Domain structures of a general phase transition

Wondratschek, H.

doi:10.1107/97809553602060000538

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 1.2, pp. 21-22 | 1 | 2 |

Section 1.2.7.4. Domain structures of a general phase transition

Hans Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

1.2.7.4. Domain structures of a general phase transition

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Up to now, the examples have been concerned either with translationengleiche or with klassengleiche transitions only. In this section, the domain structure of a general transition will be considered, i.e. a transition where $[{\cal H}]$ is a general subgroup of $[{\cal G}]$ . General subgroups are not listed in this volume but have to be derived from the maximal subgroups of each single step of the group–subgroup chain between $[{\cal G}]$ and $[{\cal H}]$ . In the following Example 1.2.7.4.2, the chain has two steps. The results obtained under the PCA and without it are different and, therefore, will be discussed in some detail. Example 1.2.7.4.2 further shows how the subgroup data of this volume can be used for the analysis of continuous or quasi-continuous phase transitions.

We start with a lemma for general subgroups which contains the results of Lemmata 1.2.7.3.3 and 1.2.7.3.4.

Lemma 1.2.7.4.1. For general subgroups, owing to the existence of the group $[{\cal M}]$ of Hermann, it always holds that $[|{\cal G}\,:\,{\cal H}| = ]$ $[|{\cal G}\,:\,{\cal M}|\cdot |{\cal M}\,:\,{\cal H}| = |{\cal P}_{{\cal G}}\,:\,{\cal P}_{{\cal H}}|\cdot |{\cal T}({\cal G}):{\cal T}({\cal H})|= i_P\cdot i_T]$ . Here [i_P] is the index of the point groups of $[{\cal G}]$ and $[{\cal H}]$ and [i_T] is the index of the translation subgroups of $[{\cal G}]$ and $[{\cal H}]$ .

Example 1.2.7.4.2.

β-Gadolinium molybdate, Gd₂(MoO₄)₃, is ferroelectric and ferroelastic. The high-temperature phase A has space group $[{\cal G}=P\overline{4}2_1m]$ , No. 113, and basis vectors a, b and c. At $[T_C\sim 433]$ K, a phase transition to a low-temperature phase B occurs with space-group type $[{\cal H}=Pba2]$ , No. 32, basis vectors $[{\bf a}'={\bf a}-{\bf b}]$ , $[{\bf b}'={\bf a}+{\bf b}]$ and $[{\bf c}'={\bf c}]$ . The group $[{\cal M}]$ , $[{\cal G}>{\cal M}>{\cal H}]$ is of the type [Cmm2] with the lattice parameters of $[{\cal H}]$ . The index of $[{\cal H}]$ in $[{\cal G}]$ is $[i=|P\overline{4}2_1m:Pba2|=4]$ . A factor of 2 stems from the reduction $[i_P=|{\cal G}:{\cal M}|=2]$ and leads to two orientation states. The other factor of 2 is caused by the loss of half of the translations, because $[i_T=|{\cal M}:{\cal H}|=2]$ .

In the continuum description, we consider the point groups and $[|\overline{4}2m:mm2|=2]$ . There is only one subgroup of $[\overline{4}2m]$ of the type [mm2] . Thus, the two orientation states belong to the same point group. The orientation state $[{\bf B}_2]$ is obtained from $[{\bf B}_1]$ by the (lost) $[\overline{4}]$ operation of $[\overline{4}2m]$ .

This is the macroscopic or continuum treatment; it is the most common treatment of domains in phase transitions. In reality, i.e. lifting the PCA, due to the orthorhombic symmetry of phase B the domains will be slightly distorted and rotated, and thus the symmetry planes of the two domain states are no longer parallel.

The full microscopic or atomistic treatment has to consider the crystal structures of phases A and B. Under the PCA, the length of [a'] and [b'] is $[(2)^{1/2}a]$ , the content of the unit cell of $[{\bf B}_1]$ is twice that of A. Because the index $[[\,i\,]=4]$ there are four domain states $[{\bf B}_1]$ to $[{\bf B}_4]$ of [Pba2] . The domain state $[{\bf B}_2]$ is obtained from $[{\bf B}_1]$ by the (lost) $[\overline{4}]$ operation of $[P\overline{4}2_1m]$ . The same holds for the pair $[{\bf B}_3]$ and $[{\bf B}_4]$ . Thus, $[{\bf B}_2]$ & $[{\bf B}_4]$ are rotated by 90° around a $[\overline{4}]$ centre in the $[(a'\ b')]$ plane with respect to the pair $[{\bf B}_1]$ & $[{\bf B}_3]$ , and the [c'] axes are antiparallel for $[{\bf B}_2]$ & $[{\bf B}_4]$ relative to those of $[{\bf B}_1]$ & $[{\bf B}_3]$ . The orientation state of the pair $[{\bf B}_1]$ & $[{\bf B}_3]$ is different from that of $[{\bf B}_2]$ & $[{\bf B}_4]$ . The two pairs $[{\bf B}_1]$ & $[{\bf B}_2]$ and $[{\bf B}_3]$ & $[{\bf B}_4]$ are shifted relative to each other by a (lost) translation of $[P\overline{4}2_1m]$ , e.g. by $[{\sf t} (1,0,0)]$ in the basis of $[P\overline{4}2_1m]$ , corresponding to $[{\sf t} ({{1}\over{2}}, {{1}\over{2}}, 0)]$ in the basis of [Pba2] .

To calculate the number of space groups [Pba2] , i.e. the number of symmetry states, one determines the normalizer of [Pba2] in $[P\overline{4}2_1m]$ . From IT A, Table 15.2.1.3 , one finds $[{\cal N}_{{\cal E}}(Pba2)=P^14/mmm]$ for the Euclidean normalizer of [Pba2] under the PCA, which includes the condition [a = b] . [P^14/mmm] is a supergroup of $[P\overline{4}2_1m]$ . Thus, $[{\cal N}_{{\cal G}}(Pba2)=({\cal N}_{{\cal E}}(Pba2)\cap{\cal G})={\cal G}]$ and $[|{\cal G}:{\cal N}_{{\cal G}}(Pba2)|=|{\cal G}:{\cal G}|=1]$ . Therefore, under the PCA all four domain states belong to one symmetry state, i.e. to one space group [Pba2] .

Analysing the group–subgroup relations between $[P\overline{4}2_1m]$ and [Pba2] with the tables of this volume, one finds only one chain $[P\overline{4}2_1m\rightarrow Cmm2\rightarrow Pba2]$ . For $[P\overline{4}2_1m]$ only one maximal subgroup of type [Cmm2] is listed, for which again only one maximal subgroup of type [Pba2] is found, in agreement with the previous paragraph.

In reality, i.e. relaxing the PCA, the observations are made at temperatures $[T_x \,\lt\, T_C]$ where the lattice parameters deviate from those of phase A and the basis no longer has tetragonal symmetry, but orthorhombic symmetry, $[a' \,\lt\, b']$ . The previous single space group now splits into two different space groups of type [Pba2] with orthorhombic metrics at [T_x] , one belonging to the pair $[{\bf B}_1]$ & $[{\bf B}_3]$ , the other to $[{\bf B}_2]$ & $[{\bf B}_4]$ . The ( $[{\bf a}']$ , $[{\bf b}']$ ) bases of these pairs are oriented perpendicular to each other and the $[{\bf c}']$ axes of their domains are antiparallel. The loss of the centring translation of [Cmm2] does not produce a new space group.

The number, two, of these space groups if the PCA is not valid can also be calculated in the usual way with the help of the normalizer. The Euclidean normalizer of [Pba2] with $[a'\ne b']$ is $[{\cal N}_{\cal E}(Pba2)=P^1mmm]$ . This is an orthorhombic group with continuous translations along the $[{\bf c}']$ direction. [P^1mmm] with $[a'\ne b']$ is not really a subgroup of $[P\overline{4}2_1m]$ because the translations of [Pba2] and thus of [Cmm2] and are not strictly translations of $[P\overline{4}2_1m]$ . The first three groups have orthorhombic lattices and the last a tetragonal one. However, by relaxing the PCA only gradually, the difference between the orthorhombic groups and the corresponding groups with tetragonal lattices is arbitrarily small. Therefore, one considers the sequence $[{\cal G}>{\cal M}={\cal N}_{{\cal G}}({\cal H})>{\cal H}]$ , i.e. $[P\overline{4}2_1m>Cmm2>Pba2]$ as a group–subgroup chain, forms the intersection $[(P^1mmm\cap P\overline{4}2_1m)]$ as if the groups would have common translations, and finds $[{\cal N}_{{\cal G}}({\cal H})=Cmm2]$ with approximately the lattice parameters of $[P\overline{4}2_1m]$ . The index $[|P\overline{4}2_1m:Cmm2|=2]$ , such that there are two space groups of type [Pba2] which are approximately subgroups of $[P\overline{4}2_1m]$ . To each of these space groups [Pba2] belong two domain states of phase B, see above.

This example shows that without the PCA, in order to cope with real observations, the terms `subgroup', `intersection of groups' etc. must not be used sensu stricto but have to be relaxed. The orthorhombic translations in this example are not group elements of $[{\cal G}]$ but are slightly modified from the original translations of $[{\cal G}]$ . All group–subgroup relations in crystal chemistry, e.g. diamond (C)–sphalerite (ZnS), as well as many phase transitions, as in this example, require such a `softened' approach.

It turns out that the transition of Gd₂(MoO₄)₃ can be considered both under the PCA (allowing exact group-theoretical arguments) and under physically realistic arguments (which require certain relaxations of the group-theoretical methods). The results are different but the realistic approach can be developed by means of an increasing deviation from the PCA, starting from idealized but unrealistic considerations.

References

International Tables for Crystallography (2006). Vol. A1. ch. 1.2, pp. 21-22