Scaled Equation of State and Critical Exponents in Magnets and Fluids

A systematic analysis in the context of the scaled equation of state has been made of available experimental data in the critical region of a number of ferromagnets [Gd, Cr${\mathrm{Br}}_3$, ${\mathrm{La}}_{0.5}$ ${\mathrm{Sr}}_{0.5}$Co${\mathrm{O}}_3$, Ni (two independent sets of data)] and fluids (C${\mathrm{O}}_2$, ${\mathrm{He}}^4$, Xe) with the following assumed form for $h\left(x\right)=H{M}^{-1\mathrm{}}{\mathrm{}\left|M\right|\mathrm{}}^{1-\delta }$: $h\left(x\right)=E_1\left[\frac{(x+x_0)}{x_0}\right]{\{1+E_2{\left[\frac{(x+x_0)}{x_0}\right]}^{2\beta }\}}^{\frac{\left[\beta \right(\delta -1)-1]}{2\beta }}$ where $h\left(x\right)\mathrm{}$ is a scaling function, $x=t{\mathrm{}\left|M\right|\mathrm{}}^{-\frac{1}{\beta }}$, $t=\frac{(T-T_c)}{T_c}$, and $x=-x_0$ is the phase boundary. A nonlinear, least-squares method was used to simultaneously determine the six parameters ($\beta$, $\delta$, $T_c$, $x_0$, $E_1$, $E_2$). Agreement between the proposed form for $h\left(x\right)\mathrm{}$ and the experimental data was found. For both the magnets and fluids, we find that $\delta \approx 4.4$; for the magnets $\beta \approx 0.37$ and for the fluids $\beta \approx 0.35$. Reasons for the considerably different values reported elsewhere for these exponents in the materials Cr${\mathrm{O}}_2$ and YFe${\mathrm{O}}_3$ are discussed.