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Fundamental aspects of symmetry and coupling for sequences in Heusler alloys
aDepartment of Earth Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EQ, UK, and bSchool of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
*Correspondence e-mail: mc43@esc.cam.ac.uk
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 ![[Im{\bar 3} m]](teximages/ra5037fi1.gif)
. Heusler structures with different stoichiometries are derived by operation of order parameters belonging to irreducible representations ![[{\rm H}^{+}_{1}]](teximages/ra5037fi2.gif)
and P1 to describe the atomic ordering configurations. Electronic instabilities are ascribed to an belonging to the centre, ![[\Gamma^{+}_{3}]](teximages/ra5037fi3.gif)
, which couples with shear strains to give tetragonal and orthorhombic distortions. An additional zone centre with ![[\Gamma^{+}_{5}]](teximages/ra5037fi4.gif)
symmetry, is typically a secondary 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 for the cubic I 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}]](teximages/ra5037fi5.gif)
. 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, Ti50Ni50−xFex, Ni2+xMn1−xGa and Ti50Pd50−xCrx 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.
Keywords: martensite; phase transitions; group theory; Heusler alloys; order parameters.
1. 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![[Salje, E. K. H. (1993). Phase Transitions in Ferroelastic and Co-Elastic Crystals. Cambridge University Press, Cambridge.]](../../../../../../logos/arrows/b_arr.gif "Salje, E. K. H. (1993). Phase Transitions in Ferroelastic and Co-Elastic Crystals. Cambridge University Press, Cambridge.")
; Carpenter et al., 1998). Most can be understood in terms of some structural or electronic instability with a driving that gives rise to the strain by coupling. Although the strength of coupling between individual strain components, ei, and the 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).
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 . 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 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 such as for atomic ordering, can be adjusted to optimize the evolution of a second, such as 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 λQ1Q2, λQ1Q22, λQ12Q2, λQ12Q22.
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′YZ) which can be derived from a parent body-centred cubic (b.c.c.) structure. Here X, X′, Y, Z represent different elements that can combine together. Ordering of atoms according to order parameters with symmetry determined by irreducible representations of 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 (Fig. 1), and give rise to four distinct subgroups. For example, the B2 structure of NiTi with ![[Pm {\bar 3}m]](teximages/ra5037fi8.gif)
has a single nonzero with H1+ symmetry. The L21 structure of Cu2MnAl, which is the classic X2YZ Heusler structure, has ![[Fm{\bar 3} m]](teximages/ra5037fi10.gif)
and two nonzero components, one with H1+ symmetry and the second with P1 symmetry. The DO3 structure of BiF3 is similar, where now X = Y. The different ordered structures form a hierarchy of structures from the 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.
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![[Fm\bar 3m]](teximages/ra5037fi168.gif)
structures ( ![[Figure 1]](ra5037fig1thm.gif) | Figure 1 Brillouin zone for structures. Atomic ordering to give structures listed in Table 1 is based on order parameters belonging to irreducible representations (irreps) at special points H and P. |
![[Figure 2]](ra5037fig2thm.gif) | Figure 2 Hierarchy of ordered structures, as specified with respect to order parameters belonging to irreps and P1. The transitions indicated by solid lines are allowed to be continuous according to Landau theory. |
2.2. 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 centre (Γ point in Fig. 1) and points along one of the 〈110〉* directions of the for structures (Σ line of Fig. 1). These are set out in Table 2 for a single reference structure with (the A2 structure in Table 1). If the transitions were driven solely by an electronic instability, such as band Jahn–Teller in Ni2MnGa (Fujii et al., 1989; Brown et al., 1999), the components would belong to irrep in most cases and the product structures would be tetragonal or orthorhombic. For example, (a,0) would give structures with space groups I4/mmm, P4/mmm, I41/amd or ![[I{\bar 4} m2]](teximages/ra5037fi17.gif)
, depending on the form of atomic order, and (a,b) would give corresponding orthorhombic structures (Table 2). A is also possible, however, and in the simplest cases would give orthorhombic structures with space groups Fmmm, Cmmm, Immm, Imma or Imm2 (Table 2).
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![[\Gamma_{3}^{+}]](teximages/ra5037fi179.gif)
, (![[\sqrt 3]](teximages/ra5037fi180.gif)
), to illustrate how this choice affects basis vectors which define the ![[Im{\bar 3}m]](teximages/ra5037fi184.gif)
![[Fm{\bar 3}m]](teximages/ra5037fi186.gif)
![[Fd{\bar 3}m]](teximages/ra5037fi187.gif)
![[F{\bar 4}3m]](teximages/ra5037fi188.gif)
![[I{\bar 4}m2]](teximages/ra5037fi189.gif)
![[{\sqrt 3}]](teximages/ra5037fi205.gif)
,0,0,0,0,0,0,0,0,0,0)By way of contrast, the driving mechanism for transitions with order parameters belonging to points along the Σ line is generally considered to involve an incipient soft mode [e.g. in Ni–Mn–Ga alloys (Stuhr et al., 1997; Mañosa et al., 2001; Moya et al., 2006) and in Ti–Pd–Cr (Shapiro et al., 2007)]. Observed repeats along [110]* of the reference structure, varying between 2 and ∼14 (110) planes, correspond to k vectors for the active representation (k-active in Table 2) of between (1/2,1/2,0) and ∼(1/14,1/14,0); k-active = (1/2,1/2,0) corresponds to the N-point, [1/2, 1/2, 0], of the for structures (Fig. 1). Taking N-4 as the active representation leads to a variety of orthorhombic or monoclinic structures depending on whether the contribution is (a,0) or (a,b), respectively. Combining (a,0) and N-4 (0,0,0,0,a,0) leads to structures with space groups Cmcm, Pmma, Pmmn, Pnma and Pmn21 as subgroups of , , , ![[Fd {\bar 3}m]](teximages/ra5037fi29.gif)
and ![[F{\bar 4}3 m]](teximages/ra5037fi30.gif)
, respectively. Combining (a,b) with the simplest N-4 components gives monoclinic structures, C2/m, P2/m, P21/m, P21/c, P21. Other combinations of nonzero components for N-4 are possible and will lead to a wide variety of predicted structures, but reported structure types appear generally to require only one nonzero component.
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 Σ2 at k vector (1/n,1/n,0) with just one component of the 12 component nonzero. The incommensurate case can be treated using the same 12 component Σ2 with just one nonzero component, by taking the k vector for the active representation to be (ξ,ξ,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 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 ordering (from Table 2, see also Table 3), the of the orthorhombic structure [(a,0)] becomes Amm2 if n = 3, Pmma or Pbam if n = 4 and Amm2 if n = 5. For odd values of n the structures obtained are either orthorhombic on a cell in Amm2 comprising 2n layers, or monoclinic in P2/m. The monoclinic structures [(a,b)] all have P2/m.
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![[\Sigma_{2}]](teximages/ra5037fi220.gif)
![[{\sqrt 2}]](teximages/ra5037fi222.gif)
![[{\sqrt 5}]](teximages/ra5037fi239.gif)
![[{\sqrt {13}}]](teximages/ra5037fi253.gif)
![[\xi]](teximages/ra5037fi273.gif)
If the modulations are treated as incommensurate, the result is a structure in Ammm(0,0,γ)0s0 (Tables 2 and 3). This represents a structure (Fig. 3a) with basic (average) orthorhombic symmetry Ammm and a modulation vector parallel to the z axis of the Ammm cell. The trailing 0s0 is to indicate that the second symmetry operator, the mirror plane perpendicular to the y axis, reverses the phase of the modulation. The lattice vectors and origin of this Ammm cell are given by the first three components of the four dimensional vectors shown in Tables 2 and 3. The repeat distance of the modulation will be 1/ξ. The variations in symmetry obtained with commensurate modulation vectors [Figs. 3(b) and 3(c)] may represent examples of the artefacts encountered when incommensurate modulations are approximated as commensurate (Janssen et al., 2006).
![[Figure 3]](ra5037fig3thm.gif) | Figure 3 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{\bar 3}m]](teximages/ra5037fi143.gif) (e.g. NiTi, Ni red, Ti blue). The basic 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 with this period. The in (b) is obtained in orthorhombic symmetry, Amm2, by extending the cell to 14 (110) planes, and the 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. |
For a structure with ordering on the basis of , the orthorhombic product structures have space groups Imm2 (n = 3), Pmma (n = 4), Imm2 (n = 5), and the monoclinic structures have 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 as the parent structure. The structure has a which is double the dimensions of the cell, so that nI = 6 (k = 1/6,1/6,0) with respect to the latter becomes nF = 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 from a parent structure with has nF = 2 (k = 1/2,1/2,0), and would correspond to the structure with nI = 4 (k = 1/4,1/4,0) in Table 2. The Pnnm structure with nF = 3 reported by Brown et al. (2002) would correspond to the structure with nI = 6 (k = 1/6,1/6,0) in Table 2, and similarly for nF = 7, nI = 14. The P2/m structure described by Brown et al. (2011) has nF = 3, k = (1/3,1/3,0) and corresponds to the structure with nI = 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 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-4 becomes M-5 so that the structural relationships acquire the more familiar form for , Pmma and P21/m structures as already set out by Barsch (2000). (a,0) gives P4/mmm, corresponding to the β′ structure of RuNb stable between ∼1030 and ∼1170 K (e.g. Dirand et al., 2012; Nó et al., 2015a, 2015b), the room-temperature structure of TiAl (Duarte et al., 2012) and the structure of NixAl1−x, x ≃ 0.64, quenched from high temperatures (Potapov et al., 1997). The low-temperature (β′′) structure of RuNb has been reported to be either orthorhombic, Cmmm (Chen & Franzen, 1989), or monoclinic, P2/m (Nó et al., 2015a,b) or P21/m (Mousa et al., 2009). All three of these structure types would have the same as some permutation of ![[\sqrt 2]](teximages/ra5037fi47.gif)
ao × ao × ao, but differing in the combination of driving order parameters.
Other sets of structures can be generated by considering k-active as having directions along several of the 〈110〉* 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,−1/3), a trigonal structure is obtained from a 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 Σ2 (Table 3). As reviewed by Otsuka & Ren (2005), various suggestions have been made for the correct of this structure, including ![[P{\bar 3}1m]](teximages/ra5037fi50.gif)
(Vatanayon & Hehemann, 1975; Goo & Sinclair, 1985), P3 (Ohba et al., 1992; Hara et al., 1997) and ![[P{\bar 3}]](teximages/ra5037fi51.gif)
(Schryvers & Potapov, 2002; Sitepu, 2003). The group theoretical treatment set out here gives 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.
For practical convenience when considering L21 Heusler compounds, Table 4 shows structures with respect to , rather than , as the parent structure. This includes, for example, the martensite structures of Ni2Mn1.44Sn0.56 and Ni2Mn1.48Sb0.52 described by Brown et al. (2006, 2010), which have Pmma and, when referring to the larger parent cell, k-active = (1/2,1/2,0). Ni2MnGa has two martensitic structures with Pnnm: k-active = (1/3,1/3,0) and (1/7,1/7,0) (Brown et al., 2002). The martensite structure of Ni1.84Mn1.64In0.52 has P2/m and k-active = (0,0,0) and (1/3,1/3,0) (Brown et al. 2011). The room-temperature structure of Ni2.19Mn0.82Ga has I4/mmm (Banik et al., 2007), which corresponds to (a,0), (0,0,0), Σ2 (0,0,0,0,0,0,0,0,0,0,0,0), k-active = (0,0,0). A limitation of using subgroups of in terms of a sequence as 1/n, n = 2, 3, 4…, however, is that the n = odd entries in Table 2 are not included. For example, a structure with k = (1/3,1/3,0) in Table 2 would have k-active = (2/3,2/3,0) if it was added to Table 4. The choice of label, 3M or 6M, 5M or 10M, etc., also depends on whether reference is being made to the repeat, with respect to the cell, or to the number of atomic layers in the repeating unit (Singh et al. 2015).
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![[\Gamma_{5}^{+}]](teximages/ra5037fi280.gif)
2.3. Primary and secondary order parameters
Inspection of Table 2 reveals that the ![[\Gamma _3^ +]](teximages/ra5037fi59.gif)
can act on its own, whereas nonzero values of components of N-4 and Σ2 are always accompanied by nonzero values of components from both and ![[\Gamma _5^ +]](teximages/ra5037fi62.gif)
. The latter can just be secondary order parameters, consequential on coupling to tetragonal and orthorhombic shear strains, et and eo (), or shear strains e4, e5, e6 (), but they could also represent primary order parameters due to separate instabilities. Similarly, is invariably accompanied by nonzero values of components of which may be secondary but could be primary from a separate, additional instability. At the heart of the diversity of martensite structures is the existence of both the fundamental electronic instability and the possibility of additional instabilities associated, for example, with the soft mode.
If the acts alone, the pattern of spontaneous strains is determined by coupling terms in the Landau free-energy expansion
![[\eqalign{G = &{1 \over 2}a_{\Gamma 3 + }\big(T - T_{{\rm c}\Gamma 3 + } \big)\big(q_{1\Gamma 3 +}^2 + q_{2\Gamma 3 +}^2 \big) \cr &+ {1 \over 3}b_{\Gamma 3 +}\big(q_{1\Gamma 3 +}^3 - 3q_{1\Gamma 3 + }q_{2\Gamma 3 + }^2 \big) + {1 \over 4}c_{\Gamma 3 +} \big(q_{1\Gamma 3 + }^2 + q_{2\Gamma 3 +}^2 \big)^2 \cr & +\lambda _{1\Gamma 3 +} e_{\rm a}\big(q_{1\Gamma 3 +}^2 + q_{2\Gamma 3 + }^2 \big) + \lambda _{2\Gamma 3 +}\big(e_{\rm t}q_{1\Gamma 3 +} - e_{\rm o} q_{2\Gamma 3 +} \big) \cr & + \lambda _{3\Gamma 3 +}\Big[\big(2e_6^2 - e_4^2 - e_5^2 \big)q_{1\Gamma 3 +} + \sqrt 3 \big(e_5^2 - e_4^2\big) q_{2\Gamma 3 +} \Big] \cr & + {1 \over 6}\big(C_{11}^{\rm o} + 2C_{12}^{\rm o} \big)e_{\rm a}^2 + {1 \over 4}\big(C_{11}^{\rm o} - C_{12}^{\rm o} \big)\big(e_{\rm t}^2 + e_{\rm o}^2 \big)\cr &+ {1 \over 2}C_{44}^{\rm o }\big(e_4^2 + e_5^2 + e_6^2 \big).}]](teximages/ra5037fd1.gif)
Here q represents components, a, b, c are standard Landau coefficients, λ's are coupling coefficients, ![[T_{{\rm c}\Gamma 3 + }]](teximages/ra5037fi68.gif)
is the ea (= e1 + e2 + e3) is the et [= (2e3 − e1 −e2)![[/{\sqrt 3}]](teximages/ra5037fi69.gif)
] is the tetragonal eo (= e1 − e2) is the orthorhombic e4, e5 and e6 are the remaining shear strains, and Co11, Co12, Co44 are elastic constants of the parent cubic structure. If the acts alone, the Landau expansion is
![[\eqalign{ G = &{1 \over 2}a_{\Gamma 5 + }\big(T - T_{{\rm c}\Gamma 5 +} \big)\big(q_{1\Gamma 5 +}^2 + q_{2\Gamma 5 +}^2 + q_{3\Gamma 5 +}^2 \big) \cr & + {1 \over 3}b_{\Gamma 5 +}(q_{1\Gamma 5 +}q_{2\Gamma 5 +}q_{3\Gamma 5 +}) \cr &+ {1 \over 4}c_{\Gamma 5 +}\big(q_{1\Gamma 5 +}^2 + q_{2\Gamma 5 +}^2 + q_{3\Gamma 5 +}^2 \big)^2 \cr & + {1 \over 4}c{^\prime}_{\Gamma 5 +}\big(q_{1\Gamma 5 +}^4 + q_{2\Gamma 5 +}^4 + q_{3\Gamma 5 +}^4 \big) \cr & + \lambda _{1\Gamma 5 + }e_{\rm a}\big(q_{1\Gamma 5 +}^2 + q_{2\Gamma 5 +}^2 + q_{3\Gamma 5 +}^2 \big) \cr & + \lambda _{2\Gamma 3 +}\big[e_{\rm t}\big(2q_{1\Gamma 5 +}^2 - q_{2\Gamma 5 +}^2 - q_{3\Gamma 5 +}^2 \big) \cr &+ \sqrt 3 e_{\rm o}\big(q_{3\Gamma 5 +}^2 - q_{2\Gamma 5 +}^2 \big) \big] \cr &+ \lambda _{3\Gamma 5 +}\big(e_{6} q_{1\Gamma 5 +} + e_{4}q_{2\Gamma 5 +} + e_{5}q_{3\Gamma 5 +} \big) \cr & + {1 \over 6}\big(C_{11}^{\rm o} + 2C_{12}^{\rm o} \big)e_{\rm a}^2 + {1 \over 4}\big(C_{11}^{\rm o} - C_{12}^{\rm o} \big)\big(e_{\rm t}^2 + e_{\rm o}^2 \big)\cr &+ {1 \over 2}C_{44}^{\rm o}\big(e_4^2 + e_5^2 + e_6^2 \big).}]](teximages/ra5037fd2.gif)
If the single is N4- or Σ2, 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 linear–quadratic, ![[\lambda eq^{2}]](teximages/ra5037fi75.gif)
, or biquadratic, ![[\lambda e^{2}q^{2}]](teximages/ra5037fi76.gif)
. 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. Carpenter et al., 1998).
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 qΓ, and an from along the Σ line out to the N point, qΣ. Biquadratic coupling, ![[\lambda q_\Gamma ^2q_\Sigma ^2]](teximages/ra5037fi77.gif)
, 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, λ, and the relative critical temperatures of the two instabilities, TcΓ and TcΣ. Linear–quadratic coupling, ![[\lambda q_\Gamma ^{}q_\Sigma ^2]](teximages/ra5037fi78.gif)
, is also allowed for some combinations, but leads to a much more restricted range of possibilities (Salje & Carpenter, 2011). In principle, TcΣ > TcΓ would be expected to give rise to a single transition from a state with qΓ = 0, qΣ = 0 to one with qΓ ≠ 0, qΣ ≠ 0 because qΣ generates a conjugate field for qΓ. Alternatively, for TcΣ < TcΓ, the sequence can be a to a structure with qΓ ≠ 0, qΣ = 0, followed by a first-order transition to a phase with qΓ ≠ 0, qΣ ≠ 0. Coupling terms between and can in principle also be linear–quadratic and biquadratic as:
![[\lambda \Big[q_{1\Gamma 3 +}\big(2q_{1\Gamma 5 +}^2 - q_{2\Gamma 5 +}^2 - q_{3\Gamma 5 +}^2 \big) + \sqrt 3 q_{2\Gamma 3 + }\big(q_{3\Gamma 5 +}^2 - q_{2\Gamma 5 +}^2 \big) \Big]]](teximages/ra5037fd3.gif)
and
![[\lambda \big(q_{1\Gamma 3 + }^2 + q_{2\Gamma 3 + }^2 \big)\big(q_{1\Gamma 5 + }^2 + q_{2\Gamma 5 + }^2 + q_{3\Gamma 5 + }^2 \big).]](teximages/ra5037fd4.gif)
Indirect coupling via shear strains would give the linear–quadratic term while coupling via the would give rise to the biquadratic term.
An example of coupling between order parameters for instabilities with two nonzero components of and one nonzero component of M5-, with respect to a parent structure, can be represented by the Landau expansion:
![[\eqalign{G = &{1 \over 2}a_{\Gamma 3 +}\big(T - T_{{\rm c}\Gamma 3 +} \big)\big(q_{1\Gamma 3 +}^2 + q_{2\Gamma 3 +}^2 \big) \cr &+ {1 \over 3}b_{\Gamma 3 +}\big(q_{1\Gamma 3 +}^3 - 3q_{1\Gamma 3 +}q_{2\Gamma 3 +}^2 \big) \cr & + {1 \over 4}c_{\Gamma 3 +}\big(q_{1\Gamma 3 +}^2 + q_{2\Gamma 3 +}^2 \big)^2 + {1 \over 2}a_{{\rm M}5 -}\big(T - T_{{\rm cM}5 -} \big)q_{{\rm M}5 -}^2 \cr &+ {1 \over 4}b_{{\rm M}5 - }q_{{\rm M}5 -}^4 + \lambda _{1\Gamma 3 +}e_{\rm a}\big(q_{1\Gamma 3 +}^2 + q_{2\Gamma 3 +}^2 \big) \cr &+ \lambda _{2\Gamma 3 +}\big(e_{\rm t}q_{1\Gamma 3 +} - e_{\rm o} q_{2\Gamma 3 +} \big) + \lambda _{3\Gamma 3 +}\Big[\big(2e_6^2 - e_4^2 - e_5^2 \big)q_{1\Gamma 3 +} \cr & + \sqrt 3 \big(e_5^2 - e_4^2 \big)q_{2\Gamma 3 +} \Big] + \lambda _{1{\rm M}5 -}e_{\rm a}q_{{\rm M}5 -}^2 \cr & + \lambda _{2{\rm M}5 -}\big(e_{\rm t} + \sqrt 3 e_{\rm o} \big)q_{{\rm M}5 -}^2 + \lambda _{3{\rm M}5 -}e_{5}q_{{\rm M}5 -}^2 \cr &+ \lambda _{6{\rm M}}\big(e_4^2 + e_6^2 \big)q_{{\rm M}5 -}^2 + \lambda _{\Gamma {\rm M}}\big(q_{1\Gamma 3 +} - \sqrt 3 q_{2\Gamma 3 +} \big)q_{{\rm M}5 - }^2 \cr &+ {1 \over 6}\big(C_{11}^{\rm o} + 2C_{12}^{\rm o} \big)e_{\rm a}^2 + {1 \over 4}\big(C_{11}^{\rm o} - C_{12}^{\rm o} \big)\big(e_{\rm t}^2 + e_{\rm o}^2 \big) \cr &+ {1 \over 2}C_{44}^{\rm o}\big(e_4^2 + e_5^2 + e_6^2 \big).}]](teximages/ra5037fd5.gif)
From Table 3, if the nonzero components of are (a, ![[-\sqrt 3]](teximages/ra5037fi85.gif)
a) and the nonzero components of M5- are (0,0,c,c,0,0), the resultant structure has Pmma symmetry (B19 structure). This has (0,0,b), i.e. the e5, as a secondary However, the same outcome could be obtained using with M5- as primaries and as secondary, or taking M5- as driving and both and 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 M with strains e will be of the form λeM2 or λe2M2. 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 an for the structural modulations and the magnetic has been given by Vasil'ev et al. (2003). A simpler form, with only the Γ-point and magnetic order parameters, is given in Vasil'ev et al. (1999).
3. Some examples of real materials
Applications of the group theoretical approach set out above can be illustrated with three specific examples, using alloys relating to NiTi, TiPd and Ni2MnGa.
3.1. NiTi, RuNb
NiTi undergoes a single transition from the B2 structure to the B19′ structure at ∼335 K, corresponding to –P21/m (Otsuka & Ren, 2005). P21/m is not a symmetry of order 2 with respect to , however, but it is a order 2 with respect to Pmma. Following Barsch (2000) and Otsuka & Ren (2005), there appear to be two instabilities and these are seen in sequence as –Pmma–P21/m in Ti50Ni50−xCux (Nam et al., 1990). Symmetry relationships are as listed in Table 3: the active representations are M5- of and of Pmma (Barsch, 2000). With respect to symmetry the two discrete electronic instabilities relate essentially to and , coupled to the M-point (zone boundary) mode.
Michal & Sinclair (1981) have given a = 2.885, b = 4.120, c = 4.622 Å, β = 96.8° for the of the monoclinic structure at room temperature, which corresponds to ∼ao × ![[{\sqrt 2} a_{\rm o}]](teximages/ra5037fi103.gif)
× , where ao is the dimension of the primitive parent cubic structure. Using an orthogonal reference system with X, Y and Z parallel to crystallographic x, y and z of the parent structure, the nonzero shear strains are ety = (2e2 − e1 − e3), e6 = e4 ≠ e5. Here ety is the tetragonal with the unique axis aligned parallel to the crystallographic y-axis. In terms of the lattice parameters of the monoclinic structure, individual strains are given by e2 = (a − ao)/ao, e1 + e3 = ![[\big[\big(b/{\sqrt 2} - a_{\rm o}\big)/ a_{\rm o}]](teximages/ra5037fi106.gif)
+ ![[\big(c/{\sqrt 2} - a_{\rm o}\big)/a_{\rm o}\big]]](teximages/ra5037fi107.gif)
, e5 = ![[ - \big(c/{\sqrt 2} - a_{\rm o}\big)/a_{\rm o}\big]]](teximages/ra5037fi109.gif)
, e4 = e6 ≃ ![[{1 \over 2}\cos\beta]](teximages/ra5037fi110.gif)
. Using ao as approximated by (abc/2)1/3, gives the values ety = −0.079, ∣e5∣ = 0.118, e6 = e4 ≃ −0.059. These three shear strains are substantially greater than any that are typically associated with transitions driven by phonon-related instabilities.
Evidence for a separate soft-mode transition in Ni–Ti alloys is revealed by the changes in transition sequences induced by addition of minor components in The transition sequence in Ti50Ni50−xFex is ––P21/m (B2–R–B19′) (Honma et al., 1980), taking the R-phase as having . In a sample with x = 3.2, a precursor is incommensurate but the R-phase itself is commensurate (Shapiro et al., 1984; Salamon et al., 1985). There is a small discontinuity in the pseudocubic lattice angle, α, at the – transition and this angle decreases to 89.3° with falling temperature (Salamon et al., 1985). The transition is thus weakly first order, with the symmetry-breaking e4 = e5 = e6 ≃ cosα, reaching a maximum value of ∼0.012, consistent with the transition being driven by softening of an acoustic phonon along the [110]* branch (Satija et al., 1984; Moine et al., 1984). Salje et al. (2008) found the same strain variation in a different sample with the same composition. The electronic and soft-mode instabilities are suppressed to different extents with increasing Fe-content such that the stability field of the R-phase expands. In principle they could combine to produce structures with commensurate or incommensurate repeat distances along [110]* but, for stoichiometric Ni–Ti, the lowest energy (P21/m) structure is not a of and has the two gamma point order parameters combined with an M-point Parlinski & Parlinska-Wojtan (2002) have shown that the latter can also be understood in terms of a soft mode.
In NiTi, the changes from (0,0,b) to (b,b,c) causing Pmma to become P21/m. The same could be primary for the second symmetry change in RuNb where the sequence is –P4/mmm [ (a,0), (0,0,0), M-5 (0,0,0,0,0,0)]–Cmmm [ (a,0), (b,0,0), M-5 (0,0,0,0,0,0)] or P2/m [ (a,b), (0,0,c), M-5 (0,0,d,e,0,0)]. The tetragonal etz [= (2e3 − e1 − e2)![[/{\sqrt 3 }]](teximages/ra5037fi128.gif)
], calculated from the lattice parameters given by Shapiro et al. (2006) for the tetragonal phase at 900 K, is 0.07 [e1 = (a − ao)/ao, e2 = (b − ao)/ao, e3 = (c − ao)/ao]. Tetragonal, etz, and orthorhombic, eo = e1 − e2, strains calculated from the orthorhombic lattice parameters given for 873 K are 0.14 and −0.02, respectively. In both cases, the same procedure as described above was used for estimating ao. The large increase in at the second transition is consistent with an electronic driving mechanism and being primary.
3.2. Ni2+xMn1−xGa
The L21 Heusler compound Ni2MnGa is cubic, , at high temperatures. Lowering of the symmetry from a parent structure in which the atoms would be disordered between all the crystallographic sites is described by two order parameters, one belonging to irrep and the second to irrep P1. It undergoes two phase transitions during cooling, at ∼260 and ∼200 K. Following Brown et al. (2002), the first is to a `pre-martensitic' structure which is incommensurate (Singh et al., 2015) but can be represented in terms of an orthorhombic structure with Pnnm and a ∼ ![[a_{\rm oF}/{\sqrt 2}]](teximages/ra5037fi133.gif)
, b ∼ ![[3a_{\rm oF}/{\sqrt 2}]](teximages/ra5037fi134.gif)
, c ∼ aoF, where aoF is the lattice parameter of the parent cubic F (Table 4; Brown et al., 2002). The driving mechanism is related to softening of the (Σ2) soft acoustic phonon at q ∼ (1/3,1/3,0) (Zheludev et al., 1995; Stuhr et al., 1997; Mañosa et al., 2001). Strains accompanying this transition are such that distortion from cubic lattice geometry is small (Brown et al., 2002; Ohba et al., 2005). Ohba et al. (2005) gave lattice parameters at 250 K as a = 5.8285, b = 5.8142, c = 5.7886 Å, which yield components e1 = (a − aoF)/aoF = 0.003, e2 = (b − aoF)/aoF = 0.001, e3 = (c − aoF)/aoF = −0.004 (with the usual approximation for ao). Expressed in symmetry-adapted forms the tetragonal and orthorhombic shear strains are etz = −0.007 and eo = 0.002, respectively.
The second transition is to a structure which may also be incommensurate but can be represented as being orthorhombic in the same Pnnm, with a ∼ , b ∼ ![[7a_{\rm oF}/{\sqrt 2}]](teximages/ra5037fi136.gif)
, c ∼ aoF (Table 4; Brown et al., 2002; Ranjan et al., 2006; Righi et al., 2006; Zheludev et al., 1996). Determining strains in the same way from the lattice parameters given by Brown et al. (2002), a = 4.2152, b = 29.3016, c = 5.5570 Å, gives et = −0.076 and eo = 0.007, respectively. The factor of 10 increase in et with respect to the pre-martensitic phase seems to be characteristic for strain coupling with the at a band Jahn–Teller transition. The two order parameters produce a large tetragonal strain from the electronic instability and multiplication of the cell dimension from the soft mode. There is also a nonzero component (a,0,0) belonging to (Table 4), but it does not appear to drive any of the instabilities and is therefore genuinely secondary.
Increasing the Ni content at the expense of Mn in Ni2+xMn1−xGa causes the transition temperatures for both transitions to increase, with slopes that give a diminishing field for the pre-martensite structure (Fig. 4, after Vasil'ev et al., 2003; Entel et al., 2014). The martensite structures also change from a 5M (k-active = (1/5,1/5,0) structure reported at x = 0.02 (Vasil'ev et al., 2003) to 7M (k-active = (1/7,1/7,0) and then to the I4/mmm structure, which has the (a,0) electronic distortion only. Linear-quadratic coupling, ![[\lambda q_{\Gamma 3 + }^{}q_{\Sigma 2}^2]](teximages/ra5037fi139.gif)
, is permitted by symmetry and, from the discussion in §2.3 above, would be expected to give rise to a single transition directly from a state with ![[q_{\Gamma 3+}]](teximages/ra5037fi140.gif)
= 0, qΣ2 = 0 to one with ≠ 0, qΣ2 ≠ 0 for TcΣ2> TcΓ3+. Instead this sequence is observed at relatively high Ni contents where TcΣ2 falls below the temperature. The implication is that linear–quadratic coupling is not a dominant factor in determining the stability of the martensitic structures. Either coupling between the two order parameters is weak or it is dominated by biquadratic terms, ![[\lambda q_{\Gamma 3 + }^2q_{\Sigma 2}^2]](teximages/ra5037fi142.gif)
, which could arise via the common The qΣ2 component presumably diminishes with increasing Ni-content since it is zero in the I4/mmm structure.
![[Figure 4]](ra5037fig4thm.gif) | Figure 4 Mn-rich portion of the Ni2+xMn1−xGa phase diagram, after Vasil'ev et al. (2003 ) and Entel et al. (2014). An approximate location for the boundary between Pnnm structures (qΓ3+ ≠ 0, qΣ2 ≠ 0) and the I4/mmm structure (qΓ3+ ≠ 0, qΣ2 = 0) is based on the data given by Banik et al. (2007, their Table 1). Tc marks the paramagnetic (PM) to ferromagnetic (FM) transition. |
3.3. Ti50Pd50−xCrx
Ti50Pd50−xCrx represents a further example of changing structural sequences with increasing doping. There is a crossover between two sequences, (B2)–Pmma (B19) and –incommensurate (IC)–incommensurate martensite (ICM), at x ∼ 4.5 (Fig. 5, following Enami et al., 1989; Schwartz et al., 1995; Shapiro et al., 2007). In contrast with Ni2+xMn1−xGa, the trend is of decreasing transition temperatures with increasing doping, and structures with qΣ2 ≠ 0 appear at relatively high values of x. The 9R structure is monoclinic (P2/m) and has a Σ2 repeat of three, while the ICM structure has IC repeat distances derived from the Σ2 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).
![[Figure 5]](ra5037fig5thm.gif) | 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). |
Linear-quadratic coupling, is again allowed by symmetry but the transition sequences with falling temperature are the same as observed for Ni2+xMn1−xGa in not complying with what would be expected from the generalized treatment of Salje & Carpenter (2011). In this system, the contributions of qΣ2 clearly increase with increasing Cr content as the transition temperature for structures with qΓ3+ ≠ 0 reduces. Other martensite materials with group–subgroup relationships need to be examined, but it appears that biquadratic coupling may be dominant in systems with band Jahn–Teller transitions.
4. 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 Carpenter & Salje (1998). Due to bilinear coupling of a symmetry breaking with the primary λesbq, transitions driven by the 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 M5-) 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.
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 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, with specific implications for the evolution of the elastic constants (e.g. Carpenter & Salje, 1998). The pattern of evolution of the at least, for the simplest case of the –P4/mmm transition in Ru–Nb, which involves only the 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 C11–C12 and C44 as the martensitic transitions are approached from above. This confirms the proximity of electronic instabilities with symmetries belonging to both and .
The pattern of evolution of both C11–C12 and C44 in Ni2MnGa ahead of and through the L21 () to IC (∼3M, Pnnm) transition (e.g. Mañosa et al., 1997; Stipcich et al., 2004) is characteristic of improper ferroelastic behaviour, implying that the driving relates predominantly to Σ2 and, hence, that is secondary. Some precursor softening of C11–C12 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 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 Σ2 and order parameters with and P1 order parameters. This coupling is biquadratic in lowest order, ![[\lambda q^{2}_{\Sigma} q^{2}_{\rm H}]](teximages/ra5037fi158.gif)
, ![[\lambda q^{2}_{\Sigma} q^{2}_{\rm P}]](teximages/ra5037fi159.gif)
, ![[\lambda q^{2}_{\Gamma} q^{2}_{\rm H}]](teximages/ra5037fi160.gif)
, ![[\lambda q^{2}_{\Gamma} q^{2}_{\rm P}]](teximages/ra5037fi161.gif)
. As a consequence, the effects of changes in the degree of atomic order are most likely to be seen as renormalization of the for the martensitic and soft-mode transitions. This is exactly analogous to the influence of Fe/Mo ordering on phase transitions in Sr2FeMoO6 (Yang et al., 2016).
5. 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 Γ-point and Σ2 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.
Footnotes
1Repeat distance is defined by the sequence of atomic displacements and may or may not correspond to a crystallographic repeat.
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