Theory of Critical-Point Scattering and Correlations. II. Heisenberg Models

The spin-spin correlation functions and the critical-scattering intensity for Heisenberg models of general spin, S=1/2 to ∞, on the sc, bcc, and fcc lattices are studied on the basis of high-temperature series expansions along the lines developed in Paper I [M. E. Fisher and R. J. Burford, Phys. Rev. 156, 583 (1967)]. Subject to increased uncertainties for low spin, it is concluded that the exponents γ=1.375_-0.01+0.02, 2ν=1.405_-0.01+0.02, and η=0.043±0.014 describe all lattices and all spin. Explicit formulas are presented for the susceptibility/zero-angle scattering χ₀(T), for the inverse correlation length κ₁(T), for the effective interaction range r₁(T), and using the Fisher-Burford approximant, for the total scattering ^χ(→k,T). The shape parameter φc attains the "universal" value φc0. 11 for large spin but shows signs of spin dependence (and lattice dependence) for low spin. At fixed k the scattering is predicted to display a maximum above Tc determined by κ₁(Tₘₐₓ)k0.10 (for S2) to 0.15. A detailed study is made of the structure dependence of the critical-point correlations 〈S₀ᶻS_→rᶻ〉c for various models. This leads to the revised, universal estimate φc0. 15 for all three cubic lattice, spin-{} Ising models. The results are compared d briefly with various experiments which support η0.05.