The scaling equation of state for a generalized classical Heisenberg ferromagnet near the critical point is derived by an expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where
d is the dimension of space. It is shown that, though infrared divergences are induced by the Goldstone modes, the equation of state is divergence free. The results are compared with previous numerical calculations. It is also shown that, for non-Ising-like systems the "linear model" cannot be exact, even at first order in $\ensuremath{\epsilon}$ (although the numerical deviations from linearity are small).
The critical exponents $\ensuremath{\gamma}$, $\ensuremath{\eta}$ and the "crossover index" $\ensuremath{\phi}$ are computed for generalized classical Heisenberg models with
n internal degrees of freedom as an exact expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ (
d is the number of space dimensions). Results are obtained to order ${\ensuremath{\epsilon}}^{2}$ for $\ensuremath{\gamma}$ and to order ${\ensuremath{\epsilon}}^{3}$ for $\ensuremath{\eta}$. The results to this order for the three-dimensional Ising case ($n=\ensuremath{\epsilon}=1$) are $\ensuremath{\gamma}=1.244$ and $\ensuremath{\eta}=0.037$.
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=21 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=21 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.
The scaling equation of state for ferromagnets with both isotropic exchange and dipolar interactions near the critical point is derived to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where
d is the dimensionality of space and is also equal to the number of spin components. The resulting scaling function is found to be rather close numerically to its pure short-range counterpart at $\ensuremath{\epsilon}=1$, although the functional forms are quite different. Several universal quantities are derived; in particular, the ratio of the amplitudes of the zero-field specific heats above and below
Tc is found to be $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=\frac{6}{5}+O(\ensuremath{\epsilon})$, compared with $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=1+O(\ensuremath{\epsilon})$ for pure short-range interactions.