International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 3.1, pp. 358396
doi: 10.1107/97809553602060000915 Chapter 3.1. Structural phase transitions^{d}Ecole Polytechnique, Route de Saclay, 91128 Palaiseau CEDEX, France,^{b}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic,^{e}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, and Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic,^{c}Earth Sciences Department, University of Cambridge, Downing Street, Cambridge CB2 3EQ, England, and ^{a}Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 182 08 Praha 8, Czech Republic Aspects of phase transitions in crystals that are of interest to crystallographers are described in this chapter. The chapter starts with a brief introduction aimed at defining the field of structural transitions and the terminology used. The theory of structural phase transitions is then described. This theory relates the symmetry characteristics of the transitions to their physical characteristics. The application of the symmetry principles that derive from this theory is illustrated by the results contained in Tables 3.1.3.1 and 3.1.4.1. The first of these two tables concerns the simple but experimentally widespread situation in which a structural transition is not accompanied by a change in the number of atoms per primitive crystal cell. The second table concerns the general case, in which the number of atoms changes, and which corresponds to the onset of superlattice reflections at the phase transition. This table provides, for a set of hypothetical transformations, the various symmetrybased predictions of the theory. The important topic of soft modes, which is related to the microscopic mechanism of a structural transition, is then discussed. The final section of the chapter is an introduction to the accompanying software package Group Informatics. 
This chapter contains six contributions describing aspects of phase transitions in crystals that are of interest to crystallographers. The first contribution (Section 3.1.1) is a brief introduction aimed at defining the field of structural transitions. This restricted field constitutes, at present, the background of the clearest set of experimental and theoretical considerations. In this section, the terminology is specified. The second section (Section 3.1.2) describes the ideas and methods of the theory of structural phase transitions. This theory relates the symmetry characteristics of the transitions to their physical characteristics. The application of the symmetry principles that derive from this theory is illustrated by the results contained in Tables 3.1.3.1 and 3.1.4.1. The first of these two tables concerns the simple but experimentally widespread situation in which a structural transition is not accompanied by a change in the number of atoms per primitive crystal cell. The second table concerns the general case, in which the number of atoms changes, and which corresponds to the onset of superlattice reflections at the phase transition. This table provides, for a set of hypothetical transformations, the various symmetrybased predictions of the theory. Section 3.1.5 is devoted to the important topic of soft modes, which is related to the microscopic mechanism of a structural transition. Finally, Section 3.1.6 is an introduction to the software package Group Informatics contained in the accompanying CDROM.
Phase transformations (the term transitions can be considered as a synonym) are experimentally recognized to exist in a large variety of systems submitted to a change in temperature or pressure: fluids, solids or mesophases, crystalline or disordered solids, metals or insulators.
This recognition is sometimes based on very obvious effects. This is, for instance, the case for the boiling or the freezing of a liquid, because the different phases, vapour, liquid, solid, differ greatly in their physical properties (e.g the difference of density between the two fluids, or the difference of mechanical hardness between the liquid and the solid). In these cases, a phase transformation appears as an abrupt and major change of the physical properties.
In other systems, solids in particular, the existence of a phase transformation is generally revealed by more subtle effects only. The nature of these effects differs from one system to another: minor discontinuities in the lattice parameters of a crystalline phase; occurence over a narrow temperature range of anomalies in certain specific physical properties; onset of a definite pattern of crystal twins etc.
Systems undergoing phase transitions constitute an important field of interest for crystallographers. This is due to the fact that, at the microscopic level, a phase transformation is generally accompanied by a change of the global or local atomic configuration. The structural data, i.e. the specification of the differences in atomic configurations between the two phases, or the study of the local ordering precursor to a transition, are thus essential, or at least important, clues to the understanding of the mechanism of the transition considered.
Conversely, the investigation of phase transitions has stimulated new developments in the techniques and concepts used by crystallographers. For instance, it has been necessary to improve the precision of goniometric measurements and the control of temperature in order to detect accurately anomalies affecting the lattice parameters across a phase transition or to study the asymmetry of diffraction spots caused by the domain structure in a `lowsymmetry' phase. On the other hand, new methods of structural determination, relying on concepts of ndimensional crystallography, had to be developed in order to study transitions to incommensurate phases.
Standard crystallographic considerations, based on the determination of the characteristics of a lattice and of a basis, appear to be most useful in the study of phase transformations between crystalline phases, due to the fact that, at a microscopic level, each phase is entirely described by its periodic crystal structure. There are a wide variety of such transformations and the task of classifying them has been attempted from several standpoints.
The most important distinction is that made between reconstructive and nonreconstructive transitions. This distinction stems from a comparison of the crystal structures of the two phases. In a reconstructive transition, the distances between certain atoms change by amounts similar to the dimension of the unit cell, and certain chemical bonds between neighbouring atoms are then necessarily broken (see Tolédano & Dmitriev, 1996, and references therein). The graphite–diamond transformation and many transformations in metals and alloys are examples of reconstructive transitions. If, instead, a transition preserves approximately the configuration of the chemical bonds between constituents, the transition is nonreconstructive.
Other classifications, which partly overlap with the preceding one, involve distinctions between diffusionless and diffusionassisted transitions (i.e those that require random hopping of atoms to achieve the change of atomic configuration) or between displacive and order–disorder transitions. Likewise, a number of transformations in metals or alloys are assigned to the class of martensitic transformations that is defined by a set of specific experimental observations (twinning behaviour, mechanical properties etc.). Finally, the distinction between ferroic and nonferroic transitions has been progressively adopted in the recent years.
Owing to an insufficient understanding of the observations, the relationships between these various classifications is not fully clear at present. It is not even clear whether the same definitions and concepts can be applied to the description of all phase transformations between solid phases. For instance, one observes in certain solids (e.g. mixed lead magnesium niobates with an average perovskite structure) very broad anomalies of the physical properties (i.e. extending over a wide range of temperatures). These systems, which have stimulated many studies in recent years, are known to be chemically and structurally heterogeneous simultaneously at several length scales. The relevance to these systems of standard concepts defined for phase transitions in homogeneous systems, in which the anomalies of the physical properties are sharp, is uncertain.
It is therefore reasonable to restrict a review of basic concepts and theories to the simple reference case of structural phase transitions. We consider this terminology, in its restricted meaning, as pertaining to the situation of only a fraction of the phase transitions that take place in solids and imply a modification of the crystal structure. These are part of nonreconstructive transitions between homogeneous crystalline phases. It is customary to specify that a structural transition only slightly alters the chemical bond lengths (by less than e.g. 0.1 Å) and their relative orientations (by less than e.g. a few degrees).
Experimentally, such transitions are characterized by small values of the heat of transformation (less than a few calories per gram), weak discontinuities in the relevant physical quantities (e.g. lattice parameters) and the occurrence of a symmetry relationship between the two phases surrounding the transition.
In the simplest case, this relationship consists of the fact that the space group of one of the phases is a subgroup of the space group of the other phase, and that there is specific correspondence between the symmetry elements of the two phases. For example, for the phase transition occurring at 322 K in triglycine sulfate (Lines & Glass, 1977), the same binary axis can be found in the two phases. Likewise, the vector defining one of the primitive translations in one phase can be a multiple of the vector defining a primitive translation in the other phase.
In a more general way, the crystal structures of the two phases considered are both slight distortions of a reference structure, termed the prototype (or parent) structure. In this case, the space groups of the two phases are both subgroups of the space group of the prototype structure, with, as in the simple case above, a specific correspondence between the symmetry elements of the two phases and of the prototype structure. A well documented example of this situation is provided by two of the three transitions occurring in barium titanate (Lines & Glass, 1977).
A subclassification of structural transitions into ferroic classes is of interest (Aizu, 1969; Tolédano & Tolédano, 1987, and references therein). Indeed the distinction of ferroic classes allows one to establish a relationship between the point symmetries of the two phases surrounding a phase transition, the observed twinning, and the nature of the physical properties mainly affected by the phase transition.
The group–subgroup relationship that exists, in the standard situation, between the space groups of the two phases adjacent to a structural transition implies that the point group of one phase is either a subgroup of the point group of the other phase or is identical to it.
If the two point groups are identical, the corresponding transition is classified as nonferroic.
In the general case, the point group of one phase (the ferroic phase) is a strict subgroup of the point group of the other phase (the prototype phase). The transition is then classified as ferroic. Originally, a somewhat more abstract definition was given (Aizu, 1969): a crystal was said to be ferroic if it can exist in two or more orientation states having equal stabilities in the absence of external forces, and when the various orientation states have crystal structures that only differ in their global spatial orientations. The latter definition, which focuses on the situation of the ferroic phase, derives from the former one: the lowering of point symmetry that accompanies the transition between the prototype phase and the ferroic phase results in the existence of various variants or twin orientations having the same structures within a global reorientation (see also Sections 3.2.1 , 3.2.3 , 3.3.7 , 3.3.10 and 3.4.1 ).
The various orientation states can coexist in a given sample and then determine a twinning pattern. Geometrical and physical considerations pertaining to twinned structures are developed in Chapters 3.2 and 3.3 of this volume. In particular, it can be shown that the structure of one orientation state can be brought to coincide with the structure of another orientation state by means of a set of geometrical transformations R which all belong to the space group of the prototype phase.
If we adopt a common frame of reference for all the orientation states of the ferroic phase, the tensors representing certain macroscopic quantities (see Chapter 1.1 ) will have different values in the different states (e.g. distinct nonzero components). If a certain macroscopic tensor has components differing in two states, a and b, these components are thus modified by the action of the geometrical transformations R which transforms (reorients) one structure into the other. Hence they are not invariant by geometrical operations belonging to the group of symmetry of the prototype phase: their value is necessarily zero in this phase.
Ferroic transitions therefore possess three characteristics:
The subclassification of ferroics into ferroic classes has a crystallographic and a physical content. The crystallographic aspect is based on the type of pointsymmetry lowering occurring at the transition, while the physical aspect focuses on the rank of the tensor (necessarily traceless) characterizing the different orientation states of the crystal in the ferroic phase and on the nature of the physical quantity (electrical, mechanical, ) related to the relevant tensor (see Chapter 1.1 ). Table 3.1.1.1 specifies this twofold classification.

Note that a given transition related to a class defined by a tensor of rank n can belong to several classes defined by tensors of higher rank: e.g. a ferroelectric transition can also be ferroelastic and will also display characteristics of a higherorder ferroic.
The pointsymmetry changes defining each class have been enumerated in various works (see for instance Aizu, 1973, and references therein).
The interest of the above classification is that it provides a guiding framework for the experimental investigations. Hence, the recognition that a transition is ferroelectric (respectively, ferroelastic) directs the investigation of the transition towards the examination of the dielectric (respectively, mechanical) properties of the system in the expectation that these will be the quantities mainly affected by the transition. This expectation is based on the fact that the dielectric polarization (respectively, the thermal strain tensor) acquires spontaneous components across the transition.
Conversely, if neither of these two classes of ferroics is involved in the transition considered, one knows that one must focus the study on components of higherrank macroscopic tensors in order to reveal the characteristic anomalies associated with the transition. Also, the knowledge of the ferroic class of a transition specifies the nature of the macroscopic tensorial quantity that must be measured in order to reveal the domain structure. For instance, ferroelastic domains correspond to different values of symmetric secondrank tensors. Aside from the spontaneous strain tensor, we can consider the dielectric permittivity tensor at optical frequencies. The latter tensor determines the optical indicatrix, which will be differently oriented in space for the distinct domains. Consequently, with suitably polarized light one should always be able to `visualize' ferroelastic domains. Conversely, such visualization will never be possible by the same method for a nonferroelastic system.
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 orderparameter 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 (`secondorder') 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 `lowsymmetry' phase (i.e. number of energetically equivalent domain orientations in this phase), enumeration of the possible symmetries of the `lowsymmetry' phase for a given symmetry of the `highsymmetry' 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.
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.
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 hightemperature 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 .
Our aim is to determine the above displacement as a function of temperature. Landau's strategy is to determine by a variational method. One considers an arbitrary displacement d of the ion. For given temperature T and pressure p (or volume V), and specified values of the components of d, there is, in principle, a definite value for the free energy F of the system. This function is a variational free energy since it is calculated for an arbitrary displacement. The equilibrium displacement is defined as the displacement that minimizes the variational free energy F. The equilibrium free energy of the system is . Note that, strictly speaking, in the case of a given pressure, one would have to consider a variational Gibbs function () in order to determine the equilibrium of the system. We will respect the current use in the framework of Landau's theory of denoting this function F and call it a free energy, though this function might actually be a Gibbs potential.
The former strategy is not very useful as long as one does not know the form of the variational free energy as a function of the components of the displacement. The second step of Landau's theory is to show that, given general assumptions, one is able to determine simply the form of in the required range of values of the functions' arguments.
The basic assumption is that of continuity of the phase transition. It is in fact a dual assumption. On the one hand, one assumes that the equilibrium displacement has components varying continuously across the transition at . On the other hand, one assumes that F is a continuous and derivable function of , which can be expanded in the form of a Taylor expansion as function of these arguments.
Invoking the continuity leads to the observation that, on either side of , is small, and that, accordingly, one can restrict the determination of the functional form of to small values of and of . F will then be equal to the sum of the first relevant terms of a Taylor series in the preceding variables.
The central property of the variational free energy which allows one to specify its form is a symmetry property. F is a function of which is invariant by the symmetry transformations of the hightemperature equilibrium structure. In other terms, an arbitrary displacement d and the displacement obtained by applying to d one of the latter symmetry transformations correspond to the same value of the free energy.
Indeed, both displacements determine an identical set of mutual distances between the positive and negative ions of the system and the free energy only depends on this `internal' configuration of the ions.
Note that, in the case considered here (Fig. 3.1.2.1), the set of symmetry transformations comprises, aside from the lattice translations, fourfold rotations around the z axis, mirror symmetries into planes and the products of these transformations. The set of rotations and reflections forms a group G of order 16, which is the crystallographic point group (or ).
Also note that this symmetry property of the free energy also holds for each degree of the Taylor expansion of F since the geometrical transformations of G act linearly on the components of d. Hence, terms of different degrees belonging to the expansion of F will not `mix', and must be separately invariant.
Let us implement these remarks in the case in Fig. 3.1.2.1. It is easy to check that by successive application to the components of d of the mirror symmetries perpendicular to the three axes, no linear combination of these components is invariant by G: each of the three former symmetry transformations reverses one component of d and preserves the two others. Linear terms are therefore absent from the expansion.
As for seconddegree terms, the same symmetry transformations preclude the existence of combinations of bilinear products of the type . Actually, one finds that the fourfold symmetry imposes that the most general form of the seconddegree contribution to the variational free energy is a linear combination of and of . Hence the Taylor expansion of F, restricted to its lowestdegree terms, is
Let us now derive the key result of the theory, namely, that either the component or the pair of components () will take nonzero values below (but not both). The meaning of this result will be clarified by symmetry considerations.
The derivation of this result relies on the fact that one, and one only, of the two coefficients in equation (3.1.2.1) must vanish and change sign at , and that the other coefficient must remain positive in the neighbourhood of .
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 seconddegree 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 thirddegree term in the expansion of F as a function of the order parameter , and to check the effective occurrence of a fourthdegree term. If we assume that this simplest form of expansion is sufficient to determine the equilibrium state of the system, the coefficient of the fourthdegree 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 lowestdegree expansion of the function is a positive constant in the vicinity of . The function , which is the zerothdegree 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.

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 oneminimum plot to a twominimum 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 squareroot 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.

(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 onedimensional. We consider one of these complications in Section 3.1.2.3.
Up to now, we have defined the order parameter as a set of degrees of freedom determining a seconddegree 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.

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 onedimensional 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 onedimensional 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 onedimensional irreducible invariant spaces. However, their set generates a twodimensional 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 threedimensional 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 highsymmetry phase.
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 twodimensional order parameter (). A simple discussion along the line developed for the onedimensional order parameter would show that the relevant form of the free energy is and that the possible stable states below are:
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 firstorder 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 fourthdegree 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 thirddegree 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.

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 firstorder 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 freeenergy 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 thirddegree 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 fourthdegree 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 hightemperature 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).

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 thirddegree 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 hightemperature 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 seconddegree 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.

In order to generalize the considerations developed in the preceding sections, we have to define, independently from any specific structure, the basic ingredients of the theory: definition of a variational set of degrees of freedom; construction of a free energy; determination of the stable states on either side of a transition temperature.
In a first step, we describe an arbitrary atomic configuration of a system by the set of densities of the particles of type (i). This set constitutes variational degrees of freedom, which can be used to construct the variational free energy of the system . The equilibrium of the system is defined by the set of functions that minimize the free energy.
The symmetry of the system at a given temperature is defined as the set of geometrical transformations that leave invariant all the . This set forms a group.
We are then in a position to define a continuous transition at by two conditions:
Hence the phase transition, though associated with a continuous change of the spatial configuration of the atoms, is associated with a sudden change of the symmetry.
As compared with the set () used in the example of Section 3.1.2.2, the variational functions have the drawback of not being small and scalar quantities, thus making an expansion of the free energy more complicated.
Without loss of generality, let us restrict ourselves to a single type of (i) particles. In order to remove the difficulty mentioned above we putwhere is the equilibrium density at . Let us denote by G the symmetry group of this equilibrium density. The sum in the second part of (3.1.2.21) is a small increment since the transition is continuous. This increment has been expanded, as in (3.1.2.20), as a function of irreducible functions with respect to the group G. Moreover, each function has been expressed as the product of a normalized function and of small scalar parameters . It is easy to convince oneself that it is possible to consider that under the action of G either the transform into each other, the being fixed coefficients, or the transform into each other, the functions being fixed. We shall adopt the second convention. The variational free energy can then be written in the form
At each temperature, the characteristics of the system are specified by the following conditions:
Note that, in , the contribution of the degrees of freedom spanning a totally symmetric representation can be ignored in the first place. One can show that such a contribution would have the same symmetry on either side of the transition. Therefore it is not crucially associated with the symmetry change defining the phase transition.
Following the ideas introduced in Section 3.1.2.2, the freeenergy form is obtained as a Taylor expansion as a function of the small parameters . Besides, each polynomial term of a given degree of this expansion is invariant by G. Indeed, F, being a scalar, is unchanged by any rotation or reflection. Among the transformations, those belonging to G have the additional property of leaving invariant the density which is, besides T and the , an argument of the function F.
It is easy to check that the grouptheoretical rules recalled at the beginning of this section determine the absence of an invariant linear term in the expansion. Moreover, these rules specify the form of the seconddegree contribution. We haveOn the other hand, the equilibrium density is
Let us first consider the system at . The equilibrium values of all the are zero. Hence, on the basis of (3.1.2.23), we conclude that all the satisfy the condition (since the seconddegree expansion must be minimum at the origin). Note that this condition cannot be strict for all . Otherwise, these coefficients would also be positive in the vicinity of , on either side of this temperature. As a consequence, the equilibrium values would all be zero and the symmetry would be unchanged. Hence, one at least of the has to vanish and to change sign at . An argument already invoked in Section 3.1.2.2 allows one to assert that one coefficient only has this property. The corresponding to the other indices keep zero equilibrium values in the vicinity of and can be ignored in the first place. We can therefore drop in (3.1.2.23) all degrees of freedom except the ones associated with the coefficient that vanishes at . The set (j ) constitutes the mdimensional order parameter of the transition considered. As this set comprises all the degrees of freedom contributing to a single seconddegree term in the free energy, it necessarily constitutes a basis for an irreducible vector space with respect to G, according to the grouptheoretical rules recalled above.
Above , due to the positivity of (we can drop the index), the equilibrium values of the are zero and the symmetry is G, identical to the symmetry at . Below , is negative and the minimum of F occurs away from the origin in the space. The symmetry of the system is defined by all the transformations leaving invariant the density: Since the contribution to the second member is small, these transformations have to be selected among those belonging to the invariance group of . The space defines a nontrivial representation of the latter group since the linear combination of the orderparameter components present in cannot be invariant by all the transformations of G. The symmetry group of the system below is therefore a subgroup F of G.
As pointed out in Section 3.1.2.2, in order to determine the minimum of F below , it is necessary to expand the free energy to degrees higher than two. The relevant expression of the free energy is thenwhere we have developed the coefficient , which is an odd function of to the lowest degree in . It can be shown that the existence of a thirddegree term depends exclusively on the nature of the representation associated with the order parameter. If the symmetry of the order parameter is such that a thirddegree term is not symmetry forbidden, the transition will be of the type analysed in Section 3.1.2.3: it will be discontinuous.
For any symmetry of the order parameter, fourthdegree terms will always be present in the freeenergy expansion (there will be at least one such term that is the square of the seconddegree term). No further general statement can be made. Depending on the form and coefficients of this term, a continuous or discontinuous transition will be possible towards one or several distinct lowsymmetry phases. The form of the term can be determined by searching the most general fourthdegree polynomial that is invariant by the set of transformations belonging to G.
In summary, in the light of the preceding considerations, the study of a phase transition according to the Landau scheme can be developed along the following lines:

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.
The phases considered being crystalline, their invariance groups, G or F, coincide with crystallographic space groups. Let us only recall here that each of these groups of infinite order is constituted by elements of the form where R is a pointsymmetry operation and t a translation. The symmetry operations R generate the point group of the crystal. On the other hand, among the translations t there is a subset forming an infinite group of `primitive' translations T generating the threedimensional Bravais lattice of the crystal.
For a space group G, there is an infinite set of unequivalent irreducible representations. An introduction to their properties can be found in Chapter 1.2 as well as in a number of textbooks. They cannot be tabulated in a synthetic manner as the betterknown representations of finite groups. They have to be constructed starting from simpler representations. Namely, each representation is labelled by a double index.

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).
The hightemperature space group G is known for GMO from Xray diffraction experiments. It is the tetragonal space group . The corresponding point group has eight elements, represented in Fig. 3.1.2.9.
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 Xray 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, Xray 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.

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 twodimensional 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).
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 fourthdegree 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 symmetryrelated 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 Xrays 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.
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 orderparameter components as well as those of these two quantities under the action of the G group.

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 orderparameter 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 fourthdegree 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.

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 lowsymmetry phase of gadolinium molybdate possesses four equally stable states differing by the values of the orderparameter 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 orderparameter 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 GMOtype 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.
In the Landau theory, presented in the preceding Section 3.1.2, symmetry considerations and thermodynamics are closely interwoven. These two aspects can be, at least to some extent, disentangled and some basic symmetry conditions formulated and utilized without explicitly invoking thermodynamics. Statements which follow directly from symmetry are exact but usually do not yield numerical results. These can be obtained by a subsequent thermodynamic or statistical treatment.
The central point of this section is Table 3.1.3.1, which contains results of symmetry analysis for a large class of equitranslational phase transitions and presents data on changes of property tensors at most ferroic phase transitions. Notions and statements relevant to these two applications are explained in Sections 3.1.3.1 and 3.1.3.2, respectively. Table 3.1.3.1 with a detailed explanation is displayed in Section 3.1.3.3. Examples illustrating possible uses of the table are given in Section 3.1.3.4.
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 lowsymmetry ferroic phase and the order parameter : The lowsymmetry group consists of all operations of the highsymmetry 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 phasetransition 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 lowsymmetry group comprises all operations of that do not change this vector. If is a real onedimensional representation, then the lowsymmetry 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 onedimensional representation determines the ferroic group unambiguously.
A real multidimensional representation can induce several lowsymmetry 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 orderparameter 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 (secondorder) and some discontinuous (firstorder) phase transitions transform according to an irreducible representation of the space group of the highsymmetry 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 (socalled physical irreducibility or Rirreducibility). This means that the matrices of Rirreducible representations (abbreviated Rireps) 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 spacegroupsymmetry 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 pointgroup symmetry descent , where G and F are point groups of space groups and , respectively. In other words, a pointgroup symmetry descent defines the set of all equitranslational spacegroup symmetry descents , where runs through all space groups with the point group G. All equitranslational spacegroup symmetry descents are available in the software GIKoBo1, 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 nonprimitive 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 pointsymmetry descent . This result is utilized in the presentation of Table 3.1.3.1.
The primary order parameter expresses the `difference' between the lowsymmetry and highsymmetry 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 & PerezMato, 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 pointgroup symmetry of the crystal: the higher this symmetry is, the smaller this number is. Lowering of pointgroup 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 symmetrybreaking 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 tetragonaltomonoclinic phase transition there is one morphic component and one relation is broken by the symmetrybreaking 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 Rirreducible 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 higherrank 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 Rirep . 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 secondrank 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 Rirep 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.

Lattice of subgroups of the group . Conjugate subgroups are depicted as a pile of cards. In the software GIKoBo1, one can pull out individual conjugate subgroups by clicking on the pile. 

Lattice of subgroups of the group . Conjugate subgroups are depicted as a pile of cards. In the software GIKoBo1, one can pull out individual conjugate subgroups by clicking on the pile. 
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 secondrank 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 nonequitranslational 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 nonferroelectric phase denotes a ferroic phase which is either nonpolar or which possesses a unique polar direction available already in the parent phase. A nonferroelastic phase exhibits no spontaneous strain.
The first systematic symmetry analysis of Landautype 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:

We note that Table 3.1.3.1 covers only those pointgroup symmetry descents that are `driven' by Rireps of G. All possible pointgroup symmetry descents are listed in Table 3.4.2.7 . Principal and secondary tensor parameters of symmetry descents associated with reducible representations are combinations of tensor parameters appearing in Table 3.1.3.1 (for a detailed explanation, see the manual of the software GIKoBo1 and Kopský, 2000). Necessary data for treating these cases are available in the software GIKoBo1 and Kopský (2001).

Example 3.1.3.4.1. Phase transition in triglycine sulfate (TGS). Assume that the space groups of both parent (highsymmetry) and ferroic (lowsymmetry) phases are known: , . The same number of formula units in the primitive unit cell in both phases suggests that the transition is an equitranslational one. This conclusion can be checked in the lattice of equitranslational subgroups of the software GIKoBo1. There we find for the lowsymmetry space group the symbol , where the vector in parentheses expresses the shift of the origin with respect to the conventional origin given in IT A (2005).
In Table 3.1.3.1, one finds that the corresponding pointgroupsymmetry descent is associated with irreducible representation . The corresponding principal tensor parameters of lowest rank are the pseudoscalar (enantiomorphism or chirality) and the vector of spontaneous polarization with one nonzero morphic component – the transition is a proper ferroelectric one. The nonferroelastic () full ferroelectric phase has two ferroelectric domain states (). Other principal tensor parameters (morphic tensor components that transform according to ) are available in the software GIKoBo1: , , , ; , , , , , , , . Property tensors with these components are listed in Table 3.1.3.3. As shown in Section 3.4.2 , all these components change sign when one passes from one domain state to the other. Since there is no intermediate group between G and F, there are no secondary tensor parameters.
Example 3.1.3.4.2. Phase transitions in barium titanate (BaTiO_{3}). We shall illustrate the solution of the inverse Landau problem and the need to correlate the crystallographic system with the Cartesian crystallophysical coordinate system. The spacegroup type of the parent phase is , and those of the three ferroic phases are , , , all with one formula unit in the primitive unit cell.
This information is not complete. To perform mode analysis, we must specify these space groups by saying that the lattice symbol P in the first case and the lattice symbol R in the third case are given with reference to the cubic crystallographic basis (), while lattice symbol C in the second case is given with reference to crystallographic basis . If we now identify vectors of the cubic crystallographic basis with vectors of the Cartesian basis by , , , where , , are three orthonormal vectors, we can see that the corresponding point groups are , , .
Notice that without specification of crystallographic bases one could interpret the point group of the space group as . Bases are therefore always specified in lattices of equitranslational subgroups of the space groups that are available in the software GIKoBo1, where we can check that all three symmetry descents are equitranslational.
In Table 3.1.3.1, we find that these three ferroic subgroups are epikernels of the Rirep with the following principal tensor components: , , , respectively. Other principal tensor parameters can be found in the main tables of the software GIKoBo1. The knowledge of the representation allows one to perform softmode analysis (see e.g. Rousseau et al., 1981).
For the tetragonal ferroelectric phase with , we find in Fig. 3.1.3.1 an intermediate group . In Table 3.1.3.1, we check that this is an epikernel of the Rirep with secondary tensor parameter . This phase is a full (proper) ferroelectric and partial ferroelastic one.
More details about symmetry aspects of structural phase transitions can be found in monographs by Izyumov & Syromiatnikov (1990), Kociński (1983, 1990), Landau & Lifshitz (1969), Lyubarskii (1960), Tolédano & Dmitriev (1996) and Tolédano & Tolédano (1987). Group–subgroup relations of space groups are treated extensively in IT A1 (2004).
In the preceding Section 3.1.3, a systematic tabulation of possible symmetry changes was provided for the class of equitranslational phase transitions. This tabulation derives from the principles described in Section 3.1.2, and relates the enumeration of the symmetry changes at structural transitions to the characteristics of the irreducible representations of the space group of the `parent' (highestsymmetry) phase adjacent to the transition. Systematic extension of this type of tabulation to the general case of transitions involving both a decrease of translational and of pointgroup symmetry has been achieved by several groups (Tolédano & Tolédano, 1976, 1977, 1980, 1982; Stokes & Hatch, 1988). The reader can refer, in particular, to the latter reference for an exhaustive enumeration of the characteristics of possible transitions. An illustration of the results obtained for a restricted class of parent phases (those associated with the point symmetry and to a simple Bravais lattice P) is presented here.
In order to clarify the content Table 3.1.4.1, let us recall (cf. Section 3.1.2) that Landau's theory of continuous phase transitions shows that the order parameter of a transition transforms according to a physically irreducible representation of the space group of the highsymmetry phase of the crystal. A physically irreducible representation is either a real irreducible representation of or the direct sum of two complexconjugate irreducible representations of . To classify the orderparameter symmetries of all possible transitions taking place between a given parent (highsymmetry) phase and another (lowsymmetry) phase, it is therefore necessary, for each parent space group, to list the various relevant irreducible representations.

Each irreducible representation of a given space group can be denoted and identified by two quantifies. The star , represented by a vector linking the origin of reciprocal space to a point of the first Brillouin zone, specifies the translational symmetry properties of the basis functions of . The dimension of is equal to the number of components of the order parameter of the phase transition considered. A given space group has an infinite number of irreducible representations. However, physical considerations restrict a systematic enumeration to only a few irreducible representations. The restrictions arise from the fact that one focuses on continuous (or almost continuous) transitions between strictly periodic crystal structures (i.e. in particular, incommensurate phases are not considered), and have been thoroughly described previously (Tolédano & Tolédano, 1987, and references therein).
Phase transitions in crystals are most sensitively detected via dynamic techniques. Two good examples are ultrasonic attenuation and internal friction. Unfortunately, while often exquisitely sensitive to subtle secondorder phase transitions [e.g. the work of Spencer et al. (1970) on BaMnF_{4}], they provide no real structural information on the lattice distortions that occur at such phase transitions, or even convincing evidence that a real phase transition has occurred (e.g. transition from one longrange thermodynamically stable ordered state to another). It is not unusual for ultrasonic attenuation to reveal a dozen reproducible anomalies over a small temperature range, none of which might be a phase transition in the usual sense of the phrase. At the other extreme are detailed structural analyses via Xray or neutron scattering, which give unambiguous lattice details but often totally miss small, nearly continuous rigid rotations of light ions, such as hydrogen bonds or oxygen or fluorine octahedra or tetrahedra. Intermediate between these techniques are phonon spectroscopies, notably infrared (absorption or reflection) and Raman techniques. The latter has developed remarkably over the past thirty years since the introduction of lasers and is now a standard analytical tool for helping to elucidate crystal structures and phase transitions investigated by chemists, solidstate physicists and materials scientists.
Landau (1937) developed a simple meanfield theory of phase transitions which implicitly assumes that each atom or ion in a system exerts a force on the other particles that is independent of the distance between them (see Section 3.1.2.2). Although this is a somewhat unphysical crude approximation to the actual forces, which are strongly dependent upon interparticle spacings, it allows the forces of all the other particles in the system to be replaced mathematically by an effective `field', and for the resulting equations to be solved exactly. This mathematical simplicity preserves the qualitative features of the real physical system and its phase transition without adding unnecessary cumbersome mathematics and had earlier been used to great advantage for fluids by Van der Waals (1873) and for magnetism by Weiss (1907). Landau's theory is a kind of generalization of those earlier theories. In it he defines an `order parameter' x, in terms of which most physical quantities of interest may be expressed via free energies. In a ferromagnet, the order parameter corresponds to the net magnetization; it is zero above the Curie temperature and increases monotonically with decreasing temperature below that temperature. In a liquid–gas phase transition the order parameter is the difference in density in the gas and liquid phases for the fluid.
Devonshire independently developed an equivalent theory for ferroelectric crystals around 1953 (Devonshire, 1954). For ferroelectrics, the order parameter is the spontaneous dielectric polarization P. In both his formalism and that of Landau, the ideas are most conveniently expressed through the free energy of the thermodynamic system: where A and C are positive quantities and B may have either sign. Scott (1999) shows that C changes sign at ferroelectrictosuperionic conducting transition temperatures. As shown in Fig. 3.1.5.1, minimization of the free energy causes the expectation value of P to go from zero above the Curie temperature to a nonzero value below. If B is positive the transition is continuous (`secondorder'), whereas if B is negative, the transition is discontinuous (`firstorder'), as shown in Fig. 3.1.5.2. The coefficient B may also be a function of pressure p or applied electric field E and may pass through zero at a critical threshold value of p or E. Such a point is referred to as a `tricritical point' and is marked by a change in the order of the transition from firstorder to secondorder. The term `tricritical' originates from the fact that in a threedimensional graph with coordinates temperature T, pressure p and applied field E, there are three lines marking the ferroelectric–paraelectric phase boundary that meet at a single point. Crossing any of these three lines produces a continuous phase transition (Fig. 3.1.5.3).

Free energy and order parameter from the Landau–Devonshire theory [equation (3.1.5.1a)] for a continuous secondorder ferroelectric phase transition [coefficient B positive in equation (3.1.5.1a)]. The insert shows the temperature dependence of the order parameter, i.e. the expectation value of the displacement . 

Free energy and order parameter from the Landau–Devonshire theory [equation (3.1.5.1a)] for a discontinuous firstorder ferroelectric phase transition [coefficient B negative in equation (3.1.5.1a)]. is the temperature (see Fig. 3.1.2.6) below which a secondary minimum appears in the free energy. 
Minimization of the free energy above leads to the dependence of spontaneous polarization P upon temperature given by for continuous transitions. In the more general case discussed by Landau, the polarization P is replaced by a generic `order parameter' with the same dependence. Cochran's contribution (1960, 1961) was to show that for continuous `displacive' (as opposed to `order–disorder') transitions, this order parameter is (or is proportional to) a normal mode of the lattice. One normal mode of the crystal must, in Cochran's theory, literally soften: the generalized force constant for this mode weakens as a function of temperature, and its frequency consequently decreases. This softmode theory provided an important step from the macroscopic description of Landau and Devonshire to a microscopic theory, and in particular, to vibrational (phonon) spectroscopy.
Cochran illustrated this theory using a `shell' model in which the electrons surrounding an ion were approximated by a rigid sphere; shell–shell force constants were treated as well as shell–core and core–core terms, in the general case. The initial application was to PbTe and other rocksalt cubic structures that undergo ferroelectric structural distortions.
For this simple case, the key equations relate the optical phonon frequencies of long wavelength to two terms: a shortrange force constant and a longrange Coulombic term. It is important that in general neither of these terms has a pathological temperature dependence; in particular, neither vanishes at the Curie temperature. Rather it is the subtle cancellation of the two terms at that produces a `soft' transverse optical phonon.
The longitudinal optical phonon frequency is positive definite and remains finite at all temperatures: where is a reduced mass for the normal mode; Ze is an effective charge for the mode, related to the valence state of the ions involved; is the highfrequency dielectric constant and is the unitcell volume, which is a function of temperature due to thermal expansion.
By comparison, the transverse optical phonon frequency can vanish accidentally when reaches a value that permits cancellation of the two terms. Note that this does not require any unusual temperature dependence of the shortrange interaction term . This description appears to satisfy all well studied ferroelectrics except for the `ultraweak' ones epitomized by TSCC (trissarcosine calcium chloride), in which the Coulombic term in (3.1.5.1b) and (3.1.5.1c) is very small and the pathological dependence occurs in . This leads to a situation in which the longitudinal optical phonon is nearly as soft as is the transverse branch.
Subsequent to Cochran's shellmodel developments, Cowley (1962, 1964, 1970) replaced this phenomenological modelling with a comprehensive manybody theory of phonon anharmonicity, in which the softmode temperature is dominated by Feynman diagrams emphasizing renormalization of phonon selfenergies due to fourphonon interactions (two in and two out). This contrasts with the threephonon interactions that dominate phonon linewidths under most conditions.
It is worth noting that the soft optical phonon branch is necessarily always observable in the lowsymmetry phase via Raman spectroscopy in all 32 pointgroup symmetries. This was first proved by Worlock (1971), later developed in more detail by Pick (1969) and follows grouptheoretically from the fact that the vibration may be regarded as a dynamic distortion of symmetry which condenses at to produce a static distortion of the same symmetry. Hence the vibration in the distorted phase has symmetry given by the product , which always contains the totally symmetric representation for any choice of . If is nondegenerate, its outer product with itself will contain only and there will be a single, totally symmetric soft mode; if is degenerate, there will be two or three soft modes of different symmetries, at least one of which is