Equation of State near the Critical Point. II. Comparison with Experiment and Possible Universality with Respect to Lattice Structure and Spin Quantum Number

Abstract

The scaling functions for the $S=\frac{1}{2}$ Ising, $S=\frac{1}{2}$ Heisenberg, and $S=\infty$ Heisenberg models are compared with experimental measurements of $h\left(x\right)\mathrm{}$ on the insulating ferromagnets Cr${\mathrm{Br}}_3$ ($S=\frac{3}{2}$) and EuO ($S=\frac{7}{2}$), with the metallic ferromagnet Ni and with the conducting alloy ${\mathrm{Pd}}_3$Fe (in both disordered and ordered states). The data agree much better with our Heisenberg model $h\left(x\right)\mathrm{}$ than with the Ising model $h\left(x\right)\mathrm{}$. Comparison with experimental data is made by using plots of scaled magnetization $M_H\equiv \frac{M}{H^{\frac{1}{\delta }}}$ vs scaled temperature ${\epsilon }_H\equiv \frac{\epsilon }{H^{\frac{1}{\beta \delta }}}$, which has the virtue that all points fall upon a single curve and log-log plots need not be resorted to. We proceed to consider the extent to which the equation of state depends upon parameters appearing in the Hamiltonian (the "universality" question). We find that our calculations indicate that $h\left(x\right)\mathrm{}$, if properly normalized, does not depend on lattice structure and is very likely independent of spin quantum number $S$. Moreover, the fact that our calculated $h\left(x\right)\mathrm{}$ agrees with data on Cr${\mathrm{Br}}_3$ (for which there exists considerable lattice anisotropy) and on EuO (for which there exists nonnegligible next-nearest-neighbor interactions) suggests the further conjecture that $h\left(x\right)\mathrm{}$ might be independent of these features of the Hamiltonian. By contrast, the change in $h\left(x\right)\mathrm{}$ on going from Ising coupling ($D=1\mathrm{}$) to Heisenberg coupling ($D=3\mathrm{}$) was quite substantial, and indeed would seem to account for the fact that earlier workers did not obtain agreement between the Ising model $h\left(x\right)\mathrm{}$ and experimental data on magnetic systems. We conclude with the working hypothesis that the scaling function depends principally on spin dimensionality $D$ and on lattice dimensionality $d$.