Critical exponents: Difference between revisions
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[[ | ==Reduced distance: <math>\epsilon</math>== | ||
<math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e. | |||
:<math>\epsilon = \left| 1 -\frac{T}{T_c}\right|</math> | |||
Note that this implies a certain symmetry when the [[Critical points|critical point]] is approached from either 'above' or 'below', which is not necessarily the case. | |||
==Heat capacity exponent: <math>\alpha</math>== | |||
The [[heat capacity]] is given by <math>C</math> | |||
:<math>\left. C\right.=C_0 \epsilon^{-\alpha}</math> | :<math>\left. C\right.=C_0 \epsilon^{-\alpha}</math> | ||
Magnetic order parameter, | |||
==Magnetic order parameter exponent: <math>\beta</math>== | |||
The magnetic order parameter, <math>m</math> is given by | |||
:<math>\left. m\right. = m_0 \epsilon^\beta</math> | :<math>\left. m\right. = m_0 \epsilon^\beta</math> | ||
==Susceptibility exponent: <math>\gamma</math>== | |||
[[Susceptibility]] | [[Susceptibility]] | ||
:<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math> | :<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math> | ||
Correlation length | ==Correlation length== | ||
:<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math> | :<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math> | ||
==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 | |||
:<math>\alpha + 2\beta + \gamma =2</math> | :<math>\alpha + 2\beta + \gamma =2</math> | ||
==Gamma divergence== | |||
When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form | When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form | ||
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where <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state]], and is usually 1.2 to 1.3. | where <math>\gamma</math> 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 | When approaching the critical point along the critical isotherm the divergence is of the form | ||
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Reduced distance: 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}
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 the reduced distance from the critical temperature, i.e.
- 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 = \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: 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}
The heat capacity 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 C}
- 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. C\right.=C_0 \epsilon^{-\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}
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
- 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. m\right. = m_0 \epsilon^\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}
- 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}}
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}}
Rushbrooke equality
The Rushbrooke equality [1] , proposed by Essam and Fisher (Eq. 38 [2]) 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}
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 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.
See also
References
- ↑ G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)
- ↑ 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)