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Modeling the Magnetocaloric Effect of Nd0.67Ba0.33Mn0.98 Fe0.02O3 by the Mean Field Theory

By Mohamed Hsini and Sadok Zemni

Submitted: July 3rd 2018Reviewed: November 15th 2018Published: July 10th 2019

DOI: 10.5772/intechopen.82559

Downloaded: 499


In this paper, we have exploited the mean field theory combined with the Bean-Rodbell model to justify the magnetocaloric effect (MCE) in Nd0.67Ba0.33Mn0.98Fe0.02O3 sample. The simulation of some magnetic properties has been investigated. Modeling magnetization curves have been successfully achieved using this model. The second-order ferromagnetic-paramagnetic (FM-PM) phase transition of our system has been verified through the value of the parameter which controls the transition nature in the Bean-Rodbell model. Theoretical and experimental expressions, which have rated the magnetic entropy change ( − ∆ S M ) under various magnetic fields, have been derived. Theoretical ( − ∆ S M ) curves have been compared to the experimental ones.


  • mean field theory
  • Bean-Rodbell
  • magnetocaloric effect
  • magnetization
  • magnetic entropy change

1. Introduction

In recent years, magnetic materials exhibiting high magnetocaloric effect (MCE) have been extensively studied experimentally and theoretically because of their intensive necessity in magnetic refrigeration (MR) [1, 2, 3]. This recent cooling technology, which is expected to replace traditional expansion/compression gas refrigeration technology, has many particular interests because of its significant economic benefits [4, 5, 6]. The magnetic entropy change (SM) is interestingly important for rating the refrigerant properties [7, 8]. Thus, numerous materials exhibiting high MCE have been widely developed in the last decades for exploitation as promising materials in MR technology [9, 10, 11, 12, 13, 14].

The mean field model [15, 16] is an efficient tool in the study of magnetic materials [17]. Currently, Amaral et al. have signaled a scaling method based on this model [18].

According to the work of Amaral et al., we have reported, in this paper, our studies on the magnetocaloric properties of the Nd0.67Ba0.33Mn0.98Fe0.02O3 sample which exhibits a second-order ferromagnetic-paramagnetic (FM-PM) phase transition [19], by scaling the experimental magnetization. We have showed, in this work, how the mean field theory may adequately model the magnetic and the magnetocaloric properties of this magnetic system, which may be applicant in MR technology.

The scaling method based on the mean field model leads us to estimate directly the saturation magnetization M0, the exchange parameter λ, the total angular momentum J, and the Lande factor gof our sample. These parameters are necessary for simulating magnetization isotherms, MHT, which are used for the calculation of the magnetic entropy change (SM) of Nd0.67Ba0.33Mn0.98Fe0.02O3 material. In addition, the second-order phase transition FM-PM of this sample is confirmed by analyzing the Bean-Rodbell model [20, 21].


2. Theoretical and experimental study

2.1 Brief overview of the experimental study

In this section, we have summarized the primary results of the structural and magnetic analysis of the manganite sample Nd0.67Ba0.33Mn0.98Fe0.02O3 reported in our precedent work [19].

This compound has been prepared by the solid-state ceramic method at 1400°C in a polycrystalline powder form. Rietveld structural analysis has showed a good crystallization of the sample which presents a pseudo-cubic structure of orthorhombic Imma distortion, with unit cell parameters a = 0.54917 (1), b = 0.77602 (1), c = 0.551955 (4) nm, and unit cell volume V = 0.235228 (3) nm3. Scanning electron microscope (SEM) analysis has indicated that the sample presented a homogeneous morphology which consists of well-formed crystal grains. The SEM analysis coupled with the EDX has confirmed that the chemical composition of the sample is close to that nominal reported by the above chemical formula (19).

The evolution of the magnetization as a function of temperature, under a 0.05 T magnetic applied field in FC and ZFC modes, is depicted in Figure 1 . This figure shows a FM-PM transition at a Curie temperature which has been estimated in the inset by determining the minimum value of the derivative magnetization versus temperature in ZFC mode at 0.05 T applied field TC=131K. However, Figure 1 shows a non-negligible monotonic decrease of the magnetization between 10 and 100 K. This indicates a canted spin state between the Nd3+and the (Mn3+, Mn4+) spin sub-lattices, with canted angle, θ, assumed to be between 0° (ferromagnetic coupling) and 180°(antiferromagnetic coupling).

Figure 1.

MversusTin 0.05 T applied magnetic field for the Nd0.67Ba0.33Mn0.98Fe0.02O3 versusTin FC and ZFC modes. The inset isdMdTversusTfor ZFC mode.

Figure 2 shows the variation of the magnetization as a function of the varied magnetic field up to 10 T, at very low temperature (10 K), for the undoped compound Nd0.67Ba0.33MnO3 and for the doped compound Nd0.67Ba0.33Mn0.98Fe0.02O3.

Figure 2.

Comparison ofMversusHatT=10Kfor Nd0.67Ba0.33MnO3 and Nd0.67Ba0.33Mn0.98Fe0.02O3 samples.

It is apparent in this figure that in spite of the intense magnetic applied field (10 T), the magnetization does not attain saturation. This is due to the presence of the magnetic moments of Nd3+(Xe4f3)which have three electrons in the 4forbital. Effectively, a comparison between magnetization of the two compounds Nd0.67Ba0.33MnO3 [19] and La0.67Ba0.33MnO3 [22] are depicted in Figure 3 . This figure shows obviously that the lanthanum compound rapidly reaches saturation even under low applied magnetic field. This is because of the non-contribution of the La3+ ion (Xe)in magnetism which has no electrons in 4f orbital. Figure 2 also indicates that a 2% iron doping proportion in Nd0.67Ba0.33Mn0.98Fe0.02O3 decreases the magnetization by 0.12μB(3.94 μBfor Nd0.67Ba0.33MnO3, whereas Nd0.67Ba0.33Mn0.98Fe0.02O3 presents 3.82 μB) under a 10 T applied magnetic field of, in a good agreement with an antiferromagnetic coupling between Mn3+and Fe3+spin sub-lattices as demonstrated by the Mössbauer spectroscopy studies [23, 24]. As knowing, the orbital momentum is quenched by the crystal field in the octahedral site of manganite for transition elements, so only the spin of Fe3+([Ar]3d5) contributes to the magnetization; therefore, we have found that the experimental value (0.12μB) is very near to that calculated (MFe3+=gSμB=0.02×2×5/2μB=0.1μB).

Figure 3.

Comparison ofMversusHatT=10KLa0.67Ba0.33MnO3 and Nd0.67Ba0.33MnO3 for samples.

2.2 Theoretical calculation

The magnetic moments of a ferromagnetic material, made under external magnetic field H, tend to align in the Hdirection. The increase of parallel magnetic moments then leads to rising magnetization. Magnetization values could be rated by the Weiss mean field theory [15, 16, 18].

In fact, Weiss has enunciated that in a ferromagnetic, an exchange interaction between magnetic moments could be created, at least in a magnetic domain, where the magnetic moments could be ordered in a same direction. This interaction may be considered as an average over all interactions between a given magnetic moment and the other Nmagnetic moments of the Weiss domain. This internal interaction contributes to an exchange field or a Weiss mean field:


where λis the exchange parameter and Mis the magnetization of the ferromagnet, given by




is the saturation magnetization,


is the Brillouin function, and


where kBis the Boltzmann constant, μBis the Bohr magnetron, Nis the number of spins, and Tis the temperature.

Eq. (2) can be written as a function of H+HexchTas follows:


Applying the reciprocal function f1of f, we can obtain the relations:


The magnetic entropy change can be expressed by the Maxwell relations [6, 25]:


Using experimental isotherm magnetization data, measured at discreet values of both applied magnetic field and temperatures, and Eq. (8a), the magnetic entropy change can be approximated as


Eqs. (7) and (8b) allow us to determine the theoretical estimation of the magnetic entropy change:


To study the nature of the magnetic transition, we have called the Bean-Rodbell model to our magnetization data. As reported earlier [26, 27, 28], system exhibiting first- or second-order phase transitions have been interpreted using this model [29]. It considers that exchange interactions adequately depend on the interatomic distances; the Curie temperature TCis expressed as follows:


where ω=vv0v0, vis the volume, v0presents the volume with no exchange interaction, and T0is the transition temperature if magnetic interactions are taking into account with no magneto-volume effects. βis the slope of the critical temperature curve on volume. The Gibbs free energy, for a ferromagnetic system, is given in Ref. [30] with the compressibility K, the magnetic entropy S, and the reduced magnetization σx=BJxas


The above free energy minimizes dG=0at


Substituting Eq. (12) into Eq. (11) and minimizing Gwith respect to σ, according to the work of Zach et al. [29] and Tishin and Spichkin [6], we can obtain the magnetic state equation:




where the parameterηchecks the order of the magnetic phase transitions. For η>1, the transition is assumed to be first order. For η<1, the second-order magnetic phase transition takes place.

After combining Eq. (2) and Eq. (13), we have got two interesting equations, Mx=M0BJx(giving simulated Mversus H) and MY=M0BJY(giving simulated Mversus T).

On the other hand, for weak values of x, the magnetization may be written as


The resolution of Eq. (15) gives easily the Curie-Weiss magnetic susceptibility:


where Tc=λCis the Curie temperature and Cis the Weiss constant.

To determine accurately the exchange constant, we use the famous law of interaction between two magnetic atoms with spins S1et S2by the Hamiltonian [12]:


given by Heisenberg, where Jis the exchange constant.

If we consider an individual atom i with its magnetic moment μiin a ferromagnetic system. This moment interacts with the external applied magnetic field Hand with the exchange field Hexch. The total interaction energy is given as


From the Heisenberg model’s viewpoint, the energy Eiof a ferromagnetic system is the sum of interaction energy of a given moment, μ, with the external field and that with all near neighbors to atom i. Let us consider that each atom has znear neighbors which can interact with spin i with the same force, i.e., that exchange parameter has the same value for all zneighbors. Then, the energy Eimay be written as


where the index k runs over all z neighbors of the atom i.

It is practical to express the Heisenberg energy in Eq. 17 in terms of the atomic moments μrather than in terms of spins S. It can be easily done if we consider that the relation between the spin and the atomic magnetic moment is μ=gμBS. So, Eq. (19) would be rewritten as


In fact, the two expressions of the energy Ei, in Eq. (18) and Eq. (20), are not similar. However, the sum term in Eq. (20) is not the same as Hexchbecause it is the interaction made by the individual spin i. But, Hexchin the spirit of the Weiss theory presents the average of all interaction terms in the total system. As a result, we should carry out such averaging over all Natoms:


By summing on k, we could obtain




Eq. (21) contains a term proportional to the magnetization, in perfect agreement with Weiss’ postulate. We can write now


from which we immediately obtain the formula for Weiss’ “effective field constant”:


Comparing this equation with Eq. (16), i.e., the phenomenological expression for λ, we obtain the solution for the critical temperature:




2.3 Mean field theory application

We begin by the determination of J, g, λ, and M0parameters, which are crucial for magnetocaloric effect simulation of Nd0.67Ba0.33Mn0.98Fe0.02O3.

  • Total angular momentum Jdetermination

To determine the total angular momentum J, we must quantify the canted spin angle, θ, between Nd3+and (Mn3+, Mn4+) spin sub-arrays, using the difference between magnetizations of Nd0.67Ba0.33MnO3 [19] and La0.67Ba0.33MnO3 [22] samples at 10T0.33μBas shown in Figure 3 . By writing the contribution of Nd3+magnetic moment network (spin-orbit coupling) under the form MNd3+=0,67JNd3+gNd3+μBcosθ=0.33μB, where JNd3+=4.5and gNd3+=0.727are, respectively, the values of angular momentum and gyromagnetic factor for free ion Nd3+as indicated in Ref. [16]. Therefore, we deduce.


Using the Hund’s rule for 4f orbital less than half full and the values of L and S indicated in Ref. [16] for Nd3+, we obtain the value of the angular momentum of Nd3+ ion incorporated in Nd0.67Ba0.33Mn0.98Fe0.02O3 sample:


As a result, the total angular momentum for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample is


  • Gyromagnetic factor (g) determination:


for all the sample g=0.67×0.96+0.65×2+0.33×2+0.02×2=2.6432.

Figure 4 shows the evolution of Mversus Hat different Tnear TCfor the Nd0.67Ba0.33Mn0.98Fe0.02O3 compound. The isothermal MHTcurves show a dependency between Mand Hat different T. Above TC, a drastic decrease of MHTis observed with an almost linear behavior indicating a paramagnetic behavior. Below TC, the curves show a nonlinear behavior with a sharp increase for low field values and a tendency to saturation, as field increases, reflecting a ferromagnetic behavior. Using Figure 4 , we could plot the evolution of HTversus 1Ttaken at constant values of magnetization M(5 emu.g−1 step) from 180 to 80 K in Figure 5 . A linear behavior of the isomagnetic curves, which are progressively shifted into higher 1Tvalues, could be observed. So, the linear relationship between HTand 1Tis preserved. To find the value of the parameter λ, it is necessary to study Hexchinduced by magnetization change. Linear fits are then kept at each isomagnetics line. Using Eq. (6), the slope of each isomagnetics line could give the suitable Hexchvalue. In Figure 6 , we have plotted Hexchvs. Mfor the Nd0.67Ba0.33Mn0.98Fe0.02O3 compound. For all materials, in the PM or antiferromagnetic domain, we can always expand increasing Hin powers of Mor Min powers of H. In this approach, we will stop at the third order, and considering that the Mis an odd function of H[24, 31], we can write


Figure 4.

Isotherm magnetizationMas a function of magnetic fieldH, measured for different temperatures with a step of 3 K for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample.

Figure 5.

HTversus1Tcurves with constant values of magnetization per curve for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample.

Figure 6.

Exchange field versus magnetization for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample, with the functionλ1M+λ3M3fit.

Then, these points in Figure 6 (Hexchversus M) should be included for the fit by Eq. (27).

However, a very small dependence on M3(λ3=0.00006T.emu1.g3) is noted for this second-order transition system. So, we can assume that HexchλM, with λ=λ1=0.40243T.emu1.g. Next, the building of the scaling plot Mversus H+HexchTis depicted in Figure 7 with black symbols. It is clear from this figure that all these curves converge into one curve which can be adjusted by Eq. (6) using MATLAB software to determine M0, J, and g. We have found a good agreement between adjusted and theoretical parameters given in Table 1 .

Figure 7.

Scaled data in magnetization versusH+HexchTand Brillouin function fit for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample.

ParametersJgM0 (emu.g−1)
Theoretical values5.6142.643
Adjusted values5.6032.68683.59

Table 1.

Theoretical and adjusted parameters of Nd0.67Ba0.33Mn0.98Fe0.02O3 sample.

The agreement between fitted and theoretical values affirms the coupling between spins indicated above.

From the formula λ=3kBTCNJJ+1g2μB2and M0=NJgμBand there adjusted values, we can estimate the value of the spin number N:


The two equalities, aand b, practically give the same spin number Nwitch verifying the validity of the mean field theory. In addition, the value of Nis near to Avogadro number NA. This implies that we can assume that molecule may be present in a same value of spin so an important order domain and the nonmagnetic molecules (impurities are very limited).

After injecting adjusted parameters λ, J, g, and M0in Eq. (6), we can get simulated Mversus Hcurves (red lines), which are plotted with the experimental ones (black symbols) in Figure 8 . This figure shows a good agreement between theoretical an experimental results. This illustrates the validity of the mean field to model the magnetization. On the other hand, Figure 9 shows that simulated Mversus Tcurves (red line) under various Hare correlated with experimental ones (black symbols) when η=0.32and T0=131K. Thus, the second-order phase transition of this compound is reconfirmed with the ηparameter value (η<1).

Figure 8.

ExperimentalMversusH(black symbols) of Nd0.67Ba0.33Mn0.98Fe0.02O3 sample and the interpolation using the mean field method (red lines).

Figure 9.

Experimental magnetization versusT(black symbols) of Nd0.67Ba0.33Mn0.98Fe0.02O3 sample and the interpolation using the Bean-Rodbell model (red lines).

Figure 10 shows simulated SMversus Tcurves (red lines) using Eq. (9b) and the experimental ones (black symbols) using the Maxwell relation from in Eq. (9a). As seen in this figure and taking account into the initial considering of Hand Mas an internal and external variable in Eq. (8a) and vice versa in Eq. (8b), SMestimated in these two considerations is little different. This aspect has been reported in the work of Amaral et al. [18]. From Figure 10 , we can estimate the full width at half maximum δTFWHM, the maximum magnetic entropy change SMmax, and the relative cooling power (RCP) which is the product of SMmaxand δTFWHM. These magnetocaloric properties are listed in Table 2 .

Figure 10.

Experimental and theoretical magnetic entropy changeSMversusTof Nd0.67Ba0.33Mn0.98Fe0.02O3 sample as a function of temperature upon different magnetic field intervals (H).

Maxwell relationδTFWHM(k)47.5454.3456.9859.1563.01
Mean field theoryδTFWHM(k)27.2833.3139.8341.5144.18

Table 2.

Comparative between δTFWHM, SMmax, and RCPcalculated graphically using Maxwell relation and mean field theory.

As shown in Table 2 , a rising of SMmaxobtained by using the mean field model could be noted. For example, it exceeds the one determined by using the classical Maxwell relation by 1.5 J.Kg−1.K−1 under 5 T applied field. Although δTFWHMdetermined by this method seems less, RCPvalues are more higher than those obtained from the Maxwell relation. As a result, the mean field model could amplify RCP. This novel method has so better performance than the classical Maxwell relation.

Considering the number of magnetic near neighbors ions, z, in our material and its critical temperature, the relation J=3kBTc2zJJ+1(Eq. (26)) allows us to find the Heisenberg exchange constant J. In the perovskite structure of Nd0.67Ba0.33Mn0.98Fe0.02O3 compound, the Mnion placed at the center of the pseudo-cubic cell has four near neighbors Nddistant from a32and six near neighbors Mndistant from aand similarly for Nd. The interaction is established between Mn-Mnand Mn-Ndor Nd-Ndand Nd-Mn.

By averaging these interactions, the relationship (19) should be written as


where zMnMn=6, zMnNd=4, and zNdNd=6


JNd=3.855; JMnNd=5.66. So, J=3×1.3807×1023×1312×16×1.745×2.745+4×5.6×6.6+6×3.855×4.8553=2.8175×1023joulesfor magnetic ion in our sample. This value explains the strength interaction between spins. Moreover, it is a crucial parameter used in the simulation with the Monte Carlo method.


3. Conclusion

In this work, we have analyzed the mean field scaling method for the Nd0.67Ba0.33Mn0.98Fe0.02O3 sample. The perspicacity saved from the usefulness of this method for a magnetic system could be of large interest. In a simple reason, we can consider that if this scaling method does not follow the mean field behavior, other methods need to be convinced in the interpretation of the system’s magnetic behavior. The mean field scaling method allows us to estimate the exchange parameter λ, the total angular momentum J, the gyromagnetic factor g, the number of spins Nof our sample, the saturation magnetization M0, and the Heisenberg exchange constant J. Some of these factors are useful in estimating some magnetic properties. The mean field and the Bean-Rodbell models allow to follow the evolution of generated magnetization curves as function as the applied field and the temperature. A good agreement between theoretical and experimental magnetizations has been noted. The dependence of the entropy change on temperature under various applied fields has been experimentally and theoretically derived. An acceptable agreement between theoretical and experimental results is observed. However, the performance of RCPhas been granted by the mean field model. Also, intervention of the Bean-Rodbell model confirms the second-order magnetic transition of our sample. Because this type of transition is needed for evaluating the MCE, a significant theoretical description of magnetic and magnetocaloric properties of the Nd0.67Ba0.33Mn0.98Fe0.02O3 sample should be taking into account and should be accordable with other models.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mohamed Hsini and Sadok Zemni (July 10th 2019). Modeling the Magnetocaloric Effect of Nd<sub>0.67</sub>Ba<sub>0.33</sub>Mn<sub>0.98</sub> Fe<sub>0.02</sub>O<sub>3</sub> by the Mean Field Theory, Magnetometers - Fundamentals and Applications of Magnetism, Sergio Curilef, IntechOpen, DOI: 10.5772/intechopen.82559. Available from:

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