Critical exponents: Difference between revisions

Revision as of 13:04, 26 November 2009

Reduced distance: $\epsilon$

$\epsilon$ is the reduced distance from the critical temperature, i.e.

\epsilon =\left|1-{\frac {T}{T_{c}}}\right|

Note that this implies a certain symmetry when the critical point is approached from either 'above' or 'below', which is not necessarily the case.

Heat capacity exponent: $\alpha$

The isochoric heat capacity is given by $C_{v}$

\left.C_{v}\right.=C_{0}\epsilon ^{-\alpha }

Theoretically one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0.1096(5)} ^[1] for the three dimensional Ising model, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = -0.0146(8)} ^[2] for the three-dimensional XY universality class. Experimentally Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0.1105^{+0.025}_{-0.027}} ^[3].

Magnetic order parameter exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta}

The magnetic order parameter, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is given by

\left.m\right.=m_{0}\epsilon ^{\beta }

Theoretically one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta =0.32653(10)} ^[1] for the three dimensional Ising model, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = 0.3485(2)} ^[2] for the three-dimensional XY universality class.

Susceptibility exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma}

Susceptibility

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \chi \right. = \chi_0 \epsilon^{-\gamma}}

Theoretically one has $\gamma =1.2373(2)$ ^[1] for the three dimensional Ising model, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = 1.3177(5)} ^[2] for the three-dimensional XY universality class.

Correlation length

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \xi \right.= \xi_0 \epsilon^{-\nu}}

Theoretically one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu = 0.63012(16)} ^[1] for the three dimensional Ising model, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu = 0.67155(27)} ^[2] for the three-dimensional XY universality class.

Rushbrooke equality

The Rushbrooke equality ^[4] , proposed by Essam and Fisher (Eq. 38 ^[5]) is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha + 2\beta + \gamma =2} .

Using the above-mentioned values one has:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.1096 + (2\times0.32653) + 1.2373 = 1.99996}

Gamma divergence

When approaching the critical point along the critical isochore (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T > T_c} ) the divergence is of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \right. C_p \sim \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is 1.0 for the Van der Waals equation of state, and is usually 1.2 to 1.3.

Epsilon divergence

When approaching the critical point along the critical isotherm the divergence is of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \right. \kappa_T \sim (p-p_c)^{-\epsilon}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} is 2/3 for the Van der Waals equation of state, and is usually 0.75 to 0.8.

References

[Campostrini2002-1] 1.0 ^1.1 ^1.2 ^1.3 Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E 65 066127 (2002)

[Campostrini2001-2] 2.0 ^2.1 ^2.2 ^2.3 Massimo Campostrini, Martin Hasenbusch, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "Critical behavior of the three-dimensional XY universality class" Physical Review B 63 214503 (2001)

[3] A. Haupt and J. Straub "Evaluation of the isochoric heat capacity measurements at the critical isochore of SF6 performed during the German Spacelab Mission D-2", Physical Review E 59 pp. 1795-1802 (1999)

[4] G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)

[5] John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)

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[5]

@@ Line 29: / Line 29: @@
 :<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math>
-Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref>  for the three dimensional Ising model,  and <math>\nu = .671 55(27)</math><ref name="Campostrini2001"> </ref>  for the three-dimensional XY universality class.
+Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref>  for the three dimensional Ising model,  and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref>  for the three-dimensional XY universality class.
 ==Rushbrooke equality==
 The Rushbrooke equality <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics  39, 842-843 (1963)]</ref> , proposed by Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics  38, 802-812 (1963)]</ref>) is given by

Critical exponents: Difference between revisions

Revision as of 13:04, 26 November 2009

Contents

Reduced distance: $\epsilon$

Heat capacity exponent: $\alpha$

Magnetic order parameter exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta}

Susceptibility exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma}

Correlation length

Rushbrooke equality

Gamma divergence

Epsilon divergence

See also

References

Navigation menu

Critical exponents: Difference between revisions

Revision as of 13:04, 26 November 2009

Reduced distance: ϵ {\displaystyle \epsilon }

Heat capacity exponent: α {\displaystyle \alpha }

Magnetic order parameter exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta}

Susceptibility exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma}

Correlation length

Rushbrooke equality

Gamma divergence

Epsilon divergence

See also

References

Navigation menu

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Reduced distance: $\epsilon$

Heat capacity exponent: $\alpha$