plasmapy.formulary.quantum.chemical_potential(n_e: Unit('1 / m3'), T: Unit('K'))

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



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


The ideal chemical potential is implicitly given by Eq. 1.2 in [Bonitz, 1998]:

\[χ = nΛ^{3} = I_{1/2}(β μ^{ideal})\]

where \(χ\) is the degeneracy parameter, \(n\) is the species number density, \(Λ\) is the thermal de Broglie wavelength, \(I_{1/2}\) is the Fermi integral with order 1/2, \(β\) is the inverse thermal energy \(β = 1/(k_B T)\), and \(μ^{ideal}\) 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, the Broyden–Fletcher–Goldfarb–Shanno algorithm is used to iteratively approach a value of \(μ^{ideal}\) which minimizes \(\lvert I_{1/2}(β μ^{ideal}) - χ \rvert\).

This function returns the dimensionless ideal chemical potential \(β μ^{ideal}\).


>>> from astropy import units as u
>>> chemical_potential(n_e=1e25***-3,T=11000*u.K)
<Quantity 283.43506297>