Forced Magnetostrictions and Magnetizations of Ni_2+xMnGa_1−x at Its Curie Temperature

Abstract

Experimental investigations into the field dependence of magnetization and the relationship between magnetization and magnetostriction in Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04) alloy ferromagnets were performed following the self-consistent renormalization (SCR) spin fluctuation theory of itinerant ferromagnetism. In this study, we investigated the magnetization of and magnetostriction on Ni_2+xMnGa_1−x (x = 0.02, 0.04) to check whether these relations held when the ratio of Ni to Ga and, the valence electron concentration per atom, e/a were varied. When the ratio of Ni to Ga was varied, e/a increased with increasing x. The magnetization results for x = 0.02 (e/a = 7.535) and 0.04 (e/a = 7.570) suggest that the critical index δ of H ∝ M^δ is around 5.0 at the Curie temperature T_C, which is the critical temperature of the ferromagnetic–paramagnetic transition. This result confirms Takahashi’s spin fluctuation theory and the experimental results of Ni₂MnGa. The spontaneous magnetization p_S slightly decreased with increasing x. For x = 0.00, the spin fluctuation parameter in k-space (momentum space; T_A) and that in energy space (T₀) were obtained. The relationship between p_eff/p_S and T_C/T₀ can also be explained by Takahashi’s theory, where p_eff indicates the effective magnetic moments. We created a generalized Rhodes-Wohlfarth plot of p_eff/p_S versus T_C/T₀ for other ferromagnets. The plot indicates that the relationship between p_eff/p_S and T₀/T_C follows Takahashi’s theory. We also measured the magnetostriction for Ni_2+xMnGa_1−x (x = 0.02, 0.04). As a result, at T_C, the plot of the magnetostriction (ΔL/L) versus M⁴ shows proportionality and crosses the origin. These magnetization and magnetostriction results were analyzed in terms of Takahashi’s SCR spin fluctuation theory. We investigated the magnetostriction at the premartensite phase, which is the precursor state to the martensitic transition. In Ni₂MnGa system alloys, the maximum value of magnetostriction is almost proportional to the e/a.

1. Introduction

Spin fluctuation theories have advanced the attempts to elucidate the physical principles of the itinerant electron system [1,2,3,4,5]. According to the self-consistent renormalization (SCR) spin fluctuation theory [1], the external magnetic field H is proportional to the third power of the magnetization M³ at the Curie temperature T_C. This relation was derived by only considering the transverse modes of the thermal spin fluctuations with respect to the direction of the static and uniform magnetic moment [6,7]. Takahashi proposed SCR theory according to zero-point spin fluctuations, which assimilate both the transverse and the longitudinal components of the fluctuations [3,4,5,8]. An outstanding characteristic of this theory is the magnetization at T_C. This theory proposed by Takahashi indicates that H is proportional to M⁵ at T_C.

The thermo-dynamical relationship between the magnetization M and the external magnetic field H can be expressed by the equation:

$$H=\frac{\partial F}{\partial M}=a\left(T\right)M+b\left(T\right)M^3+c\left(T\right)M^5+\cdots$$

(1)

where F indicates the spin fluctuation free energy. This appears as Equation (2.59) in Takahashi [8].

As T→T_C, the magnetic susceptibility $\chi \left(T\right)$ comes infinite. Therefore,

$$\underset{T\to T_C}{\lim }a\left(T\right)=\underset{T\to T_C}{\lim }\frac{H}{M}=\underset{T\to T_C}{\lim }\frac{1}{\chi }=0$$

(2)

Then, the first expansion coefficient at T_C is a(T_C) = 0.

According to the Rhodes-Wohlfarth theory [9], the third expansion coefficient b(T) in Equation (1) remains finite at T = T_C. Therefore, the following formula is satisfied at T_C:

$$H=b(T_C)M^3+c(T_C)M^5+\cdots$$

(3)

Under the Takahashi theory, b(T) vanishes at T_C, as shown in Equation (3.51) in Takahashi [8].

As a result, the M dependence of the magnetic fields H can be explained by the equation:

$$H=c(T_C)M^5$$

(4)

In Equation (4), higher terms are ignored because their magnitudes are smaller than that of the third term. In conclusion, an H ∝ M⁵ relation was obtained.

MnSi [3], CoS₂ [10], Fe_xCo_1−xSi [11], and Ni [12] follow the relationship provided in Equation (4). The Heusler isotropic ferromagnetic alloy Ni₂MnGa also follows this relation in a cubic austenite phase [12]. For Ni₂MnGa, the critical index δ of H ∝ M^δ at T_C is δ 4.70 ± 0.5 [12,13].

Takahashi proposed that magnetostriction can be observed due to the itinerant spin fluctuations around T_C [8] because the magnetostriction is calculated from the spin fluctuation free energy. The relationship between the magnetostriction and the magnetization at T_C [8] in Equation (6.101) was explained using the formula

$$\frac{{\omega }_h\left(\sigma ,t_C\right)}{{\omega }_0}=K\times A\left(0,t_C\right)\times \frac{{\sigma }^4}{{\sigma }_0^4}$$

(5)

where t_C is a relative Curie temperature; σ and σ₀ are the magnetization in a magnetic field and the spontaneous magnetization, respectively; ω₀ is the nonmagnetic volume contribution; w_h(σ, t_C) is the relative magnetic volume-striction at T_C; K has a constant value in an isothermal state; and A(0, t_C) indicates the amplitude of the thermal spin fluctuations at T_C. Equation (5) indicates that the magnetostriction is proportional to M⁴ at T_C. Kittel mentioned that the volume strain ΔV/V is three times the value of ΔL/L [14]. Accordingly, volume magnetostriction (ΔV/V) discussions were applied to the results of the magnetostriction ΔL/L in this experimental study.

For quondam research, an investigation into MnSi, which is famed for its weak itinerant magnetism, was completed [15]. The magnetostriction ΔL/L versus the square of the magnetization M² was analyzed. Around T_C = 30 K, the plot strayed from linearity. Takahashi proposed that around T_C, the magnetostriction is not proportional to the square of the magnetization. ΔL/L is proportional to M⁴ through the origin at T = 29 K around T_C [8]. In a previous study, we investigated the magnetostriction property of a polycrystalline Ni₂MnGa alloy using the self-consistent renormalization (SCR) theory of itinerant ferromagnets [13]. The magnetostriction was found to be proportional to the fourth power of magnetization. At the Curie temperature, magnetostriction crossed the point of origin. These results are in line with Takahashi’s spin fluctuation theory. In this study, we investigated Ni_2+xMnGa_1−x (x = 0.02, 0.04) alloys and studied the effect of varying alloy composition (ratio of Ni and Ga atoms) on magnetostriction. We found that the valence electron concentration per atom, i.e., the ratio e/a, increases with increasing x. The e/a values were 7.50, 7.535, and 7.570 for x = 0.00, 0.02, and 0.04, respectively. The spin fluctuation parameter in wave number space (momentum space) T_A and that in energy space T₀ were obtained from the results of the magnetization measurement. We discuss the relation between p_eff/p_S and T_C/T₀ compared with that shown in other itinerant ferromagnets by means of a generalized Rhodes-Wohlfarth plot [8]. We also investigated the e/a dependences of the maximum magnetostriction around the premartensitic–austenitic transition for Ni₂MnGa-type alloys. Researchers have studied the correlation between magnetostriction and the valence electron concentration e/a, which is related to the energy of the electron system [16,17,18]. In our prior study, we measured the properties of Ni₂Mn_1−xCr_xGa [16]. In these alloys, the e/a was smaller than 7.50, which is the value for Ni₂MnGa. In this study, we measured Ni_2+xMnGa_1−x (x = 0.02, 0.04) alloys for which the e/a is larger than 7.50 and investigated the e/a dependence of the maximum magnetostriction in the premartensite phase.

Rizal et al. investigated the magnetic property of nanostructured Fe-Co alloys [19]. At room temperature, a strong correlation was found between the saturated magnetization and the lattice constant of the Fe-Co alloy. For Ni₂MnGa-type Heusler alloys, the correlations between e/a and the magnetization (magnetic moment) or magnetostriction have been the subject of several investigations undertaken by varying alloy composition. Accordingly, in this article, we focused on the e/a dependences of the magnetostrictions.

2. Materials and Methods

The polycrystalline samples of Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04) were prepared by arc melting the constituent elements—4N Ni, 3N Mn, and 6N Ga—several times in an Ar atmosphere. Each ingot was melted several times in order to ensure good homogeneity. The products from the arc melting process were sealed in an evacuated silica tube and solution heat-treatments were applied at 1123 K for three days. After these treatments, the sample was quenched in water. The measurement of permeability was performed in alternating current (AC) magnetic fields with a frequency of 73 Hz and a maximum field of ±10 Oe. The AC magnetic fields were measured using a gaussmeter 410 (Lakeshore Cryotronix Inc., Westerville, OH, USA). The sample size chosen for the experimental investigations was 3.0 × 3.0 × 4.0 mm. The magnetostriction was measured by means of a strain gauge [13]. The magnetostriction ΔL/L was measured parallel to the external magnetic field H—the same approach used in the experimental investigation of MnSi [15]. A helium-free superconducting magnet at the Center for Advanced High Magnetic Field Science, Osaka University, Japan was used for the magnetostriction measurements up to 5 T.

The magnetization measurements were performed using a solenoid-type pulsed-field magnet at Ryukoku University, Japan [13]. The absolute value of the magnetization was calibrated with the use of a sample of pure Ni of the same size. The same bulk sample was used in the permeability, magnetization, and magnetostriction measurements in order to compare the results. The data for magnetostriction and magnetization were the results of measurements with increasing magnetic fields beginning with a zero field.

We also used a water-cooled magnet in a steady field up to 1.6 T, which was installed in Ryukoku University, and studied the magnetostriction in order to investigate the temperature dependence of the magnetostriction around the premartensite phase.

3. Results and Discussions

3.1. Magnetic Field Dependence of the Magnetization

For the Ni₂MnGa alloy, martensitic transitions occurred at the temperature T_MS of 195 K [16]. The alloy Ni₂MnGa also has a premartensite phase. This is a precursor (intermediate) state to the martensitic transition. In the premartensite phase, the alloy has a 3M modulated structure [20]. The austenitic–premartensitic transition occurs at the premartensitic temperature T_P of 260 K. Above T_P, a cubic L2₁ type austenite phase is realized. The Curie temperature T_C is 375 K, which is much higher than T_MS and T_P. The ferromagnetic–paramagnetic transition at T_C occurs in the cubic austenite phase, and the magnetic anisotropy constant K₁ in the austenite phase is 1/10 smaller than that in the martensite phase. The K₁ value at 150 K in the martensite phase was of the magnitude 4.0 × 10⁶ erg/cm³, and K₁ at 293 K in the austenite phase was 0.30 × 10⁶ erg/cm³ [16]. The magnitude of K₁ of Ni₂MnGa in the austenite phase is comparable to that of Fe. Therefore, Ni₂MnGa was decided to be an isotropic ferromagnet in the austenite phase. The value of T_M for Ni_2+xMnGa_1−x increased with increasing Ni concentration x. The value of T_P also increased with increasing x for x ≤ 0.04. Above T_P = 265 K for x = 0.02 and 275 K for x = 0.04, a cubic L2₁ type austenite phase is realized. Figure 1 plots the permeability μ for x = 0.02 (Figure 1a) and x = 0.04 (Figure 1b) during heating in a zero external magnetic field. The derivative of μ with respect to temperature, dμ/dT, is also shown in Figure 1. The T_C could not be defined from the μ-T curve because the divergence derived from Equation (2) was not found. Therefore the T_C was defined as a temperature where the absolute value of the gradient of the μ-T curve, dμ/dT is maximum. The Curie temperatures T_C were found to be 372 K and 366 K for x = 0.02 and 0.04, respectively, as obtained from the peaks of dμ/dT in Figure 1.

We measured the magnetization of Ni_2+xMnGa_1−x around T_C for the purpose of ascertaining the critical index δ of M^δ−1 versus H/M. We plotted figures of M^δ−1 versus H/M for δ = 3.0, 4.7, and 5.0; these are shown in Figure 2, Figure 3 and Figure 4, respectively. The result for δ = 3 is comparable to Moriya’s theory [1], that for δ = 5 is comparable to Takahashi’s theory [8], and that for δ = 4.7 is comparable to the former result [12]. M^δ−1 versus H/M with δ = 4.7 in Figure 3 and δ = 5.0 in Figure 4 show good linearity through the origin at T_C, denoted by the filled circles. The results suggest that for x = 0.02 and 0.04, the critical index δ is 4.7–5.0, which conforms to Takahashi’s theory [8] and the result found for Ni₂MnGa [12,13]. These relations held when the ratio of Ni to Ga and e/a were varied. H ∝ M⁵ behavior was also observed for MnSi [21] and Fe [22]. Therefore, Takahashi’s theory was again shown to be acceptable for use in analyzing magnetization in terms of itinerant electron ferromagnetism in Ni₂MnGa system alloys.

3.2. Basic Magnetic and Itinerant Spin Fluctuation Parameters and Generalized Rhodes–Wohlfarth Plot

In this subsection, we obtain the basic and spin fluctuation parameters and discuss itinerant magnetism by means of a generalized Rhodes-Wohlfarth plot of p_eff/p_S versus T_C/T₀.

The induced magnetization M [8] (Equation (3.61)) is written as:

$${\left(\frac{M}{M_S}\right)}^4=1.20\times {10}^6\times \frac{T_C^2}{T_A^3}\times \frac{H}{M}$$

(6)

where M_S = N₀p_Sμ_B represents a spontaneous magnetization in a ground state; N₀ is a molecular number; p_S = gS, where g is the Landé’s g-factor and S is spin angular momentum; and T_A is the spin fluctuation parameter in wave number space (momentum space). T_A was obtained when experimental values were inserted into Equation (6), where the magnetic field H is in units of kOe and the magnetization M is in units of Am²/kg, which is equal to emu/g.

The spontaneous magnetic moment p_S (μ_B) is expressed as:

$$p_S^2=\frac{20T_0}{T_A}\times C_{\frac{4}{3}}\times {\left(\frac{T_C}{T_0}\right)}^{\frac{4}{3}}\hspace{1em}{C}_{\frac{4}{3}}=1.006089\cdots$$

(7)

where T₀ is the width of the spin fluctuation spectrum in the energy scale. This appears as Equation (3.61) in Takahashi [8].

From Equation (7), T₀ can be obtained using the formula:

$$T_0=\frac{8147.2\times T_C^4}{T_A^3\times p_S^6}$$

(8)

Table 1. The spontaneous magnetic moment p_S and the characteristic temperatures T_C, T_A, and T₀ for Ni_2+xMnGa_1−x.

x	pS (μB)	TC (K)	TA (K)	T0 (K)
0.00	3.93	375	563	245
0.02	3.79	372	566	288
0.04	3.64	366	567	345

provides the measured spontaneous magnetic moment p_S and the characteristic temperatures T_C, calculated T_A, and T₀ for Ni_2+xMnGa_1−x. As for Ni₂MnGa, the measured p_S of 3.93 μ_B is comparable to the theoretical band calculation result at the experimental lattice constant of the L2₁ cubic austenite phase, p_S, at 3.94 μ_B [23]. With increasing Ni fraction, the p_S value decreased. This behavior appears for Ni_2+xMn_1−xGa [24] and Ni_xFe_1−x Invar alloys [25]. T₀ increased with increasing x. This is presumably because, in Equation (8), the right side varies with the sixth power of p_S, so T₀ varies even when T_A does not change.

Takahashi also derived a formula [8], shown in Equation (3.47), for the relationship between p_S, T_C, T₀, and the effective magnetic moment p_eff as follows:

$$\frac{p_{eff}}{p_S}\approx 1.4\times {\left(\frac{T_0}{T_C}\right)}^{\frac{2}{3}}$$

(9)

As for Ni₂MnGa, p_eff is 4.75 [24,26]. Equation (9) can be rewritten as:

$$k_m=\left(\frac{p_{eff}}{p_S}\right)\times {\left(\frac{T_C}{T_0}\right)}^{\frac{2}{3}}$$

(10)

When k_m is 1.4, Equation (10) is equal to Equation (9). For Ni₂MnGa, a value of 1.61 for k_m was obtained by substituting p_S, T_C, and T₀ from Table 1 and p_eff of 4.75 into Equation (10) [26]. The values of k_m for notable atoms, alloys, and compounds are Ni 1.41 [12], MnSi 1.88 [21], Ni₃Al 1.06 [27], Y(Co_0.85Al_0.15)₂ 1.08 [28], ZrZn₂ 0.74 [29], UCoGe 1.74 [8], and UGe₂ 1.61 [8]; these were calculated from the values listed in

Table 2. Basic magnetic parameters and k_m, as obtained from Equation (10).

	TC (K)	peff (μB)	pS (μB)	peff/pS	TA (K)	T0 (K)	TC/T0	km	Reference
Ni2MnGa	375	4.75 *	3.93	1.21	563	245	1.53	1.61	This work, [26] *
Ni	623	3.3	0.6	5.5	1.76 × 104	4.83 × 103	0.129	1.41	[12]
MnSi	30	2.2	0.4	5.3	2.08 × 103	231	0.13	1.88	[21]
Ni3Al	41.5	1.3	0.075	17.3	3.09 × 104	3.59 × 103	0.016	1.06	[27]
Y(Co0.85Al0.15)2	26	2.15	0.138	15.6	0.726	1.41	0.018	1.08	[28]
ZrZn2	17	1.44	0.12	12	8.83 × 103	321	0.053	0.74	[29]
UCoGe	2.4	1.93	0.039	49.5	5.92 × 103	362	0.0065	1.74	[8]
UGe2	52.6	3.00	1.41	2.13	442	92.2	0.571	1.61	[8]
NpFe4P12	23	1.55	1.35	1.15	285	16.4	1.40	1.44	[30]

. Actinide 5f compound NpFe₄P₁₂ was also analyzed using the Takahashi theory and a k_m value of 1.44 was found [30]. Table 2 provides the k_m values and the magnetic moments and characteristic temperatures relating to spin fluctuation. Figure 5 is a plot of log(p_eff/p_S) versus log(T_C/T₀) for Ni₂MnGa, Ni, and notable alloys and compounds using the data in Table 2. The dotted line indicates the line of Equation (10) when k_m is 1.4. Figure 5 clearly shows that the relation between p_eff/p_S and T_o/T_C can be explained by Equation (9). In Figure 3.3. in Takahashi [8], UGe₂ had the largest value of T_C/T₀. In Figure 5 of this article, we added Ni, Ni₂MnGa, and NpFe₄P₁₂. The T_C/T₀ value of Ni₂MnGa was almost the same as that of NpFe₄P₁₂. The magnetic alloys and compounds that were analyzed by means of Equation (9) under the Takahashi theory were magnets with T_C values lower than room temperature. Notably, the ferromagnetic alloy Ni₂MnGa, which has a T_C higher than room temperature, can be explained by Figure 5 and Equation (6).

The notable point from Table 2 and Figure 5 is that the p_eff/p_S value of Ni₂MnGa is smaller than those of other alloys and compounds. The effective moment p_eff was calculated from the Curie constant, C = Nμ_eff²/3k_B = Np_eff²μ_B²/3k_B = Nμ_B²p_C(p_C + 2)/3k_B. The term p_C refers to the effective moment deduced from the Curie constant C. The spontaneous magnetic moment μ is p_S (μ_B) at 0 K. The p_c/p_S was one for local moment ferromagnetism and was larger than one for itinerant ferromagnetism. For Ni₂MnGa, p_eff was 4.75, as shown in Table 2; therefore, a p_c value of 3.85 was obtained from the equation p_eff² = p_C(p_C + 2). Then, the p_C/p_S value was 0.98. As a result, p_C/p_S was a little smaller than one. Webster et al. compared the magnetic moment obtained by saturation magnetization measurement, p_sat = 4.17 [26]. Then, p_sat/p_S was 0.92. In this work, the magnetization of Ni₂MnGa in the magnetic field of 5.0 T at 5 K was 4.10 μ_B/f.u. Therefore, p_sat/p_S was 0.96. The Heusler compounds of CoMnSb and NiMnSb both possess the property of p_C/p_S < 1 [31]. Ott et al. proposed a simple molecular field model considering both local moments and spin-polarized itinerant electrons to explain p_C/p_S < 1 [31]. They introduced an enhanced temperature-independent Pauli susceptibility and explained that the Curie constant decreases if the interactions between local magnetic moments and holes is antiferromagnetic. Webster mentioned that in the paramagnetic phase, only the Mn atoms carry a magnetic moment [26]. It is supposed that in the paramagnetic phase, a large moment is induced by the electrons around the Mn atom at the Mn site. Conversely, at the Ni site, the spins fluctuate at high temperature in the paramagnetic phase. Therefore, it is supposed that the magnetic moment p_c at high temperature in a paramagnetic phase is smaller than the spontaneous magnetization p_S and the saturation moment p_sat at 5 K.

3.3. Magnetization and Temperature Dependences Force Magnetostrictions

We recorded magnetostriction measurements to conduct an investigation into the magnetization dependence of forced magnetostriction. In our earlier study, the magnetostriction of Ni₂MnGa was found to be proportional to the M⁴ of the magnetization and clearly passed through the origin at T_C [14]. In this study, we investigated Ni_2+xMnGa_1−x (x = 0.02, 0.04) to check whether these relations held when the ratio of Ni to Ga and e/a were varied. We plotted figures of magnetostriction ΔL/L versus M^δ for δ = 2.0 and 4.0. The result for δ = 2.0 indicates a relation under Moriya’s theory [1,15], and that for δ = 4.0 indicates a relation under Takahashi’s theory [8]. Figure 6 is a plot of magnetostriction ΔL/L versus M² for x = 0.02 (Figure 6a) and x = 0.04 (Figure 6b). The dotted lines are fitted linear plots. For the magnetostriction at T_C indicated by the filled circles, the M² linearity behavior was only observed for large magnetostriction and large magnetization area. Moreover, the dotted straight lines did not pass through the origin. These behaviors are comparable to the results for MnSi [15] and our former result for Ni₂MnGa [13]. We also investigated ΔL/L versus M⁴ dependence, as shown in Figure 7. The plot of ΔL/L versus M⁴ indicates good linearity passing through the origin at T_C, as indicated by the filled circles for both samples.

Table 3. The coefficients and standard deviations of the linear fitted plots obtained by means of the least squares method at T_C for the magnetostriction ΔL/L by the equation ΔL/L = A + kM^δ for δ = 2 or 4, as shown in Figure 5 and Figure 6, respectively. Both A and k are constants.

	δ = 2		δ = 4
x	0.02	0.04	0.02	0.04
A	3.60 × 10−5	1.65 × 10−7	3.29 × 10−5	−7.06 × 10−7
Standard deviation of A	±1.20 × 10−6 (3.3% of A)	±1.73 × 10−7 105% of A)	±1.04 × 10−6 (3.2% of A)	±2.72 × 10−7 (38% of A)
y0 = A/(Strain at 5 T)	58%	0.3%	53%	1.2%
k	−7.62 × 10−8	−3.93 × 10−11	−7.58 × 10−8	−4.11 × 10−11
Standard deviation of k	±1.2 × 10−9 (1.5% of k)	±2.08 × 10−13 (0.5% of k)	±1.03 × 10−9 (1.4% of k)	±3.36 × 10−13 (0.8% of k)

provides the coefficients A and k of the fitted linear plots given by the equation ΔL/L = A + kM^δ for δ = 2 or 4 at T_C. The standard deviations of the linear fitted lines at T_C for magnetostriction ΔL/L versus M² and ΔL/L versus M⁴ are shown in Figure 6 and Figure 7, respectively, and are also listed in Table 3. The errors of the coefficient k were within ±2% for both values of δ. The proportions of the coefficient A and the magnetostriction at 5 T (ΔL/L ≃ −60 × 10⁻⁶), y₀, were greater than 50% and less than 1.2% for δ = 2 and 4, respectively. This analysis indicates that the magnetostriction can be represented by the equation ΔL/L = kM⁴ at T_C, as presented in Figure 7. As a result, the relation between magnetostriction and magnetization confirmed that the magnetostriction is proportional to the fourth power of the magnetization, as derived from Takahashi’s theory, even when the ratio of Ni to Ga and e/a were varied.

The magnetostrictions at 5 T were 50 × 10⁻⁶, 58 × 10⁻⁶, and 61 × 10⁻⁶ for x = 0.00, 0.02, and 0.04, respectively. With increasing x, the magnetostriction increased. In our former investigation of the magnetostriction of Ni₂Mn_1−xCr_xGa (x ≤ 0.25), the magnitude of the magnetostriction increased when the premartensite transition temperature T_P and T_C were closer, as shown in Sakon et al. [16]. For Ni_2+xMnGa_1−x, the T_P values were 258 K, 265 K, and 275 K for x = 0.00, 0.02, and 0.04, respectively. The T_C values were 375 K, 372 K, and 366 K for x = 0.00, 0.02, and 0.04, respectively. With increasing x, the T_P shifted to higher temperatures and the T_C shifted to lower temperatures. We supposed that the magnetostriction of Ni_2+xMnGa_1−x has the same properties as that of Ni₂Mn_1−xCr_xGa.

Finally, we discuss the e/a dependences of the maximum magnetostriction around the premartensitic–austenitic transition for Ni₂MnGa-type alloys. Around the premartensitic transition temperature T_P, large magnetostriction has been observed [16,17]. Detailed explanations of the premartensitic transition and premartensite phase have been previously presented [16,17,18]. In our former investigation [16], we examined the magnetostrictions for Ni₂Mn_1−xCr_xGa (x = 0.00, e/a = 7.50; x = 0.15, e/a = 7.46) around T_P and T_M. With increasing x, T_P and e/a decreased; accordingly, the maximum value of the magnetostriction decreased. We assumed that if e/a increases, T_P and the magnetostriction increase. Matsui et al. experimentally investigated the Ni₂MnGa-type alloys with e/a > 7.50 [17,18]. Among these alloys, Ni_51.7Mn_24.3Ga_24.0 with T_P = 285 K and e/a = 7.59 showed large magnetostriction with strain 550 × 10⁻⁶ [17,18]. In this study, we decided to increase the concentration of Ni and decrease that of Ga because the e/a values of Ni and Ga are 10 and 3, respectively, in order to increase the e/a value of alloys to be above 7.50. Therefore, we prepared Ni_2+xMnGa_1−x alloys with x = 0.02, producing e/a = 7.535, and x = 0.04, producing e/a = 7.570. Figure 8 shows the temperature dependencies of the magnetostriction at 1.6 T. The values were 250 × 10⁻⁶ and 380 × 10⁻⁶ for x = 0.02 and 0.04, respectively. Figure 9 shows the e/a dependences of the maximum magnetostriction for Ni₂MnGa-type alloys. The maximum value of magnetostriction was almost proportional to the valence electron concentration per atom, e/a, and we also clarified the correlation between the magnetostriction and the e/a.

The softening of the lattice around T_P was investigated using ultrasonic measurements [32,33]. Seiner et al. investigated the magnetostriction around T_P for a single crystal of Ni₂MnGa [33]. They suggested a model based on adaptive concept of premartensite, explaining the softening of c₄₄ and apparent c′ stiffening prior to the martensitic transformation and discussed the magneto-elastic coupling effect by means of these magnetostriction, and ultrasonic measurements results under magnetic fields. This consideration only involves the softening of the elastic constant. Our experimental results indicate that the e/a and the magnetostriction are correlated and investigation by means of the itinerant electron magnetism is needed to better understand the fundamental origin of the magnetostriction. Future experimental and fundamental theoretical studies are needed to investigate the magneto-elastic coupling effect precisely, for example, with spectroscopy measurements for investigations of electron band structure and with itinerant electron magnetism theories.

4. Conclusions

Experimental investigations of the field dependence of magnetization and the relationship between magnetization and magnetostriction for Ni_2+xMnGa_1−x (x = 0.00, 0.02, 0.04) alloy ferromagnets were performed in accordance with the self-consistent renormalization (SCR) spin fluctuation theory of itinerant ferromagnetism. In this study, we investigated the magnetization of and the magnetostriction on Ni_2+xMnGa_1−x (x = 0.02, 0.04) to check whether these relations held when the ratio of Ni to Ga and e/a were varied. When the ratio of Ni to Ga varied, the valence electron concentration per atom, e/a, increased with increasing x. The magnetization results for x = 0.02 (e/a = 7.535) and 0.04 (e/a = 7.570) suggest that the critical index δ of H ∝ M^δ is around 5.0 at the Curie temperature T_C, which is the critical temperature of the ferromagnetic–paramagnetic transition. This result confirms Takahashi’s spin fluctuation theory and the experimental results obtained for Ni₂MnGa. The spontaneous magnetization p_S slightly decreased with increasing x. For x = 0.00, the obtained spin fluctuation parameter in k-space (momentum space) T_A and that in energy space T₀ were 563 K and 245 K, respectively. The relationship between p_eff/p_S and T_C/T₀ can be explained by Takahashi’s theory, where p_eff indicates the effective magnetic moments. We produced a generalized Rhodes-Wohlfarth plot of p_eff/p_S versus T_C/T₀ values including those of other ferromagnets. The plot indicates that the relation between p_eff/p_S and T₀/T_C follows Takahashi’s theory. We also measured the magnetostriction for Ni_2+xMnGa_1−x (x = 0.02, 0.04). At T_C, the plot of the magnetostriction ΔL/L versus M⁴ showed proportionality and crossed the origin. These magnetization and magnetostriction results were analyzed in the context of Takahashi’s SCR spin fluctuation theory. Further, we investigated the magnetostriction at the premartensite phase, which is the precursor state to the martensitic transition. In Ni₂MnGa system alloys, the maximum value of magnetostriction is almost proportional to e/a.