# Magnetocaloric Properties and Critical Behaviour of the Sm\({}_{2}\)Ni\({}_{17}\) Compound
## 3 Results and Discussion
### Magnetocaloric Properties of Sm\({}_{2}\)Ni\({}_{17}\)  
When a magnetic material is under an applied magnetic field, its temperature is affected, this phenomenon is called the MCE. Calculating the change in magnetic entropy  
Figure 3: Magnetization versus \(\mu_{0}H\) (0-5 T) with temperatures ranging from 120 to 170 K for Sm\({}_{2}\)Ni\({}_{17}\). Figure 2: Variation in \(M\) versus \(T\) at 0.1 T for the Sm\({}_{2}\)Ni\({}_{17}\) alloy. The inset is the \(dM/dT\) versus \(T\) plot. (\(\Delta S_{\text{M}}\)) is important to study the magnetocaloric effect because it provides a quantitative measure of the effect's magnitude in a given material, which is essential for optimizing the design and performance of magnetocaloric devices for refrigeration and energy conversion applications. The integration of the Maxwell relation is used to calculate the change in magnetic entropy that occurs as a result of the application of a magnetic field:  
\[\Delta S_{\text{M}}=\mu_{0}\,\int_{0}^{H}\,\left(\frac{\partial M}{\partial T} \right)_{H}dH\]  
From the experimental measurements \(M(H)\) (Figure 3), we derive \(\Delta S_{\text{M}}\) versus temperature using the following expression [55]:  
\[|\Delta S_{\text{M}}|=\mu_{0}\underset{i}{\sum}\frac{M_{i}-M_{i+1}}{T_{i+1}-T_ {i}}\,\Delta H_{i}\]  
where \(M_{i+1}\) and \(M_{i}\) refer to the magnetization at \(T_{i+1}\) and \(T_{i}\), respectively under an increased \(\Delta H_{i}\). Figure 4 shows the magnetic entropy variation as a function of temperature and the applied magnetic field, especially near the FM-PM transition. It is obvious that the maximum \(\Delta S_{\text{M}}\) occurs near the Curie temperature, with the peak value heavily reliant on the strength of the magnetic field. Although the maximum \(\Delta S_{\text{M}}\) is moderate for the studied compounds, the width at mid-height is relatively large \(\sim\)45 K for \(\mu_{0}H=5\) T.  
The graphical method commonly used to determine magnetocaloric parameters, such as the magnetic entropy maximum (\(\Delta S_{\text{M}}^{max}\)), width at half-maximum (\(\delta T^{FWHM}\)), and temperature-averaged entropy change (\(TEC\)), are generally not precise enough. Therefore, in the following section, we use a phenomenological model that allows for more accurate estimates of these parameters. In particular, this model allows for the prediction of a material's magnetocaloric effect from a single thermomagnetic measurement at low-applied magnetic fields, thus enhancing its utility. A phenomenological model is used to simulate the relationship between magnetization and temperature. This simulation is performed assuming adiabatic conditions, under an applied magnetic field, and the model is tuned to closely match the experimental data. Figure 4: \(\Delta S_{\text{M}}\) versus temperature for Sm\({}_{2}\)Ni\({}_{17}\) with different applied \(\mu_{0}H\). \(|\Delta S_{\text{M}}|\) curves are deduced from magnetic isotherms using the Maxwell relation. The relationship between magnetization, temperature changes, and \(T_{\rm C}\) can be described as follows [56]:  
\[M(T)=\frac{(M_{F}-M_{P})}{2}\tanh[\alpha(T_{\rm C}-T)]+T.\frac{dM}{dT}|_{T<T_{ \rm C}}+\beta \tag{1}\]  
The parameters \(\alpha\) and \(\beta\) can be determined using the following equations:  
\[\alpha=\frac{2(\frac{dM}{dT}|_{T<T_{\rm C}}-\frac{dM}{dT}|_{T_{\rm C}})}{M_{F} -M_{P}}\]  
\[\beta=\frac{M_{F}+M_{P}}{2}-\frac{dM}{dT}|_{T<T_{\rm C}}.T_{\rm C}\]  
where M\({}_{F}\) represents the initial magnetization value around \(T_{\rm C}\) and M\({}_{P}\) represents the final magnetization value at the same transition. In Figure 5, it is evident that the magnetization variations in Sm\({}_{2}\)Ni\({}_{17}\), as modelled and measured against temperature, show strong agreement. The experimental data are represented by (\(\ast\)) in the figure, while the simulated magnetization is shown by the red line.