Equation of State near the Critical Point. I. Calculation of the Scaling Function forS=12andS=∞Heisenberg Models Using High-Temperature Series Expansions

Abstract

In recent years there have been many measurements of the scaling-law equation of state for different materials, and the "scaling function" so obtained has generally been fit by an empirical equation involving the selection of several adjustable parameters. We propose a method for calculating, directly from high-temperature series expansions, the function $h(x)$ that determines the scaling-law equation of state $H={M}^{\ensuremath{\delta}}h(x)$. Previously, $h(x)$ has been calculated only for the $S=\frac{1}{2}$ Ising model, but the method is not generalizable to the case of the Heisenberg model because it relies upon the use of low-temperature expansions as well, and these are not known for the Heisenberg model. First we calculate $h(x)$ for the Ising model (bcc, fcc, and simple cubic lattices) in order to assess the utility and credibility of our method. Our Ising model $h(x)$ agrees well with the previous calculation that used both high-and low-temperature expansions. Next we calculate $h(x)$ in its entire region of definition for the $S=\frac{1}{2}$ Heisenberg model (fcc and bcc lattices) and the $S=\ensuremath{\infty}$ Heisenberg models (fcc lattice), where $S$ denotes the spin quantum number. The accuracy of our resulting expressions is limited by the finite number of known terms in the corresponding high-temperature series expansions, but it is generally of the order of a few percent. In Paper II the scaling functions calculated here are compared with experiment and with the predictions of the universality hypothesis.