Exploring the negative thermal expansion and magnetocaloric effect in Fe2(Hf,Ti) Laves phase materials

The transition-metal based Laves phase materials represent an extended family of alloys with rich and fascinating physical properties. In this work, we have investigated the negative thermal expansion and magnetocaloric effect in arc-melted and melt-spun Fe 2 Hf 1-x Ti x ( x = 0.15, 0.27, 0.30, 0.33, 0.36, 0.40) alloys. For x = 0.30 – 0.40, two hexagonal phases with different compositions share the same P 6 3 /mmc lattice symmetry, but have slightly different lattice parameters. The saturation magnetization and Curie temperature both follow a decreasing trend with the average unit-cell volume. For Fe 2 Hf 0.6 Ti 0.4 melt spinning improves the saturation magnetization from 48.7 to 59.6 Am 2 /kg and the magnetic entropy change from 0.46 to 0.54 J/kgK at a magnetic field change of 2 T. These enhanced values are attributed to an improved homogeneity caused by a suppression of phase segregation during rapid solidification. We have utilized neutron powder diffraction and M ¨ ossbauer spectroscopy to illustrate the correlation between the magnetic order and the negative thermal expansion in single-phase Fe 2 Hf 0.85 Ti 0.15 . The magnetic moments of Fe align below 400 K in the a-b plane and a moment change for the Fe atoms is responsible for the large volumetric coefficient of thermal expansion of (cid:0) 25 × 10 (cid:0) 6 K (cid:0) 1 over a wide temperature range of 300 – 400 K.


Introduction
The magnetocaloric effect (MCE) is the caloric response of the lattice on the application or removal of a magnetic field and can be characterized by the magnetic entropy change and the adiabatic temperature change [1,2].In terms of the energy efficiency and environmental safety of MCE-based magnetic refrigeration, the exploration of new MCE materials and the understanding of the mechanism controlling the magnetic phase transition have attracted much attention in recent decades [3,4].Many magnetocaloric materials undergoing a magnetostructural or a magnetoelastic transition have been studied, such as Mn-Fe-P-Si [5,6], La(Fe,Si) 13 [7,8], Ni-Mn-based Heusler alloys [9,10], MnCoGe [11], Fe 2 (Hf,Ta) [12][13][14] and Mn 2 Sb [15][16][17].Due to the strong spin-lattice coupling, some of the magnetocaloric materials, such as La-Fe-Si-Co [18], MnCoGe [19], and Fe-based Laves phase materials [20,21], are also considered as potential candidates for negative thermal expansion (NTE) materials, which are in high demand in modern industry for the adjustment of the thermal expansion in materials [22,23].Among them magnetic Fe-based Laves phase materials combine a spontaneous magnetization with a magnetoelastic lattice volume contribution based on two different Fe atomic sites, which combine the advantages of a high thermal conductivity and good mechanical properties for NTE and magnetocaloric applications [20,14].
MCE materials with a large magnetic entropy change generally show a large change in magnetic moment.It is well known that itinerant electron systems with a large magnetic moment have a larger unit-cell volume.Therefore, a steep change in magnetization for itinerant electron systems is generally linked to a steep change in volume, except for magnetic materials that rather change c/a ratios, which may yet occur without volume change.An interesting example is the Fe 2 (Hf,Ta) Laves phase compound, which shows a sharp magnetization jump at the ferromagnetic-to-antiferromagnetic phase transition, accompanied by a negative thermal expansion effect [24].This sharp transition is due to a frustration effect, in which the magnetic moment of the Fe atom at the 2a site of the hexagonal P6 3 /mmc structure (space group 194) disappears below the transition temperature [25].This leads to a large adiabatic temperature change of about 3.4 K in the presence of a relatively small change in applied magnetic field of 1.5 T [14].The tuneable transition temperature, large adiabatic temperature change and small thermal hysteresis make these compounds attractive for magnetocaloric applications [13,14].On the other hand, the linear NTE coefficient of − 16.3 × 10 − 6 K − 1 over a broad temperature window of 105 K (222-327 K) is attributed to the asynchronous ferromagnetic ordering of the Fe moments at the 6h and 2a sites revealed by Electron Spin Resonance [20].A similar magnetoelastic transition has also been observed in Fe-based Fe 2 (Sc,Ti) [21], Fe 2 (Hf,Ti) [26] and Fe 2 (Hf,Nb) [27] Laves phase materials.So far however, there have only been a limited number of systematic reports on the mechanism responsible for the magnetoelastic coupling in Fe-based Laves phase materials.
In this work, Mössbauer spectroscopy, neutron powder diffraction and DFT calculations are employed to investigate the correlation between the negative thermal expansion and the magnetic behaviour in Fe 2 (Hf,Ti) compounds.In addition, we propose a materials fabrication method to enhance the magnetocaloric effect in Fe 2 (Hf,Ti) Laves phase materials by supressing phase segregation using melt spinning technique.

Experimental methods
Polycrystalline Fe 2 Hf 1-x Ti x (x = 0.15, 0.27, 0.30, 0.33, 0.36, 0.40) compounds were prepared from high-purity elements (Fe 99.98%, Hf 99.95%, Ti 99.99%) by arc melting.Samples with a total mass of 5 g were melted four or five times, and the button-shaped alloys were flipped over after each melting.Four different heat treatments were tried for x = 0.36: (a) annealing at 1173 K for 24 h and quenching; (b) annealing at 1273 K for 134 h and quenching; (c) annealing at 1323 K for 24 h and quenching; (d) annealing at 1373 K for 24 h and quenching.All these heat treatments resulted in comparable lattice parameters, saturation magnetizations and transition temperatures (see Fig. S1 in Supplementary Material).Accordingly, only as-cast samples are reported for most compositions (with some exceptions noted in the manuscript).For Fe 2 Hf 1-x Ti x (x = 0.36, 0.40) compounds, the melt-spinning technique is used to improve homogeneity.The crucible of melt-spinning device is a commercial quartz tube with a melting point of 1883 K.The melting point of Fe 2 Hf 1-x Ti x is estimated to be about 1673 K.The melt-spun ribbons were produced under argon atmosphere at different surface speeds of the copper wheel.By comparing the melt-spun ribbons with different heat treatments (see Fig. S2 in Supplementary Material), those without a heat treatment are characterized for the discussion.
X-ray diffraction (XRD) data were collected with a Panalytical X-Pert PRO using Cu-K α radiation and an Anton Paar TTK 450 temperature chamber.The lattice structure was analysed by Rietveld refinement using Fullprof [28].The magnetic properties at low temperature (4-370 K) were measured using a superconducting quantum interference device (SQUID) magnetometer model MPMS-XL, equipped with the reciprocating sample option.Magnetisation measurements at high-temperature (315-650 K) were carried out using a vibrating sample magnetometer (VSM) model LakeShore 7307 equipped with a high-temperature furnace (model 73,034).The ferromagnetic transition temperature T C is determined from the minimum in the temperature derivative of the magnetization as a function of temperature at an applied magnetic field of 1 T. The magnetic entropy change was calculated from the M-T curves obtained at different magnetic fields using the Maxwell relations.Differential scanning calorimetry (DSC) measurements were performed in a commercial TA-Q2000 DSC calorimeter (at a sweep rate of 10 K/min).The microstructure was analysed by scanning electron microscopy (SEM) using a FEI Quanta FEG 450 equipped with energy dispersive X-ray spectroscopy (EDS).Transmission 57 Fe Mössbauer spectra were collected at different temperatures in the range from 4.2 to 300 K with conventional constant-acceleration or sinusoidal velocity spectrometers using a 57 Co(Rh) source.The velocity calibration was performed using an α-Fe foil at room temperature.The source and the absorbing samples were kept at the same temperature during the cryogenic measurements.The Mössbauer spectra were fitted using the Mosswinn 4.0 program [29].Powder neutron diffraction (ND) data at different temperatures were collected at a wavelength of 1.6672 Å using the ND apparatus PEARL [30] at the research reactor of Delft University of Technology.The crystal structures and the atomic occupancies were refined using the Rietveld refinement method implemented in the Fullprof software package [31][32][33].
Density functional theory (DFT) modelling was performed with the Vienna ab initio simulation package (VASP) [34,35].Exchange interactions were taken into account by using the generalized gradient approximation (GGA) by Perdew, Burke, and Ernzerhof (PBE) [36].For all calculations, the projector augmented-wave (PAW) [37,38] pseudopotentials were used with the valence electron configuration 5p5d6s for Hf, 3s3p3d4s for Ti and 3d4s for Fe.All calculations were performed for a 1×1×2 supercell.The structural degrees of freedom were fully relaxed on a gamma centred k-grid of 11×11×3.The k-space integrations were performed using the second order Methfessel-Paxton method [39] with a smearing width of 0.05 eV.The lattice parameters and atomic positions were optimized with the criteria that the atomic force is weaker than 0.1 meV/Å and an energy convergence of 1 μeV.The kinetic energy cutoff was set at 400 eV.

Results and discussion
Fig. 1(a) shows the XRD patterns for arc-melted Fe 2 Hf 1-x Ti x (x = 0.15, 0.27, 0.30, 0.33, 0.36, 0.40), denoted as TiX alloys (where X corresponds to the Ti fraction x).With increasing Ti substitution, the peaks become broader, which is seen in the enlarged region in Fig. 1(b) and the increased full width at half-maximum (FWHM) of ( 110) and (003) peaks in Fig. 2(a).All samples crystalize in the hexagonal MgZn 2type structure with P6 3 /mmc symmetry (space group 194), where the Fe atoms occupy the 2a and 6h sites and the Hf/Ti atoms occupy the 4f site.According to the Rietveld refinement results in Fig. 1(c) the patterns for the Ti0.15 and Ti0.27 samples can be fitted best as single-phase hexagonal.The other four samples, for example the Ti0.36 sample in Fig. 1 (d) can be fitted best by a two-phase model.These two phases share the same space group 194, but with slightly different lattice parameters.The refined lattice parameters are given in Fig. 2(d-f).The fraction of the Tirich phase P1 increases with increasing Ti content.The average unit-cell volume, based on the fraction of the two phases P1 and P2, decreases linearly with Ti content.
To confirm the presence of phase segregation, the backscattered electron SEM images of the Ti0.36 sample are shown in Fig. 2(b).The dark region refers to the Ti-rich phase, corresponding to P1.The chemical compositions of the matrix and Ti-rich phase are determined by EDS and are found to be Fe 65.9 Hf 25.5 Ti 8.6 and Fe 66.3 Hf 19.7 Ti 14.0 , respectively.It is found that this phase segregation is difficult to avoid by different heat treatments (see Fig. S1 in Supplementary Material).It is worthwhile to note that in Hf 0.6 Ti 0.4 Fe 2 a similar XRD peak broadening is attributed to a phase coexistence of AFM and FM states [26], where the homogeneity is supported by SEM mapping at the scale of 100 µm.A variation in chemical composition at a smaller length scale or in other areas of the samples cannot be excluded [40].From the insert picture in Fig. 2(c), the annealed Ti0.36 sample (annealed at 1273 K for 134 hours) exhibits significant compositional fluctuations, as indicated from the line-scan profile across the Ti-rich boundaries.The chemical composition of the matrix and the Ti-rich phases are Fe 65.4 Hf 27.0 Ti 7.6 and Fe 65.3 Hf 18.3 Ti 16.3 , respectively.
The M-T curves in Fig. 3(a) show that the magnetization decreases continuously with increasing temperature for the arc-melted Ti0.15 sample.With decreasing temperature it follows the same trajectory without any thermal hysteresis, which suggests a second-order phase transition from the ferromagnetic to the paramagnetic phase.Fig. 3(b)   T C initially decreases linearly with the average unit-cell volume and then stabilizes around 310 K, as seen in Fig. 3(e).Substitution of Hf by Ti decreases the atomic spacing of the Fe atoms and leads to a suppression of the FM order, so that T C can be tuned between 310 and 390 K by changing the Ti content.A series of M-H curves was measured at 5 K, as shown in Fig. 4(d).All samples reach a magnetic saturation at the highest field of 5 T. A significant decrease in saturation magnetization is observed for the Ti0.40 sample due to the lower saturation magnetization from the dominant Ti-rich phase (67%).We employed spinpolarized DFT calculations for the formation energy, lattice parameters and magnetic moment, as shown in Fig. 3(g-i).Two magnetic configurations: (i) ferromagnetic (FM) structure, with the Fe moments at the 2a and 6h sites orientated in the a-b plane and (ii) antiferromagnetic (AFM) structure, with at the 6h site ferromagnetically aligned intralayer Fe magnetic moments and antiferromagnetically aligned interlayer Fe magnetic moments with unordered Fe magnetic moments at the 2a site [13] are considered in our DFT calculations (see Fig. S3 in Supplementary Material).Comparing the formation energy of the two magnetic configurations, the FM structure is the more favourable one.As the Ti content increases, the energy difference decreases (in Fe 2 Ti the energy difference is merely 6 meV/atom) because the FM is destabilized by the decrease in lattice parameters due to the smaller atomic size of Ti compared to Hf [41].The total magnetic moment also decreases with increasing Ti content.It is worth noting that the magnetic moment of Fe at the 2a site is more sensitive to the Ti content than that of Fe at the 6h site.This stronger instability of the Fe magnetic moment at 2a site is also reported in Ti 0.70 Sc 0.30 Fe 2 , as observed by neutron diffraction [42].The lattice parameter decreases linearly with increasing Ti content.For single-phase samples, the magnitude of the calculated lattice parameter and magnetic moment are in agreement with the values obtained from XRD and SQUID, as shown in Fig. 3(h) and Fig. 3(i).The spin-polarized density of states (DOS) for Fe 2 Hf 1-x Ti x (x = 0.125, 0.25, 0.375) is shown in Fig. S4.For an increasing Ti substitution the lattice shrinks and the Fermi energy moves towards a lower energy, resulting in a lower magnetization and a lower T C .The substitution of the Ti atom for the Hf atom brings a peak in the DOS close to the Fermi level, which results in a significant impact on the magnetic interactions.
Phase segregation has also been reported in other Fe-based Laves   phase materials such as Fe 2 (Hf,Nb) [27,40], where the cause of phase segregation is attributed to the positive mixing energy calculated by DFT.However, this is ruled out for Fe 2 (Hf,Ti), since the formation energy for both the AFM and FM state is about − 9 eV according to the DFT calculation in Fig. 3(g).For Fe 2 (Hf,Ti), the large difference in atom size (7.5%) between Hf (159 pm) and Ti (147 pm) could be responsible for the phase segregation in high-Ti content samples, since a similar phase segregation was observed in the perovskite oxides Gd 1-x Sr x FeO 3-δ [43], which was caused by the large size mismatch of 10% for Gd and Sr ions.
Phase segregation is a way to relieve local stresses due to a size mismatch.
As a rapid solidification technology, melt-spinning process is found to suppress phase segregation in Fe 2 Hf 1-x Ti x samples with a high-Ti content (x = 0.30, 0.33, 0.36, 0.40).Fig. 4(a) shows the homogeneous microstructure for the melt-spun Ti0.36 sample on the wheel side prepared with a wheel surface speed of 17 m/s.For melt-spun ribbons, the free side of the ribbon has a lower cooling rate than the wheel side of the ribbon [44].Therefore, for the Ti0.36 (17 m/s) sample on the free side, the Ti-rich phase can be seen at the grain boundaries, while it is not observed in Ti0.36 (22 m/s) sample.From Fig. 4(b-c) the ribbons are found to be well crystalline with a grain size of about 1-3 µm.Line scanning across the grain boundaries shows that the Ti0.36 (22 m/s) sample is more homogeneous than the Ti0.36 (17 m/s) sample and thus the magnetic transition is sharper than the latter (see Fig. S5(a) in Supplementary Material).Compared to the arc-melting technique, the improved homogeneity in melt-spun samples can be attributed to the fact that the high solidification rate of the melt-spinning process suppresses phase segregation.Therefore, a wheel speed of 30 m/s is used for the melt-spun Ti0.40 sample.
The XRD pattern of the melt-spun Ti0.40 (30 m/s) sample shows a single hexagonal MgZn 2 -type phase (see Fig. S6 in Supplementary Material).Fig. 5(a) and 5(b) show the M-T and M-H curves for the arcmelted and melt-spun Ti0.40 samples.The T C (283 K) of the melt-spun Ti0.40 sample is lower than that of the arc-melted sample (309 K).This can be explained by the fact that the Ti in the melt-spun sample is more homogeneously distributed than in the arc-melted sample, so in the latter only part of the Ti plays a role in reducing T C .The increased amount of ferromagnetic contribution at high temperature in melt-spun sample can be ascribed to the anti-site disorder of Fe atoms introduced by the melt-spinning process, which is also observed in (La,Ce)(Fe,Si) 13 [44] and melt-spun Fe 2 (Hf,Ta) [45].On the other hand, M s increases by 22% from 48.7 Am 2 /kg in the arc-melted sample to 59.6 Am 2 /kg in the melt-spun sample due to the more homogeneous microstructure.The magnetic entropy change ΔS m shown in Fig. 5(e) and 5(f) is calculated from the isofield magnetization curves in Fig. 5(c

S7(e) in Supplementary Material
).The Arrott-plots of the arc-melted Ti0.15 and Ti0.27 samples in Fig. S7(c) and S7(f) (Supplementary Material) prove that the phase transitions are of second-order, which is consistent with the broad exothermic and endothermic peaks in the DSC curves (see Fig. S8 in Supplementary Material).The |ΔS m | of Fe 2 Hf 1-x Ti x for a magnetic field change of 2 T is comparable to other Fe-based Laves phase materials with a SOMT, with 0.4 J/kgK in Sc 0.35 Ti 0.65 Fe 1.95 [46], but it is smaller than Fe-based Laves phase materials with a FOMT, with 1.4 J/kgK in Sc 0.35 Ti 0.65 Fe 1.95 [46] and 2.3 J/kgK in Fe 2 Hf 0.86 Ta 0.14 [13].
To analyse the nature of the magnetoelastic transition in this system, powder neutron diffraction (ND) and Mössbauer spectroscopy were used to study the single-phase arc-melted Ti0.15 sample.To eliminate possible strains due to thermal gradients during arc-melting, the sample was quenched into water after annealing at 1273 K for 32 h.A detailed Rietveld analysis was performed to refine the crystal structure and magnetic structure from the neutron diffraction patterns at different temperatures.The diffraction patterns recorded at 573 K (PM phase) and 4 K (FM phase) are shown in Fig. 6(a) and 6(b).The diffraction pattern measured at 573 K is characteristic for the nuclear Bragg scattering (PM phase) and was fitted to the MgZn 2 -type phase with P6 3 /mmc symmetry.The ND pattern obtained at 4 K shows no additional reflections compared to the paramagnetic diffraction pattern.The analysis of the magnetic reflections indicates that the corresponding magnetic structure is defined by the propagation vector k = (0,0,0), which means that the magnetic unit cell coincides with the crystallographic one [13].The arc-melted Ti0.15 sample shows a ferromagnetic ordering of Fe moments below 400 K, which is in good agreement with the T C (392 K) from SQUID measurements.The FM ordering only increases the intensity at the nuclear peaks.The magnetic contributions are most pronounced at the (100), ( 101) and (002) peaks.The magnetic Fe moments were independently refined for the two sites.The best fit corresponds to magnetic moments that lie within the basal plane of the hexagonal lattice.The corresponding FM spin configuration is shown in Fig. 6(c).A summary of the crystallographic parameters and the refined Fe moments at different temperatures is shown in Table 1.The total moment of 3.14 µ B /f.u.(calculated by 3/2 M Fe(6h) + 1/2 M Fe(2a) ) obtained from ND at 4 K agrees well with the 3.19 µ B /f.u.calculated by DFT.
The temperature dependence of the lattice constants a and c, and the unit-cell volume V are displayed in Fig. 6(d,e).The lattice constant c increases continuously, while a only increases slowly from 4 to 300 K and, then decreases from 300 to 400 K and finally increases again from 400 to 573 K.As a result, V decreases abnormally from 300 to 400 K.The drop in lattice constant a is about 0.15%, about half that of Hf 0.86 Ta 0.14 Fe 2 [13].The unit-cell volume of the FM phase is larger than that of the PM phase, and the spontaneous volume magnetostriction accompanying the second-order magnetic phase transition is estimated to be about ΔV/V = 0.25%.The negative thermal expansion observed from 300 K (171.63Å 3 ) to 400 K (171.20 Å 3 ) leads to a negative thermal over a temperature range of 100 K, in agreement with [26].This negative thermal expansion coefficient is comparable to commercial NTE materials like ZrW 2 O 8 (α v = − 27.3 × 10 − 6 K − 1 , 0-300 K) [47] and PbTiO 3 (α v = − 19.9 × 10 − 6 K − 1 , 298-763 K) [48] and other Fe-based Laves phase materials like Hf 0.80 Nb 0.20 Fe 2 (α v = − 15 × 10 − 6 K − 1 , 150-240 K) [27], and Hf 0.80 Ta 0.20 Fe 2 (α v = − 57.6 × 10 − 6 K − 1 , 250-340 K) [20].
The Mössbauer spectra of arc-melted Ti0.15 and melt-spun Ti0.40 samples are shown in Fig. 7(a-f).The area ratio of the components corresponding to the Fe(6h) and Fe(2a) sites was set to 3:1 [40,49].Table 2 summarizes the fitted parameters for the Mössbauer spectra at different temperatures.The small linewidth (0.4 mm/s) for all sites (except for the relaxing phase) indicates a good quality of fit for the spectra.The similar isomer shift and quadrupole splitting for Fe at the 2a and 6h sites indicates that both sites have a similar electronic state [50].For the arc-melted Ti0.15 sample, the spectrum exhibits a paramagnetic feature at 400 K, which is consistent with the neutron diffraction results.The spectra at 4.2, 120 and 300 K contain two magnetic sub-spectra, attributed to the Fe(6h) and Fe(2a) sites.The hyperfine fields for the Ti0.15 sample at 4 K are 19.6 T and 17.2 T at the 6h and 2a sites, respectively.Using a proportionality factor of 14.2 T/µ B [51] yields magnetic moments for the Fe(6h) site and the Fe(2a) site of 1.4 µ B and 1.2 µ B , respectively.This is in reasonable agreement with the ND results (see Table 1).At 350 K, the spectrum consists of a sextet magnetic sub-spectrum and a quadrupole doublet, which is considered to be the Q.Shen et al. relaxing phase, with a coexistence of the ferromagnetic and the paramagnetic phase (45%) [52].This coexistence of FM and PM phases is consistent with the ND data at 350 K.
The temperature dependence of the hyperfine field at the 2a and 6h sites is shown in Fig. 8(a).It is clear that the compound exhibits a ferromagnetic-to-paramagnetic transition between 350 and 400 K, which is consistent with the neutron diffraction results in Fig. 8(b).The magnetic hyperfine fields of Fe atoms at the 2a and 6h sites decrease simultaneously with increasing temperature, which is different from Fe 2 Hf 0.8 Ta 0.2 [49] and Fe 2 Hf 0.9 Nb 0.1 [40].For Hf 0.8 Ta 0.2 Fe 2 , the hyperfine field of Fe at the 2a site decreases to zero at the critical temperature of the first-order FM-AFM transition and the hyperfine field of Fe at the 6h site decreases to zero at the Curie temperature.The wide NTE window for (Hf,Ta)Fe 2 is attributed to the asynchronous FM ordering of Fe moments at 6h and 2a sites revealed by electron spin resonance [20].As in the case in Fe 2 (Hf,Ti), the simultaneous decrease of the hyperfine field of Fe at the 2a and 6h sites indicates a second-order transition.Fig. 8(c) and 8(d) show the interatomic distances of Fe atoms derived from ND.The interlayer spacing of Fe 6h -Fe 6h , Fe 6h -Fe 2a and Fe 6h -Hf/Ti 4f increases linearly with increasing temperature, while a sharp decrease of 0.36% in the intralayer spacing of Fe 6h -Fe 6h is observed within the NTE temperature range.This phenomenon could originate from the fact that the shortest Fe 6h -Fe 6h distance controls the magnetic interactions in Fe 2 (Hf,Ti).A similar trend is also reported for the temperature dependence of magnetism in Sc 0.4 Ti 0.6 Fe 2 [21] and the pressure-dependant magnetism in Mn 3 Ga 0.95 N 0.94 [53].
To reveal the relation between negative thermal expansion and the magnetic order, the spontaneous volume magnetostriction is calculated by subtracting the thermal expansion of the nonmagnetic phase from the experimental one.The magnitude of the spontaneous volume magnetostriction can be defined as ω s = Vexp− Vnm Vnm × 100%, where V exp and V nm (shown in Fig. 6(e)) are the unit-cell volumes of experimental and the non-magnetic phases, respectively [21,54].Fig. 8(e) shows dω s /dT and dM/dT as a function of temperature.Both dω s /dT and dM/dT show the same trend, indicating a strong coupling between magnetism and lattice in this system.A sharp magnetic transition indicates a strong contribution of the spontaneous magnetostriction to the thermal contraction, which contributes to the occurrence of the NTE, as also reported in Mn 3 Ge [55] and Fe 2 Zr 0.7 Nb 0.3 [56].Based on the ND results, the shortest in-plane Fe 6h -Fe 6h distance controls the magnetic interactions of Fe 2 (Hf, Ti), so the rate at which the in-plane Fe magnetic moments decrease with temperature governs the negative thermal expansion.
A linear correlation between the square of the Fe magnetic moment

Table 1
Fitted neutron diffraction parameters of the Ti0.15 sample obtained at different temperatures.The lattice structure corresponds to the P6 3 /mmc symmetry (space group 194) with the Fe atoms at the 2a site with position (0, 0, 0) and the 6h site with position (x, 2x, 1/4) and the Hf/Ti atoms at the 4f site with position (1/3, 2/3, z). and ω s is found in Fig. 8(f).The magnetic-lattice coupling can quantitatively be described by: ω s (T) = kCM(T) 2 [57], where k and C are the compressibility and the magneto-volume coupling constant, and M is the magnetic moment.The constant kC for the Fe 2 Hf 0.85 Ti 0.15 sample can be obtained from the slope of the curve in Fig. 8(f).For Fe 2 Hf 0.85 Ti 0.15 , the constant kC at 4.2 K is calculated to be 1.48 × 10 − 10 cm 2 /A 2 .This value is of the same order of magnitude as those for Fe 2 Hf 0.83 Ta 0.17 (1.32 × 10 − 10 cm 2 /A 2 ) [41] and La(Fe 0.86 Al 0.24 ) 13 (1.79 × 10 − 10 cm 2 /A 2 ) [58], which are potential MCE materials with a strong magnetoelastic coupling.Short-range disorder, either caused by phase segregation in the Fe 2 (Hf,Ti) system ascribed to the large difference in Hf and Ti atoms sizes or due to rapid solidification, may account for the second-order phase transition in this magnetoelastic material.

Conclusions
In summary, we have studied the crystal structure, microstructure, and magnetic properties of arc-melted and melt-spun Fe 2 Hf 1-x Ti x (x = 0.15, 0.27, 0.30, 0.33, 0.36, 0.40) alloys, which are derived from two isostructural HfFe 2 and TiFe 2 Laves phase materials.Surprisingly, a phase segregation is found in samples with x = 0.30-0.40,which we attribute to the 7.5% size difference between Hf and Ti.The high cooling rate achieved by melt-spinning at a wheel speed of 30 m/s yields homogeneous alloys.The magnetic transition temperature decreases linearly with the average unit-cell volume.The saturation magnetization suppressed by phase segregation can be recovered from 48.7 Am 2 /kg to 59.6 Am 2 /kg by melt spinning and thus the magnetocaloric effect can be enhanced from 0.46 to 0.54 J/kg K under a magnetic field change of 2 T due to the improved homogeneity.ND results reveal that the magnetic moments lie within the a-b plane, and the reduction in magnetic moment   of the Fe atoms is consistent with the abnormal decrease in the lattice parameter a with increasing temperature.We observe a negative thermal expansion (− 25 × 10 − 6 K − 1 ) for the single-phase Fe 2 Hf 0.85 Ti 0.15 compound over a wide temperature range of 300-400 K.The relation between the negative thermal expansion and the magnetism is discussed by quantitatively analysing the correlation between the spontaneous magnetostriction and magnetic moment of the Fe atoms.Our results provide a significant insight into the magnetoelastic coupling in these magnetocaloric materials and provide an experimental route to solve common fabrication problems in Fe-based Laves phase materials.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 2 .
Fig. 2. (a) Full width at half-maximum (FWHM) of (110) and (003) peaks for arc-melted Fe 2 Hf 1-x Ti x (x = 0.15, 0.27, 0.30, 0.33, 0.36, 0.40).Backscattered electron SEM images of (b) arc-melted Ti0.36 sample without any heat treatment and (c) arc-melted Ti0.36 sample annealed at 1273 K for 134 h.(d, e, f) Ti-content dependence of the lattice parameters a and c and the unit-cell volume V.The inset in (f) is the fraction of the Ti-rich P1 phase as a function of the Ti content.

Fig. 3 .
Fig. 3. (a) M-T curves for the Ti0.15 sample.(b) M-T curves of Fe 2 Hf 1-x Ti x upon heating in a magnetic field of 0.01 T and (c) M-T curves of Fe 2 Hf 1-x Ti x upon heating in a magnetic field of 1 T. (d) M-H curves for Fe 2 Hf 1-x Ti x at a temperature of 5 K. (e) T C as a function of the average unit-cell volume of Fe 2 Hf 1-x Ti x .(f) M s as a function of average unit-cell volume of Fe 2 Hf 1-x Ti x .DFT calculation results for the Fe 2 Hf 1-x Ti x alloys with: (g) formation energy for the ferromagnetic (FM) and antiferromagnetic (AFM) phase, (h) calculated magnetic moment, and (i) calculated lattice parameters a and c as a function of the Ti content.For the single-phase samples (arc-melted Ti0.15, Ti0.27 and melt-spun Ti0.40), the experimental magnetic moment and lattice parameters are also given for comparison in (h) and (i), respectively.
) and 5(d) using the Maxwell relation: ΔS m (ΔH, T) = , where we choose µ 0 H 0 = 0 T. For the arc-melted Ti0.40 sample, |ΔS m | broadens asymmetrically towards the low-temperature region as the magnetic field is increased, whereas the shape of |ΔS m | becomes symmetric for the melt-spun sample.The magnetic entropy change |ΔS m | under a magnetic field change of 2 T for arc-melted and melt-spun Ti0.40 sample is 0.46 J/kgK at 306 K and 0.54 J/kg K at 280 K, respectively.The melt-spun Ti0.40 sample generates a 20% higher magnetic entropy change.The value of |ΔS m | for arc-melted Ti0.15 and Ti0.27 samples is calculated to be 1.1 and 1.0 J/kgK, respectively (see Fig. S7(b) and Fig.

Fig. 6 .
Fig. 6.Powder neutron diffraction (ND) patterns for the arc-melted Ti0.15 sample (a) at 573 K and (b) at 4 K.The top row of marked Bragg peak positions is from the nuclear contribution.The second row in (b) corresponds to the magnetic contribution.(c) Refined magnetic moment configuration.Temperature dependence of refined parameters: (d) lattice constants a and c and (e) experimental and non-magnetic unit-cell volume (V exp and V nm ) from ND as a function of temperature for the arc-melted Ti0.15 sample.V nm is calculated by the extrapolation of the high temperature data for the non-magnetic PM phase by ignoring the expected lowtemperature levelling of the thermal expansion.The hatched zone in (d) indicates the NTE window.

Fig. 8 .
Fig. 8. (a) Temperature dependence of the hyperfine field of the arc-melted Ti0.15 sample, derived from Mössbauer spectroscopy.(b) Temperature dependence of magnetic moment at the 2a and 6h site of the arc-melted Ti0.15 sample derived from ND (Table 1).(c-d) Interatomic distance as a function of temperature.(e) dω s /dT and dM/dT (at a magnetic field of 1 T) as a function of temperature.(f) ω s as a function of the square of the Fe magnetic moment.