Scaled Equation of State and Critical Exponents in Magnets and Fluids

Abstract

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 $h(x)=H{M}^{\ensuremath{-}1}{|M|}^{1\ensuremath{-}\ensuremath{\delta}}$: $h(x)={E}_{1}[\frac{(x+{x}_{0})}{{x}_{0}}]{{1+{E}_{2}{[\frac{(x+{x}_{0})}{{x}_{0}}]}^{2\ensuremath{\beta}}}}^{\frac{[\ensuremath{\beta}(\ensuremath{\delta}\ensuremath{-}1)\ensuremath{-}1]}{2\ensuremath{\beta}}}$ where $h(x)$ is a scaling function, $x=t{|M|}^{\ensuremath{-}\frac{1}{\ensuremath{\beta}}}$, $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, and $x=\ensuremath{-}{x}_{0}$ is the phase boundary. A nonlinear, least-squares method was used to simultaneously determine the six parameters ($\ensuremath{\beta}$, $\ensuremath{\delta}$, ${T}_{c}$, ${x}_{0}$, ${E}_{1}$, ${E}_{2}$). Agreement between the proposed form for $h(x)$ and the experimental data was found. For both the magnets and fluids, we find that $\ensuremath{\delta}\ensuremath{\approx}4.4$; for the magnets $\ensuremath{\beta}\ensuremath{\approx}0.37$ and for the fluids $\ensuremath{\beta}\ensuremath{\approx}0.35$. 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.