Chapter 3.1. Structural phase transitions

In the study of structural phase transitions, the crystallographer is often confronted by an ambiguous situation. Small changes in atomic positions determine structures having different space groups, and the data are generally compatible with several possible symmetry assignments. In order to make a choice, the crystallographer must be able to rely on some theoretical substrate, which will allow him to discard certain of the possible assignments.

The relevant theoretical framework in this field is the thermodynamical and symmetry considerations that form the Landau theory of phase transitions. In this chapter, we describe the ideas and results of this theory.

In the next section, we give an introduction to the main ideas of the theory by using an example consisting of a simple speculative type of structural phase transition. In Section 3.1.2.3, we discuss various situations of experimental interest relative to the thermodynamical aspect of the theory: first and second order of the transition, metastable states and thermal hysteresis. In Section 3.1.2.4, we provide a brief description, in two steps, of the general arguments constituting the foundation of the theory. In Section 3.1.2.5, we discuss the case of a structural transition actually occurring in nature and having a greater complexity than the speculative case considered in Section 3.1.2.2. In this section we also analyse the relationship between the ferroic character of a transition (see Section 3.1.1) and its order-parameter symmetry.

The Landau theory of phase transitions is a phenomenological theory. It does not aim to establish that a phase transition exists in a given system. The existence of a transition is an experimental fact considered as a starting point of the theory. The explanatory power of the theory is to establish the overall consistency of the microscopic characteristics of the transition (space symmetry and structural changes, anomalies in the phonon spectrum etc.) and the results of the measurement of various relevant macroscopic quantities of thermal, dielectric, optical or mechanical nature.

The continuous (`second-order') character of the transition plays an essential role in working out the general foundations of the theory. However, though its strict field of validity is that of continuous transitions, the theory also satisfactorily applies to a large fraction of discontinuous transitions.

The Landau theory defines two basic concepts: the order parameter (OP) and the transition free energy (LFE). It is worth pointing out that these concepts keep their usefulness in the modern statistical theory of critical phenomena, even though these phenomena do not generally comply with the results of Landau's theory. From the symmetry properties of the order parameter and of the Landau free energy, it is possible to infer, on the one hand, a certain number of observable symmetry characteristics of the system: degeneracy of the `low-symmetry' phase (i.e. number of energetically equivalent domain orientations in this phase), enumeration of the possible symmetries of the `low-symmetry' phase for a given symmetry of the `high-symmetry' phase. On the other hand, macroscopic physical quantities can be classified as functions of their symmetries with respect to the order parameter. This classification leads to that of the various types of anomalous behaviours that can be induced by the occurrence of a phase transition.

In order to give an intuitive approach to the basic arguments of the Landau theory, and to its use, we first analyse an artificially simplified example of a crystalline phase transition.

3.1.2.2.1. Description of a prototype example

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Fig. 3.1.2.1 represents a unit cell of a speculative crystalline structure with a simple tetragonal Bravais lattice, in which a phase transition is assumed to take place. Negative ions (filled circles) occupy the vertices of the tetragonal cell (lattice constants ). A positive ion is at the centre of the cell.

Figure 3.1.2.1 | top | pdf | Model of a structural transition. The filled circles at the vertices of the cell are singly charged negative ions and the empty circle at the centre is a singly charged positive ion. d is an arbitrary displacement of the central ion.

This configuration is assumed to be the equilibrium state of the system above the temperature of the transition (see Fig. 3.1.2.2). Below , equilibrium is assumed to correspond to a structure that only differs from the high-temperature structure by the fact that lies out of the centre of the cell in an unspecified direction. Hence the latter equilibrium is characterized by the magnitude and direction of the displacement of the central ion. At high temperature, the equilibrium corresponds to .

Figure 3.1.2.2 | top | pdf | (a) Variation of the free energy as function of the amplitude of the displacement of the central ion in Fig. 3.1.2.1. (b) Typical temperature dependence in the vicinity of of the coefficient of a second-degree term in the Landau expansion (3.1.2.1) whenever this coefficient is strictly positive at : one can see that this positivity is also valid slightly above and below .

3.1.2.2.5. Stable state below T_c and physical anomalies induced by the transition

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We have seen that either or the couple () of components of the displacement constitute the order parameter of the transition and that the free energy needs only to be expanded as a function of the components of the order parameter. Below the transition, the corresponding coefficient is negative and, accordingly, the free energy, limited to its second-degree terms, has a maximum for and no minimum. Such a truncated expansion is not sufficient to determine the equilibrium state of the system. The stable state of the system must be determined by positive terms of higher degrees. Let us examine first the simplest case, for which the order parameter coincides with the component.

The same symmetry argument used to establish the form (3.1.2.1) of the Landau free energy allows one straightforwardly to assert the absence of a third-degree term in the expansion of F as a function of the order parameter , and to check the effective occurrence of a fourth-degree term. If we assume that this simplest form of expansion is sufficient to determine the equilibrium state of the system, the coefficient of the fourth-degree term must be positive in the neighbourhood of . Up to the latter degree, the form of the relevant contributions to the free energy is therefore

In this expression, , which is an odd function of since it vanishes and changes sign at , has been expanded linearly. Likewise, the lowest-degree expansion of the function is a positive constant in the vicinity of . The function , which is the zeroth-degree term in the expansion, represents the normal `background' part of the free energy. It behaves smoothly since it does not depend on the order parameter. A plot of for three characteristic temperatures is shown in Fig. 3.1.2.4.

Figure 3.1.2.4 | top | pdf | Plots of the Landau free energy as a function of the order parameter, for values of the temperature above or below or coincident with . The shape of the plot changes qualitatively from a one-minimum plot to a two-minimum plot.

The minima of F, determined by the set of conditions occur above for , as expected. For they occur for

This behaviour has a general validity: the order parameter of a transition is expected, in the framework of Landau's theory, to possess a square-root dependence as a function of the deviation of the temperature from .

Note that one finds two minima corresponding to the same value of the free energy and opposite values of . The corresponding upward and downward displacements of the ion (Fig. 3.1.2.1) are distinct states of the system possessing the same stability.

Other physical consequences of the form (3.1.2.2) of the free energy can be drawn: absence of latent heat associated with the crossing of the transition, anomalous behaviour of the specific heat, anomalous behaviour of the dielectric susceptibility related to the order parameter.

The latent heat is , where is the difference in entropy between the two phases at . We can derive S in each phase from the equilibrium free energy using the expressionHowever, since F is a minimum for , the second contribution vanishes. Hence

Since both and () are continuous at , there is no entropy jump , and no latent heat at the transition.

Several values of the specific heat can be considered for a system, depending on the quantity that is maintained constant. In the above example, the displacement d of a positive ion determines the occurrence of an electric dipole (or of a macroscopic polarization P). The quantity , which is thermodynamically conjugated to , is therefore proportional to an electric field (the conjugation between quantities and is expressed by the fact that infinitesimal work on the system has the form – cf. Sections 1.1.1.4 and 1.1.5 ). Let us show that the specific heat at constant electric field has a specific type of anomaly.

This specific heat is expressed by Using (3.1.2.6), we find Hence above and below the specific heat is a different, smoothly varying function of temperature, determined by the background free energy and by the smooth variation of the coefficient. Fig. 3.1.2.5(a) reproduces the anomaly of the specific heat, which, on cooling through , has the form of an upward step.

Figure 3.1.2.5 | top | pdf | (a) Qualitative temperature dependence of the specific heat at a continuous transition. (b) Temperature dependence of the susceptibility at a continuous transition.

Finally, let us consider the anomaly of the susceptibility , which, in the case considered, is proportional to the dielectric susceptibility of the material. It is defined as

In order to calculate , it is necessary to examine the behaviour of the system in the presence of a small field, , conjugated to the order parameter. In this case, the appropriate thermodynamical potential whose minimum determines the equilibrium of the system is not F but . Minimizing G with respect to leads to

For small values of , the solution of this equation must tend towards the equilibrium values . Deriving these solutions with respect to , we obtainThe susceptibility goes to infinity when from either side of the transition (Fig. 3.1.2.5b). The set of anomalies in and described in this paragraph represents the basic effects of temperature on quantities that are affected by a phase transition. They constitute the `canonical signature' of a phase transition of the continuous type.

Certain complications arise in the cases where the transition is not strictly continuous, where the order parameter is coupled to other degrees of freedom, and where the order parameter is not one-dimensional. We consider one of these complications in Section 3.1.2.3.

3.1.2.2.6. Symmetry considerations

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3.1.2.2.6.1. Order-parameter symmetry

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Up to now, we have defined the order parameter as a set of degrees of freedom determining a second-degree contribution to the free energy, the coefficient of which has a specific temperature dependence proportional to . Actually, the order parameter can also be defined on the basis of its specific symmetry characteristics.

Let us consider the manner by which the components (, , ) transform when we apply to the crystal each of the 16 symmetry operations of the group . Table 3.1.2.1 specifies the results of these transformations.

Table 3.1.2.1 | top | pdf | Transformation of the components of under the symmetry operations of group G E G I

In the first place, we note that is transformed either into itself or into (). If we consider this coordinate as the basis vector of a one-dimensional vector space, we can conclude that this vector space (i.e. the space formed by the set of vectors that are linear combinations of the basis) is invariant by all the transformations of the group G. Such a space, containing obviously no space of smaller dimension, is, according to the definitions given in Chapter 1.2 , a one-dimensional irreducible invariant space with respect to the group G.

Each of the components () is not transformed into a proportional component by all the elements of G. Certain of these elements transform into , and conversely. Hence and are not, separately, bases for one-dimensional irreducible invariant spaces. However, their set generates a two-dimensional vector space that has the property to be invariant and irreducible by all the transformations of G.

Note that the set of the three components (, , ) carries a three-dimensional vector space which, obviously, has the property to be invariant by all the transformations of G. However, this vector space contains the two invariant spaces carried respectively by and by (). Hence it is not irreducible.

In conclusion, from a symmetry standpoint, the order parameter of a phase transition is a set of degrees of freedom that carries an irreducible vector space (an irreducible representation) with respect to the action of the group G, the latter group being the symmetry group of the high-symmetry phase.

3.1.2.2.6.2. Degeneracy of the low-symmetry phase

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We had noted above that the structure is invariant by G in the stable state of the system above . When , the structure becomes invariant by a smaller set of transformations. Let us enumerate these transformations for each possible stable state of the system below .

When the order parameter coincides with , we determined, below , two stable states, . The crystalline structures determined by these displacements of the ion parallel to the z axis are both invariant by the same set of eight symmetry transformations. These comprise the cyclic group of order 4 generated by the fourfold rotation around z, and by the reflections in planes containing this axis. This set is the group , a subgroup F of G. The transition is thus accompanied by a lowering of the symmetry of the system.

Also note that the two states are transformed into each other by certain of the symmetry operations such as the mirror symmetry `lost' below . These two states correspond to the same value of the free energy [the minimum value determined in equation (3.1.2.3)]: they are equally stable. This can also be checked by applying to the system the mirror symmetry . This transformation keeps unchanged the value of F since the free energy is invariant by all the transformations belonging to G (to which belongs). The state is, however, not preserved, and is transformed into ().

We have not determined explicitly the stable states of the system in the case of a two-dimensional order parameter (). A simple discussion along the line developed for the one-dimensional order parameter would show that the relevant form of the free energy is and that the possible stable states below are:

(i) , ; (ii) , ; (iii) and (iv) .

Like the case of , there is a lowering of the crystal symmetry below . In the four cases, one finds that the respective symmetry groups of the structure are (i) ; (ii) ; (iii) ; (iv) .

States (i) and (ii) correspond to each other through one of the `lost' transformations of G (the rotations by ). They therefore possess the same free energy and stability. The second set of states (iii) and (iv) also constitute, for the same reason, a pair of states with the same value of the equilibrium free energy.

Note that the symmetry groups associated with equally stable states are conjugate relative to G, that is they satisfy the relationship , with g belonging to G.

Expression (3.1.2.2) for the free energy, discussed in the preceding section, only contains terms of even degrees as a function of the order parameter. We have stressed that this property derives from symmetry considerations. Let us provisionally ignore the symmetry constraints and assume that the phase transition in a given system is described by a free energy containing a term of degree three as a function of the order parameter. The stable state of this system at each temperature is determined by the minimum of F. The extrema of this function are provided by the solutions of which are

A straightforward analysis based on (3.1.2.13) and (3.1.2.15) shows that, depending on the range of temperatures, the free energy has one of the forms schematically represented in Fig. 3.1.2.6(a). This form changes at three characteristic temperatures.

The remarkable physical consequences of this sequence of shapes are the following.

In the first place, it appears that the equilibrium value of the order parameter changes discontinuously at . The free energy (3.1.2.13) therefore provides us with a model of discontinuous phase transitions. Referring to equation (3.1.2.6), we can see that a discontinuity of the order parameter is necessarily associated with a nonzero latent heat (or entropy) for the transition. More precisely, the downward jump experienced, on heating, by the equilibrium value of the order parameter corresponds to an endothermal transition. Such a transition is also termed (following Ehrenfest's classification) a first-order transition, since the entropy, which is a first derivative of the free energy, is discontinuous.

On the other hand, the occurrence of metastable states, in certain temperature ranges, generates thermal hysteresis. Indeed, on cooling from above , the system is likely to remain in the state down to the temperature , even though between and this state is not the stable state of the system. Conversely, on heating, the system will remain in a state up to , even though this state does not constitute the stable state of the system between and . Hence, the variations of the order parameter will schematically vary as in Fig. 3.1.2.6(b), the temperature dependence below the discontinuity being determined by equation (3.1.2.15). Likewise, the susceptibility will vary as in Fig. 3.1.2.6(c).

The form (3.1.2.13) of the free energy is not the only model form for discontinuous transitions. Another canonical form iswhere , and are positive coefficients. The negative coefficient of the fourth-degree term has the effect of introducing more than one minimum in a certain temperature range. Fig. 3.1.2.7 shows the different shapes of the plot of over different temperature ranges. The situation is similar to the one already discussed in the presence of a third-degree contribution to the free energy. It corresponds to a discontinuous transition associated with a latent heat as well as to the existence of a range of thermal hysteresis.

Figure 3.1.2.7 | top | pdf | Plots of the free energy as a function of the order parameter for various temperatures in the framework of the model of a discontinuous transition associated with equation (3.1.2.16). The temperature decreases from right to left, the transition being reached for the temperature corresponding to the third plot.

Two relevant questions arise from consideration of the above models of first-order transitions.

In the first place, one can object to the use of a polynomial expansion of the free energy in cases involving a discontinuity of the order parameter while the assumption of continuity of the phase transition has been used as an essential substrate of the argument developed. However, the approach will clearly keep its validity if a transition, though discontinuous, involves `small' discontinuties. The criterion for estimating if a discontinuity is small relies on the comparison between the atomic displacements involved by the transition and the distances between atoms in the structure. If the displacements are a small fraction of the distances between atoms, then the method used can be considered as valid. Indeed, the total free energy of the system depends on the distance between atoms, because this distance controls the strength of the interaction energy within the system. Hence, the transition only changes in a minor way the value of the system's free energy.

On the other hand, one has to check that there are systems of physical interest for which the crystallographic symmetry allows free-energy forms of the type (3.1.2.13), (3.1.2.16). Indeed, the crystallographic symmetry relative to the example in Section 3.1.2.2 was such that the presence of a third-degree term in the Landau free energy was excluded.

Such verification is not necessary for the free energy of type (3.1.2.16). This free energy is only characterized by a specific sign of the coefficient of the fourth-degree term, a circumstance that is not defined by symmetry considerations.

By contrast, an actual crystallographic model of a transition described by (3.1.2.13), which involves a term of degree three, is required to support the relevance of the corresponding model. Such a model is provided, for instance, by a crystal the high-temperature phase of which has a rhombohedral symmetry (e.g. ), and which undergoes a transition corresponding to an atomic displacement d perpendicular to the ternary axis (Fig. 3.1.2.8).

Figure 3.1.2.8 | top | pdf | Schematic representation of the displacement associated with the order parameter in a crystal having trigonal (rhombohedral) symmetry.

If we refer the components to a rectangular frame of coordinates, the matrices representing the mode of transformation of the components () under application of the generating elements of the group have a form reproduced in existing tables. It is easy to check that the form of the Landau free energy resulting from a search of invariant polynomials of successive degrees is

We note that the form of the free energy of this system, determined by its symmetry, involves a third-degree term. Let us show that the thermodynamic properties corresponding to this form are qualitatively identical to the ones derived from the canonical free energy (3.1.2.13). In this view, let us put and . The free energy takes the form

For such a free energy, it is remarkable that for the directions of the extrema, which are determined by , are independent of the value of . These directions form two sets which we denote A (, , ) and B (, , ). If we replace in equation (3.1.2.18) by one of these values, we obtainthe sign in front of the coefficient being for the A set of angles and − for the B set. We are therefore brought back to a form close to the canonical one [equation (3.1.2.13)]. Note that for , the stable second minimum of the free energy [equation (3.1.2.15)] corresponded to , i.e. to . Hence in (3.1.2.19), being a positive modulus, the second stable minimum will correspond to a negative coefficient for . Depending on the sign of , the direction of this minimum will either be the set A or the set B of values.

Let us summarize the results obtained in the study of the specific models described in the preceding sections. We have shown that an order parameter (e.g. or ) is a set of scalar degrees of freedom that allows the description of the symmetry and physical changes accompanying the phase transition in a system. The equilibrium values of the n components of the order parameter are zero for and not all zero for . The n components define a vector space that is an irreducible invariant space by the group G, which is the crystallographic symmetry group of the high-temperature phase. A variational free energy F associated with the transition, and termed the Landau free energy, can be expanded as a function of the components of the order parameter. The second-degree term of this expansion has a coefficient that vanishes and changes sign at the transition temperature. The form of F is determined by the symmetry properties of the order parameter, i.e. by the mode of transformation of the components of the order parameter into each other when the operations of G are applied to them. The specific form of F determines the symmetries of the potentially stable phases below , the degeneracy of these phases, the temperature dependencies of relevant physical quantities and, in certain cases, the thermodynamic order of the phase transition.

In this section, let us briefly outline the arguments used in order to formulate the theory in a general framework.

These arguments rely in part on the properties of the irreducible representations of a group. The reader can refer to Chapter 1.2 for a comprehensive presentation of irreducible representations. We will consider here representations of a group G carried by a set of degrees of freedom (), and use essentially the following properties.

Let us examine the particular ingredients needed to apply Landau's theory to an example of structural transitions, i.e. a transition between crystalline phases.

3.1.2.5.2. The example of gadolinium molybdate, Gd₂(MoO₄)₃

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Gadolinium molybdate (GMO) is a substance showing one complication with respect to the example in Section 3.1.2.2. Like the prototype example already studied, it possesses below its phase transition an electric dipole (and a spontaneous polarization) resulting from the displacement of ions. However, one does not observe the expected divergence of the associated susceptibility (Fig. 3.1.2.5).

3.1.2.5.2.1. Experimental identification of the order-parameter symmetry

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The high-temperature space group G is known for GMO from X-ray diffraction experiments. It is the tetragonal space group . The corresponding point group has eight elements, represented in Fig. 3.1.2.9.

Figure 3.1.2.9 | top | pdf | Rotations/reflections belonging to the point group of gadolinium molybdate.

The k vector labelling the irreducible representation associated with the order parameter can be directly deduced from a comparison of the diffraction spectra above and below . We have seen that the difference of the two stable structures surrounding the transition is specified by the equilibrium density: One can show, using the properties of the irreducible representations of the space groups, that the Fourier transform of the difference of densities given above is proportional to , i.e. this Fourier transform is nonzero only for a K vector equal to the k vector indexing the order parameter. The implication of this property is that below , the X-ray spectrum of the system will contain additional reflections whose locations in reciprocal space are defined by the vector of the order parameter. Experimentally, the vectors joining the Bragg spots existing in both phases to the closest superlattice spots only appearing below are the vectors k defining partly the irreducible representation that specifies the symmetry properties of the order parameter.

In GMO, X-ray diffraction measurements show that superlattice spots appear below at one of the four equivalent k vectorsThe operations of the point group transform these vectors into each other. The star of k is therefore reduced to a single vector. On the other hand, consultation of available tables provides us with the possible representations necessary to construct the representation of the order parameter (the entries of the tables being the k vector determined and the space group G). There are three unequivalent , which are reproduced in Table 3.1.2.2.

Table 3.1.2.2 | top | pdf | Matrices defining the irreducible representations of for   G

The ambiguity in the symmetry of the order parameter has now to be lifted. In this approach, the method is to work out for each the symmetries G of the phases that are possibly stable below . One then compares the results with the observed space group below , which, for GMO, is the orthorhombic space group .

The group F of interest is the invariance group of the density difference [equation (3.1.2.27)]. Note that this difference can be considered as a `vector' with components in the irreducible space of the order parameter. In each irreducible space, the action of the elements of G on a vector is represented by the set of matrices reproduced in Table 3.1.2.2. Let us first examine in this table. Clearly, the matrices relative to , , and rotate by any vector of the two-dimensional space carrying the representation. These elements will not leave any direction unmoved and consequently they will not belong to F. The other elements either preserve any vector (and they then obviously belong to F) or they reverse any direction. However, in the latter case, the product of any two of these elements belongs to F.

Summarizing these remarks, we obtain a single possible group F consisting of the elements , , , and by the infinite translation group generated by the vectors and . The symbol for this space group is .

A similar inspection yields for the representation the group and for three possible groups (, and ). Comparison with the experimental observation, recalled above, allows one to identify unambiguously the appropriate representation as . In conclusion, the irreducible representation associated with the order parameter of the transition in GMO can be denoted . Its k vector is , and its `small representation' is . The number of components of the order parameter is two, equal to the dimension of , which itself is equal to the product of the number of vectors in the star of k (one) and of the dimension of (two).

3.1.2.5.2.2. Construction of the free energy and stable states

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Denote by (, ) the two components of the order parameter. The Landau free energy can be constructed by selecting the homogeneous polynomials of different degrees that are invariant by the distinct matrices of . There are four such matrices. It is easy to check that the most general form of fourth-degree polynomial invariant by the action of these four matrices isA discussion of the minima of this free energy can be made according to the same method as in Section 3.1.2.3, by putting , . One then finds that, in accordance with the symmetry considerations developed in Section 3.1.2.5.2.1, there is a single possible symmetry below . The equilibrium state of the system corresponds to an angle whose value depends on the values of the coefficients in the expansion. The modulus has the standard temperature dependence .

As in the model/example described in Section 3.1.2.2, below there are several stable states having the same free energy. Indeed, one can easily check in expression (3.1.2.29) that if (, ) is an absolute minimum of the free energy (3.1.2.29), the states (, ), (, ), (, ) are symmetry-related minima corresponding to the same value of the equilibrium free energy.

The intensities of the diffraction `superlattice' spots, being proportional to the square of the atomic displacement , vary linearly as a function of temperature. On the other hand, the diverging susceptibility associated with the order parameter is related to a rapid increase of the diffuse scattering of X-rays or neutrons at the location of the superlattice spots in reciprocal space. Hence, consistent with the macroscopic measurements, it is not related to a divergence of the dielectric susceptibility.

3.1.2.5.2.3. Macroscopic behaviour of GMO

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In GMO, macroscopic quantities are degrees of freedom that are distinct from the order parameter. Indeed, their symmetry properties are different, since any lattice translation will leave them invariant, while this is not the case for the order parameter (see Section 3.1.2.5.2.1). Nevertheless, certain of the macroscopic quantities behave singularly at the transition. These degrees of freedom can be decomposed, as shown in Section 3.1.2.4, as the sum of irreducible degrees of freedom. Having a symmetry different from that of the order parameter, they were neglected in the first step of the description of the phase transition. In a more detailed description, they have to be taken into account.

Let us, for instance, consider the component of the dielectric polarization of GMO, as well as the component of the strain tensor which represents a shear in the xy plane of the crystal. The matrices in Table 3.1.2.3 recall the mode of transformation of the order-parameter components as well as those of these two quantities under the action of the G group.

Table 3.1.2.3 | top | pdf | Action of the generators of on the order parameter and on the polarization and strain components   E 1 1 1 1 1 1 1 1 1 1

We can complete the expression of the free energy of the system by adding to F in (3.1.2.29) the contributions of the preceding degrees of freedom up to the second degree (which, as will be seen, is comparable to the fourth degree used for the order parameter). The resulting expression is provided by (3.1.2.30) below, in which we have neglected a bilinear term in and as this term does not change the qualitative result we want to establish. where F is provided by equation (3.1.2.29). At equilibrium, the derivatives of with respect to and vanish. These conditions yield

As stressed in Section 3.1.2.5.2.2, the equilibrium direction in the order-parameter space corresponds to the trivial angle. Hence below , resulting in the fact that nonzero values of and will onset below the transition temperature. Besides, the form (3.1.2.31) indicates that the two macroscopic quantities considered, which are proportional to the square of the order parameter, are expected to vary linearly as a function of temperature below . Note that terms such as are of the same order of magnitude as fourth-degree terms of the order parameter.

We can also determine the behaviour of the dielectric susceptibility , by calculating the variations of the equilibrium value of as a function of an applied electric field E parallel to the polarization. We proceed as in Section 3.1.2.2, and minimize the potential with respect to the order parameter and to the polarization. In order to obtain the qualitative behaviour of , we simplify the free energy by considering a single component of the order parameter. We also neglect the shear strain component . The set of simplified equations yields the following expression of the susceptibility: We find an upward step of the dielectric susceptibility on cooling. Likewise, consideration of the `elastic' susceptibility relative to the shear strain component would determine an upward step of the elastic compliance (Fig. 3.1.2.10). The more usually measured elastic constant, which is the inverse of the compliance, undergoes a downward step on cooling.

Figure 3.1.2.10 | top | pdf | Temperature dependence of the macroscopic susceptibility (or elastic compliance, ) in gadolinium molybdate. Compare with the `normal' behaviour in Fig. 3.1.2.5(b).

We have seen in the preceding paragraph that the low-symmetry phase of gadolinium molybdate possesses four equally stable states differing by the values of the order-parameter components. Equation (3.1.2.31) shows that two of the states are associated with the same sign of the polarization or of the shear strain , while the two other states possess opposite values of and . According to the definitions given in Section 3.1.1, gadolinium molybdate belongs to the category of ferroelectrics as well as to that of ferroelastics.

The example of GMO clearly shows that the ferroic classification is less informative than the one based on the order-parameter symmetry. The latter determines the full symmetry change (orientational and translational), while the former only specifies the orientational symmetry change. On the other hand, the ferroic classification is not informative about the physical behaviour as a function of temperature. Thus, the model ferroelectric in Section 3.1.2.2 has a diverging dielectric susceptibility at , while a GMO-type ferroelectric keeps a finite susceptibility. The ferroic classification has nevertheless the advantage of specifying the nature of the macroscopic quantities that are expected to behave anomalously at the transition, and are therefore worth measuring.

A basic role is played in symmetry considerations by the relation between the space group of the high symmetry parent or prototype phase, the space group of the low-symmetry ferroic phase and the order parameter : The low-symmetry group consists of all operations of the high-symmetry group that leave the order parameter 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 but the transformation properties of , which are expressed by the representation of . The order parameter with components can be treated as a vector in a -dimensional carrier space of the representation , and the low-symmetry group comprises all operations of that do not change this vector. If is a real one-dimensional representation, then the low-symmetry group consists of those operations for which the matrices [or characters ] of the representation equal one, . This condition is satisfied by one half of all operations of (index of in is two) and thus the real one-dimensional representation determines the ferroic group unambiguously.

A real multidimensional representation can induce several low-symmetry groups. A general vector of the carrier space of is invariant under all operations of a group , called the kernel of representation , which is a normal subgroup of comprising all operations for which the matrix is the unit matrix. Besides that, special vectors of – specified by relations restricting values of order-parameter components (e.g. some components of equal zero, some components are equal etc.) – may be invariant under larger groups than the kernel . These groups are called epikernels of (Ascher & Kobayashi, 1977). The kernel and epikernels of represent potential symmetries of the ferroic phases associated with the representation . 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 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 of R-irreducible representations (abbreviated R-ireps) can contain only real numbers. (Physically irreducible matrix representations are denoted by instead of the symbol 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 of a space group is specified by a vector of the first Brillouin zone, and by an irreducible representation of the little group of k, denoted . 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 , 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 vector (this situation is conveniently denoted as the point). Then there is no change of translational symmetry at the transition. In this case, the group is called an equitranslational or translationengleiche (t) subgroup of , and this change of symmetry will be called an equitranslational symmetry descent . An equitranslational phase transition is a transition with an equitranslational symmetry descent .

Any ferroic space-group-symmetry descent uniquely defines the corresponding symmetry descent , where G and F are the point groups of the space groups and , respectively. Conversely, the equitranslational subgroup of a given space group is uniquely determined by the point-group symmetry descent , where G and F are point groups of space groups and , respectively. In other words, a point-group symmetry descent defines the set of all equitranslational space-group symmetry descents , where runs through all space groups with the point group G. All equitranslational space-group symmetry descents are available in the software GIKoBo-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 for all space groups with the point group G by a simple process called engendering (Jansen & Boon, 1967). The translation subgroup of is a normal subgroup and the point group G is isomorphic to a factor group . This means that to every element there correspond all elements of the space group with the same linear constituent g, the same non-primitive translation and any vector of the translation group (see Section 1.2.3.1 ). If a representation of the point group G is given by matrices , then the corresponding engendered representation of a space group with vector assigns the same matrix to all elements of .

From this it further follows that a representation of a point group G describes transformation properties of the primary order parameter for all equitranslational phase transitions with point-symmetry descent . This result is utilized in the presentation of Table 3.1.3.1.

The primary order parameter expresses the `difference' between the low-symmetry and high-symmetry structures and can be, in a microscopic description, identified with spontaneous displacements of atoms (frozen in soft mode) or with an increase of order of molecular arrangement. To find a microscopic interpretation of order parameters, it is necessary to perform mode analysis (see e.g. Rousseau et al., 1981; Aroyo & Perez-Mato, 1998), which takes into account the microscopic structure of the parent phase.

Physical properties of crystals in a continuum description are described by physical property tensors (see Section 1.1.1.2 ), for short property tensors [equivalent expressions are matter tensors (Nowick, 1995; Wadhawan, 2000) or material tensors (Shuvalov, 1988)]. Property tensors are usually expressed in a Cartesian (rectangular) coordinate system [in Russian textbooks called a crystallophysical system of coordinates (Sirotin & Shaskolskaya, 1982; Shuvalov, 1988)] which is related to the crystallographic coordinate system (IT A , 2005) by convention (see IEEE Standard on Piezoelectricity, 1987; Sirotin & Shaskolskaya, 1982; Shuvalov, 1988). In what follows, Cartesian coordinates will mean coordinates in the crystallophysical system and tensor components will mean components in this coordinate system.

As explained in Section 1.1.4 , the number of independent components of property tensors depends on the point-group symmetry of the crystal: the higher this symmetry is, the smaller this number is. Lowering of point-group symmetry at ferroic phase transitions is, therefore, always accompanied by an increased number of independent components of some property tensors. This effect manifests itself by the appearance of morphic (Strukov & Levanyuk, 1998) or spontaneous tensor components, which are zero in the parent phase and nonzero in the ferroic phase, and/or by symmetry-breaking increments of nonzero components in the ferroic phase that break relations between these tensor components which hold in the parent phase. Thus, for example, the strain tensor has two independent components in a tetragonal phase and four independent components in a monoclinic phase. In a tetragonal-to-monoclinic phase transition there is one morphic component and one relation is broken by the symmetry-breaking increment .

Changes of property tensors at a ferroic phase transition can be described in an alternative manner in which no symmetrybreaking increments but only morphic terms appear. As we have seen, the transformation properties of the primary order parameter are described by a -dimensional R-irreducible matrix representation of the group G. One can form linear combinations of Cartesian tensor components that transform according to the same representation . These linear combinations will be called components of a principal tensor parameter of the ferroic phase transition with a symmetry descent . Equivalent designations are covariant tensor components (Kopský, 1979a) or symmetry coordinates (Nowick, 1995) of representation of group G. Unlike the primary order parameter of a ferroic phase transition, a principal tensor parameter is not uniquely defined since one can always form further principal tensor parameters from Cartesian components of higher-rank tensors. However, only the principal tensor parameters formed from components of one, or even several, property tensors up to rank four are physically significant.

A principal tensor parameter introduced in this way has the same basic properties as the primary order parameter: it is zero in the parent phase and nonzero in the ferroic phase, and transforms according to the same R-irep . However, these two quantities have different physical nature: the primary order parameter of an equitranslational phase transition is a homogeneous microscopic distortion of the parent phase, whereas the principal tensor parameter describes the macroscopic manifestation of this microscopic distortion. Equitranslational phase transitions thus possess the unique property that their primary order parameter can be represented by principal tensor parameters which can be identified and measured by macroscopic techniques.

If the primary order parameter transforms as a vector, the corresponding principal tensor parameter is a dielectric polarization (spontaneous polarization) and the equitranslational phase transition is called a proper ferroelectric phase transition. Similarly, if the primary order parameter transforms as components of a symmetric second-rank tensor, the corresponding principal tensor parameter is a spontaneous strain (or spontaneous deformation) and the equitranslational phase transition is called a proper ferroelastic phase transition.

A conspicuous feature of equitranslational phase transitions is a steep anomaly (theoretically an infinite singularity for continuous transitions) of the generalized susceptibility associated with the primary order parameter, especially the dielectric susceptibility near a proper ferroelectric transition (see Section 3.1.2.2.5) and the elastic compliance near a proper ferroelastic transition (see e.g. Tolédano & Tolédano, 1987; Tolédano & Dmitriev, 1996; Strukov & Levanyuk, 1998).

Any symmetry property of a ferroic phase transition has its pendant in domain structure. Thus it appears that any two ferroic single domain states differ in the values of the principal tensor parameters, i.e. principal tensor parameters ensure tensor distinction of any two ferroic domain states. If, in particular, the principal order parameter is polarization, then any two ferroic domain states differ in the direction of spontaneous polarization. Such a ferroic phase is called a full ferroelectric phase (Aizu, 1970). In this case, the number of ferroic domain states equals the number of ferroelectric domain states. Similarly, if any two ferroic domain states exhibit different spontaneous strain, then the ferroic phase is a full ferroelastic phase. An equivalent condition is an equal number of ferroic and ferroelastic domain states (see Sections 3.4.2.1 and 3.4.2.2 ).

The principal tensor parameters do not cover all changes of property tensors at the phase transition. Let be a -dimensional matrix R-irep of G with an epikernel (or kernel) L which is an intermediate group between F and G, in other words, L is a supergroup of F and a subgroup of G, This means that a vector of the -dimensional carrier space of is invariant under operations of L. The vector specifies a secondary order parameter of the transition, i.e. is a morphic quantity, the appearance of which lowers the symmetry from G to L (for more details on secondary order parameters see Tolédano & Tolédano, 1987; Tolédano & Dmitriev, 1996). Intermediate groups (3.1.3.1) can be conveniently traced in lattices of subgroups displayed in Figs. 3.1.3.1 and 3.1.3.2.

Figure 3.1.3.1 | top | pdf | Lattice of subgroups of the group . Conjugate subgroups are depicted as a pile of cards. In the software GIKoBo-1, one can pull out individual conjugate subgroups by clicking on the pile. All conjugate subgroups are given explicitly in Table 3.4.2.7 .

Figure 3.1.3.2 | top | pdf | Lattice of subgroups of the group . Conjugate subgroups are depicted as a pile of cards. In the software GIKoBo-1, one can pull out individual conjugate subgroups by clicking on the pile. All conjugate subgroups are given explicitly in Table 3.4.2.7 .

One can form linear combinations of Cartesian tensor components that transform according to . These combinations are components of a secondary tensor parameter which represents a macroscopic appearance of the secondary order parameter .

If a secondary tensor parameter is a spontaneous polarization and no primary order parameter with this property exists, the phase transition is called an improper ferroelectric phase transition (Dvořák, 1974; Levanyuk & Sannikov, 1974). Similarly, an improper ferroelastic phase transition is specified by existence of a secondary tensor parameter that transforms as components of the symmetric second-rank tensor (spontaneous strain) and by absence of a primary order parameter with this property. Unlike proper ferroelectric and proper ferroelastic phase transitions, which are confined to equitranslational phase transitions, the improper ferroelectric and improper ferroelastic phase transitions appear most often in non-equitranslational phase transitions. Classic examples are an improper ferroelectric phase transition in gadolinium molybdate (see Section 3.1.2.5.2) and an improper ferroelastic phase transition in strontium titanate (see Section 3.1.5.2.3). Examples of equitranslational improper ferroelectric and ferroelastic symmetry descents can be found in Table 3.1.3.2.

Secondary tensor parameters and corresponding susceptibilities exhibit less pronounced changes near the transition than those associated with the primary order parameter (see e.g. Tolédano & Tolédano, 1987; Tolédano & Dmitriev, 1996; Strukov & Levanyuk, 1998).

In tensor distinction of domains, the secondary tensor parameters play a secondary role in a sense that some but not all ferroic domain states exhibit different values of the secondary tensor parameters. This property forms a basis for the concept of partial ferroic phases (Aizu, 1970): A ferroic phase is a partial ferroelectric (ferroelastic) one if some but not all domain states differ in spontaneous polarization (spontaneous strain). A non-ferroelectric phase denotes a ferroic phase which is either non-polar or which possesses a unique polar direction available already in the parent phase. A non-ferroelastic phase exhibits no spontaneous strain.

The first systematic symmetry analysis of Landau-type phase transitions was performed by Indenbom (1960), who found all equitranslational phase transitions that can be accomplished continuously. A table of all crystallographic point groups G along with all their physically irreducible representations, corresponding ferroic point groups F and related data has been compiled by Janovec et al. (1975). These data are presented in an improved form in Table 3.1.3.1 together with corresponding principal tensor parameters and numbers of ferroic, ferroelectric and ferroelastic domain states. This table facilitates solving of the following typical problems: