Theory of Critical-Point Scattering and Correlations. I. The Ising Model

The theory of the correlations and critical scattering of two- and three-dimensional nearest-neighbor Ising models is discussed critically. A distinction is drawn between (T), the true inverse range of exponential decay of the correlations, and ₁ (T), the effective range determined from the low-angle scattering intensity. Ten to eleven terms of appropriate high-temperature series exapnsions for and ₁ are determined for the square and simple cubic lattices, and shorter series are given for the triangular, fcc, and bcc lattices. For the former lattices, the complete correlation expansions are obtained to the same order. It is shown that and ₁ vary as (T-{T₀) }^ when TT₂, with =1 for dimensionality d=2, but =0. 64300. 0025914 for d=3. The asymptotic decay of correlation at T=T₂ is found to be 1{r^d-2+}, where is related to the exponent of the divergence of the susceptibility by (2-) =, Numerical values are =14 for d=2 and =0. 0560. 008118 for d=3. The relative scattering intensity ^ as a function of wave number k is given to high accuracy for all T>~T₂ by ^ (k, T) (a{r₁}) ^2{- ({₁a) }^2+^2a^2K^2 (k) ^{2}}{ ({₁a) }^2+a^2K^2 (k) }, where (i) a is the lattice spacing, (ii) a^2K^2=2d1-q^-1 (ik) (ka) ^2, the sum being over the q nearest-neighbor lattice sites, (iii) r₁ (T) is a slowly-varying decreasing function near T₂, (iv) =1+12^2, and (v) (T) is slowly varying with a magnitude at T₂ of 0. 03 for d=2 and of 0. 06 to 0. 09 for d=3. Explicit formulas are given for ₁, r₁, and as functions of T. The correlations and the scattering are isotropic near T₂. The critical scattering isotherm is curved for low k according to ^{}^-1k^2- and it intersects the isotherms for T>T₂. Correspondingly, ^ (k, T) exhibits a maximum for fixed k, at a temperature above T₂; for d=2 the maxima are very well marked, but for d=3 they are smaller and occur closer to T₂. The theory is compared favorably with recent neutron-scattering experiments on pure beta-brass.