chemical_potential

plasmapy.formulary.quantum.chemical_potential(n_e: Quantity, T: Quantity) → Quantity

Calculate the ideal chemical potential.

Parameters:

• n_e (Quantity) – Electron number density.

• T (Quantity) – The temperature.

Returns:

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

Return type:

Quantity

Raises:

• TypeError – If argument is not a Quantity.

• UnitConversionError – If argument is in incorrect units.

• ValueError – If argument contains invalid values.

Warns:

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

Notes

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 \(β μ^{ideal}\).

Examples

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