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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.^[1][2] 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.^[3][4]

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. The Saha equation only holds for weakly ionized plasmas for which the Debye length is small. 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:

$${\displaystyle {\frac {n_{i+1}n_{e}}{n_{i}}}={\frac {2}{\lambda ^{3}}}{\frac {g_{i+1}}{g_{i}}}\exp \left}$$

• ${\displaystyle n_{i}}$ is the density of atoms in the i-th state of ionization, that is with i electrons removed.
• ${\displaystyle g_{i}}$ is the degeneracy of states for the i-ions
• ${\displaystyle \varepsilon _{i}}$ is the energy required to remove i electrons from a neutral atom, creating an i-level ion.
• ${\displaystyle n_{e}}$ is the electron density
• ${\displaystyle \lambda }$ is the thermal de Broglie wavelength of an electron
  $${\displaystyle \lambda \ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {h^{2}}{2\pi m_{e}k_{\text{B}}T}}}}$$

  
• ${\displaystyle m_{e}}$ is the mass of an electron
• ${\displaystyle T}$ is the temperature of the gas
• ${\displaystyle h}$ is Planck's constant

The expression ${\displaystyle (\varepsilon _{i+1}-\varepsilon _{i})}$ 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:

$${\displaystyle {\frac {n_{e}^{2}}{n-n_{e}}}={\frac {2}{\lambda ^{3}}}{\frac {g_{1}}{g_{0}}}\exp \left}$$

${\displaystyle \varepsilon }$

${\displaystyle x=n_{1}/n}$

$${\displaystyle {\frac {x^{2}}{1-x}}=A={\frac {2}{n\lambda ^{3}}}{\frac {g_{1}}{g_{0}}}\exp \left}$$

This gives a quadratic equation that can be solved in closed form:

$${\displaystyle x^{2}+Ax-A=0}$$

For small ${\displaystyle A}$, ${\displaystyle x\approx A^{1/2}\propto n^{-1/2}}$, so that the ionization decreases with density.

As a simple example, imagine a gas of monatomic hydrogen atoms, set ${\displaystyle g_{0}=g_{1}}$ and let ${\displaystyle \varepsilon =13.6}$ electron-volts = 158,000 Kelvin, the ionization energy of hydrogen from its ground state. Let ${\displaystyle n=2.69\times 10^{25}\mathrm {m} ^{-3}}$which is the Loschmidt constant, or particle density of Earth's atmosphere at standard pressure and temperature. At ${\displaystyle T=300}$ K, the ionization is essentially none: ${\displaystyle x=5\times 10^{-115}}$and there would almost certainly be no ionized atoms in the volume of Earth's atmosphere. ${\displaystyle x}$ increases rapidly with ${\displaystyle T}$, reaching 0.35 for ${\displaystyle T=\mathrm {20\,000\,K} }$. There is substantial ionization even though this ${\displaystyle T}$ is much less than the ionization energy (although this depends somewhat on density). This is a common occurrence. Physically, it stems from the fact that at a given temperature, the particles have a distribution of energies, including some with several times ${\displaystyle T}$. These high energy particles are much more effective at ionizing atoms. In Earth's atmosphere, ionization is actually governed not by the Saha equation but by very energetic cosmic rays, largely muons. These particles are not in thermal equilibrium with the atmosphere, so they are not at its temperature and the Saha logic does not apply.

Particle densities

The Saha equation is useful for determining the ratio of particle densities for two different ionization levels. The most useful form of the Saha equation for this purpose is