Partial Test of the Universality Hypothesis: The Case of Different Coupling Strengths in Different Lattice Directions

High-temperature series expansions are used to examine the dependence of critical-point exponents upon lattice anisotropy (different interaction strengths in different directions in the lattice). The two-spin correlation function C₂(→r) is calculated to tenth order in (1kBT) for the Ising Hamiltonian ${H}l anis={-}{J}xy{Σ}{xy}{〈ij〉}{S}ᵢᶻ{S}ⱼᶻ{-}{J}z{Σ}{z}{〈ij〉}{S}ᵢᶻ{S}ⱼᶻ$ for a wide range of anisotropy parameters $R{≡}{{J}z}{{J}xy}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${{{→}}{r}}ᵢⱼ$ has no $z$ component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for $R=0$, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R{→}{∞}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}₂({{→}}{r})$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent γ and the correlation length exponent ν) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).