Statistical Equilibrium

The aim here is to use the statistical equilibrium equation to calculate the new level populations given the radiation field that has just been calculated in the main program. The statistical equilibrium equations can be written (according to Athay [3]) as:

$$n_i \sum_{i \neq j} C_{i,j} + \sum_{j<i} \left( n_iA_{i,j}-\l... ...ndex{statistical equilibrium} \right)\overline{J}_{j,i}\right)$$

where $i$ and $j$ label the levels and take any valid level number. The $n_i$ are the level populations. The $A_{i,j}$ and $B_{i,j}$ are, respectively the Einstein A and B coefficients^4.13.   The $\overline{J}_{i,j}$ are the mean isotropic radiation field for any given transition and the $C_{i,j}$ are the collision coefficients. This can be expanded into its individual components to give:

$$\begin{array}{l} \overbrace{\sum_{j \neq i}n_jC_{j,i}}^{A} +\... ...ce{\sum_{j>i}n_iB_{i,j}\overline{J}_{j,i}}_{H}=0\\ \end{array}$$

where, again, $i$ and $j$ take values to cover all levels for which there are available collision coefficients. The list below describes the different sections of this equation:

A

Molecules entering level $i$ per second due to molecules leaving all other levels due to collisions with ${\rm H_2}$ molecules.

B

Molecules entering level $i$ due to spontaneous emission from all higher levels.

C

Molecules entering level $i$ due to stimulated emission from all higher levels.

D

The number of molecules that enter level $i$ due to absorption in all lower energy levels.

E

The number of molecules that leave level $i$ for any of the other levels due to collisions with ${\rm H_2}$ molecules.

F

Molecules leaving level $i$ due to spontaneous emission into all lower levels.

G

Molecules leaving level $i$ due to stimulated emission into all other lower energy levels.

H

Molecules leaving level $i$ due to absorption into all higher levels.

This can also be written (Matthews [20]) as:

$$n_i\sum_{i\neq j} \left( C_{i,j}+B_{i,j}\overline{J}_{i,j} \r... ...>i}n_j\left(C_{j,i}+A_{j,i}+B_{j,i}\overline{J}_{i,j} \right)=0$$

This is the form that the program uses for its calculations.