Date Published: December 01, 2018
Publisher: International Union of Crystallography
Author(s): Michael A. Carpenter, Christopher J. Howard.
The combinations of phase transitions which occur in Heusler alloys in terms of order parameters and symmetry have been analysed using a group theoretical approach. It is shown how this approach can be applied to relevant examples.
Ferroelastic phase transitions in functional oxides are accompanied by symmetry-breaking shear strains which typically fall in the range ∼0.1–5% (Salje, 1993 ▸; Carpenter et al., 1998 ▸). Most can be understood in terms of some structural or electronic instability with a driving order parameter that gives rise to the strain by coupling. Although the strength of coupling between individual strain components, ei, and the order parameter, Q, is a material property, its form, λeiQ, λeiQ2, λei2Q, λei2Q2 …, depends on symmetry and is determined by rigorous group theoretical rules. The same symmetry rules apply to coupling between two or more order parameters in materials with multiple instabilities, and the form of this coupling determines how, for example, multiferroic materials may respond to an external electric or magnetic field. As set out for the cases of transitions in perovskites driven by combinations of octahedral tilting, ferroelectric displacements, atomic ordering and cooperative Jahn–Teller distortions, the group theory program ISOTROPY (Stokes et al., 2007 ▸) has allowed such relationships to be tabulated even for the most complex cases (Howard & Stokes, 1998 ▸, 2004 ▸, 2005 ▸; Stokes et al., 2002 ▸; Carpenter & Howard, 2009 ▸).
Applications of the group theoretical approach set out above can be illustrated with three specific examples, using alloys relating to NiTi, TiPd and Ni2MnGa.
Differences in the symmetry properties of martensitic structures define distinct patterns of thermodynamic behaviour and are not simply matters of form or representation. The most obvious way to distinguish between them is by observing variations in the elastic constants, as set out more generally, for example, by Carpenter & Salje (1998 ▸). Due to bilinear coupling of a symmetry breaking shear strain with the primary order parameter, λesbq, transitions driven by the order parameter will show pseudoproper ferroelastic softening of C11–C12 and those driven by will show pseudoproper ferroelastic softening of C44 as temperature reduces towards the transition point. Transitions driven by a Σ2 (or ) order parameter will be improper ferroelastic with stepwise softening in either or both of C11–C12 and C44 below the transition point due to coupling of the form λesbq2.
Group theoretical analysis of order parameters relating to atomic ordering, electronic instabilities and soft-mode behaviour has been used to specify the symmetry relationships which can lead to a wide variety of structures in alloys with multiple premartensitic and martensitic phase transitions.