International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 3.1, pp. 338340

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.
References
Aizu, K. (1969). Possible species of ferroelastic crystals and of simultaneously ferroelectric and ferroelastic crystals. J. Phys. Soc. Jpn, 27, 387–396.Aizu, K. (1973). Second order ferroic states. J. Phys. Soc. Jpn, 34, 121–128.
Lines, M. E. & Glass, A. M. (1977). Principles and applications of ferroelectrics and related materials. Oxford University Press.
Tolédano, J.C. & Tolédano, P. (1987). The Landau theory of phase transitions. Singapore: World Scientific.
Tolédano, P. & Dmitriev, V. (1996). Reconstructive phase transitions. Singapore: World Scientific.