In physics, the Saha ionization equation is an expression that relates the ionization state of a gas in thermal equilibrium 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 physicist Meghnad Saha in 1920. It is discussed in many textbooks on statistical physics and plasma physics.
Description For a gas at a high enough temperature (here measured in energy units, i.e. keV or J) and/or density, the thermal collisions of the atoms will ionize some of the atoms, making an ionized gas. When several or more of the electrons that are normally bound to the atom in orbits around the atomic nucleus are freed, they form an independent electron gas cloud co-existing with the surrounding gas of atomic ions and neutral atoms. With sufficient ionization, the gas can become the state of matter called plasma. The Saha equation describes the degree of ionization for any gas in thermal equilibrium as a function of the temperature, density, and ionization energies of the atoms. For a gas composed of a single atomic species, the Saha equation is written: n i + 1 n e n i = 2 λ th 3 g i + 1 g i exp [ − ε i + 1 − ε i k B T ] {\displaystyle {\frac {n_{i+1}n_{\text{e}}}{n_{i}}}={\frac {2}{\lambda _{\text{th}}^{3}}}{\frac {g_{i+1}}{g_{i}}}\exp \left[-{\frac {\varepsilon _{i+1}-\varepsilon _{i}}{k_{\text{B}}T}}\right]} where:
n i {\displaystyle n_{i}} is the number density of atoms in the i-th state of ionization, that is with i electrons removed.
g i {\displaystyle g_{i}} is the degeneracy of state for the i-ions.
ε i {\displaystyle \varepsilon _{i}} is the energy required to remove i electrons from a neutral atom, creating an i-level ion.
n e {\displaystyle n_{\text{e}}} is the electron density
k B {\displaystyle k_{\text{B}}} is the Boltzmann constant
λ th {\displaystyle \lambda _{\text{th}}} is the thermal de Broglie wavelength of an electron λ th = d e f h 2 π m e k B T {\displaystyle \lambda _{\text{th}}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {h}{\sqrt {2\pi m_{\text{e}}k_{\text{B}}T}}}}
m e {\displaystyle m_{\text{e}}} is the mass of an electron
T {\displaystyle T} is the temperature of the gas
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