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

Abstract

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 $\kappa \mathrm{}\left(T\right)\mathrm{}$, the true inverse range of exponential decay of the correlations, and ${\kappa }_1\mathrm{}\left(T\right)\mathrm{}$, the effective range determined from the low-angle scattering intensity. Ten to eleven terms of appropriate high-temperature series exapnsions for $\kappa$ and ${\kappa }_1$ 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 $\kappa$ and ${\kappa }_1$ vary as ${(T-T_0)}^{\nu }$ when $T\to T_c$, with $\nu =1\mathrm{}$ for dimensionality $d=2\mathrm{}$, but $\nu =0.6430\mathrm{}\pm 0.0025\simeq \frac{9}{14}$ for $d=3\mathrm{}$. The asymptotic decay of correlation at $T=T_c$ is found to be $\frac{1}{r^{d-2+\eta }}$, where $\eta$ is related to the exponent $\gamma$ of the divergence of the susceptibility by $(2-\eta )\nu =\gamma$, Numerical values are $\eta =\frac{1}{4}$ for $d=2\mathrm{}$ and $\eta =0.056\mathrm{}\pm 0.008\simeq \frac{1}{18} \mathrm{for} d=3\mathrm{}$. The relative scattering intensity $\hat{\chi }$ as a function of wave number k is given to high accuracy for all $T>~T_c$ by $\hat{\chi }(\mathbf{k},T)\simeq \left(\frac{a}{r_1}\right)\frac{{}_{}{}^{2-\eta }{}_{}{}^{}[{\left({\kappa }_{1}a\right)}^2+{\phi }^2{a}^2{K}^2(\mathbf{k}\left)\right]_{}^{\frac{\eta }{2}}{}_{}{}^{}}{[{\left({\kappa }_{1}a\right)}^2+\psi a^2{K}^2(\mathbf{k}\left)\right]},$ where (i) $a$ is the lattice spacing, (ii) $a^2{K}^2=2d[1-q^{-1\mathrm{}}\Sigma \exp \left(i\mathbf{k}·\mathbf{r}\right)]\simeq {\mathrm{}\left(\mathrm{ka}\right)\mathrm{}}^2$, the sum being over the $q$ nearest-neighbor lattice sites, (iii) $r_1\mathrm{}\left(T\right)\mathrm{}$ is a slowly-varying decreasing function near $T_c$, (iv) $\psi =1+\frac{1}{2}\eta {\phi }^2$, and $\left(\mathrm{v}\right) \phi \mathrm{}\left(T\right)\mathrm{}$ is slowly varying with a magnitude at $T_c$ of 0.03 for $d=2\mathrm{}$ and of 0.06 to 0.09 for $d=3\mathrm{}$. Explicit formulas are given for ${\kappa }_1$, $r_1$, and $\phi$ as functions of $T$. The correlations and the scattering are isotropic near $T_c$. The critical scattering isotherm is curved for low $k$ according to ${\hat{\chi }}^{-1\mathrm{}}\sim k^{2-\eta }$ and it intersects the isotherms for $T>\mathrm{}{T}_c$. Correspondingly, $\hat{\chi }(\mathbf{k},T)$ exhibits a maximum for fixed k, at a temperature above $T_c$; for $d=2\mathrm{}$ the maxima are very well marked, but for $d=3\mathrm{}$ they are smaller and occur closer to $T_c$. The theory is compared favorably with recent neutron-scattering experiments on pure beta-brass.