plasmapy.formulary.quantum.chemical_potential(n_e: Unit("1 / m3"), T: Unit("K")) -> Unit(dimensionless)

Calculate the ideal chemical potential.


beta_mu – The dimensionless ideal chemical potential. That is the ratio of the ideal chemical potential to the thermal energy.

Return type:



~astropy.units.UnitsWarning – If units are not provided, SI units are assumed.


The ideal chemical potential is given by [1]:

\[\chi_a = I_{1/2}(\beta \mu_a^{ideal})\]

where \(\chi\) is the degeneracy parameter, \(I_{1/2}\) is the Fermi integral with order 1/2, \(\beta\) is the inverse thermal energy \(\beta = 1/(k_B T)\), and \(\mu_a^{ideal}\) is the ideal chemical potential.

The definition for the ideal chemical potential is implicit, so it must be obtained numerically by solving for the Fermi integral for values of chemical potential approaching the degeneracy parameter. Since values returned from the Fermi_integral are complex, a nonlinear Levenberg-Marquardt least squares method is used to iteratively approach a value of \(\mu\) which minimizes \(I_{1/2}(\beta \mu_a^{ideal}) - \chi_a\)

This function returns \(\beta \mu^{ideal}\) the dimensionless ideal chemical potential.

Warning: at present this function is limited to relatively small arguments due to limitations in the mpmath package’s implementation of polylog, which PlasmaPy uses in calculating the Fermi integral.


[1]Bonitz, Michael. Quantum kinetic theory. Stuttgart: Teubner, 1998.


>>> from astropy import units as u
>>> chemical_potential(n_e=1e21***-3,T=11000*u.K)  # doctest: +SKIP
<Quantity 2.00039985e-12>