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. 526-527

## Section 3.4.3.6.4.1. Explanation of Table 3.4.3.6

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

#### 3.4.3.6.4.1. Explanation of Table 3.4.3.6

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Table 3.4.3.6 presents representative domain pairs of all classes of ferroelastic domain pairs for which compatible domain walls exist. The first five columns concern the domain pair. In subsequent columns, each row splits into two rows describing the orientation of two associated perpendicular equally deformed planes and the symmetry properties of the four domain twins that can be formed from the given domain pair. We explain the meaning of each column in detail.

 Table 3.4.3.6| top | pdf | Ferroelastic domain pairs and twins with compatible domain walls
 : symmetry of domain state ; : switching operation, ; : twinning group, group extension of by ; Axis h: intersection of compatible walls; Equation: component B expressed as a function of strain components or lattice parameters (see end of table); Wall normals: coordinates of normals n1 and n2 of two perpendicular compatible walls, subscript e: wall is charged (see Explanation); : obliquity, for numbers (n) see end of table; : extended layer-group symmetry of the twin and the wall; : non-trivial twinning operation of the twin; : layer-group symmetry of the twin and the wall, twin law of the ferroelastic twin; Classification: classification of the twin and the wall (see Table 3.4.4.3).
Axis hEquationWall normals nClassification
(a)   (1)   1
SI
(a)   (1) SI
1
(a)   (1) SR
SR
(2) SR
SR
(2) SR
SR
(b) (3)   1 AR
1 AR
(b) (3)   1 AR
1 AR
(c)   (4)   1 AR
1 AR
(c) (4) 1 AR
1 AR
(c)   (4)   1 AR
1 AR
(c) (4) 1 AR
1 AR
(d)   (5)   1
SI
(d)   (5) SI
1
(e)   (6)   1
SI
(e)   (6) SI
1
(f)   (7)   1
SI
(f)   (7) SI
1
(g)   (8) SI
1
(2) SI

(b)   (3)   AI
AI
(b) (3)   AI
AI
(c)   (4)   AI
AI
(c) (4)   AI
AI
(c)   (4)   AI
AI
(c) (4)   AI
AI
(d)   (5) SI
1
(d)   (5)   1
SI
(e)   (6) SI
1
(e)   (6)   1
SI
(f)   (6) SI
1
(f)   (6)   1
SI
(h)   (9)   1
SI
(2) SR
SR
(b) (3)   AR
AR
(c)   (4)   AR
AR
(c) (4)   AR
AR
(d)   (5) SR
SR
(e)   (6) SR
SR
(f)   (6) SR
SR
(11) SR
SR
(11) SR
SR
(10) SR
SR
(10) SR
SR
(13) SR
SR
(k)   (12)   1
SI
(k)   (12) SI
1
(11) SR
SR
(11) SR
SR
(10) SR
SR
(10) SR
SR
(10) SI

(11)
SI
(10)
SI
(10) SI

(13) SR
SR
(k)   (12) SI
1
(k)   (12)   1
SI
(k)   (12) SI
1
(k)   (12)   1
SI
(10) SR
SR
(10) SR
SR
(k)   (12) SR
SR
(14)   1
SI
(14) SI
1
(14) SI
1
(14)   1
SI
(14) SR
SR
(14) SR
SR
(14) SR
SR
(14)
SI
(14) SR
SR
(14)
SR
(14) SR
SI
(14) SR
SR
(15)   1
SI
(15) SI
1
(15) SI
1
(15)   1
SI
(15) SR
SR
(15) SR
SR
(15) SR
SR
(15) SR
SR
(15)
SI
(15) SI

(15) SR
SR
 Expressions for obliquity as a function of spontaneous strain components and lattice parameters
as a function of spontaneous strain components as a function of lattice parameters a, b, c; , ,
Expression
(1)
(2)
(3)
(4)

(5)

(6)
(7)
(8)

(9)
(10)
(11)
(12)

(13)
(14)
(15)
 Expressions for component B of wall normal as a function of spontaneous strain components and lattice parameters
EquationB as a function of spontaneous strain components B as a function of lattice parameters a, b, c; , ,
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(k)

The first three columns specify domain pairs.

 : point-group symmetry (stabilizer in ) of the first domain state in a single-domain orientation. : switching operations (if available) that specify the domain pair . Subscripts specify the orientation of the symmetry operations in the Cartesian coordinate system of . Subscripts and denote a Cartesian coordinate system rotated about the z axis through 120 and 240°, respectively, from the Cartesian coordinate axes x and y. Diagonal directions are abbreviated: , , , . Where possible, mirror planes and 180° rotations are chosen such that the two perpendicular permissible walls have crystallographic orientations. : twinning group of the domain pair . For the pair with and , where the twinning group does not specify the domain pair unambiguously, we add after in parentheses a switching operation or that defines the domain pair. Axis: axis of ferroelastic domain pair around which single-domain states must be rotated to establish a contact along a compatible domain wall. This axis is parallel to the intersection of the two compatible domain walls given in the column Wall normals and its direction is defined by a vector product of normal vectors and of these walls. The letter B denotes components of h which depend on spontaneous strain. Equation: a reference to an expression, given at the end of the table, for the direction of the axis, where the component B in the column Axis is expressed as functions of spontaneous strain components, and the matrices above these expressions give the form of the absolute' spontaneous strain. Wall normals: orientation of equally deformed planes. As explained above, each plane represents two mutually reversed compatible domain walls. Numbers or parameters B, C given in parentheses can be interpreted either as components of normal vectors to compatible walls or as intercepts analogous to Miller indices: Planes of compatible domain walls and [see equations (3.4.3.55)] pass through the origin of the Cartesian coordinate system of and have normal vectors and . It is possible to find a plane with the same normal vector but not passing through the origin, e.g. . Then parameters A, B and C can be interpreted as the reciprocal values of the oriented intercepts on the coordinate axes cut by this plane, . In analogy with Miller indices, the symbol is used for expressing the orientation of a wall. However, parameters A, B and C are not Miller indices, since they are expressed in an orthonormal and not a crystallographic coordinate system. A left square bracket [ in front of two equally deformed planes signifies that the two domain walls (domain twins) associated with one equally deformed plane are crystallographically equivalent (in ) with two domain walls (twins) associated with the perpendicular equally deformed plane, i.e. all four compatible domain walls (domain twins) that can be formed from domain pair are crystallographically equivalent in (see Fig. 3.4.3.8). The subscript e indicates that the wall carries a nonzero polarization charge, . This can happen in ferroelectric domain pairs with spontaneous polarization not parallel to the axis of the pair. If one domain wall is charged then the perpendicular wall is not charged. In a few cases, polarization and/or orientation of the domain wall is not determined by symmetry; then it is not possible to specify which of the two walls is charged. In such cases, a subscript or indicates that one of the two walls is charged and the other is not. : reference to an expression, given at the end of the table, in which the shear angle (in radians) is given as a function of the absolute' spontaneous strain components, defined in a matrix given above the equations. : symmetry of the `twin pair'. The meaning of this group and its symbol is explained in the next section. This group specifies the symmetry properties of a ferroelastic domain twin and the reversed twin with compatible walls of a given orientation and with domain states , and , . This group can be used for designating a twin law of the ferroelastic domain twin. : one non-trivial twinning operation of the twin and the wall. An underlined symbol with a star symbol signifies an operation that inverts the wall normal and exchanges the domain states (see the next section). : layer-group symmetry of the ferroelastic domain twin and the reversed twin with compatible walls of a given orientation. Contains all trivial and non-trivial symmetry operations of the domain twin (see the next section). Classification: symbol that specifies the type of domain twin and the wall. Five types of twins and domain walls are given in Table 3.4.4.3. The letter S denotes a symmetric domain twin (wall) in which the structures in two half-spaces are related by a symmetry operation of the twin, A denotes an asymmetric twin where there is no such relation. The letters R (reversible) and I (irreversible) signify whether a twin and reversed twin are, or are not, crystallographically equivalent in .