Formula Free Electron Gas in 3d Fermi temperature Charge carrier density
$$T_{\text F} ~=~ \frac{\hbar^2}{2\class{brown}{m} \, k_{\text B}} \, (3\pi^2 \, n)^{2/3}$$ $$T_{\text F} ~=~ \frac{\hbar^2}{2\class{brown}{m} \, k_{\text B}} \, (3\pi^2 \, n)^{2/3}$$ $$n ~=~ \frac{1}{3\pi^2} \, \left( \frac{ 2\class{brown}{m} \, k_{\text B} \, T_{\text F} }{ \hbar^2 } \right)^{3/2}$$ $$\class{brown}{m} ~=~ \frac{\hbar^2}{2 k_{\text B} \, T_{\text F}} \, (3\pi^2 \, n)^{2/3}$$
Fermi temperature
$$ T_{\text F} $$ Unit $$ \mathrm{K} $$ Fermi temperature is used to compare Fermi energy with thermal energy. Typical value is \( 50 \, 000 \, \text{K} \), which is well above the melting temperature of most elements. The Fermi temperature is related to the Fermi energy via the Boltzmann constant: \( T_{\text F} = \frac{E_{\text F}}{k_{\text B}}\).
Charge carrier density
$$ n $$ Unit $$ \frac{1}{\mathrm{m}^3} $$ Charge carrier density is the number \(N\) of charges per volume \(V\): \( n = N/V \). Since the free Fermi gas is mostly used to describe the free electrons, \(n\) gives the electron density.
Mass
$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$ Mass of a particle of the Fermi gas. This can be for example the (effective) mass of the electron.
Reduced Planck's constant
$$ \hbar $$ Unit $$ \mathrm{Js} $$ Action quantum is a physical constant (of quantum mechanics) and has the value: \( \hbar ~=~ \frac{h}{2\pi} ~=~ 1.054 \, 571 \, 817 \,\cdot\, 10^{-34} \, \text{Js} \).
Boltzmann Constant
$$ k_{\text B} $$ Unit $$ \frac{\mathrm J}{\mathrm K} = \frac{\mathrm{kg} \,\mathrm{m}^2}{\mathrm{s}^2 \, \mathrm{K}} $$ This physical constant is often used in statistical physics and thermodynamics. It has the value: \( k_{\text B} ~\approx~ 1.380 \,\cdot\, 10^{-23} \, \frac{\text J}{\text K} \).