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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 517-518

Section Equally deformed planes of a ferroelastic domain pair

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: 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.


Fousek, J. (1971). Permissible domain walls in ferroelectric species. Czech. J. Phys. B, 21, 955–968.
Fousek, J. & Janovec, V. (1969). The orientation of domain walls in twinned ferroelectric crystals. J. Appl. Phys. 40, 135–142.
Nye, J. F. (1985). Physical Properties of Crystals. Oxford: Clarendon Press.
Sapriel, J. (1975). Domain-wall orientations in ferroelastics. Phys. Rev. B, 12, 5128–5140.

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