Saha Ionization Equation

Equivalent forms

\frac{n_{i+1}}{n_i} = \frac{2 g_{i+1}}{g_i n_e}\left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} e^{-\chi/k_B T}

A chemical-equilibrium law for 'atom ⇌ ion + electron' — it turned stellar spectra into thermometers and launched modern astrophysics.

Symbol	Name	SI unit	Dimension	Range
$n_{i+1}/n_i$	Ionization ratiooutput Ratio of ionized to neutral atom density	dimensionless	1	0 – 1000
$T$	Temperature Gas (plasma) temperature	K	Θ	3000 – 30000
$n_e$	Electron density Free-electron number density	m⁻³	L⁻³	1000000000000000000 – 1e+24
$χ$	Ionization energy Energy to remove the electron	J	M L² T⁻²	8e-19 – 4e-18

Unit systems

SI:: n in $m^{-3}$ , $\chi in J$ , T in K
natural:: energies in eV, $k_B = 1$
CGS:: n in $cm^{-3}$

Where it holds

Local thermodynamic equilibrium (LTE); breaks down in low-density nebulae and stellar coronae where radiation, not collisions, controls the level populations.

Discovery

Meghnad Saha · 1920

Meghnad Saha derived his ionization equation in 1920, applying statistical mechanics to stellar atmospheres and founding quantitative astrophysics — explaining the Harvard spectral sequence O–B–A–F–G–K–M.

Try this

Why does a star's spectrum reveal its temperature, not just its elements?

Two stars made of identical hydrogen can show wildly different spectral lines. Saha's equation explains it: temperature sets how ionized the gas is, and ionization — not composition — controls the lines.

Research status: stable

Common misconceptions

Weak Balmer lines in hot O stars do NOT mean little hydrogen — the hydrogen is simply too ionized to absorb; Saha disentangles abundance from ionization state.

Derivation

Treat ionization as the reaction A ⇌

A^{+} + e^{-} in

thermal equilibrium.

Equate chemical potentials using the ideal-gas (Maxwell-Boltzmann) partition functions for each species; the free electron contributes a translational partition function

(2\pi m_e k_B T/h^{2})^{3/2}

per unit volume and a spin factor 2.

The Boltzmann factor

e^{-\chi /k_B T}

weights the ionization-energy cost

\chi

, yielding the Saha relation.

Limiting cases

k_B T ≪

\chi

⟶ Exponential suppression: gas almost fully neutral

k_B T ≳

\chi

⟶ Gas becomes strongly ionized; spectral lines of the neutral atom vanish

Low electron density n_e⟶ Ionization fraction rises (fewer electrons to recombine) — why stellar atmospheres ionize more than dense lab plasmas

Boltzmann Entropy→Maxwell-Boltzmann Speed Distribution→Canonical Partition Function→Planck Radiation Law→