Issue 
A&A
Volume 530, June 2011



Article Number  A18  
Number of page(s)  12  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/201116511  
Published online  29 April 2011 
New effective recombination coefficients for nebular N ii lines^{⋆}
^{1}
Department of AstronomySchool of Physics, Peking University, Beijing 100871, PR China
email: fangx@vega.bac.pku.edu.cn
^{2}
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
^{3}
The Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, PR China
Received: 14 January 2011
Accepted: 22 March 2011
Aims. In nebular astrophysics, there has been a longstanding dichotomy in plasma diagnostics between abundance determinations using the traditional method based on collisionally excited lines (CELs), on the one hand, and (optical) recombination lines/continuum, on the other. A number of mechanisms have been proposed to explain the dichotomy. Deep spectroscopy and recombination line analysis of emission line nebulae (planetary nebulae and H ii regions) in the past decade have pointed to the existence of another previously unknown component of cold, Hdeficient material as the culprit. Better constraints are needed on the physical conditions (electron temperature and density), chemical composition, mass, and spatial distribution of the postulated Hdeficient inclusions in order to unravel their astrophysical origins. This requires knowledge of the relevant atomic parameters, most importantly the effective recombination coefficients of abundant heavy element ions such as C ii, O ii, N ii, and Ne ii, appropriate for the physical conditions prevailing in those cold inclusions (e.g. T_{e} ≤ 1000 K).
Methods. Here we report new ab initio calculations of the effective recombination coefficients for the N ii recombination spectrum. We have taken into account the density dependence of the coefficients arising from the relative populations of the finestructure levels of the ground term of the recombining ion (^{2}P° _{1/2} and ^{2}P° _{3/2} in the case of N iii), an elaboration that has not been attempted before for this ion, and it opens up the possibility of electron density determination via recombination line analysis. Photoionization crosssections, bound state energies, and the oscillator strengths of N ii with n ≤ 11 and l ≤ 4 have been obtained using the closecoupling Rmatrix method in the intermediate coupling scheme. Photoionization data were computed that accurately map out the nearthreshold resonances and were used to derive recombination coefficients, including radiative and dielectronic recombination. Also new is including the effects of dielectronic recombination via highn resonances lying between the ^{2}P° _{1/2} and ^{2}P° _{3/2} levels. The new calculations are valid for temperatures down to an unprecedentedly low level (approximately 100 K). The newly calculated effective recombination coefficients allow us to construct plasma diagnostics based on the measured strengths of the N ii optical recombination lines (ORLs).
Results. The derived effective recombination coefficients are fitted with analytic formulae as a function of electron temperature for different electron densities. The dependence of the emissivities of the strongest transitions of N ii on electron density and temperature is illustrated. Potential applications of the current data to electron density and temperature diagnostics for photoionized gaseous nebulae are discussed. We also present a method of determining electron temperature and density simultaneously.
Key words: atomic data / line: formation / Hii regions / ISM: atoms / planetary nebulae: general
Tables 3–15 are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/530/A18
© ESO, 2011
1. Introduction
The principal means of electron temperature and density diagnostics and heavyelement abundance determinations in nebular plasmas has, until recently, been measurement of collisionally excited lines (CELs). The emissivities of CELs have, however, an exponential dependence on electron temperature and, consequently, so do the heavy element abundances deduced from them. An alternative method of determining heavy element abundances is to divide the intensities of ORLs emitted by heavy element ions with those by hydrogen. Such ratios are only weakly dependent on temperature. In planetary nebulae (PNe), abundances of C, N, and O derived from ORLs have been shown to be systematically larger than those derived from CELs typically by a factor of 2. Discrepancies of much larger magnitudes, say by more than a factor of 5, are also found for a small fraction of PNe (about 10%; Liu et al. 1995, 2000, 2001, 2006b, 2004; Luo et al. 2001). In the most extreme case, the abundance discrepancy factor (ADF) reaches a record value of 70 (Liu et al. 2006). There is strong evidence that nebulae contain another component of metalrich, cold plasma, probably in the form of H deficient inclusions embedded in the diffuse gas (Liu et al. 2000). The existence of cold inclusions provides a natural explanation to the longstanding dichotomy of abundance determinations and plasma diagnostics (Liu 2003, 2006a,b). The prerequisite for reliable determinations of recombination line abundances is accurate effective recombination coefficients for heavy element recombination lines. In this paper, we present new effective recombination coefficients for the recombination spectrum of N ii.
Radiative recombination coefficients of N ii have been given by Péquignot et al. (1991). Nussbaumer & Storey (1984) tabulate dielectronic recombination coefficients of N ii obtained from a model in which resonance states are represented by boundstate wave functions. Escalante & Victor (1990) calculate effective recombination coefficients for C i and N ii lines using an atomic model potential approximation for transition probabilities and recombination crosssections. They then add the contribution from dielectronic recombination using the results of Nussbaumer & Storey (1984).
In the most recent work of N ii by Kisielius & Storey (2002), they follow the approach of Storey (1994) who, in dealing with O ii, uses a unified method for the treatment of radiative and dielectronic recombination by directly calculating recombination coefficients from photoionization crosssections of each initial state. They also incorporate improvements introduced by Kisielius et al. (1998) in their work on Ne ii. The N^{+} photoionization crosssections were calculated in LScoupling using the ab initio methods developed for the Opacity Project (Seaton 1987; Berrington et al. 1987) and the Iron Project (Hummer et al. 1993), hereafter referred to as the OP methods. The calculations employed the Rmatrix formulation of the closecoupling method, and the resultant crosssections are of higher quality. In the photoionization calculations of Kisielius & Storey (2002), for the energy ranges in which the resonances make main contributions to the total recombination, an adaptive energy mesh was used to map out all the strong resonances near thresholds. This approach has also been adopted in our current calculations, and it will be described in more details in a later section. In the work of Kisielius & Storey (2002), transition probabilities for all lowlying bound states were also calculated using the closecoupling method, so that the boundbound and boundfree radiative data used for calculating the recombination coefficients formed a selfconsistent set of data and were expected to be significantly more accurate than those employed in earlier work.
Here we report new calculations of the effective recombination coefficients for the N ii recombination spectrum. Hitherto few high quality atomic data were available to diagnose plasmas of very low temperatures ( ≤ 1000 K), such as the cold, Hdeficient inclusions postulated to exist in PNe (Liu et al. 2000). We have calculated the effective recombination coefficients for the N^{+} ion down to an unprecedentedly low temperature (about 100 K). At such low temperatures, dielectronic recombination via highn resonances between the N^{2+} ground ^{2}P° _{1/2} and ^{2}P° _{3/2} finestructure levels contributes significantly to the total recombination coefficient. We include such effects in our calculations. We also take into account the density dependence of the coefficients through the level populations of the finestructure levels of the ground state of the recombining ion (^{2}P° _{1/2,3/2} in the case of N^{2+}). That opens up the possibility of electron density determinations via recombination line analysis. With the exception of the calculations of Kisielius & Storey (1999) on O iii recombination lines, all previous work on nebular recombination lines has been in LScoupling and therefore it tacitly assumed that the levels of the ground state are populated in proportion to their statistical weights. Photoionization crosssections, bound state energies and oscillator strengths of N ii with n ≤ 11 and l ≤ 4 have been obtained using the Rmatrix method in the intermediate coupling scheme. The photoionization data are used to derive recombination coefficients, including contributions from radiative and dielectronic recombination. The results are applicable to PNe, H ii regions and nova shells for a wide range of electron temperature and density.
2. Atomic data for N^{+}
2.1. The N^{+} term scheme
The principal series of N ii is 2s^{2}2p(^{2}P°)nl, which gives rise to singlet and triplet terms. Also interspersed are members of the series 2s2p^{2}(^{4}P)nl (n = 3, 4) giving rise to triplet and quintet terms. Higher members of this series lie above the first ionization limit, and hence may give rise to lowtemperature dielectronic recombination. There are a few members of the 2s2p^{2}(^{2}D)nl and 2s2p^{2}(^{2}S)nl series located above the first ionization threshold which also give rise to resonance structures in the photoionization crosssections for singlets and triplets. For photon energies above the second ionization threshold, the main resonance structures are due to the 2s2p^{2}(^{2}D)nl series with some interlopers from the 2s2p^{2}(^{2}S)nl and 2s2p^{2}(^{2}P)nl series (Kisielius & Storey 2002).
2.2. New Rmatrix calculation
We have carried out a new calculation of bound state energies, oscillator strengths and photoionization crosssections for N ii states with n ≤ 11 using the OP methods (Hummer et al. 1993). The N^{2+} target configuration set was generated with the general purpose atomic structure code SUPERSTRUCTURE (Eissner et al. 1974) with modifications of Nussbaumer & Storey (1978). The target radial wave functions of N^{2+} were then generated with another atomic structure code AUTOSTRUCTURE^{1}, which, developed from SUPERSTRUCTURE and capable of treating collisions, is able to calculate autoionization rates, photoionization crosssections, etc. The original theory of AUTOSTRUCTURE is described by Badnell (1986). The wave functions of the nineteen target terms were expanded in terms of the 21 electron configurations 1s^{2}2s^{2}2p, 1s^{2}2s2p^{2}, 1s^{2}2p^{3}, 1s^{2}2ss, 1s^{2}2sp, 1s^{2}2sd, 1s^{2}2s2ps, 1s^{2}2s2pp, 1s^{2}2s2pd, 1s^{2}2ps, 1s^{2}2pp, 1s^{2}2pd, 1s^{2}2sd^{2}, 1s^{2}2pd^{2}, 1s^{2}2ss, 1s^{2}2sd, 1s^{2}2s2ps, 1s^{2}2s2pp, 1s^{2}2s2pd, 1s^{2}2s2pf, 1s^{2}2pp, where 1s, 2s and 2p are spectroscopic orbitals and and (l = 0 − 2 and l′ = 0 − 3) are correlation orbitals. The oneelectron radial functions for the 1s, 2s and 2p orbitals were calculated in adjustable ThomasFermi potentials, while the radial functions for the remaining orbitals were calculated in Coulomb potentials of variable nuclear charge, Z_{nl} = 7  λ_{nl}  . The potential scaling parameters λ_{nl} were determined by minimizing the sum of the energies of the eight energetically lowest target states in our model. We obtained for the potential scaling parameters: λ_{1s} = 1.4279, λ_{2s} = 1.2840, λ_{2p} = 1.1818, , , , , , and .
Comparison of energies (in Ry) for the N^{2+} target states.
The N^{2+} target terms.
In Table 1, we compare experimental target state energies (Eriksson 1983) as well as values calculated by Kisielius & Storey (2002) with our results for the eight lowest target terms that belong to the three lowest configurations, 2s^{2}2p, 2s2p^{2} and 2p^{3}. We use the experimental target energies in the calculation of the Hamiltonian matrix of the (N + 1) electron system and in the calculation of energy levels, oscillator strengths and photoionization crosssections of N ii. We use nineteen target terms in our Rmatrix calculation where we add selected terms from the seven configurations 1s^{2}2ss, 1s^{2}2sp, 1s^{2}2sd, 1s^{2}2s2p s, 1s^{2}2s2p p, 1s^{2}2s2p d and 1s^{2}2p^{2} in order to increase the dipole polarizability of the terms of the 2s^{2}2p and 2s2p^{2} configurations of the N^{2+} target. These additional terms provide the main contributions to the dipole polarizability of the 1s^{2}2s^{2}2p ^{2}P° and 1s^{2}2s2p^{2} ^{4}P states and significant contributions to the polarizability of 1s^{2}2s2p^{2} ^{2}D, ^{2}S and ^{2}P. The chosen target terms are listed in Table 2, which shows that our calculated target energies are in slightly worse agreement with experiment than those of Kisielius & Storey (2002) although it should be noted that their energies were calculated in LScoupling, whereas ours are weighted averages of finestructure level energies. The configuration interaction in our target is less extensive than in theirs due to the additional computational constraints imposed by an intermediate coupling calculation as opposed to an LScoupling one. We do, however, consider that our set of target states represents the polarizability of the important N^{2+} states better than does the target of Kisielius & Storey (2002).
2.3. Energy levels of N^{+}
Experimental energy levels for N^{+} have been given by Eriksson (1983) for members of the series 2s^{2}2p(^{2}P°) nl with n ≤ 16 and l ≤ 4, for the series 2s2p^{2}(^{4}P^{e}) nl with n ≤ 12 and l ≤ 4 and for the series 2s2p^{2}(^{2}D^{e}) 3l, 4p although some levels are missing. Energy levels for the states ^{3}P^{e}_{2,1,0} belonging to the equivalent electron configuration 2s^{2}2p^{4} are also presented. We use these experimental data as a benchmark for our N ii energy level calculation.
The current calculations of energy levels include only the states belonging to the configurations 2s^{2}2p(^{2}P$\begin{array}{}\mathrm{\u25e6}\\ {\mathit{J}}_{\mathrm{C}}\end{array}$) nl, 2s2p^{2}(^{4}P$\begin{array}{}\mathrm{e}\\ {\mathit{J}}_{\mathrm{C}}\end{array}$) nl and possibly 2s2p^{2}(^{2}D$\begin{array}{}\mathrm{e}\\ {\mathit{J}}_{\mathrm{C}}\end{array}$) nl with ionization energies less than E_{0} (corresponding to n = 11 in the principal series of N ii) and total orbital angular momentum quantum number L ≤ 8, where J_{C} is the total angular momentum quantum number of the core electrons. Every single energy level calculated by ab initio methods is further identified with the help of the experimental energy level tables of Eriksson (1983), and is used in preference to quantum defect extrapolation from experimentally known lower states. The bound state energies for the lowest levels are calculated using Rmatrix codes, with an effective principal quantum number range set to be 0.5 ≤ ν ≤ 10.5, and the codes are run with a searching step of δν = 0.01. We assume that the subsequent iteration converges on a final energy when Δ E < 1 × 10^{5} Ryd. We only keep the final energy levels with n ≤ 11 and l ≤ 4, and delete the ng levels with n > 6, because of numerical instabilities in the codes. In total, 377 levels are obtained. The photoionization crosssections from these bound states are calculated later using outer region Rmatrix codes.
For states with 11 < n ≤ n_{d}, and l ≤ 3, where calculated energies exist for lower members of the series, a quantum defect has been calculated for the highest known member (usually with n = 11), and this quantum defect is used to determine the energies of all higher terms.
Finally, if neither of the above methods can be used, the state is assumed to have a zero quantum defect.
2.4. Boundbound radiative data
Radiative transition probabilities are taken from three sources,

(1)
Ab initio calculations: we have computed values of weightedoscillator strengths, gf, for all transitions between bound states with ionization energies less than or equal to E_{0} (corresponding to n = 11 in the principal series of N^{+}), with total orbital angular momentum quantum number L ≤ 8, and with total angular momentum quantum number J ≤ 6. The data are calculated in the intermediate coupling scheme, so there are transitions between states of different total spins. Twoelectron transitions, which involve a change of core state, are also included.

(2)
Coulomb approximation: for pairs of levels where oscillator strengths are not computed by the ab initio method, but where one or both of the states have a nonzero quantum defect, the dipole radial integrals required for the calculation of transition probabilities are calculated using the Coulomb approximation. Details are given by Storey (1994).

(3)
Hydrogenic approximation: for pairs of levels with a zero quantum defect hydrogenic dipole radial integrals are calculated, using direct recursion on the matrix elements themselves as described by Storey & Hummmer (1991).
2.5. Photoionization crosssections and recombination coefficients
The recombination coefficient for each level nls L [K] ^{π} _{J} is calculated directly from the photoionization crosssections for that state. There are three approximations in which the photoionization data are obtained.

(1)
Photoionization crosssections are computed for all the377 states with ionization energy less than or equalto E_{0}, L ≤ 8 and J ≤ 6. We obtain the recombination coefficient directly by integrating the appropriate Rmatrix photoionization crosssections.

(2)
Coulomb approximation: as in the boundbound case, the Coulomb approximation is used for states where no OP data are available, but which have a nonzero quantum defect. The calculation of photoionization data using Coulomb functions has been described by Burgess & Seaton (1960) and Peach (1967).

(3)
For all states for which the Rmatrix photoionization crosssections have not been calculated explicitly, the hydrogenic approximation to the photoionization crosssections is evaluated, using the routines of Storey & Hummer (1991) to generate radiative data in hydrogenic systems.
2.6. Energy mesh for N ii photoionization crosssections
The photoionization crosssections for N ii generated by the Opacity Project (OP) method were based on a quantum defect mesh with 100 points per unit increase in the effective quantum number derived from the next threshold. In contrast to the OP calculations, we use a variable step mesh for photoionization crosssection calculations for a particular energy region above the ^{2}P° _{3/2} threshold, which is appropriate to dielectronic recombination in nebular physical conditions. This energy mesh delineates all resonances to a prescribed accuracy (Kisielius et al. 1998; Kisielius & Storey 2002). This detailed consideration of the energy mesh was undertaken for the region from the 2s^{2}2p (^{2}P° _{3/2}) limit up to 0.160 Ryd (n = 5) below the 2s2p^{2} (^{4}P _{1/2}) limit, since this region contains the main contribution to the total recombination at the temperatures of interest for the triplet and singlet series.
From 0.160 Ryd below the 2s2p^{2} (^{4}P _{1/2}) limit to 0.0331 Ryd below the 2s2p^{2} (^{4}P _{1/2}) limit, we use a quantum defect mesh with an increment of 0.01 in effective principal quantum number. The energy 0.160 Ryd corresponds to a principal quantum number of five relative to the next threshold, and 0.0331 Ryd corresponds to eleven. In total about 600 points are used in this region. For the region from 0.0331 Ryd (n = 11) below the 2s2p^{2} (^{4}P _{1/2}) limit up to the 2s2p^{2} (^{4}P _{5/2}) limit, we use the Gailitis average (Gailitis 1963). We use the method for this part of photoionization calculations because of the very dense resonances in this narrow energy region. About 100 points are used.
In the region from the 2s2p^{2} (^{4}P _{5/2}) limit up to 0.0331 Ryd (n = 11) below the 2s2p^{2} (^{2}D _{3/2}) limit, a quantum defect mesh is again used, with an increment of 0.01 in effective principal quantum number. Here 0.0331 Ryd corresponds to a principal quantum number of eleven relative to the next threshold, 2s2p^{2} (^{2}D _{3/2}). There are about 800 points used in this region. For the energetically lowest region between the two ground finestructure levels of N iii, 2s^{2}2p (^{2}P° _{1/2}) and 2s^{2}2p (^{2}P° _{3/2}), we use linear extrapolation from a few points lying right above the 2s^{2}2p (^{2}P° _{3/2}) threshold. About 100 points are used in this region.
During the calculation of photoionization crosssections, we check for every bound states to make sure that the crosssection data of different energy areas all join smoothly. Figure 1 shows the photoionization crosssections calculated from the five lowest levels ^{3}P^{e} _{0}, ^{3}P^{e} _{1}, ^{3}P^{e} _{2}, ^{1}D^{e} _{2} and ^{1}S^{e} _{0} belonging to the ground configuration 1s^{2}2s^{2}2p^{2} of N^{+}. Photoionization crosssections calculated for the five different energy regions have been joined together.
Fig. 1 The data calculated by different methods for the five energy regions have been joined together. The insert zooms in a particular energy area, and the mesh points used for the photoionization calculations are shown in the inset. 
3. Calculation of N^{+} population
3.1. The N^{+} populations
The calculation of populations is a three stage process to compute departure coefficients, b(J_{C};nl K J), defined in terms of populations by $\left(\frac{\mathit{N}\mathrm{(}{\mathit{J}}_{\mathrm{C}}\mathrm{;}\mathit{nl}\hspace{0.17em}\mathit{K}\hspace{0.17em}\mathit{J}\mathrm{)}}{{\mathit{N}}_{\mathrm{e}}{\mathit{N}}_{\mathrm{+}}\mathrm{\left(}{\mathit{J}}_{\mathrm{C}}\mathrm{\right)}}\right)\mathrm{=}{\left(\frac{\mathit{N}\mathrm{(}{\mathit{J}}_{\mathrm{C}}\mathrm{;}\mathit{nl}\hspace{0.17em}\mathit{K}\hspace{0.17em}\mathit{J}\mathrm{)}}{{\mathit{N}}_{\mathrm{e}}{\mathit{N}}_{\mathrm{+}}\mathrm{\left(}{\mathit{J}}_{\mathrm{C}}\mathrm{\right)}}\right)}_{\mathit{S}}\mathit{b}\mathrm{(}{\mathit{J}}_{\mathrm{C}}\mathrm{;}\mathit{nl}\hspace{0.17em}\mathit{K}\hspace{0.17em}\mathit{J}\mathrm{)}\mathit{,}$(1)where J_{C} is an N^{2 + } core state and the subscript S refers to the value of the ratio given by the Saha and Boltzmann equations, and N_{e} and N_{ + }(J_{C}) are the number densities of electrons and recombining ions, respectively. We distinguish two boundaries in principal quantum number, n_{d} and n_{l}. For n ≤ n_{d} collisional processes are negligible compared to radiative decays and can be omitted from the calculation of the populations. For higher n a full collisionalradiative treatment of the populations is necessary as described by Hummer & Storey (1987) and Storey & Hummer (1995) with some additions to treat dielectronic recombination. The boundary at n = n_{l} is defined such that for n > n_{l} the redistribution of population due to lchanging collisions is rapid enough to assume that the populations obey the Boltzmann distribution for a given n and hence that b_{nl} = b_{n} for all l.
The three stages of the calculation are as follows:

Stage 1:
a calculation ofb(J_{C};n) is made for all n < 1000, using the techniques and atomic rate coefficients described by Hummer & Storey (1987) and Storey & Hummer (1995) with the addition of laveraged autoionization and dielectronic capture rates computed with AUTOSTRUCTURE for states of (^{2}P° _{3/2}) parentage that lie above the ionization limit. For n > 1000, we assume b_{n} = 1. The results of this calculation provide the values of b for n > n_{l} and the initial values for n ≤ n_{l} for Stage 2.

Stage 2:
a calculation of b(J_{C};nl K J) is made for all n ≤ n_{l} using the same collisionalradiative treatment as in Stage 1 but now resolved by total J. The combined results of Stages 1 and 2 provide the values of b for n > n_{d}.

Stage 3:
for the energies less than that which corresponds to n = n_{d} in the principal series, departure coefficients b(J_{C};nl K J) are computed for states of all series. Since only spontaneous radiative decays link these states, the populations are obtained by a stepwise solution from the energetically highest to the lowest state.
3.2. Dielectronic recombination within the ^{2}P° parents
Within the ^{2}P° parents, the contribution by dielectronic recombination to the total recombination is shown to become significant at very low temperatures ( ≤ 250 K), due to recombination into highlying bound states of the ^{2}P$\begin{array}{}\mathrm{\u25e6}\\ \mathrm{3}\mathit{/}\mathrm{2}\end{array}$ parent from the ^{2}P$\begin{array}{}\mathrm{\u25e6}\\ \mathrm{1}\mathit{/}\mathrm{2}\end{array}$ continuum states. We incorporate this lowtemperature process into our calculation of the N^{+} populations. Figure 2 is a schematic diagram illustrating N ii dielectronic capture, autoionization and radiative decays. The electrons captured to the highn autoionizing levels decay to lown bound states through cascades, and optical recombination lines are emitted. Radiative transitions which change parent, such as (^{2}P$\begin{array}{}\mathrm{\u25e6}\\ \mathrm{3}\mathit{/}\mathrm{2}\end{array}$)n_{1}l_{1}–(^{2}P$\begin{array}{}\mathrm{\u25e6}\\ \mathrm{1}\mathit{/}\mathrm{2}\end{array}$)n_{0}l_{0}, are included for those states for which Rmatrix calculated values are present. Higher states are treated by hydrogenic or Coulomb approximations which do not allow parent changing transitions.
In Fig. 2, three multiplets of N ii are presented as examples: V3 2s^{2}2p3p ^{3}D^{e}–2s^{2}2p3s ^{3}P°, the strongest 3p–3s transition, V19 2s^{2}2p3d ^{3}F°–2s^{2}2p3p ^{3}D^{e}, the strongest 3d–3p transition, and V39 2s^{2}2p4f G[7/2,9/2]^{e}–2s^{2}2p3d ^{3}F°, the strongest 4f − 3d transition. The results for these three multiplets are analysed in Sect. 4.3.
Fig. 2 Schematic figure showing the lowtemperature ( ≤ 250 K) dielectronic recombination of N ii through the finestructure autoionizing levels between the two lowest ionization thresholds of N iii ^{2}P° _{1/2} and ^{2}P° _{3/2}. The electrons captured to the highn autoionizing levels decay to lown bound states through cascades and optical recombination lines (ORLs) are thus emitted. Here Multiplets V3, V19 and V39 are presented as examples. 
3.3. The ^{2}P° parent populations
The contribution to the total recombination coefficient of a state depends on the relative populations of the ^{2}P° _{1/2} and ^{2}P° _{3/2} parent levels, which generally dominate the populations of the recombining ion N^{2+} under typical nebular physical conditions. The relative populations of the two finestructure levels deviate from the statistical weight ratio, 1:2, which is assumed in all work hitherto on this ion. The deviation affects the populations of the high Rydberg states, and consequently total dielectronic recombination coefficients at lowdensity and lowtemperature conditions.
We model the N^{2+} populations with a five level atom comprising the two levels of the ^{2}P° term and the three levels of the ^{4}P term, although it should be noted that the populations of the three ^{4}P levels are almost negligible in the nebular conditions considered here (Sect. 4.5). The relative populations are assumed to be determined only by collisional excitation, collisional deexcitation and spontaneous radiative decay. Transition probabilities were taken from Fang et al. (1993) and thermally averaged collision strengths from Nussbaumer & Storey (1979) and Butler & Storey (priv. comm.).
Figure 3 shows the fractional populations of the N^{2+} ^{2}P${}^{\mathrm{\u25e6}}\hspace{0.17em}_{\mathrm{1}\mathit{/}\mathrm{2}}$ and ^{2}P${}^{\mathrm{\u25e6}}\hspace{0.17em}_{\mathrm{3}\mathit{/}\mathrm{2}}$ finestructure levels at several electron temperatures and as a function of electron density, ranging from 10^{2} to 10^{6} cm^{3}, applicable to PNe and H ii regions. The fractional populations vary significantly below 10^{4} cm^{3} and converge to the thermalized values at higher densities.
Fig. 3 Fractional populations of the N^{2+} ^{2}P${}^{\mathrm{\u25e6}}\hspace{0.17em}_{\mathrm{1}\mathit{/}\mathrm{2}}$ and ^{2}P${}^{\mathrm{\u25e6}}\hspace{0.17em}_{\mathrm{3}\mathit{/}\mathrm{2}}$ parent levels. The red solid and blue dashed curves represent the populations of the ^{2}P° _{1/2} and ^{2}P° _{3/2} levels, respectively. Four temperature cases, 200 K, 1000 K, 5000 K and 10 000 K are shown. 
3.4. The Cases A and B
Baker & Menzel (1938) define the Cases A and B with reference to the recombination spectrum of hydrogen. In N ii, there are five lowlying levels belonging to the ground configuration 2s^{2}2p^{2}, ^{3}P^{e} _{0,1,2}, ^{1}D^{e} _{2} and ^{1}S^{e} _{0}. Just as in Kisielius & Storey (2002), we define two cases for N ii. In Case A, all emission lines are assumed to be optically thin. In Case B, lines terminating on the three lowest levels ^{3}P^{e} _{0,1,2} are assumed to be optically thick and no radiative decays to these levels are permitted when calculating the population structure. The latter case is generally a better approximation for most nebulae.
4. Results and discussion
4.1. Effective recombination coefficients
The population structure of N^{+} has been calculated for electron temperature log T_{e} [K] = 2.1 ~ 4.3, with a step of 0.1 in logarithm, and for the electron density N_{e} = 10^{2} ~ 10^{6} cm^{3}, also with a step of 0.1 in logarithm. Constrained by the range of the exponential factors involved in the calculation of the departure coefficients using the SahaBoltzmann equation, calculation of the effective recombination coefficients starts from 125 K (log T_{e} = 2.1). For electron densities greater than 10^{6} cm^{3}, as pointed out by Kisielius & Storey (2002), it is necessary to include lchanging collisions for n < 11. This is however beyond the scope of the current treatment.
In Tables 3−6 we present the effective recombination coefficient, α_{eff}(λ), in units of cm^{3} s^{1}, for strongest N ii transitions with valence electron orbital angular momentum quantum number l ≤ 5, at electron densities N_{e} = 10^{2}, 10^{3}, 10^{4} and 10^{5} cm^{3}, respectively, in Case B. The effective recombination coefficient is defined such that the emissivity ϵ(λ), in a transition of wavelength λ is given by $\mathit{\u03f5}\mathrm{\left(}\mathit{\lambda}\mathrm{\right)}\mathrm{=}{\mathit{N}}_{\mathrm{e}}{\mathit{N}}_{\mathrm{+}}{\mathit{\alpha}}_{\mathrm{eff}}\mathrm{\left(}\mathit{\lambda}\mathrm{\right)}\frac{\mathit{hc}}{\mathit{\lambda}}\mathrm{[}{\mathrm{erg}\mathrm{cm}}^{3}{\mathrm{s}}^{1}\mathrm{]}\mathit{.}$(2)Transitions included in the tables are selected according to the following criteria:

(1)
λ ≥ 912 Å;

(2)
α_{eff}(λ) ≥ 1.0 × 10^{14} cm^{3} s^{1} at T_{e} = 1000 K for all N_{e}’s, and ≥ 1.0 × 10^{15} cm^{3} s^{1} at all T_{e}’s and N_{e}’s;

(3)
all finestructure components are presented for multiplets from the 3d − 3p and 3p − 3s configurations. For the 4f − 3d configurations, a few selected multiplets are listed, but only V38 and V39 includes all the individual components. These transitions fall in the visible part of the spectrum and among them are the strongest recombination lines of N ii.
In Tables 3−6, the wavelengths of all the 4−3 and 3−3 transitions and majority of the 5−4 and 5−3 transitions are experimentally known. All the wavelengths of the 6−5 and 6−4 transitions are predicted. Our calculated wavelengths, derived from the experimental energies, agree with the experimentally known wavelengths within 0.001%. Our predicted wavelengths for experimentally unknown transitions agree with those predicted by Hirata & Horaguchi (1995) within 0.007% except for one 6d−3p transition wavelength which differs by 0.24 Å. However, this transition is spectroscopically less important compared to the 4−3 and 3−3 ones.
In these tables, we use the paircoupling notation $\mathit{L}\mathrm{[}\mathit{K}{\mathrm{]}}_{\mathit{J}}^{\mathit{\pi}}$ for the states belonging to the (^{2}P°) nf and ng configurations, as in Eriksson (1983). As shown by Cowan (1981), paircoupling is probably appropriate for the states of intermediatel (l = 3,4). The same notation is adopted for the (^{2}P°) nh configurations. For states belonging to lowl (l ≤ 2) configurations, LScoupling notation ${}^{\mathrm{2}\mathit{S}\mathrm{+}\mathrm{1}}\mathit{L}_{\mathit{J}}^{\mathit{\pi}}$ is used.
4.2. Effective recombination coefficient fits
We fit the effective recombination coefficients as a function of electron temperature in logarithmic space with analytical expressions for selected transitions, using a nonlinear leastsquare algorithm. Tables 7−14 present fit parameters and maximum deviations δ[%] for four densities, N_{e} = 10^{2}, 10^{3}, 10^{4} and 10^{5} cm^{3}. Only strongest optical transitions are presented, including multiplets V3, V5, V19, V20, V28, V29, V38 and V39. As the dependence of recombination coefficient on T_{e} at electron temperatures below 10 000 K is different from that at high temperatures (10 000 ~ 20 000 K in our case), we use different expressions for the two temperature regimes.
Fig. 4 Analysis fit to the effective recombination coefficients for the N ii V3 3p ^{3}D_{3}–3s ^{3}P$\begin{array}{}\mathrm{\u25e6}\\ \mathrm{2}\end{array}$λ5679.56 line, at N_{e} = 1000 cm^{3}. Subfigure a) shows the data fit for the lowtemperature regime T_{e} < 10 000 K, and Subfigure b) for the hightemperature regime 10 000 ≤ T_{e} ≤ 20 000 K. In both Subfigures, solid lines are fitting equations (Eq. (3) for Subfigure a) and Eq. (4) for b)), and plus signs “+” are the calculated data for all the temperatures. 
For the lowtemperature regime, T_{e} < 10 000 K, effective recombination coefficients are dominated by contribution from radiative recombination α_{rad}, which has a relatively simple dependence on electron temperature, α_{rad} ∝ T_{e}^{−a}, where a ~ 1. At low temperatures, dielectronic recombination through lowlying autoionizing states are also important for ions such as C ii, N ii, O ii, Ne ii (Storey 1981, 1983; Nussbaumer & Storey 1983, 1984, 1986, 1987). Considering the fact that direct radiative recombination rate is nearly a linear function of temperature in logarithm, and the deviation introduced by dielectronic recombination, we use a fiveorder polynomial expression to fit the effective recombination coefficient, $\mathit{\alpha}\mathrm{=}{\mathit{a}}_{\mathrm{0}}\mathrm{+}{\mathit{a}}_{\mathrm{1}}\hspace{0.17em}\mathit{t}\mathrm{+}{\mathit{a}}_{\mathrm{2}}\hspace{0.17em}{\mathit{t}}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{3}}\hspace{0.17em}{\mathit{t}}^{\mathrm{3}}\mathrm{+}{\mathit{a}}_{\mathrm{4}}\hspace{0.17em}{\mathit{t}}^{\mathrm{4}}\mathrm{+}{\mathit{a}}_{\mathrm{5}}\hspace{0.17em}{\mathit{t}}^{\mathrm{5}}\mathit{,}$(3)where α = log _{10} α_{eff} + 15 and t = log _{10} T_{e}, and a_{0}, a_{1}, a_{2}, a_{3}, a_{4} and a_{5} are constants.
For the hightemperature regime, 10 000 ≤ T_{e} ≤ 20 000 K, the contribution from dielectronic recombination, α_{DR}, can significantly exceed that of direct radiative recombination (Burgess 1964). Dielectronic recombination coefficient α_{DR} has a complex exponential dependence on T_{e} (Seaton & Storey 1976; Storey 1981), α_{DR} ∝ ${\mathit{T}}_{\mathrm{e}}^{\mathrm{}\mathrm{3}\mathit{/}\mathrm{2}}$ exp( − E/k T_{e}), where E is the excitation energy of an autoionizing state, to which a free electron is captured, relative to the ground state of the recombining ion (N^{2+} in our case) and k is the Boltzmann constant. The expression adopted for this temperature regime is, $\mathit{\alpha}\mathrm{=}\mathrm{(}{\mathit{b}}_{\mathrm{0}}\mathrm{+}{\mathit{b}}_{\mathrm{1}}\hspace{0.17em}\mathit{t}\mathrm{+}{\mathit{b}}_{\mathrm{2}}\hspace{0.17em}{\mathit{t}}^{\mathrm{2}}\mathrm{+}{\mathit{b}}_{\mathrm{3}}\hspace{0.17em}{\mathit{t}}^{\mathrm{3}}\mathrm{+}{\mathit{b}}_{\mathrm{4}}\hspace{0.17em}{\mathit{t}}^{\mathrm{4}}\mathrm{)}\mathrm{\times}{\mathit{t}}^{{\mathit{b}}_{\mathrm{5}}}\mathrm{\times}\mathrm{exp}\mathrm{(}{\mathit{b}}_{\mathrm{6}}\hspace{0.17em}\mathit{t}\mathrm{)}\mathit{,}$(4)where α = log _{10} α_{eff} + 15 and t = T_{e} [K] /10^{4}, the reduced electron temperature, and b_{0}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5} and b_{6} are constants.
In order to make the data fits accurate for the hightemperature regime, 10 000 ≤ T_{e} ≤ 20 000 K, where the original calculations are carried out for only four temperature cases (log T_{e} [K] = 4.0, 4.1, 4.2 and 4.3), nine more temperature cases are calculated. For the temperature region log T_{e} [K] = 3.9 ~ 4.0, two more temperature cases are also calculated, so that the data fits near 10 000 K are accurate enough. Figure 4 is an example of the fit to the effective recombination coefficients of the N ii V3 2s^{2}2p3p ^{3}D$\begin{array}{}\mathrm{e}\\ \mathrm{3}\end{array}$–2s^{2}2p3s ^{3}P$\begin{array}{}\mathrm{\u25e6}\\ \mathrm{2}\end{array}$λ5679.56 transition.
By using different expressions for the two temperature regimes, we manage to control the maximum fitting errors to well within 0.5 per cent.
4.3. Relative intensities within N ii multiplets
As mentioned in the Sect. 3.4 above, the populations of the ground finestructure levels ^{2}P° _{1/2,3/2} of the recombining ion N^{2+} vary with electron density under typical nebular conditions. The variations are reflected in the relative intensities of the resultant recombination lines of N ii, which arise from upper levels with the same orbital angular momentum quantum number l but of different parentage, i.e., ^{2}P° _{1/2} and ^{2}P° _{3/2} in the current case. A number of such recombination lines have been observed in photoionized gaseous nebulae including PNe and H ii regions, and their intensity ratios can thus be used for density diagnostics.
As the relative populations of ^{2}P° _{1/2,3/2} vary with N_{e}, so do the fractional intensities of individual finestructure components within a given multiplet of N ii. The most prominent N ii multiplets in the optical include: V3 2s^{2}2p3p ^{3}D^{e}–2s^{2}2p3s ^{3}P°, V19 2s^{2}2p3d ^{3}F°–2s^{2}2p3p ^{3}D^{e} and V39 2s^{2}2p4f G[7/2,9/2]^{e}–2s^{2}2p3d ^{3}F°.
4.3.1. 2s^{2}2p3p ^{3}D^{e}–2s^{2}2p3s ^{3}P° (V3)
The fractional intensities of finestructure components of Multiplet V3, 2s^{2}2p3p ^{3}D^{e}–2s^{2}2p3s ^{3}P°, are presented in Fig. 5. The strongest component is λ5679.56, which forms from core ^{2}P° _{3/2} capturing an electron plus cascades from higher states, while the second strongest component λ5666.63 can form, in addition, from recombination of core ^{2}P° _{1/2}.
For the target N iii, the population of the finestructure level ^{2}P° _{3/2} relative to ^{2}P° _{1/2} increases with electron density N_{e} due to collisional excitation, and consequently, so does the intensity of the λ5679.56 line relative to the λ5666.63 line. Their intensity ratio peaks around N_{e} = 2000 cm^{3}, the critical density N_{c} of the level ^{2}P° _{3/2}, and then decreases as N_{e} increases further. The relative intensities of all components converge to constant values at high densities ( ≥ 10^{5} − 10^{6} cm^{3}), as the relative populations of the ground finestructure levels of the target N iii approach the Boltzmann distribution.
The line ratio I(λ5679.56)/I(λ5666.63) thus serves as a density diagnostic for nebulae of low and intermediate densities, N_{e} ≤ 10^{5} cm^{3}. At very low electron temperatures, where kT_{e} is comparable to the ^{2}P° _{1/2}–^{2}P° _{3/2} energy separation, the sensitivity to density in the components of V3 is reduced. This arises because, at very low temperatures, the states (^{2}P^{o} _{3/2}) nl are populated more significantly by dielectronic capture from the (^{2}P° _{1/2}) κl continuum than by direct recombination on N^{2+} (^{2}P° _{3/2}). The density dependence of the population distribution between the ^{2}P° _{1/2} and ^{2}P° _{3/2} is then of less importance.
Fig. 5 Fractional intensities of components of Multiplet V3: 2s^{2}2p3p ^{3}D^{e}–2s^{2}2p3s ^{3}P°. The numbers in brackets following the wavelength labels are the total angular momentum quantum numbers J_{2} − J_{1} of the upper to lower levels of the transition. Components from upper levels of the same total angular momentum quantum number J are represented by same colour and line type. Four temperature cases, log _{10}T_{e} = 2.5, 3.0, 3.5 and 4.0 K, are presented. 
4.3.2. 2s^{2}2p3d ^{3}F°–2s^{2}2p3p ^{3}D^{e} (V19)
The fractional intensities of finestructure components of Multiplet V19, 2s^{2}2p3d ^{3}F°–2s^{2}2p3p ^{3}D^{e}, are presented in Fig. 6. The strongest component, λ5005.15, forms exclusively from recombination of target ^{2}P° _{3/2} plus cascades, while the second and third strongest components of almost identical wavelengths, λ5001.48 and λ5001.14, can form, in addition, from recombination of the ground target ^{2}P° _{1/2}.
At very low densities of about 10^{2} cm^{3}, ^{2}P° _{1/2} dominates the population of N iii, and this is manifested by the intensity of the λ5005.15 line being lower than the λ5001.48 line and than λ5001.14 by a further amount. As electron density increases, the intensity of the λ5005.15 line relative to the λ5005.48/15 lines increases and peaks around 2000 cm^{3}. At densities above 10^{5} cm^{3}, the fractional populations of all components converge to constant values. The trends are similar to Multiplet V3 discussed above.
The intensity ratio I(λ5005.15)/I(λ5001.48 + λ5001.14) serves as another potential density diagnostic. In reality, however, given the closeness in wavelength of the λ5005.15 line to the [O iii] λ5007 nebular line, which is often several orders of magnitude (3 − 4) brighter, accurate measurement of λ5005.15 line is essentially impossible.
4.3.3. 2s^{2}2p4f G[7/2,9/2]^{e}–2s^{2}2p3d ^{3}F° (V39)
The fractional intensities of finestructure components of Multiplet V39, 2s^{2}2p4f G[7/2,9/2]^{e}–2s^{2}2p3d ^{3}F°, are presented in Fig. 7. The strongest component λ4041.31 forms exclusively from recombination of target ^{2}P° _{3/2} plus cascades from higher states, while the second and third strongest components, λ4035.08 and λ4043.53, which have comparable intensities, can form, in addition, from recombination of target ^{2}P° _{1/2}.
The behaviour of the intensity of the λ4041.31 line relative to those of the λ4035.08 and λ4043.53 lines as a function of electron density is quite similar to those of their counterparts of Multiplets V3 and V19 discussed above.
The line ratios I(λ4041.31)/I(λ4035.08) and I(λ4041.31)/ I(λ4043.53) can in principle serve as additional density diagnostics. There are however complications in their applications:

(1)
All finestructure components of Multiplet V39 2s^{2}2p4f G[7/2,9/2]^{e}–2s^{2}2p3d ^{3}F° are extremely faint. The strongest component λ4041.31 is 2 − 3 times fainter than λ5679.56, the strongest component of V3, while the latter is typically one thousand times fainter than Hβ in a real nebula.

(2)
The λ4041.31 line is blended with the O ii recombination line λ4041.29 of Multiplet V50c 2p^{2}4f F[2]° _{5/2}–2p^{2}3d ^{4}F^{e} _{5/2}, while the λ4035.08 line is blended with the O ii lines λ4035.07 of Multiplet V50b 2p^{2}4f F[3]° _{5/2}–2p^{2}3d ^{4}F^{e} _{5/2} and λ4035.49 of Multiplet V50b 2p^{2}4f F[3]° _{7/2}–2p^{2}3d ^{4}F^{e} _{5/2}.
Fig. 8 Loci of the N ii recombination line ratios I(λ5679.56)/I(λ5666.63) and I(λ5679.56)/I(λ4041.31) for different T_{e}’s and N_{e}’s. 
4.4. Plasma diagnostics
Unlike the UV and optical CELs, whose emissivities have an exponential dependence on T_{e} (Osterbrock & Ferland 2006), emissivities of heavy element ORLs have only a relatively weak, powerlaw dependence on T_{e}. The dependence varies for lines originating from levels of different orbital angular momentum quantum number l. Thus the relative intensities of ORLs can also be used to derive electron temperature, provided very accurate measurements can be secured (Liu 2003; Liu et al. 2004; Tsamis et al. 2004). In the case of N ii, the intensity ratio of λ5679.56 and λ4041.31 lines, the strongest components of Multiplets V3 3p ^{3}D^{e}–3s ^{3}P° and V39 4f G[9/2]^{e}–3d ^{3}F°, respectively, has a relatively strong temperature dependence, and thus can serve as a temperature diagnostic. As shown in Sect. 4.3 above, the N ii line ratio I(λ5679.56)/I(λ5666.63) is a good density diagnostic. Combining the two line ratios thus allows one to determine T_{e} and N_{e} simultaneously. Figure 8 shows the loci of N ii recombination line ratios I(λ5679.56)/I(λ5666.63) and I(λ5679.56)/I(λ4041.31) for different electron temperatures and densities. With high quality measurements of the two line ratios, one can readout T_{e} and N_{e} directly from the diagram.
Ions such as N ii and O ii have a rich optical recombination line spectrum. Rather than relying on specific line ratios, it is probably beneficial and more robust to determine T_{e} and N_{e} by fitting all lines with a good measurement and free from blending simultaneously. Details about this approach and its application to photoionized gaseous nebulae will be the subject of a subsequent paper.
4.5. Population of excited states of N^{2+}
In the current calculations of the effective recombination coefficients of N ii, we have assumed that only the ground finestructure levels of the recombining ion, N^{2+} 2s^{2}2p ^{2}P° _{1/2,3/2}, are populated. This is a good approximation under typical nebular conditions. The first excited spectral term of N^{2+}, 2s2p^{2} ^{4}P, lies 57 161.7 cm^{1} above the ground term (Eriksson 1983), and the population of this term is 1.75 × 10^{6} relative to that of the ground term ^{2}P° even at the highest temperature and density considered in the current work, T_{e} = 20 000 K and N_{e} = 10^{6} cm^{3}. Recombination from the 2s2p^{2} ^{4}P term is thus completely negligible.
4.6. Total recombination coefficients
Calculations presented in the current work are carried out in intermediate coupling in Case B, representing a significant improvement compared to Kisielius & Storey (2002), in which the calculations are entirely in LScoupling. In our calculations, we have also considered the fact that the ground term of N^{2+} comprises two finestructure levels, ^{2}P° _{1/2} and ^{2}P° _{3/2}, and the populations of those two levels deviate from the Boltzmann distribution under typical nebular densities. We thus treat the recombination of the highl states using the closecoupling photoionization data incorporating the population distribution among the ^{2}P$\begin{array}{}\mathrm{\u25e6}\\ \mathit{J}\end{array}$ levels. The critical electron density, at which the rates of collisional deexcitation and radiative decay from ^{2}P° _{3/2} to ^{2}P° _{1/2} are equal, is approximately 2000 cm^{3}. At this density, our calculation shows that the populations of the ^{2}P° _{1/2} and ^{2}P° _{3/2} levels differ from their Boltzmann values by approximately 34% at T_{e} = 10 000 K, The difference given by Kisielius & Storey (2002) at this density is 30%.
In Table 15, we compare our direct recombination coefficients with those calculated by Nahar (1995) and by Kisielius & Storey (2002). The calculations of Nahar (1995) and Kisielius & Storey (2002) are both in LScoupling, and the N ii states are not Jresolved. Their direct recombination coefficients are all to spectral term ${}^{\mathrm{2}\mathit{S}\mathrm{+}\mathrm{1}}\mathit{L}^{\mathit{\pi}}$. Our present calculations are in intermediate coupling, and recombinations are all to Jresolved levels. In order to compare to their results, we sum the direct recombination coefficients to all the finestructure levels belonging to individual spectral terms.
At 1000 K, the differences between the results of Kisielius & Storey (2002) and ours are less than 10% for most cases, except for the state 2s2p^{3} ^{3}S°. For this state, our direct recombination coefficient is 50% larger than that of Kisielius & Storey (2002).
At this temperature (1000 K is about 0.1 eV), we believe we have found out the exact energy positions for all the resonances below 0.1 Ryd above the ionization threshold of N iii 2p ^{2}P° _{1/2}. This region contains most of the important resonances that dominate the total recombination rate. In our photoionization calculations, all resonances from those of widths as narrow as 10^{9} Ryd to those of widths as wide as 10^{4} Ryd, are properly resolved using a highly adaptive energy mesh. There are typically about 22 points sampling each resonance. In the calculation by Kisielius & Storey (2002), the number is about ten, while in Nahar (1995) a fixed interval of 0.0004 Ryd is used in this energy range.
The three lowlying resonances, ^{3}P, ^{3}D and ^{3}F belonging to the 2s2p^{2}(^{4}P) 3d configuration, are situated between 0.075 and 0.085 Ryd above the ionization threshold of ^{2}P° _{1/2} (Kisielius & Storey 2002). For the state 2s2p^{3} ^{3}S°, one of the main sources of recombination is from the term ^{3}P belonging to the 2s2p^{2}(^{4}P) 3d configuration. There are three finestructure resonance levels of the ^{3}P term with quantum numbers J = 0 − 2. The full widths of these three resonances are 1.02 × 10^{4} Ryd for ^{3}P$\begin{array}{}\mathrm{e}\\ \mathrm{0}\end{array}$, 1.35 × 10^{5} Ryd for ^{3}P$\begin{array}{}\mathrm{e}\\ \mathrm{1}\end{array}$ and 1.42 × 10^{5} Ryd for ^{3}P$\begin{array}{}\mathrm{e}\\ \mathrm{2}\end{array}$. The steps of energy mesh adopted for the three resonances are: 4.64 × 10^{6} Ryd for ^{3}P$\begin{array}{}\mathrm{e}\\ \mathrm{0}\end{array}$, 6.14 × 10^{7} Ryd for ^{3}P$\begin{array}{}\mathrm{e}\\ \mathrm{1}\end{array}$ and 6.46 × 10^{7} Ryd for ^{3}P$\begin{array}{}\mathrm{e}\\ \mathrm{2}\end{array}$.
Our calculations are carried out entirely in intermediate coupling. This leads to a high recombination rate to the 2s2p^{3} ^{3}S° state, produced by radiative intercombination transitions (transitions between levels of different total spins) from levels above the ionization threshold to the 2s2p^{3} ^{3}S° level. The widths of such intercombination transitions are usually much narrower than those of allowed transitions. For example, the resonance level ^{5}P^{e} _{1} belonging to the configuration 2s2p^{2}(^{4}P) 3d lies about 0.051 Ryd above the ionization threshold, and it can decay to the level 2s2p^{3} ^{3}S° _{1} via an intercombination transition. The width of this resonance is 2.49 × 10^{9} Ryd, and the energy interval of the photoionization mesh is set to 1.13 × 10^{10} Ryd. Intercombination transitions were not considered in Kisielius & Storey (2002), given the calculations were in LScoupling.
At 1000 K, the differences between the calculations of Nahar (1995) and ours are smaller than 10%, except for states belonging to the 2s2p^{3} configuration.
At 10 000 K, the differences between the calculations of Kisielius & Storey (2002) and ours are all better than 10%. The agreement for the state 2s2p^{3} ^{3}S° is particularly good.
At this temperature, the differences between the results of Nahar (1995) and ours are within 15%, except for states ^{3}S°, ^{3}P° and ^{3}D° belonging to the configuration 2s2p^{3}, where the differences are larger than 30%. The large discrepancies are likely to be caused by the coarse energy mesh adopted by Nahar (1995) for the photoionization calculations, leading to the recombination rates to states belonging to the 2s2p^{3} configuration being underestimated.
In Table 15, we compare our total direct recombination coefficients, which are the sum of all the direct recombination coefficients to individual atomic levels with n ≤ 35, with those of Nahar (1995) and Kisielius & Storey (2002). At 1000 K, our total recombination coefficient is 13 per cent lower than that of Kisielius & Storey (2002). That is probably because the sum only reaches up to n = 35. At 10 000 K, our total recombination coefficient is higher than the other two.
5. Conclusion
Effective recombination coefficients for the N^{+} recombination line spectrum have been calculated in Case B for a wide range of electron density and temperature. The results are fitted with analytical formulae as a function of electron temperature for different electron densities, to an accuracy of better than 0.5%.
The high quality basic atomic data adopted in the current work, including photoionization crosssections, boundbound transition probabilities, and bound state energy values, were obtained from Rmatrix calculations for all bound states with n ≤ 11 in the intermediate coupling scheme. All major resonances near the ionization thresholds were properly resolved. In calculating the N ii level populations, we took into account the fact that the populations of the ground finestructure levels of the recombining ion N^{2+} deviate from the Boltzmann distribution. Finestructure dielectronic recombination, which occurs through high Rydberg states lying between the doublet ^{2}P$\begin{array}{}\mathrm{\u25e6}\\ \mathrm{1}\mathit{/}\mathrm{2}\mathit{,}\hspace{0.17em}\mathrm{3}\mathit{/}\mathrm{2}\end{array}$ thresholds and is very effective at low temperatures ( ≤ 250 K), was also included in the current investigation. The calculations extend to l ≤ 4.
The effective recombination coefficients for the N ii recombination spectrum presented in the current work represent recombination processes under typical nebular conditions. The sensitivity of individual lines within a multiplet to the density and temperature of the emitting medium opens up the possibility of electron temperature and density diagnostics and abundance determinations which were not possible with earlier theory.
AUTOSTRUCTURE is developed by the Department of Physics at the University of Strathclyde, Glasgow, Scotland. The code is available from the website http://amdpp.phys.strath.ac.uk/autos
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All Tables
All Figures
Fig. 1 The data calculated by different methods for the five energy regions have been joined together. The insert zooms in a particular energy area, and the mesh points used for the photoionization calculations are shown in the inset. 

In the text 
Fig. 2 Schematic figure showing the lowtemperature ( ≤ 250 K) dielectronic recombination of N ii through the finestructure autoionizing levels between the two lowest ionization thresholds of N iii ^{2}P° _{1/2} and ^{2}P° _{3/2}. The electrons captured to the highn autoionizing levels decay to lown bound states through cascades and optical recombination lines (ORLs) are thus emitted. Here Multiplets V3, V19 and V39 are presented as examples. 

In the text 
Fig. 3 Fractional populations of the N^{2+} ^{2}P${}^{\mathrm{\u25e6}}\hspace{0.17em}_{\mathrm{1}\mathit{/}\mathrm{2}}$ and ^{2}P${}^{\mathrm{\u25e6}}\hspace{0.17em}_{\mathrm{3}\mathit{/}\mathrm{2}}$ parent levels. The red solid and blue dashed curves represent the populations of the ^{2}P° _{1/2} and ^{2}P° _{3/2} levels, respectively. Four temperature cases, 200 K, 1000 K, 5000 K and 10 000 K are shown. 

In the text 
Fig. 4 Analysis fit to the effective recombination coefficients for the N ii V3 3p ^{3}D_{3}–3s ^{3}P$\begin{array}{}\mathrm{\u25e6}\\ \mathrm{2}\end{array}$λ5679.56 line, at N_{e} = 1000 cm^{3}. Subfigure a) shows the data fit for the lowtemperature regime T_{e} < 10 000 K, and Subfigure b) for the hightemperature regime 10 000 ≤ T_{e} ≤ 20 000 K. In both Subfigures, solid lines are fitting equations (Eq. (3) for Subfigure a) and Eq. (4) for b)), and plus signs “+” are the calculated data for all the temperatures. 

In the text 
Fig. 5 Fractional intensities of components of Multiplet V3: 2s^{2}2p3p ^{3}D^{e}–2s^{2}2p3s ^{3}P°. The numbers in brackets following the wavelength labels are the total angular momentum quantum numbers J_{2} − J_{1} of the upper to lower levels of the transition. Components from upper levels of the same total angular momentum quantum number J are represented by same colour and line type. Four temperature cases, log _{10}T_{e} = 2.5, 3.0, 3.5 and 4.0 K, are presented. 

In the text 
Fig. 6 Same as Fig. 5 but for Multiplet V19: 2s^{2}2p3d ^{3}F°–2s^{2}2p3p ^{3}D^{e}. 

In the text 
Fig. 7 Same as Fig. 5 but for Multiplet V39: 2s^{2}2p4f G[7/2,9/2]^{e}–2s^{2}2p3d ^{3}F°. 

In the text 
Fig. 8 Loci of the N ii recombination line ratios I(λ5679.56)/I(λ5666.63) and I(λ5679.56)/I(λ4041.31) for different T_{e}’s and N_{e}’s. 

In the text 
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