Reducible , , 4 4 0 1
, , 4 1 0 4
, , , , 8 2 8 4
, , , , 8 2 0 4
, , , , 8 2 0 4
, , Reducible , , , , , , 16 4 8 4
, 3 1 0 3
, , , 6 2 0 3
, , 6 2 0 3
, , 6 2 6 3
, , , 6 2 0 3
, , 6 2 0 3
, Reducible , , , , 12 4 0 3
, Reducible , , , , 12 4 6 3
, 6 6 0 6
, , 6 2 0 6
, Reducible , , , , 12 4 6 6
, Reducible , , , , 12 4 6 6
, , , , , , , , 12 2 12 6
, , , , 12 12 0 6
, Reducible , , , , 12 4 0 6
Reducible , , , , 12 4 0 12
, , , , , , , , 12 2 0 12
, , , , , , , , 24 8 24 12
, , , , , , , , , , , , , 24 4 24 12
, Reducible , , , , , , , , , , 24 8 6 12
, , , , , , , , , , , , , , 24 4 12 12
, , , , 24 24 0 24
1 , , , , , , , , 48 48 48 24
, y, z, , , , , , , , , , .
• : this point group is a proper subgroup of G given in the first column and expresses the symmetry of the ferroic phase in the first single-domain state . In accordance with IT A (2005), five groups are given in two orientations (bold and normal type). Subscripts of generators in the group symbol specify their orientation in the Cartesian (rectangular) crystallophysical coordinate system of the group G (see Tables 3.4.2.5 and 3.4.2.6, and Figs. 3.4.2.3and 3.4.2.4). In the cubic groups, the direction of the body diagonal is denoted by abbreviated symbols: (all positive), , , . In the hexagonal and trigonal groups, axes x′, y′ and x′′, y′′ of a Cartesian coordinate system are rotated about the z axis through 120 and 240°, respectively, from the crystallophysical Cartesian coordinate axes x and .

Symmetry groups in parentheses are groups conjugate to under G (see Section 3.2.3.2 ). These are symmetry groups (stabilizers) of some domain states different from (for more details see Section 3.4.2.2.3).

• : physically irreducible representation of the group G. This specifies the transformation properties of the principal tensor parameter of the phase transition in a continuum description and transformation properties of the primary order parameter of the equitranslational phase transitions in the microscopic description. The letters A, B signify one-dimensional representations, and letters E and T two- and three-dimensional irreducible representations, respectively. Two letters T indicate that the symmetry descent can be accomplished by two non-equivalent three-dimensional irreducible representations (see Table 3.1.3.2 ). Reducible' denotes a reducible representation of G. In this case, there are always several non-equivalent reducible representations inducing the same descent [for more detailed information see the software GIKoBo-1 and Kopský (2001)].

Knowledge of enables one to determine for all ferroic transitions property tensors and their components that are different in all principal domain states, and, for equitranslational transitions only, microscopic displacements and/or ordering of atoms and molecules that are different in different basic (microscopic) domain states (for details see Section 3.1.3 , especially Table 3.1.3.1 , and Section 3.1.2 ).

• : the normalizer of in G (defined in Section 3.2.3.2.4 ) determines subgroups conjugate to in G and specifies which domain states have the same symmetry (stabilizer in G). The number of subgroups conjugate to in G is [see equation (3.4.2.36)] and the number of principal domain states with the same symmetry is [see equation (3.4.2.35)]. There are three possible cases:

(i) . There are no subgroups conjugate to and the symmetry group (stabilizer of in G) of all principal domain states is equal to for all ; hence domain states cannot be distinguished by their symmetry. The group is a normal subgroup of (see Section 3.2.3.2 ). This is always the case if there are just two single-domain states , , i.e. if the index of in G equals two, .

(ii) . Then any two domain states , have different symmetry groups (stabilizers), , i.e. there is a one-to-one correspondence between single-domain states and their symmetries, . In this case, principal domain states can be specified by their symmetries . The number of different groups conjugate to is equal to the index .

(iii) . Some, but not all, domain states , have identical symmetry groups (stabilizers) . The number of domain states with the same symmetry group is = = [see equation (3.4.2.35)], . The number of different groups conjugate to is equal to the index [see equation (3.4.2.36)] and in this case . It always holds that [see equation (3.4.2.37)].

• : twinning group of a domain pair (, ). This group is defined in Section 3.4.3.2. It can be considered a colour (polychromatic) group involving c colours, where , and is, therefore, defined by two groups and , and its full symbol is . In this column only is given, since appears in the second column of the table.

If the group symbol of contains generators with the star symbol, , which signifies transposing operations of the domain pair (, ), then the symbol denotes a dichromatic (black-and-white') group signifying a completely transposable domain pair. In this special case, just the symbol containing stars specifies the group unequivocally.

The number in parentheses after the group symbol of is equal to the number of twinning groups equivalent with .

In the continuum description, a twinning group is significant in at least in two instances:

 (1) A twinning group specifies the distinction of two domain states and , where (see Sections 3.4.3.2 and 3.4.3.4). (2) A twinning group may assist in signifying classes of equivalent domain pairs (orbits of domain pairs). In most cases, to a twinning group there corresponds just one class of equivalent domain pairs (an orbit) G(); then a twinning group can represent this class of equivalent domain pairs. Nevertheless, in some cases two or more classes of equivalent domain pairs have a common twinning group. Then one has to add a switching operation to the twinning group, (see the end of Section 3.4.3.2). In this way, classes of equivalent domain pairs G(, ) are denoted in synoptic Tables 3.4.2.7 and 3.4.3.6.

Twinning groups given in column thus specify all G-orbits of domain pairs. The number of G-orbits and representative domain pairs for each orbit are determined by double cosets of group (see Section 3.4.3.2). Representative domain pairs from each orbit of domain pairs are further analysed in synoptic Table 3.4.3.4 (non-ferroelastic domain pairs) and in synoptic Table 3.4.3.6 (ferroelastic domain pairs).

The set of the twinning groups given in this column is analogous to the concept of a complete twin defined as an edifice comprising in addition to an original crystal (domain state ) as many twinned crystals (domain states ) as there are possible twin laws' (see Curien & Le Corre, 1958). If a traditional definition of a twin law [a geometrical relationship between two crystal components of a twin', see Section 3.3.2 and Koch (2004); Curien & Le Corre (1958)] is applied sensu stricto to domain twins then one gets the following correspondence:

 (i) a twin law of a non-ferroelastic domain twin is specified by the twinning group (see Section 3.4.3.3 and Table 3.4.3.4); (ii) two twin laws of two compatible ferroelastic domain twins, resulting from one ferroelastic single-domain pair , are specified by two layer groups associated with the twinning group of this ferroelastic single-domain pair (see Section 3.4.3.4 and Table 3.4.3.6).

• n: number of principal single-domain states, the finest sub­division of domain states in a continuum description, [see equation (3.4.2.11)].

• : number of principal domain states with the same symmetry group (stabilizer), [see equation (3.4.2.35)]. If , then the group does not specify the first single-domain state . The number of subgroups conjugate with is .

• : number of ferroelectric single-domain states, , where is the stabilizer (in G) of the spontaneous polarization in the first domain state [see equation (3.4.2.32)]. The number of principal domain states compatible with one ferroelectric domain state (degeneracy of ferroelectric domain states) equals [see equation (3.4.2.33)].

Aizu's classification of ferroelectric phases (Aizu, 1969; see Table 3.4.2.3): , fully ferroelectric; , partially ferroelectric; , non-ferroelectric, the parent phase is polar and the spontaneous polarization in the ferroic phase is the same as in the parent phase; , non-ferroelectric, parent phase is non-polar.

• : number of ferroelastic single-domain states, , where is the stabilizer (in G) of the spontaneous strain in the first domain state [see equation (3.4.2.28)]. The number of principal domain states compatible with one ferroelastic domain state (degeneracy of ferroelastic domain states) is given by [see equation (3.4.2.29)].

Aizu's classification of ferroelastic phases (Aizu, 1969; see Table 3.4.2.3): , fully ferroelastic; , partially ferroelastic; , non-ferroelastic.

Example 3.4.2.5. Orthorhombic phase of perovskite crystals.  The parent phase has symmetry and the symmetry of the ferroic orthorhombic phase is . In Table 3.4.2.7, we find that , i.e. the phase is fully ferroelectric. Then we can associate with each principal domain state a spontaneous polarization. In column there are four twinning groups. As explained in Section 3.4.3, these groups represent four twin laws' that can be characterized by the angle between the spontaneous polarization in single-domain state and , . If we choose along the direction [110] ( does not specify unambiguously this direction, since !), then the angles between and , representing the twin law' for these four twinning groups , , , , are, respectively, 60, 120, 90 and 180°.

### References

International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, 5th ed., edited by Th. Hahn. Heidelberg: Springer.
Aizu, K. (1969). Possible species of `ferroelastic' crystals and of simultaneously ferroelectric and ferroelastic crystals. J. Phys. Soc. Jpn, 27, 387–396.
Curien, H. & Le Corre, Y. (1958). Notation des macles à l'aide du symbolisme des groupes de couleurs de Chubnikov. Bull. Soc. Fr. Mineral. Cristallogr. 81, 126–132.
Koch, E. (2004). Twinning. In International Tables for Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, 3rd ed., edited by E. Prince, ch. 1.3. Dordrecht: Kluwer Academic Publishers.
Kopský, V. (2001). Tensor parameters of ferroic phase transitions I. Theory and tables. Phase Transit. 73, 1–422.