Magneto-structural coupling in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ni_xZn_{1-x}Cr_2O_4$$\end{document}NixZn1-xCr2O4

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ZnCr_2O_4$$\end{document}ZnCr2O4 compound is well Known to show the frustration of the spin structure. At 12 K, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ZnCr_2O_4$$\end{document}ZnCr2O4 distorts to break symmetry of the degenerated frustrated spin states by the spin-Peierls-like phase transition, accompanying with the antiferromagnetic ordering. On the other hand, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NiCr_2O_4$$\end{document}NiCr2O4 undergoes a Jahn–Teller phase transition at a temperature of 310 K, differing from the low temperature ferrimagnetic transition temperature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_c$$\end{document}Tc of about 60 K. It is also reported that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NiCr_2O_4$$\end{document}NiCr2O4 shows another magnetic phase transition at about 30 K. These two phase transitions accompanying with the lattice change can be understood by the magneto-elastic interactions. Two interactions, the Jahn–Teller interaction and the spin-Peierls-like interaction are co-exist in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ni_xZn_{1-x}Cr_2O_4$$\end{document}NixZn1-xCr2O4 system. In this report the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ni_xZn_{1-x}Cr_2O_4$$\end{document}NixZn1-xCr2O4 compounds with x = 0.8, 0.6 and 1 are investigated by the X-ray diffraction measurements. From these measurements the crystal structures are determined. The full width at half maximum and integrated intensity give the fruitful information for magnetic elastic interactions.

12 K. Since Cr 3+ ion in cubic crystal is not a JT ion, that is, the ground state of Cr 3+ has no orbital degeneracy, ZnCr 2 O 4 distorts to break the symmetry of the degenerated frustrated spin states by the spin-Peierls-like phase transition. Experimental results of low temperature X-ray diffraction (LTXRD), Xue et al. (2008) showed that below the transition temperature, the profile of (800) Bragg reflection splits into two peaks with lower angle peak having intensity twice that of the higher peak. The crystal distorted from cubic to tetragonal with c < a with space group I4 1 /amd. However, frustration driven magnetostructural coupling is not expected in the ferrimagnetic spinel NiCr 2 O 4 . This is because Ni 2+ -O-Cr 3+ interaction can be stronger than the frustrated interaction between the Cr 3+ ions (Reehuis et al. 2015;Suchomel et al. 2012). Furthermore, JT ion Ni 2+ can cause tetragonal distortions that should further alleviate frustration in the Cr 3+ sublattice. Low temperature structures and magnetic spin structures of NiCr 2 O 4 have been studied by many authors (Suchomel et al. 2012;Ishibashi and Yasumi 2007;Tomiyasu and Kagomiya 2004). At 310 K co-operative JT distortion occurs with lowering the structural symmetry from cubic (Fd3m) to tetragonal (I4 1 /amd). Measurement of heat capacity, magnetic susceptibility and X-ray diffraction (XRD) show that the JT phase transition of NiCr 2 O 4 compound occurs at T S = 310 K with the elongated NiO 4 tetrahedron along the c-axis that is c > a giving rise to the tetragonal structure with space group Fd3m to I4 1 /amd (Klemme and O'neill 2000;Klemme and van Miltenburg 2002;Ueno et al. 1999;Kino et al. 1972;Prince 1961). In the tetragonal phase the lattice constants ratio c/a > 1, in contrast with ZnCr 2 O 4 . Further distortion of tetragonal NiCr 2 O 4 to orthorhombic phase occurs due to the ferrimagnetic ordering at T c = 60K. Bertaut and Dulac observed that both ferrimagnetic and antiferromagnetic ordering of NiCr 2 O 4 simultaneously occur at T c = 65 K Dulac 1972, 1980). M.R. Suchomel et al. however, observed that at the ferrimagnetic transition of 65 K, the tetragonal (440) spectrum split into the orthorhombic (800) and (080) reflections by high-resolution synchrotron X-ray diffraction (Suchomel et al. 2012). Ishibashi and Yasumi (2007) reported another magnetic transition in NiCr 2 O 4 at 31 K. Tomiyasu and Kagomiya also reported this temperature of T N = 31 K corresponds to the ordering of the antiferromagnetic component of the magnetic structure of NiCr 2 O 4 (Tomiyasu and Kagomiya 2004). Klemme and Miltenburg also observed anomaly in the specific heat at this temperature, but no changes in the average structure of NiCr 2 O 4 have been observed at T N = 31 K (Klemme and O'neill 2000). Among these three phase transitions, JT transition and the ferrimagnetic order accompany with crystal distortion from tetragonal to orthorhombic structure are well confirmed. The phase transition at 31 K is not yet well confirmed. This magneto-structural transition at 31 K should be necessary for further investigations. With the substitution of nonmagnetic Zn 2+ ion in addition to magnetic Ni 2+ ion in NiCr 2 O 4 , the magnetic interactions can be weakened and also the geometrical frustration can be enhanced. Study on Ni x Zn 1−x Cr 2 O 4 system shows that the JT and magnetic phase transitions depend on the Ni concentration. It was observed by Kino et al. (1972) that when x > 0.6 JT phase transition occurs at the higher temperature than magnetic phase transition. In Ni x Zn 1−x Cr 2 O 4 system, ZnCr 2 O 4 and Ni 0.5 Zn 0.5 Cr 2 O 4 were already published in Ref. 8. Ni 0.5 Zn 0.5 Cr 2 O 4 shows the transition temperatures at T S = 20 K and T c = 16.5 K respectively. In the present article, we have investigated the phase transitions of Ni x Zn 1−x Cr 2 O 4 system with x = 0.6, 0.8 and 1 by measuring the low temperature X-ray diffraction. Not only the measurement of diffraction spectrum with Rietveld fitting but also the temperature dependence of full width at half maximum (FWHM), integrated intensity (I.I.) and the lattice spacing d of the powder specimen give the information about the phase transition. Especially we are interested in the magnetoelastic interactions and their effects with doping the nonmagnetic Zn 2+ ion in NiCr 2 O 4 . We would also like to investigate the phase transition at about 30 K in NiCr 2 O 4 . This phase transition is not yet well understood.

Experiment
X-ray diffraction measurement for the powder specimens were performed using RINT 2500 system, Rigaku Company. The X-ray beam was generated by a rotating Cu anode. The whole profile of the reflection were measured with a step size of 0.20° and stepcounting time of 1 s above and below the transition temperature for calculating the lattice constant by Rietveld refinements using " RIETAN-2000" (Izumi andIkeda 2000). The temperature variation of the several short spectrum were taken at the higher angle. The FWHM and I.I. were obtained from the observed spectrum. Pseudo-Voight function is used to analyzes the short spectrum. The grain sizes of the particle are measured by the scanning electron microscope (SEM) FEI Company, Model inspect S50 with Tungsten filament.

Sample preparations
The powder samples Ni x Zn 1−x Cr 2 O 4 with x = 0.6, 0.8 and 1 were prepared from ZnO (≥ 99 %), NiO (≥ 99 %), Cr 2 O 3 (≥ 99 %) powders. The powders were mixed in appropriate proportions in an agate mortar under acetone for 3 h. The mixtures were then pre-sintered at 800 °C for 5 h. Resultant mixtures were reground and hand mixed for 3 h under acetone. The mixture were pressed into pellets and sintered at 1200 °C (Klemme and van Miltenburg 2004) for 5 h (increasing 10 °C/min). Powder X-ray diffraction (XRD) pattern at room temperature confirmed that the final product has a cubictype spinel structure with single phase for 0.6 and 0.8 and tetragonal spinel structure with single phase for x = 1. Also XRD indicates no impurities for the samples x = 0.6 and 0.8. But the XRD indicated small impurity phases for the sample NiCr 2 O 4 . This is due to the raw materials NiO or Cr 2 O 3 and Cu of the sample holder.
For the morphology test the samples Ni x Zn 1−x Cr 2 O 4 with x = 0.6, 0.8 were prepared from the above mentioned powders. But the mixtures were pre-sintered at 1200 °C for 4 h with an increase in temperature 8 °C/min. Resultant mixtures were reground and hand mixed for 2 h under polyvinyl alcohol. The mixture was Pressed into pellets and sintered at 1500 °C for 4 h (increasing 8 °C/min) and they were polished to make the mirror surface.

Results and discussions
NiCr 2 O 4 compound has a cubic spinel structure with the space group Fd3m to above 310 K. As described in introduction, it has been reported that NiCr 2 O 4 shows the three phase transitions, that is at 310 K the JT distortion, at 65 K, the ferrimagnetic order and at 30 K antiferromagnetic ordering (Suchomel et al. 2012). To investigate these three phase transitions LTXRD was performed from 300 K down to 10 K. So it was not possible to observe the phase transition at 310 K. The whole profiles of reflection peaks were measured at 300, 100 and 15 K. The extra peak due to NiO or Cr 2 O 3 and Cu are subtracted and then observed diffraction spectra were refined by the Rietveld refinements using "RIETAN-2000" software. The results of the Rietveld analysis at 300, 100 and 15 K are listed in Table 1(a-c). At 300 and 100 K the crystal structure is tetragonal with space group I4 1 /amd. The refinement precisions are given by the value of S, 2.3491 and 1.1836 respectively. At 300 K, just below the phase transition temperature 310 K, the crystal structure transition should be transient. So Rietveld fitting gives rather worse S value than one at 100 K. These lattice parameters are completely identical with those obtained by Suchomel et al. (2012). At 15 K the crystal structure is orthorhombic with space group Fddd and the refinement precision is 1.29. The temperature variations of the several short spectra were measured between 10 and 300 K. The diffraction intensity I can be expressed by the following Debye-Waller equation Xue et al. (2008).
where I 0 is the scattered intensity from the rigid lattices, θ B is the scattering angle, M is the mass of the atom and ω is the frequency of the oscillation and E R is the recoilfree energy. In this temperature range the spectrum was split into two peaks due to the tetragonal distortion. The spectrum was fitted to a Pseudo-Voigit function and the temperature dependence of the lattice spacing d1 (440) and d2 {(404) and (044)} were obtained as shown in Fig. 1a, b. At the ferrimagnetic transition at 65 K, M. R. Suchomel et al. observed the tetragonal (440) spectrum that split into orthorhombic (800) and (080) reflections as measured by high-resolution synchrotron X-ray powder diffraction. Unfortunately we did not observe the split of the tetragonal (440) reflection. The FWHM of tetragonal (440) and (044) reflection, however, increases abruptly below about 60 K as shown in Fig. 2, suggesting the structure distortion. The FWHM stopped to increase at 20 K. The I.I. also shows the abrupt decreasing below about 60 K, shown in Fig. 3a, b and also below about 25 K again slightly increased. These results of FWHM and I.I. suggest that at about 60 K, the structure distortion, might be occured from tetragonal to orthorhombic. As shown in Table 1c the Rietveld fitting gives the orthorhombic lattice constants at 15 K. At about 25 K the symmetry of the structure does not change but the lattice constant such as c-axis, changes its values due to the magneto-elastic interaction during the antiferromagnetic components orders. Tomiyasu and Kagomiya proposed the spin structure of NiCr 2 O 4 from their neutron magnetic scattering and the magnetization measurements. Below T c = 75 K, the longitudinal component of the spins along  404) and (044)} reflection decreases below about 60 K. Below about 25 K it seems to be stopped to decrease. But the data around 25 K are rather scattering, so we will not discuss it anymore. The size of Ni 0.8 Zn 0.2 Cr 2 O 4 particles were checked by SEM. The distribution of particle size is large i.e., average particle size is about 1.85 μm. The smallest size of the particle is about 0.35 μm and the largest size of the particle is about 3.59 μm.

Crystal structure
As discussed in the previous section, three phase transitions in NiCr 2 O 4 are almost confirmed, though the third phase transition at about 30 K is still not well understood. With doping the nonmagnetic Zn 2+ ion in place of the magnetic Ni 2+ ion in  (044) Tables 2, 3 and 4 respectively. At room temperatures, the prepared Ni 0.8 Zn 0.2 Cr 2 O 4 spinel compounds are rather homogeneous compound in the space group Fd3m with lattice parameter 8.31642 ± 0.001 Å and the refinement precision is given by the value of S = 1.2468. The overall quality of the fitting is fairly good. Profiles of (θ − 2θ) scans of (440) Bragg reflection at 200 K (T > T S ), 150 K (T ≤ T S ) and 30 K (T < T S ) are shown in Fig. 4. These profiles contain CuK α2 in the right shoulder of the peaks. At 150 K the profile of (440) already split into two peaks. This phase transition undergoes by JT interaction and also spin-Peierls-like interaction and as shown in Table 2, the symmetry of the crystal lowers from Fd3m to Fddd. The lattice constants obtained by the Rietveld refinement which are listed in Tables 2,  3     (440) reflection is measured at various temperatures between 15 and 300 K. The spectrum was fitted to a Pseudo-Voigit function and the temperature dependence of the lattice constants is obtained and shown in Figs. 6, 7 and 8 and also I.I. is shown in Fig. 9. The FWHM, of the spectrum, together with that of the NiCr 2 O 4 (440) reflection and Ni 0.6 Zn 0.4 Cr 2 O 4 (440) reflection are shown in Fig. 10 ZnCr 2 O 4 (c/a < 1) separately occurs so as to minimize the crystal distortion energy. Then as M. Kataoka and J. Kanamori has already mentioned in their theoretical work on Cu 1−x Ni x Cr 2 O 4 system Kataoka and Kanamori (1972), the elongated c-axis of NiCr 2 O 4 will align along one of a-axes of ZnCr 2 O 4 , say a-or b-axis which are also elongated in ZnCr 2 O 4 . If the alignment of c-axis prefers to a particular axis, say a-axis, a and b axes are not equal, so the result is orthorhombic. When the distorted NiCr 2 O 4 with c/a > 1 and the distorted ZnCr 2 O 4 with c/a < 1 are mixed in Ni 0.8 Zn 0.2 Cr 2 O 4 compound below the structural transition temperature, the FWHM can be expected to increase. In Fig. 8 (c− < a, b >) vs. T is plotted. Here < a, b > is the average values of lattice constants a and b obtained from the reflection (440) measurement. The figure shows that the structure transition seems to be at 160 K which corresponds to the peak temperature of FWHM. With decreasing temperature, (c− < a, b >) increases and is goes to saturate at low temperatures, while the FWHM decreases quickly. This result can be understood as follows. At the transition temperatures, the c-axis of local NiCr 2 O 4 structure aligns rather randomly and with lowering the temperature the alignment becomes more stable to one of a-axes. In present Rietveld fitting, the S value is rather large in the distorted phase. The FWHM in Ni 0.8 Zn 0.2 Cr 2 O 4 starts to increase rather sharply below about 30 K (Fig. 10). This behavior is similar to one in NiCr 2 O 4 below about 60 K which corresponds to the ferrimagnetic ordering temperature. In NiCr 2 O 4 , the ferrimagnetic transition accompany with the crystal distortion from the tetragonal symmetry to the orthorhombic. Ni 0.8 Zn 0.2 Cr 2 O 4 is however, orthorhombic structure above the ferrimagnetic transition. So only the lattice constants can be changed with spin order. As shown in Fig. 7, the lattice constants a or b do not show the drastic change at the ferrimagnetic order, but the c axis as shown in Fig. 6 shows the sudden drop at the ordering temperature. The I.I. of Ni 0.8 Zn 0.2 Cr 2 O 4 in Fig. 9 also shows the sudden increase at about 25 K. The Rietveld fitting S value at 15 K shows a large value of 2.15. But when we analyze the spectrum measured at 15 K by using the model of Fddd and I4 1 /amd symmetry mixing, S value is 1.81, not so small but much smaller than the S = 2.15 obtained assuming only I4 1 /amd structure model for the same spectrum at 15 K. In the distorted phase between 30 and 160 K, it is better to analyze the spectrum by using the model of The phase transition of NiCr 2 O 4 at 60 K is the ferrimagnetic order accompanied with the structural change from tetragonal to orthorhombic phase. So, the structural phase transition is due to the magneto-elastic interaction. The structural phase transition of Ni 0.8 Zn 0.2 Cr 2 O 4 at 160 K may also be due to the magneto-elastic coupling, that is, the collaboration of JT interaction due to the Ni 2+ ion and spin-Peierls-like interaction due to the Cr 2+ ion. The phase transition of Ni 0.8 Zn 0.2 Cr 2 O 4 at 25 K is the ferrimagnetic transition, but it also accompany with the structure change. But it already has the orthorhombic symmetry. So it also does not change the symmetry and just changes the lattice constants, similar to the phase transition which occurs at 30 K in NiCr 2 O 4 . In NiCr 2 O 4 , the ferrimagnetic transition occurs at 60 K, but it accompany with the crystal symmetry change. In NiCr 2 O 4 the antiferromagnetic transition without the crystal symmetry change occurs at 30 K. In Ni 0.8 Zn 0.2 Cr 2 O 4 another phase transition also occurs at about 16 K which should be antiferromagnetic transition similar to one at 30 K in NiCr 2 O 4 .
Ni 0.6 Zn 0.4 Cr 2 O 4 The particle sizes of the Ni 0.6 Zn 0.4 Cr 2 O 4 were also checked by SEM. The distribution of particle size is large. The average particle size is about 1.92 μm. The smallest size of the particle is about 0.78 μm and the largest size of the particle is 3.46 μm. The ionic radius of Zn 2+ is about 0.75 Å which is larger than ionic radius of Ni 2+ (0.70 Å). So the average size of the particle for the sample Ni 0.6 Zn 0.4 Cr 2 O 4 will be larger than the Ni 0.8 Zn 0.2 Cr 2 O 4 . Experimental results show the same as were expected. X-ray diffraction spectrum does not show any clear split due to the crystal distortion. Profiles of (θ − 2θ) scans of (440) Bragg reflection at 300 K (T > T S ), 100 K (T > T S ), 30 K (T ∼ = T S ) and 15 K (T < T S ) are shown in Fig. 11. These profiles contain CuK α2 in the right shoulder of the peaks.At temperature 30 K (T ∼ = T S ) the peak is rather broad. But FWHM shows the maximum at about 30 K, as shown in Fig. 12. In the case of Ni 0.8 Zn 0.2 Cr 2 O 4 , FWHM shows the very sharp peak. This peak may be attributed to the crystal structure transition. The broad peak at about 30 K in Ni 0.6 Zn 0.4 Cr 2 O 4 may also be due to the structural phase transition. The crystal structure below the structural phase transition temperature T S can be the tetragonal, I4 1 /amd or the orthorhombic, Fddd, but not yet determined. The most possible structure should be the mixed structure of I4 1 /amd and Fddd which can produce the broad peak of FWHM. The temperature dependence of the FWHM also shows another anomaly at about 20 K. This can be the magnetic phase transition such as the paramagnetic to ferrimagnetic state. The third phase transition which can be observed in NiCr 2 O 4 and in Ni 0.6 Zn 0.4 Cr 2 O 4 cannot be seen in the present measurements down to 12 K in Ni 0.6 Zn 0.4 Cr 2 O 4 . On the other hand, the lattice spacing d value for (440) reflection increases below about 80 K with decreasing temperature as shown in Fig. 13. This result must correspond to the FWHM results, that is, showing the change in crystal structure at about 30 K. It can be understood as follows. The reflection of (440) can split into two reflections (440) and {(404), (044)}. This tetragonal structure can be c/a > 1, similar to Ni 0.8 Zn 0.2 Cr 2 O 4 compound. The integrated intensity of {(404), (044)} reflection can be twice that of (440) reflection. The sample Ni 0.6 Zn 0.4 Cr 2 O 4 is rather inhomogeneous. So the reflection peaks of (440) and {(404), (044)} are broad. Even in the tetragonal phase the reflection peak shows one broad peak as shown in Fig. 11. The behavior of this broad peak can be determined by the large intensity of {(404), (044)} reflections. The d value obtained from the (440) reflections starts to increase below about 80 K with decreasing temperature. In the case of Ni 0.8 Zn 0.2 Cr 2 O 4 compound, the I.I. also shows the anomaly at the phase transition of T S and T c . But in Ni 0.6 Zn 0.4 Cr 2 O 4 compound, the I.I. does not show clear anomaly at these phase transition temperatures. This result can be understood by  Fig. 15. c axis is along (001), a and b are perpendicular direction to (001). In ferrimagnetic spin structure, the spins align along (001) direction, but slightly canted and the perpendicular components of c axis order antiferromagnetically.