Cooperative Jahn–Teller effect and the role of strain in the tetragonal-to-cubic phase transition in MgxCu1 − xCr2O4

In the MgxCu1 − xCr2O4 solid solution, progressive substitution of the Jahn–Teller and d 9 Cu2+ cation with the spherical and closed-shell Mg2+ cation results in a gradual reduction of the splitting of a and c unit-cell parameters, until transformation occurs from the tetragonal I41/amd to the cubic archetype spinel structure. Tetragonal members of the series transform to cubic at high temperature, and the transition temperature decreases with increasing Mg content. The phase transition is first order in character for Cu-rich samples and evolves towards second-order character at intermediate compositions.


Introduction
Complex AB 2 O 4 oxides with the spinel structure comprise a family of materials, which exhibit a wide range of electronic, magnetic and optical properties through the variation of cations on tetrahedral (A) and octahedral (B) sites. The archetypical spinel structure belongs to the space group Fd " 3 3m (No. 227) and is usually described as a pseudocubic closepacked array of O atoms with the A and B cations occupying one eighth of the tetrahedral sites and one half of the octahedral sites, respectively. Such occupancy of the interstitial sites results in an fcc unit cell which is 2 Â 2 Â 2 times that of the basic ccp oxygen array. One of the characteristics of the spinel structure is its flexibility in the range of possible cations and cation charge combinations, making it a structure adopted by over a hundred compounds. In fact, within the spinel space group, the fractional coordinates of the octahedral and tetrahedral sites are fixed at special positions (A on 8a at 0,0,0; B on 16d at 5/8,5/8,5/8), while the O atoms are on 32e with coordinates uuu. This means that if the relative sizes of the A and B cations change, their positions remain the same but the oxygen array expands or contracts to accommodate them and maintain the same symmetry throughout.
The most common distortion of the spinel structure is by far the tetragonal distortion, whereby one of the cubic axes would become compressed or elongated with respect to the other two. If no additional symmetry breaking occurs, the tetragonal distortion alone decreases the symmetry from Fd " 3 3m to I4 1 /amd (No. 141). The c/a ratio is normally used as a parameter of tetragonal distortion. A phase transition from the cubic to the tetragonal structure may be induced by a sufficient concentration of non-spherical, Jahn-Teller (JT) ions, such as Cu 2+ or Mn 3+ , causing a cooperative distortion. Although less common, tetrahedral Cu 2+ on the A site can display JT activity. The degeneracy of the partially occupied t 2 levels is broken by compressing the tetrahedron and thereby lowering the symmetry, as in copper chromite, CuCr 2 O 4 , which is a tetragonally distorted spinel with unit-cell parameters ratio c/a < 1 (Fig. 1, left panel). Cu 2+ cations can be stabilized in flattened tetrahedral environments because of the preference of Cr 3+ ions to occupy the octahedral sites. The cooperative nature of the crystal distortion can be rationalized in terms of elastic interactions among locally distorted tetrahedra, as a consequence of coupling of electronic states to bulk deformation via elastic strain. On heating, CuCr 2 O 4 undergoes a first-order structural transition from the tetragonal distorted spinel structure to the archetypal cubic spinel structure at 853 K (Yé et al., 1994;Kennedy & Zhou, 2008). The structural distortion in CuCr 2 O 4 is large and the transition temperature high, in particular if considering that CuO 4 tetrahedra are not directly linked but separated from each other by non-JT ions. Nonetheless, enhancement of the ground-state JT splitting and of lattice distortion have been explained by considering the electronic and elastic coupling of Cu 2+ and Cr 3+ (Atanasov et al., 1993;Reinen et al., 1988). The relevance of strain associated with the Jahn-Teller distortion is curiously showed in NiCr 2 O 4 by the observation that the crystals literally jump off a flat surface when they pass through the transition point (Crottaz et al., 1997), due to the large and abrupt change in shear strain.
MgCr 2 O 4 forms in the cubic spinel structure (Fig. 1, right panel). At room temperature (RT), the Mg-rich (x > 0.6) members of the Mg x Cu 1 À x Cr 2 O 4 solid solution are cubic, whereas the Cu-rich members (x < 0.43) are tetragonal. A twophase region separates the cubic and tetragonal phases (Shoemaker & Seshadri, 2010;De et al., 1983).
In this work, the effects on the crystal structure and on the tetragonal-to-cubic phase transition of progressive substitution of the Jahn-Teller and d 9 Cu 2+ cation with the spherical and closed-shell Mg 2+ cation in the Mg x Cu 1 À x Cr 2 O 4 solid solution have been studied. Given the relevance of strain in determining the structure-electronic properties relation, in situ high-temperature (HT) single-crystal diffraction data are analysed in terms of the evolution of symmetry-adapted strains for samples with different compositions along the joint Mg x Cu 1 À x Cr 2 O 4 . Observed variations of spontaneous strains accompanying phase transitions are expected to provide detailed insights into the transition mechanisms.

Synthesis and crystal growth
Single crystals of cubic Mg-rich and tetragonal Cu-rich chromites belonging to the series Mg x Cu 1 À x Cr 2 O 4 were grown by flux decomposition methods. The synthesis of single crystals of the Mg end-member was conducted on the basis of the strategy reported by Lenaz et al. (2004) Perspective views of the crystal structures of CuCr 2 O 4 (left) and MgCr 2 O 4 (right) along the a-axis of the I4 1 /amd cell. CuO 4 tetrahedra are drawn in green, MgO 4 tetrahedra in grey. In CuCr 2 O 4 , the contraction along the c direction due to JT flattening of tetrahedral sites is given by a change of tetrahedral angles. Relevant geometrical parameters are reported. Mg x Cu 1 À x Cr 2 O 4 solid solution, the method reported by Yé et al. (1994) for growing crystals of CuCr 2 O 4 has been adapted to different Cu/Mg stoichiometric ratios. Starting compounds were CuO (Fluka, > 99%), MgO (Carlo Erba, > 99%) and K 2 Cr 2 O 7 (Carlo Erba, > 99%). K 2 Cr 2 O 7 transforms into Cr 2 O 3 , which, when freshly formed, is highly reactive towards copper oxide. Potassium dichromate acts as a reactive flux for the crystal growth and an excess amount was therefore added according to where n was chosen to be equal to 0.2 mol. B 2 O 3 (1 wt%) was added to the mixture in order to increase the homogeneity of the solution and hence improve the quality of the crystals. Different CuO/MgO stoichiometric ratios were used to obtain spinels with nominal compositions: CuCr 2 O 4 (x = 0), Mg 0.05 Cu 0.95 Cr 2 O 4 (x = 0.05), Mg 0.1 Cu 0.9 Cr 2 O 4 (x = 0.1), Mg 0.4 Cu 0.6 Cr 2 O 4 (x = 0.4). The mixtures were submitted to the following heating cycle: (1) heating from RT to 1093 K at 100 K h À1 ; (2) soaking at 1093 K for 24 h; (3) cooling at 30 K h À1 . Given the high refractory properties of MgO, stage (2) was prolonged for 115 h in the case of the mixture with x = 0.4. After the thermal runs, the residues were washed with boiling water and the single crystals removed from the solidified flux.

Single-crystal XRD at room temperature
Several crystals were isolated from each synthesis residue. They were checked for crystal quality by analysing X-ray diffraction profiles. The selected crystals, labelled Cu100, Cu90, Cu82, Cu57, Cu47 and Mg100 on the basis of their actual compositions as determined from structure refinements and electron-microprobe analyses (see xx2.4 and 2.5), were submitted to single-crystal diffraction analysis at RT using a Bruker-AXS APEX diffractometer equipped with a CCD detector. Data collections were carried out with operating conditions 50 kV and 30 mA and graphite-monochromated Mo K radiation ( = 0.7107 Å ). The Bruker SMART system of programs was used for preliminary crystal lattice determination and X-ray data collection. A total of 4800 frames (resolution: 512 Â 512 pixels) were collected with eight different goniometer settings using the !-scan mode (scan width: 0.3 !; exposure time: 5-20 s per frame, depending on the size and relative scattering power of the crystals analysed; detector-sample distance: 60 mm). Complete data collection was achieved up to sin / ca 0.95 Å À1 .
All the tetragonal crystals of the series were twinned, as expected given the synthesis conditions which imply the use of high temperature, where the cubic phase is stable, and subsequent transformation to tetragonal on cooling. I4 1 /amd is a maximal nonisomorphic t-subgroup of Fd " 3 3m and the formation of ferroelastic domains, including transformation twinning, is inevitable during the phase transition. The corresponding ferroelastic species according to Aizu's notation (Aizu, 1969) is m " 3 3mF4=mmm, where 'F' stands for ferroic, and separates the parent point group (m " 3 3m) from the derived point group (4/mmm). As m " 3 3m and 4/mmm are of the order 48 and 16, respectively, there are three possible orientation states in the tetragonal phase. In our study, three twin components were found to be present in crystal Cu100, while two components were detected in the other tetragonal crystals.
The Bruker program SAINT+ was used for the data reduction, including intensity integration, background and Lorentz-polarization corrections. Intensity data from the twin components present in tetragonal crystals were integrated taking into account the superposition affecting some diffraction spots. Final unit-cell parameters were obtained by the Bruker GLOBAL least-squares orientation matrix refinement procedure, based on the positions of all measured reflections. The semi-empirical absorption correction of Blessing (1995), based on the determination of transmission factors for equivalent reflections, was applied using the Bruker programs SADABS or, for twinned crystals, TWINABS (Sheldrick,   2003). Details of room-temperature data collection by the CCD diffractometer are reported in Table 1.

Single-crystal XRD at high temperature
Crystals Cu100, Cu90, Cu82, Cu54 (not measured at RT by the CCD diffractometer) and Mg100 were submitted to in situ high-temperature single-crystal diffraction investigations using a Philips PW1100 four-circle diffractometer with pointcounter detector. Crystals Cu57 and Cu47 were not used for the HT study due to their low diffracted intensities. Operating conditions were 55 kV and 30 mA with graphite-monochromated Mo K radiation ( = 0.7107 Å ). Horizontal and vertical apertures of the point counter detector were 2.0 and 1.5 , respectively. High-temperature measurements were performed by using a home-made U-shaped microfurnace, which has been in use in our laboratory for over 15 years. It makes use of a Pt-Pt/Rh resistance, which allows temperatures up to 1273 K to be achieved, and is equipped with a K-type thermocouple. Temperature calibration (calibration curve R 2 = 0.9994) is regularly done by known melting points of several pure compounds and by the transition temperature of quartz (Carpenter, Salje, Graeme-Barber, Wruck et al., 1998).
Reported temperatures are precise to within AE5 K in the whole temperature range. The design of the furnace limits the angular excursion of the ! circle to ca 27.5 (sin / ca 0.65 Å À1 with Mo K radiation). As routinely done for HT measurements using this system, the selected crystals were inserted into quartz capillaries (0.3-0.5 mm Ø, depending on the dimensions of the crystals) and kept in position by means of quartz wool in order to avoid any mechanical stress. Unit-cell parameters were measured from RT up to 1173 K at regular steps. At each working temperature, the orientation matrix was updated by centring 24 reflections selected in the range of sin / ca 0.2-0.34 Å À1 , and accurate lattice param-  Table 2 Tetragonal unit-cell parameters of Mg x Cu 1 À x Cr 2 O 4 crystals at different temperatures.
Standard deviations are in parentheses and refer to the last significant digits.

Table 3
Cubic unit-cell parameters of Mg x Cu 1 À x Cr 2 O 4 crystals at different temperatures.
Standard deviations are in parentheses and refer to the last significant digits. Some intermediate data have been omitted in the table but are present in the graphs. (2) 8.3420 (5) Tables 2 and 3 for tetragonal and cubic phases, respectively) were derived from a least-squares procedure based on the Philips LAT routine over up to 60 d*spacings, each measured from the positions of all reflection pairs at AE in the range of sin / 0.073-0.628 Å À1 . For each crystal, complete datasets of diffracted intensities were collected at different temperatures, both below and above the phase transition temperature, using the same operating conditions as reported above. Intensity data were measured in the sin / range 0.05-0.628 Å À1 in the !-scan mode (2.0 scan width; 0.05 s À1 scan speed). Only diffraction spots belonging to the orientation matrix of the main twin component were measured, thus including overlapping reflections. Three standard reflections were collected every 200 measured reflections. X-ray diffraction intensities were obtained by measuring step-scan profiles and analysing them by the Lehmann & Larsen (1974) I /I method, as modified by Blessing et al. (1974). Intensities were corrected for absorption using the semi-empirical '-scan method of North et al. (1968). Relevant parameters for data collected at different temperatures are reported in Table 4. Some reflections, representative of different classes, were also scanned periodically (!/2 scan mode; 2.0 scan width; 0.1 s À1 scan speed) to check for the crystallinity of the sample.  Table 4 Details on data collections and structure refinements of Mg x Cu 1 À x Cr 2 O 4 crystals at HT.

Structure refinements
All structure refinements were carried out by full-matrix least-squares using SHELXL97 (Sheldrick, 2008). Equivalent reflections were averaged, and the resulting internal agreement factors R int are reported in Table 1 for all the datasets collected at RT, and in Table 4 for datasets collected at HT. The atomic scattering curves were taken from International Tables for X-ray Crystallography (Ibers & Hamilton, 1974). For datasets collected at RT by the CCD diffractometer, contributions from the different twin components were taken into account by using the HKLF-5 format in SHELXL97 and including the BASF parameter in the refinement. For all structure refinements, structure factors were weighted according to w = 1/[ 2 (F 2 o ) + (AP) 2 + BP], where P ¼ ðF 2 o þ 2F 2 c Þ=3, and A and B were chosen for every crystal to produce a flat analysis of variance in terms of F 2 c , as suggested by the program. An extinction parameter x was refined to correct the structure factors according to the equation: F o = F c k½ 1 þ 0:001xF 2 c 3 = sin 2 À1=4 (where k is the overall scale factor). In addition to x and k, atomic positions, anisotropic displacement parameters and site occupancy at the A site (for terms with intermediate composition) were refined simultaneously. The Cu/Mg ratios obtained from unconstrained refinements of site occupancy were close to the nominal compositions and were then confirmed by electron microprobe analyses performed on the same crystals at the end of the HT experiments (see x2.5). Final difference-Fourier maps were featureless. Values of the conventional agreement indices, R 1 and R all , as well as the goodness of fit (S) are reported in Tables 1 and 4 for RT and HT datasets, respectively, whereas interatomic distances and selected geometrical parameters are reported in Table 5 for the RT datasets and in Tables 6-10 for the HT datasets. Atomic fractional coordinates, anisotropic displacement parameters U ij and lists of observed and calculated structure factors are available in the CIF files of supporting information.

Electron probe microanalyses (EPMA)
At the end of the diffraction experiments, all the crystals used in the present study were embedded in epoxy resin, polished and analysed by electron microprobe. The chemical compositions were measured with a Jeol JXA-8200 electron microprobe, fully automated with 5 crystals and 5 wavelength dispersive spectrometers. The polished samples were coated with about 10 nm of amorphous carbon to avoid charging of the surface and studied at acceleration voltages of 15 kV and probe current of 15 nA. The analytical standards used for the calibration of the energy position of the analyzed elements were Cu 2 O, MgO and Cr 2 O 3 for Cu, Mg and Cr, respectively. For each sample, 10 to 15 points were measured and the averaged chemical compositions, as well as the corresponding standard deviations, are reported in Table 5.

Unit-cell parameters and geometry of tetrahedra at RT
Refinements of X-ray diffraction data reveal a high sensitivity of the crystal structure to the amount of Cu 2+ present. Variation of unit-cell parameters as a function of composition at room temperature across the Mg x Cu 1 À x Cr 2 O 4 join. For ease of comparison, the a parameter in the low-temperature tetragonal phase has been scaled and displayed in the pseudocubic setting a pc ¼ a ffiffi ffi 2 p . The vertical size of the symbols exceeds the uncertainties in unit-cell parameters. Table 5 Electron microprobe analyses and selected geometrical parameters for Mg x Cu 1 À x Cr 2 O 4 crystals at RT.

Cu100
Cu90 Cu82 Cu57 Cu47 Mg100 x (site occupancy) 0 0.103 (1) 0.183 (1) 0.442 (5) 0.504 (2) 1 x (EPMA) † 0 (0) 0.103 (21)  The unit-cell parameters are reported as a function of Mg content in Fig. 2. They are expressed in terms of the cubic unit cell itself (MgCr 2 O 4 ) or of the reduced pseudocubic cell (I4 1 /amd structures of the Cu-rich samples). Samples with x 0.53 are isostructural with CuCr 2 O 4 , thus the tetragonal region seems slightly larger than previously reported by De et al. (1983), in agreement with recent data of Shoemaker & Seshadri (2010). Across the solid solution, starting from tetragonal CuCr 2 O 4 , replacement of Cu by Mg is accompanied by an increase in the c-axis and a decrease in the a-axis lengths, and hence leads to a gradual decrease of the tetragonal distortion.
The influence of the Jahn-Teller effect can be better estimated by looking at the geometry of the tetrahedra. These are flattened and, with respect to the ideal tetrahedron, display four smaller and two angles larger than 109.47 (see Fig. 1 for visual reference). The distortion of the tetrahedra is large, with ÁO-Cu-O = 12.94 in the Cu end-member. The distortion of the tetrahedra in the tetragonal phase is also evident in the behaviour of the OÁ Á ÁO edges. In Fig. 3 (Table 5) is related to the difference in ion size between Cu 2+ and Mg 2+ . Homogeneity of the solid solution is quite good. EPMA spot analyses reveal a rather narrow composition range within each sample (Table 5), with Cu54 and Cu47 showing the highest e.s.d.s. When looking at the equivalent atomic displacement parameters (ADPs; Fig. 4a), crystals with intermediate compositions show slightly higher values than those of the two end-members due to some static disorder, with an overall behaviour that is common for solid solutions. Interestingly and as already reported previously (e.g. Kennedy & Zhou, 2008), in all Cu-bearing crystals, the Cu/Mg site is the one showing the highest displacement parameters. This is mainly due to an elongated displacement ellipsoid towards the c-axis (Fig. 4b).
The R max /R min ratio of the principal axes of the thermal ellipsoid is 2.67 for the tetrahedral cation in Cu100 and decreases almost linearly with increasing Mg content, with Cu47 slightly deviating from this trend likely due to some compositional heterogeneity. However, the behaviour  Table 6 Selected geometrical parameters for CuCr 2 O 4 (Cu100) at HT. Standard deviations are in parentheses and refer to the last significant digits. OAV = Octahedral Angle Variance; OQE = Octahedral Quadratic Elongation; TAV = Tetrahedral Angle Variance; TQE = Tetrahedral Quadratic Elongation (Robinson et al., 1971  observed for the average structure by XRD does not necessarily allow to differentiate the distinct cation coordinations of Mg 2+ and Cu 2+ if they are different on the local length scale.

High-temperature behaviour
The temperature dependence of the lattice parameters for CuCr 2 O 4 and all the intermediate compounds of the series Mg x Cu 1 À x Cr 2 O 4 is shown in Fig. 5. Heating the samples results in a gradual reduction of the splitting of a and c unitcell parameters. Variation of the unit-cell parameters with temperature for CuCr 2 O 4 is in good agreement with previously reported data (Kennedy & Zhou, 2008) and shows (Fig. 5a) a large first-order jump above 818 K, when the tetragonal splitting is abruptly lost. By inspection of the variations of the lattice parameters for the samples of intermediate compositions (Figs. 5b-d), it is possible to note how the gradual substitution of Cu for Mg causes a reduction of the initial a-c splitting and of the discontinuity at the transition, and a shift of the transition temperature towards lower temperatures.
The evolution of lattice parameters of cubic phases have been fitted with straight lines, yielding the following thermal expansion a coefficients: Cu100: 7.1 (3) Â 10 À5 K À1 ; Cu90: 6.9 (3) Â 10 À5 K À1 ; Cu82: 7.1 (3) Â 10 À5 K À1 ; Cu54: 5.9 (2) Â 10 À5 K À1 . The reported values are all very similar and in good agreement with values reported for most cubic spinels. Variations of the cubic reference parameters, a 0 , for determination of spontaneous strains of the tetragonal phase were obtained by extrapolation to lower temperatures of these fits. The a and c parameters of the tetragonal phase do not converge symmetrically into the extrapolated values of a 0 for the cubic phase because of a volume expansion associated with the transition. The temperature dependence of the angles within the (Cu,Mg)O 4 tetrahedron is illustrated in Fig. 6 for all the analysed Cu-rich samples. Heating results in a gradual reduction of the compression of the tetrahedron but, clearly, the compression remains significant until near the transition temperature. The variation with temperature of the O-(Cu,Mg)-O angles for the samples of intermediate composition mimics the change in the lattice parameters, the discontinuity of distortion at the transition decreases when Mg is added.
The nature of JT transitions, cooperative or order-disorder, at the solid state has been debated at length and both scenarios observed in several phases (for a review see, Variation of atomic displacement parameters as a function of composition: (a) isotropic ADPs of Cu/Mg (blue diamonds), Cr (green diamonds) and O (red diamonds); (b) anisotropic APDs U 11 (circles) and U 33 (triangles) for the Cu/Mg site. e.g., Kugel & Khomskii, 1982;Goodenough, 1998;Bersuker, 2006). Similarly, in copper spinel, two mechanisms for the JT transition can be proposed: in the first case, upon increasing T, all tetrahedra transform from flattened in the JT distorted spinel to ideal in the cubic spinel; in the second case, the energy-lowering cation coordination distortions persist above the structural transition temperature and CuO 4 tetrahedra are always locally JT distorted. In the latter hypothesis, the structural transition has to be regarded as an order-disorder transition, at which local JT distortions become spatially uncorrelated, although do not disappear; the average crystal structure, which would in turn result in regular tetrahedra, accounts for this disorder through increased thermal displacement parameters.
In the case of JT transitions in systems with octahedrally coordinated cations, as in the case of manganite perovskites, large ADPs for O atoms are observed in the disordered phase along the cation-O bonds, reflecting the largest static bond length distribution (i.e. a mixture of long and short bonds) in

Figure 7
Variation as a function of temperature of isotropic atomic displacement parameters of Cu/Mg ( transition to be driven by cooperative distortion, although the presence of an order-disorder component cannot be excluded on the basis of average structure information only.

Spontaneous strain in CuCr 2 O 4
Instability of the electronic structure is the driving mechanism for a Jahn-Teller transition. However, the change in the structural state appears overtly as changes in lattice parameters, and these can be described formally in terms of macroscopic strains. Variations of spontaneous strains accompanying a phase transition can be used to quantify the associated order parameters and are expected to provide detailed insights into the mechanisms of the transition itself. Strain parameters have been calculated by using the equations given by , who reviewed the use of spontaneous strain to measure order parameters associated with phase transitions in minerals. In this case, there are two strains, the symmetry-breaking tetragonal shear strain, e t ¼ ð1= ffiffi ffi 3 p Þð2e 3 À e 1 À e 2 Þ, where e 2 ¼ e 1 ¼ ða À a 0 Þ=a 0 , e 3 ¼ ðc À a 0 Þ=a 0 , and the volume strain, Values for the reference parameter, V 0 , were obtained by fitting a straight line to data for the unit-cell volume above the transition and extrapolating to lower temperatures. The resulting strains e 2 t and V s are reported in Figs. 8(a) and 8(b) as a function of T.
Symmetry rules determine the nature of coupling between the strain and order parameters. The order parameter q JT and the tetragonal strain e t transform as À þ 3 of Fd3m, the active representation for the transition to I4 1 /amd, giving coupling of the form e t q JT . V s , which does not break the cubic symmetry of the high-temperature phase, transforms as the identity representation and is proportional to q 2 JT . Therefore, the expected relationships between strain components are: e t / V 1=2 / q JT . Two strains show a linear dependence when plotted as e 2 t versus V s (Fig. 8c), consistent with these symmetry considerations.
The simultaneous linear and quadratic coupling of strain components to the Jahn-Teller order parameter implies a renormalization of the Landau expansion in order parameter for free energy (Landau & Lifshitz, 1958): the transition temperature is renormalized by coupling between the Jahn-Teller order parameter and the symmetry-breaking strain e t , while the fourth-order term of the expansion contains contributions from coupling of the square of q JT with the volume strain.
With increasing Mg content, the magnitudes of the strains all decrease (Figs. 8a and 8b). A decrease in the discontinuity at the transition and a trend towards linear behaviour are also fairly evident.
The Fd " 3 3m $ I4 1 =amd transition is required to be first order in character due to the existence of a third-order term in q JT . However, the strain variations with temperatures are not well represented by the standard solution for the order parameter q JT ¼ 3 4 q 0;JT 1 þ 1 À 4 9 where T tr is the transition temperature and T Ã c the renormalized critical temperature. Rather, the data for Cu100, Cu90  Symmetry-adapted strains calculated from lattice parameters for Mg x Cu 1 À x Cr 2 O 4 with x = 0, 0.10, 0.18, 0.46. (a) The symmetry-adapted tetragonal strains follow the classical pattern of a first-order phase transition driven by a single-order parameter. (b) Volume strain, V s , data have been fit with standard solutions to a Landau expansion assuming that V s scales with the square of the order parameter. On this basis the transition is first order in character at x = 0, 0.10, 0.18 and second order at x = 0.46. (c) Strain-strain relationships: tetragonal strains and volume strains vary linearly with each other at each composition and, within experimental uncertainty, extrapolate to the origin. Same symbols as for Fig. 6. and Cu82 are well represented by the standard solution for a 246 solution with negative fourth-order coefficient (reported here, from Carpenter, Salje, Graeme-Barber, Wruck et al., 1998) q 2 JT ¼ 3 2 q 2 0;JT 1 þ 1 À 3 4 T À T Ã c T tr À T Ã c 1=2 ( )

Figure 9
Variation of T tr and T Ã c as a function of Mg content. Magnetic driven structural transition temperature (Nè el temperature) for MgCr 2 O 4 (Kemei et al., 2013) is also shown for comparison.