# Magnetocaloric Properties and Critical Behaviour of the Sm\({}_{2}\)Ni\({}_{17}\) Compound
## 3 Results and Discussion
### Critical Phenomenon and Spin Interaction  
#### 3.4.1 Arrott Plot  
The characteristics of the FM-PM transition in this sample were evaluated using the Arrott plots. Figure 7 displays the Arrott plots obtained from the M(\(\mu_{0}\) H) curves by plotting the squared magnetization (\(M^{2}\)) against \(\mu_{0}H/M\). Based on the Banerjee criterion [64], if the Arrott plot slope of the linear curve near the Curie temperature is negative, the FM-PM transition is a first-order magnetic (FOM) transition; conversely, a second-order magnetic transition occurs when the plot has a positive slope. Therefore, the Arrott plots of the Sm\({}_{2}\)Ni\({}_{17}\) alloy imply an SOMT. The Arrott plot method is utilized to derive the critical exponents and the Curie temperature, assuming \(\gamma\) = 1.0 and \(\beta\) = 0.5. This case corresponds to the mean-field model (MFM) critical exponents. The proposed method has been widely explained in the literature [65], involving plotting \(M^{2}\) versus \(\mu_{0}H/M\) in the high-field zone exhibiting linear behaviour, and at T\({}_{C}\), where the isotherm is a line intersecting the origin. In contrast, the high magnetic field curves for the Sm\({}_{2}\)Ni\({}_{17}\) compound shown in Figure 7 do not exhibit parallel behaviour near the transition temperature. This finding indicates that the MFM is inadequate to describe the magnetic phase transition of our compound; therefore, to identify the universality class of the studied compound we use a method based on modified Arrott plots (MAP). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Compound** & \(\mu_{0}\Delta H\) **(T)** & \(\Delta S_{max}\) **J(K-kg)\({}^{-1}\)** & **RCP J-kg\({}^{-1}\)** & **Ref.** \\ \hline Sm\({}_{2}\)Ni\({}_{17}\) & 5 & 0.7 & \(\sim\)37 & This Work \\ Er\({}_{2}\)Ni\({}_{17}\) & 5 & \(\sim\)0.6 & \(\sim\)24 & [39] \\ SmNi\({}_{5}\) & 5 & 5 & 26.85 & [58] \\ Pr\({}_{2}\)Co\({}_{7}\) & 1.5 & 1.1 & 18.5 & [59] \\ Pr\({}_{5}\)Co\({}_{19}\) & 1.5 & 5 & 18.2 & [60] \\ Fe\({}_{65}\)Ni\({}_{35}\) & 3 & 0.35 & \(\sim\)18 & [61] \\ Gd\({}_{2}\)Fe\({}_{2}\) & 1.5 & 0.79 & 13.3 & [62] \\ PrCo\({}_{2}\)Cu\({}_{3}\) & 4 & 0.7 & \(\sim\)128 & [63] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of \(RCP\) and \(\Delta S_{max}\) of Sm\({}_{2}\)Ni\({}_{17}\) and other materials. Figure 6: (**a**) Experimental and simulated magnetic entropy change versus temperature for Sm\({}_{2}\)Ni\({}_{17}\) under several magnetic fields. (**b**) Maximum magnetic entropy change plotted against the applied magnetic field. (**c**) Relative cooling power plotted against applied magnetic field. (**d**) Temperature-averaged entropy change versus applied magnetic field for \(\Delta T_{H-C}\) = 5 K.  
To examine other possible exponent pairs (\(\beta\) and \(\gamma\)) the generalized Arrott equation must be used [66]. To determine the appropriate model to describe this phase transition we will use the \(\gamma\) and \(\beta\) couples of the three-dimensional theoretical model commonly used in the literature, such as the tricritical-MFM (\(\beta\) = 0.25 and \(\gamma\) = 1.0, Figure 8a), the three-dimensional Ising model (\(\beta\) = 0.325 and \(\gamma\) = 1.241, Figure 8b), the three-dimensional Heisenberg model (\(\beta\) = 0.365 and \(\gamma\) = 1.386, Figure 8c) and the three-dimensional XY model (\(\beta\) = 0.345 and \(\gamma\) = 1.316, Figure 8d) to make the MAPs (\(M^{(1/\beta)}\) versus \((\mu_{0}H/M)^{1/\gamma}\)). None of the above universality classes seem suitable to directly describe the critical properties of our compound. Since we are dealing with a problem that has two parameters, we employ a method that involves performing multiple iterations, as described below: In the high field data in the three-dimensional Heisenberg model, the initial values of \(M_{S}(T)\) and \(\chi_{0}^{-1}(T)\) are determined by linear extrapolation, where \(M_{S}(T)\) and \(\chi_{0}^{-1}(T)\) are the \(Y\) and \(X\) intercepts, respectively.