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. 516-517

Section 3.4.3.6.1. Spontaneous strain

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

3.4.3.6.1. 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. 3.4.3.4[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. 3.4.3.4[link]). This is a preferred approach.

    [Figure 3.4.3.4]

    Figure 3.4.3.4 | top | pdf |

    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. 3.4.3.4[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(3.4.3.44) ]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(3.4.3.45) ]

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 3.4.3.6.  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. 3.4.2.2[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(3.4.3.46) ]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 3.4.2.1[link]), [{\bf u}^{(2)}= \left(\matrix{u_{22} &0 &0 \cr 0 &u_{11} &0 \cr 0 & 0 &u_{33}}\right).\eqno(3.4.3.47) ]

The average spontaneous strain is, according to equation (3.4.3.45[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(3.4.3.48) ]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 (3.4.3.44[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 &&(3.4.3.49)\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 &&(3.4.3.50)}%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).

References

Aizu, K. (1970b). Determination of the state parameters and formulation of spontaneous strain for ferroelastics. J. Phys. Soc. Jpn, 28, 706–716.
Carpenter, M. A., Salje, E. K. H. & Graeme-Barber, A. (1998). Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals. Eur. J. Mineral. 10, 621–691.
Salje, E. K. H. (1990). Phase Transitions in Ferroelastic and Co-elastic Crystals, 1st ed. Cambridge University Press.
Schlenker, J. L., Gibbs, G. V. & Boisen, M. B. (1978). Strain-tensor components expressed in terms of lattice parameters. Acta Cryst. A34, 52–54.
Wadhawan, V. K. (2000). Introduction to Ferroic Materials. The Netherlands: Gordon and Breach.








































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