Saha ionization equation

The Saha ionization equation was developed by the Indian astrophysicist Meghnad Saha in 1920. For a gas at a high enough temperature, the thermal collisions of the atoms will ionize some of the atoms. One or more of the electrons that are normally bound to the atom in orbits around the atomic nucleus will be ejected from the atom and will form an electron gas that co-exists with the gas of atomic ions and neutral atoms. This state of matter is called a plasma. The Saha equation describes the degree of ionization of this plasma as a function of the temperature, density, and ionization energies of the atoms.

For a gas composed of a single atomic specie, the Saha equation is written:

<math>\frac{n_{i+1}n_e}{n_i} = \frac{2}{\Lambda^3}\frac{g_{i+1}}{g_i}\exp\left[-\frac{(\epsilon_{i+1}-\epsilon_i)}{k_BT}\right]</math>

where:

  • <math>n_i\,</math> is the density of atoms in the i-th state of ionization, that is with i electrons removed.
  • <math>g_i\,</math> is the degeneracy of states for the i-ions
  • <math>\epsilon_i\,</math> is the energy required to remove an electron from an (i-1)-level ion, creating an i-level ion.
  • <math>n_e\,</math> is the electron density
  • <math>\Lambda\,</math> is the thermal de Broglie wavelength of an electron
<math>\Lambda \equiv \sqrt{\frac{h^2}{2\pi m_ek_BT}}</math>

In the case where only one level of ionization is important, we have <math>n_1=n_e</math> and defining the total density n  as <math>n=n_0+n_1</math>, the Saha equation simplifies to:

<math>\frac{n_e^2}{n-n_e} = \frac{2}{\Lambda^3}\frac{g_1}{g_0}\exp\left[\frac{-\epsilon}{k_BT}\right]</math>

where <math>\epsilon</math> is the energy of ionization.

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

<math>\frac{Z_r}{N_r} = \frac{Z_{r+1}Z_e}{N_{r+1}N_e}</math>,

where Z denotes the partition function, N represents the number of atoms of the gas (divide by volume to get the number density), and the subscript r refers to a particular ionization state, r+1 refers to the next higher ionization state, and e refers to an electron. This form of the Saha equation arises from the equillibrium condition for the chemical potentials:

<math>\mu_r = \mu_{r+1} + \mu_e\,</math>

This equation simply states that the potential for an atom of ionization state r to ionize is the same as the potential for an electron and an atom of ionization state r+1; the potentials are equal, therefore the system is in equillibrium and no net change of ionization will occur.

[edit]

External links