3 Schematic model

We illustrate the relevant processes in the formation of the resonance, intercombination, and forbidden lines with a simplified level scheme (cf. Mewe & Schrijver 1978a) consisting of the following levels denoted by short labels: g: ground level 1 $^{1}{\rm S}_{0}$; $m^{\prime}$: upper metastable level 2¹S₀ of the two-photon transition; 1$^{\prime}$: upper level 2¹P₁ of the resonance line; m: upper metastable level 2³S₁ of the forbidden (f) line; p_k (k=1,2,3: levels 2 $^3{\rm P}_k$ (2 $^3{\rm P}_1$ is the upper level of the by far the strongest component (y) of the intercombination line i, and 2 $^3{\rm P}_2$ is the upper level of the weaker component (x)); c: continuum level which lumps together all higher levels to represent the cascades from excitation and recombination processes.

The electron collisional rate coefficient (in cm³ s^-1) for transition $j \to k$ is written as:

$$% C_{jk} = 8.63\times 10^{-6} {\gamma_{jk}\over {w_j \sqrt{T_... ...}}} {\rm exp}\Bigl(-{{\Delta E_{jk}}\over {kT_{\rm e}}}\Bigr),$$ (3)

where $\Delta E_{jk}$ is the excitation energy, $T_{\rm e}$ is the electron temperature in K, $\gamma_{jk}$ the collision strength averaged over a Maxwellian electron energy distribution, and w_j the statistical weight of the initial level j.

The total rate coefficients for the formation of the forbidden (f or z notation) and intercombination (i or x+y) line can be written as (e.g., Mewe & Schrijver 1978a, Eqs. (18-30)):

$$BR_{mg} \Bigl[ C_{gm} + \sum_{k=0}^2 BR_{p_km} C_{gp_k}\Bigr],$$ (4)

$$\sum_{k=1}^2 BR_{p_kg} \Bigl[ C_{gp_k} + BR_{mp_k} C_{gm} \Bigr],$$ (5)

with the various branching ratios:

$$% BR_{mg} = {A_{mg}\over {A_{mg} + n_{\rm e} S_{mp}^{\prime}}},$$ (6)

$$% S_{mp}^{\prime} = \sum_{k=1}^2 C_{mp_k}BR_{p_kg}^{\prime},$$ (7)

$$% BR_{p_km} = {A_{p_km}\over {A_{p_km} + A_{p_kg}}} \ \ (BR_{p_0m} \equiv 1),$$ (8)

$$% BR_{p_kg}^{\prime} = {A_{p_kg}\over {A_{p_kg} + A_{p_km}}} \ \ (BR_{p_0g}^{\prime} \equiv 0),$$ (9)

$$% BR_{p_kg} = {A_{p_kg}\over {A_{p_kg} + A_{p_km}BR_{mg}}} \ \ (BR_{p_0g} \equiv 0),$$ (10)

$$% BR_{mg} = {A_{mg}\over {A_{mg} + n_{\rm e} S_{mp}}},$$ (11)

$$% BR_{mp_k} = {{n_{\rm e} C_{mp_k}}\over {A_{mg} + n_{\rm e} S_{mp}}},$$ (12)

$$% S_{mp} = \sum_{k=0}^2 C_{mp_k}.$$ (13)

Analogously, we can derive the total rate coefficients for the formation of the resonance (r or w) line or two-photon radiation (2ph) by substituting $m \rightarrow m^{\prime}$, $p,p_k \rightarrow 1^{\prime}$ and performing no summation:

$$% I_r = BR_{1^{\prime}g} \Bigl[ C_{g1^{\prime}} + BR_{m^{\prime}1^{\prime}} C_{gm^{\prime}} \Bigr],$$ (14)

$$% I_{2ph} = BR_{m^{\prime}g} \Bigl[ C_{gm^{\prime}} + BR_{1^{\prime}m} C_{g1^{\prime}}\Bigr],$$ (15)

and changing all BR's etc. appropriately.

We note that the collision coefficients C_jk include also the n > 2 cascades. The radiative transition probabilities are tabulated in the Table 2 of PaperI and the effective collision strengths $\gamma$ in their Tables 9-13 and in their Fig. 4. We assume in this section that the electron density is so low that the collision de-excitation rate $n_{\rm e}C_{p_km}$ can be neglected with respect to the spontaneous radiative rate A_pkm (e.g., for C V $n_{\rm e}C_{p_km} < A_{p_km}$ for $n_{\rm e} < 10^{15}$ cm^-3). This can be easily taken into account by the substitution

$$% A_{p_km} \rightarrow A_{p_km} + n_{\rm e} C_{p_km}.$$ (16)

Further, all collision processes that couple the singlet and triplet system are neglected in this schematic model. It turns out that at high density the coupling $m,p_k \leftrightarrow m^{\prime},1^{\prime}$ cannot be neglected.

In the full calculations used in this work (see Sect. 5), the coupling between the singlet and triplet system has been taken into account, as well as the collisional de-excitation.

If we take also into account the contribution from radiative and dielectronic recombinations of the hydrogen-like ion we substitute

$$% C_{gk} \rightarrow C_{gk} + \alpha^{\prime}_{ck},$$ (17)

where k = m or k = p_k, respectively, and

$$% \alpha^{\prime}_{ck} = (N_{\rm H}/N_{\rm He}) \alpha_{ck},$$ (18)

where $\alpha_{ck}$ is the total radiative and dielectronic recombination rate coefficient including cascades and $N_{\rm H}/N_{\rm He}$ is the abundance ratio of hydrogen-like to helium-like ions (e.g. taken from Arnaud & Rothenflug 1985; Mazzotta et al. 1998). The recombination coefficients are given in Tables 3-8 in Paper I. However, for a collision-dominated plasma, the recombination generally gives only a minor effect (i.e. far less than few percents), but are nevertheless introduced in the full calculations in Sect. 5.

Mewe & Schrijver (1978a) took also into account the effect of a stellar radiation field (also called photo-excitation) with effective radiation temperature $T_{\rm rad}$. This can be done by substituting in the above equations:

$$% n_{\rm e} C_{mp_k} \rightarrow n_{\rm e} C_{mp_k} + B_{mp_k},$$ (19)

$$% n_{\rm e} S_{mp} \rightarrow n_{\rm e} S_{mp} + B_{mp},$$ (20)

where $B_{mp} = \sum_{k=0}^2 B_{mp_k}$ and

$$% B_{mp_k} = {{W A_{p_km} (w_{p_k}/w_m)}\over {{\rm exp}\Bigl({{\Delta E_{mp_k}}\over {kT_{\rm rad}}}\Bigr) - 1}},$$ (21)

is the rate (in s^-1) of absorption $m \rightarrow p_k$ and W is the dilution factor of the radiation field (as a special case we take $W={1\over 2}$ close to the stellar surface). We have checked that the radiation field is so low that the induced emission rate $B_{p_km}=(w_m/w_{p_k})\,B_{mp_k}$ is negligible respect to the spontaneous radiative rate A_pkm. Nevertheless, in the full calculations used in this work (see Sect. 5), we have taken into account induced emission as well as photo-excitation.

Mewe & Schrijver (1978a) considered also inner-shell ionization of the Lithium-like ion which can give an important contribution to the forbidden line in a transient plasma (Mewe & Schrijver 1978b) such as a supernova remnant. However, in the present calculations we neglect this because we consider plasmas in ionization equilibrium.

Finally, Mewe & Schrijver (1978a, 1978c) have considered also excitation 2³S$\to$ 2³P by proton collisions using approximations of Coulomb-Born results from Blaha (1971). In a new version (SPEX90) of our spectral code SPEX (Kaastra et al. 1996), which contains for the H- and He-like ions an improvement of the known MEKAL code (Mewe et al. 1985, 1995a), proton collisions are taken into account based on Blaha's results. Test calculations with SPEX90 show that for an equilibrium plasma in all practical cases proton excitation is negligible compared to electron excitation.

In the case where $N_{\rm H}/N_{\rm He} \gg 1$, recombination is dominant e.g., for photo-ionized plasmas it turns out that the ratios $R\equiv f/i$ are for collisional and photo-ionized plasmas comparable in the same density range (cf. also Mewe 1999), but the ratios $G\equiv (i+f)/r$ are very different, e.g., $G \sim 1$ for a collisional plasma and a factor ${\sim}2{-}4$ larger for a photo-ionized plasma where the resonance line is relatively much weaker.