# Magnetocaloric Properties and Critical Behaviour of the Sm\({}_{2}\)Ni\({}_{17}\) Compound
## 3 Results and Discussion
### Critical Phenomenon and Spin Interaction
#### 3.4.2 Kouvel-Fisher Plot  
We have utilized the Kouvel-Fisher (KF) technique as our next approach in order to derive more precise sets of \(T_{C}\), \(\gamma\), and \(\beta\) in the scaling analysis. The functions employed in this method are described by the following two equations [68]:  
\[M_{S}(T)\left(\frac{dM_{S}(T)}{dT}\right)^{-1}=\frac{T-T_{C}}{\beta} \tag{6}\]  
\[\chi_{0}^{-1}(T)\left(\frac{d\chi_{0}^{-1}(T)}{dT}\right)^{-1}=\frac{T-T_{C}}{\gamma} \tag{7}\]  
By applying the KF approach, \(M_{S}(T)\left(\frac{dM_{S}(T)}{dT}\right)^{-1}\) and \(\chi_{0}^{-1}(T)\left(\frac{d\chi_{0}^{-1}(T)}{dT}\right)^{-1}\) exhibit linear behaviour as a function of \(T\), with slopes \(1/\beta\) and \(1/\gamma\), respectively. The linear fitting shown in Figure 10 using Equation (6) provides \(T_{C}\) = 160.5(2) K and \(\beta\) = 0.25(1). In the same way, Equation (7) gives \(T_{C}\) = 159.7(3) K and \(\gamma\) = 1.39(6), which is in agreement with those obtained from the MAP. In Table 2, we have summarized the critical exponents of Sm\({}_{2}\)Ni\({}_{17}\) and those corresponding to the well-known universality classes. One can see that \(T_{C}\) and the critical exponent's values derived using the KF method are in agreement with those derived utilizing the MAP. This implies that the obtained values are consistent and unequivocal. The final critical exponents, obtained through the KF method, were used to plot the final Arrott plots shown in Figure 11.  
Figure 9: **(left)** The spontaneous magnetization versus temperature and (right) the inverse of initial susceptibility versus temperature. The solid lines represent the fit. The resulting plot revealed a series of parallel lines in the strong field region and confirmed the critical isotherm passing through the origin. These observations provide visual confirmation of the validity of the critical exponents. However, to further verify the accuracy of the \(\beta\) and \(\gamma\) values, a method based on the scaling equation is employed at the end of this section. The determination of the third critical exponent \(\delta\) involves critical isotherm analysis, requiring the magnetization \(M(\mu_{0}H)\) at \(T_{\rm C}=160\) K as \(ln(M)\) versus \(ln(\mu_{0}H)\) to be plotted, as illustrated in Figure 12. The equation relating \(M\) and \(\mu_{0}H\) at \(T=T_{\rm C}\) is [69]:  
\[ln(M)=ln(D)+\frac{1}{\delta}ln(\mu_{0}H) \tag{8}\]  
where \(D\) is the amplitude and the exponent \(\delta\) describes the curvature of \(M(\mu_{0}H)\) at \(T_{\rm C}\). Based on Equation (8), a linear line with a slope of \(1/\delta\) can be derived through linear regression of \(ln(M)\) as a function of \(ln(H)\) at the critical temperature. As depicted in the inset of Figure 12, our analysis using this method yielded a value of \(\delta=6.64(2)\). By applying the Widom scaling equation (\(\delta=\frac{T}{\beta}+1\)) [70], with \(\beta\) and \(\gamma\) derived from the KF and MAP plot, \(\delta=6.50(4)\) and \(6.36(6)\) was obtained, respectively. These values are consistent with the critical isothermal analysis, demonstrating that the critical exponents obtained are both precise and dependable within the bounds of experimental precision. Figure 11: The final MAP using critical exponents, obtained through the KF method. Figure 10: The Kouvel–Fisher (KF) plot displays the temperature-dependent behaviour of spontaneous magnetization \(M_{s}(T,0)\) and the inverse of susceptibility \(\chi_{0}^{-1}(T,0)\), plotted against temperature. Finally, the two methods give values close to \(\delta\) justifying the validity of this value and confirming that the FM-PM phase transition does not belong to any universality class. It is crucial to check if the obtained \(T_{\rm C}\) and critical exponents produce a scaling equation of state for the system.