Physical review. B, Condensed matter

Generalized Ornstein-Zernike equation

Gerald L. Jones

Published Nov 1, 1984• 0 citations

Abstract

A previously proposed generalization of the usual Ornstein-Zernike equation for near-critical correlations is examined. It is shown that near the critical point, to leading order, this equation is compatible with realistic critical behavior in both spatial and thermal variables. The scaling predictions of this equation are compared to known results for $d=2$ and $d=3$ Ising models. The generalized equation changes the usual Ornstein-Zernike values of various quantities in qualitatively the correct way; however, quantitative discrepancies remain.

DOI: 10.1103/physrevb.30.5279

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Physical review. B, Condensed matter

Generalized Ornstein-Zernike equation

Gerald L. Jones

Published Nov 1, 1984• 0 citations

Publisher

Abstract

d=2

and

d=3

Ising models. The generalized equation changes the usual Ornstein-Zernike values of various quantities in qualitatively the correct way; however, quantitative discrepancies remain.

DOI: 10.1103/physrevb.30.5279

Related papers

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William J. Camp, Michael Ellis Fisher

The asymptotic decay of the general two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})$ at high temperatures is analyzed on the basis of the $d$ -dimensional, spin-\textonehalf{} Ising model in general field $H$ . For $H\ensuremath{\ne}0$ the Ornstein-Zernike form $\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}$ is found for general $\stackrel{^}{A}$ and $\stackrel{^}{B}$. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as ${H}^{2}$ and in zero field only the higher order decay $\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}$ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.The asymptotic decay of the general two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})$ at high temperatures is analyzed on the basis of the $d$-dimensional, spin-\textonehalf{} Ising model in general field $H$. For $H\ensuremath{\ne}0$ the Ornstein-Zernike form $\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}$ is found for general $\stackrel{^}{A}$ and $\stackrel{^}{B}$. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as ${H}^{2}$ and in zero field only the higher order decay $\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}$ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.

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Critical Scattering Parameters for General Fields and Temperatures

Monique Combescot, Michel Droz, J. Michael Kosterlitz

The critical scattering parameters have been calculated for an Ising-like system in a field, below and above ${T}_{c}$ up to second order in $\ensuremath{\epsilon}(=4\ensuremath{-}d)$. The corrections to the Ornstein-Zernike form in the small- and large-$q\ensuremath{\xi}$ limit are presented and their relevance to experiments is discussed.The critical scattering parameters have been calculated for an Ising-like system in a field, below and above ${T}_{c}$ up to second order in $\ensuremath{\epsilon}(=4\ensuremath{-}d)$. The corrections to the Ornstein-Zernike form in the small- and large-$q\ensuremath{\xi}$ limit are presented and their relevance to experiments is discussed.

Physical Review LettersSep 16, 197411 citations

d=2

and

d=3

Ising models. The generalized equation changes the usual Ornstein-Zernike values of various quantities in qualitatively the correct way; however, quantitative discrepancies remain.

The asymptotic decay of the general two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})$ at high temperatures is analyzed on the basis of the

d

-dimensional, spin-\textonehalf{} Ising model in general field

H

. For $H\ensuremath{\ne}0$ the Ornstein-Zernike form $\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}$ is found for general $\stackrel{^}{A}$ and $\stackrel{^}{B}$. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as

{H}^{2}

and in zero field only the higher order decay $\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}$ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.

{T}_{c}

than above. Numerical comparison with series-expansion results agree surprisingly well. A powerful technique of evaluating diagrams using the Fourier transform of the propagator is also presented.

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>{T}_{c}

. 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={T}_{c}

and for $T\ensuremath{\gtrless}{T}_{c}$ with

H=0

The critical scattering parameters have been calculated for an Ising-like system in a field, below and above

{T}_{c}

up to second order in $\ensuremath{\epsilon}(=4\ensuremath{-}d)$. The corrections to the Ornstein-Zernike form in the small- and large-$q\ensuremath{\xi}$ limit are presented and their relevance to experiments is discussed.