Investigation of the Critical Behavior, Magnetocaloric Effect and Hyperfine Structure in the Fe72Nb8B20 Powders

Microstructure as well as magnetic, thermal and magnetocaloric properties of the mechanically alloyed Fe72Nb8B20 powders have been investigated by means of Mössbauer spectrometry, differential scanning calorimetry (DSC), and magnetic measurements. The Mössbauer spectrometry results showed the formation of nanostructured Fe(B) and Fe(Nb) solid solutions, Fe2B boride, and an amorphous phase. The endothermic and exothermic peaks that are observed in the DSC curves might be related to the Curie temperature, and the crystallization of the amorphous phase, respectively. The critical exponent values around the magnetic phase transition of the amorphous phase (TC = 480 K), are deduced from the modified Arrott plots, Kouvel−Fisher curves and critical isotherm examination. The calculated values (β = 0.457 ± 0.012, γ = 0.863 ± 0.136 and δ = 3.090 ± 0.004) are near to those of the mean field model, revealing a dominating role of magnetic order arising due to long-range ferromagnetic interactions, as the critical exponents are mean-field-like. The maximum entropy change and the refrigerant capacity values are 1.45 J/kg·K and 239 J/kg, respectively, under a magnetic field of 5 T.


Introduction
Magnetic refrigeration (MR) is a promising alternate to the conventional refrigeration, and a developing technology that enhances energy efficiency and environmental respect as it uses clean energy. MR is established on the magnetocaloric effect (MCE) which represents a thermal reaction or a temperature variation of certain magnetic solids under the solicitation/removal of a magnetic field in an adiabatic condition [1]. Indeed, the application of a magnetic field gives rise to the alignment of the magnetic moments of a solid parallel to it and hence, to the increase of the temperature owing to the released thermal energy. Consequently, the magnetic entropy is reduced. By removing the magnetic field, the sample cooled down due to the random orientation of the magnetic moments, and the entropy increased [2].
Many researches have been devoted to nanoscale magnetic materials magnetic materials owing to a large MCE in the superparamagnetic nanostructured materials [3,4]. According to their magnetic

Experimental Details
Fe 72 Nb 8 B 20 (wt. %) powders were ball-milled for 50 h. The experimental details are reported in reference [26]. The local 57 Fe environment was studied by Mössbauer spectrometry in transmission geometry, at 300 K, by means of a 57 Co source diffused in an Rh matrix. Setaram DSC131 evo apparatus (DSC) (Setaram Instrumentation, Lyon, France) was used to examine the thermal behavior within the temperature range 323−973 K, under argon atmosphere, by using different heating rates. The hysteresis loops were measured, at room temperature, with a Lakeshore 7404 vibrating sample magnetometer (VSM) (LakeShore, Westerville, Oh, USA) under an applied magnetic field of 1.5 T. Magnetization versus temperature comparisons were performed on a BS2 magnetometer developed at the Néel Institute. The demagnetization field effect might have been neglected because the sample was used in powder form. The demagnetization factor D = 0.027 was determined from the slope of the M(H app ) curve near zero field. The corrected magnetic field after subtraction of the demagnetization field is H = H app − H D = H app DM, where H app is the applied magnetic field.

Hyperfine Structure
Mössbauer spectrometry of 57 Fe enables the examination of the iron sites by the determination of the local Fe environment, composition variations and the spreading of Fe atoms. The coexistence of crystalline and amorphous magnetic phases is evidenced by the presence of sharp and widened magnetic lines, respectively, in the Mössbauer spectra of the Fe 72 Nb 8 B 20 powders (Figure 1). The presence of nonmagnetic B and/or Nb atoms in the neighborhood of Fe atoms leads to the atomic disorder which is manifested by the enlargement of the Mössbauer lines. In order to identify the different Fe sites, the Mössbauer spectra were fitted by a least-squares MOSFIT program [27], by using two magnetic sextets (SS1, SS2), one paramagnetic doublet (SS3) and a hyperfine field distribution (HFD). The obtained hyperfine parameters magnetic field (B hf ), isomer shift (IS), quadrupolar splitting/shift (QS/2ε), line width (Γ) are presented in Table 1. The IS is related to α-Fe at room temperature. Mössbauer spectrometry of 57 Fe enables the examination of the iron sites by the determination of the local Fe environment, composition variations and the spreading of Fe atoms. The coexistence of crystalline and amorphous magnetic phases is evidenced by the presence of sharp and widened magnetic lines, respectively, in the Mössbauer spectra of the Fe72Nb8B20 powders ( Figure 1). The presence of nonmagnetic B and/or Nb atoms in the neighborhood of Fe atoms leads to the atomic disorder which is manifested by the enlargement of the Mössbauer lines. In order to identify the different Fe sites, the Mössbauer spectra were fitted by a least-squares MOSFIT program [27], by using two magnetic sextets (SS1, SS2), one paramagnetic doublet (SS3) and a hyperfine field distribution (HFD). The obtained hyperfine parameters magnetic field (Bhf), isomer shift (IS), quadrupolar splitting/shift (QS/2ε), line width (Γ)) are presented in Table 1. The IS is related to α-Fe at room temperature.  The hyperfine parameters of the sextet SS1 (Bhf = 33 T and IS = 0.011 mm/s) can be associated with an Fe-rich Fe(B) solid solution containing a very small concentration of boron atoms. Indeed, existence of one B atom as the nearest neighbor (nn) of an Fe atom raises IS by about 0.07 mm/s. In such Fe(B) sites, the average number of B atoms (nB) can be estimated from the linear relationship between the hyperfine magnetic field (Bhf) and the number of B atoms [28]: Bhf = 33.6-2.7nB. Accordingly, the number of B atoms in the sextet SS1 is about nB = 0.22. The sextet SS2 with B =23.2 T and IS = 0.062 mm/s, is ascribed to the Fe2B boride phase. The paramagnetic doublet with IS = − 0.203 mm/s and QS = 0.996 mm/s can be linked to an Fe(Nb) solid solution, since the existence of one Nb atom as the first or second nn of an Fe atom diminishes IS by 0.04 mm/s [29]. The HFD is due to the existence of numerous non−equivalent Fe surroundings where the Fe atoms are mainly surrounded by B atoms in their neighborhoods. The HFD can be linked to a B-rich FeB amorphous matrix. These results agree well with XRD findings (not shown here) [26].  The hyperfine parameters of the sextet SS1 (B hf = 33 T and IS = 0.011 mm/s) can be associated with an Fe-rich Fe(B) solid solution containing a very small concentration of boron atoms. Indeed, existence of one B atom as the nearest neighbor (nn) of an Fe atom raises IS by about 0.07 mm/s. In such Fe(B) sites, the average number of B atoms (n B ) can be estimated from the linear relationship between the hyperfine magnetic field (B hf ) and the number of B atoms [28]: B hf = 33.6-2.7n B . Accordingly, the number of B atoms in the sextet SS1 is about n B = 0.22. The sextet SS2 with B =23.2 T and IS = 0.062 mm/s, is ascribed to the Fe 2 B boride phase. The paramagnetic doublet with IS = −0.203 mm/s and QS = 0.996 mm/s can be linked to an Fe(Nb) solid solution, since the existence of one Nb atom as the first or second nn of an Fe atom diminishes IS by 0.04 mm/s [29]. The HFD is due to the existence of numerous non−equivalent Fe surroundings where the Fe atoms are mainly surrounded by B atoms in their neighborhoods. The HFD can be linked to a B-rich FeB amorphous matrix. These results agree well with XRD findings (not shown here) [26].

Thermal Analysis
The continuous heating DSC curves with several heating rates (5, 10, 15 and 20 K/min) are shown in Figure 2. The DSC scans exhibit two endothermic peaks at about~389 K and 428 K that can be associated to the magnetic transition (T C ) of the amorphous phase, since those of α−Fe (1043 K) and Fe 2 B (1015 K) are higher [30]. The existence of two T C might be related to the impurity phases and/or the distribution of Curie transitions in the highly disordered amorphous matrix. Similar results have been observed in other ball-milled powder alloys [31,32]. The obtained values are analogous to those of B containing alloys [17,18]. The broad exothermic peak in the temperature range 650−800 K can be attributed to the crystallization of the amorphous phase. The apparent activation energy under continuous heating conditions can be calculated by means of the Kissinger peak displacement method [33]: ln(β/T 2 ) = −E A /RT + const., where β is the constant heating rate, R is the gas constant and T stands for the crystallization peak temperature. The activation energy E A = 342 ± 10 kJ/mol has been estimated from the linear fit of ln(β/T 2 ) versus 1/T plot. This value can be linked to a grain growth process. A slightly different value of about 324 ± 35 kJ/mol has been found in the 80h ball-milled Fe 74 Nb 6 B 20 powders [34]. Those discrepancies might be related to the milling conditions and the obtained phases. The continuous heating DSC curves with several heating rates (5, 10, 15 and 20 K/min) are shown in Figure 2. The DSC scans exhibit two endothermic peaks at about ~389 K and 428 K that can be associated to the magnetic transition (TC) of the amorphous phase, since those of α−Fe (1043 K) and Fe2B (1015 K) are higher [30]. The existence of two TC might be related to the impurity phases and/or the distribution of Curie transitions in the highly disordered amorphous matrix. Similar results have been observed in other ball-milled powder alloys [31,32]. The obtained values are analogous to those of B containing alloys [17,18]. The broad exothermic peak in the temperature range 650−800 K can be attributed to the crystallization of the amorphous phase. The apparent activation energy under continuous heating conditions can be calculated by means of the Kissinger peak displacement method [33]: ln(β/T 2 ) = −EA/RT + const., where β is the constant heating rate, R is the gas constant and T stands for the crystallization peak temperature. The activation energy EA = 342 ± 10 kJ/mol has been estimated from the linear fit of ln(β/T 2 ) versus 1/T plot. This value can be linked to a grain growth process. A slightly different value of about 324 ± 35 kJ/mol has been found in the 80h ball-milled Fe74Nb6B20 powders [34]. Those discrepancies might be related to the milling conditions and the obtained phases.  Figure 3 displays the hysteresis loops recorded at room temperature for the ball-milled and heattreated powders after DSC analysis. The hysteresis loops show the same trend. They are saturated and exhibit a sigmoidal shape type. For the as-milled powders, the coercivity is 0.0302 T and the saturation magnetization is 92 emu/g. However, after DSC heat treatment, both the coercivity and saturation magnetization increased to ~0.0397 T and 181 emu/g, respectively. The increase in coercivity may be attributed to a higher number of non-magnetic phases and/or Fe2B boride. However, the increase in saturation magnetization might be correlated to the formation of α−Fe nanocrystals. Figure 4 displays the magnetization as a function of temperature, M(T), measured in a magnetic field of 0.05 T. TC of the amorphous phase that corresponds to the minimum of δM/δT, was found to be 480 K. This value is higher than that observed in the DSC curves by about 100 K. The measured TC depends on the compositional heterogeneity, strain distribution, sample shape and/or the determination method, in particular in several constituent alloys [35]. During heating, the sensitivity of a reaction is related to its energy evolved as well as to the mass of the sample. In the M(T) curve, TC is usually determined from the drop of magnetization or the inflection point method, whereas,  Figure 3 displays the hysteresis loops recorded at room temperature for the ball-milled and heat-treated powders after DSC analysis. The hysteresis loops show the same trend. They are saturated and exhibit a sigmoidal shape type. For the as-milled powders, the coercivity is 0.0302 T and the saturation magnetization is 92 emu/g. However, after DSC heat treatment, both the coercivity and saturation magnetization increased to~0.0397 T and 181 emu/g, respectively. The increase in coercivity may be attributed to a higher number of non-magnetic phases and/or Fe 2 B boride. However, the increase in saturation magnetization might be correlated to the formation of α−Fe nanocrystals. Figure 4 displays the magnetization as a function of temperature, M(T), measured in a magnetic field of 0.05 T. T C of the amorphous phase that corresponds to the minimum of δM/δT, was found to be 480 K. This value is higher than that observed in the DSC curves by about 100 K. The measured T C depends on the compositional heterogeneity, strain distribution, sample shape and/or the determination method, in particular in several constituent alloys [35]. During heating, the sensitivity of a reaction is related to its energy evolved as well as to the mass of the sample. In the M(T) curve, T C is usually determined from the drop of magnetization or the inflection point method, whereas, DSC detects T C as a heat flow variation owing to the small quantity of energy accompanying the ferromagnetic-to-paramagnetic phase transition. Hence, the endothermic reaction that happens below T C represents the absorbed energy during heating to induce randomization of the magnetic dipoles. Furthermore, the presence of many phases should impact the modification of the magnetization around T C . This later depends on the exchange interaction between the magnetic moments, which in turn depends on the distance between the magnetic atoms. Consequently, T C is dependent on the composition of the amorphous phase. For example, in the Fe 80.5 Nb 7 B 12.5 melt-spun ribbons T C was found to be 363 K [18]. In the amorphous Fe 100-x B x alloys (10≤ x ≤35 at. % B), T C of the amorphous phase increased with the augmentation of the boron content from 480 K for x = 10 up to 820 K for x = 28, and then decreased [36].

Magnetic Properties
Materials 2020, 13, x FOR PEER REVIEW 5 of 13 DSC detects TC as a heat flow variation owing to the small quantity of energy accompanying the ferromagnetic-to-paramagnetic phase transition. Hence, the endothermic reaction that happens below TC represents the absorbed energy during heating to induce randomization of the magnetic dipoles. Furthermore, the presence of many phases should impact the modification of the magnetization around TC. This later depends on the exchange interaction between the magnetic moments, which in turn depends on the distance between the magnetic atoms. Consequently, TC is dependent on the composition of the amorphous phase. For example, in the Fe80.5Nb7B12.5 melt-spun ribbons was found to be 363 K [18]. In the amorphous Fe100-xBx alloys (10≤ x ≤35 at. % B), of the amorphous phase increased with the augmentation of the boron content from 480 K for x = 10 up to 820 K for x = 28, and then decreased [36].    DSC detects TC as a heat flow variation owing to the small quantity of energy accompanying the ferromagnetic-to-paramagnetic phase transition. Hence, the endothermic reaction that happens below TC represents the absorbed energy during heating to induce randomization of the magnetic dipoles. Furthermore, the presence of many phases should impact the modification of the magnetization around TC. This later depends on the exchange interaction between the magnetic moments, which in turn depends on the distance between the magnetic atoms. Consequently, TC is dependent on the composition of the amorphous phase. For example, in the Fe80.5Nb7B12.5 melt-spun ribbons was found to be 363 K [18]. In the amorphous Fe100-xBx alloys (10≤ x ≤35 at. % B), of the amorphous phase increased with the augmentation of the boron content from 480 K for x = 10 up to 820 K for x = 28, and then decreased [36].     which has been previously identified as . One also observes that ΔSM increases as the applied magnetic field rises and attains 1.45 J/kg · K under 5 T. Different values of −ΔSM and ( Table 2) are obtained for certain Fe-Nb-B alloys [17,18,[37][38][39]. Those differences might be accredited to the experimental conditions such as the fabrication method, alloy composition, particle size and shape, structure, phase nature, matrix interactions, neighboring particles, etc. Those parameters have a deep effect on the magnetic behavior of a material.

Magnetocaloric Effect
where M i and M i+1 are the experimental data of the magnetization at T i and T i+1 , respectively, under the magnetic field H C , the temperature dependence of change in magnetic entropy − ∆S M (T) is presented in Figure 6. The magnetic entropy change versus temperature shows a peak, which has been previously identified as T C . One also observes that ∆S M increases as the applied magnetic field rises and attains 1.45 J/kg·K under 5 T. Different values of −∆S M and T C ( Table 2) are obtained for certain Fe-Nb-B alloys [17,18,[37][38][39]. Those differences might be accredited to the experimental conditions such as the fabrication method, alloy composition, particle size and shape, structure, phase nature, matrix interactions, neighboring particles, etc. Those parameters have a deep effect on the magnetic behavior of a material. where and are the experimental data of the magnetization at and , respectively, under the magnetic field HC, the temperature dependence of change in magnetic entropy − ΔSM(T) is presented in Figure 6. The magnetic entropy change versus temperature shows a peak, which has been previously identified as . One also observes that ΔSM increases as the applied magnetic field rises and attains 1.45 J/kg · K under 5 T. Different values of −ΔSM and ( Table 2) are obtained for certain Fe-Nb-B alloys [17,18,[37][38][39]. Those differences might be accredited to the experimental conditions such as the fabrication method, alloy composition, particle size and shape, structure, phase nature, matrix interactions, neighboring particles, etc. Those parameters have a deep effect on the magnetic behavior of a material.    A linear dependence has been found between the maximum entropy change and log(H). The Equation is: The linear fitting is ∆S M (H) = 1.1192 log H + 0.3386 (R 2 = 0.9694). This tendency indicates that when increasing the applied magnetic field, the maximum entropy change increases (with a factor below that corresponding to magnetic field change).
The refrigerant capacity (RC) associated with the entropy variation represents a way to evaluate the magnetocaloric efficacy of materials. RC denotes the transferred quantity of warmth between the warm and cold tanks [40]. RC is determined experimentally from ∆S M (T) and the full width at half maximum (δT FWHM ) of the peak entropy, since it is defined as follows: The temperatures T 1 and T 2 are defined by δT FWHM of ∆S M (T) peak; as an example, T 1 = 325 K and T 2 = 650 K at 1 T. RC reaches 239 J/kg for a magnetic field change of 5 T.

Critical Behavior
The universal behavior of materials can be studied by the critical exponents (β, γ and δ) related to the phase transitions (Table 3). Four distinct conventional models [11] can be used to estimate the critical exponents β, and δ from the M(H) curves such as the: (i) mean field model related to long-range mean field theory, (ii) Heisenberg model correlated to short-range interactions, (iii) 3D-Ising model, and (iv) tricritical mean field model. The exponent β is correlated to the variation of the spontaneous magnetization as a function of temperature (M S ≈ (T − T C ) β ). It describes the ordered moment growth for T < T C ; γ is connected to the temperature dependence of the initial magnetic susceptibility against of the temperature ((χ o ) −1 ≈ (T − T C ) γ ). It defines the divergence of χ o at T C , and δ is associated to with the critical isothermal magnetization. It designates the curvature of the isothermal magnetization curves M(H) at T C . The exponents β, γ and δ have been evaluated by using the modified Arrott plots (MAP) [41], Kouvel-Fisher plots (K-F)( [42,43] and critical isotherm (CI) methods according to evaluated by using Equations: ε = (T-T C )/ T C is the reduced temperature; M o , h o , and D are the critical amplitudes. The modified Arrott plots around T C of the amorphous phase are presented in Figure 7. In order to determine the model that defines the system, it is necessary to evaluate the relative slope RS = S(T)/S(T c ) which is defined by the relationship between the slope at each temperature T, S(T), and the slope at T C , S(T C ). RS is obtained from the linear fit of the high field area of each curve ( Figure 8). Accordingly, the phase transition in the ball-milled Fe 72 Nb 8 B 20 powders can be described by the mean field model because the relative slope RS is close to the unit. Accordingly, the phase transition in the ball-milled Fe72Nb8B20 powders can be described by the mean field model because the relative slope RS is close to the unit.  The exponents β, γ can be deduced by fitting of M S (T,0) and χ −1 0 (T, 0) curves by means of Equations (5) and (6), respectively ( Figure 9). The determined values β = 0.457 ± 0.012 and γ = 0.863 ± 0.136 are reasonable and comparable to those of the mean field (Table 3). T C is approximately 40 K higher than that obtained from the M(T) curve. Those divergences might be correlated to the determination method. Moreover, the Kouvel-Fisher (K-F) method can be used to evaluate the critical exponents' β and γ from the slopes 1/β and 1/γ of M S (T)(dM S (T)/dT) −1 and χ −1 plots versus temperature, respectively ( Figure 10).   (Table 3). is approximately 40 K higher than that obtained from the M(T) curve. Those divergences might be correlated to the determination method. Moreover, the Kouvel-Fisher (K-F) method can be used to evaluate the critical    One notes that β = 0.432 ± 0.015 and γ = 1.002 ± 0.093 values are also close to those of the mean field model. Likewise, the Widom scaling relationship permits the determination of the third exponent δ since it is related to β and γ exponent values through the subsequent Equation [44]: By using the critical exponents β and γ that are deduced from the K-F method, the obtained δ value, δ = 2.888 ± 0.124 (Table 3), is similar to that estimated from the CI curves (Figure 11), δ = 3.090 ± 0.004. The scaling hypothesis confirms the reliability of the critical exponents and [45]:  One notes that β = 0.432 ± 0.015 and γ = 1.002 ± 0.093 values are also close to those of the mean field model. Likewise, the Widom scaling relationship permits the determination of the third exponent δ since it is related to β and γ exponent values through the subsequent Equation [44]: By using the critical exponents β and γ that are deduced from the K-F method, the obtained δ value, δ = 2.888 ± 0.124 (Table 3), is similar to that estimated from the CI curves (Figure 11), δ = 3.090 ± 0.004. The scaling hypothesis confirms the reliability of the critical exponents and T C [45]: The regular analytic functions f + and f − are undertaken for T > T C and T < T C , respectively. Figure 12 displays the M|ε| -β as a function of H|ε| -(β+γ) are plotted in the vicinity of the T c . The accuracy of the predicted critical exponents and T C is confirmed by the presence of two distinct branches below and above TC.  The regular analytic functions f+ and f-are undertaken for T > TC and T < TC, respectively. Figure  12 displays the M|ε| -β as a function of H|ε| -(β+γ) are plotted in the vicinity of the Tc. The accuracy of the predicted critical exponents and is confirmed by the presence of two distinct branches below and above TC.

Conclusions
Partially amorphous Fe72Nb8B20 powders have been prepared by MA. The MCE, critical behavior, thermal, hyperfine structure and magnetic properties have been investigated. The milling process leads to nanocomposite type structure where nanocrystalline α−Fe(B), Fe(Nb) and Fe2B phases and embedded into an amorphous matrix. The detected endothermic and exothermic peaks in the DSC scans are related to the Curie temperature and crystallization of the amorphous phase,

Conclusions
Partially amorphous Fe 72 Nb 8 B 20 powders have been prepared by MA. The MCE, critical behavior, thermal, hyperfine structure and magnetic properties have been investigated. The milling process leads to nanocomposite type structure where nanocrystalline α−Fe(B), Fe(Nb) and Fe 2 B phases and embedded into an amorphous matrix. The detected endothermic and exothermic peaks in the DSC scans are related to the Curie temperature and crystallization of the amorphous phase, respectively. The saturation magnetization and the coercivity increase after the crystallization. The critical exponent's values (β = 0.457 ± 0.012, γ = 0.863 ± 0.136 and δ = 3.090 ± 0.004) around T C = 480 K, are near to those of the mean field model, with a dominating role of magnetic order arising due to long-range ferromagnetic interactions, as the critical exponents are mean-field-like. The maximum entropy change and the refrigerant capacity values are of about 1.45 J/kg·K and 239 J/kg, respectively, for an applied magnetic field of 5 T. These alloys, as magnetocaloric materials, are candidates to work in magnetic refrigeration devices (high temperature span applications) after consolidation in optimized geometries.