(updated for 2013)

We will use the notation from Spitzer (1978). See also Draine, Ch. 3. We represent the density of a state *j* as

, where

- n: particle density
- j: quantum state
- X: element
- (r): ionization state
- For example,

In his book, Spitzer defines something called “Equivalent Thermodynamic Equilibrium” or “ETE”. In ETE, gives the “equivalent” density in state *j*. The true (observed) value is . He then defines the ratio of the true density to the ETE density to be

.

This quantity approaches 1 when collisions dominate over ionization and recombination. For LTE, for all levels. The level population is then given by the Boltzmann equation:

,

where and are the energy and statistical weight (degeneracy) of level j, ionization state r. The exponential term is called the “Boltzmann factor”‘ and determines the relative probability for a state.

The term “Maxwellian” describes the velocity distribution of a 3-D gas. “Maxwell-Boltzmann” is a special case of the Boltzmann distribution for velocities.

Using our definition of b and dropping the “r” designation,

Where is the frequency of the radiative transition from k to j. We will use the convention that , such that .

To find the fraction of atoms of species excited to level j, define:

as the particle density of in all states. Then

Define , the “partition function” for species , to be the denominator of the RHS of the above equation. Then we can write, more simply:

to be the fraction of particles that are in state *j. *By computing this for all *j* we now know the distribution of level populations for ETE.

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