Critical exponents

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 $\alpha =0.1096(5)$ for the three dimensional Ising model, and $\alpha =-0.0146(8)$ for the three-dimensional XY universality class. Experimentally $\alpha =0.1105_{-0.027}^{+0.025}$.

Magnetic order parameter exponent: $\beta$

The magnetic order parameter, $m$ is given by

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

Theoretically one has $\beta =0.32653(10)$ for the three dimensional Ising model, and $\beta =0.3485(2)$ 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 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 = 1.2373(2)} 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)} 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 $\nu =0.63012(16)$ 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)} for the three-dimensional XY universality class.

Rushbrooke equality

The Rushbrooke equality , proposed by Essam and Fisher (Eq. 38 ) 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

$\left.\right.C_{p}\sim \kappa _{T}\sim (T-T_{c})^{-\gamma }\sim (p-p_{c})^{-\gamma }$

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.

See also

• Universality classes