**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) nm<sup>3</sup> . 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 Tð Þ <sup>C</sup> ¼ 131 K . 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 Nd<sup>3</sup><sup>þ</sup> and the (Mn<sup>3</sup>þ, Mn<sup>4</sup>þ) spin sub-lattices, with canted angle, θ, assumed to be between 0° (ferromagnetic coupling) and 180°(antiferromagnetic coupling).

#### **Figure 1.**

M *versus* T *in 0.05 T applied magnetic field for the Nd0.67Ba0.33Mn0.98Fe0.02O3 versus* T *in FC and ZFC modes. The inset is* dM dT *versus* T *for ZFC mode.*

*Modeling the Magnetocaloric Effect of Nd0.67Ba0.33Mn0.98 Fe0.02O3 by the Mean Field Theory DOI: http://dx.doi.org/10.5772/intechopen.82559*

**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.

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 Nd<sup>3</sup><sup>þ</sup>( Xe ½ � 4f <sup>3</sup> Þ which have three electrons in the 4f orbital. 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 ½ �Þ 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 0.12μ<sup>B</sup> (3.94 μ<sup>B</sup> for 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 Mn<sup>3</sup><sup>þ</sup> and Fe3<sup>þ</sup> 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<sup>þ</sup> ([Ar]3d5 ) contributes to

### **Figure 2.** *Comparison of* M *versus* H *at* T ¼ 10 K *for Nd0.67Ba0.33MnO3 and Nd0.67Ba0.33Mn0.98Fe0.02O3 samples.*

**Figure 3.** *Comparison of* M *versus* H *at* T ¼ 10 K *La0.67Ba0.33MnO3 and Nd0.67Ba0.33MnO3 for samples.*

the magnetization; therefore, we have found that the experimental value (0.12μB) is very near to that calculated MFe3<sup>þ</sup> ¼ gSμ<sup>B</sup> ¼ 0*:*02 � 2 � 5*=*2 μ<sup>B</sup> ¼ 0*:*1μ<sup>B</sup> ð ).

### **2.2 Theoretical calculation**

The magnetic moments of a ferromagnetic material, made under external magnetic field ð Þ *H* , tend to align in the *H* direction. 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 N magnetic moments of the Weiss domain. This internal interaction contributes to an exchange field or a Weiss mean field:

$$
\overrightarrow{\mathbf{H}}\_{\text{W}} = \overrightarrow{\mathbf{H}}\_{\text{exch}} = \lambda \,\overrightarrow{\mathbf{M}} \tag{1}
$$

∂S ∂H � �

∂S ∂M � �

> *∂M ∂T* � �

> > H2 Mj H1

H

Eqs. (7) and (8b) allow us to determine the theoretical estimation of the mag-

f �1

To study the nature of the magnetic transition, we have called the Bean-Rodbell model to our magnetization data. As reported earlier [26–28], 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

v0 , v is the volume, v0 presents the volume with no exchange

interaction, and T0 is the transition temperature if magnetic interactions are taking into account with no magneto-volume effects. β is 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

� �NkBTCσ<sup>2</sup> � HgJμBN<sup>σ</sup> <sup>þ</sup>

J J þ 1

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

kB

H þ 9 5

" #

<sup>d</sup><sup>ω</sup> <sup>¼</sup> <sup>0</sup> � � at

ðH2 H1

change can be approximated as

netic entropy change:

where <sup>ω</sup> <sup>¼</sup> <sup>v</sup>�v0

magnetization σð Þ¼ x BJð Þ x as

the magnetic state equation:

<sup>Y</sup> <sup>¼</sup> <sup>1</sup>

<sup>T</sup> 3T0

with

**15**

<sup>G</sup> ¼ � <sup>3</sup> 2

The above free energy minimizes dG

ΔSMð Þ¼ T*;*Δ*H*

*DOI: http://dx.doi.org/10.5772/intechopen.82559*

<sup>Δ</sup>SMð Þ <sup>T</sup> H1!H2 ¼ � <sup>ð</sup><sup>M</sup><sup>j</sup>

distances; the Curie temperature TC is expressed as follows:

J J þ 1

> <sup>ω</sup> <sup>¼</sup> <sup>3</sup> 2

J J þ 1

� �<sup>σ</sup> <sup>þ</sup> gJμ<sup>B</sup>

T

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

T

<sup>¼</sup> <sup>∂</sup><sup>M</sup> ∂T � �

¼� <sup>∂</sup><sup>H</sup> ∂T � �

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

> dH ≈ ∑*n*

H

M

ð Þ� <sup>M</sup> *<sup>∂</sup>*Hexch

*∂T* � �

TC ¼ T0ð Þ 1 þ βω (10)

1

� �NkBKTCσ<sup>2</sup> (12)

σð Þ¼ Y BJð Þ Y (13)

ð Þ 2J <sup>þ</sup> <sup>1</sup> <sup>4</sup> � <sup>1</sup> ½ � 2 Jð Þ <sup>þ</sup> <sup>1</sup> <sup>4</sup> T0ησ<sup>3</sup>

2Kω<sup>2</sup> � TS (11)

(14)

M � �dM (9b)

ð Þ Mnþ<sup>1</sup> � Mn <sup>H</sup> Tnþ<sup>1</sup> � Tn

(8a)

(8b)

ΔHn (9a)

where λ is the exchange parameter and M is the magnetization of the ferromagnet, given by

$$\mathbf{M} = \mathbf{M}\_0 \mathbf{B}\_l(\mathbf{x}) \tag{2}$$

where

$$\mathbf{M}\_0 = \mathbf{N} \mathbf{J} \mathbf{g} \mu\_\mathbf{B} \tag{3}$$

is the saturation magnetization,

$$\mathbf{B}\_{\mathbb{I}}(\mathbf{x}) = \frac{\mathbf{2J} + \mathbf{1}}{\mathbf{2J}} \coth\left(\frac{\mathbf{2J} + \mathbf{1}}{\mathbf{2J}} \mathbf{x}\right) - \frac{\mathbf{1}}{\mathbf{2J}} \coth\left(\frac{\mathbf{x}}{\mathbf{2J}}\right) \tag{4}$$

is the Brillouin function, and

$$\mathbf{x} = \frac{\mathbf{J} \mathbf{g} \mu\_{\mathrm{B}}}{\mathbf{k}\_{\mathrm{B}}} \left( \frac{\mathbf{H} + \mathbf{H}\_{\mathrm{exch}}}{\mathbf{T}} \right) \tag{5}$$

where kB is the Boltzmann constant, μ<sup>B</sup> is the Bohr magnetron, N is the number of spins, and T is the temperature.

Eq. (2) can be written as a function of <sup>H</sup>þHexch T as follows:

$$\begin{aligned} \mathbf{M}(\mathbf{H}, \mathbf{T}) &= \mathbf{f}\left(\frac{\mathbf{H} + \mathbf{H}\_{\mathrm{exch}}}{\mathbf{T}}\right) = \mathbf{M}\_0 \left[\frac{\mathbf{2J} + \mathbf{1}}{\mathbf{2J}} \coth\left(\frac{\mathbf{2J} + \mathbf{1}}{\mathbf{2J}} \frac{\mathbf{J} \mathbf{g} \mu\_{\mathrm{B}}}{\mathbf{k}\_{\mathrm{B}}} \left(\frac{\mathbf{H} + \mathbf{H}\_{\mathrm{exch}}}{\mathbf{T}}\right)\right)\right. \\ &\left. - \frac{\mathbf{1}}{\mathbf{2J}} \coth\left(\frac{\mathbf{1}}{\mathbf{2J}} \frac{\mathbf{J} \mathbf{g} \mu\_{\mathrm{B}}}{\mathbf{k}\_{\mathrm{B}}} \left(\frac{\mathbf{H} + \mathbf{H}\_{\mathrm{exch}}}{\mathbf{T}}\right)\right)\right] \end{aligned} \tag{6}$$

Applying the reciprocal function f�<sup>1</sup> of f, we can obtain the relations:

$$\frac{\mathbf{H}}{\mathbf{T}} = \mathbf{f}^{-1}(\mathbf{M}) - \frac{\mathbf{H}\_{\text{exch}}}{\mathbf{T}}; \mathbf{H} = \mathbf{T}\mathbf{f}^{-1}(\mathbf{M}) - \lambda\mathbf{M}.\tag{7}$$

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

*Modeling the Magnetocaloric Effect of Nd0.67Ba0.33Mn0.98 Fe0.02O3 by the Mean Field Theory DOI: http://dx.doi.org/10.5772/intechopen.82559*

$$\left(\frac{\partial \mathbf{S}}{\partial \mathbf{H}}\right)\_{\mathbf{T}} = \left(\frac{\partial \mathbf{M}}{\partial \mathbf{T}}\right)\_{\mathbf{H}}\tag{8a}$$

$$
\left(\frac{\partial \mathbf{S}}{\partial \mathbf{M}}\right)\_{\mathrm{T}} = -\left(\frac{\partial \mathbf{H}}{\partial \mathbf{T}}\right)\_{\mathrm{M}}\tag{8b}
$$

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

$$
\Delta \mathbf{S}\_{\mathbf{M}}(\mathbf{T}, \Delta H) = \int\_{\mathbf{H}\_{\mathrm{I}}}^{\mathbf{H}\_{\mathrm{2}}} \left( \frac{\partial \mathbf{M}}{\partial T} \right)\_{\mathrm{H}} \mathrm{d}\mathbf{H} \approx \sum\_{\mathbf{n}} \frac{(\mathbf{M}\_{\mathrm{n}+1} - \mathbf{M}\_{\mathrm{n}})\_{\mathrm{H}}}{\mathbf{T}\_{\mathrm{n}+1} - \mathbf{T}\_{\mathrm{n}}} \Delta \mathbf{H}\_{\mathrm{n}} \tag{9a}
$$

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

$$\Delta \mathbf{S}\_{\mathbf{M}}(\mathbf{T})\_{\mathbf{H}\_1 \rightarrow \mathbf{H}\_2} = -\int\_{\mathbf{M}|\_{\mathbf{H}\_1}}^{\mathbf{M}|\_{\mathbf{H}\_2}} \left( \mathbf{f}^{-1}(\mathbf{M}) - \left( \frac{\partial \mathbf{H}\_{\text{exch}}}{\partial T} \right)\_{\mathbf{M}} \right) d\mathbf{M} \tag{9b}$$

To study the nature of the magnetic transition, we have called the Bean-Rodbell model to our magnetization data. As reported earlier [26–28], 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 TC is expressed as follows:

$$\mathbf{T\_{C}} = \mathbf{T\_{0}}(\mathbf{1} + \mathbf{\beta}\mathbf{o}) \tag{10}$$

where <sup>ω</sup> <sup>¼</sup> <sup>v</sup>�v0 v0 , v is the volume, v0 presents the volume with no exchange interaction, and T0 is the transition temperature if magnetic interactions are taking into account with no magneto-volume effects. β is 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 σð Þ¼ x BJð Þ x as

$$\mathbf{G} = -\frac{3}{2} \left( \frac{\mathbf{J}}{\mathbf{J} + \mathbf{1}} \right) \mathbf{N} \mathbf{k}\_{\mathbf{B}} \mathbf{T}\_{\mathbf{C}} \sigma^2 - \mathbf{H} \mathbf{g} \mathbf{J} \mu\_{\mathbf{B}} \mathbf{N} \sigma + \frac{\mathbf{1}}{2\mathbf{K}} \alpha^2 - \mathbf{T} \mathbf{S} \tag{11}$$

The above free energy minimizes dG <sup>d</sup><sup>ω</sup> <sup>¼</sup> <sup>0</sup> � � at

$$\alpha = \frac{\mathfrak{Z}}{2} \left( \frac{\mathfrak{J}}{\mathfrak{J} + 1} \right) \text{Nk}\_{\mathsf{B}} \text{KT}\_{\mathsf{C}} \sigma^2 \tag{12}$$

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

$$\sigma(\mathbf{Y}) = \mathbf{B}\_{\mathsf{I}}(\mathbf{Y}) \tag{13}$$

with

$$\mathbf{Y} = \frac{\mathbf{1}}{\mathbf{T}} \left[ \mathbf{3T}\_0 \left( \frac{\mathbf{J}}{\mathbf{J} + \mathbf{1}} \right) \sigma + \frac{\mathbf{g} \mathbf{J} \mu\_\mathbf{B}}{\mathbf{k}\_\mathbf{B}} \mathbf{H} + \frac{\mathbf{9} \left( \mathbf{2J} + \mathbf{1} \right)^4 - \mathbf{1}}{\mathbf{2} \left[ \mathbf{2} \left( \mathbf{J} + \mathbf{1} \right) \right]^4} \mathbf{T}\_0 \eta \sigma^3 \right] \tag{14}$$

where the parameter η checks 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, M xð Þ¼ M0BJð Þ x (giving simulated M versus H) and M Yð Þ¼ M0BJð Þ Y (giving simulated M versus T).

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

$$\mathbf{M} = \mathbf{M}\_0 \mathbf{g} \mu\_\mathbf{B} \frac{\mathbf{H}}{\mathbf{k} \mathbf{T}} \frac{\mathbf{J} + \mathbf{1}}{\mathbf{3}} + \lambda \mathbf{M}\_0 \mathbf{g} \mu\_\mathbf{B} \frac{\mathbf{1}}{\mathbf{k} \mathbf{T}} \frac{\mathbf{J} + \mathbf{1}}{\mathbf{3}} \mathbf{M} = \chi \mathbf{H} \tag{15}$$

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

$$\chi = \frac{\text{NJ}(\text{J} + \text{1}) \text{g}^2 \frac{\mu\_{\text{B}}^2}{\text{\%} \text{k}\_{\text{B}}}}{\text{T} - \lambda \text{NJ}(\text{J} + \text{1}) \text{g}^2 \frac{\mu\_{\text{B}}^2}{\text{\%} \text{k}\_{\text{B}}}} = \frac{\text{C}}{\text{T} - \text{T}\_{\text{c}}} \tag{16}$$

where Tc ¼ λC is the Curie temperature and C is the Weiss constant.

To determine accurately the exchange constant, we use the famous law of interaction between two magnetic atoms with spins S1 ! et S2 ! by the Hamiltonian [12]:

$$\mathbf{H} = -\mathbf{2J}\overrightarrow{\mathbf{S}}\_1\overrightarrow{\mathbf{S}}\_2\tag{17}$$

interaction made by the individual spin i. But, H!

*DOI: http://dx.doi.org/10.5772/intechopen.82559*

should carry out such averaging over all N atoms:

exch <sup>¼</sup> 2J

exch <sup>¼</sup> 2J

<sup>g</sup>μ<sup>B</sup> ð Þ<sup>2</sup> <sup>∑</sup>

<sup>g</sup>μ<sup>B</sup> ð Þ<sup>2</sup>

Ei ¼ �μ ! <sup>i</sup> H !

> H !

for λ, we obtain the solution for the critical temperature:

Tc <sup>¼</sup> *<sup>λ</sup><sup>C</sup>* <sup>¼</sup> 2Jz

for magnetocaloric effect simulation of Nd0.67Ba0.33Mn0.98Fe0.02O3.

magnetic moment network (spin-orbit coupling) under the form

• Total angular momentum Jð Þ determination

*z k*¼1 μ ! k

> 1 <sup>N</sup> <sup>z</sup> <sup>∑</sup> *N i* μ ! i � �

\* +

H !

By summing on k, we could obtain

H !

with Weiss' postulate. We can write now

(then,)

constant":

Therefore

**17**

**2.3 Mean field theory application**

presents the average of all interaction terms in the total system. As a result, we

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

<sup>¼</sup> 2J <sup>g</sup>μ<sup>B</sup> ð Þ<sup>2</sup>

<sup>þ</sup> 2Jz <sup>g</sup>μ<sup>B</sup> ð Þ<sup>2</sup> N M !

Eq. (21) contains a term proportional to the magnetization, in perfect agreement

exch <sup>¼</sup> 2Jz <sup>g</sup>μ<sup>B</sup> ð Þ<sup>2</sup> N M !

from which we immediately obtain the formula for Weiss' "effective field

<sup>λ</sup> <sup>¼</sup> 2Jz <sup>g</sup>μ<sup>B</sup> ð Þ<sup>2</sup>

<sup>g</sup>μ<sup>B</sup> ð Þ<sup>2</sup> N

<sup>J</sup> <sup>¼</sup> 3kBTc

We begin by the determination of J, g, λ, and M0 parameters, which are crucial

To determine the total angular momentum Jð Þ, we must quantify the canted spin angle, θ, between Nd3<sup>þ</sup>and (Mn<sup>3</sup>þ, Mn<sup>4</sup>þ) spin sub-arrays, using the difference between magnetizations of Nd0.67Ba0.33MnO3 [19] and La0.67Ba0.33MnO3 [22] samples at 10 T 0*:*33μ<sup>B</sup> ð Þ as shown in **Figure 3**. By writing the contribution of Nd<sup>3</sup><sup>þ</sup>

Comparing this equation with Eq. (16), i.e., the phenomenological expression

NJ Jð Þ <sup>þ</sup> <sup>1</sup> <sup>g</sup><sup>2</sup>μ<sup>B</sup>

3kB

2

2zJ Jð Þ <sup>þ</sup> <sup>1</sup> (26)

!

1 <sup>N</sup> <sup>∑</sup> *N i* ∑ *Z k*¼1 μ ! i*,*k � � (21)

<sup>¼</sup> 2J <sup>g</sup>μ<sup>B</sup> ð Þ<sup>2</sup>

1 <sup>N</sup> z M! *,*

<sup>N</sup> (24)

exch in the spirit of the Weiss theory

(22)

(23)

(25)

given by Heisenberg, where **J** is the exchange constant.

If we consider an individual atom i with its magnetic moment μ ! <sup>i</sup> in a ferromagnetic system. This moment interacts with the external applied magnetic field H! and with the exchange field H! exch. The total interaction energy is given as

$$\mathbf{E}\_{\mathbf{i}} = -\overrightarrow{\boldsymbol{\mu}}\_{\mathbf{i}} \left( \overrightarrow{\mathbf{H}} + \overrightarrow{\mathbf{H}}\_{\text{exch}} \right) \tag{18}$$

From the Heisenberg model's viewpoint, the energy Ei of 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 z near neighbors which can interact with spin i with the same force, i.e., that exchange parameter has the same value for all z neighbors. Then, the energy Ei may be written as

$$\mathbf{E}\_{\mathbf{i}} = -\overrightarrow{\boldsymbol{\mu}}\_{\mathbf{i}} \, \overrightarrow{\mathbf{H}} - \sum\_{k=1}^{Z} \mathbf{2J} \, \overrightarrow{\mathbf{S}}\_{\mathbf{i}} \, \overrightarrow{\mathbf{S}}\_{\mathbf{k}} \tag{19}$$

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 μ ! rather 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 !¼ �gμ<sup>B</sup> S ! . So, Eq. (19) would be rewritten as

$$\mathbf{E}\_{\mathbf{i}} = -\overrightarrow{\boldsymbol{\mu}}\_{\mathbf{i}} \left( \overrightarrow{\mathbf{H}} + \frac{\mathbf{Z} \mathbf{J}}{\left( \mathbf{g} \boldsymbol{\mu}\_{\mathbf{B}} \right)^{2}} \sum\_{k=1}^{x} \overrightarrow{\boldsymbol{\mu}}\_{\mathbf{k}} \right) \tag{20}$$

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 H! exch because it is the *Modeling the Magnetocaloric Effect of Nd0.67Ba0.33Mn0.98 Fe0.02O3 by the Mean Field Theory DOI: http://dx.doi.org/10.5772/intechopen.82559*

interaction made by the individual spin i. But, H! exch in 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 N atoms:

$$\overrightarrow{\mathbf{H}}\_{\text{exch}} = \left\langle \frac{\mathbf{2J}}{\left(\mathbf{g}\mu\_{\text{B}}\right)^{2}} \sum\_{k=1}^{x} \overrightarrow{\mu}\_{\text{k}} \right\rangle = \frac{\mathbf{2J}}{\left(\mathbf{g}\mu\_{\text{B}}\right)^{2}\mathbf{N}} \left\langle \sum\_{i}^{N} \sum\_{k=1}^{Z} \overrightarrow{\mu}\_{i,\text{k}} \right\rangle \tag{21}$$

By summing on k, we could obtain

$$\overrightarrow{\mathbf{H}}\_{\text{exch}} = \frac{\mathbf{2J}}{\left(\mathbf{g}\mu\_{\text{B}}\right)^{2}\mathbf{N}} \frac{\mathbf{1}}{\mathbf{N}} \mathbf{z} \left\langle \sum\_{i}^{N} \overrightarrow{\mu}\_{i} \right\rangle = \frac{\mathbf{2J}}{\left(\mathbf{g}\mu\_{\text{B}}\right)^{2}\mathbf{N}} \frac{\mathbf{1}}{\mathbf{N}} \mathbf{z} \left\langle \overrightarrow{\mathbf{M}} \right\rangle\_{i}$$

(then,)

$$\mathbf{E}\_{\mathbf{i}} = -\overrightarrow{\boldsymbol{\mu}}\_{\mathbf{i}} \left( \overrightarrow{\mathbf{H}} + \frac{2 \mathbf{J} \mathbf{z}}{(\mathbf{g}\mu\_{\mathbf{B}})^2 \mathbf{N}} \overrightarrow{\mathbf{M}} \right) \tag{22}$$

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

$$
\overrightarrow{\mathbf{H}}\_{\text{exch}} = \frac{\mathbf{2J}\mathbf{z}}{(\mathbf{g}\mu\_{\text{B}})^2 \mathbf{N}} \overrightarrow{\mathbf{M}} \tag{23}
$$

from which we immediately obtain the formula for Weiss' "effective field constant":

$$\lambda = \frac{2\text{Jz}}{(\text{g}\mu\_{\text{B}})^2 \text{N}} \tag{24}$$

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

$$\mathbf{T\_c = \lambda C = \frac{2\mathbf{Jz}}{\left(\mathbf{g}\mu\_\mathrm{B}\right)^2 \mathbf{N}}} \frac{\mathbf{NJ(J+1)g^2\mu\_\mathrm{B}}^2}{\mathbf{3k\_B}} \tag{25}$$

Therefore

$$\mathbf{J} = \frac{\mathbf{3k\_B T\_c}}{2\mathbf{zJ(J+1)}} \tag{26}$$

#### **2.3 Mean field theory application**

We begin by the determination of J, g, λ, and M0 parameters, which are crucial for magnetocaloric effect simulation of Nd0.67Ba0.33Mn0.98Fe0.02O3.

• Total angular momentum Jð Þ determination

To determine the total angular momentum Jð Þ, we must quantify the canted spin angle, θ, between Nd3<sup>þ</sup>and (Mn<sup>3</sup>þ, Mn<sup>4</sup>þ) spin sub-arrays, using the difference between magnetizations of Nd0.67Ba0.33MnO3 [19] and La0.67Ba0.33MnO3 [22] samples at 10 T 0*:*33μ<sup>B</sup> ð Þ as shown in **Figure 3**. By writing the contribution of Nd<sup>3</sup><sup>þ</sup> magnetic moment network (spin-orbit coupling) under the form

MNd3<sup>þ</sup> ¼ 0*,* 67 JNd3<sup>þ</sup> gNd3<sup>þ</sup> μ<sup>B</sup> cos θ ¼ 0*:*33μB, where JNd3<sup>þ</sup> ¼ 4*:*5 and gNd3<sup>þ</sup> ¼ 0*:*727 are, respectively, the values of angular momentum and gyromagnetic factor for free ion Nd3<sup>þ</sup> as indicated in Ref. [16]. Therefore, we deduce.

$$\cos \theta = \frac{0.33}{0.67 \text{(Nd}^{3+}\text{)} \text{ g(Nd}^{3+})} = \frac{0.33}{0.67 \times 4.5 \times 0.727} = 0.15, \text{ so, } \theta = 81.34^{\circ} \text{ }^{\circ}$$

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<sup>þ</sup>, we obtain the value of the angular momentum of Nd3+ ion incorporated in Nd0.67Ba0.33Mn0.98Fe0.02O3 sample:

$$\mathbf{J(Nd^{3+}) = 0.67 \times |L - S \times \cos\theta| = 0.67 \times \left| 6 - \frac{3}{2} \times 0.15 \right| = 3.869$$

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

low field values and a tendency to saturation, as field increases, reflecting a ferro-

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

*DOI: http://dx.doi.org/10.5772/intechopen.82559*

*<sup>T</sup> curves with constant values of magnetization per curve for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample.*

<sup>T</sup> is preserved. To find the value of the parameter λ, it is necessary to study

isomagnetics line. Using Eq. (6), the slope of each isomagnetics line could give the

Then, these points in **Figure 6** (Hexch versus M) should be included for the fit by

However, a very small dependence on M3 (λ<sup>3</sup> ¼ �0*:*00006 T*:*emu�<sup>1</sup> ð Þ *:*<sup>g</sup>

noted for this second-order transition system. So, we can assume that Hexch ≈λM,

*Exchange field versus magnetization for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample, with the function* <sup>λ</sup>1*<sup>M</sup>* <sup>þ</sup> <sup>λ</sup>3*M*<sup>3</sup>

Nd0.67Ba0.33Mn0.98Fe0.02O3 compound. For all materials, in the PM or antiferromagnetic domain, we can always expand increasing H in powers of *M* or *M* in powers of *H*. In this approach, we will stop at the third order, and considering that

<sup>T</sup> values, could be observed. So, the linear relationship between <sup>H</sup>

Hexch <sup>¼</sup> <sup>λ</sup>1M <sup>þ</sup> <sup>λ</sup>3M<sup>3</sup> (27)

at constant values of magnetization M (5 emu.g�<sup>1</sup> step) from 180 to 80 K in **Figure 5**. A linear behavior of the isomagnetic curves, which are progressively

Hexch induced by magnetization change. Linear fits are then kept at each

suitable Hexch value. In **Figure 6**, we have plotted Hexch vs. M for the

the *M* is an odd function of *H* [24, 31], we can write

<sup>T</sup> versus <sup>1</sup>

3 ) is

<sup>T</sup> taken

T

magnetic behavior. Using **Figure 4**, we could plot the evolution of <sup>H</sup>

shifted into higher <sup>1</sup>

and <sup>1</sup>

**Figure 5.** *H <sup>T</sup> versus* <sup>1</sup>

Eq. (27).

**Figure 6.**

*fit.*

**19**

$$\begin{aligned} \mathbf{J} &= \mathbf{J} \left( \mathbf{N} \mathbf{d}^{\mathbf{3}+} \right) + \mathbf{S} \left( \mathbf{M} \mathbf{n}^{\mathbf{3}+} \right) + \mathbf{S} \left( \mathbf{M} \mathbf{n}^{4+} \right) - \mathbf{S} \left( \mathbf{F} \mathbf{e}^{\mathbf{3}+} \right) \\ &= \mathbf{3.869} + 0.65 \times \mathbf{2} + 0.33 \times \mathbf{1.5} - 0.02 \times \mathbf{2.5} = \mathbf{5.614}. \end{aligned}$$

• Gyromagnetic factor (g) determination:

$$\log\left(\text{Nd}^{3+}\right) = 1 + \frac{\text{J}(\text{J}+1) + \text{Soso}\\$\text{(Soso}\\\text{0}+1) - \text{L}(\text{L}+1)}{\text{Zl}(\text{J}+1)}\\= 1 + \frac{\text{5.754} \times \text{6.754} + \text{0.246} \times \text{1.246} - \text{6} \times \text{7}}{2 \times \text{5.754} \times \text{6.754}}\\= 0.96... + \text{0.246} - \text{0.246} \times \text{0.246} - \text{0.246} \times \text{0.246} + \text{0.246} \times \text{0.246} - \text{0.246} \times \text{0.246} + \text{0.246} \times \text{0.246} - \text{0.246} \times \text{0.246} + \text{0.246} \times \text{0.246}$$

for all the sample g ¼ 0*:*67 � 0*:*96 þ 0*:*65 � 2 þ 0*:*33 � 2 þ 0*:*02 � 2 ¼ 2*:*6432.

**Figure 4** shows the evolution of M versus H at different T near TC for the Nd0.67Ba0.33Mn0.98Fe0.02O3 compound. The isothermal M Hð Þ *;* T curves show a dependency between M and H at different T. Above TC, a drastic decrease of *M H*ð Þ *; T* is observed with an almost linear behavior indicating a paramagnetic behavior. Below TC, the curves show a nonlinear behavior with a sharp increase for

**Figure 4.**

*Isotherm magnetization M as a function of magnetic field H, measured for different temperatures with a step of 3 K for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample.*

*Modeling the Magnetocaloric Effect of Nd0.67Ba0.33Mn0.98 Fe0.02O3 by the Mean Field Theory DOI: http://dx.doi.org/10.5772/intechopen.82559*

**Figure 5.** *H <sup>T</sup> versus* <sup>1</sup> *<sup>T</sup> curves with constant values of magnetization per curve for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample.*

low field values and a tendency to saturation, as field increases, reflecting a ferromagnetic behavior. Using **Figure 4**, we could plot the evolution of <sup>H</sup> <sup>T</sup> versus <sup>1</sup> <sup>T</sup> taken at constant values of magnetization M (5 emu.g�<sup>1</sup> step) from 180 to 80 K in **Figure 5**. A linear behavior of the isomagnetic curves, which are progressively shifted into higher <sup>1</sup> <sup>T</sup> values, could be observed. So, the linear relationship between <sup>H</sup> T and <sup>1</sup> <sup>T</sup> is preserved. To find the value of the parameter λ, it is necessary to study Hexch induced 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 Hexch value. In **Figure 6**, we have plotted Hexch vs. M for the Nd0.67Ba0.33Mn0.98Fe0.02O3 compound. For all materials, in the PM or antiferromagnetic domain, we can always expand increasing H in powers of *M* or *M* in powers of *H*. In this approach, we will stop at the third order, and considering that the *M* is an odd function of *H* [24, 31], we can write

$$\mathbf{H}\_{\text{exch}} = \lambda\_1 \mathbf{M} + \lambda\_3 \mathbf{M}^3 \tag{27}$$

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

However, a very small dependence on M3 (λ<sup>3</sup> ¼ �0*:*00006 T*:*emu�<sup>1</sup> ð Þ *:*<sup>g</sup> 3 ) is noted for this second-order transition system. So, we can assume that Hexch ≈λM,

#### **Figure 6.**

*Exchange field versus magnetization for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample, with the function* <sup>λ</sup>1*<sup>M</sup>* <sup>þ</sup> <sup>λ</sup>3*M*<sup>3</sup> *fit.*

with <sup>λ</sup> <sup>¼</sup> <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*40243 T*:*emu�<sup>1</sup>*:*g. Next, the building of the scaling plot M versus HþHexch <sup>T</sup> is 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**.

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 *M* versus *T* curves (red line) under various *H* are correlated with experimental ones

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

*DOI: http://dx.doi.org/10.5772/intechopen.82559*

*Experimental M versus H (black symbols) of Nd0.67Ba0.33Mn0.98Fe0.02O3 sample and the interpolation using*

*Experimental magnetization versus T (black symbols) of Nd0.67Ba0.33Mn0.98Fe0.02O3 sample and the*

**Figure 8.**

**Figure 9.**

**21**

*interpolation using the Bean-Rodbell model (red lines).*

*the mean field method (red lines).*

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

From the formula <sup>λ</sup> <sup>¼</sup> 3kBTC NJ Jð Þ þ1 g2μ<sup>B</sup> <sup>2</sup> and M0 ¼ NJgμ<sup>B</sup> and there adjusted values, we can estimate the value of the spin number N:

$$\begin{cases} \text{N} = \frac{\text{3k}\_{\text{B}}\text{T}\_{\text{C}}}{\text{J}(\text{J}+1)\text{g}^{2}\mu\_{\text{B}}^{2}} = \frac{3 \times 1,30807 \times 131.10^{-23}}{0,4024 \times 5,603 \times (5,603+1) \times 2,498^{2} \times \left(9,274.10^{-24}\right)^{2}} \approx 6.10^{23} \text{ (a)}\\\text{N} = \frac{\text{M}\_{\text{0}}}{\text{J}\text{g}\mu\_{\text{B}}} = \frac{83,592}{5.603 \times 2,6432 \times 9,274.10^{-24}} \approx 6.10^{23} \text{ (b)} \end{cases} (\text{s})$$

The two equalities, að Þ and bð Þ, practically give the same spin number N witch verifying the validity of the mean field theory. In addition, the value of N is 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 M0 in Eq. (6), we can get simulated *M* versus *H* curves (red lines), which are plotted with the experimental

**Figure 7.** *Scaled data in magnetization versus <sup>H</sup>*þ*Hexch <sup>T</sup> and Brillouin function fit for Nd0.67Ba0.33Mn0.98Fe0.02O3 sample.*


**Table 1.**

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

*Modeling the Magnetocaloric Effect of Nd0.67Ba0.33Mn0.98 Fe0.02O3 by the Mean Field Theory DOI: http://dx.doi.org/10.5772/intechopen.82559*

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 *M* versus *T* curves (red line) under various *H* are correlated with experimental ones

#### **Figure 8.**

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

#### **Figure 9.**

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

neighbors Mn distant from a and similarly for Nd. The interaction is established

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

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

S Mn ð Þ¼ *;* Fe S Mn<sup>3</sup><sup>þ</sup> � � <sup>þ</sup> S Mn<sup>4</sup><sup>þ</sup> � � � S Fe<sup>3</sup><sup>þ</sup> � � ¼ 0*:*65 � 2 þ 0*:*33 � 1*,* 5 � 0*,* 02 � 2*:*5 ¼ 1*:*3 þ 0*:*495 � 0*:*05 ¼ 1*:*745;

<sup>¼</sup> <sup>2</sup>*:*<sup>8175</sup> � <sup>10</sup>�<sup>23</sup> joules for magnetic ion in our sample. This value explains the strength interaction between spins. Moreover, it is a crucial parameter used in the

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 Jð Þ, the gyromagnetic factor g, the number of spins N of our sample, the saturation magnetization *M*0, 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 RCP has 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

1 zMnS Sð Þþ þ1 zMn�NdJMn�Ndð Þþ JMn�Ndþ1 zNdJ Jð Þ þ1 3 j k

<sup>2</sup> � <sup>1</sup> <sup>6</sup>�1*:*745�2*:*745þ4�5*:*6�6*:*6þ6�3*:*855�4*:*<sup>855</sup>

3 h i

between Mn-Mn and Mn-Nd or Nd-Nd and Nd-Mn.

where zMn�Mn ¼ 6, zMn�Nd ¼ 4, and zNd�Nd ¼ 6

JNd <sup>¼</sup> <sup>3</sup>*:*855; JMn�Nd <sup>¼</sup> <sup>5</sup>*:*66. So, J <sup>¼</sup> <sup>3</sup>�1*:*3807�10�23�<sup>131</sup>

<sup>J</sup> <sup>¼</sup> 3kBTc 2 �

*DOI: http://dx.doi.org/10.5772/intechopen.82559*

simulation with the Monte Carlo method.

should be accordable with other models.

**3. Conclusion**

**23**

#### **Figure 10.**

*Experimental and theoretical magnetic entropy change* �*ΔSM versus T of Nd0.67Ba0.33Mn0.98Fe0.02O3 sample as a function of temperature upon different magnetic field intervals (ΔH).*


#### **Table 2.**

*Comparative between <sup>δ</sup>TFWHM,* �*ΔSMmax, and RCP calculated graphically using Maxwell relation and mean field theory.*

(black symbols) when η ¼ 0*:*32 and T0 ¼ 131 K. Thus, the second-order phase transition of this compound is reconfirmed with the η parameter value (η , 1).

**Figure 10** shows simulated �ΔSM versus T curves (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 H and M as an internal and external variable in Eq. (8a) and vice versa in Eq. (8b), �ΔSM estimated 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 <sup>δ</sup>TFWHM, the maximum magnetic entropy change �ΔSMmax, and the relative cooling power RCP <sup>ð</sup> ) which is the product of �ΔSMmax and δTFWHM. These magnetocaloric properties are listed in **Table 2**.

As shown in **Table 2**, a rising of �ΔSMmax obtained 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�<sup>1</sup> .K�<sup>1</sup> under 5 T applied field. Although δTFWHM determined by this method seems less, RCP values 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 <sup>¼</sup> 3kBTc 2zJ Jð Þ <sup>þ</sup><sup>1</sup> (Eq. (26)) allows us to find the Heisenberg exchange constant J. In the perovskite structure of Nd0.67Ba0.33Mn0.98Fe0.02O3 compound, the Mn ion placed at the center of the pseudo-cubic cell has four near neighbors Nd distant from <sup>a</sup> ffiffi 3 p <sup>2</sup> and six near

*Modeling the Magnetocaloric Effect of Nd0.67Ba0.33Mn0.98 Fe0.02O3 by the Mean Field Theory DOI: http://dx.doi.org/10.5772/intechopen.82559*

neighbors Mn distant from a and similarly for Nd. The interaction is established between Mn-Mn and Mn-Nd or Nd-Nd and Nd-Mn.

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

$$\mathbf{J} = \frac{\mathbf{3k\_B T\_c}}{\mathbf{2}} \times \frac{\mathbf{1}}{\left\lfloor \frac{\mathbf{z\_{Mn}S(S+1) + z\_{Mn-Nd}\mathbf{J}\_{Mn-Nd}(\mathbf{J}\_{Mn-Nd}+\mathbf{1}) + z\_{Nd}\mathbf{J}(\mathbf{J}+\mathbf{1})}{\mathbf{3}} \right\rfloor}}$$

where zMn�Mn ¼ 6, zMn�Nd ¼ 4, and zNd�Nd ¼ 6

$$\begin{aligned} \mathbf{S(Mn,Fe)} &= \mathbf{S(Mn^{3+})} + \mathbf{S(Mn^{4+})} - \mathbf{S(Fe^{3+})}\\ = \mathbf{0.65} \times \mathbf{2} + \mathbf{0.33} \times \mathbf{1}, \mathbf{5} - \mathbf{0}, \mathbf{02} \times \mathbf{2.5} &= \mathbf{1.3} + \mathbf{0.495} - \mathbf{0.05} = \mathbf{1.745}; \end{aligned}$$

JNd <sup>¼</sup> <sup>3</sup>*:*855; JMn�Nd <sup>¼</sup> <sup>5</sup>*:*66. So, J <sup>¼</sup> <sup>3</sup>�1*:*3807�10�23�<sup>131</sup> <sup>2</sup> � <sup>1</sup> <sup>6</sup>�1*:*745�2*:*745þ4�5*:*6�6*:*6þ6�3*:*855�4*:*<sup>855</sup> 3 h i

<sup>¼</sup> <sup>2</sup>*:*<sup>8175</sup> � <sup>10</sup>�<sup>23</sup> joules for 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.
