Fundamental aspects of symmetry and order parameter coupling for martensitic transition sequences in Heusler alloys

Abstract

<jats:p>Martensitic phase transitions in which there is a group–subgroup relationship between the parent and product structures are driven by combinations of soft-mode and electronic instabilities. These have been analysed from the perspective of symmetry, by considering possible order parameters operating with respect to a parent structure which has space group Im{\bar 3} m. Heusler structures with different stoichiometries are derived by operation of order parameters belonging to irreducible representations {\rm H}^{+}_{1}and P1 to describe the atomic ordering configurations. Electronic instabilities are ascribed to an order parameter belonging to the Brillouin zone centre, \Gamma^{+}_{3}, which couples with shear strains to give tetragonal and orthorhombic distortions. An additional zone centre order parameter, with \Gamma^{+}_{5} symmetry, is typically a secondary order parameter but in some cases may drive a transition. Soft-mode instabilities produce commensurate and incommensurate structures for which the order parameters have symmetry properties relating to points along the Σ line of the Brillouin zone for the cubic <jats:italic>I</jats:italic> lattice. The electronic and soft-mode order parameters have multiple components and are coupled in a linear–quadratic manner as \lambda q_{\Gamma}q_{\Sigma}^{2}. As well as providing comprehensive tables setting out the most important group–subgroup relationships and the order parameters which are responsible for them, examples of NiTi, RuNb, Ti<jats:sub>50</jats:sub>Ni<jats:sub>50−<jats:italic>x</jats:italic> </jats:sub>Fe<jats:italic> <jats:sub>x</jats:sub> </jats:italic>, Ni<jats:sub>2+<jats:italic>x</jats:italic> </jats:sub>Mn<jats:sub>1−<jats:italic>x</jats:italic> </jats:sub>Ga and Ti<jats:sub>50</jats:sub>Pd<jats:sub>50−<jats:italic>x</jats:italic> </jats:sub>Cr<jats:italic> <jats:sub>x</jats:sub> </jats:italic> are used to illustrate practical relevance of the overall approach. Variations of the elastic constants of these materials can be used to determine which of the multiple order parameters is primarily responsible for the phase transitions that they undergo.</jats:p>