4 Results and discussion

4.1 Total and effective recombination coefficients

The population structure of Ne⁺ has been calculated for the electron temperatures $T_{\rm e}$ = 1000, 2000, 3000, 5000, 7500, 10000, 12500, 15000, 20000 K, and for the electron densities $N_{\rm e}$ = 10², 10⁴, 10⁵, 10⁶ cm^-3. For electron densities greater than 10⁶ cm^-3, l-changing collisions would have to be included for n < 15, which is beyond the scope of the current approximation.

 

Table 2:  Total recombination coefficients [10^-12 cm³ s^-1]

In Table 2, we give the total recombination coefficients for $\ion{Ne}{^{2+}} + {\rm e}^-$ for the above range of electron temperatures and densities. Total recombination coefficients were obtained by summing recombination coefficients to the metastable and ground states of Ne⁺.

In Tables 3 and 4 are given the effective recombination coefficients $\alpha_{\rm eff}(\lambda)$, for the strongest recombination lines of Neii. The effective recombination coefficient is defined such that the emissivity $\epsilon(\lambda)$, in a transition of wavelength $\lambda$ is

 

$$\epsilon(\lambda) = N_{\rm e} N_+ \alpha_{\rm eff}(\lambda) ... ...ver{\lambda}} \; \; \; \; [{\rm erg\, cm}^{-3}\, {\rm s}^{-1}].$$ (7)

Results are not tabulated for $N_{\rm e} = 10^2$ cm^-3 and $N_{\rm e} = 10^5$ cm^-3 as the recombination coefficients are not significantly different to those at 10⁴ cm^-3; typically they agree within two percent. In the tables, data are given for Cases A and B, as appropriate, but data for Case B are tabulated only if the recombination coefficients differ by more than one percent from the Case A values. Also tabulated is the air wavelength of the multiplet, which is calculated from the centres of gravity of the two terms involved. Most of the multiplets listed have considerable fine-structure and the reader is referred to Persson (1971) for a full list of fine-structure transitions and their wavelengths, although we tabulate the relative strengths and wavelengths of some of the strongest transitions (see Sect. 4.2 below). Transitions are included in the tables according to the following criteria:

(1) All the components of the 3d-3p and 3p-3s transition arrays are given irrespective of intensity, including those with ¹D and ¹S parentage. These transitions fall in the visible part of the spectrum and among them are the strongest recombination lines of Neii.

(2) For other transition arrays, only transitions with effective recombination coefficients greater than 10^-14 cm³ s^-1 at at least one temperature are tabulated. However, we did not include transitions having l > 2 because the LS-coupling scheme becomes inappropriate for such transitions.

(3) Only transitions giving rise to lines with wavelengths greater than 91.2 nm are tabulated.

In Table 5, fit parameters and maximum deviations from the calculated data are given for the effective recombination coefficients at $N_{\rm e} = 10^4$ cm^-3. The coefficients are fitted by a least-squares algorithm to the functional form

 

$$\alpha_{\rm eff} = 10^{-14}\, a t^{f} \left(1 \!+\! b \left(... ...c \left(1\!-\!t\right)^2\! +\! d\left(1\!-\!t\right)^3\right),$$ (8)

where $t = T_{\rm e}{\rm{[K]}}/10^4$, and a, b, c, d and f are constants. Fitting is valid for the whole temperature range studied for all lines except those denoted by asterisk in the last column.

 

Table 5:  Fitting coefficients and maximum fitting errors ($\%$) for effective recombination coefficients. Electron density $N_{\rm e} = 10^4$ [cm^-3]. Asterisks denote lines for which fitting is valid from $T_{\rm e} = 2000$ K

 

Table 5: continued

 

Table 5: continued

 

Table 5: continued

One can notice that the fits are typically accurate to within a few tenths of one percent for all lines in the main series, with the exception being the lines originating from the 4p (²P$^{\rm{o}}$) and 3p (²P$^{\rm{o}}$) states. Here we have a maximum fitting error $< 2\%$. The fitting of the lines from the second and third series (denoted by an asterisk) is valid for the temperature range $T_{\rm e} = 2000 - 20\,000$ K, and the fitting error is typically within $1.5 \%$ and not exceeding $3.5 \%$. For these lines, direct radiative recombination and dielectronic recombination are of equal importance at low temperatures. For this reason, the effective recombination coefficients have a more complex dependence on temperature.

The value of parameter a is constrained to have the value of 10¹⁴$\alpha_{\rm eff}(\lambda)$ at t=1. An approximate fit to the effective recombination coefficients at a different value of $N_{\rm e}$ can be obtained by replacing a with $\alpha_{\rm eff}(\lambda, t=1)$ at that density.

4.2 Relative line strengths

 

Table 6:  The factors $b(J_{\rm i},J_{\rm f})$, defined by Eq.(10), and wavelengths $\lambda$ (in air) for the strongest multiplets of NeII

To get the effective line recombination coefficient for the transitions between the initial level $S L_{\rm i} J_{\rm i}$ and the final level $S L_{\rm f} J_{\rm f}$ of the multiplet $S L_{\rm i} - S L_{\rm f}$, one should use the relation

 

$$\alpha_{\rm eff}\left(SL_{\rm i}J_{\rm i} \rightarrow SL_{\r... ...L_{\rm i} \rightarrow SL_{\rm f}\right) b(J_{\rm i},J_{\rm f}),$$ (9)

where the factors $b(J_{\rm i},J_{\rm f})$ can be calculated using the expression

 

$$b(J_{\rm i},J_{\rm f}) = \frac{(2J_{\rm i}+1)(2J_{\rm f}+1)}... ...f} & 1 \\ L_{\rm f} & L_{\rm i} & S \\ \end{array} \right\}^2,$$ (10)

where { } is the six-j symbol defined, for example, by Brink & Satchler (1968).

Values of $\alpha_{\rm eff}\left(SL_{\rm i} \rightarrow SL_{\rm f}\right)$may be obtained either from Tables 3 and 4 or by using Eq. (8) and data from Table 5. In Table 6, we tabulate the factors $b(J_{\rm i},J_{\rm f})$ and wavelengths of lines in the strongest multiplets of Neii. The multiplets were chosen to satisfy the condition that their effective recombination coefficients are greater than 10^-13 cm³ s^-1 at temperature 5000 K. Only transitions within the 3d-3p and 3p-3s arrays having ³P parentage fulfil this condition.

The air wavelengths of the transitions presented in Table 6 were taken from Persson (1971).

4.3 Population of excited states of Ne²⁺

In most calculations of recombination coefficients in nebular plasmas it is assumed that only the ground state of the recombining ion, 2s²2p^4 3P in the case of Ne²⁺, is populated. This is a reliable approximation in the case of astrophysical objects having relatively low electron density. In our case, there are two additional states 2s²2p^4 1D and ¹S having energies 25840.8 cm^-1 and 55750.6 cm^-1 (see Persson et al. 1991) above ground state. In our photoionization calculation, we obtained partial cross-sections from the states of Ne²⁺ to the ground state 2s²2p^4 3P and to the ¹D and ¹S states of the same configuration. These data enable us to obtain coefficients for direct recombination leading to series other than those of ³P parentage.

We generalised the model for calculation of the population structure for Ne⁺ described in Sect. 3.1 to include the three lowest terms of the Ne²⁺ ion. To determine the population numbers of these states, we used a model which included collisional excitation, collisional de-excitation and radiative decay among these three states. For temperatures of $T_{\rm e} \leq 20\,000$ K and electron densities of $N_{\rm e} \leq 10^6$ cm^-3 we found the population of excited ¹D and ¹S states to be very small (a maximum of $0.6\%$ for ¹D and $0.00045\%$ for ¹S). The extension of the full recombination calculation from one with only the ground term populated to a more extended one did not cause any substantial changes in the effective recombination coefficients. In general, only very small changes ($\leq 0.5\%$) were detected for strong lines. Changes of up to $2\%$ occured for some lines originating from states with ¹D parentage but these lines were weak. For this reason, we present data which include recombination only to the ground state of the Ne²⁺ ion. Nevertheless, the effects of population in excited states in Ne²⁺ ion could become significant when densities become higher.

4.4 Discussion

The calculations described here were carried out entirely in LS-coupling. The work of Persson (1971) indicates that LS-coupling is no longer a good approximation for f- and g- states of Ne⁺, and that the states should be described by an alternative coupling scheme. A full treatment of the high l states also needs to recognise the fact that the ground term of Ne²⁺ comprises three levels, ³P₂, ³P₁ and ³P₀ and that the populations of the ³P$_{\rm J}$ levels may well differ from those given by the Boltzmann distribution, so that a correct treatment of the recombination to the high l states will require intermediate coupling photoionization data and incorporation of the population distribution among the ³P$_{\rm J}$ levels.

We therefore do not present any transitions with l > 2 even though sometimes such multiplets have relatively strong lines with $\alpha_{\rm eff} \gt 10^{-14}$ cm³ s^-1. These transitions will be the subject of a subsequent paper.

Examining the data in Tables 3 and 4, one can see that for transitions from the 4d doublet states, the strongest lines are those with a final principal quantum number n=4, rather than n=3 as one might expect. For the transitions from the upper states 4d (²D) and (²P), there are no lines in the Tables terminating in a 3p state whereas lines terminating in 3p$^{\prime}$ state are present. A similar situation was observed by Storey (1994) in the Oii case. Closer examination of energy levels and oscillator strengths for these cases indicates that there exists a strong interaction between 3d$^{\prime}$ states and the 4d levels of main series. This interaction redistributes the oscillator strength leading to weakness of the 4d-3p lines.

  

Figure 2: Comparison of total recombination coefficients (in 10-12 cm3 s-1) for Ne2+$\; +\; {\rm e}^-$. The solid line represents the present calculation, the long-dashed line the radiative recombination coefficients of Péquignot et al. (1991), the short-dashed line the dielectronic recombination coefficients of Nussbaumer & Storey (1987). The dotted line is the sum of the radiative and dielectronic recombination coefficients

The present results for the total recombination coefficients (Table 2) can be compared with the radiative recombination coefficients (RR) for Neii given by Péquignot et al. (1991), and the dielectronic recombination (DR) coefficients from Nussbaumer & Storey (1987). The comparison is shown in Fig. 2, which also shows the sum of the RR and DR results. The difference between our data and the sum of the RR and DR coefficients increases with temperature while the difference between our data and the RR coefficient decreases. This is due to the different temperature behaviour and different magnitude of the dielectronic component of the recombination coefficients that is obtained in the present work.

  

Figure 3: Photoionization cross-section (in Mb) from the ground state 2P$^{\rm{o}}$ of the Ne+ ion (top) and oscillator strength distribution corresponding to the three lowest resonances obtained from these cross-section (solid line) and calculated by Nussbaumer & Storey (1987) (dashed line). In the top figure, the upper case letters denote transitions to the ground state from the doubly excited states: A - 3d$^{\prime \prime}$ 2P, B - 5d$^{\prime}$ 2P, C - 5d$^{\prime}$ 2D, D - 5g$^{\prime}$ 2P, E - 5s$^{\prime}$ 2D. In the bottom figure, the lower case letters denote transitions from states: a - 3d$^{\prime \prime}$ 2P, b - 5d$^{\prime}$ 2P, c - 5d$^{\prime}$ 2D

We cannot separate the radiative and dielectronic part of the total recombination coefficient in our method, so instead we consider the DR coefficients for the ground state 2s²2p⁵ (²P$^{\rm{o}}$) obtained in our work with those of Nussbaumer & Storey (1987), as a means of examining the importance of the resonance contributions to the recombination. In Fig. 3, we show a part of the photoionization cross-section for this state including the five lowest resonances. In the same figure we give the oscillator strength between the ground state and the continuum, binned into equal energy intervals, from the present photoionization data and from the work of Nussbaumer & Storey (1987). In the method of Nussbaumer & Storey (1987), resonances are treated as bound states and have no energy width, so we assign the oscillator strength in a particular resonance to one energy bin.

  

Figure 4: Calculated coefficients (in 10-12 cm3 s-1) of dielectronic recombination (DR) to the ground state 2P$^{\rm{o}}$ of Ne+. The top figure shows DR coefficients obtained from the present data corresponding to the A, B, and C transitions of Fig. 3 and their total (solid line). The bottom figure shows DR coefficients obtained using data from Nussbaumer & Storey (1987) and corresponding to the a, b, and c transitions of Fig. 3 and their total (solid line)

There are significant differences in resonance positions (especially for the first resonance) and oscillator strengths between the two sets of data. Our resonance A representing state 3d$^{\prime \prime}$ ²P is closer to the ionization threshold and has a larger f-value than the corresponding resonance a of Nussbaumer & Storey (1987). This causes a more rapid rise and a larger value of the dielectronic component of the recombination compared to the DR data of Nussbaumer & Storey (1987). In Fig. 4 we show the recombination coefficients derived from these oscillator strength data. The relative positions of other resonances a rising from the series 2s²2p$^4\;(^1$D)nl are similar although the corresponding oscillator strengths are several time smaller in our calculation. For this reason the dielectronic contributions arising from these resonances are significantly larger in the Nussbaumer & Storey (1987) calculation (see lines B, C and b, c in Fig. 4). These three resonances contribute $50\%$ of the total DR coefficient to the ground state, with the remainder coming from higher resonances and cascading from upper states. We conclude from these comparisons that there are significant differences in resonance positions and areas between the present results and those of Nussbaumer & Storey (1987), and that in our calculation the dielectronic component is significantly smaller than in that work. We believe the current work to be significantly more accurate than that of Nussbaumer & Storey (1987).