3 Calculation of Ne⁺ populations

3.1 The Ne⁺ populations

The calculation of the population structure is a two-stage process. The first stage involves the calculation of a purely hydrogenic model to determine the departure coefficients b_nl, related to the populations N_nl by

 

$$\left( {{N_{nl}}\over{N_{\rm e} N_+}} \right) = \left( {{N_{nl}}\over{N_{\rm e} N_+}} \right)_{\rm S} \ b_{nl},$$ (6)

where the subscript S refers to the value of the ratio given by the Saha equation, and $N_{\rm e}$ and N₊ are the number densities of electrons and recombining ions respectively. The details of the atomic rate coefficients and the numerical techniques employed in this calculation have been fully described elsewhere (Hummer & Storey 1987, 1992).

The second stage starts from the hydrogenic results for $n \gt n_{\rm d}$, and then solves for the populations of the states from $n = n_{\rm d}$ to n = 16 in descending order. The populations of states with $n \leq 15$ are determined by matrix inversion. More details are given by Storey (1994).

3.2 The Cases A and B

Baker & Menzel (1938) defined the Cases A and B with reference to the recombination spectrum of hydrogen. In Neii, there is only one low-lying term, 2s²2p^{5 2}P$^{\rm{o}}$.Accordingly, two cases can be defined for Neii. In Case A, all emission lines are assumed to be optically thin. In Case B, lines terminating on the ²P$^{\rm{o}}$ term are assumed to be thick and no radiative decays to this state are permitted when calculating the population structure. Since the calculations are made entirely in LS-coupling, Cases A and B differ only for the doublet series.