2 Atomic data for Ne⁺

2.1 The Ne⁺ term scheme

The principal series of Neii is 2s²2p$^4\,(^3$P)nl, which gives rise to doublet and quartet terms. Also interspersed among the bound states are members of the series 2s²2p$^4\,(^1$D)nl and 2s²2p⁴/(¹S)nl, which only give rise to doublet terms. The latter two series also have members lying above the first ionization limit, which appear as resonances in the photoionization of the true bound states and hence may give rise to low-temperature dielectronic recombination.

The tables of Nussbaumer & Storey (1984), show that this process mainly affects the two doublet terms in the ground electronic configuration, 2s²2p⁵ (²D$^{\rm{o}}$, ²P$^{\rm{o}}$). The calculation of recombination coefficients and the role of dielectronic recombination are discussed in Sect. 2.5.

We use the OP methods to calculate the bound-bound and bound-free radiative data for Neii assuming LS-coupling. Consequently, there are no radiative transitions between quartet and doublet states. The data extend up to principal quantum number n = 15 and to total atomic orbital angular momentum quantum number L = 6. We therefore partition the calculation of level populations into several distinct regimes, according to quantum numbers and energy. We define E₀ as the ionization energy in the principal series of Ne⁺ corresponding to n = 15; (E₀ = 0.0178 Ry).

We also define the principal quantum number $n = n_{\rm d}$, such that for $n < n_{\rm d}$, the population structure is determined solely by radiative processes. Collisionally induced transitions can be neglected. The approximations that are used for different values of n are described fully in Storey (1994). We give only a brief summary here.

(1) $n \gt n_{\rm d}$: The rate of l-changing collisions is more rapid than the rate of radiative decay. Populations in this regime are taken from a purely hydrogenic calculation of departure coefficients, b_nl, using the method described by Hummer & Storey (1987), which makes full allowance for all collisional effects. Only states that belong to the principal series of Ne⁺ are included.

(2) $15 < n \leq n_{\rm d}$: Collisional effects are no longer important, so populations are now determined only by radiative processes, but no accurate atomic data are available. Various approximate methods are used to calculate energy levels, radiative transition probabilities and recombination coefficients. Only states that belong to the principal series of Ne⁺ are included.

(3) States with ionization energy less than or equal to E₀: All atomic terms in this energy regime are included in the calculation of populations, irrespective of parentage. An energy ordered list of terms is set up and it is assumed that their populations are determined solely by recombination and radiative cascading from all accessible higher states.

2.2 New R-matrix calculation

We have carried out a new calculation of bound state energies, oscillator strengths and photoionization cross-sections for Neii states with $n \leq 15$ using the suite of programmes developed for the Opacity Project (Seaton 1987; Berrington et al. 1985) and the Iron Project (Hummer et al. 1993). The Ne²⁺ target state wave functions were calculated with the general purpose atomic structure code SUPERSTRUCTURE (Eissner et al. 1974) with the modifications of Nussbaumer & Storey (1978). The wave functions of the six target terms were expanded in terms of the electron configurations 1s²2s²2p⁴, 1s²2s2p⁵, 1s²2p⁶, 1s²2s²2p³${\bar 3} l$,1s²2s2p⁴${\bar 3} l$, 1s²2p⁵${\bar 3} l$, 1s²2s²2p²${\bar 3} l$${\bar 3} l^{\prime}$,1s²2s2p³${\bar 3} l$${\bar 3} l^{\prime}$, 1s²2p⁴${\bar 3} l$${\bar 3} l^{\prime}$, where 1s, 2s and 2p are spectroscopic orbitals and ${\bar 3} l$, ${\bar 3} l^{\prime}$ ($l, l^{\prime} = 0, 1, 2$) are correlation orbitals. The one-electron radial functions were calculated in adjustable Thomas-Fermi potentials, with the potential scaling parameters $\lambda_{nl}$ determined by minimising the sum of the energies of six target states. In our case, we obtained for the scaling parameters: $\lambda_{\rm 1s} = 1.45892$, $\lambda_{\rm 2s} = 1.12996$, $\lambda_{\rm 2p} = 1.08051$, $\lambda_{\bar{3}{\rm s}} = -1.01396$,$\lambda_{\bar{3}{\rm p}} = -0.94108$, $\lambda_{\bar{3}{\rm d}} = -1.14250$, with the negative values having the significance detailed by Nussbaumer & Storey (1978).

 

Table 1:  Comparison of calculated and experimental energies (in Ry) for the Ne²⁺ target states

In Table 1, we compare experimental target state energies with our calculated values. We use the experimental values for the target energies to obtain the Hamiltonian matrix of the (N+1) electron system in our calculation of energy levels, oscillator strengths and photoionization cross-sections of Neii.

2.3 Energy levels

Experimental energy levels for Ne⁺ have been given by Persson (1971) for members of the series 2s²2p⁴ (³P, ¹D, ¹S)$\,nl$ with $n \leq 9$ and $l \leq 4$, although some levels are missing. For states where no data are given by Persson, energies have been estimated in the following ways.

Our new calculation of energy levels includes all terms 2s$^{\rm p}$2p$^{\rm q}\;(S_{\rm c} L_{\rm c})\;nl$ (SL) with ionization energy less than E₀ and $L \leq 6$, where $S_{\rm c}$ and $L_{\rm c}$ are the total angular momentum quantum numbers of the core electrons. Energies calculated by ab initio methods have been used in preference to quantum defect extrapolation from experimentally known lower terms because they allow, albeit approximately, for the presence of perturbations of the principal series by members of other series. Such perturbations can significantly alter energy levels and the radiative properties of the states.

Secondly, for states with $15 < n \leq n_{\rm d}$, and $l \leq 5$, where calculated energies exist for lower members of the series, a quantum defect has been calculated for the highest known member (usually with n=15), and this quantum defect has been used to determine the energies of all higher terms.

Finally, if neither of the above methods can be used, the term is assumed to have hydrogenic energy.

2.4 Bound-bound radiative data

Radiative transition probabilities are taken from three sources:

(1) Ab initio calculation: We have computed values of (gf) for all the bound terms with ionization energy less than or equal to E₀, and with $L \leq 6$. The data are in LS-coupling and in the electric dipole approximation, so there are no transitions between states of different total spin, but two-electron transitions, which involve a change of core state are included.

(2) Coulomb approximation: For pairs of terms which were not computed by the method described in (1), but where one or both of the states have a non-zero quantum defect, the dipole radial integrals required for the calculation of transition probabilities are calculated using the Coulomb approximation. Details are given in Storey (1994).

(3) Hydrogenic approximation: For pairs of terms with zero quantum defect hydrogenic dipole radial integrals are calculated, either using the expressions of Gordon (1929) in terms of hypergeometric functions, or using direct recursion on the matrix elements themselves as described by Storey & Hummer (1991). More details on determining transition probabilities were presented by Storey (1994).

2.5 Photoionization cross-sections and recombination coefficients

The recombination coefficient for each term SL, or orbital nl is calculated directly from the photoionization cross-section for that state. As in the bound-bound case, there are three approximations in which the photoionization data are obtained.

(1) Photoionization cross-sections were computed for all terms with ionization energy less than or equal to E₀ and $L \leq 5$.In general, the cross-section for each state consists of a background contribution which declines monotonically with increasing ejected electron energy, and resonance contributions. These cross-sections can in principle be convolved with a thermal distribution of free-electron energies to obtain a recombination coefficient which incorporates both "radiative" and "dielectronic" recombination. There are two potential problems with this approach. The first is a shortcoming of the theory, in that the photoionization cross-sections have been calculated using first-order perturbation theory. This approach does not give the correct behaviour of the cross-section in the vicinity of a resonance whose radiative width is greater than its autoionization width (Seaton & Storey 1976), and overestimates the contribution of such a resonance to the recombination coefficient. This problem can only be overcome in general by including radiative channels in the original scattering calculation (Bell & Seaton 1985), and this cannot yet be done for complex ions. We discuss this problem further in Sect. 2.6 below.

The second problem is that in many cases treated in the OP, the free-electron energy mesh on which the cross-section was calculated was too coarse to accurately describe the narrower resonance features. The contribution of a resonance to the recombination coefficient depends on the area under it, which is also a measure of the oscillator strength between the initial state and the resonance. The width of a resonance, however, depends on the strength of the interaction with the accessible continuum states, and is independent of the area under it. As a result, a resonance may contribute significantly to the recombination to a particular state, but still be very narrow. Such resonances may be poorly described or missed altogether by calculation on a coarse energy mesh. We discuss the treatment of this problem further in Sect. 2.6 below.

(2) Coulomb approximation: As in the bound-bound case, the Coulomb approximation is used for terms where no OP data is available, but which have a non-zero quantum defect. The calculation of photoionization cross-section data using Coulomb functions has been described by Burgess & Seaton (1960) and Peach (1967), whose tables are used here.

(3) For the remaining states, hydrogenic photoionization cross-sections are used, calculated using the methods and computer codes of Storey & Hummer (1991).

2.6 Energy mesh for Neii photoionization cross-sections

In many cases treated by the OP, the free electron energy mesh on which the photoionization cross-sections are calculated, was too coarse to accurately map narrow resonance features. The photoionization cross-sections for Neii generated by the OP were based on a quantum defect mesh with 100 points per unit increase in the effective quantum number derived from the next threshold. Figure 1 (bottom) shows a section of the OP cross-section from the ground state of Neii (Cunto et al. 1993), in which resonance peaks are truncated by the coarse mesh.

  

Figure 1: Comparison of the photoionization cross-section (in Mb) for the ground state 2s22p5 2P$^{\rm{o}}$ of Neii calculated in this work (top) and by the OP (bottom)

In contrast to the OP calculations, we use a variable step energy mesh for photoionization cross-sections that delineates all resonances to a prescribed accuracy. The first step is to employ quantum defect theory (Seaton 1983) to determine the positions and widths of all resonances in a specific energy region. Since the quantum defect methods rely on functions that vary slowly with electron energy, only a coarse energy mesh is required for this step. Resonances are due to poles in the scattering matrix ${\mathcal S}$ described by a matrix $\chi$ which varies slowly with energy and contains functions pertaining to both open (o) and closed (c) channels:

 

$${\mathcal S} = \chi_{\rm oo} - \chi_{\rm oc} \left[ \chi_{\rm cc} - \exp (-2 \pi i \nu \right]^{-1} \chi_{\rm co}.$$ (1)

Diagonalization of the matrix $\chi_{\rm cc}$ yields the complex effective quantum numbers, $\nu$ related to the complex quantum defect $\mu_{\rm c}$ 

$$\mu_{c} = \alpha + i \beta$$ (2)

by $\nu = n-\mu_{\rm c}$. The complex energy of the ${\mathcal S}$ matrix pole $\varepsilon ({\rm pole})$ is related to $\alpha$ and $\beta$ by  

$$\varepsilon ({\rm pole}) = \varepsilon_{\rm c} - \left( n - \alpha \right)^{-2} - \frac{i}{2} \gamma,$$ (3)

where $\gamma = 4 \beta \left( n - \alpha \right)^{-3}$ is the autoionization width of resonance and $\varepsilon_{\rm c}$ is the appropriate target energy.

In the second step, we determine a new fine energy mesh of variable step length $\Delta \varepsilon$ assuming a Lorentzian profile for each resonance:

 

$$F(\varepsilon ,\gamma) = \frac{\left( \frac{\gamma}{2} \righ... ...ilon ({\rm pole})\right)^2 + \left(\frac{\gamma}{2}\right)^2}.$$ (4)

To calculate recombination coefficients, we integrate the photoionization cross-sections using a 3-point Simpson's rule. If we specify the accuracy required for each three-point integration as $\Delta$, we can find the necessary energy interval for a given resonance from the known error associated with Simpson's rule

 

$$\Delta \varepsilon = \frac {90 \Delta}{F^{(4)}(\xi)},$$ (5)

where $F^{(4)}(\xi)$ is the maximum value of the 4th derivative of the Lorentzian function in the given interval. This leads to an iterative procedure in which the step length and the fourth derivative are recalculated until the integration error over the next two points is below the specified value. It is repeated for all the resonances and the smallest value of $\Delta \varepsilon$ obtained is used for the next two steps.

Such detailed consideration of the energy mesh was undertaken for the regions from the 2s²2p⁴(³P) limit up to 0.0178 Ry below the 2s²2p⁴(¹D) limit and from the 2s²2p⁴(¹D) limit up to 0.0178 Ry below the 2s²2p⁴(¹S) limit, since these regions contain the main contribution to the recombination at the temperatures of interest. The energy 0.0178 Ry corresponds to a principal quantum number of fifteen relative to the next threshold. In the region above the 2s²2p⁴(¹S) threshold, a quantum defect mesh was used.

One problem that arose was that of interlopers from higher series. In the region below a particular threshold, the quantum defect method outlined above does not give information about resonances that come from higher thresholds, these having been eliminated by the use of contracted matrices (Seaton 1983). Below the (¹D) threshold and in the range of our variable step energy mesh calculation, there are resonances arising from the (¹S)3d ²P, ²S and (¹S)3p ²P$^{\rm{o}}$ states. The positions and widths of these resonances were determined from a preliminary calculation of photoionization cross-sections from a suitable bound state. These data were then added to the list of resonance information used to generate the final energy mesh for the detailed photoionization calculations.

In Fig. 1, we demonstrate the difference in resolution between the OP cross-section (Cunto et al. 1993) and the one calculated by our method for the ground state ²P$^{\rm{o}}$ of Neii. The OP cross section in the energy region between the first and second ionization threshold consists of 124 points based on a quantum defect mesh whilst the latter is based on an energy mesh of 2600 points. In the OP cross-section, the resonances are usually described by three or four points, and some of them are missing altogether. As one can also see from Fig. 1, there is also a significant difference in the peak heights of the resonances between the two calculations.

These differences lead to a significant change in the calculated area under the two cross-sections leading in turn to a difference in the derived recombination coeffcients. In the above case, the area under the OP curve is about $15\%$ smaller than is obtained from our data. This difference becomes even more pronounced when we consider only the area under particular resonances removing the effect of the background. In the case of the low-lying resonance at $\varepsilon = 0.0514\; {\rm Ry}$ which is very significant in determining the recombination coefficient, the area under our cross-section is $\rm 0.130~{Mb}\,\cdot\,{Ry}$ whereas the OP data give $\rm 0.085~{Mb} \,\cdot\, {Ry}$. The importance of the correct delineation of resonances is greatest for recombination to states whose parent is an excited state of Ne²⁺, since their photoionization cross-sections generally have little or no background contribution.

The unified approach suffers from a shortcoming whereby, the cross-section in the vicinity of resonance whose autoionization width is comparable or smaller than its radiative width overestimates its contribution to the recombination coefficient. To address this problem of radiative damping of resonances, we have compared total radiative decay probabilities (calculated with SUPERSTRUCTURE) with autoionization probabilities calculated with the quantum defect methods described above. As a result, the resonances corresponding to the states 2s²2p⁴(¹D)nf,ng (²L, L > 3) were eliminated from the list of resonances used to generate the energy mesh. As a result, a coarse energy mesh was used in the vicinity of these very narrow reonances and they were absent from the calculated photoionization cross-section.