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 low-density linked-cluster expansion of the ferromagnetic Ising model $F(H,T)=\ensuremath{-}mH\ensuremath{-}\frac{1}{2}qJ\ensuremath{-}{k}_{B}T\ensuremath{\Sigma}\stackrel{\ensuremath{\infty}}{l=1}{L}_{l}(u){\ensuremath{\mu}}^{l}$ ($u={e}^{\frac{\ensuremath{-}4J}{{k}_{B}T}},\ensuremath{\mu}={e}^{\frac{\ensuremath{-}2\mathrm{mH}}{{k}_{B}}T}$) is numerically analyzed. We find that all the polynomials
Ll are positive between 0 and ${u}_{c}={e}^{\frac{\ensuremath{-}4J}{{k}_{B}{T}_{c}}}$,
Tc being the critical temperature, and that
uc can be calculated with good accuracy as the limit point of the zeros of these polynomials
Ll. Given
uc, we compute
Ll(uc) and find, numerically, their asymptotic behavior to be ${c}_{0}{l}^{\ensuremath{-}s}$. It is then shown that the singular part of the free energy per spin, as $H\ensuremath{\rightarrow}0$, is given by $F(H,{T}_{c})\ensuremath{\sim}{(\ensuremath{-}\mathrm{ln}\ensuremath{\mu})}^{s\ensuremath{-}1}\ensuremath{\sim}{H}^{s\ensuremath{-}1}$.
s is found to be about 2.061 in two dimensions and about 2.19 in three dimensions. Hence, the critical $M\ensuremath{-}H$ isotherm is found and the critical index $\ensuremath{\delta}$ is calculated in fair agreement with the previous estimates. An analytic expression for the corresponding critical amplitude is also found. The form of
F given above is based on numerical work and is not to be regarded as proven analytically.
Simple but rigorous derivations are presented for the inequalities2−η≤dγ′/(2β+γ′)≤dγ′/(2−α′),2−η≤d(δ−1)/(δ+1),γ≤(2−η)ν.(Proofs of the first two inequalities have been given by Buckingham and Gunton.1) Here d is the dimensionality and α′, β, γ, γ′ and δ are the exponents characterizing a ferromagnet near its critical point,2 while η and ν describe the decay of the corresponding spin-spin correlation functions2 Γ2(r)=〈S0zSrz〉, according to ∑ |r|<RΓ2(r)∼R2−η(T=Tc,H=0,R→∞),κ1∼(T−Tc)ν,(T→Tc,H=0),where the inverse range of correlation κ1 may be defined by κ1−2=[Σrr2Γ2(r)]/[ΣrΓ2(r)]. The corresponding exponent ηE for the energy-energy correlation function (essentially Γ4(r, a)−[Γ(a)]2, where Γ4(r, a)=〈S0zSazSrzSr+az〉) is proved to satisfy 2−ηE≤dα′,2−ηE≤dαc/(1+ζ+αc),where the specific heat CM at T=Tc diverges with the magnetization as M−αc while the energy derivative | ∂U/∂M |Tc varies as Mζ. (Mean Field or classical values are αc=0, ζ=1.) The proofs are based on the ``intuitively obvious'' properties: (a) positivity, namely, Γ1=〈S0z〉≥0,Γ(r)−Γ12≥0,Γ4(r, a)−[Γ2(a)]2≥0, and (b) monotonic increase of Γ1, Γ2 and Γ4 under increase of magnetic field and decrease of temperature. These properties are known3 to be rigorously valid for Ising models of arbitrary spin, lattice structure and ferromagnetic coupling (Jij≥0). Their validity for real magnets together with the experimental observation δ≤4.7 leads to the significant conclusion η≥0.05.
An argument is presented for a certain asymptotic form for the correlation functions of two local dynamical variables when the system is near its critical point and in the one phase region. If
A(r) and
B(r) are local dynamical variables and $\ensuremath{\omega}$ stands for the variables specifying the thermodynamic state, it is found that for $\ensuremath{\omega}$ near the critical point ${\ensuremath{\omega}}_{c}$ and for
r large enough, $〈A(\mathbf{r})B(0)〉\ensuremath{-}〈A(\mathbf{r})〉〈B(0)〉\ensuremath{\simeq}\frac{{d}_{A}(\ensuremath{\omega}){d}_{B}(\ensuremath{\omega}){e}^{\ensuremath{-}\ensuremath{\kappa}(\ensuremath{\omega})}}{{r}^{b(\ensuremath{\omega})}}$, where $\ensuremath{\kappa}(\ensuremath{\omega})$ and $b(\ensuremath{\omega})$ are the same for a large class of
A and
B and $\ensuremath{\kappa}(\ensuremath{\omega})\ensuremath{\rightarrow}0$, $b(\ensuremath{\omega})\ensuremath{\rightarrow}b\ensuremath{\ge}0 \mathrm{as} \ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}$. For the Ising model of any dimension in zero field and below the critical temperature, the additional assumption of a single correlation length implies the relationship ${\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta}=2\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}$, which is often conjectured for the critical exponents.