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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 515-528

Section Ferroelastic domain pairs

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: Ferroelastic domain pairs

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A ferroelastic domain pair consists of two domain states that have different spontaneous strain. A domain pair [({\bf S}_1,{\bf S}_j) ] is a ferroelastic domain pair if the crystal family of its twinning group [K_{1j}] differs from the crystal family of the symmetry group [F_1] of domain state [{\bf S}_1], [{\rm Fam}K_{1j}\neq {\rm Fam}F_1.\eqno(]

Before treating compatible domain walls and disorientations, we explain the basic concept of spontaneous strain. Spontaneous strain

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A strain describes a change of crystal shape (in a macroscopic description) or a change of the unit cell (in a microscopic description) under the influence of mechanical stress, temperature or electric field. If the relative changes are small, they can be described by a second-rank symmetric tensor [\bf u] called the Lagrangian strain. The values of the strain components [u_{ik},] [i,k=1,2,3] (or in matrix notation [u_{\mu},] [\mu=1, \ldots, 6]) can be calculated from the `undeformed' unit-cell parameters before deformation and `deformed' unit-cell parameters after deformation (see Schlenker et al., 1978[link]; Salje, 1990[link]; Carpenter et al., 1998[link]).

A spontaneous strain describes the change of an `undeformed' unit cell of the high-symmetry phase into a `deformed' unit cell of the low-symmetry phase. To exclude changes connected with thermal expansion, one demands that the parameters of the undeformed unit cell are those that the high-symmetry phase would have at the temperature at which parameters of the low-symmetry phase are measured. To determine these parameters directly is not possible, since the parameters of the high-symmetry phase can be measured only in the high-symmetry phase. One uses, therefore, different procedures in order to estimate values for the high-symmetry parameters under the external conditions to which the measured values of the low-symmetry phase refer (see e.g. Salje, 1990[link]; Carpenter et al., 1998[link]). Three main strategies are illustrated using the example of leucite (see Fig.[link]):

  • (i) The lattice parameters of the high-symmetry phase are extrapolated from values measured in the high-symmetry phase (a straight line [a_0] in Fig.[link]). This is a preferred approach.


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    Temperature dependence of lattice parameters in leucite. Courtesy of E. K. H Salje, University of Cambridge.

  • (ii) For certain symmetry descents, it is possible to approximate the high-symmetry parameters in the low-symmetry phase by average values of the lattice parameters in the low-symmetry phase. Thus for example in cubic [\rightarrow] tetragonal transitions one can take for the cubic lattice parameter [a_0=(2a+c)/3] (the dotted curve in Fig.[link]), for cubic [\rightarrow] orthorhombic transitions one may assume [a_0=(abc)^{1/3}], where [a, b, c] are the lattice parameters of the low-symmetry phase. Errors are introduced if there is a significant volume strain, as in leucite.

  • (iii) Thermal expansion is neglected and for the high-symmetry parameters in the low-symmetry phase one takes the lattice parameters measured in the high-symmetry phase as close as possible to the transition. This simplest method gives better results than average values in leucite, but in general may lead to significant errors.

Spontaneous strain has been examined in detail in many ferroic crystals by Carpenter et al. (1998[link]).

Spontaneous strain can be divided into two parts: one that is different in all ferroelastic domain states and the other that is the same in all ferroelastic domain states. This division can be achieved by introducing a modified strain tensor (Aizu, 1970[link]b), also called a relative spontaneous strain (Wadhawan, 2000[link]): [{\bf{u}}_{(s)}^{(i)}={\bf{u}}^{(i)}-{\bf{u}}_{(s)}^{({\rm av})}, \eqno( ]where [{\bf{u}}_{(s)}^{(i)}] is the matrix of relative (modified) spontaneous strain in the ferroelastic domain state [{\bf R}_i], [{\bf{u}}^{(i)}] is the matrix of an `absolute' spontaneous strain in the same ferroelastic domain state [{\bf R}_i] and [{\bf{u}}_{(s)}^{({\rm av})} ] is the matrix of an average spontaneous strain that is equal to the sum of the matrices of absolute spontaneous strains over all [n_a ] ferroelastic domain states, [{\bf{u}} ^{({\rm av})}={{1}\over{n_a}}\sum_{j=1}^{n_a}{\bf{u}}^{(j)}. \eqno( ]

The relative spontaneous strain [{\bf b}_{(s)}^{(i)}] is a symmetry-breaking strain that transforms according to a non-identity representation of the parent group G, whereas the average spontaneous strain is a non-symmetry breaking strain that transforms as the identity representation of G.

Example  We illustrate these concepts with the example of symmetry descent [4_z/m_zm_xm_{xy}\supset 2_xm_ym_z] with two ferroelastic domain states [{\bf R}_1] and [{\bf R}_2] (see Fig.[link]). The absolute spontaneous strain in the first ferroelastic domain state [{\bf R}_1] is [{\bf u}^{(1)}= \left(\matrix{{{a-a_0}\over{a_0}} &0 &0 \cr 0 &{{b-a_0}\over{a_0}} &0 \cr 0 &0 &{{c-c_0}\over{c_0}} }\right) = \left(\matrix{u_{11} &0 &0 \cr 0 &u_{22} &0 \cr 0 &0 &u_{33}}\right),\eqno( ]where [a,b,c] and [a_0,b_0,c_0] are the lattice parameters of the orthorhombic and tetragonal phases, respectively.

The spontaneous strain [{\bf u}^{(2)}] in domain state [{\bf R}_2 ] is obtained by applying to [{\bf{u}}^{(1)}] any switching operation that transforms [{\bf R}_1] into [{\bf R}_2] (see Table[link]), [{\bf u}^{(2)}= \left(\matrix{u_{22} &0 &0 \cr 0 &u_{11} &0 \cr 0 & 0 &u_{33}}\right).\eqno( ]

The average spontaneous strain is, according to equation ([link]), [{\bf u}^{(\rm av)}=\textstyle{{1}\over{2}} \left(\matrix{u_{11}+u_{22} &0 &0 \cr 0 &u_{11}+u_{22} &0 \cr 0 &0 &u_{33}+u_{33} }\right). \eqno( ]This deformation is invariant under any operation of G.

The relative spontaneous strains in ferroelastic domain states [{\bf R}_1 ] and [{\bf R}_2] are, according to equation ([link]), [\eqalignno{{\bf u}_{(s)}^{(1)}&={\bf u}^{(1)}-{\bf u}^{(\rm av)}= \pmatrix{{{1}\over{2}}(u_{11}-u_{22}) &0 &0 \cr 0 &-{{1}\over{2}}(u_{11}-u_{22}) &0 \cr 0 &0 &0},&\cr &&(\cr{\bf u}_{(s)}^{(2)}&={\bf u}^{(2)}-{\bf u}^{(\rm av)}= \pmatrix{-{{1}\over{2}}(u_{11}-u_{22}) &0 &0 \cr 0 &{{1}\over{2}}(u_{11}-u_{22}) &0 \cr 0 &0 &0}. &\cr &&(}%fd3.4.3.50 ]

Symmetry-breaking nonzero components of the relative spontaneous strain are identical, up to the factor [{{1}\over{2}}], with the secondary tensor parameters [\lambda_{b}^{(1)}] and [\lambda_{b}^{(2)}] of the transition [4_z/m_zm_xm_{xy}\supset2_xm_ym_z] with the stabilizer [I_{4_z/m_zm_xm_{xy}}({\bf R}_1)=] [I_{4_z/m_zm_xm_{xy}}({\bf R}_2) =] [m_xm_ym_z]. The non-symmetry-breaking component [u_{33}] does not appear in the relative spontaneous strain.

The form of relative spontaneous strains for all ferroelastic domain states of all full ferroelastic phases are listed in Aizu (1970[link]b). Equally deformed planes of a ferroelastic domain pair

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We start with the example of a phase transition with the symmetry descent [G=4_z/m_zm_xm_{xy}\supset 2_xm_ym_z], which generates two ferroelastic single-domain states [{\bf R}_1] and [{\bf R}_2] (see Fig.[link]). An `elementary cell' of the parent phase is represented in Fig.[link](a) by a square [B_0E_0C_0F_0 ] and the corresponding domain state is denoted by [{\bf R}_0].


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Two ways of constructing a ferroelastic domain twin. (a) Formation of ferroelastic single-domain states [{\bf R}_1, {\bf R}_2] from the parent phase state [{\bf R}_0] and then rotating away these single-domain states through an angle [\pm\textstyle{1\over 2}\omega] about the domain-pair axis H so that disoriented ferroelastic domain states [{\bf R}_1^+] and [{\bf R}_2^-] meet along one of two perpendicular planes of equal deformation p or [p^\prime]. (b) Formation of a ferroelastic twin from one ferroelastic domain state [{\bf R}_1^+] by a simple shear deformation with a shear angle (obliquity) [\omega]. For more details see the text.

In the ferroic phase, the square [B_0E_0C_0F_0] can change either under spontaneous strain [{\bf u}^{(1)}] into a spontaneously deformed rectangular cell [B_1E_1C_1F_1] representing a domain state [{\bf R}_1 ], or under a spontaneous strain [{\bf u}^{(2)}] into rectangular [B_2E_2C_2F_2] representing domain state [{\bf R}_2]. We shall use the letter [{\bf R}_0] as a symbol of the parent phase and [{\bf R}_1, ] [{\bf R}_2] as symbols of two ferroelastic single-domain states.

Let us now choose in the parent phase a vector [{\buildrel{\longrightarrow}\over{HB_0}} ]. This vector changes into [{\buildrel{\longrightarrow}\over{HB_1}} ] in ferroelastic domain state [{\bf R}_1] and into [{\buildrel{\longrightarrow}\over{HB_2}} ] in ferroelastic domain state [{\bf R}_1]. We see that the resulting vectors [{\buildrel{\longrightarrow}\over{HB_1}}] and [{\buildrel{\longrightarrow}\over{HB_2}} ] have different direction but equal length: [|{\buildrel{\longrightarrow}\over{HB_1}}|= |{\buildrel{\longrightarrow}\over{HB_2}}| ]. This consideration holds for any vector in the plane p, which can therefore be called an equally deformed plane (EDP). One can find that the perpendicular plane [p'] is also an equally deformed plane, but there is no other plane with this property.

The intersection of the two perpendicular equally deformed planes p and [p'] is a line called an axis of the ferroelastic domain pair [({\bf R}_1,{\bf R}_2)] (in Fig.[link] it is a line at H perpendicular to the paper). This axis is the only line in which any vector chosen in the parent phase exhibits equal deformation and has its direction unchanged in both single-domain states [{\bf R}_1 ] and [{\bf R}_2] of a ferro­elastic domain pair.

This consideration can be expressed analytically as follows (Fousek & Janovec, 1969[link]; Sapriel, 1975[link]). We choose in the parent phase a plane p and a unit vector [{\bf v}(x_1,x_2,x_3) ] in this plane. The changes of lengths of this vector in the two ferroelastic domain states [{\bf R}_1] and [{\bf R}_2] are [u^{(1)}_{ik}x_ix_k ] and [u^{(2)}_{ik}x_ix_k], respectively, where [u^{(1)}_{ik} ] and [u^{(2)}_{ik}] are spontaneous strains in [{\bf R}_1] and [{\bf R}_2], respectively (see e.g. Nye, 1985[link]). (We are using the Einstein summation convention: when a letter suffix occurs twice in the same term, summation with respect to that suffix is to be understood.) If these changes are equal, i.e. if [u^{(1)}_{ik}x_ix_k=u^{(2)}_{ik}x_ix_k,\eqno(]for any vector [{\bf v}(x_1,x_2,x_3)] in the plane p, then this plane will be an equally deformed plane. If we introduce a differential spontaneous strain[\Delta u_{ik}\equiv u^{(2)}_{ik}-u^{(1)}_{ik}, \quad i, k=1,2,3, \eqno( ]the condition ([link]) can be rewritten as [\Delta u_{ik}x_ix_j=0. \eqno(]This equation describes a cone with the apex at the origin. The cone degenerates into two planes if the determinant of the differential spontaneous strain tensor equals zero, [{\rm det}\Delta u_{ik}=0. \eqno( ]If this condition is satisfied, two solutions of ([link]) exist: [Ax_1+Bx_2+Cx_3=0, \quad A{^\prime}{x_1}+B{^\prime}{x_2}+C{^\prime}{x_3}=0. \eqno( ]These are equations of two planes p and [p^\prime] passing through the origin. Their normal vectors are [{\bf n}=[ABC]] and [{\bf n'}=[A'B'C']]. It can be shown that from the equation [\Delta u_{11}+\Delta u_{22}+\Delta u_{33}=0, \eqno(]which holds for the trace of the matrix [{\rm det}\Delta u_{ik}], it follows that these two planes are perpendicular: [AA^\prime+BB^\prime+CC^\prime=0. \eqno( ]

The intersection of these equally deformed planes ([link]) is the axis h of the ferroelastic domain pair [({\bf R}_1,{\bf R}_2) ].

Let us illustrate the application of these results to the domain pair [({\bf R}_1,{\bf R}_2)] depicted in Fig.[link](b) and discussed above. From equations ([link]) and ([link]), or ([link]) and ([link]) we find the only nonzero components of the difference strain tensor are[\Delta u_{11}=u_{22}-u_{11}, \quad \Delta u_{22}=u_{11}-u_{22}. \eqno( ]Condition ([link]) is fulfilled and equation ([link]) is [\Delta u_{11}x_1^2+\Delta u_{22}x_2^2=(u_{22}-u_{11})x_1^2+(u_{11}-u_{22})x_2^2=0. \eqno( ]There are two solutions of this equation: [x_1=x_2, \quad x_1=-x_2. \eqno(]These two equally deformed planes p and [p^\prime] have the normal vectors [{\bf n}=[\bar 110] ] and [{\bf n}'=[110]]. The axis h of this domain pair is directed along [001].

Equally deformed planes in our example have the same orientations as have the mirror planes [m_{\bar{x}y}] and [m_{xy}] lost at the transition [4_z/m_zm_xm_{xy} \supset m_xm_ym_z]. From Fig.[link](a) it is clear why: reflection [m_{\bar{x}y}], which is a transposing operation of the domain pair ([{\bf R}_1,{\bf R}_2]), ensures that the vectors [{\buildrel{\longrightarrow}\over{HB_1}}] and [{\buildrel{\longrightarrow}\over {HB_2}} ] arising from [{\buildrel{\longrightarrow}\over{HB_0}}] have equal length. A similar conclusion holds for a 180° rotation and a plane perpendicular to the corresponding twofold axis. Thus we come to two useful rules:

Any reflection through a plane that is a transposing operation of a ferroelastic domain pair ensures the existence of two planes of equal deformation: one is parallel to the corresponding mirror plane and the other one is perpendicular to this mirror plane.

Any 180° rotation that is a transposing operation of a ferroelastic domain pair ensures the existence of two equally deformed planes: one is perpendicular to the corresponding twofold axis and the other one is parallel to this axis.

A reflection in a plane or a 180° rotation generates at least one equally deformed plane with a fixed prominent crystallographic orientation independent of the magnitude of the spontaneous strain; the other perpendicular equally deformed plane may have a non-crystallographic orientation which depends on the spontaneous strain and changes with temperature. If between switching operations there are two reflections with corresponding perpendicular mirror planes, or two 180° rotations with corresponding perpendicular twofold axes, or a reflection and a 180° rotation with a corresponding twofold axis parallel to the mirror, then both perpendicular equally deformed planes have fixed crystallographic orientations. If there are no switching operations of the second order, then both perpendicular equally deformed planes may have non-crystallographic orientations, or equally deformed planes may not exist at all.

Equally deformed planes in ferroelastic–ferroelectric phases have been tabulated by Fousek (1971[link]). Sapriel (1975[link]) lists equations ([link]) of equally deformed planes for all ferro­elastic phases. Table[link] contains the orientation of equally deformed planes (with further information about the walls) for representative domain pairs of all orbits of ferroelastic domain pairs. Table[link] lists representative domain pairs of all ferroelastic orbits for which no compatible walls exist. Disoriented domain states, ferroelastic domain twins and their twin laws

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To examine another possible way of forming a ferroelastic domain twin, we return once again to Fig.[link](a) and split the space along the plane p into a half-space [Q_1 ] on the negative side of the plane p (defined by a negative end of normal [{\bf n}]) and another half-space [Q_2] on the positive side of p. In the parent phase, the whole space is filled with domain state [{\bf R}_0] and we can, therefore, treat the crystal in region [Q_1] as a domain [{\bf D}_1({\bf R}_0,Q_{1}) ] and the crystal in region [Q_2] as a domain [{\bf D}_2({\bf R}_0,Q_2) ] (we remember that a domain is specified by its domain region, e.g. [Q_1], and by a domain state, e.g. [{\bf R}_1 ], in this region; see Section[link]).

Now we cool the crystal down and exert the spontaneous strain [{\bf u}^{(1)} ] on domain [{\bf D}_1({\bf R}_0,Q_1)]. The resulting domain [{\bf D}_1({\bf R}_1,Q_1^{-})] contains domain state [{\bf R}_1 ] in the domain region [Q_1^{-}] with the planar boundary along [(\overline{B_1C_1})] (the overbar `−' signifies a rotation of the boundary in the positive sense). Similarly, domain [{\bf D}_2({\bf R}_0,Q_2) ] changes after performing spontaneous strain [{\bf u}^{(2)}] into domain [{\bf D}_2({\bf R}_2,Q_2^{+})] with domain state [{\bf R}_2] and the planar boundary along [(\overline{B_2C_2})]. This results in a disruption in the sector [B_1AB_2] and in an overlap of [{\bf R}_1] and [{\bf R}_2] in the sector [C_1AC_2].

The overlap can be removed and the continuity recovered by rotating the domain [{\bf D}_1({\bf R}_1,Q_1^{-})] through angle [\omega /2 ] and the domain [{\bf D}_2({\bf R}_2,Q_2^{+})] through [-\omega /2] about the domain-pair axis A (see Fig.[link]a and b). This rotation changes the domain [{\bf D}_1({\bf R}_1,Q_1^{-})] into domain [{\bf D}_1({\bf R}_1^{+},Q_1) ] and domain [{\bf D}_2({\bf R}_2,Q_2^{-})] into domain [{\bf D}_1({\bf R}_2^{-},Q_2)], where [{\bf R}_1^{+}] and [{\bf R}_2^{-}] are domain states rotated away from the single-domain state orientation through [\omega /2] and [-\omega /2], respectively. Domains [{\bf D}_1({\bf R}_1,Q_1)] and [{\bf D}_1({\bf R}_2,Q_2) ] meet without additional strains or stresses along the plane p and form a simple ferroelastic twin with a compatible domain wall along p. This wall is stress-free and fulfils the conditions of mechanical compatibility.

Domain states [{\bf R}_1^{+}] and [{\bf R}_2^{-}] with new orientations are called disoriented (misoriented) domain states or suborientational states (Shuvalov et al., 1985[link]; Dudnik & Shuvalov, 1989[link]) and the angles [\omega /2 ] and [-\omega/2] are the disorientation angles of [{\bf R}_1^{+}] and [{\bf R}_2^{-}], respectively.

We have described the formation of a ferroelastic domain twin by rotating single-domain states into new orientations in which a stress-free compatible contact of two ferroelastic domains is achieved. The advantage of this theoretical construct is that it provides a visual interpretation of disorientations and that it works with ferroelastic single-domain states which can be easily derived and transformed.

There is an alternative approach in which a domain state in one domain is produced from the domain state in the other domain by a shear deformation. The same procedure is used in mechanical twinning [for mechanical twinning, see Section[link] and e.g. Cahn (1954[link]); Klassen-Neklyudova (1964[link]); Christian (1975[link])].

We illustrate this approach again using our example. From Fig.[link](b) it follows that domain state [{\bf R}_2^{-} ] in the second domain can be obtained by performing a simple shear on the domain state [{\bf R}_1^{+}] of the first domain. In this simple shear, a point is displaced in a direction parallel to the equally deformed plane p (in mechanical twinning called a twin plane) and to a plane perpendicular to the axis of the domain pair (plane of shear). The displacement [{\bf q}] is proportional to the distance d of the point from the domain wall. The amount of shear is measured either by the absolute value of this displacement at a unit distance, [s=q/d], or by an angle [\omega] called a shear angle (sometimes [2\omega] is defined as the shear angle). There is no change of volume connected with a simple shear.

The angle [\omega] is also called an obliquity of a twin (Cahn, 1954[link]) and is used as a convenient measure of pseudosymmetry of the ferroelastic phase.

The high-resolution electron microscopy image in Fig.[link] reveals the relatively large shear angle (obliquity) [\omega] of a ferroelastic twin in the monoclinic phase of tungsten trioxide (WO3). The plane (101) corresponds to the plane p of a ferro­elastic wall in Fig.[link](b). The planes [(\bar1 01) ] are crystallographic planes in the lower and upper ferroelastic domains, which correspond in Fig.[link](b) to domain [{\bf D}_1({\bf R}_1^{+},Q_1)] and domain [{\bf D}_2({\bf R}_2^{-},Q_2) ], respectively. The planes [(\bar1 01)] in these domains correspond to the diagonals of the elementary cells of [{\bf R}_1^{+}] and [{\bf R}_2^{-}] in Fig.[link](b) and are nearly perpendicular to the wall. The angle between these planes equals [2\omega], where [\omega] is the shear angle (obliquity) of the ferroelastic twin.


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High-resolution electron microscopy image of a ferroelastic twin in the orthorhombic phase of WO3. Courtesy of H. Lemmens, EMAT, University of Antwerp.

Disorientations of domain states in a ferroelastic twin bring about a deviation of the optical indicatrix from a strictly perpendicular position. Owing to this effect, ferroelastic domains exhibit different colours in polarized light and can be easily visualized. This is illustrated for a domain structure of YBa2Cu3O7−δ in Fig.[link]. The symmetry descent G = [4_z/m_zm_xm_{xy}\supset m_xm_ym_z =] [F_1=] [F_2] gives rise to two ferroelastic domain states [{\bf R}_1] and [{\bf R}_2 ]. The twinning group [K_{12}] of the non-trivial domain pair [({\bf R}_1,{\bf R}_2)] is [K_{12}[m_xm_ym_z]= J_{12}^{\star}=m_xm_ym_z \cup 4_z^{\star}\{2_xm_ym_z\} = 4_z^{\star}/m_zm_xm_{xy}^{\star}. \eqno( ]The colour of a domain state observed in a polarized-light microscope depends on the orientation of the index ellipsoid (indicatrix) with respect to a fixed polarizer and analyser. This index ellipsoid transforms in the same way as the tensor of spontaneous strain, i.e. it has different orientations in ferroelastic domain states. Therefore, different ferroelastic domain states exhibit different colours: in Fig.[link], the blue and pink areas (with different orientations of the ellipse representing the spontaneous strain in the plane of of figure) correspond to two different ferroelastic domain states. A rotation of the crystal that does not change the orientation of ellipses (e.g. a 180° rotation about an axis parallel to the fourfold rotation axis) does not change the colours (ferroelastic domain states). If one neglects disorientations of ferroelastic domain states (see Section[link]) – which are too small to be detected by polarized-light microscopy – then none of the operations of the group [F_1=] [F_2=m_xm_ym_z] change the single-domain ferroelastic domain states [{\bf R}_1], [{\bf R}_2], hence there is no change in the colours of domain regions of the crystal. On the other hand, all operations with a star symbol (operations lost at the transition) exchange domain states [{\bf R}_1] and [{\bf R}_2 ], i.e. also exchange the two colours in the domain regions. The corresponding permutation is a transposition of two colours and this attribute is represented by a star attached to the symbol of the operation. This exchange of colours is nicely demonstrated in Fig.[link] where a −90° rotation is accompanied by an exchange of the pink and blue colours in the domain regions (Schmid, 1991[link], 1993[link]).


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Ferroelastic twins in a very thin YBa2Cu3O7−δ crystal observed in a polarized-light microscope. Courtesy of H. Schmid, Université de Geneve.

It can be shown (Shuvalov et al., 1985[link]; Dudnik & Shuvalov, 1989[link]) that for small spontaneous strains the amount of shear s and the angle [\omega] can be calculated from the second invariant [\Lambda_2] of the differential tensor [\Delta u_{ik}]: [\eqalignno{s&=2\sqrt{-\Lambda_2}, &(\cr \omega&=\sqrt{-\Lambda_2 }, &(}%fd3.4.3.63 ]where [\Lambda_2 = \left|\matrix{\bigtriangleup u_{11} &\bigtriangleup u_{12} \cr \bigtriangleup u_{21} &\bigtriangleup u_{22}}\right| + \left|\matrix{\bigtriangleup u_{22} &\bigtriangleup u_{23} \cr \bigtriangleup u_{32} &\bigtriangleup u_{33}}\right| + \left|\matrix{\bigtriangleup u_{11} &\bigtriangleup u_{13} \cr \bigtriangleup u_{31} &\bigtriangleup u_{33}}\right|.\eqno( ]

In our example, where there are only two nonzero components of the differential spontaneous strain tensor [see equation ([link])], the second invariant [\Lambda_2] = [-(\Delta u_{11}\Delta u_{22}) ] = [-\,(u_{22}-u_{11})^2] and the angle [\omega] is [\omega=\pm|u_{22}-u_{11}|.\eqno( ]In this case, the angle [\omega] can also be expressed as [\omega=] [\pi /2–2\,{\rm arctan}\,a/b], where a and b are lattice parameters of the orthorhombic phase (Schmid et al., 1988[link]).

The shear angle [\omega] ranges in ferroelastic crystals from minutes to degrees (see e.g. Schmid et al., 1988[link]; Dudnik & Shuvalov, 1989[link]).

Each equally deformed plane gives rise to two compatible domain walls of the same orientation but with opposite sequence of domain states on each side of the plane. We shall use for a simple domain twin with a planar wall a symbol [({\bf R}_1^{+}|\bf{n}|{\bf R}_2^{-})] in which n denotes the normal to the wall. The bra–ket symbol [(\,\,|] and [|\,\,)] represents the half-space domain regions on the negative and positive sides of [{\bf n}], respectively, for which we have used letters [Q_1] and [Q_2], respectively. Then [({\bf R}_1^{+}|] and [|{\bf R}_2^{-})] represent domains [{{\bf D}_1}({\bf R}_1^{+},Q_1) ] and [{{\bf D}_2}({\bf R}_2^{-},Q_2)], respectively. The symbol [({\bf R}_1^{+}|{\bf R}_2^{-})] properly specifies a domain twin with a zero-thickness domain wall.

A domain wall can be considered as a domain twin with domain regions restricted to non-homogeneous parts near the plane p. For a domain wall in domain twin [({\bf R}_1^{+}|{\bf R}_2^{-})] we shall use the symbol [[{\bf R}_1^{+}|{\bf R}_2^{-}] ], which expresses the fact that a domain wall of zero thickness needs the same specification as the domain twin.

If we exchange domain states in the twin [({\bf R}_1^{+}|{\bf n}|{\bf R}_2^{-}) ], we get a reversed twin (wall) with the symbol [({\bf R}_2^{-}|{\bf n}|{\bf R}_1^{+}) ]. These two ferroelastic twins are depicted in the lower right and upper left parts of Fig.[link], where – for ferroelastic–non-ferroelectric twins – we neglect spontaneous polarization of ferroelastic domain states. The reversed twin [{\bf R}_2^{-}|{\bf n}'|{\bf R}_1^{+}] has the opposite shear direction.


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Exploded view of four ferroelastic twins with disoriented ferroelastic domain states [{\bf R}_1^+, {\bf R}_2^-] and [{\bf R}_1^-, {\bf R}_2^+ ] formed from a single-domain pair [({\bf S}_1,{\bf S}_2)] (in the centre).

Twin and reversed twin can be, but may not be, crystallographically equivalent. Thus e.g. ferroelastic–non-ferroelectric twins [({\bf R}_1^{+}|{\bf n}|{\bf R}_2^{-}) ] and [({\bf R}_2^{-}|{\bf n}|{\bf R}_1^{+})] in Fig.[link] are equivalent, e.g. via [2_z], whereas ferroelastic–ferroelectric twins [({\bf S}_1^{+}|{\bf n}|{\bf S}_3^{-}) ] and [({\bf S}_3^{-}|{\bf n}|{\bf S}_1^{+})] are not equivalent, since there is no operation in the group [K_{12}] that would transform [({\bf S}_1^{+}|{\bf n}|{\bf S}_3^{-})] into [({\bf S}_3^{-}|{\bf n}|{\bf S}_1^{+}) ].

As we shall show in the next section, the symmetry group [{\sf T}_{12}({\bf n}) ] of a twin and the symmetry group [{\sf T}_{21}({\bf n})] of a reverse twin are equal, [{\sf T}_{12}({\bf n}) = {\sf T}_{21}({\bf n}).\eqno( ]

A sequence of repeating twins and reversed twins [\ldots {\bf R}_1^{+}|{\bf n}|{\bf R}_2^{-}|{\bf n}|{\bf R}_1^{+}|{\bf n}|{\bf R}_2^{-} |{\bf n}|{\bf R}_1^{+}|{\bf n}|{\bf R}_2^{-}|{\bf n}|{\bf R}_1^{+}|{\bf n}|{\bf R}_2^{-}\ldots\eqno( ]forms a lamellar ferroelastic domain structure that is very common in ferroelastic phases (see e.g. Figs.[link] and[link]).

Similar considerations can be applied to the second equally deformed plane [p^\prime] that is perpendicular to p. The two twins and corresponding compatible domain walls for the equally deformed plane [p^\prime] have the symbols [({\bf R}_1^{-}|{\bf n}^\prime|{\bf R}_2^{+})] and [({\bf R}_2^{-}|{\bf n}^\prime|{\bf R}_1^{+})], and are also depicted in Fig.[link]. The corresponding lamellar domain structure is [\ldots{\bf R}_1^{-}|{\bf n}^\prime|{\bf R}_2^{+}|{\bf n}^\prime|{\bf R}_1^{-}|{\bf n}^\prime|{\bf R}_2^{+} |{\bf n}^\prime|{\bf R}_1^{-}|{\bf n}^\prime|{\bf R}_2^{+}|{\bf n}^\prime|{\bf R}_1^{-}|{\bf n}^\prime|{\bf R}_2^{+}\ldots\,.\eqno( ]

Thus from one ferroelastic single-domain pair [({\bf R}_1, {\bf R}_2) ] depicted in the centre of Fig.[link] four different ferroelastic domain twins can be formed. It can be shown that these four twins have the same shear angle [\omega] and the same amount of shear s. They differ only in the direction of the shear.

Four disoriented domain states [{\bf R}_1^{-}, {\bf R}_1^{+}] and [{\bf R}_2^{-}, {\bf R}_2^{+}] that appear in the four domain twins considered above are related by lost operations (e.g. diagonal, vertical and horizontal reflections), i.e. they are crystallographically equivalent. This result can readily be obtained if we consider the stabilizer of a disoriented domain state [{\bf R}_1^{+}], which is [I_{4/mmm}({\bf R}_1^{+})=2_z/m_z ]. Then the number [n_a^{\rm dis}] of disoriented ferroelastic domain states is given by [n^{\rm dis}_a=[G:I_g({\bf R}_1^{+})]=|4_z/m_zm_xm_{xy}|:|2_z/m_z]=16:4=4.\eqno( ]All these domain states appear in ferroelastic polydomain structures that contain coexisting lamellar structures ([link]) and ([link]).

Disoriented domain states in ferroelastic domain structures can be recognized by diffraction techniques (e.g. using an X-ray precession camera). The presence of these four disoriented domain states results in splitting of the diffraction spots of the high-symmetry tetragonal phase into four or two spots in the orthorhombic ferroelastic phase. This splitting is schematically depicted in Fig.[link]. For more details see e.g. Shmyt'ko et al. (1987[link]), Rosová et al. (1993[link]), and Rosová (1999[link]).


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Splitting of diffraction spots at a tetragonal-to-orthorhombic transition. (a) Fragment of the reciprocal lattice (stars) of the parent tetragonal phase. (b) Superposition of the reciprocal lattices of the four ferroelastic domains states in Fig.[link]: [{\bf R}_1^+] (reciprocal lattice black squares), [{\bf R}_1^-] (reciprocal lattice black circles), [{\bf R}_2^+] (reciprocal lattice white squares) and [{\bf R}_2^-] (reciprocal lattice white circles).

Finally, we turn to twin laws of ferroelastic domain twins with compatible domain walls. In a ferroelastic twin, say [({\bf R}_1^{+}|{\bf n}|{\bf R}_2^{-}) ], there are just two possible twinning operations that interchange two ferroelastic domain states [{\bf R}_1^{+}] and [{\bf R}_2^{-} ] of the twin: reflection through the plane of the domain wall ([m^{\star}_{\bar{x}y} ] in our example) and 180° rotation with a rotation axis in the intersection of the domain wall and the plane of shear ([2^{\star}_{x y} ]). These are the only transposing operations of the domain pair [({\bf R}_1,{\bf R}_2)] that are preserved by the shear; all other transposing operations of the domain pair [({\bf R}_1,{\bf R}_2)] are lost. (This is a difference from non-ferroelastic twins, where all transposing operations of the pair become twinning operations of a non-ferroelastic twin.)

Consider the twin [({\bf S}_1^{+}|{\bf n}|{\bf S}_3^{-})] in Fig.[link]. By non-trivial twinning operations we understand transposing operations of the domain pair [({\bf S}_1^{+},{\bf S}_3^{-}) ], whereas trivial twinning operations leave invariant [{\bf S}_1^{+} ] and [{\bf S}_3^{-}]. As we shall see in the next section, the union of trivial and non-trivial twinning operations forms a group [{\sf T}_{1^{+}2^{-}}({\bf n}) ]. This group, called the symmetry group of the twin [({\bf S}_1^{+}|{\bf n}|{\bf S}_3^{-}) ], comprises all symmetry operations of this twin and we shall use it for designating the twin law of the ferroelastic twin, just as the group [J_{1j}^{\star}] of the domain pair [({\bf S}_1,{{\bf S}_j)} ] specifies the twin law of a non-ferroelastic twin. This group [{\sf T}_{1^{+}2^{-}}({\bf n}) ] is a layer group (see Section[link]) that keeps the plane p invariant, but for characterizing the twin law, which specifies the relation of domain states of two domains in the twin, one can treat [{\sf T}_{1^{+}2^{-}}({\bf n})] as an ordinary (dichromatic) point group [T_{1^{+}2^{-}}({\bf n})]. Thus the twin law of the domain twin [({\bf S}_1^{+}|{\bf n}|{\bf S}_3^{-})] is designated by the group [T_{1^{+}3^{-}}({\bf n})=2_{xy}^{\star}m_{x\bar y}^{\star}m_z=T_{3^{-}1^{+}}{({\bf n})}, \eqno( ]where ([link]) expresses the fact that a twin and the reversed twin have the same symmetry, see equation ([link]). We see that this group coincides with the symmetry group [J_{1^{+}2^{-}} ] of the single-domain pair [({\bf S}_1,{\bf S}_3)] (see Fig.[link]b).

The twin law of two twins [({\bf S}_1^{-}|{\bf n^\prime}|{\bf S}_3^{+}) ] and [({\bf S}_3^{+}|{\bf n^\prime}|{\bf S}_1^{-})] with the same equally deformed plane [p^\prime] is expressed by the group [T_{1^{-}3^{+}}({\bf n^\prime})=m_z= T_{3^{-}1^{+}}({\bf n}^\prime), \eqno( ]which is different from the [T_{1^{+}3^{-}}({\bf n})] of the twin [({\bf S}_1^{+}|{\bf n}|{\bf S}_3^{-})].

Representative domain pairs of all orbits of ferroelastic domain pairs (Litvin & Janovec, 1999[link]) are listed in two tables. Table[link] contains representative domain pairs for which compatible domain walls exist and Table[link] lists ferroelastic domain pairs where compatible coexistence of domain states is not possible. Table[link] contains, beside other data, for each ferroelastic domain pair the orientation of two equally deformed planes and the corresponding symmetries of the corresponding four twins which express two twin laws. Ferroelastic domain pairs with compatible domain walls, synoptic table

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As we have seen, for each ferroelastic domain pair for which condition ([link]) for the existence of coherent domain walls is fulfilled, there exist two perpendicular equally deformed planes. On each of these planes two ferroelastic twins can be formed; these two twins are in a simple relation (one is a reversed twin of the other), have the same symmetry, and can therefore be represented by one of these twins. Then we can say that from one ferroelastic domain pair two different twins can be formed. Each of these twins represents a different `twin law' that has arisen from the initial domain pair. All four ferroelastic twins can be described in terms of mechanical twinning with the same value of the shear angle [\omega]. Explanation of Table[link]

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Table[link] 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.

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Ferroelastic domain pairs and twins with compatible domain walls

[F_1]: symmetry of domain state [{\bf S}_1]; [g_{1j}]: switching operation, [g_{1j}{\bf S}_1={\bf S}_j]; [K(F_1,g_{1j})]: twinning group, group extension of [F_1] by [g_{1j}]; 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[link]); [\omega]: obliquity, for numbers (n) see end of table; [{\sf \overline J}_{1j} ]: extended layer-group symmetry of the twin and the wall; [{\underline t}_{1j}^{\star}]: non-trivial twinning operation of the twin; [{\sf T}_{1j}]: 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[link]).

[F_1][g_{1j}][K(F_1, g_{1j} )]Axis hEquationWall normals n[\omega][{\sf \overline J}_{1j}][\underline t^{\star}_{1j}][{\sf T}_{1j}]Classification
[1] [2^{\star}_z] [2^{\star}_z] [[B\bar{1}0]] (a)   [[001]] (1) [2^{\star}_z]   1 [{\rm AR}^{\star}]
[[1B0]_e] [\underline2^{\star}_z] [\underline 2^{\star}_z] [\underline 2^{\star}_z] SI
[1] [m^{\star}_z] [m^{\star}_z] [[B\bar{1}0]] (a)   [[001]_e] (1) [{\underline m}^{\star}_z] [{\underline m}^{\star}_z] [{\underline m}^{\star}_z] SI
[[1B0]] [m^{\star}_z]   1 [{\rm AR}^{\star}]
[\bar{1}] [m^{\star}_z, 2^{\star}_z] [2^{\star}_z/m^{\star}_z] [[B\bar{1}0]] (a)   [[001]] (1) [2_z^{\star}/\underline{m}^{\star}_z] [\underline{m}^{\star}_z] [\underline{m}^{\star}_z] SR
[[1B0]] [\underline2^{\star}_z/m^{\star}_z] [\underline2^{\star}_z] [\underline2^{\star}_z] SR
[2_z] [2^{\star}_x,2^{\star}_y] [2^{\star}_x2^{\star}_y2_z] [[001]]     [[100]] (2) [2^{\star}_x\underline2^{\star}_y\underline2_z] [\underline2^{\star}_y] [\underline2^{\star}_y] SR
[[010]] [\underline2^{\star}_x2^{\star}_y\underline2_z] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[2_z] [m^{\star}_x,m^{\star}_y] [m^{\star}_xm^{\star}_y2_z] [[001]]     [[100]] (2) [\underline{m}^{\star}_xm^{\star}_y\underline2_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[010]] [m^{\star}_x\underline{m}^{\star}_y\underline2_z] [\underline{m}^{\star}_y] [\underline{m}^{\star}_y] SR
[2_z] [4^{\star}_z,4_z^{3 \star}] [4^{\star}_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [\bar4^{\star}_z,\bar4_z^{*3}] [\bar{4}^{\star}_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [3_z,6_z^5] [6_{z}] [[001]] (c)   [[1B0]] (4) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[3_z^2,6_z] [6_z] [[001]] (c) [[1B0]] (4) [\underline2_z] 1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [\bar3_z^5,\bar6_z] [6_{z}/m_z] [[001]] (c)   [[1B0]] (4) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[\bar3_z,\bar6_z^5] [6_z/m_z] [[001]] (c) [[1B0]] (4) [\underline2_z] 1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_x] [2^{\star}_{xy},4_z] [4_z2_x2_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[1\bar{1}B]_e] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[2_x] [m^{\star}_{xy},\bar4_z] [\bar4_z2_xm_{xy}] [[\bar{B}B2]] (d)   [[110]_e] (5) [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[[1\bar{1}B]] [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[2_x] [2^{\star}_{x'},3_z^2] [3_z2_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[[\sqrt{3}1B]_e] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SI
[2_x] [m^{\star}_{x'},\bar3_z^5] [\bar{3}_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]_e] (6) [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SI
[[\sqrt{3}1B]] [m^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[2_x] [2^{\star}_{y'},6_z] [6_z2_x2_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (7) [2^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[[\bar1\sqrt{3}B]_e] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SI
[2_x] [m^{\star}_{y'},\bar6_z] [\bar{6}_z2_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]_e] (7) [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SI
[[\bar1\sqrt{3}B]] [m^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[2_{xy}] [m^{\star}_{x},\bar4_z^{3}] [\bar4_zm_x2_{xy}] [[0B\bar{1}]] (g)   [[100]_e] (8) [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] SI
[[01B]] [m^{\star}_{x}]   1 [{\rm AR}^{\star}]
[m_z] [m^{\star}_x,2^{\star}_y] [m^{\star}_x2^{\star}_ym_z] [[001]]     [[100]_e] (2) [\underline{m}^{\star}_x\underline2^{\star}_ym_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_ym_z] SI
[[010]] [m_x^{\star}2_y^{\star}m_z]   [m_z] [{\rm AR}^{\star}]
[m_{z}] [4_z,\bar4_z^3] [4_z/m_{z}] [[001]] (b)   [[1B0]_{e0}] (3) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[4_z^3,\bar4_z] [4_z/m_z] [[001]] (b) [[1B0]_{e0}] (3) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_z] [3_z,\bar6^5_z] [\bar6_{z}] [[001]] (c)   [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[3^2_z,\bar6_z] [\bar6_z] [[001]] (c) [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_z] [\bar3_z,6_z^5] [6_{z}/m_z] [[001]] (c)   [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[\bar3_z^5,6_z] [6_z/m_z] [[001]] (c) [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_x] [m^{\star}_{xy},4_z] [4_zm_xm_{xy}] [[\bar{B}B2]] (d)   [[110]_e] (5) [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[[1\bar{1}B]] [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{xy},\bar4_z] [\bar{4}_zm_x2_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[1\bar{1}B]_e] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[m_x] [m^{\star}_{x'},3_z^2] [3_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]_e] (6) [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SI
[[\sqrt{3}1B]] [m^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{x'},\bar3_z^5] [\bar3_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[[\sqrt{3}1B]_e] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SI
[m_x] [m^{\star}_{y'},6_z] [6_zm_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]_e] (6) [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SI
[[\bar1\sqrt{3}B]] [m^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{y'},\bar6_z] [\bar{6}_zm_x2_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (6) [2^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[[\bar1\sqrt{3}B]_e] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SI
[m_{xy}] [2^{\star}_{x},\bar4^3_z] [\bar{4}_z2_xm_{xy}] [[0B\bar{1}]] (h)   [[100]] (9) [2^{\star}_{x}]   1 [{\rm AR}^{\star}]
[[01B]_e] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] SI
[2_z/m_z] [m^{\star}_x,m^{\star}_y] [m^{\star}_xm^{\star}_ym_z] [[001]]     [[100]] (2) [\underline{m}^{\star}_xm^{\star}_ym_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_ym_z] SR
[[010]] [m^{\star}_x\underline{m}^{\star}_ym_z] [\underline{m}^{\star}_y] [\underline2^{\star}_x\underline{m}^{\star}_ym_z] SR
[2_z/m_z] [4^{\star}_z,4^{3 \star}_z] [4^{\star}_z/m_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[2_z/m_z] [3_z,6_z^5] [6_{z}/m_z] [[001]] (c)   [[1B0]] (4) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[3_z^2,6_z] [6_z/m_z] [[001]] (c) [[1B0]] (4) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[2_x/m_x] [m^{\star}_{xy},4_z] [4_z/m_zm_xm_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}/\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar{1}B]] [\underline2^{\star}_{xy}/m^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[2_x/m_x] [m^{\star}_{x'},3_z^2] [\bar{3}_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}/\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}1B]] [\underline2^{\star}_{x'}/m^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[2_x/m_x] [m^{\star}_{y'},6_z] [6_z/m_zm_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (6) [2^{\star}_{y'}/\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[[\bar1\sqrt{3}B]] [\underline2^{\star}_{y'}/m^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[2_x2_y2_z] [2^{\star}_{x\bar{y}}, 2^{\star}_{xy}] [4^{\star}_z2_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [2^{\star}_{xy}\underline2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{x\bar{y}}] [\underline2^{\star}_{x\bar{y}}] SR
[[1\bar10]] [\underline2^{\star}_{xy}2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[2_x2_y2_z] [m^{\star}_{x\bar{y}},m^{\star}_{xy}] [\bar{4}^{\star}_z2_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [\underline{m}^{\star}_{xy}m^{\star}_{x\bar{y}}\underline2_z] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar10]] [m^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}\underline2_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}] SR
[2_x2_y2_z] [2^{\star}_{x'},2^{\star}_{y'}] [6_z2_x2_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}\underline2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[[\sqrt{3}10]] [\underline2^{\star}_{x'}2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[2_x2_y2_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}10]] [{m}^{\star}_{x'}\underline{m}^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[2_{x\bar{y}}2_{xy}2_z] [m^{\star}_{x},m^{\star}_{y}] [\bar{4}^{\star}_zm_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[100]] (13) [\underline{m}^{\star}_{x}m^{\star}_{y}\underline2_z] [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] SR
[[010]] [m^{\star}_{x}\underline{m}^{\star}_{y}\underline2_z] [\underline{m}^{\star}_{y}] [\underline{m}^{\star}_{y}] SR
[2_{x\bar{y}}2_{xy}2_z] [2^{\star}_{xz},4_y] [4_z3_{p}2_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[2_{x\bar{y}}2_{xy}2_z] [m_{xz}^{\star},\bar4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [\underline{m}_{xz}^{\star}] [\underline{m}_{xz}^{\star}] [\underline{m}_{xz}^{\star}] SI
[[\bar{1}B1]] [m_{xz}^{\star}]   1 [{\rm AR}^{\star}]
[m_xm_y2_z] [m^{\star}_{x\bar{y}},m^{\star}_{xy}] [4^{\star}_zm_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [m^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}\underline2_z] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar10]] [\underline{m}^{\star}_{x\bar{y}}{m}^{\star}_{xy}\underline2_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}] SR
[m_xm_y2_z] [2^{\star}_{x\bar{y}},2^{\star}_{xy}] [\bar{4}^{\star}_zm_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [2^{\star}_{xy}\underline2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{x\bar{y}}] [\underline2^{\star}_{x\bar{y}}] SR
[[1\bar10]] [\underline2^{\star}_{xy}2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[m_xm_y2_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}10]] [{m}^{\star}_{x'}\underline{m}^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[m_xm_y2_z] [2^{\star}_{x'},2^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}\underline2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[[\sqrt{3}10]] [\underline2^{\star}_{x'}2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[m_x2_ym_z] [m^{\star}_{x'},2^{\star}_{y'}] [\bar{6}_zm_x2_y] [[001]]     [[\bar1\sqrt{3}0]_e] (10) [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SI
[[\sqrt{3}10]] [{m}^{\star}_{x'}2^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[2_xm_ym_z] [m^{\star}_{x\bar{y}},2^{\star}_{xy}] [4_z/m_zm_xm_{xy}] [[001]]     [[110]] (11) [2^{\star}_{xy}m^{\star}_{x\bar{y}}m_z]   [m_z] [{\rm AR}^{\star}]
[[1\bar10]_e] [\underline2^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}m_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline2^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}m_z] SI
[2_xm_ym_z] [m^{\star}_{y'},2^{\star}_{x'}] [\bar{6}_z2_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}m^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[[\sqrt{3}10]_e] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] [\underline{m}_{y'}] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] SI
[2_xm_ym_z] [m^{\star}_{x'},2^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]_e] (10) [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SI
[[\sqrt{3}10]] [m^{\star}_{x'}2^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[m_{x\bar{y}}m_{xy}2_z] [2^{\star}_{x},2^{\star}_{y}] [\bar{4}^{\star}_z2^{\star}_xm_{xy}] [[001]]   [\Bigl[] [[100]] (13) [2^{\star}_{x}\underline2^{\star}_{y}\underline2_z] [\underline2^{\star}_{y}] [\underline2^{\star}_{y}] SR
[[010]] [\underline2^{\star}_{x}2^{\star}_{y}\underline2_z] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] SR
[m_{x\bar{y}}m_{xy}2_z] [m^{\star}_{xz},\bar4_y] [\bar{4}_z3_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]_e] (12) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar{1}B1]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[m_{x\bar{y}}m_{xy}2_z] [2^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[m_{x\bar{y}}2_{xy}m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}(m^{\star}_{xz})] [[B2\bar{B}]] (k)   [[101]_e] (12) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar{1}B1]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[m_{x\bar{y}}2_{xy}m_z] [2^{\star}_{xz},\bar4_y] [m_z\bar{3}_{p}m_{xy}(2^{\star}_{xz})] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[m_xm_ym_z] [m^{\star}_{xy},m^{\star}_{x\bar{y}}] [4^{\star}_z/m_zm_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (10) [m^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}m_z] [\underline{m}^{\star}_{xy}] [\underline2^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}m_z] SR
[[1\bar10]] [\underline{m}^{\star}_{x\bar{y}}{m}^{\star}_{xy}m_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}\underline{2}^{\star}_{xy}m_z] SR
[m_xm_ym_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SR
[[\sqrt{3}10]] [m^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] [\underline{m}^{\star}_{y'}] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] SR
[m_{xy}m_{\bar{x}y}m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}/\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar{1}B1]] [\underline2^{\star}_{xz}/m^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z] [2^{\star}_{xz},4_y] [4_z3_{p}2_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[4_z] [m^{\star}_{xz},\bar{4}_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]_e] (14) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar101]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[\bar{4}_z] [m^{\star}_{xz},\bar4_y] [\bar{4}_z3_{p}m_{xy}] [[010]]     [[101]] (14) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar101]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[\bar{4}_z] [2^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2_{xz}^{\star}] [\underline2_{xz}^{\star}] [\underline2_{xz}^{\star}] SI
[4_z/m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}/\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [\underline2^{\star}_{xz}/m^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z2_x2_{xy}] [2^{\star}_{xz},2^{\star}_{x\bar{z}}] [4_z3_{p}2_{xy}] [[010]]   [\Bigl[] [[101]] (14) [2^{\star}_{xz}\underline2^{\star}_{x\bar{z}}\underline2_{y}] [\underline2^{\star}_{\bar{x}z}] [\underline2^{\star}_{\bar{x}z}] SR
[[\bar101]] [\underline2^{\star}_{xz}2^{\star}_{x\bar{z}}\underline2_y] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z2_x2_{xy}] [m^{\star}_{xz},m^{\star}_{x\bar{z}}] [m_z\bar{3}_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline{m}^{\star}_{x\bar{z}}] SR
[4_zm_xm_{xy}] [m^{\star}_{x\bar{z}},2^{\star}_{xz}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}m^{\star}_{x\bar{z}}m_y]   [m_y] [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SI
[\bar{4}_z2_xm_{xy}] [m^{\star}_{xz},m^{\star}_{x\bar{z}}] [\bar{4}_z3_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline{m}^{\star}_{x\bar{z}}] SR
[\bar{4}_zm_x2_{xy}] [m_{x\bar{z}}^{\star},2_{xz}^{\star}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}m^{\star}_{x\bar{z}}m_y]   [m_y] [{\rm AR}^{\star}]
[[\bar101]] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SR
[\bar{4}_z2_xm_{xy}] [2^{\star}_{xz},2^{\star}_{x\bar{z}}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}2^{\star}_{x\bar{z}}2_y] [\underline2^{\star}_{x\bar{z}}] [\underline2^{\star}_{x\bar{z}}] SR
[[\bar101]] [\underline2^{\star}_{xz}\underline{2}^{\star}_{x\bar{z}}2_y] [\underline{2}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{2}^{\star}_{x\bar{z}}2_y] SI
[4_z/m_zm_xm_{xy}] [m^{\star}_{xz},m^{\star}_{\bar{x}z}] [m_z\bar{3}_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{\bar{x}z}m_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}\underline{2}^{\star}_{\bar{x}z}m_y] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SR
[3_{p}] [2^{\star}_{x},3_r] [2_z 3_{p}] [[01\bar1]]     [[100]] (15) [2^{\star}_x]   1 [{\rm AR}^{\star}]
[[011]_e] [\underline2^{\star}_x] [\underline2^{\star}_x] [\underline2^{\star}_x] SI
[3_{p}] [m^{\star}_{x},\bar3_r] [m_z \bar3_{p}] [[01\bar1]]     [[100]_e] (15) [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SI
[[011]] [m^{\star}_x]   1 [{\rm AR}^{\star}]
[3_{p}] [2^{\star}_{xy},4_y] [4_z 3_{p} 2_{xy}] [[1\bar10]]     [[001]_e] (15) [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[[110]] [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[3_{p}] [m^{\star}_{xy},\bar4_y] [\bar4_z3_{p}m_{xy}] [[1\bar10]]     [[001]] (15) [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[110]_e] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[\bar3_{p}] [m^{\star}_{x},3_r] [m_z\bar3_{p}] [[01\bar1]]     [[100]] (15) [2^{\star}_x/\underline{m}^{\star}_x] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[011]] [\underline2^{\star}_x/m^{\star}_x] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[\bar3_{p}] [m^{\star}_{xy},4_y] [m_z\bar3_{p}m_{xy}] [[1\bar10]]     [[001]] (15) [\underline2^{\star}_{xy}/m^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[[110]] [2^{\star}_{xy}/\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[3_{p} 2_{x\bar{y}}] [2^{\star}_{x},2^{\star}_{yz}] [4_z3_{p} 2_{xy}] [[01\bar1]]     [[100]] (15) [2^{\star}_x\underline2^{\star}_{yz}\underline2_{y\bar{z}}] [\underline2^{\star}_{yz}] [\underline2^{\star}_{yz}] SR
[[011]] [\underline2^{\star}_x2^{\star}_{yz}\underline2_{y\bar{z}}] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[3_{p} 2_{x\bar{y}}] [m^{\star}_{x},m^{\star}_{yz}] [m_z\bar3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [\underline{m}^{\star}_xm_{yz}^{\star}\underline2_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[011]] [m^{\star}_x\underline{m}^{\star}_{yz}\underline2_{y\bar{z}}] [\underline{m}^{\star}_{yz}] [\underline{m}^{\star}_{yz}] SR
[3_{p} m_{x\bar{y}}] [2^{\star}_{x},m^{\star}_{yz}] [\bar4_z3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [m^{\star}_{yz}m_{y\bar{z}}2^{\star}_x]   [m_{y\bar{z}}] [{\rm AR}^{\star}]
[[011]_e] [\underline{m}^{\star}_{yz}m_{y\bar{z}}\underline2^{\star}_x] [\underline{m}^{\star}_{yz}] [\underline{m}^{\star}_{yz}m_{y\bar{z}}\underline2^{\star}_x] SI
[3_{p} m_{x\bar{y}}] [m^{\star}_x,2^{\star}_{yz}] [m_z\bar3_{p}m_{xy}] [[01\bar1]]     [[100]_e] (15) [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] SI
[[011]] [m^{\star}_x2^{\star}_{yz}m_{y\bar{z}}]   [m_{y\bar{z}}] [{\rm AR}^{\star}]
[\bar3_{p}m_{x\bar{y}}] [m^{\star}_x,m^{\star}_{yz}] [m_z \bar3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [\underline{m}^{\star}_xm^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] SR
[[011]] [m^{\star}_x\underline{m}^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_{yz}] [\underline2^{\star}_x\underline{m}^{\star}_{yz}m_{y\bar{z}}] SR

Expressions for obliquity [\omega] as a function of spontaneous strain components and lattice parameters

 [\omega] as a function of spontaneous strain components [\omega] as a function of lattice parameters a, b, c; [\alpha = \angle (b;c)], [\beta = \angle (a;c)], [\gamma = \angle (a;b)]
Expression[\pmatrix{q&v&u\cr v&r& t\cr u&t&s\cr}] [Scheme scheme1]
(1) [\omega= 2 \sqrt{t^2+u^2}] [\omega=\left|\arccos {{\displaystyle \sqrt{1-\cos^2\alpha -\cos^2\beta -\cos^2\gamma +2\cos\alpha \cos\beta \cos\gamma }}\over{\displaystyle \sin\gamma }} \right|]
(2) [\omega= 2|v|] [\omega= |\pi /2 -\gamma |]
(3) [\omega=\sqrt{(q-r)^2+4v^2}] [\omega=\left |\arcsin {{\displaystyle \sqrt{(2ab\cos\gamma)^2+(b^2-a^2)}}\over{\displaystyle b^2-a^2}}\right|]
(4) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4v^2}] [\omega= \left|\pi/2-\psi_1- \psi_2\right|]
    [\psi _1={\rm arccotan}{{\displaystyle c^2(a^2+b^2–2d^2)-(a^2-b^2)^2+D(a^2-b^2-d^2)}\over {\displaystyle (D-b^2+d^2)\sqrt{4a^2d^2-(a^2-b^2-d^2)^2}}}]
    [\psi_2={\rm arccotan}{{\displaystyle b^2(a^2-b^2)+d^2(a^2-b^2)-D(a^2-b^2+d^2)}\over {\displaystyle (D-b^2+d^2)\sqrt{4a^2d^2-(a^2-b^2-d^2)^2}}}]
(5) [\omega=\sqrt{(q-r)^2+2t^2}] [\omega=\left|\arcsin {{\displaystyle c^2(a^2+b^2)\sin^2 \alpha -b^2(a+c\cos\alpha)(2Da+c\cos\alpha)}\over {\displaystyle \sqrt{c^2(a^2+b^2) \sin^2\alpha +b^2(a+c\cos\alpha )^2} \sqrt{ c^2(a^2+b^2)\sin^2\alpha+b^2(2Da+c\cos\alpha)^2}}}\right|]
    [D={{\displaystyle ac\cos\alpha }\over {\displaystyle b^2-a^2}}]
(6) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4t^2}] [\omega=\left|\arcsin {{\displaystyle (4a^2-b^2)\left[\left(1-{ {c\cos\beta }\over{a+b}}\right)b\cos\beta -{{c}\over{2}}\right]+{{3cb^2\sin^2\beta }\over{2}}}\over {\displaystyle \sqrt{4a^2-b^2(1–9\sin^2\beta )} \sqrt {(ac\sin\beta)^2+(4a^2-b^2)\left[1-{{c\cos\beta }\over{a+b}})b-{{c\cos\beta }\over{2}}\right]^2}}}\right| \quad(^*)]
(7) [\omega=2|t|] [\omega=|\pi /2 -\alpha |]
(8) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4t^2}] [\omega=\left|\arcsin {{\displaystyle 3b^2c-c(4a^2-b^2)\sin^2\alpha +2 b^2D\sqrt{4a^2-b^2}\cos\alpha }\over {\displaystyle \sqrt{b^2+(4a^2-b^2)\sin^2\alpha }\sqrt{ 9b^2c^2+(4a^2-b^2)(c^2\sin^2\alpha +4b^2D^2)+12b^2Dc\sqrt{4a^2-b^2}}}} \right|\quad(^*)]
    [D={{\displaystyle 2ac\cos\alpha }\over {\displaystyle b^2-a^2}} ]
(9) [\omega=2\sqrt{q^2+f^2}] [\omega=\left|\arccos {{\displaystyle \sqrt{1-\cos^2\alpha -\cos^2\beta -\cos^2\gamma +2\cos\alpha \cos\beta \cos\gamma }}\over{\displaystyle \sin\alpha }}\right|]
(10) [\omega={{\sqrt3}\over{2}} |q-r| ] [\omega=\left|\arcsin{{\displaystyle b^2-a^2}\over{\displaystyle a\sqrt{2b^2+a^2}}}\right|]
(11) [\omega=|q-r| ] [\omega=\left|\arcsin {{\displaystyle a^2-b^2}\over{\displaystyle b^2+a^2}}\right|]
(12) [\omega=\sqrt{(q-s)^2+2v^2}] [\omega=\left|\arcsin {{\displaystyle c^2(D\cos\gamma -1)-a^2\sin^2\gamma }\over {\displaystyle \sqrt{c^2+a^2 \sin^2\gamma} \sqrt{ 4c^2(D^2-D\cos\gamma )+c^2+a^2\sin^2\gamma }}}\right|]
    [D={{\displaystyle 2a^2\cos\gamma }\over {\displaystyle c^2-a^2}} ]
(13) [\omega=2|v| ] [\omega= |\pi /2 -\gamma |]
(14) [\omega=|q-s|] [\omega=\left|\arcsin {{\displaystyle a^2-c^2}\over{\displaystyle c^2+a^2}}\right|]
(15) [\omega=2\sqrt2 |v|] [\omega=\left|\arcsin {{\displaystyle \sqrt 2\cos\alpha }\over {\displaystyle 1+\cos\alpha}} \right|]

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 [\pmatrix{q&v&u\cr v&r& t\cr u&t&s\cr}]B as a function of lattice parameters a, b, c; [\alpha = \angle (b;c)], [\beta = \angle (a;c)], [\gamma = \angle (a;b)]
(a) [B={{\displaystyle t}\over{\displaystyle u}}]  
(b) [B={{\displaystyle 2v+\sqrt{(q-r)^2+4v^2}}\over{\displaystyle q-r}}] [B={{\displaystyle -2ab\cos\gamma +\sqrt{(2ab\cos\gamma)^2+(b^2-a^2)}}\over{\displaystyle b^2-a^2}}]
(c) [B={{\displaystyle (q-r)+2\sqrt{3}v+4\sqrt{(q-r)^2+4v^2}}\over{\displaystyle \sqrt3(r-q)+2v}}] [B=2{{\displaystyle a^2-c^2-\sqrt{(a^2-c^2)^2-(a^2-b^2)(b^2-c^2)}}\over{\displaystyle a^2-b^2}}-1]
(d) [B={{\displaystyle 2t}\over{\displaystyle q-r}}]  
(e) [B={{\displaystyle 4t}\over{\displaystyle r-q}}]  
(f) [B={{\displaystyle 4t}\over{\displaystyle q-r}}]  
(g) [B={{\displaystyle 4t}\over{\displaystyle r-q}}]  
(h) [B={{\displaystyle -u}\over {\displaystyle v}}]  
(k) [B={{\displaystyle 2v}\over{\displaystyle s-v}}] [B={{\displaystyle 2a^2\cos\gamma }\over {\displaystyle c^2-a^2}}]

The first three columns specify domain pairs.

  • [F_1]: point-group symmetry (stabilizer in [K_{1j}]) of the first domain state [{\bf S}_1] in a single-domain orientation.

  • [g_{1j}]: switching operations (if available) that specify the domain pair [({\bf S}_1,{\bf S}_j =g_{1j}{\bf S}_1)]. Subscripts [x,y,z] specify the orientation of the symmetry operations in the Cartesian coordinate system of [K_{1j}]. Subscripts [x^\prime, y^\prime ] and [x^{\prime\prime}, y^{\prime\prime}] 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: [p=[111]], [q=[\bar 1\bar 11] ], [r=[1\bar 1\bar 1]], [s=[\bar 1 1 \bar 1]]. Where possible, mirror planes and 180° rotations are chosen such that the two perpendicular permissible walls have crystallographic orientations.

  • [K_{1j}]: twinning group [K(F_1,g_{1j})] of the domain pair [({\bf S}_1,{\bf S}_j)]. For the pair with [F_1=m_{x\bar y}2_{xy}m_{z} ] and [K=m\bar3 m], where the twinning group does not specify the domain pair unambiguously, we add after [K_{1j}] in parentheses a switching operation [2^{\star}_{xz}] or [m^{\star}_{xz}] 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 [{\bf h}] is defined by a vector product [{\bf h}={\bf n}_1 \times {\bf n}_2 ] of normal vectors [{\bf n}_1] and [{\bf n}_2] 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 [{\bf h}] 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 [Ax_1+Bx_2+Cx_3=0] and [A{^\prime}x_1+B{^\prime}x_2+C{^\prime}x_3=0 ] [see equations ([link])] pass through the origin of the Cartesian coordinate system of [K_{1j}] and have normal vectors [{\bf n}_1=[ABC]] and [{\bf n}_2=[A{^\prime}B{^\prime}C{^\prime}]]. It is possible to find a plane with the same normal vector [[ABC]] but not passing through the origin, e.g. [Ax_1+Bx_2+Cx_3=1]. 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, [[x_1/(1/A)] + [x_2/(1/B)]+[x_3/(1/C)] =1]. In analogy with Miller indices, the symbol [(ABC)] 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 [K_{1j}]) 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 [({\bf S}_1,{\bf S}_j) ] are crystallographically equivalent in [K_{1j}] (see Fig.[link]).

    The subscript e indicates that the wall carries a nonzero polarization charge, [\hbox{Div }{\bf P}\neq0]. 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 [e0] or [0e] indicates that one of the two walls is charged and the other is not.

  • [\omega]: reference to an expression, given at the end of the table, in which the shear angle [\omega] (in radians) is given as a function of the `absolute' spontaneous strain components, defined in a matrix given above the equations.

  • [{\sf \overline J}_{1j}]: 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 [{\bf S}_1^+], [{\bf S}_j^-] and [{\bf S}_1^+], [{\bf S}_j^-]. This group can be used for designating a twin law of the ferroelastic domain twin.

  • [{\underline t}_{1j}^{\star}]: one non-trivial twinning operation of the twin [{\bf S}_1[ABC]{\bf S}_j] 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).

  • [{\sf T}_{1j}]: 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[link]. 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 [K_{1j}].

Example The rhombohedral phase of perovskite crystals.  Examples include PZN-PT and PMN-PT solid solutions (see e.g. Erhart & Cao, 2001[link]) and BaTiO3 below 183 K. The phase transition has symmetry descent [m\bar 3m\supset 3m].

In Table[link] we find that there are eight domain states and eight ferroelectric domain states. In this fully ferroelectric phase, domain states can be specified by unit vectors representing the direction of spontaneous polarization. We choose [{\bf S}_1\equiv [111]] with corresponding symmetry group [F_1=3_pm_{z\bar y}].

From eight domain states one can form [7\times8=56] domain pairs. These pairs can be divided into classes of equivalent pairs which are specified by different twinning groups. In column [K_{1j}] of Table[link] we find three twinning groups:

  • (i) The first twin law [\bar3_p^{\star}m_{x\bar y} ] characterizes a non-ferroelastic pair (Fam[\bar3_p^{\star}m_{x\bar y} =] Fam[3_p^{\star}m_{x\bar y}]) with inversion [\bar 1] as a twinning operation of this pair. A representative domain pair is [({\bf S}_1,g_{12}{\bf S}_1=] [{\bf S}_2) =] [([111],[\bar1\bar1\bar1])], domain pairs consist of two domain states with antiparallel spontaneous polarization (`180° pairs'). Domain walls of low energy are not charged, i.e. they are parallel with the spontaneous polarization.

  • (ii) The second twinning group [K_{13}=\bar4 3m ] characterizes a ferro­elastic domain pair (Fam[\bar4 3m=] [m\bar 3m] [\neq] Fam[F_1=] [\bar3_pm_{z\bar y}]). In Table[link], we find [g_{13}^{\star}=2^{\star}_x], which defines the representative pair [([111],[1\bar 1\bar 1])] (`109° pairs'). Orientations of compatible domain walls of this domain pair are [(100)] and [(011)_e] (this wall is charged). All equivalent orientations of these compatible walls will appear if all crystallographically equivalent pairs are considered.

  • (iii) The third twinning group [K_{14}=m\bar 3m ] also represents ferroelastic domain pairs with representative pair [([111],m^{\star}_x[111])=] [([111],[\bar 111]) ] (`71° pairs') and compatible wall orientations [(100)_e ] and (011). We see that for a given crystallographic orientation both charged and non-charged domain walls exist; for a given orientation the charge specifies to which class the domain wall belongs.

These conclusions are useful in deciphering the `domain-engineered structures' of these crystals (Yin & Cao, 2000[link]). Ferroelastic domain pairs with no compatible domain walls, synoptic table

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Ferroelastic domain pairs for which condition ([link]) for the existence of coherent domain walls is violated are listed in Table[link]. All these pairs are non-transposable pairs. It is expected that domain walls between ferroelastic domain states would be stressed and would contain dislocations. Dudnik & Shuvalov (1989[link]) have shown that in thin samples, where elastic stresses are reduced, `almost coherent' ferroelastic domain walls may exist.

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Ferroelastic domain pairs with no compatible domain walls

[F_1] is the symmetry of [{\bf S}_1], [g_{1j} ] is the switching operation, [K_{1j}] is the twinning group. Pair is the domain pair type, where ns is non-transposable simple and nm is non-transposable multiple (see Table[link]). [ v= z], [p=[111]], [q=[\bar 1\bar 11]], [r =[1\bar 1\bar 1]], [s=[\bar 1 1 \bar 1]] (see Table[link] and Fig.[link]).

[F_1][g_{1j} ][K_{1j} ]Pair
[1 ] [4_z ] [4_z ] ns
[1 ] [\bar4_z] [\bar4_z ] ns
[1 ] [3_v] [3_v] ns
[1 ] [\bar3_v] [\bar3_v] ns
[1 ] [6_z ] [6_z ] ns
[1 ] [\bar6_z] [\bar6_z ] ns
[\bar1 ] [4_z], [4_z^3] [4_z/m_z ] ns
[\bar1 ] [3_v], [3_v^2] [\bar3_v] ns
[\bar1 ] [6_z], [6_z^5] [6_z/m_z ] ns
[2_z ] [3_p], [3_p^2] [2_z3_p ] nm
[2_z ] [\bar3_p], [\bar3_p^5 ] [m_z\bar3_p] nm
[2_{xy} ] [3_p], [3_p^2] [4_z3_p2_{xy} ] nm
[2_{xy} ] [\bar3_p], [\bar3_p^5 ] [m_z\bar3_pm_{xy}] nm
[m_z ] [3_p], [3_p^2] [m_z3_p^2 ] nm
[m_{xy} ] [3_p], [3_p^2] [\bar4_z3_pm_{xy} ] nm
[m_{xy} ] [4_x], [4_x^3] [m_z\bar3_pm_{xy}] nm
[2_z/m_z ] [3_p], [3_p^2] [m_z\bar3_p] nm
[2_{xy}/m_{xy} ] [3_p], [3_p^2] [m_z\bar3_pm_{xy}] nm
[2_x2_y2_z ] [3_p], [3_p^2] [2_z3_p ] ns
[2_x2_y2_z ] [\bar3_p], [\bar3_p^5 ] [m_z\bar3_p ] ns
[m_xm_y2_z ] [3_p], [3_p^2 ] [m_z\bar3_p ] nm
[m_xm_ym_z ] [3_p], [3_p^2 ] [m_z\bar3_p ] ns

Example Ferroelastic crystal of langbeinite.  Langbeinite K2Mg2(SO4)3 undergoes a phase transition with symmetry descent [23\supset 222] that appears in Table[link]. The ferroelastic phase has three ferroelastic domain states. Dudnik & Shuvalov (1989[link]) found, in accord with their theoretical predictions, nearly linear `almost coherent' domain walls accompanied by elastic stresses in crystals thinner than 0.5 mm. In thicker crystals, elastic stresses became so large that crystals were cracking and no domain walls were observed.

Similar effects were reported by the same authors for the partial ferroelastic phase of CH3NH3Al(SO4)2·12H2O (MASD) with symmetry descent [\bar 3m\supset mmm], where ferroelastic domain walls were detected only in thin samples.


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