Physical Review Letters

Critical Scattering Parameters for General Fields and Temperatures

Monique Combescot, Michel Droz, J. Michael Kosterlitz

Published Sep 16, 1974• 11 citations

Abstract

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.

DOI: 10.1103/physrevlett.33.705

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Critical Scattering in a Field and Below Tc

Howard B. Tarko, Michael Ellis Fisher

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Two point correlation function for general fields and temperatures in the critical region

Monique Combescot, Michel Droz, J. Michael Kosterlitz

Physical review. B, Solid stateJun 1, 197520 citations

Theory of critical point scattering and correlations. III. The Ising model below Tc and in a field

Howard B. Tarko, Michael Ellis Fisher

Physical review. B, Solid stateFeb 1, 1975237 citations

Generalized Ornstein-Zernike equation

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Examination of the phenomenological scaling functions for critical scattering

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Physical review. B, Solid stateJul 1, 1975120 citations

Behavior of Two-Point Correlation Functions at High Temperatures

William J. Camp, Michael Ellis Fisher

Physical Review LettersJan 11, 197153 citations

Physical Review Letters

Critical Scattering Parameters for General Fields and Temperatures

Monique Combescot, Michel Droz, J. Michael Kosterlitz

Published Sep 16, 1974• 11 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevlett.33.705

Related papers

Critical Scattering in a Field and Below Tc

Howard B. Tarko, Michael Ellis Fisher

Series expansions for spin-spin correlations of simple-cubic, bcc, and square lattice Ising ferromagnets for general field and temperature have been analyzed. To within the attainable precision (of \ifmmode\pm\else\textpm\fi{} 1 to \ifmmode\pm\else\textpm\fi{} 5%) the exponents ${\ensuremath{\nu}}^{\ensuremath{'}}$ and ${\ensuremath{\nu}}^{c}$ are consistent with scaling predictions. Deviations of the scattering from Ornstein-Zernike forms are significantly larger quantitatively than above ${T}_{c}$ ; the second-moment correlation lengths verify $\frac{{{\ensuremath{\xi}}_{1}}^{+}}{{{\ensuremath{\xi}}_{1}}^{\ensuremath{-}}(T)}=1.96\ifmmode\pm\else\textpm\fi{}0.03$ as $T\ensuremath{\rightarrow}{{T}_{c}}^{\ifmmode\pm\else\textpm\fi{}}$.Series expansions for spin-spin correlations of simple-cubic, bcc, and square lattice Ising ferromagnets for general field and temperature have been analyzed. To within the attainable precision (of \ifmmode\pm\else\textpm\fi{} 1 to \ifmmode\pm\else\textpm\fi{} 5%) the exponents ${\ensuremath{\nu}}^{\ensuremath{'}}$ and ${\ensuremath{\nu}}^{c}$ are consistent with scaling predictions. Deviations of the scattering from Ornstein-Zernike forms are significantly larger quantitatively than above ${T}_{c}$; the second-moment correlation lengths verify $\frac{{{\ensuremath{\xi}}_{1}}^{+}}{{{\ensuremath{\xi}}_{1}}^{\ensuremath{-}}(T)}=1.96\ifmmode\pm\else\textpm\fi{}0.03$ as $T\ensuremath{\rightarrow}{{T}_{c}}^{\ifmmode\pm\else\textpm\fi{}}$.

Physical Review LettersOct 8, 197373 citations

Two point correlation function for general fields and temperatures in the critical region

Monique Combescot, Michel Droz, J. Michael Kosterlitz

A detailed calculation, to order ${\ensuremath{\epsilon}}^{2} (\ensuremath{\epsilon}=4\ensuremath{-}d)$, of the two-point-correlation function of an Ising-like system in the whole critical region is presented. The scaling function is shown to be a cut-off-independent function of two variables which is universal in the context of a sharp cut off. Explicit asymptotic expansions in the large and small momentum (relative to the inverse correlation length) are given. Particular attention is paid to the corrections from the Ornstein-Zernike theory. These corrections are two orders of magnitude larger below ${T}_{c}$ 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.A detailed calculation, to order ${\ensuremath{\epsilon}}^{2} (\ensuremath{\epsilon}=4\ensuremath{-}d)$, of the two-point-correlation function of an Ising-like system in the whole critical region is presented. The scaling function is shown to be a cut-off-independent function of two variables which is universal in the context of a sharp cut off. Explicit asymptotic expansions in the large and small momentum (relative to the inverse correlation length) are given. Particular attention is paid to the corrections from the Ornstein-Zernike theory. These corrections are two orders of magnitude larger below ${T}_{c}$ 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.

Physical review. B, Solid stateJun 1, 197520 citations

Theory of critical point scattering and correlations. III. The Ising model below Tc and in a field

Howard B. Tarko, Michael Ellis Fisher

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 $H=0$ .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 $H=0$.

Physical review. B, Solid stateFeb 1, 1975237 citations

Generalized Ornstein-Zernike equation

Gerald L. Jones

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.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.

Physical review. B, Condensed matterNov 1, 19840 citations

Examination of the phenomenological scaling functions for critical scattering

Craig A. Tracy, Barry M. McCoy

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.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.

Physical review. B, Solid stateJul 1, 1975120 citations

Behavior of Two-Point Correlation Functions at High Temperatures

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.

Physical Review LettersJan 11, 197153 citations

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.

{T}_{c}

; the second-moment correlation lengths verify $\frac{{{\ensuremath{\xi}}_{1}}^{+}}{{{\ensuremath{\xi}}_{1}}^{\ensuremath{-}}(T)}=1.96\ifmmode\pm\else\textpm\fi{}0.03$ as $T\ensuremath{\rightarrow}{{T}_{c}}^{\ifmmode\pm\else\textpm\fi{}}$.

{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

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.

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 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.