Saha ionization equation

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.

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:$${\displaystyle {\frac {n_{i+1}n_{\text{e}}}{n_{i}}}={\frac {2}{\lambda _{\text{th}}^{3}}}{\frac {g_{i+1}}{g_{i}}}\exp \left[-{\frac {\varepsilon _{i+1}-\varepsilon _{i}}{k_{\text{B}}T}}\right]}$$where:

The expression ${\textstyle (\varepsilon _{i+1}-\varepsilon _{i})}$ is the energy required to ionize the species from state ${\displaystyle i}$ to state ${\displaystyle i+1}$.

In the case where only one level of ionization is important, we have ${\textstyle n_{1}=n_{\text{e}}}$ for H⁺; defining the total density H/H⁺ as ${\textstyle n=n_{0}+n_{1},}$ the Saha equation simplifies to:$${\displaystyle {\frac {n_{\text{e}}^{2}}{n-n_{\text{e}}}}={\frac {2}{\lambda _{\text{th}}^{3}}}{\frac {g_{1}}{g_{0}}}\exp \left[{\frac {-\varepsilon }{k_{\text{B}}T}}\right]}$$where ${\displaystyle \varepsilon }$ is the energy of ionization. We can define the degree of ionization ${\textstyle x=n_{1}/n}$ and find$${\displaystyle {\frac {x^{2}}{1-x}}=A={\frac {2}{n\lambda _{\text{th}}^{3}}}{\frac {g_{1}}{g_{0}}}\exp \left[{\frac {-\varepsilon }{k_{\text{B}}T}}\right]}$$This gives a quadratic equation that can be solved (in closed form):$${\displaystyle x^{2}+Ax-A=0,x=(A{\sqrt {(}}1+{\tfrac {4}{A}})-A)/2}$$For small ${\textstyle A(T),}$ low temperature, ${\textstyle x\approx A^{1/2},\propto n^{-1/2},}$ so that the ionization decreases with higher number density (factors 10 in both plots).

Note that except for weakly ionized plasmas, the plasma environment affects the atomic structure with the subsequent lowering of the ionization potentials and the "cutoff" of the partition function. Therefore, ${\displaystyle \varepsilon _{i}}$ and ${\displaystyle g_{i}}$ depend, in general, on ${\displaystyle T}$ and ${\displaystyle n_{\text{e}}}$ and solving the Saha equation is only possible iteratively.

As a simple example, imagine a gas of monatomic hydrogen, set ${\displaystyle g_{0}=g_{1}}$ and let ${\displaystyle \varepsilon }$ = 13.6 eV (158000 K), the ionization energy of hydrogen from its ground state. Let ${\displaystyle n}$ = 2.69×10²⁵ m⁻³, which is the Loschmidt constant (n_L for N_A), or particle density of Earth's atmosphere at standard pressure and temperature. At ${\displaystyle T}$ = 300 K, the ionization is essentially none: ${\displaystyle x}$ = 5×10⁻¹¹⁵ and there would almost certainly be no ionized atoms in the volume of Earth's atmosphere. But ${\displaystyle x}$ increases rapidly with ${\displaystyle T}$, reaching 0.35 for ${\displaystyle T}$ = 20000 K. There is substantial ionization even though this ${\textstyle k_{B}T}$ 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 ${\textstyle k_{B}T.}$ These high energy particles are much more effective at ionizing atoms.