Theory of critical point scattering and correlations. III. The Ising model belowTcand in a field

Abstract

Power-series expansions of the spin-pair correlation functions of the square, sc, and bcc Ising lattices have been obtained using the semi-invariant approach for general field $H$ and temperature $T$. In three dimensions Páde-approximant analyses indicate $2v^{\prime }=1.285\mathrm{}\pm 0.020$, in agreement with scaling, but $2v^c\simeq 0.835\pm 0.02$ which is some (1-3)% above the scaling prediction $v^c=\frac{v}{\beta \delta }\simeq 0.823$. However, ratio techniques reveal that this discrepancy can be attributed to significant confluent critical singularities. Cubic and quintic parametric representations of the critical equation of state, and corresponding expressions for the correlation length, ${\xi }_1(H, T)$, are developed, which are considerable improvements over the linear model. The universality and spherical symmetry of the critical scattering intensity $\hat{\chi }(\overrightarrow{\mathrm{k}}, H, T)$ is confirmed to within (1-2)% by estimating suitable invariant combinations of amplitudes. The deviations from Ornstein-Zernike theory for general $H$ and $T$ are found to be considerably greater than for $H=0\mathrm{}$, $T>\mathrm{}{T}_c$. Complete parametric scaling representations of $\hat{\chi }(\overrightarrow{\mathrm{k}}, H, T)$ are developed for three dimensions; corresponding scaling approximants are constructed in two dimensions but only for $T=T_c$ and for $T\gtrless T_c$ with $H=0\mathrm{}$.