Thermal Strain and Magnetization Studies of the Ferromagnetic Heusler Shape Memory Alloys Ni 2 MnGa and the Effect of Selective Substitution in 3 d Elements on the Structural and Magnetic Phase

© 2013 Sakon et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Thermal Strain and Magnetization Studies of the Ferromagnetic Heusler Shape Memory Alloys Ni2MnGa and the Effect of Selective Substitution in 3d Elements on the Structural and Magnetic Phase


Introduction
Ferromagnetic shape memory alloys (FSMAs) have been extensively studied as potential candidates for smart materials.Among FSMAs, Ni2MnGa is the most familiar alloy [1].It has a cubic L21 Heusler structure (space group Fm 3 m) with the lattice parameter a = 5.825 Å at room temperature, and it orders ferromagnetically at the Curie temperature TC ≈ 365 K [2,3].Upon cooling from room temperature, a martensite transition occurs at the martensite transition temperature TM ≈ 200 K. Below TM, a superstructure forms because of lattice modulation [4,5].For the Ni-Mn-Ga Heusler alloys, TM varies from 200 to 330 K by nonstoichiometrically changing the concentration of composite elements.
Several studies on Ni-Mn-Ga alloys address the martensite transition and correlation between magnetism and crystallographic structures [6][7][8][9][10][11][12][13][14][15][16][17][18].Ma et al. studied the crystallography of Ni50+xMn25Ga25-x alloys (x = 2-11) by powder X-ray diffraction and optical microspectroscopy [7].In the martensite phase, typical microstructures were observed for x < 7. The martensite variants exhibit configurations typical of self-accommodation arrangements.The TEM image of Ni54Mn25Ga21 indicates that the typical width of a variant is about 1 m.The interaction between the magnetism and crystallographic rearrangements was discussed in Refs.[1,8,17,18].The memory strain was observed in single crystal Ni2MnGa and polycrystalline Ni53.6Mn27.1Ga19.3[10].As for the magnetism, the magnetic anisotropy constant KU in martensite phase is 1.17  10 -5 J/m 3 , which is forth larger than that in austenite phase (0.27  10 -5 J/m 3 ) [1].Manosa et al. suggested that the martensitic transition take place in the ferromagnetic phase, and the decrease in magnetization observed at intermediated fields (0< B < 1 T) is due to the strong magnetic anisotropy of the martensite phase in association with the multi-domain structure of the martensite state [8].Likhachev et al. stated that the magnetic driving force responsible for twin boundary motion is practically equal to the magnetic anisotropy constant KU [17].The magnetization results indicate that the martensite Ni-Mn-Ga alloys have higher magnetocrystalline anisotropy.This is because lower initial permeability or lower magnetization at low fields than the cubic austenite phase.Furthermore, the magnetization results indicate that the coercivity and saturation field at martensite phase are higher than those of the cubic austenite phase [11][12][13][14][15]. Zhu et al. investigated the lattice constant change c/c of -4.8 % by means of X-ray diffraction study around martensite transition temperature [11].Chernenko et al. also studied about the magnetization and the X-ray powder diffractions and clear changes were found at martensite temperature for both measurements [12].Murray et al. studied the polycrystalline Ni-Mn-Ga alloys [18].The magnetization step at TM is also observed and this is a reflection of the magnetic anisotropy in the tetragonal martensite phase.In the martensite phase, strong magnetic anisotropy exists.Then the magnetization that reflects the percentage of the magnetic moments parallel to the magnetic field is smaller than that in the austenite phase where the magnetic anisotropy is not strong in the weak magnetic field.Therefore the magnetization step is observed at TM. NMR experiments indicate Mn-Mn indirect exchange via the faults in Mn-Ga layers interchange caused by excessive Ga [13].This result indicates the exchange interaction between Mn-Mn magnetic moments is sensitive with the lattice transformation.Then the magnetism changes from soft magnet in the austenite phase to hard magnet in the martensite phase, which is due to higher magnetic anisotropy.
To use Ni-Mn-Ga alloys as advanced materials for actuators, polycrystalline materials are useful because of their robustness.Moreover, in daily use, magnetic actuators should be used around room temperature (300 K).Therefore, we selected the Ni52Mn25Ga23 alloy, which shows ferromagnetic transition at the Curie temperature TC, about 360 K, and the martensite transformation occurs around 330 K.
The purpose of this study is to investigate the correlation between magnetism and crystallographic structures as it relates to the martensite transition of Ni52Mn12.5Fe12.5Ga23,Ni2Mn0.75Cu0.25Ga,Ni2MnGa0.88Cu0.12 and Ni52Mn25Ga23, which undergoes the martensite transition below TC [6,7].Especially, we focused on the physical properties in magnetic fields.We performed in this study that by using the polycrystalline samples, it is possible to provide information on the easy axis of the magnetization in the martensite structure with temperature dependent strain measurements under the constant magnetic fields.In this paper, thermal strain, permeability, and magnetization measurements were performed for polycrystalline Ni52Mn12.5Fe12.5Ga23,Ni2Mn0.75Cu0.25Ga,Ni2MnGa0.88Cu0.12 and Ni52Mn25Ga23 in magnetic fields (B), and magnetic phase diagrams (B-T phase diagram) were constructed.The results of thermal strain in a magnetic field and magnetic-field-induced strain yield information about the twin boundary motion in the fields.From the permeability and magnetization measurements, the magnetic anisotropy constant KU can be calculated.The experimental results were compared with those of other Ni-Mn-Ga single crystalline or polycrystalline alloys, and correlations between magnetism and martensite transition were found.

Experimental details
The Ni52Mn12.5Fe12.5Ga23and Ni2Mn0.75Cu0.25Gaalloys were prepared by the arc melting of 99.9 % pure Ni, 99.99 % pure Mn, and Cu, 99.95 % pure Fe, and 6N pure Ga in an argon atmosphere.To obtain homogenized samples, the reaction products were sealed in double evacuated silica tubes, which were annealed at 1123 K for 3 days, and quenched into cold water.The samples obtained for both alloys were polycrystalline.The Ni2MnGa0.88Cu0.12alloy was prepared by the arc melting of 99.99 % pure Ni, 99.99 % pure Mn, and Cu, and 99.9999 % pure Ga in an argon atmosphere.To obtain the homogenized sample, the reaction product was sealed in double evacuated silica tubes, which was annealed at 1123 K for 3 days, and quenched into cold water.The obtained sample was polycrystalline.From the x-ray powder diffraction, 14M (P2/m) martensitic structure and D022 tetragonal structure were mixed at 298 K [19].The lattice parameter of tetragonal structure is a = 3.8920 Å and c = 6.5105Å.
The Ni52Mn25Ga23 alloy was prepared by arc melting 99.99% pure Ni, 99.99% pure Mn, and 99.9999% pure Ga in an argon atmosphere.To obtain a homogenized sample, the reaction product was sealed in double-evacuated silica tubes, and then annealed at 1123 K for 3 days and quenched in cold water.The obtained sample was polycrystalline.From x-ray powder diffraction, the 14M (P2/m) martensite structure and the D022 tetragonal structure were mixed at 298 K.The lattice parameters of the 14M structure were a = 4.2634 Å, b = 5.5048 Å, c = 29.5044Å, and β = 85.863°, and those of the D022 structure were a = 3.8925 Å, and c = 6.5117Å.The size of the sample was 2.0 mm × 2.0 mm × 4.0 mm.The measurements in this study were performed at atmospheric pressure, P = 0.10 MPa.Thermal strain measurements were performed using strain gauges (Kyowa Dengyo Co., Ltd., Chofu, Japan).Electrical resistivity of the strain gauges was measured by the fourprobe method.The relationship between strain,  and deviation of electrical resistivity, R, is given by where KS is the gauge factor (KS = 1.98) and R0 is the electrical resistivity above TR.The strain gauge was fixed parallel to the longitudinal axis of the sample.
Thermal strain measurements were performed using a 10 T helium-free cryocooled superconducting magnet at the High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University.The magnetic field was applied along the longitudinal axis of the sample.The thermal strain is denoted by the reference strain at the temperature just above TM.
Magnetization measurements were performed using a Bitter-type water-cooled pulsed magnet (inner bore: 26 mm; total length: 200 mm) at Akita University.The magnetic field was applied along the longitudinal axis of the sample.The values of magnetization were corrected using the values of spontaneous magnetization for 99.99% pure Ni.The magnetic permeability measurements were performed in AC fields with the frequency f = 73 Hz and the maximum field Bmax = 0.0050 T using an AC wave generator WF 1945B (NF Co., Ltd., Yokohama, Japan) and an audio amp PM17 (Marantz Co. Ltd., Kawasaki, Japan) at Akita University with the same magnet we used for the magnetization measurement, having the compensating high homogeneity magnetic field.AC fields were applied along the longitudinal axis of the sample.

Results and discussions
3.1.Ni52Mn12.5Fe12.5Ga23and Ni2Mn0.75Cu0.25Ga Figure 1 shows the temperature dependence of the linear thermal expansion of Ni52Mn12.5Fe12.5Ga23 in static magnetic fields.When cooling from 310 K (Ferro-A phase), the alloy shrinks gradually in zero magnetic fields.Small elongation was observed at 288 K.Then, sudden shrinking occurs below 286 K, which indicates transformation from austenite phase to martensite phase.We define the martensitic transformation temperature TM as the midpoint of the steep decrease in the cooling measurement.The TM of this alloy is 284 K.The reason of small elongation at 288 K is considered that L21 and 14M structures coexist each other.Therefore apertures between L21 and 14M structures were originated and small expansion occured.As for Ni2+xMn1-xGa alloys, small elongation was observed just above TM [21].As shown in reference 20, the phase below TM is Ferro-M.When heating from 270 K, expansion occurs at about TR = 288 K, which indicates reverse martensitic transformation.Small elongations just above the temperatures of TM and TR were also observed in polycrystalline Ni2+xMn1-xGa (0.16  x  0.20) [21].
TM and TR gradually changed with increasing magnetic fields.The strain at TM and TR was about −2.5  10 −3 (−0.25 %) and was almost the same as that in magnetic fields.Kikuchi et al. performed the x-ray diffraction experiments of Ni50+XMn12.5Fe12.5Ga25-X[20].The x-ray patterns at room temperature (T = 300 K, austenite phase) for the samples of 0  x  2.0 were indexed with the L21 Heusler structure.In the x-ray diffraction pattern at room temperature of the sample with x =2.0, a very weak reflection from a  phase was observed, where the  phase has a disordered fcc structure.The lattice parameter a of x = 2.0 was found to be 5.7927 Å [22].On the other hand, for 3.0 x  , the martensite phase appeared at room temperature.The martensitic structure of x = 3.0 was indexed as a monoclinic structure with 14M (7R) structure.The lattice parameters of the sample were determined as a We also estimated the strain of Ni52Mn12.5Fe12.5Ga23(x = 2.0) at TM using the lattice parameter of x = 2.0 in the austenite phase and that of x = 3.0 in the martensite phase.In the austenite phase, for the L21 cubic structure, the lattice parameter a was 5.7927 Å [22] (2) The linear strain of a polycrystal is one-third of the volume strain [24].Therefore, we estimate the linear strainΔεas, ( This estimated value is approximately comparable to the strain value Δε = −0.25 % of Ni52Mn12.5Fe12.5Ga23obtained from this experimental study. Figures 2 (a) and (b) show the temperature dependence of magnetic permeability and linear thermal expansion of Ni2Mn0.75Cu0.25Ga in zero magnetic fields, respectively.When cooling from a high temperature, it shrinks and the permeability increases at about TM = 308 K.The permeability at austenite phase is very low as compared with that at the martensite phase.
These results indicate that the region above TM or TR is Para-A and the region below TM or TR is Ferro-M.When heating from a low temperature, the expansion occurs at about TR = 316 K, which indicates reverse martensitic transformation.The strain at TM or TR is about 3.0  10 −3 (0.30 %).This value is higher than that of Ni52Mn12.5Fe12.5Ga23.Kataoka et al. studied the x-ray powder diffraction of Ni2Mn1-xCuxGa2 (0  x  0.40) [23].In the vicinity of martensitic transformation, the strain exhibits complicated behavior; when cooling from 342 K, it shrinks gradually and rapid shrinking occurs at TM = 308 K, subsequently, exhibiting elongation; repetition of small elongation and shrinking was observed between 303 K and 291 K; in addition, it shrinks linearly below 291 K.When heating from 257 K, the repetition of small elongation and shrinking was observed between 307 K and 311 K. Thereafter, it shrinks by 9.0  10 −4 and exhibits elongation.This sequential phenomenon has been observed in single crystalline Ni2.19Mn0.81Ga[21].In particular, steep shrinking occurs before elongation due to reverse martensitic transformation during heating.As for polycrystalline Ni2+xMn1−xGa (0.16  x  0.20), the shape of the small elongation or small shrinking due to the large change of the strain associated with martensitic transformation is broader than that of the single crystalline alloy.In our study, Ni2Mn0.75Cu0.25Gashowed steep shrinking before elongation during heating from a low temperature, which is similar to that of single crystalline Ni2.19Mn0.81Ga.It is possible that the Ni2Mn0.75Cu0.25Gacrystal is oriented to some extent.
The x-ray diffraction measurement of Ni2Mn0.75Cu0.25Gaindicates that cubic L21 phase and the 14M phase coexist in the martensite phase.The reason for the repetition of small elongation and shrinking in Figure .2 (b) is supposed to be this complex structure.
Figure 3 shows the temperature dependence of the linear thermal expansion of Ni2Mn0.75Cu0.25Ga in static magnetic fields.TM and TR gradually changed with increasing magnetic fields.
Next, we compared the two samples.The linear thermal coefficients  of Ni52Mn12.5Fe12.5Ga23and Ni2Mn0.75Cu0.25Ga in zero magnetic fields obtained in this study are shown in Table 1.In the austenite phase,  of Ni52Mn12.5Fe12.5Ga23 is much lower than that of Ni2Mn0.75Cu0.25Ga,which means that Ni52Mn12.5Fe12.5Ga23 is harder than Ni2Mn0.75Cu0.25Ga. is higher in the martensite phase than in the austenite phase of Ni52Mn12.5Fe12.5Ga23.This is probably due to the 14M martensitic structure.
T. Sakon Figure 4 shows the magnetic phase diagram of the thermal expansion of Ni52Mn12.5Fe12.5Ga23 in static magnetic fields.TM and TR gradually changed with increasing magnetic fields like Ni2+xMn1-xGa alloys.The shifts of TM and TR in magnetic fields were estimated as dTM/dB  0.5 K/T and dTR/dB  0.5 K/T, respectively.The shifts of TM and TR can be explained by the difference of the magnetization between austenite phase and martensitic phase.Afterwards we discuss about the correlation between magnetization and the shift of TM.
Figure 5 shows the magnetic phase diagram of the thermal expansion of Ni2Mn0.75Cu0.25Ga in static magnetic fields.TM and TR gradually changed with increasing magnetic fields such as in the Ni2+xMn1-xGa or Ni52Mn12.5Fe12.5Ga23alloys.The shifts of TM and TR in magnetic fields were estimated as dTM/dB  1.2 K/T and dTR/dB  1.1 K/T, respectively.These ratios are within measurement errors.The magnetic phase diagrams constructed from the thermal expansion measurements of this study are shown in Figures 4 and 5.
Figure 6 (a) shows the magnetization of Ni52Mn12.5Fe12.5Ga23 in a pulsed magnetic field.Below 250 K in the Ferro-M state, the M-B curves resemble each other, and this is consistent with the results in reference 7.In the Ferro-A state, the magnetization at 300 K is lower than that in the Ferro-M state.Figure.6 (b) shows the high-field magnetization in a pulsed magnetic field.At 90 K, steep increase in magnetization occurs when magnetic field is applied.Above 2 T, the magnetization increases gradually.The magnetization at 300 K, which is above TM and TR, is also ferromagnetic.The magnetization above 5 T is almost flat.This property is quite different from that at T = 90 K.The magnetism of the austenite phase appears to be similar to a localized ferromagnetic state, because the magnetization value is constant in high magnetic fields.Figures 7 (a) and (b) show the magnetization of Ni2Mn0.75Cu0.25Ga in a pulsed magnetic field.These measurements were performed after zero-field cooling processes at 323 K in the austenite phase.Below TM, the magnetization shows ferromagnetic properties, whereas above TM it exhibits paramagnetic properties.This is consistent with the permeability result shown in Figure .2. Below TM, for instance, at 300 K, a steep increase occurred around zero fields and a spin-flop like behavior was shown below 0.06 T. Usually, magnetic alloys such as FeCl3 show spin-flop behavior, and a linear extrapolation line at the canted magnetic moments phase crosses the origin point of the coordinate axis in the M-B graph.However, in Figure 7 (a), the M-B graph shows that the linear extrapolation line at the canted magnetic moments phase did not cross the origin point at 300 K.It is possible that steep increase just above the zero fields was due to the localized magnetic moments on the Mn atoms, for example, 3.8-4.2B/Mn atom which was obtained by the neutron scattering experiments of Ni2+xMn1-xGa alloys [2,[24][25].The magnetic moments on Ni atoms are considerably low, such as 0.2 B/Ni atom for Ni2+x Mn1-xGa alloys [2,[24][25], and therefore, it is possible that the Ni moments that were arranged in a canted-like formation get ordered by the mutual correlations between external magnetic fields and internal magnetic fields due to the Mn moments.

Ni2MnGa0.88Cu0.12
Figure 8 shows the temperature dependence of magnetic permeability.When heating from 310 K, the signal gradually increased.A slightly peak was observed at 338 K and a sudden decrease occurred around 342 K.When cooling from a high temperature, the permeability shows a sharp peak at about 337 K.A dip was observed around 324 K. Figure 9 shows the linear thermal expansion.When heating from 305 K, slight expansion was observed at zero magnetic fields.Around 343 K, a sharp expansion was observed.Considering the results of a previous study [19], this is due to the reverse martensitic transition and TR = 343 K, which is defined as the midpoint temperature of the transition.When cooling from 360 K, a sudden shrinking was observed at 336 K. Considering the lattice structure, the martensitic transition temperature TM is 336 K.When cooling below 336 K, the linear expansion shows a dip and below 320 K, the value of linear expansion is nearly constant.Mentioned above, the permeability measurement also shows a dip between 336 K and 320 K.As for the Then the ratio T  / TC = 0.012, which indicates the increase of the permeability of Ni2MnGa0.88Cu0.12occurs within a narrower temperature range than that of pure Fe.
Concerning the permeability and linear expansion results, dips and drastic changes, the magnetism and the lattice affects one another.The permeability of the austenite phase is very low as compared with that at the martensite phase.The results of the permeability and the linear expansion measurements indicate that the region above TM is a paramagnetic austenite phase (Para-A) and the region below TM is a ferromagnetic martensite phase (Ferro-M).
T. Sakon The contraction at TM under zero fields is about 1.3  10 −3 (0.13 %).As for Ni52Mn12.5Fe12.5Ga23and Ni2Mn0.75Cu0.25Ga, the contraction occurs at martensite temperature [27].The strain at TM of polycrystalline Ni52Mn12.5Fe12.5Ga23was estimated as 0.14 % contraction.This value is almost the same as that of Ni2MnGa0.88Cu0.12.After zero field measurements of the linear expansion, measurements in a magnetic field were performed from 3 T to 10 T. With increasing field, TM and TR are gradually increased.The shifts of TM and TR around zero magnetic fields were estimated as dTM/dB =1.3 K/T and dTR/dB = 1.5 K/T, as shown in Figure 3.This behavior is the same as that of the Ni2+xMn1-xGa ferromagnetic alloys.The typical temperature TL, which was defined as the kink point of the linear expansion for heating processes in Figure 9, also gradually increases with increasing fields.A noteworthy fact is that the dip of linear expansion measurements in magnetic fields is larger than that in zero fields.The variation of the strain between zero fields and non-zero field was observed for Ni2.19Mn0.81Ga[28].The contraction in magnetic fields was larger than that in zero fields.The reason is considered that the magnetic moments of Mn and Ni atoms are aligned parallel to the magnetic field just below TM and the 14M and/or D022 tetragonal lattices are rearranged by the magnetic moments.Therefore the rearrangement of these lattices due to magnetic fields occurred in high magnetic fields.Figure 11 (a) shows the magnetization of Ni2MnGa0.88Cu0.12 in a pulsed magnetic field up to 10 T. The unit of the magnetization M, J/μ0kgT in SI unit system is equal to emu/g in CGS unit system.The hysteresis of the M-B curve is considerably small.In other magnetic material, for example Gd3Ga5O12, the magnetocaloric effect was reported [21].They performed the magnetization measurements at initial temperature 4.2 K, then the magnetic contribution to heat capacity is comparable to the lattice heat capacity.In our experiment, the temperature change of the sample due to the magnetocaloric effect is considered within 1 K.This is due that these experiments were performed around room temperature, then the lattice heat capacity is much larger than the heating or cooling power by the magnetocaloric effect.Figure 11 (b) shows the magnetization of Ni2MnGa0.88Cu0.12 in a pulsed magnetic field up to 2.2 T. The M-B curves with increasing field processes are shown.The M-B curves show ferromagnetic behavior below 333 K.The prominent decrease of magnetization occurred between 333 K and 336 K. Figure 12 shows the temperature dependence of the magnetization M-T at 0.5 T and 1 T, which were obtained by magnetization measurements in pulsed magnetic fields.A sudden decrease is apparent between 333 K and 336 K for each field.This temperature region corresponds to the sharp increase of the permeability when heating from low temperature in Figure 8, and just below TM, which was obtained by the linear expansion measurement in Figure 9.The M-T curve shows a shallow depression between 310 K and 330 K, which corresponds to the dip of the permeability and the linear expansion results.Magnetic Field (T)  Figure 13 shows the Arrott plot of Ni2MnGa0.88Cu0.12.The spontaneous magnetization at 289 K in a Ferro-M state is 47.1 J/μ0kgT.The obtained TC of the martensite phase by Arrott plots in Figure 13 is 340 K, which is almost the same as TM = 337 K.This is consistent with the x-T phase diagram of Ni2MnGa1-xCux, which is obtained in reference 19.

T. Sakon
Figure 14 shows the magnetization of Ni2MnGa0.88Cu0.12 in a pulsed high magnetic field up to 18.6 T. The difference of the magnetization between 333 K and 336 K is clearly seen.In high magnetic fields, an almost linear increase can be seen for each M-B curve.Ni2Mn0.75Cu0.25Gaalso shows the difference of the magnetization between 302 K and 305 K, which is little lower than TC = 307 K or TM = 308 K [27].It is noticeable that the Arrott plots of Ni2MnGa0.88Cu0.12left a space between 333 K and 336 K, and Ni2Mn0.75Cu0.25Gaalso left a space between 302 K and 303 K.The spontaneous magnetizations of Ni2MnGa0.88Cu0.12are 33.4J/μ0kgT at 333 K and 16.7 J/μ0kgT at 336K, which was obtained by the Arrott plot shown in Figure 13.As for Ni2Mn0.75Cu0.25Ga, the spontaneous magnetizations are 40.0J/μ0kgT at 302 K and 28.3 J/μ0kgT at 303 K.

Ni52Mn25Ga23
Figure 15 shows the temperature dependence of permeability.When heating from 300 K, permeability increases gradually.As shown in Figure 15, permeability increases above 330 K and suddenly decreases around 360 K.When cooling from a high temperature, permeability shows a sudden increase at about 356 K and decreases at 325 K.The sudden changes in Magnetic Field (T) Ni 2 MnGa 0.88 Cu 0.12 permeability indicate that the ferrromagnetic transition occurs around 358 K.The temperature dependence of permeability for Ni52Mn25Ga23 is similar to that for Ni52Mn12.5Fe12.5Ga23,which shows a transition of a ferromagnetic-martensite (Ferro-M) phase to a ferromagneticaustenite (Ferro-A) phase [29].The step around 330 K (heating process) and 325 K (cooling process) reflects stronger magnetic anisotropy in the tetragonal martensite phase [8,18].Polycrystalline Ni49.5Mn28.5Ga22,Ni50Mn28Ga22 and Ni52Mn12.5Fe12.5Ga23alloys also indicate the magnetization (or permeability) step at TM [9,18,27] below the field of 10 mT. Figure 16 (a) shows the linear thermal strain of Ni52Mn25Ga23.Solid lines are the experimental data and dotted lines are the extrapolated lines.At zero magnetic fields, the memory strain was observed, as polycrystalline Ni53.6Mn27.1Ga19.3[10].When heating from 300 K, slight strain is observed first at zero magnetic fields.Around 334 K, a sharp strain is observed.The results of previous studies [6,7] suggest that this is because of the reverse martensite transition TR = 334 K, which is defined as the midpoint temperature of the transition.When cooling from 370 K, a sudden decrease is observed at 328 K. Given the lattice structure, the martensite transition temperature TM is 328 K.The permeability at the Ferro-M phase is very low compared with that at the Ferro-A phase.The results of permeability and linear strain measurements indicate that the region above TM is a Ferro-A phase and that below TM is a Ferro-M phase.The permeability measurement results indicate that the ferromagnetic transition from the paramagnetic-austenite (Para-A) phase to the Ferro-A phase occurs around 358 K (see Figure 15).On the other hand, the linear strain does not show noticeable anomaly at the ferromagnetic transition around 358 K.

T. Sakon
When cooling from 370 K, the thermal strain shows a peak at 329 K.This may be attributed to the intermingling of the L21 austenite lattices and the M14 martensite lattices at the martensite transition.The sequential phenomenon is observed in single crystalline Ni2.19Mn0.81Ga[31].Zhu et al. suggests that the small satellite peaks in heat flow plot, which flanks the central peak indicates the structural transition takes place in multiple steps [11].The contraction at TM under zero fields is about 0.5  10 −3 (0.05%).As for other Heusler alloys, Ni52Mn12.5Fe12.5Ga23and Ni2Mn0.75Cu0.25Ga, the contraction occurs at martensite temperature [27].The strain at TM of polycrystalline Ni52Mn12.5Fe12.5Ga23was estimated as 0.14% contraction.This value is larger than that of Ni52Mn25Ga23.After zero field measurements of the linear strain, measurements in (a) magnetic fields from 1 T to 10 T were performed.The strain at TM under the magnetic field of 1 T was estimated as 0.10% contraction, which is twice that under zero magnetic field (0.05%).These results indicate that the magnetic fields influence the structural phase transition.After these thermal cycles in magnetic fields, the thermal strain in zero fields was 0.05 % contraction, which is as same as the first cycle in zero fields.Around 358 K, which is the ferromagnetic transition temperature, no anomaly was observed in the magnetic fields.Figure 16 (b) shows the magnetic field dependence of the strain at TM.At zero field, the strain is 3.6 × 10 −4 .On the other hand, the strain in a magnetic field is about 7.1 × 10 −4 , which is almost twice that in zero field.Ullakko et al. measured the magnetic-field-induced strain of a Ni2MnGa single crystal [1].The strain at TM in zero field was 2 × 10 −4 .This is only a small fraction compared with the lattice constant change for c-axis from the austenite to martensite phases, which was / c c  = 6.56%.It is proposed that the strain accommodation is occurred by different twin variant orientations.As shown in Figure .16 (b), the thermal strain under the magnetic field of 1 T was 7.2 × 10 −4 , indicating the field aligned some of the twin variants.
In the martensite phase, the magnetic moment in the magnetic easy direction was coupled with the strain along the short c-axis of the martensite variant structure.As a result, under the applied magnetic field, the variant rearrangement occurs with the assistance of twin boundary motion, such that the magnetic easy axis is parallel to the applied field.Therefore, Magnetic Field (T) the total magnetic free energy is minimal.The variant rearrangement results in field influence on the thermal expansion as shown in Figure 16(b).
Variation in the strain between zero field and non-zero field was observed for Ni2.19Mn0.81Gaand Ni2.20Mn0.80Gapolycrystalline samples [30].The change in the sample length by means of the thermal strain measurements at the martensite phase transition was 0.04 % for Ni2.19Mn0.81Gaand 0.12 % for Ni2.20Mn0.80Ga.The thermal strain for Ni2.19Mn0.81Ga in the presence of 1.4 T magnetic field, the change was increased to 0.13 %, which means 3.2 times increase of the strain.The increase of the strain was 2.6 times (0.31 % strain) for Ni2.20Mn0.80Ga.The variation in the strain between zero fields and non-zero field was also observed for Ni49.6Mn27.3Ga23.1 polycrystalline samples [31].With increasing measuring magnetic fields, the difference in the strain increased.Aksoy et al. proposed that the strain increase is due to increase of the preferred alignment of the short c axis along the applied field, and, high twin boundary mobility in Ni-Mn-Ga is expected to be the main case of the alignment, although the martensite variant nucleation with preferred c axis orientation in the external field already just at the martensite transition temperature is also the influence of the shrinkage [31].Further they mentioned that, when a sample was cooled from the austenite down to the martensite phase in zero fields, no preferred orientation is given to the variant growth during nucleation, whether the easy axis is a long axis or a short axis.When a magnetic field is applied in the austenite phase and the sample is cooled down through TM in the constant field, a preferred growth direction is provided to the variants.Consequently, the variants with easy axis along the applied field direction nucleate more and more.If the easy axis is short axis, the sample length decreases.Then the contraction at TM is observed in thermal strain measurements.
As for Ni2MnGa single crystal, in zero-field cooling process, strains of nearly 0.02 % have been observed at TM = 276 K [1].The strain at transformation in 1.0 T is 0.145 %, indicating that the field has aligned some of the twin variants.Now we compare the strain and the magnetization results of Ni2+xMn1-xGa alloys [28].For the alloys which showed increase of the strain for x = 0.18 and 0.20, the TM and Tc are almost same temperature.Consequently, the magnetization change is large.For these composition alloys, clear hysteresis in the magnetization was observed, which indicates first order magnetic transition.From these results, it is supposed that the magnetic field influences the orientation of the easy c axis along the magnetic field.As for Ni52Mn25Ga23, The magnetization change is large at TM, as shown in Figure 22.The permeability in Figure 15 shows clear change and hysteresis, which indicates the first order transition.It is also supposed that the magnetic field influences the orientation of the easy c axis along the magnetic field, and then the variant rearrangement was occurred.Consequently, the variation in the strain between zero fields and non-zero field was observed.
Figure 17 shows the magnetic-field-induced strain at 300 K (Ferro-M phase) in a static magnetic field.When increasing the magnetic field from zero fields, a sudden contraction occurs up to 1 T. Above 1 T, a gradual contraction is observed.When decreasing the magnetic field from 10 T, a modicum of strain occurs.Below 1 T, a sudden strain is observed.The magnetic-field-induced strain at 10 T is −100 ppm or −0.010%, which is considerably smaller than the contraction value at TM.The sudden contraction between 0 and 1 T when increasing the field is supposed to be related to the temperature dependences of the linear strains between zero fields and above 1 T and below TM.The variant rearrangement results in a magnetic-field-induced strain, which is the origin of the magnetostriction shown in Figure 17.The reason of smallness of the magnetic field induced strain is supposed that; when the sample is cooled down from the austenite phase to the martensite phase in a constant field, variant arrangement is occurred and the contraction is occurred, as mentioned above.In zero fields, cooling from the austenite phase to the martensite phase, the variant arrangement is fixed.When the magnetic field is applied with constant temperature, the variant rearrangement is considered to be difficult.Therefore the magnetic field induced strain is smaller than the strain at TM. in the linear strain measurements.
T. Sakon The magnetic-field-induced strain of the polycrystalline Ni50Mn28Ga22 alloy was reported by Murray et al. [18].They mentioned that the strain in the martensite phase below TM is an order of magnitude smaller than that of a single crystal of the stoichiometric compounds [1].They attributed this to the polycrystalline nature of the material or to the presence of impurities that impede twin boundary motion.The field-induced strain of Ni50Mn28Ga22 increases on cooling from the austenite phase, leading to an abrupt increase with the appearance of the twin variant below TM.On heating from the martensite phase, an abrupt increase occurs in the field-induced strain around TM.They suggest that this is caused by lattice softening near TM.As for the thermal strain of Ni52Mn25Ga23, shown in Figure 16 (a), peaks appear for both TM and TR in zero field and all values of the magnetic field.The peak at TR, associated with heating, is larger than that at TM, associated with cooling.These peaks indicate that the lattice expands abruptly.Dai et al. studied the elastic constants of a Ni0.50Mn0.284Ga0.216single crystal using the ultrasonic continuous-wave method [31].C11, C33, C66, and C44 modes were investigated; every mode indicated abrupt softening around TM.This lattice softening appears to be affected by the abrupt expansion just above TM when cooling from the austenite phase.Figure 19 shows the magnetization curves of Ni52Mn25Ga23 in a pulsed magnetic field up to 2.2 T. The unit of magnetization M is J/μ0kgT in the SI unit system or emu/g in the CGS unit system (both having identical numerical values).The M-B curves were measured from low temperature.The hysteresis of the M-B curve is considerably small.The magneto caloric effects in other magnetic materials were also reported; for example, Levitin et al. reported for Gd3Ga5O12 [32].They performed magnetization measurements at an initial temperature of 4.2 K, where the magnetic contribution to heat capacity is comparable to the lattice heat capacity.In our experiment, the temperature change of the sample due to the magneto caloric effect is considered to be within 1 K.This is because these experiments were performed around room temperature, where the lattice heat capacity is much larger than the heating or cooling power by the magneto caloric effect.Figure 20 shows magnetization in a magnetic field up to 15 T. In high magnetic fields, an almost linear increase can be seen for each M-B curve.In particular, as for the M-B curve below 334 K, the high magnetic field susceptibility is quite small.Figure 21 shows the Arrott plot of Ni52Mn25Ga23.The spontaneous magnetization at 294 K in a Ferro-M phase is 55.0 J/μ0kgT.The Curie temperature of the austenite phase TCA determined by Arrott plots in Figure 21 is 358 K.This is consistent with the x-T phase diagram of Ni50+xMn25Ga25-x [6,7].In high magnetic fields, an almost linear increase can be seen for each M-B curve.Ni2Mn0.75Cu0.25Gaalso shows the difference in magnetization between 302 and 305 K, which is somewhat lower than TC = 307 K or TM = 308 K [27].Note that the Arrott plots of Ni52Mn25Ga23 left a space between 333 and 335 K, and Ni2Mn0.75Cu0.25Galeft a space between 302 and 303 K.The spontaneous magnetizations of Ni52Mn25Ga23 are 42.2J/μ0kgT at 333 K and 34.2 J/μ0kgT at 335 K, which were obtained by the Arrott plot shown in Figure 21.As for Ni2Mn0.75Cu0.25Ga, the spontaneous magnetizations are 40.0J/μ0kgT at 302 K and 28.3 J/μ0kgT at 303 K. Figure 22 shows the temperature dependence of the magnetization M-T at 0.1, 0.5, and 1 T, which was obtained by magnetization measurements in pulsed magnetic fields.Open circles are the spontaneous magnetizations, which was obtained by the Arrott plot method.A sudden decrease is apparent between 333 and 336 K for each field, and also the spontaneous magnetization.This temperature region corresponds to the sharp increase in permeability when heating from low temperature in Figure 15, and just below TR, which was obtained by the linear strain measurement in Figure 16 (a).

T. Sakon
The M-T curve in Figure 22 can be seen as the combination of two single-phase M-T curves.One corresponds to the martensite phase, and the other corresponds to the austenite phase.The obtained Curie temperatures in the martensite phase and the austenite phase are TCM = 333.5  0.5 K and TCA = 358.0 0.5 K.This is due to the difference of the ferromagnetic interactions for both structural phases.These analyses of magnetic properties in Ni51.9Mn23.2Ga24.9 were also reported in reference 11.
It is well known that the tetragonal martensite Ni-Mn-Ga has higher magnetocrystalline anisotropy in association with the multi-dominant structure of the martensite phase.Consequently, lower initial permeability and higher coercivity than the cubic austenite Ni-Mn-Ga alloys can occur [8,[11][12][13]15].The martensite transition occurs in the ferromagnetic phase, and the decrease in magnetization is observed at intermediate fields for 0 < B < 0.5 T, as shown in Figure 22.This property is also shown by magnetization in many Ni-Mn-Ga alloys (e.g., Ni49.5Mn25.4Ga25.1)and Ni-Mn-Sn alloys (e.g., Ni50Mn35Sn15) [8,33,34].Consequently, at low field, the austenitic Ni-Mn-Ga (with softer ferromagnetism) shows an abrupt increase in M, while the martensite Ni-Mn-Ga (with harder ferromagnetism) shows gradual increase in M with the field.On the other hand, the martensite Ni-Mn-Ga (in lowtemperature phase) has higher saturation magnetization (typically, Ms increases with decreasing temperature) than the austenite Ni-Mn-Ga.As a result, at very high field or saturation field (>1 T), magnetization of the martensite is higher than that of the austenite, as shown in Figures 20 and 22.As for other Ni-Mn-Ga alloys, Kim et al. reported magnetization in a Ni2.14Mn0.84Ga1.02single crystal, which shows a transition from the Ferro-A phase to Ferro-M phases with 14M structure [14].The magnetization curve in Ni2.14Mn0.84Ga1.02 at 290 K, just below the martensite transition temperature, sharply bend at the critical field, BS = 0.6 T, and above 0.6 T, the magnetization slightly increases with increasing fields.On the other hand, a bend in the magnetization is not clear.We defined the critical field BS in Ni52Mn25Ga23 as the field where the magnetization Arrott plot was off from the extrapolated linear line, which is illustrated by the dotted line in Figure 21, and obtained BS as 0.84 T, which is of the same order as that in Ni2.14Mn0.84Ga1.02.The magnetization is the same as that in Ni52Mn25Ga23.The magnetic anisotropy constant KU in a Ni2MnGa single crystal is 1.17 × 10 5 J/m 3 (11.7 × 10 5 erg/cm 3 ) in the martensite phase and 2.7 × 10 4 J/m 3 (2.7 × 10 5 erg/cm 3 ) in the austenite phase [1], indicating that the magnetic anisotropy is about four times larger in the martensite phase than that in the austenite phase.The Zeeman energy and/or magnetocrystalline anisotropy energy that is sufficient to induce motion of the twin boundary is denoted as MSBS/2 = KU [1].Kim et al. also mentioned that the magnetocrystalline anisotropy energy is of the order of 10 5 J/m 3 [14].The spontaneous magnetization in Ni52Mn25Ga23 at 333 K, just below TR is 42.2 J/μ0kgT, which was obtained by the Arrott plot in Figure 22.When using this value as MS, the magnetocrystalline anisotropy energy in the martensite phase of Ni52Mn25Ga23 is MSBS/2 = KU = 1.04 × 10 5 J/m 3 , which is on the same order as that in the martensite phase of Ni2MnGa.These magnetic properties were also shown for Ni51.9Mn23.2Ga24.9[11], Ni49.5Mn25.4Ga25.1 [12], and Ni54Mn21Ga25 [13].The relationship between magnetism and TM in magnetic fields is discussed for Ni2MnGatype Heusler alloys.Table 1 shows the spontaneous magnetizations and dTM/dB values of Ni2+xMn1-xGa, Ni52Mn12.5Fe12.5Ga23,Ni2Mn0.75Cu0.25Ga,Ni2MnGa0.88Cu0.12,and Ni52Mn25Ga23.As for Ni2+xMn1-xGa alloys, shifts in TM in magnetic fields were observed by magnetization measurements [2,[26][27][28].TM and TC of Ni2MnGa (x = 0) are 200 and 360 K, respectively.The region above TM is the Ferro-A phase.The sample with x = 0 of Ni2+xMn1-xGa shows phase transition from the Ferro-A to Ferro-M phases at TM.The sample with x = 0.19 shows ferromagnetic transition and martensite transition at TM.For x = 0, the shift in TM is estimated as dTM/dB = 0.2 K/T [35] and for x = 0.19, dTM/dB = 1 .0K/T [36].The shift in TM for x = 0.19 is higher than that for x = 0.These results indicate that the shift in TM for the alloy that shows Para-A to Ferro-M phase transition is larger than that for the alloy that shows Ferro-A to Ferro-M phase transition.The values of dTM/dB are roughly proportional to the change in spontaneous magnetization, (MM -MA)/MM, as shown in Table 1.This indicates that the magnetic moments influence the martensite transition; in other words, the structural transition and the TM increase in accordance with the magnetic fields are proportional to the difference between the magnetization of the austenite phase and that of the martensite phase.Therefore, it is considered that the alloys, in which TM and TC are close to each other, show a larger shift in TM in magnetic fields.

T. Sakon
Khovailo et al. discussed the correlation between the shifts in TM for Ni2+xMn1-xGa (0  x  0.19) using theoretical calculations [37,38].The experimental values of this shift for Ni2+xMn1-xGa (0  x  0.19) are in good agreement with the theoretical calculation results.In general, in a magnetic field, the Gibbs free energy is lowered by the Zeeman energy −MB that enhances the motive force of the martensite phase transition.Thus, TM of the ferromagnetic Heusler alloys Ni52Mn12.5Fe12.5Ga23,Ni2Mn0.75Cu0.25Ga,and Ni2MnGa0.88Cu0.12 in recent studies [27,39,40] and Ni52Mn25Ga23 in this study are considered to have shifted in accordance with the magnetic fields because high magnetic fields are favorable for ferromagnetic martensite phases.
Chernenko et al. studied the temperature dependence of both the saturation magnetic field values and the x-ray powder diffraction patterns of Ni-Mn-Ga alloys and analyzed with the theoretical consideration [12].The theory proposes that the free energy for ferromagnetic martensite phase, exposed to an external magnetic field, is expressed as three terms.First term is the magnetic anisotropy energy.The second and third terms describe the magnetostatic and the Zeeman energy, respectively.The c/a ratio was expressed as where Hs indicates the saturation magnetic field.M denotes the absolute value of the magnetization.D1 and D2 denote the diagonal matrix elements, and  is the dimensionless magnetoelastic parameter.The linear dependence of the magnetic anisotropy constant on the tetragonal distortion of the cubic crystal lattice arising in the course of the martensite transition.
In order to apply this theory to our present work, it is considered that further theoretical consideration is needed for apply this theory for analyzing the influence between the martensite variant structure and the magnetic field, which is reflected by the Zeeman term.

Thermal expansion
When cooling from austenite phase, steep decrease due to the martensitic transformation was obtained for both alloys.TM and TR increase gradually with increasing magnetic fields.

Magnetization and permeability
Ni52Mn12.5Fe12.5Ga23----The M-B curves indicate that the property of the Ferro-M phase is different from the Ferro-A phase.The Ferro-A phase is considered to be a more localized ferromagnetic phase as compared with Ferro-M phase.
Ni2Mn0.75Cu0.25Ga----The permeability abruptly changes around TM.The permeability below TM is about one-tenth times higher than that above TM.The Arrott plot of magnetization indicates that TC of the martensite phase is 307 K, which is almost the same as TM = 308 K.
3. The values of dTM/dB are roughly proportional to the change of the spontaneous magnetization (MM -MA)/MM.TM of the ferromagnetic Heusler alloys Ni52Mn12.5Fe12.5Ga23and Ni2Mn0.75Cu0.25Ga in the magnetic field is considered to be shifted in accordance with the magnetic fields and proportional to the difference between the magnetization of austenite phase with that of martensite phase.

Thermal expansion
When cooling from austenite phase, a steep decrease due to the martensitic transition was obtained.TM and TR increase gradually with increasing magnetic fields.The shift of TM was estimated as dTM/dB = 1.3 K/T.

Magnetization and permeability
The permeability abruptly changes and shows the clear peak around TM.The permeability below TM is about one-tenth than that above TM.The temperature dependence of the magnetization also shows a clear decrease around TM.The Arrott plot of magnetization indicates that TC of the martensite phase is 340 K, which is almost the same as TM = 337 K, which was obtained by the linear expansion.
3. The values of dTM/dB are roughly proportional to the change of the spontaneous magnetization (MM -MA)/MM in Ni2MnGa type Heusler alloys.TM of the ferromagnetic Heusler alloy Ni2MnGa0.88Cu0.12 in the magnetic field is considered to be shifted in accordance with the magnetic fields and proportional to the difference between the magnetization of austenite and martensite phase.

Ni52Mn25Ga23
Thermal strain, permeability, and magnetization measurements were performed on the Heusler alloy Ni52Mn25Ga23.
1. Thermal strain: When cooling from the austenite phase, a steep decrease in the thermal strain is obtained because of the martensite transition.TM and TR increase gradually with increasing magnetic fields.The shifts in TM and TR in a magnetic field are estimated as dTM/dB = 0.46 K/T and dTR/dB = 0.43 K/T, respectively.

V
. The distance between Mn-Mn atoms was / 2 a = 4.0961 Å, and the volume of the unit cell was VA = 0961) 3 = 68.72Å 3 .Furthermore, the volume VM in the martensite phase was estimated and compared with VA in the same area.In the 14M (7R) martensite phase, a = 4.2495 Å in the basal plane, is parallel to one of the a axis in the L21 structure, and is of the same unit.The other axis in the martensite phase corresponds to one of the a axis in the L21 structure of the Mn-Mn ridge in the basal plane ( 2 ) b  .The c axis is almost normal ( = 93.36°) to the basal plane and the seven Mn-Mn cycles at c = 29.340Å.Therefore, the volume, M = a × (c/7) × ( 2 ) b  × sin = 4.2495×4.1914× (1.4142×2.7211)×sin93.36°= 68.55×0.9983= 68.43Å 3 .

Figure 2 .Figure 3 .
Figure 2. (a).Temperature dependence of the magnetic permeability  of Ni2Mn0.75Cu0.25Ga in zero magnetic fields.The magnetic permeability measurement was perforemed in AC fields with f = 73 Hz and Bmax = 1.0 mT.Zero means the permeability .(b) Temperature dependences of linear thermal expansion of Ni2Mn0.75Cu0.25Ga.The strain was defined by the difference from the sample length at 340 K.

Ni 2
Mn 0.75 Cu 0.25 Ga Thermal Strain and Magnetization Studies of the Ferromagnetic Heusler Shape Memory Alloys Ni2MnGa and the Effect of Selective Substitution in 3d Elements on the Structural and Magnetic Phase 87 T. Sakon

Figure 6 .
Figure 6.(c) shows an Arrott plot, i.e., M 2 vs B/M, of the magnetization of Ni52Mn12.5Fe12.5Ga23.The spontaneous magnetizations at 90 K and 250 K in a Ferro-M state are 70.1 J/0kgT and 63.1 J/μ0kgT, respectively.The spontaneous magnetization at 300 K in a Ferro-A state is 52.7 J/μ0kgT.
( T/( J/ 0 kgT)) Ni 52 Mn 12.5 Fe 12.5 Ga 23 90 K 150 K 250 K 300 K Thermal Strain and Magnetization Studies of the Ferromagnetic Heusler Shape Memory Alloys Ni2MnGa and the Effect of Selective Substitution in 3d Elements on the Structural and Magnetic Phase 91

Figure 7 (
Figure 7 (c)  shows the Arrott plot of Ni2Mn0.75Cu0.25Ga.The spontaneous magnetization at 300 K in a Ferro-M state is 42.4 J/μ0kgT.The obtained TC of the martensite phase is 307 K, which is almost the same as TM = 308 K and this is consistent with the x-T phase diagram of Ni2Mn1-xCuxGa, which is obtained experimental and theoretical calculations[23].
( T/( J/ 0 kgT))Thermal Strain and Magnetization Studies of the Ferromagnetic Heusler Shape Memory Alloys Ni2MnGa and the Effect of Selective Substitution in 3d Elements on the Structural and Magnetic Phase 93 permeability of pure Fe, a large peak was observed just below TC = 1040 K[26].The half width of the peak T  is about 100 K and the ratio T  / TC = 0.095.Meanwhile, the half width of the peak T  of the permeability of Ni2MnGa0.88Cu0.12 is about 4 K and TC = 345 K.

Figure 8 .
Figure 8. Temperature dependence of the magnetic permeability  of Ni2MnGa0.88Cu0.12 in AC fields with f = 73 Hz and Bmax = 0.0050 T. The origin of the vertical axis is the reference point when the sample is empty in the pick up coil of the magnetic permeability measurement system.

Figure 9 .
Figure 9. Temperature dependences of the linear thermal expansion of Ni2MnGa0.88Cu0.12, in static magnetic fields.

2 Figure 10 .
Figure 10.Magnetic phase diagram of Ni2MnGa0.88Cu0.12.Filled circles indicate the martensitic transition temperature TM.Filled triangles indicate reverse martensitic temperature TR.Crosses indicate the typical temperature TL.
Strain and Magnetization Studies of the Ferromagnetic Heusler Shape Memory Alloys Ni2MnGa and the Effect of Selective Substitution in 3d Elements on the Structural and Magnetic Phase 97 T. Sakon

Figure 15 .
Figure 15.Temperature dependence of the magnetic permeability  of Ni52Mn25Ga23 in AC fields with f = 73 Hz and Bmax = 0.0050 T. The origin of the vertical axis is the reference point when the sample is empty in the pickup coil of the magnetic permeability measurement system.

Figure 16 .
Figure 16.(a).Temperature dependence of the linear thermal strain of Ni52Mn25Ga23 in static magnetic fields.The dotted lines are the extrapolated lines of the thermal strain.(b).Magnetic field dependence of the strain at the martensite transition temperature obtained from the thermal strain in Figure.16 (a).

Figure 17 .
Figure 17.Magnetostriction of Ni52Mn25Ga23 at 300 K in a static magnetic field up to 10 T.

Figure 18 Figure 18 .
Figure18shows the magnetic phase diagram of Ni52Mn25Ga23.With increasing field, TM and TR gradually increase.The shifts in TM and TR around zero magnetic field are estimated as dTM/dB = 0.46 K/T and dTR/dB = 0.43 K/T, which are similar to those of the Ni52Mn12.5Fe12.5Ga23alloy (dTM/dB = 0.5 K/T)[27].

Figure 19 .
Figure 19.Magnetization of Ni52Mn25Ga23 in a pulsed magnetic field up to 2.2 T.The M-B curves show ferromagnetic behavior below 356 K.It is clear that the field dependence of the magnetization at the Ferro-A phase above TR = 334 K is different from that at the Ferro-M phase below TR.At the Ferro-M phase, magnetization increases with magnetic fields.On the other hand, at the Ferro-A phase between 334 and 356 K, a sudden increase in magnetization occurs between 0 and 0.1 T.

Figure 20 .Figure 21 .
Figure 20.Magnetization of Ni52Mn25Ga23 in a pulsed magnetic field up to 15 T.

Figure 22 .
Figure 22.Temperature dependence of the magnetization of Ni52Mn25Ga23.Open circles are the spontaneous magnetizations, which was obtained by the Arrott plot method.Dotted lines are extrapolated lines of the spontaneous magnetization plots.TCM and TCA indicate the martensite Curie temperature and the austenite Curie temperature, respectively.

2 .
Magnetization and permeability: Permeability abruptly changes around TM and TR.Permeability below TM is about one-third that above TM.The temperature dependence of the magnetization also shows a clear discontinuity around TM.The Arrott plot of magnetization indicates that TC is 358 K.The sudden decrease in magnetization at the temperature of the martensite transition and the M-B curve indicate the magnetism of the hard Ferro-M phase and the soft Ferro-A phase.3. The dTM/dB values are roughly proportional to the change in spontaneous magnetization [(MM -MA)/MM] in Ni2MnGa-type Heusler alloys.The TM of the ferromagnetic Heusler alloy Ni52Mn25Ga23 in the magnetic field is considered to be shifted in accordance with the magnetic fields and proportional to the difference in magnetization between the austenite and martensite phases.