The critical behavior of a system of
n-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. To first order in $\ensuremath{\epsilon}$ the correlation and susceptibility exponents in the isotropic limit are $2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}$, while the anisotropy or crossover exponent is $\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}$. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ these expansions agree with exact spherical-model results. For
n=3 and
d=3 series expansions indicate $\ensuremath{\varphi}\ensuremath{\simeq}1.2$ (compared with $\ensuremath{\gamma}\ensuremath{\simeq}1.38$).
The critical behaviour of an anisotropic n-vector model with purely Ising-like coupling is studied on the FCC lattice by means of high-temperature series. The model includes the s= 1 / 2 (n=1) and s= infinity (n=3) Ising models as special cases and exhibits rigorously mean-field behaviour at n= infinity . The crossover region 1/n to 0, t identical to (1-v/v c ) to 0, is examined in detail. The susceptibility exponent gamma remains Ising-like for all finite n. The critical temperature and critical susceptibility amplitude vary as 1/n.