To superspace and beyond

Abstract

Much progress has been made in understanding the structure-property relationship over the past few decades. This is partly due to our improved ability to measure the subtle structural distortions that are either linked to the development of a specific physical property or that lead to its suppression. A very significant contribution in this field has also been the development of the language required to formally describe such symmetrybreaking events that characterize the order parameters that drive these phase transitions. However, until now, the tools available have only allowed for the exploration of the symmetry implications of order parameters corresponding to commensurate modulations. In this issue, Stokes & Campbell (2017) present an algorithm for computing the allowed subgroups arising from any number of incommensurate order parameters and for identifying the resulting (3 + d)-dimensional superspace groups. They also describe a practical implementation of this algorithm for cases consisting of up to three arbitrary independent modulations. This work provides key tools for understanding the structureproperty relationships of the many technologically important materials that display incommensurate modulations in their atomic and/or magnetic structure.While Landau theory has been widely used since the 1960s to describe second-order phase transitions (Cowley, 1980), often the starting point has been to use an order parameter that is phenomenological. That is to say, the order parameter is related to an observation of the physical property, such as the magnitude of the measured electrical or magnetic polarization. However, this need not be the case, and indeed formally each order parameter that contributes to the phase transition (leading to a child structure) must transform as an irreducible representation (Campbell et al., 2006) of the highsymmetry (parent) structure. Or put another way, it must be an allowed excitation of the parent structure, which in the harmonic approximation will transform as an irreducible representation of the parent space group. With the use of invariant analysis (Hatch & Stokes, 2003), this allows a full expansion of the free energy to be performed about the parent structure in terms of the the allowed excitation(s) beyond just the harmonic (second) order, provided that the secondary 'symmetry-allowed' couplings of the primary order parameter can be elucidated.As crystallographers, experimentally what we determine is the positions of the atoms above and below a phase transition. If we are to pursue the analysis described above, then we must identify the set of atomic displacements that lead to a phase transition from parent to child structure. One may then decompose these symmetry-breaking distortions into order parameters of the irreducible representations of the parent space group. This process may simply be thought of as a change of basis to one which is orthogonal in the symmetry space of the parent structure. A substantial bottleneck to all of this analysis is hence not just ou...

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