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:\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 is the
number density of atoms in the
i-th state of ionization, that is with
i electrons removed. • g_i is the
degeneracy of state for the
i-ions. • \varepsilon_i is the energy required to remove
i electrons from a neutral atom, creating an
i-level ion. • n_\text{e} is the
electron density • k_\text{B} is the
Boltzmann constant • \lambda_\text{th} is the
thermal de Broglie wavelength of an electron \lambda_\text{th} \ \stackrel{\mathrm{def}}{=}\ \frac{h}{\sqrt{2\pi m_\text{e} k_\text{B} T}} • m_\text{e} is the
mass of an electron • T is the
temperature of the gas • h is the
Planck constant The expression (\varepsilon_{i+1}-\varepsilon_i) is the energy required to ionize the species from state i to state i+1. In the case where only one level of ionization is important, we have n_1=n_\text{e} for H+; defining the total density H/H+ as n=n_0+n_1, the Saha equation simplifies to:\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 \varepsilon is the energy of ionization. We can define the degree of ionization x=n_1/n and find\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):x^2+Ax-A=0, x=(A\sqrt(1+\tfrac{4}{A})-A)/2For small A(T), low temperature, 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, \varepsilon_i and g_i depend, in general, on T and n_\text{e} and solving the Saha equation is only possible
iteratively. or at
ICP conditions? As a simple example, imagine a gas of monatomic hydrogen, set g_0=g_1 and let , the ionization energy of hydrogen from its ground state. Let , which is the
Loschmidt constant (n
L for
NA), or particle density of Earth's atmosphere at standard pressure and temperature. At , the ionization is essentially none: and there would almost certainly be no ionized atoms in the volume of Earth's atmosphere. But x increases rapidly with T, reaching 0.35 for . There is substantial ionization even though this k_BT 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 k_BT. 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 of
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. n_0=p_{atm}/(k_BT_{std})).|none|frame Rigorously, the Saha equation is only valid for dilute gases, due to the underlying
ideal gas assumption used in its derivation. For dense gases this assumption is no longer valid, because particle interactions becoming significant modifies the chemical potential of the species. And the
compressibility of ionized gas and plasma. Hence, the Saha ionization framework has been extended to deal with systems that are denser than the ideal gas limit p/
RT
mole/m3], by incorporating corrections for these non-ideal interactions into the thermodynamic potential. This correction leads to improved estimates for the degree of ionization in the
corona of the Sun. == Particle densities ==