plasmapy.formulary.ionization.Saha(g_j, g_k, n_e: Unit("1 / m3"), E_jk: Unit("J"), T_e: Unit("K")) -> Unit(dimensionless)

The Saha equation, derived in statistical mechanics, gives an approximation of the ratio of population of ions in two different ionization states in a plasma. This approximation applies to plasmas in thermodynamic equilibrium where ionization and recombination of ions with electrons are balanced.

\[\frac{N_j}{N_k} = \frac{1}{n_e} \frac{g_j}{4 g_k a_0^{3}} \left( \frac{k_B T_e}{\pi E_H} \right)^{\frac{3}{2}} \exp\left( \frac{-E_{jk}}{k_B T_e} \right)\]

Where \(k_B\) is the Boltzmann constant, \(a_0\) is the Bohr radius, \(E_H\) is the ionization energy of Hydrogen, \(N_j\) and \(N_k\) are the population of ions in the j and k states respectively. This function is equivalent to Eq. 3.47 in Drake.

  • T_e (Quantity) – The electron temperature.
  • g_j (int) – The degeneracy of the ‘j’th ionization state.
  • g_k (int) – The degeneracy of the ‘k’th ionization state.
  • E_jk (Quantity) – The energy difference between ionization states j and k.
  • n_e (Quantity) – The electron number density of the plasma.

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

  • TypeError – The T_e, E_jk, or n_e is not a Quantity and cannot be converted into a ~astropy.units.Quantity.
  • UnitConversionError – If the T_e, E_jk, or n not in appropriate units.

ratio – The ratio of population of ions in ionization state j to state k.

Return type:



>>> import astropy.units as u
>>> T_e = 5000 * u.K
>>> n = 1e19 * u.m ** -3
>>> g_j = 2
>>> g_k = 2
>>> E_jk = 1 * u.Ry
>>> Saha(g_j, g_k, n, E_jk, T_e)
<Quantity 3.299...e-06>
>>> T_e = 1 * u.Ry
>>> n = 1e23 * u.m ** -3
>>> Saha(g_j, g_k, n, E_jk, T_e)
<Quantity 1114595.586...>


For reference to this function and for more information regarding the Saha equation, see chapter 3 of R Paul Drake’s book, “High-Energy-Density Physics: Foundation of Inertial Fusion and Experimental Astrophysics” (DOI: 10.1007/978-3-319-67711-8_3).