Magnetic refrigeration is gaining increasing attention due to its potential to address environmental concerns and energy efficiency in refrigeration technology. The necessity for magnetic refrigeration arises from the limitations and environmental impact of conventional refrigeration methods, particularly those based on vapor compression cycles and refrigerants with high global warming potential [1] - [3] . Conventional refrigeration systems contribute significantly to greenhouse gas emissions through the use of hydrofluorocarbon refrigerants, which are known contributors to climate change. Additionally, these systems often suffer from energy inefficiency and require periodic maintenance, leading to increased operating costs [4] - [6] . Magnetic refrigeration offers a sustainable alternative by utilizing the magnetocaloric effect (MCE) in certain materials, which allows for efficient and environmentally friendly cooling without the need for harmful refrigerants. This technology has the potential to significantly reduce energy consumption and greenhouse gas emissions in various applications, including household refrigerators, air conditioning systems, and industrial cooling processes [7] - [9] .

The modelling of the MCE plays a crucial role in advancing the understanding and optimization of magnetocaloric materials for various applications. By simulating the behavior of MCE materials under different magnetic fields, temperatures, and compositions, researchers can predict their thermodynamic properties and performance characteristics with high precision [10] . This predictive capability is essential for designing efficient magnetocaloric cooling systems and optimizing their energy efficiency. Furthermore, simulations enable researchers to explore the underlying mechanisms governing the MCE, such as magnetic phase transitions and lattice dynamics, providing valuable insights into the material’s behavior at the atomic and molecular levels [11] . Additionally, they allow for the rapid screening of potential magnetocaloric candidates, accelerating the discovery of novel materials with enhanced refrigeration capabilities and reduced environmental impact. Moreover, simulation-based approaches facilitate the design of tailored magnetic structures and configurations to optimize the magnetocaloric response, leading to the development of more compact and cost-effective cooling devices. Overall, they are indispensable tools for advancing the field of magnetic refrigeration and realizing its full potential in addressing global energy and environmental challenges.

Particularly, the mean-field model serves as a powerful tool in simulating MCE, offering insights into the thermodynamic behavior of magnetocaloric materials under varying magnetic fields [12] - [14] . In this model, the magnetic interactions within the material are approximated by a mean field, simplifying the complex interactions between individual magnetic moments. By employing the mean-field approach, researchers can predict key thermodynamic properties such as the magnetic entropy change and adiabatic temperature change, which are crucial for evaluating the cooling performance of magnetocaloric materials [15] . Additionally, this model allows for the investigation of phase transitions and critical phenomena associated with the MCE, shedding light on the underlying physics governing these processes [16] . Recent advancements in computational techniques have further enhanced the accuracy and applicability of mean-field simulations, enabling researchers to explore a wide range of magnetocaloric materials and operating conditions [17] . These simulations have contributed to the discovery of novel magnetocaloric compounds and the optimization of existing materials for practical cooling applications [18] . Moreover, mean-field modelling provides valuable guidance for the design and engineering of magnetocaloric devices with enhanced efficiency and reliability, driving forward the development of next-generation magnetic refrigeration technologies. In applications way, manganites play a crucial role in the MCE due to their unique magnetic and structural properties [19] . These materials undergo a magnetostructural transition, leading to significant changes in entropy under the influence of a magnetic field, making them ideal candidates for magnetic refrigeration. Recent studies have investigated the MCE in various manganite compounds, highlighting their potential for efficient cooling applications [20] [21] .

In this work we will apply the mean-field theory to model the MCE in La_0.5Sm_0.2Sr_0.3Mn_1-xFe_xO₃ compounds with x = 0 (LSSMO) and x = 0.05 (LSSMFO). In their work, Khouloud et al. [22] investigated the structural, magnetic, and magnetocaloric properties of La_0.5Sm_0.2Sr_0.3Mn_1-xFe_xO₃ compounds within the range of x from 0 to 0.05. X-ray diffraction analysis reveals the compounds maintain a rhombohedral structure throughout the studied range of x. The substitution of Fe for Mn induces a decrease in the magnetic moment and Curie temperature with increasing Fe content. The compounds exhibit significant magnetocaloric effects, characterized by the maximum magnetic entropy change, which decreases with increasing Fe content.

The Brillouin function, $B_J\left(x\right)$, is commonly used to describe the magnetization M behavior of a ferromagnetic material at finite temperatures T and applied fields H:

$M\left(H,T\right)=M_0{B}_J\left(x\right)$(1)

Where: $M_0$ is the saturation magnetization, $B_J\left(x\right)=\frac{2J+1}{2J}\coth \left(\frac{2J+1}{2J}\frac{Jg{\mu }_B}{k_B}\left(\frac{H+H_{exch}}{T}\right)\right)-\frac{1}{2J}\coth \left(\frac{1}{2J}\frac{Jg{\mu }_B}{k_B}\left(\frac{H+H_{exch}}{T}\right)\right)$ with $x=\frac{Jg{\mu }_B}{k_B}\left(\frac{H+H_{exch}}{T}\right)$, $H_{exch}=\lambda M$ [23] is the exchange magnetic with $\lambda$ is the exchange parameter, J is the total angular momentum, g the Lande factor, ${\mu }_B$ is the Bohr magnetron and $k_B$ is the Boltzmann.

By incorporating the expression $M\left(T,H\right)=f\left(\frac{H+H_{exch}}{T}\right)$ and utilizing the inverse function $f^{-1}$ of f, we derive the following relationship:

$\frac{H}{T}=f^{-1}\left(M\right)-\frac{H_{exch}}{T}$(2)

The magnetic entropy can be expressed by Maxwell relations as ${\left(\frac{\partial S}{\partial H}\right)}_T={\left(\frac{\partial M}{\partial T}\right)}_H$ or ${\left(\frac{\partial S}{\partial M}\right)}_T=-{\left(\frac{\partial H}{\partial T}\right)}_M$. Consequently, between two magnetic fields H₁ and H₂, the magnetic entropy change is given by:

$-\Delta S_M\left(T\right)={\displaystyle {\int }_{{M|}_{H_1}}^{{M|}_{H_2}}\left(f^{-1}\left(M\right)-{\left(\frac{\partial \lambda \left(T\right)}{\partial T}\right)}_{M}M\right)\text{d}M}$(3)

The isotherms $M\left(H,T\right)$ of the LSSMO and LSSMFO compounds were analyzed, and the evolution of $\frac{H}{T}$ versus $\frac{1}{T}$ at constant magnetization values was plotted in Figure 1 . A linear relationship of the plots was observed, with isomagnetics curves shifting towards higher temperature values. Linear fits were applied to these curves to determine the exchange mean-field parameter, obtained from the slopes representing $H_{exch}$.

In the paramagnetic domain or materials with ordered domains like antiferromagnetic, the expansion of M in powers of H or H in powers of M was performed up to the third order, considering the magnetization as an odd function of the field: $H_{exch}={\lambda }_{1}M+{\lambda }_3{M}^3$.

Then, the adjustment of $H_{exch}$ versus M in Figure 2 is carried out. It has been observed a very small dependence on $M^3$ which could be negligeable and only ${\lambda }_1$ may be kept. Thus $H_{exch}={\lambda }_{1}M\approx \lambda M$, with $\lambda =1.63$ and 1.46 T.emu⁻¹.g for LSSMO and LSSMFO samples, respectively.

The next step of this approach involves constructing a scaling plot of M vs. $\frac{H+H_{exch}}{T}$, as depicted in Figure 3 with black symbols. Essentially, these curves, converging into a single curve, are fitted with Equation (1) to ascertain M₀, J, and g. Figure 3 displays certain data points that noticeably deviate from the scaling function. These points correspond to the magnetic domain region, characterized by low fields and temperatures below the Curie temperature. The different adjusted parameters of the LSSMO and LSSMFO are summarized in Table 1 .

The numerical resolution of Equation (1) with the given adjusted parameters λ, J, g and M₀ has been achieved using MATLAB software. Generated (blue lines) and experimental (black symbols) isotherms $M\left(H,T\right)$ are correlated as depicted in Figure 4 .

Figure 5 illustrates the comparison between the mean-field generated $-\Delta S_M\left(T\right)$ curves (red lines) obtained using Equation (3) and the corresponding experimental results (solid symbols) obtained using the Maxwell relation. A clear agreement is observed between the generated and experimental $-\Delta S_M\left(T\right)$ curves.

In fact, the applicability of Maxwell relation may be limited in complex systems where deviations from ideal behavior occur, such as in strongly correlated electron systems or systems with competing magnetic interactions. On the other hand, the mean-field model is a simplified theoretical approach that approximates the behavior of a system by assuming that each magnetic moment interacts with an effective average field. This model is often more tractable and can provide reasonable estimates of magnetic properties in many cases, especially for systems where interactions between magnetic moments are relatively weak and can be treated perturbatively. However, it may fail to capture the effects of fluctuations and correlations that are important in strongly interacting systems.

To assess the efficiency of a refrigerant substance, a proposed metric known as the Temperature-averaged Entropy Change (TEC) is suggested [24] [25] :

$\text{TEC}=\frac{1}{\Delta T_{H-C}}\max \left\{{\displaystyle {\int }_{T_{mid}-\frac{\Delta T_{H-C}}{2}}^{T_{mid}+\frac{\Delta T_{H-C}}{2}}\left|\Delta S_M\right|\text{d}T}\right\}$(4)

Where $\Delta T_{H-C}$ represents the specified range of temperatures(in this study $\Delta T_{H-C}=2,5$ and 10 K) and $T_{mid}$ is the temperature maximizing the integration.

TEC is based on the temperature range within which the material is designed or capable of delivering the maximum isothermal entropy change. This metric aims to quantify the magnitude of the thermodynamic effect across a practical temperature spectrum, thereby indicating the material’s effectiveness in practical applications. Greater performance is achieved with larger TECs at wider $\Delta T_{H-C}$ ranges. Figure 6 illustrates the change in TECs as the field changes for various $\Delta T_{H-C}$ values. The values decrease as the temperature span $\Delta T_{H-C}$ increases across all samples.

The data in Figure 6 shows that TEC values stabilize across different $\Delta T_{H-C}$ (2 K, 5 K, 10 K) as the magnetic field changes, indicating that the magnetic field strength, rather than temperature differences, primarily influences TEC. This stabilization suggests that once the magnetic field reaches a certain strength, its effect on entropy change dominates, becoming less dependent on temperature variations. This phenomenon likely arises from the material’s intrinsic properties, with the magnetic field inducing a state where entropy change due to magnetic effects is predominant. At higher magnetic fields, the TEC values plateau, indicating a saturation effect where further increases in the magnetic field do not significantly alter TEC, making the material less sensitive to temperature differences. The significant TEC values obtained further bolster the suitability of these materials for application in magnetic refrigeration technology. Under 1 T field, TEC(10) = 1 and 0.57 J.kg⁻¹.K⁻¹ for the LSSMO and LSSMFO compounds, respectively. These values are near to the ones of Gd (2.91 J.kg⁻¹.K⁻¹) [26] .

In conclusion, isotherms $M\left(H,T\right)$ of LSSMO and LSSMFO compounds, were analyzed. the linear fits of $\frac{H}{T}$ versus $\frac{1}{T}$ at constant magnetization determine the exchange mean-field $H_{exch}$, which is approximated as $\lambda M$. A scaling plot of M versus $\frac{H+H_{exch}}{T}$ is constructed and fitted to sort out the total angularmomentum J, the Lande factor, g and the saturation magnetization M₀. The comparison between mean-field generated isotherms $M\left(H,T\right)$ and magnetic entropy change $-\Delta S_M\left(T\right)$ curves with experimental results show a strong agreement. A proposed metric, Temperature-averaged Entropy Change (TEC), indicates the materials’ effectiveness in practical applications. The change in TECs with varying fields and specified range of temperatures $\Delta T_{H-C}$ values further support the materials’ suitability for magnetic refrigeration technology.

Experimental data were kindly provided by Abdouli K.