Saha equation

The Saha equation (also Saha ionization equation or Eggert-Saha equation ) describes the dependence of the degree of ionization of a gas on the temperature in thermodynamic equilibrium ; If the degree of ionization reaches a significant level, one no longer speaks of a gas, but of a plasma .

The equation was derived in 1920 by the then 27-year-old Indian astrophysicist Meghnad Saha from the Boltzmann statistics and is important for the physics of stars. John Eggert provided preliminary work in 1919 through a publication in the Physikalische Zeitschrift. Saha read this paper in India and was able to improve it significantly.

The Saha equation can be read in such a way that precisely those atoms are ionized for which the thermal energy of the electrons is higher than the ionization energy according to the Boltzmann distribution .

formulation

For pure gases the Saha equation is

${\displaystyle {\frac {n_{\rm {e}}\cdot n_{i+1}}{n_{i}}}={\frac {2}{\lambda ^{3}}}{\frac {Z_{i+1}}{Z_{i}}}\exp \left[-{\frac {(\epsilon _{i+1}-\epsilon _{i})}{k_{\rm {B}}T}}\right]}$

With:

• ${\displaystyle n_{i}\,}$the particle density of the ionized gas (where i is the number of missing electrons = ionization level)
• ${\displaystyle n_{\rm {e}}\,}$the electron density
• ${\displaystyle Z_{i}\,}$the partition function of the atom / ion of the i- th level
• ${\displaystyle \epsilon _{i+1}-\epsilon _{i}\,}$the ionization energy required to remove another electron from an ion (from i to i +1).
• ${\displaystyle \lambda \,}$the thermal wavelength (of an electron) with ${\displaystyle \lambda \equiv {\sqrt {\frac {h^{2}}{2\pi m_{\rm {e}}k_{\rm {B}}T}}}}$
  • ${\displaystyle h\,}$the Planck constant
  • ${\displaystyle m_{\rm {e}}\,}$the mass of an electron (constant)
  • ${\displaystyle k_{\rm {B}}T\,}$of thermal energy
• Alternatively, as the partition function of the free electron are interpreted (the factor 2 represents then the spin - degeneracy of the electron) ${\displaystyle {\frac {2}{\lambda ^{3}}}\,}$ ${\displaystyle Z_{\rm {e}}\,}$
• ${\displaystyle k_{\rm {B}}\,}$the Boltzmann constant
• ${\displaystyle T\,}$the absolute temperature of the gas.

Individual evidence

1. ^ Ionization in the solar chromosphere. Philosophical Magazine Series 6, 40 (1920), No. 238, pp. 472-488