A one-parameter scaling function
s(x) of $x=k\ensuremath{\xi}$, where $\ensuremath{\xi}$ is the correlation length and
k the wave number of a density or concentration fluctuations in a fluid, is derived from a phenomenological field theoretic approach. On the basis of a positive-definite spectral function, we exploit the requirements of a three-particle threshold and the correct asymptotic approach as $x\ensuremath{\rightarrow}\ensuremath{\infty}$. The resulting
s(x) is compared with the Fisher-Burford approximant obtained from numerical studies of the three-dimensional Ising model.
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 P\'ade-approximant analyses indicate $2{v}^{\ensuremath{'}}=1.285\ifmmode\pm\else\textpm\fi{}0.020$, in agreement with scaling, but $2{v}^{c}\ensuremath{\simeq}0.835\ifmmode\pm\else\textpm\fi{}0.02$ which is some (1-3)% above the scaling prediction ${v}^{c}=\frac{v}{\ensuremath{\beta}\ensuremath{\delta}}\ensuremath{\simeq}0.823$. 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, ${\ensuremath{\xi}}_{1}(H, T)$, are developed, which are considerable improvements over the linear model. The universality and spherical symmetry of the critical scattering intensity $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ 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,
T>Tc. Complete parametric scaling representations of $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ are developed for three dimensions; corresponding scaling approximants are constructed in two dimensions but only for
T=Tc and for $T\ensuremath{\gtrless}{T}_{c}$ with
H=0.
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 $\ensuremath{\kappa}(T)$, the true inverse range of exponential decay of the correlations, and ${\ensuremath{\kappa}}_{1}(T)$, the effective range determined from the low-angle scattering intensity. Ten to eleven terms of appropriate high-temperature series exapnsions for $\ensuremath{\kappa}$ and ${\ensuremath{\kappa}}_{1}$ 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 $\ensuremath{\kappa}$ and ${\ensuremath{\kappa}}_{1}$ vary as ${(T\ensuremath{-}{T}_{0})}^{\ensuremath{\nu}}$ when $T\ensuremath{\rightarrow}{T}_{c}$, with $\ensuremath{\nu}=1$ for dimensionality
d=2, but $\ensuremath{\nu}=0.6430\ifmmode\pm\else\textpm\fi{}0.0025\ensuremath{\simeq}\frac{9}{14}$ for
d=3. The asymptotic decay of correlation at
T=Tc is found to be $\frac{1}{{r}^{d\ensuremath{-}2+\ensuremath{\eta}}}$, where $\ensuremath{\eta}$ is related to the exponent $\ensuremath{\gamma}$ of the divergence of the susceptibility by $(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}=\ensuremath{\gamma}$, Numerical values are $\ensuremath{\eta}=\frac{1}{4}$ for
d=2 and $\ensuremath{\eta}=0.056\ifmmode\pm\else\textpm\fi{}0.008\ensuremath{\simeq}\frac{1}{18} \mathrm{for} d=3$. The relative scattering intensity $\stackrel{^}{\ensuremath{\chi}}$ as a function of wave number k is given to high accuracy for all
T> Tc by $\stackrel{^}{\ensuremath{\chi}}(\mathbf{k},T)\ensuremath{\simeq}\left(\frac{a}{{r}_{1}}\right)\frac{^{2\ensuremath{-}\ensuremath{\eta}}[{({\ensuremath{\kappa}}_{1}a)}^{2}+{\ensuremath{\varphi}}^{2}{a}^{2}{K}^{2}(\mathbf{k})]^{\frac{\ensuremath{\eta}}{2}}}{[{({\ensuremath{\kappa}}_{1}a)}^{2}+\ensuremath{\psi}{a}^{2}{K}^{2}(\mathbf{k})]},$ where (i)
a is the lattice spacing, (ii) ${a}^{2}{K}^{2}=2d[1\ensuremath{-}{q}^{\ensuremath{-}1}\ensuremath{\Sigma}\mathrm{exp}(i\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{r})]\ensuremath{\simeq}{(\mathrm{ka})}^{2}$, the sum being over the
q nearest-neighbor lattice sites, (iii)
r1(T) is a slowly-varying decreasing function near
Tc, (iv) $\ensuremath{\psi}=1+\frac{1}{2}\ensuremath{\eta}{\ensuremath{\varphi}}^{2}$, and $(\mathrm{v}) \ensuremath{\varphi}(T)$ is slowly varying with a magnitude at
Tc of 0.03 for
d=2 and of 0.06 to 0.09 for
d=3. Explicit formulas are given for ${\ensuremath{\kappa}}_{1}$,
r1, and $\ensuremath{\varphi}$ as functions of
T. The correlations and the scattering are isotropic near
Tc. The critical scattering isotherm is curved for low
k according to ${\stackrel{^}{\ensuremath{\chi}}}^{\ensuremath{-}1}\ensuremath{\sim}{k}^{2\ensuremath{-}\ensuremath{\eta}}$ and it intersects the isotherms for
T>Tc. Correspondingly, $\stackrel{^}{\ensuremath{\chi}}(\mathbf{k},T)$ exhibits a maximum for fixed k, at a temperature above
Tc; for
d=2 the maxima are very well marked, but for
d=3 they are smaller and occur closer to
Tc. The theory is compared favorably with recent neutron-scattering experiments on pure beta-brass.
In the scaling limit $k\ensuremath{\rightarrow}0$, $\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $y=k\ensuremath{\xi}$ is fixed, the
k-dependent susceptibility $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)$ can, according to the scaling hypotheses of Kadanoff and Fisher, be written as $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)={\ensuremath{\xi}}^{\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}}X_{\ifmmode\pm\else\textpm\fi{}}(y)+o({\ensuremath{\xi}}^{\frac{\ensuremath{\gamma}}{\ensuremath{\nu}}})$. We exactly compute the scale functions $X_{\ifmmode\pm\else\textpm\fi{}}(y)$ for the two-dimensional Ising model in zero magnetic field. We then compare the various phenomenological scale functions (Ornstein-Zernike pole approximate, Fisher approximate, Fisher-Burford approximate, Tarko-Fisher approximates, etc.) with the exact $X_{\ifmmode\pm\else\textpm\fi{}}(y)$ for the two-dimensional Ising model. This comparison provides insight into those regions of $y=k\ensuremath{\xi}$ where these phenomenological scale functions are applicable. Such insight is important since the region of experimentally accessible
y is rather limited. We then use these results to examine the method of data analysis used in critical scattering experiments. We conclude that no experiment to date unambiguously and directly establishes that the critical exponent $\ensuremath{\eta}$ is greater than zero.
The zero-field, two-point correlation function of an
n-vector system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions is calculated to order ${\ensuremath{\epsilon}}^{2}$ for
T> Tc. The scaling function is obtained as a closed, cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector
q, the leading variation is ${{\stackrel{^}{E}}_{1}}^{n,d}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{{\stackrel{^}{E}}_{2}}^{n,d}(q)t$, where $\ensuremath{\alpha}$ is the specific-heat exponent; thence the maximum in the scattering above
Tc is located, in good agreement with high-
T series-expansion estimates.
A new approximant is proposed for the critical correlation function
g(y) of $y=k\ensuremath{\xi}$, where $\ensuremath{\xi}$ is the correlation length and
k the wave number of an order-parameter fluctuation. The approach involves truncating the spectral function associated with the three-term Fisher-Langer approximant in a manner suggested by the requirement of a three-particle threshold. The resulting
g(y) is accurate to better than 0.03% everywhere for the two-dimensional Ising model. Results for the three-dimensional case are consistent with $\ensuremath{\epsilon}$-expansion and high-temperature series results.