Saha ionization equation

The Saha ionization equation, also known as the Saha–Langmuir equation, is an expression that relates the ionization state of an element to the temperature and pressure. The equation is a result of combining ideas of quantum mechanics and statistical mechanics and is used to explain the spectral classification of stars. The expression was developed by the Indian astrophysicist Meghnad Saha in 1920, and later (1923) by Irving Langmuir.

For a gas at a high enough temperature, the thermal collisions of the atoms will ionize some of the atoms. One or more of the electrons that are normally bound to the atom in orbits around the atomic nucleus will be ejected from the atom and will form an electron gas that co-exists with the gas of atomic ions and neutral atoms. This state of matter is called a plasma. The Saha equation describes the degree of ionization of this plasma as a function of the temperature, density, and ionization energies of the atoms. The Saha equation only holds for weakly ionized plasmas for which the Debye length is large. This means that the screening of the Coulomb interaction of ions and electrons by other ions and electrons is negligible. The subsequent lowering of the ionization potentials and the "cutoff" of the partition function is therefore also negligible.

For a gas composed of a single atomic species, the Saha equation is written:

where:

The expression ${\displaystyle (\epsilon _{i+1}-\epsilon _{i})}$ is the energy required to remove the ${\displaystyle (i+1)^{th}}$ electron. In the case where only one level of ionization is important, we have ${\displaystyle n_{1}=n_{e}}$ and defining the total density n  as ${\displaystyle n=n_{0}+n_{1}}$, the Saha equation simplifies to:

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