3 Recombination coefficients

We base our calculation of the recombination coefficients on a model of the O²⁺ ion including the two series 1s²2s²2p(²P$_{1/2}^\mathrm{o})$nl and 1s²2s²2p(²P$_{3/2}^\mathrm{o}$)nl. At low electron densities $N_{\mathrm e}$, the population of the ground levels $^2{\mathrm P}_{1/2}^\mathrm{o}$ and $^2{\mathrm P}_{3/2}^\mathrm{o}$ in the recombining ion O³⁺ differs significantly from the Boltzmann distribution. The critical density, defined as the electron density at which the rate of collisional de-excitation of ²P$_{3/2}^\mathrm{o}$ is equal to the rate of radiative de-excitation, is 7330 cm^-3 at a temperature of 10⁴ K. Thus for typical nebular densities the relative populations of the levels of the two series (²P$_{1/2}^\mathrm{o})$nl and (²P$_{3/2}^\mathrm{o}$)nl will depend on density (as well as temperature $T_{\mathrm e}$).

To calculate recombination coefficients $\alpha(5\mathrm{g}\,J\pi)$ and effective recombination coefficients for the lines originating from the 5g ($J\pi$) levels, we consider the contribution to the 5g level population made both by direct recombination from the two parent levels and by cascades within the same series and between the two series. For sufficiently high orbital angular momentum of the valence electron, jj-coupling prevails and the population of a given level depends only on the population of other levels with the same parent and on direct recombination from that parent state. The 4f and 5g levels lie at intermediate orbital angular momentum where neither LS- nor jj-coupling are good descriptions. We calculate the recombination coefficients to the 5g ($J\pi$) levels from the expression  

$$\alpha(5\mathrm{g}\,J\pi) = \mathcal{N}_1\, \alpha_1^\math... ...mathrm{g})\; \omega\left(^2\mathrm{P}^\mathrm{o}_{3/2}\right)},$$ (1)

where the $\omega$ are statistical weights, with $\omega(^2{\mathrm P}_{1/2}^\mathrm{o}) = 2$, $\omega(^2{\mathrm P}_{3/2}^\mathrm{o}) = 4$ and $\omega(5\mathrm{g})=18$. $\mathcal{N}_1$ and $\mathcal{N}_2$ are the fractions of the O³⁺ population in the $^2\mathrm{P}^\mathrm{o}_{1/2}$ and $^2\mathrm{P}^\mathrm{o}_{3/2}$ states respectively and $$\mathcal{N}_1 + \mathcal{N}_2 = 1$$.

In Eq. (1), the $\alpha_i^\mathrm{dir}(J\pi)$(i=1,2) are the direct radiative recombination coefficients to the $5\mathrm{g}\;(J\pi)$ level. These coefficients were computed from the appropriate photoionization cross-sections obtained from the R-matrix calculations performed in the Breit-Pauli approximation described in Sect. 2.

The third and fourth terms in Eq. (1) are the cascade contributions. The coefficients $\alpha_\mathrm{H}^\mathrm{casc}(5\mathrm{g})$ are the combined cascade contributions to all 5g levels calculated in a hydrogenic approximation. These coefficients were obtained using the methods described by Storey & Hummer (1995) and include full allowance for all radiative and collisional processes between excited states, and as a result depend upon both temperature and density. We assume that the fraction of the cascading represented by the coefficient $\alpha_\mathrm{H}^\mathrm{casc}(5\mathrm{g})$ that falls on a particular level 5g ($J\pi$) is proportional to its statistical weight and also that there are cascade contributions from states of both parentage. The fraction from each parent is given by the coefficients g$_1^\mathrm{casc}(J\pi)$ and g$_2^\mathrm{casc}(J\pi)$, which are determined from the photoionization cross-sections from the state 5g ($J\pi$) at the two parent thresholds.

3.1 Evaluation of the coefficients

The coefficients $\alpha(5\mathrm{g}\,J\pi)$ in Eq. (1) can be computed as follows. We have fitted the coefficients $\alpha_i^\mathrm{dir}(J\pi)$ as a function of temperature by:  

$$\alpha_i^{\rm dir}(J\pi) = 10^{-15}a_i \, t^{{\rm b}_i} \l... ...i t^2 \right) \;\;\; \mathrm{[cm}^3\ \mathrm{s}^{-1}\mathrm{]},$$ (2)

where $t = T~\mathrm{[K]}/10^4$. The fitting coefficients a_i, b_i, c_i and d_i are presented in Table 3.

  

Table 3: Fitting coefficients for $\alpha_i^\mathrm{dir}(J\pi)$, the direct recombination to the 5g levels of O²⁺

The cascade contributions, $\alpha_\mathrm{H}^\mathrm{casc}(5\mathrm{g})$, can be fitted with a maximum error of $2\%$ by a product of a function of density and a function of temperature as follows:  

$$\alpha_\mathrm{H}^\mathrm{casc}(5\mathrm{g}) = 0.69808 \ 1... ...m{d} Y^3 \right)\;\; \mathrm{[cm}^3\ \mathrm{s}^{-1}\mathrm{]},$$ (3)

where $t = T_\mathrm{e}~\mathrm{[K]}/10^4$ and $Y = \log\left(N_\mathrm{e}\right)-2$. The values of the fitting parameters are, for the temperature dependence, $a_\mathrm{t} = -0.94619$, $b_\mathrm{t} = -0.20470$,$c_\mathrm{t} = 0.02837$, and for the density dependance, $a_\mathrm{d} =-0.0086442$, $b_\mathrm{d} = -0.0013913$, $c_\mathrm{d} = -0.0005993$. The fits given in Eq. (2) and Eq. (3) are valid for $5000\leq T_\mathrm{e}\mathrm{[K]}\leq 20\,000$ and $10^2\leq N_\mathrm{e}\mathrm{[cm}^{-3}\mathrm{]}\leq10^6$, and the fitting error is less than $0.03 \%$ in these ranges.

The fractional populations $\mathcal{N}_1$ and $\mathcal{N}_2$ can be expressed as follows:  

$$\mathcal{N}_1 = \frac{1}{1+X}$$ (4)

and  

$$\mathcal{N}_2 = \frac{X}{1+X},$$ (5)

where X is the ratio of the population of the upper level $N_2 \equiv N(^2{\mathrm P}_{3/2}^\mathrm{o})$ relative to the population of the lower level $N_1 \equiv N(^2{\mathrm P}_{1/2}^\mathrm{o})$. This ratio can be calculated from the relation:  

$$X = \frac{N_2}{N_1} = \frac{N_{\mathrm e} C_{12}}{N_{\mathrm e} C_{21} + A_{21}},$$ (6)

where $A_{21} = 5.166 \ 10^{-4} \; \mathrm{s}^{-1}$ is the magnetic dipole transition probability, calculated from the O³⁺ target wave functions described in Sect. 2, using the experimental energy difference. C₁₂ is the excitation rate for the transition $^2{\mathrm P}_{1/2}^\mathrm{o}-^2{\mathrm P}_{3/2}^\mathrm{o}$ and C₂₁ is the de-excitation rate for this transition. The excitation rate can be expressed in form:  

$$C_{12}= \frac{8.63 \ 10^{-6}} {\omega (^2\mathrm{P}_{1/2}^... ...e}^{1/2}} \, \Upsilon_{12} \, \exp(- \Delta E /k T_\mathrm{e}),$$ (7)

and the de-excitation rate is:  

$$C_{21} = \frac{8.63 \ 10^{-6}} {\omega(^2{\mathrm P}_{3/2}^\mathrm{o}) T_{\mathrm e}^{1/2}} \, \Upsilon_{12},$$ (8)

where the energy difference between the two lowest levels $\mathrm{P}_{1/2}^\mathrm{o}$ and $\mathrm{P}_{3/2}^\mathrm{o}$ of the recombining ion is $\Delta E =$ 0.0035166 Ry, the electron temperature $T_{\mathrm e}$ is expressed in K and the Boltzmann constant is $1/k = 1.57888 \ 10^5$ [KRy^-1].

The collision strengths $\Upsilon_{12}$ were taken from the R-matrix calculations of Blum & Pradhan (1992). These can be fitted to

 

$$\Upsilon_{12} = a_\Upsilon + b_\Upsilon \ln(t) + c_\Upsilon / t + d_\Upsilon/t^2 ,$$ (9)

where $t = T_{\mathrm e}[{\mathrm K}]/10\,000$, $a_\Upsilon = 3.6292$, $b_\Upsilon = -0.5728$, $c_\Upsilon = -1.4231$ and $d_\Upsilon = 0.2101$. This fitting is valid for the temperature range $T_{\mathrm e} = 4000 - 20\,000$ K, and the maximum fitting error is less than $0.25 \%$.

  

Table 4: The fractions of the cascade contributions from each series to the recombination coefficients of the 5g levels of O²⁺

Finally, the fractions $g_1^\mathrm{casc}(J\pi)$ and $g_2^\mathrm{casc}(J\pi)$ can be obtained from Table 4.

In order to calculate recombination coefficients for any 5g level of O²⁺ at a specific electron temperature $T_{\mathrm e}$and electron density $N_{\mathrm e}$, one uses the expressions for the coefficients from Eqs. (1), (2) and (3), the data from Tables 3 and 4 and the recombining ion level population determined from Eqs. (4-9).