3 Recombination coefficients

An accurate treatment of the radiative-cascade problem requires detailed calculations of recombination into excited states. Few detailed photoionization calculations had been performed for n > 2 states, before the work of the OP. It has been noted (Storey [1982]) that, in C ⁺, the dominant contribution to the recombination coefficient of the 2s2p^2 2D$\rm ^e$ state comes from the 2s2p(³P^o)3d ²F^o resonance. It is not only an important source of recombination for this state, but for the whole ion, particularly at low temperatures when the average free electron energy lies close to threshold. The 2s2p(³P^o)3d ²P^o resonance, although higher in energy is also a significant source of recombination.

In general, the theoretically calculated energies of states are higher than the experimental values. For low temperature recombination work, the placement of resonances is critical. A change in the position of the resonances can lead to a large change in the the value of the total recombination coefficient if the average free electron energy is lower than the resonance energy.

Figure 3: The effect of moving the 2F$^{\rm o}$ and 2P$^{\rm o}$ resonances on the recombination coefficient (in 10-12 cm3 s-1) of the 2s2p2 2D$\rm ^e$ state. The solid lines represent recombination coefficients calculated with experimental energies for the resonances, the dotted lines - with calculated energies. The top figure shows the effect on the total recombination coefficient of moving both resonances; the centre figure shows the contribution only from the 2F$^{\rm o}$ resonance; the lower figure shows the contribution only from the 2P$^{\rm o}$ resonance

The experimentally determined positions of the ²F^o and ²P^o resonances (Moore [1970]) are 3301 cm^-1 and 5523 cm^-1 above the 2s² (¹S$\rm ^e$) threshold. This compares with the theoretically calculated values of 3778 cm^-1 and 6172 cm^-1. We evaluate the resonance contributions to the total recombination coefficient using their experimental rather than the theoretical energies by, in effect, moving these two resonances.

This is done by fitting the photoionization cross-section in the energy region around the resonances using a Fano profile (Fano [1961]). For a resonance superimposed on a constant background the profile is given by

$$F(x) = \beta + \alpha\frac{(x+q)^{2}}{(1+x^{2})} .$$ (1)

The value of $\beta$ can be estimated from the minimum point, x = -q, since then $F(x) = \beta$.

This form does not allow for any energy dependence in the background, and therefore can only be used to fit a narrow energy range. The theoretical position and width of the resonance are used to start an iterative procedure to determine the quantities q, $\beta$ and $\alpha$. A new profile is then generated, differing only in the energy of the peak of the profile, and integrated to give a new value of the contribution to the recombination coefficient from the resonance and the adjacent energy region. Using the new profile gives a modified recombination coefficient for the state with the resonance contributions evaluated at their experimental positions. This method was applied to all cross-sections in which the two resonances appeared, for all states $n \le 10$. In Fig. 3, we compare the recombination coefficients for the 2s2p^2 2D$\rm ^e$ term computed with the two lowest lying resonances in their calculated and experimentally determined positions. There are significant changes to the recombination coefficients as a result, particularly at low electron temperatures. For example, the placement of the 2s2p(³P^o)3d ²F^o, ²P^oresonances at their experimental positions brings about a $63\%$ increase in the recombination coefficient of the 2s2p^2 2D$\rm ^e$ state at 1000 K.

The total recombination coefficient of a state depends, in general, on the relative populations of the C ²⁺ 2s²(¹S) and 2s2p(³P^o) states. For the temperature and density range under consideration here, the fraction of the population in the 2s2p(³P^o) state is always less than 0.01% and so we ignore recombination from this state in the present calculation.