Equitranslational phase transitions and their order parameters

Janovec, V.; Kopský, V.

doi:10.1107/97809553602060000642

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

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International Tables for Crystallography (2006). Vol. D. ch. 3.1, p. 350

Section 3.1.3.1. Equitranslational phase transitions and their order parameters

V. Janovec^b ^* and V. Kopský^e

3.1.3.1. Equitranslational phase transitions and their order parameters

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A basic role is played in symmetry considerations by the relation between the space group $[\cal G]$ of the high symmetry parent or prototype phase, the space group $[\cal F]$ of the low-symmetry ferroic phase and the order parameter $[\eta]$ : The low-symmetry group $[\cal F]$ consists of all operations of the high-symmetry group $[\cal G]$ that leave the order parameter $[\eta]$ invariant. By the term order parameter we mean the primary order parameter , i.e. that set of degrees of freedom whose coefficient of the quadratic invariant changes sign at the phase-transition temperature (see Sections 3.1.2.2.4 and 3.1.2.4.2).

What matters in these considerations is not the physical nature of $[\eta]$ but the transformation properties of $[\eta]$ , which are expressed by the representation $[\Gamma_{\eta}]$ of $[\cal G]$ . The order parameter $[\eta]$ with $[d_{\eta}]$ components can be treated as a vector in a $[d_{\eta}]$ -dimensional carrier space $[V_{\eta}]$ of the representation $[\Gamma_{\eta}]$ , and the low-symmetry group $[\cal F]$ comprises all operations of $[\cal G]$ that do not change this vector. If $[\Gamma_{\eta}]$ is a real one-dimensional representation, then the low-symmetry group $[{\cal F}]$ consists of those operations $[g \in {\cal G}]$ for which the matrices $[D^{(\eta)}(g)]$ [or characters $[\chi_{\eta}(g)]$ ] of the representation $[\Gamma_{\eta}]$ equal one, $[D^{(\eta)}(g)=\chi_{\eta}(g)=1]$ . This condition is satisfied by one half of all operations of $[\cal G]$ (index of $[\cal F]$ in $[\cal G]$ is two) and thus the real one-dimensional representation $[\Gamma_{\eta}]$ determines the ferroic group $[\cal F]$ unambiguously.

A real multidimensional representation $[\Gamma_{\eta}]$ can induce several low-symmetry groups. A general vector of the carrier space $[V_{\eta}]$ of $[\Gamma_{\eta}]$ is invariant under all operations of a group $[\hbox{Ker} \ \Gamma_{\eta}]$ , called the kernel of representation $[\Gamma_{\eta}]$ , which is a normal subgroup of $[\cal G]$ comprising all operations $[g\in {\cal G}]$ for which the matrix $[D^{(\eta)}(g)]$ is the unit matrix. Besides that, special vectors of $[V_{\eta}]$ – specified by relations restricting values of order-parameter components (e.g. some components of $[\eta]$ equal zero, some components are equal etc.) – may be invariant under larger groups than the kernel $[\hbox{Ker}\ \Gamma_{\eta}]$ . These groups are called epikernels of $[\Gamma_{\eta}]$ (Ascher & Kobayashi, 1977). The kernel and epikernels of $[\Gamma_{\eta}]$ represent potential symmetries of the ferroic phases associated with the representation $[\Gamma_{\eta}]$ . Thermodynamic considerations can decide which of these phases is stable at a given temperature and external fields.

Another fundamental result of the Landau theory is that components of the order parameter of all continuous (second-order) and some discontinuous (first-order) phase transitions transform according to an irreducible representation of the space group $[\cal G]$ of the high-symmetry phase (see Sections 3.1.2.4.2 and 3.1.2.3). Since the components of the order parameter are real numbers, this condition requires irreducibility over the field of real numbers (so-called physical irreducibility or R-irreducibility). This means that the matrices $[D^{(\eta)}(g)]$ of R-irreducible representations (abbreviated R-ireps) can contain only real numbers. (Physically irreducible matrix representations are denoted by $[D^{(\alpha)}]$ instead of the symbol $[\Gamma_{\alpha}]$ used in general considerations.)

As explained in Section 1.2.3 and illustrated by the example of gadolinium molybdate in Section 3.1.2.5, an irreducible representation $[\Gamma_{{\bf k},m}]$ of a space group is specified by a vector $[\bf k]$ of the first Brillouin zone, and by an irreducible representation $[\tau_m({\bf k})]$ of the little group of k, denoted $[G({\bf k})]$ . It turns out that the vector k determines the change of the translational symmetry at the phase transition (see e.g Tolédano & Tolédano, 1987; Izyumov & Syromiatnikov, 1990; Tolédano & Dmitriev, 1996). Thus, unless one restricts the choice of the vector $[\bf k]$ , one would have an infinite number of phase transitions with different changes of the translational symmetry.

In this section, we restrict ourselves to representations with zero $[\bf k]$ vector (this situation is conveniently denoted as the $[\Gamma]$ point). Then there is no change of translational symmetry at the transition. In this case, the group $[\cal F]$ is called an equitranslational or translationengleiche (t) subgroup of $[\cal G]$ , and this change of symmetry will be called an equitranslational symmetry descent $[{\cal G} \Downarrow^t {\cal F}]$ . An equitranslational phase transition is a transition with an equitranslational symmetry descent $[{\cal G}\Downarrow^t{\cal F}]$ .

Any ferroic space-group-symmetry descent $[{\cal G} \Downarrow {\cal F}]$ uniquely defines the corresponding symmetry descent $[G \Downarrow F]$ , where G and F are the point groups of the space groups $[\cal G]$ and $[\cal F]$ , respectively. Conversely, the equitranslational subgroup $[\cal F]$ of a given space group $[\cal G]$ is uniquely determined by the point-group symmetry descent $[G \Downarrow F]$ , where G and F are point groups of space groups $[\cal G]$ and $[\cal F]$ , respectively. In other words, a point-group symmetry descent $[G \Downarrow F]$ defines the set of all equitranslational space-group symmetry descents $[{\cal G} \Downarrow^t {\cal F}]$ , where $[\cal G]$ runs through all space groups with the point group G. All equitranslational space-group symmetry descents $[{\cal G} \Downarrow^t {\cal F}]$ are available in the software GI $[\star]$ KoBo-1, where more details about the equitranslational subgroups can also be found.

Irreducible and reducible representations of the parent point group G are related in a similar way to irreducible representations with vector $[{\bf k}=\bf{0}]$ for all space groups $[\cal G]$ with the point group G by a simple process called engendering (Jansen & Boon, 1967). The translation subgroup $[{\bf T}_{G}]$ of $[\cal G]$ is a normal subgroup and the point group G is isomorphic to a factor group $[{\cal G}/{\bf T}_{G}]$ . This means that to every element $[g \in G]$ there correspond all elements $[\{g|{\bf t}+{\bf u}_{G}(g)\}]$ of the space group $[\cal G]$ with the same linear constituent g, the same non-primitive translation $[{\bf u}_{G}(g)]$ and any vector $[\bf{t}]$ of the translation group $[{\bf T}_{G}]$ (see Section 1.2.3.1 ). If a representation of the point group G is given by matrices [D(g)] , then the corresponding engendered representation of a space group $[\cal G]$ with vector $[{\bf k}={\bf 0}]$ assigns the same matrix [D(g)] to all elements $[\{g|{\bf t}+{\bf u}_{G}(g)\}]$ of $[\cal G]$ .

From this it further follows that a representation $[\Gamma_{\eta}]$ of a point group G describes transformation properties of the primary order parameter for all equitranslational phase transitions with point-symmetry descent $[G \Downarrow F]$ . This result is utilized in the presentation of Table 3.1.3.1.

References

Ascher, E. & Kobayashi, J. (1977). Symmetry and phase transitions: the inverse Landau problem. J. Phys. C: Solid State Phys. 10, 1349–1363.Google Scholar

Izyumov, Yu. A. & Syromiatnikov, V. N. (1990). Phase transitions and crystal symmetry. Dordrecht: Kluwer Academic Publishers.Google Scholar

Jansen, L. & Boon, M. (1967). Theory of finite groups. Applications in physics. Symmetry groups of quantum mechanical systems. Amsterdam: North-Holland.Google Scholar

Tolédano, J.-C. & Tolédano, P. (1987). The Landau theory of phase transitions. Singapore: World Scientific.Google Scholar

Tolédano, P. & Dmitriev, V. (1996). Reconstructive phase transitions. Singapore: World Scientific.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.1, p. 350