2.3. Solving the system of statistical equilibrium equations

Let us now proceed to set the system of equations to calculate the relative population ratios.

For notation purposes, let us define:

(15)

so that eq. (2) reads (taking into account the possible effect of fluorescence):

(16)

or equivalently in matrix notation:

(17)

After rewriting the system to have the solution expressed in terms of the populations relative to the ground level n₁ we are left with:

(18)

Eq. (18) is the linear system of equations that is solved by PopRatio.