plasmapy.formulary.ionization.ionization_balance(n: Quantity, T_e: Quantity) Quantity[source]

Return the average ionization state of ions in a plasma assuming that the numbers of ions in each state are equal.

Z_bal is the estimate average ionization level of a plasma in thermal equilibrium that balances the number density of ions in two different ionization states. Z_bal is derived from the Saha equation with the assumptions that the atoms are of a single species, are either hydrogenic or completely ionized, and that there is a balance between ionization and recombination, meaning that the number of atoms in either state are equal. The Saha equation and therefore Z_bal are more accurate when the plasma is at a high density and temperature.

\[Z\_bal = \sqrt{\frac{k_B T_e}{E_H}} \sqrt{\ln{\frac{1}{4 n a_{0}^3} (\frac{k_B T_e}{π E_H})^{3/2}}} - \frac{1}{2}\]

Where \(k_B\) is the Boltzmann constant, \(a_0\) is the Bohr radius, and \(E_H\) is the ionization energy of Hydrogen

Aliases: Z_bal_

  • T_e (Quantity) – The electron temperature.

  • n (Quantity) – The electron number density of the plasma.


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



>>> import astropy.units as u
>>> T_e = 5000 * u.K
>>> n = 1e19 * u.m**-3
>>> ionization_balance(n, T_e)
<Quantity 0.274...>
>>> T_e = 50 * u.eV
>>> n = 1e10 * u.m**-3
>>> ionization_balance(n, T_e)
<Quantity 12.615...>

Z – The average ionization state of the ions in the plasma assuming that the numbers of ions in each state are equal.

Return type: