Electron Properties#

Below we show the equations used for the calculation of electron gas properties.

Fermi Integral#

The Fermi integral \(\mathcal F_{p}\) is defined as

\[\mathcal F_p [\eta] = \frac{1}{\Gamma( p + 1) } \mathcal I_{p} [\eta] = \frac{1}{\Gamma(p + 1) } \int_0^{\infty} dx \frac{x^p}{1 + e^{x - \eta} },\]

\[\Gamma (s) = \int_0^{\infty} dx x^{s - 1} e^{-x} dx.\]

The Fermi-Dirac integral satisfies the relation

\[\frac{d}{dx} \mathcal F_{p}(x) = \mathcal F_{p - 1}(x),\]

and here are some useful values of \(\Gamma(x)\)

\[\Gamma \left(- \frac{5}{2} \right ) = -\frac{8\sqrt{\pi}}{15}, \quad \Gamma \left( \frac {5}{2} \right ) = \frac{3\sqrt{\pi} }{4},\]

\[\Gamma \left( - \frac{3}{2} \right ) = \frac{4\sqrt{\pi}}{3}, \quad \Gamma \left ( \frac {3}{2} \right ) = \frac{ \sqrt{\pi} }{2},\]

\[\Gamma \left (- \frac{1}{2} \right ) = - 2 \sqrt{\pi}, \quad \Gamma \left ( \frac {1}{2} \right ) = \sqrt{\pi}.\]

Thermodynamics Electron Gas#

For future reference we define some of thermodynamics quantities of the unpolarized paramagnetic electron gas with dimensionless chemical potential \(\eta = \beta \mu\) and spin degeneracy \(g = 2\).

de Broglie wavelength#

The de Broglie wavelength is given by

\[\Lambda_e = \sqrt{\frac{ 2\pi \hbar^2 \beta}{m_e}}.\]

Grand Potential#

The grand thermodynamic potential \(\Omega\) is given by

\[\Omega = - \frac{g \beta^{-1} }{\Lambda^3} \int d^3r \mathcal F_{3/2}[\eta] = - \frac 43 \frac{g \beta^{-1} }{\sqrt{\pi} \Lambda^3} \int d^3r \mathcal I_{3/2}\left [ \eta \right ]\]

Number of Particles#

The total number of electrons \(N_e\) is

\[N_e = \frac{g}{\Lambda^3} \int d^3r \mathcal F_{1/2}[\eta] = \frac{g}{\Lambda^3} \frac{2}{\sqrt{\pi} } \int d^3r \mathcal I_{1/2}[\eta]\]

Pressure#

The pressure is given by the thermodynamic formula \(PV = - \Omega\)

\[PV = \frac{g\beta^{-1}}{\Lambda^3} \int d^3r \mathcal F_{3/2}[\eta] = \frac{4}{3 \sqrt{\pi} } \frac{g\beta^{-1} }{\Lambda^3} \int d^3r\mathcal I_{3/2}[\eta]\]

Internal Energy#

The internal energy is given by

\[E = \frac{3}{2} \frac{g \beta^{-1} }{\Lambda^3} \int d^3r \mathcal F_{3/2}[\eta] = \frac{2}{\sqrt{\pi} } \frac{g}{\beta\Lambda^3} \int d^3r \mathcal I_{3/2}[\eta]\]

which is equal to \(E = 3/2 PV\).

Free Energy#

The Free energy is

\[F = \frac{g }{\beta \Lambda^3} \int d^3r \left ( \eta \mathcal F_{1/2}[\eta] - \mathcal F_{3/2}[\eta] \right ) = \frac{2g}{ \beta \sqrt{\pi} \Lambda^3} \int d^3r \left ( \eta \mathcal I_{1/2}[\eta] - \frac 23 \mathcal I_{3/2}[\eta] \right ).\]

Notice that \(F\) is a functional of the density \(N/V\) not of \(n(\mathbf{r})\), because that is integrated out.

Entropy#

The entropy is

\[S = \frac{g}{\Lambda^3} \int d^3r \left (\frac 52 \mathcal F_{3/2}[\eta] - \eta \mathcal F_{1/2}[\eta] \right ) = \frac{2 g}{\sqrt{\pi} \Lambda^3 } \int d^3r \left ( \frac 53 \mathcal I_{3/2}[\eta] - \eta \mathcal I_{1/2}[\eta] \right ).\]

Dimensionless Parameters#

Coupling parameters#

\[\Gamma_e = \frac{\bar{e}^2\beta}{a_e}, \qquad r_s = a_e/a_0\]

where \(a_e\) is the Wigner-Seitz radius of the electron gas and \(a_0 = \hbar^2/m_e \bar{e}^2\) is the Bohr radius.

Fermi Energy#

The Fermi energy of a non-interacting electron gas is calculated from the Fermi wave number \(k_F = (3 \pi^2 n)^{1/3}\)

\[E_{\textrm F} = \frac{\hbar^2k_F^2}{2m_e} = \frac{\Lambda^2}{\beta} \left( \frac{3\sqrt{\pi}}{8} n \right )^{2/3}.\]

Degeneracy Parameter#

The above equation leads immediately to the degeneracy parameter

\[\Theta = \frac{k_BT}{E_{\textrm{F}}} = \left( \frac{3 \sqrt{\pi}}{8} \frac{n}{\Lambda^3} \right )^{-2/3} = \left( \frac 32 \mathcal I_{1/2}[\eta] \right )^{-2/3}.\]

Relativistic Parameter#

Relativistic effect are given by

\[x_F = \frac{\hbar k_F}{ m_e c}\]

Warm Dense Matter Parameter#

\[\mathcal W(n_e, \beta) = \mathcal S(\Gamma_e) \mathcal S(\Theta)\]

where \(\mathcal S(x) = 2/(1/x + x)\).

Landau Length#

This is given by

\[l_{\textrm{L}} = 4\pi \bar{e}^2 \beta.\]

Thomas-Fermi Wavelength#