There is an alternate approach to considering the process of phase transformations known as “representational analysis”, or sometimes called “irreducible representational analysis”. This is yet another subject where I am far from an expert. The history of representational analysis comes more from a spectroscopy background than from one in structural science, but in recent years there have been major advances in reconciling the nomenclature of subgroup-supergroup relationships and representational analysis. The Bilbao Crystallographic Server (§28.4) includes a number of tools for representational analysis, but Harold Stokes and Dorian Hatch have spent a lifetime developing tools and nomenclature for representational analysis under the collective name of ISOTROPY. Thanks to the work of Stokes, Hatch, and especially Branton Campbell, as well as others, there is now a Brigham Young University website, called the ISOTROPY Software Suite, https://iso.byu.edu, which provides a number of useful tools for applying representational analysis to a structure. This is discussed further below in §28.5.
Typical applications for representational analysis (RA) will be to describe coordinated displacement of atoms, or groupings of occupancy changes, or placing magnetic spins on atoms. Lattice strain and rotational distortions of atom groups are also possible. To provide a quick overview of RA, the symmetry of a material is classified in terms of symmetry building blocks (matrices) that cannot be further simplified, known as irreducible representations (irreps to those in the business). These irreps will be used to generate RA distortion modes which describe the relaxing the symmetry of the model by changing parameters by a formula from the value assigned to the RA mode. Representational analysis will relate two structural models, one with higher symmetry and one with lower symmetry, by a series of RA distortion modes, which each describe a set of shifts that can be applied to the model. Note that the two structural models linked by RA will be a parent-subgroup relationship, but there can be a number of subgroup links between the pair.
Rather than vary a single atomic parameter is done in conventional crystallographic analysis, a RA mode will usually group together collective changes. For example, a RA mode might move some atoms down while moving others up. Frequently, the changes associated with an RA mode make much more chemical sense than discrete changes to crystallographic parameters, just as in spectroscopy one thinks of vibrational modes as groups of atom moving with “bends” and “stretches” rather than thinking about the individual coordinate changes. However, the number of crystallographic structural degrees of freedom will always match the total number of RA distortion modes. So, as an example if in our lower symmetry setting, we have only two atoms can be moved and those two atoms are fixed to lie on a plane, but can each be moved in two different directions, e.g. so x and y can be changed for each atom, then there are 4 free atomic parameters, so we will also have 4 RA modes. The RA modes are formulated so that if all distortion modes are set to values of 0, the structure is in the highest symmetry setting, but as the value changes from 0, the model is reduced in symmetry. Thus, the RA modes values provide a continuous way to describe how a structure is lowered in symmetry, even though the structure will potentially move between several discrete subgroups. The Landau theory order parameter will be a vector of RA distortion modes values.
Representational analysis provides a mechanism for exploring changes in symmetry. As well, it is useful for categorizing potential changes in structure. However, in the end, the goal of crystallography is to describe a material’s structure and for that a space group assignment is needed.