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

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.


Introduction
Ferroelastic phase transitions in functional oxides are accompanied by symmetry-breaking shear strains which typically fall in the range $0.1-5% (Salje, 1993;. 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, e i , and the order parameter, Q, is a material property, its form, e i Q, e i Q 2 , e i 2 Q, e i 2 Q 2 . . . , 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 ISSN 2052-5206 complex cases (Howard & Stokes, 1998Stokes et al., 2002;Carpenter & Howard, 2009).
Martensitic transitions in which there is a group/subgroup relationship between parent and product structures, such as in the cases of Heusler compounds and shape memory alloys based on NiTi, may appear to be different because of the much larger shear strains involved (typically ! 10%), but they are still essentially ferroelastic. Multiple instabilities are also characteristic and the relevant order parameters relate to atomic ordering, band Jahn-Teller effects, magnetic ordering, superconductivity and soft modes. This leads to a great diversity of structures and structure-property relationships with potential for device applications. Exactly the same group theoretical constraints apply as for perovskite superstructures, and these determine the form of coupling of different order parameters with strain, permissible couplings between different order parameters and the full range of possible structures which might result.
The primary objective of the present paper is to present a group theoretical treatment of martensitic materials which can be derived from the simplest b.c.c. parent structure with space group Im " 3 3m. It has been notoriously difficult to distinguish between structure types on the basis of diffraction observations alone when the distinctions involve subtle differences in screw axes or glide planes. The software package ISOTROPY produces lists of allowable space groups which are definitive for subgroup structures and can be used to resolve such ambiguities. In addition, strain fields are long ranging so that the interaction length of the order parameter(s) is (are) also long ranging. As a consequence, critical fluctuations tend to be suppressed and the resulting changes in physical properties are expected to evolve according to mean field behaviour. Landau theory therefore provides a rigorous and quantitative framework for representing the thermodynamic and structural evolution of martensitic phases with single or multiple instabilities in response to changing temperature, pressure, stress, magnetic field and electric field. Finally, it is well understood that particular properties of interest can be engineered or tuned by changing other properties. In other words, one order parameter, such as for atomic ordering, can be adjusted to optimize the evolution of a second, such as magnetic moment, to produce, say, a desirable magnetocaloric response. These interactions will differ according to the form of allowed coupling between two (or more) order parameters, as Q 1 Q 2 , Q 1 Q 2 2 , Q 1 2 Q 2 , Q 1 2 Q 2 2 .
2. Group theoretical analysis 2.1. Parent structures Table 1, after Graf et al. (2011), lists the generic stoichiometry and structures of Heusler-type phases (XX 0 YZ) which can be derived from a parent body-centred cubic (b.c.c.) structure. Here X, X 0 , Y, Z represent different elements that can combine together. Ordering of atoms according to order parameters with symmetry determined by irreducible representations of space group Im " 3 3m are also given [using the notation of Miller & Love (1967) here and throughout the rest of the paper]. These belong to the special points P, [1/2,1/2, 1/2], and H, [0,1,0], of the Brillouin zone (Fig. 1), and give rise to four distinct subgroups. For example, the B2 structure of NiTi with space group Pm " 3 3m has a single nonzero order parameter with H þ 1 symmetry. The L2 1 structure of Cu 2 MnAl, which is the classic X 2 YZ Heusler structure, has Table 1 Derivative structures based on a body centred cubic parent structure with space group Im " 3 3m (after Graf et al., 2011).

P1
Unit-cell edge with respect to Im " 3 3m

Figure 1
Brillouin zone for Im " 3 3m structures. Atomic ordering to give subgroup structures listed in Table 1 is based on order parameters belonging to irreducible representations (irreps) at special points H and P. space group Fm " 3 3m and two nonzero order parameter components, one with H þ 1 symmetry and the second with P1 symmetry. The DO 3 structure of BiF 3 is similar, where now X = Y. The different ordered structures form a hierarchy of subgroup structures from the Im " 3 3m parent, as set out in Fig. 2. Solid lines in this figure represent phase transitions which are allowed by symmetry to be thermodynamically continuous according to Landau although, because they require rearrangement of atoms, would be expected to be slow.

Martensite structures
The ferroelastic transitions which give rise to martensitic phases are characterized primarily by two effects, substantial shear strains and the development of large unit cells. Both depend on the symmetry of the driving order parameter(s) and their coupling with strain. Most of the observed product structures appear to be understandable in terms of separate order parameters which have symmetry properties related to the Brillouin zone centre (À point in Fig. 1) and points along one of the h110i* directions of the reciprocal lattice for Im " 3 3m structures (AE line of Fig. 1). These are set out in Table 2 for a single reference structure with space group Im " 3 3m (the A2 structure in Table 1). If the transitions were driven solely by an electronic instability, such as band Jahn-Teller in Ni 2 MnGa (Fujii et al., 1989;Brown et al., 1999), the order parameter components would belong to irrep À þ 3 in most cases and the product structures would be tetragonal or orthorhombic. For example, À þ 3 (a,0) would give structures with space groups I4/mmm, P4/mmm, I4 1 /amd or I " 4 4m2, depending on the form of atomic order, and À þ 3 (a,b) would give corresponding orthorhombic structures (Table 2). A À þ 5 order parameter is also possible, however, and in the simplest cases would give orthorhombic structures with space groups Fmmm, Cmmm, Immm, Imma or Imm2 (Table 2).
Repeat distances along [110]* (with respect to the cubic I lattice) 1 are observed to be incommensurate in some cases but are commonly referred to in terms of a commensurate repeat, n, such as 3, 5 and 7 for 3M, 5M and 7M structures, where n corresponds to the number of atomic layers parallel to (110) involved in a particular sequence of atomic displacements.The layers may be slightly displaced according to a conventional sinusoidal modulation or, as illustrated for example by Otsuka et al. (1993), displaced (shuffled) in consequence of the stacking characteristics of these nearly close-packed planes. In either case, we can describe the situation using irrep AE 2 at k vector (1/n,1/n,0) with just one component of the 12 component order parameter nonzero. The incommensurate case can be treated using the same 12 component AE 2 order parameter with just one nonzero component, by taking the k vector for the active representation to be (,,0). Otsuka et al. (1993) introduced a new description in which 3M, 5M and 7M were relabelled as 6M, 10M and 14M because they chose to describe the structures on centred unit cells. In the 5M/10M structure, for example, the (110) layers have a sequence of five shuffles that must occur twice in the unit cell to achieve a B-centred rather than primitive (in the case of a primitive starting structure) cell. An earlier nomenclature, for at least some of these martensites, is based on the number of (110) layers, in most cases a larger number, needed to complete a stacking sequence for these nearly close-packed atomic layers.
Considering the example of a parent structure with Pm " 3 3m ordering (from Table 2, see also Table 3  Hierarchy of ordered structures, as specified with respect to order parameters belonging to irreps H þ 1 and P1. The transitions indicated by solid lines are allowed to be continuous according to Landau theory. Table 2 Symmetry relationships, order parameters and unit-cell configurations for selected subgroups of space group Im " 3 3m, as derived using the group theory program ISOTROPY (Stokes et al., 2007).
Two orientations have been given in some cases for À þ 3 , (a,0) and (a,a ffiffi ffi 3 p ), to illustrate how this choice affects basis vectors which define the unit cell of the subgroup structure.
Labels in the last column are taken from the literature, including, in particular, from Otsuka et al. (1993). For a structure with ordering on the basis of Fm " 3 3m, the orthorhombic product structures have space groups Imm2 (n = 3), Pmma (n = 4), Imm2 (n = 5), and the monoclinic structures have space group P2/m (n = even) or C2/m (n = odd). Comparison of these with known structures needs to take account of the fact that the values of n in Table 2 refer to Im " 3 3m as the parent structure. The Fm " 3 3m structure has a unit cell which is double the dimensions of the Im " 3 3m cell, so that n I = 6 (k = 1/6,1/6,0) with respect to the latter becomes n F = 3 (k = 1/3,1/3,0) with respect to the former. The Pmma structure reported by Brown et al. (2006) as the product of a phase transition from a parent structure with space group Fm " 3 3m has n F = 2 (k = 1/2,1/2,0), and would correspond to the structure with n I = 4 (k = 1/4,1/4,0) in Table 2. The Pnnm structure with n F = 3 reported by Brown et al. (2002) would correspond to the structure with n I = 6 (k = 1/6,1/6,0) in Table 2, and similarly for n F = 7, n I = 14. The P2/m structure described by Brown et al. (2011) has n F = 3, k = (1/3,1/3,0) and corresponds to the structure with n I = 6, k = (1/6,1/6,0) in Table 2. Table 3 contains the same information as Table 2 for the specific case of a Pm " 3 3m parent structure, in a slightly different format that might prove to be more practicable when considering B2 structures such as NiTi and NiAl or TiAl and RuNb. The zone boundary irrep N À 4 becomes M À 5 so that the structural relationships acquire the more familiar form for Pm " 3 3m, Pmma and P2 1 /m structures as already set out by Barsch (2000). À þ 3 (a,0) gives P4/mmm, corresponding to the 0 structure of RuNb stable between $1030 and $1170 K (e.g. Dirand et al., 2012;Nó et al., 2015aNó et al., , 2015b, the roomtemperature structure of TiAl (Duarte et al., 2012) and the structure of Ni x Al 1Àx , x ' 0.64, quenched from high temperatures (Potapov et al., 1997). The low-temperature ( 00 ) structure of RuNb has been reported to be either orthorhombic, Cmmm (Chen & Franzen, 1989), or monoclinic, P2/m (Nó et al., 2015a,b) or P2 1 /m (Mousa et al., 2009). All three of these structure types would have the same unit cell as some permutation of ffiffi ffi 2 p a o Â ffiffi ffi 2 p a o Â a o , but differing in the combination of driving order parameters.
Other sets of structures can be generated by considering kactive as having directions along several of the h110i * directions, instead of just one. For example, if there are three equivalent directions, (1/3,1/3,0), (1/3,0,1/3), (0,1/3,À1/3), a trigonal structure is obtained from a Pm " 3 3m parent. This is the R-phase observed in Ni-Ti and Au-Cd alloys (e.g. Otsuka & Ren, 2005;Zolotukin et al., 2012), and can be generated with (a,0,0,0,a,0,0,0,0,0,a,0) as components of the AE 2 order parameter (Table 3). As reviewed by Otsuka & Ren (2005), various suggestions have been made for the correct space group of this structure, including P " 3 31m (Vatanayon & Hehemann, 1975;Goo & Sinclair, 1985), P3 (Ohba et al., 1992;Hara et al., 1997) and P " 3 3 (Schryvers & Potapov, 2002;Sitepu, 2003). The group theoretical treatment set out here gives space group P " 3 3 for the particular combination of order parameters listed in Table 3. If there are just two equivalent directions, (1/3,1/3,0), (1/3,0,1/3), tetragonal structures will result, but these have not been explored further. Examples of the graphics output from ISODISTORT (Stokes et al., 2017). (a) An incommensurate modulation with k vector (0.143,0.143,0) applied to a parent structure in Pm " 3 3m (e.g. NiTi, Ni red, Ti blue). The basic space group for the distorted structure is Ammm, and the figure shows, as well as the parent cell, the cell corresponding to this basic (average) symmetry. Note that the basic symmetry is orthorhombic. The modulation vector is along the z axis of the Ammm cell, and the period is 1/0.143, i.e. approximately seven (110) planes. (b) and (c) show results obtained from applying a commensurate modulation, k vector (1/7,1/7,0). It can be seen that, though the displacements have a period of seven (110) planes, the atomic arrangement precludes the construction of a simple unit cell with this period. The unit cell in (b) is obtained in orthorhombic symmetry, Amm2, by extending the cell to 14 (110) planes, and the unit cell in (c) by resorting to the monoclinic symmetry P2/m. The symmetries in (b) and (c), and especially the monoclinic symmetry in (c), may be artefacts arising from commensurate choices for the modulation vector k.

Primary and secondary order parameters
Inspection of Table 2 reveals that the À þ 3 order parameter can act on its own, whereas nonzero values of components of N À 4 and AE 2 are always accompanied by nonzero values of components from both À þ 3 and À þ 5 . The latter can just be secondary order parameters, consequential on coupling to tetragonal and orthorhombic shear strains, e t and e o (À þ 3 ), or  Table 4 Symmetry relationships, order parameters and unit cell configurations for selected subgroups of space group Fm " 3 3m.
Note that components of the k-active vector are a factor of two larger here than for the same structures in Table 2, due to the fact that the parent Fm " 3 3m structure has a unit cell with dimensions twice those of the Im " 3 3m parent cell. For the same reason, the lattice vectors listed to describe the origin and basis are halved relative to those shown in Table 2. Finally, we note that the origin of space group Fm " 3 3m is at (1/2,1/2,1/2) with respect to the Im " 3 3m cell.
If the À þ 3 order parameter acts alone, the pattern of spontaneous strains is determined by coupling terms in the Landau free-energy expansion Á : Here q represents order parameter components, a, b, c are standard Landau coefficients, 's are coupling coefficients, T cÀ3þ is the critical temperature, e a (= e 1 + e 2 + e 3 ) is the volume strain, e t [= (2e 3 À e 1 Àe 2 )= ffiffi ffi 3 p ] is the tetragonal shear strain, e o (= e 1 À e 2 ) is the orthorhombic shear strain, e 4 , e 5 and e 6 are the remaining shear strains, and C o 11 , C o 12 , C o 44 are elastic constants of the parent cubic structure. If the À þ 5 order parameter acts alone, the Landau expansion is Á : If the single order parameter is N À 4 or AE 2 , the equivalent Landau expansion requires six or 12 components, respectively, though the space groups of real structures so far identified can be understood with just one or two nonzero values. The generality of couplings with strain is that they must be linearquadratic, eq 2 , or biquadratic, e 2 q 2 . For each of the three cases, the relationship(s) between individual strains and the driving order parameter(s) can be found by applying the equilibrium condition, @G/@e = 0, in the usual way (e.g. . In materials with multiple instabilities, coupling between the separate order parameters can be direct or indirect via the common strain. The simplest generalization here is for coupling between a zone centre order parameter, q À , and an order parameter from along the AE line out to the N point, q AE . Biquadratic coupling, q 2 À q 2 AE , is always allowed between two order parameters with different symmetries and a wide variety of sequences of structures and phase transitions can result (Salje & Devarajan, 1986). The important parameters are the strength of coupling, , and the relative critical temperatures of the two instabilities, T cÀ and T cAE . Linear-quadratic coupling, q À q 2 AE , is also allowed for some combinations, but leads to a much more restricted range of possibilities (Salje & Carpenter, 2011). In principle, T cAE > T cÀ would be expected to give rise to a single transition from a state with q À = 0, q AE = 0 to one with q À 6 ¼ 0, q AE 6 ¼ 0 because q AE generates a conjugate field for q À . Alternatively, for T cAE < T cÀ , the sequence can be a second-order transition to a structure with q À 6 ¼ 0, q AE = 0, followed by a first-order transition to a phase with q À 6 ¼ 0, q AE 6 ¼ 0. Coupling terms between À þ 3 and À þ 5 can in principle also be linear-quadratic and biquadratic as: Indirect coupling via shear strains would give the linearquadratic term while coupling via the volume strain would give rise to the biquadratic term. An example of coupling between order parameters for instabilities with two nonzero components of À þ 3 and one nonzero component of M À 5 , with respect to a parent Pm " 3 3m structure, can be represented by the Landau expansion: Á :

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From Table 3, if the nonzero components of À þ 3 are (a, À ffiffi ffi 3 p a) and the nonzero components of M À 5 are (0,0,c,c,0,0), the resultant structure has Pmma symmetry (B19 structure). This has À þ 5 (0,0,b), i.e. the shear strain e 5 , as a secondary order parameter. However, the same outcome could be obtained using À þ 5 with M À 5 as primaries and À þ 3 as secondary, or taking M À 5 as driving and both À þ 3 and À þ 5 as secondary. Treatment of magnetic transitions is beyond the scope of the present work but all the same symmetry and strain coupling arguments would apply. The only fundamental difference is that the coupling of a magnetic order parameter M with strains e will be of the form eM 2 or e 2 M 2 . It follows that pseudoproper ferroelastic softening will not be observed if the transition is driven by the magnetic instability. A Landau expansion which includes strain as a driving order parameter, an order parameter for the structural modulations and the magnetic order parameter has been given by Vasil'ev et al. (2003). A simpler form, with only the À-point and magnetic order parameters, is given in Vasil'ev et al. (1999).

Figure 5
Simplified phase diagram showing the variation of transition temperatures for B2-incommensurate (IC) and B2-B19, IC-9R, IC-incommensurate martensite (ICM) transitions at the Pd-rich end of the TiPd-TiCr solid solution. The first-order martensitic transition occurs in stoichiometric TiPd at $810 K (Matveeva et al., 1982;Enami & Nakagawa et al., 1993). Vertical dashed lines are approximate composition limits for different martensitic phases observed at room temperature, based on observations of Enami et al. (1989) and Schwartz et al. (1995). structure has IC repeat distances derived from the AE 2 order parameter over a range between $3 and $5. This pattern is similar to that of other Ti-Pd alloys with V, Mn, Fe, Ce or Ni as the additional, minor component (Enami & Nakagawa, 1993).
Linear-quadratic coupling, q À3þ q 2 AE2 is again allowed by symmetry but the transition sequences with falling temperature are the same as observed for Ni 2+x Mn 1Àx Ga in not complying with what would be expected from the generalized treatment of Salje & Carpenter (2011). In this system, the contributions of q AE2 clearly increase with increasing Cr content as the transition temperature for structures with q À3+ 6 ¼ 0 reduces. Other martensite materials with groupsubgroup relationships need to be examined, but it appears that biquadratic coupling may be dominant in systems with band Jahn-Teller transitions.

Patterns of elastic anomalies due to strain-order parameter coupling
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 . Due to bilinear coupling of a symmetry breaking shear strain with the primary order parameter, e sb q, transitions driven by the À þ 3 order parameter will show pseudoproper ferroelastic softening of C 11 -C 12 and those driven by À þ 5 will show pseudoproper ferroelastic softening of C 44 as temperature reduces towards the transition point. Transitions driven by a AE 2 (or M À 5 ) order parameter will be improper ferroelastic with stepwise softening in either or both of C 11 -C 12 and C 44 below the transition point due to coupling of the form e sb q 2 .
In some previous Landau expansions produced to describe the electronic and soft mode instabilities with order parameters belonging separately to zone centre and zone boundary irreps, strain itself was used as the driving order parameter for the electronic part (e.g. Entel et al., 2006;Vasil'ev et al., 2003). In other words, the expectation was for a true-proper, as opposed to pseudo-proper, ferroelastic transition, with specific implications for the evolution of the elastic constants (e.g. . The pattern of evolution of the shear modulus, at least, for the simplest case of the Pm " 3 3m-P4/mmm transition in Ru-Nb, which involves only the À þ 3 order parameter, is of nonlinear softening as the transition point is approached from both sides (Dirand et al., 2012;Nó et al., 2015a,b). This fits with pseudoproper behaviour which, in turn, suggests that it is the change in electronic structure and not the strain that provides the driving order parameter.
The compilation of temperature-dependent single-crystal elastic constants given by Otsuka & Ren (2005, their Fig. 38) for Ni-Ti-Fe and Ni-Ti-Cu alloys shows softening of both C 11 -C 12 and C 44 as the martensitic transitions are approached from above. This confirms the proximity of electronic instabilities with symmetries belonging to both À þ 3 and À þ 5 .
The pattern of evolution of both C 11 -C 12 and C 44 in Ni 2 MnGa ahead of and through the L2 1 (Fm " 3 3m) to IC ($3M, Pnnm) transition (e.g. Mañ osa et al., 1997;Stipcich et al., 2004) is characteristic of improper ferroelastic behaviour, implying that the driving order parameter relates predominantly to AE 2 and, hence, that À þ 3 is secondary. Some precursor softening of C 11 -C 12 has been reported by Stipcich et al. (2004), however, and this was enhanced following heat treatments (Seiner et al., 2013). A driving role clearly can exist for À þ 3 but with a strength that depends on the structural state of the sample. The additional factor controlling this strength is most likely the degree of atomic order, as could be expressed in terms of coupling of AE 2 and À þ 3 order parameters with H þ 1 and P1 order parameters. This coupling is biquadratic in lowest order, q 2 AE q 2 H , q 2 AE q 2 P , q 2 À q 2 H , q 2 À q 2 P . As a consequence, the effects of changes in the degree of atomic order are most likely to be seen as renormalization of the critical temperature for the martensitic and soft-mode transitions. This is exactly analogous to the influence of Fe/Mo ordering on phase transitions in Sr 2 FeMoO 6 (Yang et al., 2016).

Conclusions
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.
Coupling between order parameters can be direct or indirect via coupling with common strains. The most significant coupling in this context is between À-point and AE 2 order parameters, with both linear-quadratic and biquadratic terms allowed. In the small number of materials considered as examples here, the characteristic sequences of transformations expected from linear-quadratic coupling are not observed, however.
Transformation sequences and phase stabilities in a given material depend on the balance of energies associated with each of the possible order parameters. The composition and degree of atomic order can be chosen so that, in principle, the different order parameters and the strength of their coupling can be engineered to produce optimal properties in functional materials.
In terms of testing models of multiple phase transitions in martensitic phases, observed patterns of elastic constants are likely to prove definitive, because of the characteristic patterns of elastic softening and stiffening in ferroelastic materials due to bilinear, linear-quadratic and biquadratic coupling with strains.