The zero-field two-point correlation function of an
n-vector system in
d dimensions is calculated to order
n1 for $T\ensuremath{\ge}{T}_{c}$ and
2<d<4. The critical-point 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 variation of the Fourier transform of the correlation function is ${\stackrel{^}{E}}_{1}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{\stackrel{^}{E}}_{2}(q)t+{\stackrel{^}{E}}_{3}(q){t}^{2}+\ensuremath{\cdots}$, where $\ensuremath{\alpha}$ is the specific-heat exponent and ${\stackrel{^}{E}}_{2}$, ${\stackrel{^}{E}}_{3}$ are of order
n1. At
d=3 the term
t2 is modified by logarithmic corrections. As $q\ensuremath{\rightarrow}0$ at fixed nonzero
t, deviations of the scaling function from the Ornstein-Zernike form ${(1+{\ensuremath{\xi}}^{2}{q}^{2})}^{\ensuremath{-}1}$ are of order $\frac{{10}^{\ensuremath{-}3}{\ensuremath{\xi}}^{4}{q}^{4}}{n}$. Results are compared with high-temperature series-expansion estimates and with the $\ensuremath{\epsilon}$-expansion results of previous work.
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 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.
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 dispersion-theory approach provides an exact representation of the two-point correlation function $G(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},t)=C{t}^{\ensuremath{-}\ensuremath{\gamma}}g({q}^{2}{\ensuremath{\xi}}^{2})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and $\ensuremath{\xi}$ is the correlation length, which describes the critical scattering near a second-order phase transition. The threshold property of the spectral weight function $F(\frac{x}{3})\ensuremath{\propto}\mathrm{Im}{g}^{\ensuremath{-}1}(\ensuremath{-}{x}^{2})$ is used to motivate an approximation to the scaling function
g(x2) based on the asymptotically exact Fisher-Langer approximant ${g}_{\mathrm{FL}}({x}^{2})=(\frac{{C}_{1}}{{x}^{2\ensuremath{-}\ensuremath{\eta}}})(1+\frac{{C}_{2}}{{x}^{\frac{(1\ensuremath{-}\ensuremath{\alpha})}{\ensuremath{\nu}}}}+\frac{{C}_{3}}{{x}^{\frac{1}{\ensuremath{\nu}}}})$, where $\ensuremath{\alpha}$, $\ensuremath{\eta}$, $\ensuremath{\nu}$ are the usual critical exponents. The approximation consists of truncating the spectral function associated with ${g}_{\mathrm{FL}}^{\ensuremath{-}1}({x}^{2})$ in a manner designed to simulate the known threshold property of the exact spectral function, namely
F(3x)=0 for $x\ensuremath{\le}3$. The new approximant is checked on the two-dimensional Ising model and shown to agree with the exact result to better than 0.03% for all
x. Near four dimensions, the agreement with exact results is also excellent. A phenomenological approach to the intermediate three-dimensional case is presented, and shown to be in good agreement with high-temperature series and $\ensuremath{\epsilon}$-expansion results. The dispersion-theory approach is also shown to provide a convenient framework for calculations within the $\ensuremath{\epsilon}$ expansion and is used to compute
g(x2) to $O({\ensuremath{\epsilon}}^{3})$, a new result.
The spin-spin correlation functions and the critical-scattering intensity for Heisenberg models of general spin,
S=21 to $\ensuremath{\infty}$, 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 $\ensuremath{\gamma}={1.375}_{\ensuremath{-}0.01}^{+0.02}$, $2\ensuremath{\nu}={1.405}_{\ensuremath{-}0.01}^{+0.02}$, and $\ensuremath{\eta}=0.043\ifmmode\pm\else\textpm\fi{}0.014$ describe all lattices and all spin. Explicit formulas are presented for the susceptibility/zero-angle scattering ${\ensuremath{\chi}}_{0}(T)$, for the inverse correlation length ${\ensuremath{\kappa}}_{1}(T)$, for the effective interaction range
r1(T), and using the Fisher-Burford approximant, for the total scattering $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)$. The shape parameter ${\ensuremath{\varphi}}_{c}$ attains the "universal" value ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 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 $\frac{{\ensuremath{\kappa}}_{1}({T}_{max})}{k}\ensuremath{\simeq}0.10 (\mathrm{for} S\ensuremath{\gtrsim}2)$ to 0.15. A detailed study is made of the structure dependence of the critical-point correlations ${〈{S}_{0}^{z}{S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉}_{c}$ for various models. This leads to the revised, universal estimate ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 15 for all three cubic lattice, spin-\textonehalf{} Ising models. The results are compared d briefly with various experiments which support $\ensuremath{\eta}\ensuremath{\gtrsim}0.05$.