Two new inequalities, (i) ${\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremath{\ge}\frac{2\ensuremath{\beta}(2\ensuremath{-}\ensuremath{\eta})}{(d\ensuremath{-}2+\ensuremath{\eta})}+2\ensuremath{\varphi}[\frac{2\ensuremath{\beta}}{(d\ensuremath{-}2+\ensuremath{\eta})}\ensuremath{-}{\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}]$ and (ii) $\frac{(\ensuremath{\delta}\ensuremath{-}1)}{\ensuremath{\delta}}\ensuremath{\ge}\frac{2(2\ensuremath{-}\ensuremath{\eta})}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}+2\ensuremath{\varphi}[\frac{2}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}\ensuremath{-}{\ensuremath{\mu}}_{\ensuremath{\varphi}}]$, are derived among critical-point exponents that describe the behavior of the two-spin correlation function ${C}_{2}(T, H, \stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})\ensuremath{\equiv}〈{s}_{0}^{z}{s}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉\ensuremath{-}〈{s}_{0}^{z}〉〈{s}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉$, subject to plausible assumptions (rigorous for Ising magnets). Here ${\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}$ and ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$ describe the divergence as $T\ensuremath{\rightarrow}{T}_{c}^{\ensuremath{-}}$ and as $H\ensuremath{\rightarrow}{0}^{+}$, respectively, of the "generalized correlation length" ${\ensuremath{\xi}}_{\ensuremath{\varphi}}(T, H)$, defined as the $2\ensuremath{\varphi}\mathrm{th}$ root of the normalized $2\ensuremath{\varphi}\mathrm{th}$ spatial moment of ${C}_{2}(T, H, \stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. Also derived are the corresponding inequalities among exponents that describe the behavior of the energy-energy correlation function. Inequality (i) is shown to lead to an inequality between primed and unprimed exponents. Moreover, if ${\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}$ is independent of $\ensuremath{\varphi}$, then (i) implies that ${\ensuremath{\nu}}^{\ensuremath{'}}\ensuremath{\ge}\frac{2\ensuremath{\beta}}{(d\ensuremath{-}2+\ensuremath{\eta})}$ and ${\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremath{\ge}(2\ensuremath{-}\ensuremath{\eta}){\ensuremath{\nu}}^{\ensuremath{'}}$, while if ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$ is independent of $\ensuremath{\varphi}$, then (ii) implies $\ensuremath{\mu}\ensuremath{\ge}\frac{2}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}$ and $\frac{(\ensuremath{\delta}\ensuremath{-}1)}{\ensuremath{\delta}}\ensuremath{\ge}(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\mu}$.
An analysis is made of the fluctuations in a finite part of an infinite,
d-dimensional ferromagnetic Ising lattice at and near the critical point. This leads to a formal proof that $\frac{2d}{(\ensuremath{\delta}+1)}$ and $\frac{2d\ensuremath{\beta}}{({\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta})}$ are lower bounds on the exponent characterizing the asymptotic decrease with distance of the two-body correlation function at the critical point, where $\ensuremath{\beta}$, $\ensuremath{\delta}$, and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ are exponents representing the asymptotic forms of the coexistence curve, critical isotherm, and isothermal susceptibility.
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.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.