2 Atomic data

Liedahl ([1999]) described the basic mechanisms of density diagnostics for X-ray photoionized plasmas from He-like ions. As he noted, a proper calculation of the population of the n=2 shell levels depends upon a number of additional levels. We propose in this article to use extensive calculations of atomic data taking into account upper level (n>2) radiative cascade contribution on n=2 shell levels for C V, N VI, O VII, Ne IX, Mg XI, and Si XIII, to give a much more precise treatment of this plasma diagnostic.

We consider in this paper, the main atomic processes involved in pure photoionized and hybrid plasmas: radiative recombination and dielectronic recombination (only important for high temperature plasmas), collisional excitation inside the n=2 shell, and collisional excitation from the ground level (important for high temperature plasmas).

2.1 Energy levels, radiative transition probabilities

Using the SUPERSTRUCTURE code (Eissner et al. [1974]), we have calculated the energy levels for the first 49 fine-structure levels ( ^2S+1L_J) for the six ions. This corresponds to the levels of the first 15 configurations (from 1s² to 1s5g). Nevertheless, for the first seven levels, we have preferred to use the Vainshtein & Safronova ([1985]) data which have a slightly better accuracy ($\sim$10^-3).

 

 

Table 1: Energy (in cm^-1) for the first 17 levels for C V, N VI, O VII, Ne IX, Mg XI and Si XIII calculated by the SUPERSTRUCTURE code (except for the first seven levels which are from Vainshtein & Safronova 1985). Here X(Y) means $X\ 10^{Y}$

i	conf	level	C V	N VI	O VII	Ne IX	Mg XI	Si XIII
1	1s2	$^{1} {\rm S}_{0}$	0.	0.	0.	0.	0.	0.
2	1s2s	$^{3} {\rm S}_{1}$	2.4114(+6)	3.3859(+6)	4.5253(+6)	7.2996(+6)	10.7358(+6)	14.8357(+6)
3	1s2p	$^{3} {\rm P}_{0}$	2.4553(+6)	3.4383(+6)	4.5863(+6)	7.3779(+6)	10.8317(+6)	14.9495(+6)
4	1s2p	$^{3} {\rm P}_{1}$	2.4552(+6)	3.4383(+6)	4.5863(+6)	7.3782(+6)	10.8325(+6)	14.9513(+6)
5	1s2p	$^{3} {\rm P}_{2}$	2.4554(+6)	3.4386(+6)	4.5869(+6)	7.3798(+6)	10.8361(+6)	14.9585(+6)
6	1s2s	$^{1} {\rm S}_{0}$	2.4551(+6)	3.4393(+6)	4.5884(+6)	7.3824(+6)	10.8385(+6)	14.9585(+6)
7	1s2p	$^{1} {\rm P}_{1}$	2.4833(+6)	3.4737(+6)	4.6291(+6)	7.4361(+6)	10.9062(+6)	15.0417(+6)
8	1s3s	$^{3} {\rm S}_{1}$	2.8239(+6)	3.9765(+6)	5.3251(+6)	8.6105(+6)	12.6824(+6)	17.5435(+6)
9	1s3p	$^{3} {\rm P}_{0}$	2.8352(+6)	3.9902(+6)	5.3441(+6)	8.6314(+6)	12.7081(+6)	17.5741(+6)
10	1s3p	$^{3} {\rm P}_{1}$	2.8352(+6)	3.9903(+6)	5.3412(+6)	8.6316(+6)	12.7087(+6)	17.5752(+6)
11	1s3p	$^{3} {\rm P}_{2}$	2.8353(+6)	3.9904(+6)	5.3414(+6)	8.6322(+6)	12.7099(+6)	17.5775(+6)
12	1s3s	$^{1} {\rm S}_{0}$	2.8401(+6)	3.9953(+6)	5.3463(+6)	8.6368(+6)	12.7136(+6)	17.5795(+6)
13	1s3d	$^{3} {\rm D}_{1}$	2.8408(+6)	3.9973(+6)	5.3497(+6)	8.6433(+6)	12.7238(+6)	17.5942(+6)
14	1s3d	$^{3} {\rm D}_{2}$	2.8408(+6)	3.9973(+6)	5.3497(+6)	8.6433(+6)	12.7239(+6)	17.5945(+6)
15	1s3d	$^{3} {\rm D}_{3}$	2.8408(+6)	3.9973(+6)	5.3498(+6)	8.6435(+6)	12.7244(+6)	17.5953(+6)
16	1s3d	$^{1} {\rm D}_{2}$	2.8411(+6)	3.9977(+6)	5.3502(+6)	8.6411(+6)	12.7251(+6)	17.5962(+6)
17	1s3p	$^{1} {\rm P}_{1}$	2.8433(+6)	4.0004(+6)	5.3534(+6)	8.6480(+6)	12.7294(+6)	17.6005(+6)

 

 

Table 2: Radiative transitions probabilities (A_ki in s^-1, i=1, 7; k=2, 17) for C V, N VI, O VII, Ne IX, Mg XI and Si XIII calculated by the SUPERSTRUCTURE code, except for marked values (a) which are from Lin et al. (1977) and (b) which are from Mewe & Schrijver (1978a). i and k correspond respectively to the lower and the upper level of the transition

		Aki (s-1)
i	k	C V	N VI	O VII	Ne IX	Mg XI	Si XIII
1	2	4.960(+01)$^{\rm a}$	2.530(+02)$^{\rm b}$	1.060(+03)$^{\rm a}$	1.100(+04)$^{\rm a}$	7.330(+04)$^{\rm a}$	3.610(+05)$^{\rm a}$
1	4	2.159(+07)	1.100(+08)	4.447(+08)	4.470(+09)	2.867(+10)	1.345(+11)
1	5	2.650(+04)$^{\rm a}$	1.030(+05)$^{\rm b}$	3.330(+05)$^{\rm a}$	2.270(+06)$^{\rm a}$	1.060(+07)$^{\rm a}$	3.890(+07)$^{\rm a}$
1	6	3.310(+05)$^{\rm a}$	9.430(+05)$^{\rm b}$	2.310(+06)$^{\rm a}$	1.000(+07)$^{\rm a}$	3.220(+07)$^{\rm a}$	8.470(+07)$^{\rm a}$
1	7	9.477(+11)	1.911(+12)	3.467(+12)	9.197(+12)	2.010(+13)	3.857(+13)
1	10	6.939(+06)	3.525(+07)	1.423(+08)	1.429(+09)	9.141(+09)	4.268(+10)
1	17	3.105(+11)	6.061(+11)	1.073(+12)	2.752(+12)	5.877(+12)	1.107(+13)
2	3	5.616(+07)	6.717(+07)	7.818(+07)	1.003(+08)	1.228(+08)	1.460(+08)
2	4	5.655(+07)	6.794(+07)	7.956(+07)	1.039(+08)	1.304(+08)	1.602(+08)
2	5	5.735(+07)	6.955(+07)	8.249(+07)	1.118(+08)	1.486(+08)	1.977(+08)
2	9	1.376(+10)	2.872(+10)	5.342(+10)	1.466(+11)	3.280(+11)	6.406(+11)
2	10	1.375(+10)	2.870(+10)	5.337(+10)	1.464(+11)	3.269(+11)	6.366(+11)
2	11	1.374(+10)	2.867(+10)	5.329(+10)	1.461(+11)	3.262(+11)	6.360(+11)
2	17	2.898(+05)	1.607(+06)	6.902(+06)	7.514(+07)	5.061(+08)	2.452(+09)
3	8	7.088(+08)	1.366(+09)	2.398(+09)	6.079(+09)	1.290(+10)	2.426(+10)
3	13	2.349(+10)	4.847(+10)	8.947(+10)	2.432(+11)	5.408(+11)	1.052(+12)
4	8	2.129(+09)	4.106(+09)	7.211(+09)	1.831(+10)	3.890(+10)	7.315(+10)
4	13	1.761(+10)	3.634(+10)	6.706(+10)	1.822(+11)	4.046(+11)	7.856(+11)
4	14	3.165(+10)	6.519(+10)	1.199(+11)	3.213(+11)	6.967(+11)	1.314(+12)
4	16	5.150(+07)	2.346(+08)	8.525(+08)	6.862(+09)	3.311(+10)	1.073(+11)
5	8	3.557(+09)	6.870(+09)	1.208(+10)	3.080(+10)	6.583(+10)	1.248(+11)
5	13	1.174(+09)	2.421(+09)	4.468(+09)	1.213(+10)	2.695(+10)	5.240(+10)
5	14	1.054(+10)	2.169(+10)	3.983(+10)	1.061(+11)	2.268(+11)	4.173(+11)
5	15	4.225(+10)	8.718(+10)	1.609(+11)	4.369(+11)	9.708(+11)	1.887(+12)
5	16	2.214(+07)	1.025(+08)	3.784(+08)	3.145(+09)	1.582(+10)	5.438(+10)
6	7	5.875(+06)	9.199(+06)	1.307(+07)	2.266(+07)	3.541(+07)	5.286(+07)
6	10	4.013(+05)	2.088(+06)	8.582(+06)	8.838(+07)	5.759(+08)	2.730(+09)
6	17	1.457(+10)	2.982(+10)	5.478(+10)	1.482(+11)	3.286(+11)	6.371(+11)
7	12	5.646(+09)	1.145(+10)	2.071(+10)	5.436(+10)	1.175(+11)	2.232(+11)
7	13	3.673(+05)	1.940(+06)	8.054(+06)	8.401(+07)	5.522(+08)	2.638(+09)
7	14	6.862(+07)	3.164(+08)	1.162(+09)	9.519(+09)	4.675(+10)	1.547(+11)
7	16	3.950(+10)	8.194(+10)	1.516(+11)	4.092(+11)	8.896(+11)	1.674(+12)

In Table 1, in order to reduce the amount of data, we only report the energy levels for the first 17 levels (n=1 to n=3 shell). The values for the others levels are available on request. The transition probabilities (A_ki in s^-1) for the "allowed'' transition (E1), are also calculated by the SUPERSTRUCTURE code; for the other transitions (M1, M2 & 2E1) the A_ki values are from Lin et al. ([1977]). In a same way, only direct radiative contributions of the first 17 levels onto the first 7 levels are given in Table 2.

2.2 Recombination coefficient rates

Blumenthal et al. ([1972]) have noted that radiative and dielectronic recombination can have a significant effect on the populations of the n=2 states in He-like ions through radiative cascades from higher levels as well as through direct recombination.

  
2.2.1 Radiative recombination (RR) coefficients rates

For radiative recombination rate coefficients, we have used the method of Bely-Dubau et al. ([1982a]). This method is based on (Z-0.5) screened hydrogenic approximation of the Burgess ([1958]) formulae, as we explain below.

For recombination of a bare nucleus of charge Z to form H-like ions, Burgess ([1958]) fitted simple power law expressions to the "exact'' theoretical hydrogenic photoionization cross-sections $\sigma_{{nl}}(E)$ (in cm²) for the n l levels ( $1\le n\le 12$ and $0\le l \le n-1$). According to Burgess "for moderately small n, the errors should be not more than about 5%. Such accuracy should be sufficient for most astrophysical applications''.

 

$$\sigma_{nl}(E) =0.55597\, \frac {Z^2}{n^2}\frac 1{2(2l+1)}$$

$$\times \left[l~\vert\sigma(nl,{\rm o}~l-1)\vert^{2}\left(\frac {I_{\rm H} Z^2}{n^2 E}\right)^{\gamma(nl,l-1)} \right .$$

$$\left . +~(l+1)~\vert\sigma(nl,{\rm o}~l+1)\vert^{2} \left(\frac{I_{\rm H} Z^2}{n^2 E}\right)^{\gamma(nl,l+1)}\right]$$ (3)

where E is the photon energy $E\ge I_{\rm H} Z^2/n^2$.

Bely-Dubau et al. ([1982a]) used this equation for He-like 1snl levels by replacing Z with (Z-0.5). The quantity (Z-0.5) was chosen to take into account the screening of the 1s orbital. To check the validity of this assumption we compared the photoionization cross sections obtained from Eq. (3) to the recent calculations of the Opacity Project by Fernley et al. ([1987]). In Fig. 2 are plotted photoionisation cross sections for 1s2s ¹S ³S, 1s2p ¹P ³P and 1s10d ¹D ³D for Z = 6, 10, 14 (continuous curves), scaled as (Z-0.5). With the exception of 1s2p ¹P, the three continuous curves can hardly be distinguished. Furthermore, the curves do not differ when passing from singlet to triplet cases. This is strong evidence that for 1snl, it is possible to use screened hydrogenic calculations. For comparison, we give the present calculation corresponding to Eq. (3) modified (empty circles).

Figure 2: Scaled photoionization cross sections $\sigma _{{\rm s}}=\sigma$( Z-0.5)2 (in cm-3s-1) as a function of E/(Z-0.5)2 (E is in Rydberg). Empty circles: photoionisation cross sections calculated in the present work; solid lines: photoionisation cross sections available in Topbase for different values of Z=6, 10, 14

The Opacity Project data were taken from the Topbase Bank (Cunto et al. [1993]). This bank includes the 1snl photoionization cross sections for $1 \le n \le 10$ and l=0,1,2. The Burgess data, $\sigma(nl,{\rm o}~l\pm 1)$ and $\gamma(nl,l\pm 1)$, are more complete since they also include $3\le l\le n-1$. Formula (3) is also more convenient since being analytic one can derive directly the radiative recombination rates (cm³s^-1) from it.

$${\alpha_{nl}(Z,T_{\rm e})~=~8.9671\ 10^{-23}~T_{\rm e}^{3/2}~Z~f_{nl}(T_{\rm e})}$$ (4)

where $T_{\rm e}$ is the electronic temperature, Z is the atomic number and

$${f_{nl}(T_{\rm e})} \!\!\! = \!\! \frac{x_{n}^3}{{n}^2}~[~{l}\vert\sigma({nl,{\rm o}~l-1})\vert^{2}~\Gamma_{\rm c}({x_n},3-\gamma({nl,l-1)})~$$

$$\vert\sigma({nl,{\rm o}~l+1})\vert^{2}~\Gamma_{\rm c}( {x_{n}},3-\gamma( {nl,l+1}))].$$ + (l+1) (5)

$${\rm with} \qquad { {x_{n}}~=~\frac{ {Z^{2}~I_{\rm H}}}{ {... ...~n}^{2}}~~~~\left(\frac{ {I_{\rm H}}}{ {k}}=157\,890 \right).}$$ (6)

The quantities $\vert\sigma( {nl,{\rm o}~l}\pm1)\vert/{ {n}}^{2}$ and $\gamma(nl,l\pm 1)$ are given in Table1 of Burgess and

$${\Gamma_{\rm c}(x,p)=\frac{{\rm e}^{x}}{x^p}~\int_{x}^{\infty}t^{(p-1)}~{\rm e}^{-t}~{\rm d}t}.$$ (7)

Finally, to transform H-like data to He-like data, we used the two following expressions for 1s² and 1snl:

$$\alpha_{\rm 1s^{2}}~=~\frac{1}{2}~\alpha_{\rm 1s} {(Z,T_{\rm e})}~~~(n=1, \textrm{ground~level})$$ (8)

$${\noindent \alpha_{{\rm 1s}~nl~({\rm LSJ})}=\frac{\rm (2J+1)}{\rm (2L+1)~(2S+1)}~\alpha_{nl}(Z,T_{\rm e})~~(n\geq2)}.$$ (9)

And we replace Z by (Z-0.5) in formula (4) and (6).

For $10<n< \infty$, we have used the Seaton ([1959]) formula (see below) which gives RR rates for each quantum number n (shell) of H-like ions. We have assumed that the l recombination for such high n is the same as for n=10. Seaton derived his formula by expanding the Gaunt factor, usually taken to be one, to third order (Menzel & Pekeris [1935]; Burgess [1958]). According to Seaton, the radiative recombination rates (in cm³s^-1) for the n shell of H-like ions can be written as:

$${\alpha_{n(Z,T)}~=~5.197\ 10^{-14}~Z~x_{n}^{3/2}~S_{n}(x_{n})}$$ (10)

$${S(x_n)~=~X_0(x_n)+\frac{0.1728}{n^{\frac{2}{3}}}~{X_1(x_n})- \frac{0.0496}{n^{\frac{4}{3}}}~X_2(x_n)}$$ (11)

$${X_0(x_n)=\Gamma_{\rm c}(x,0)}\\$$ (12)

$${X_{1}(x_{n})=\Gamma_{\rm c}\!\left(x,\frac{1}{3}\right)-2~\Gamma_{\rm c}\!\left(x,-\frac{2}{3}\right)}\\$$ (13)

$${X_2(x_n)\! =\! \Gamma_{\rm c}\!\left(x,\frac{2}{3}\right)\! ... ... +\! \frac{2}{3}~\Gamma_{\rm c}\!\left(x,-\frac{4}{3}\right)}.$$ (14)

Next, we have computed the effects of cascades from n>2 levels on each 1s2l level (1s2s  ³ S₁, ¹ S₀; 1s2p ³ P₀, ³ P₁, ³ P₂, ¹ P₁; n=2 shell levels). The present study has shown that the radiative recombination (RR) is slowly convergent with n, thus the first 49 levels ($n\leq 5$) are considered as fine-structure levels (LSJ), the levels from n=6 to n=10 (l=9) shells are separated in LS term (Bely-Dubau et al. [1982a], [1982b]), and finally levels from n=11 to $n=\infty$ are taken into account inside n=10. Figure 3 shows the scaled direct plus upper (n>2) level radiative cascade RR rates $\alpha ^{\rm s}=T^{1/2}$ $\alpha /(Z-0.5)^{2}$ versus $T^{\rm s}=T/(Z-0.5)^{2}$ for 1s2l levels (Z= 8, 10, 12 and 14), and for comparison the direct RR contribution. T is in Kelvin. This points out the importance of the cascade contribution at low temperature. The $\alpha^{\rm s}$ curves are very well superposed and thus allows us to deduce the RR rate coefficients for other Z, as for example Z= 9, 11, 13.

Tables 3, 4, 5, 6, 7 and 8 report separately the direct and the cascade contribution to the RR rate coefficients for each 1s2l level.

Figure 3: Scaled total radiative recombination rates (upper curves: direct plus cascade contribution from n>2 levels) $\alpha ^{\rm s}=T^{1/2}$ $\alpha /(Z-0.5)^{2}$($\times$1012 cm3s-1) versus $T^{\rm s}=T/(Z-0.5)^{2}$($\times$10-4) towards each n=2 level (Plus, star, circle and cross are respectively for Z=8, 10, 12, 14), and for comparison the direct contribution (lower curve in each graph). T is in Kelvin

We checked that the calculated rates summed over $n\ge 2$ and l, added to the rate of the 1s² (ground level) level, are similar to the total RR rates calculated by Arnaud & Rothenflug ([1985]), Pequignot et al. ([1991]), Mazzotta et al. ([1998]), Jacobs et al. ([1977]) (for He-like Fe ion) and Nahar ([1999]) (for O VII). Since these authors used hydrogenic formulae, the RR rate coefficient depends on which screening value was used. As already noted, we have taken for our calculations a screening of 0.5 which is a realistic screening of the atomic nuclei by the 1s inner electron. Most probably, some of these authors have used a (Z-1) scaling. For example for C V, a screening of unity implies a lower value by some 20% with respect to the value obtained with a screening of 0.5.

2.2.2 Dielectronic recombination (DR) coefficient rates

For the low temperature range (photoionized plasma) considered in this paper the dielectronic recombination can be neglected. However at high temperatures, the contribution of DR is no longer negligible. Therefore, we have calculated DR coefficients rates (direct plus upper (n>2) level radiative cascade contribution).

We used the same method as Bely-Dubau et al. ([1982a]). The AUTOLSJ code (including the SUPERSTRUCTURE code) was run with 42 configurations belonging to 1snl, 2snl and 2pnl, with $n\le 5$. All the fine-structure radiative and autoionization probabilities were calculated. For low Z ions, it was necessary to do an extrapolation to higher n autoionizing levels. Specifically, we extrapolate autoionization probabilities, as 1/n³, while keeping the radiative probabilities constant. This extrapolation is not perfectly accurate, and we can estimate that the RD for C, N and O might be slightly over or under estimated.

In Tables 3, 4 ,5, 6, 7, and 8, the DR rates are reported for Z=6, 7, 8, 10, 12, 14 over a wide range of temperature.

  
2.3 Electron excitation rate coefficients

The collisional excitation (CE) rate coefficient (in cm³s^-1) for each transition is given by:

$${C_{ij}(T_{\rm e})=\frac{8.60 \ 10^{-6}}{g_i~T^{1/2}}\exp\left(-\frac{\Delta E_{ij}}{k~T}\right)~\Upsilon_{ij}(T_{\rm e})}.$$ (15)

Where $\Delta E_{{ij}}$ is the energy of the transition, g_i is the statistical weight of the lower level of the transition, and $\Upsilon_{{ij}}$ is the so-called effective collision strength of the transition $i\to j$.

 

 

Table 3: Radiative and dielectronic recombination rates (respectively RR and DR) calculated in this work (in cm³s^-1) for each n=2 level of C V

$T_{\rm e}$	$^{3} {\rm S}_{1}$	$^{3} {\rm P}_{0}$	$^{3} {\rm P}_{1}$	$^{3} {\rm P}_{2}$	$^{1} {\rm S}_{0}$	$^{1} {\rm P}_{1}$
5.0(+04)	2.43 $(-13)^{\rm a}$	7.13(-14)	2.14(-13)	3.57(-13)	8.09(-14)	2.14(-13)
	8.97 $(-13)^{\rm b}$	2.51(-13)	7.51(-13)	1.25(-12)	1.69(-14)	6.87(-13)
	0$^{\rm c}$	0	0	0	0	0
1.0(+05)	1.71(-13)	4.85(-14)	1.46(-13)	2.43(-13)	5.69(-14)	1.46(-13)
	5.78(-13)	1.43(-13)	4.30(-13)	7.16(-13)	1.09(-14)	3.89(-13)
	0	0	0	0	0	0
2.0(+05)	1.20(-13)	3.20(-14)	9.60(-14)	1.60(-13)	3.99(-14)	9.60(-14)
	3.58(-13)	7.80(-14)	2.34(-13)	3.89(-13)	6.70(-15)	2.09(-13)
	4.32(-19)	1.34(-20)	3.81(-20)	5.13(-20)	1.33(-19)	5.61(-19)
5.0(+05)	7.35(-14)	1.71(-14)	5.12(-14)	8.54(-14)	2.45(-14)	5.12(-14)
	1.75(-13)	3.19(-14)	9.58(-14)	1.60(-13)	3.30(-15)	8.38(-14)
	2.75(-15)	4.72(-16)	1.28(-15)	1.55(-15)	6.77(-16)	4.56(-15)
1.0(+06)	4.96(-14)	9.77(-15)	2.93(-14)	4.89(-14)	1.65(-14)	2.93(-14)
	9.64(-14)	1.53(-14)	4.60(-14)	7.61(-14)	1.80(-15)	3.95(-14)
	3.72(-14)	8.94(-15)	2.42(-14)	2.85(-14)	8.17(-15)	6.92(-14)
2.0(+06)	3.24(-14)	5.10(-15)	1.53(-14)	2.55(-14)	1.08(-14)	1.53(-14)
	4.98(-14)	7.10(-15)	2.12(-14)	3.53(-14)	9.00(-16)	1.79(-14)
	8.57(-14)	2.32(-14)	6.26(-14)	7.31(-14)	1.80(-14)	1.68(-13)

$^{\rm a}$

RR Direct contribution.

$^{\rm b}$

RR Upper level radiative cascade contribution from the n>2 levels.

$^{\rm c}$

DR Direct plus upper level radiative cascade from the n>2 levels contributions (when the value is equal to zero this means that the DR rate is negligible compared to the RR rates).

Note: a+b+c represent the total recombination rates.

 

 

Table 4: Same as Table 4 for N VI

$T_{\rm e}$	$^{3} {\rm S}_{1}$	$^{3} {\rm P}_{0}$	$^{3} {\rm P}_{1}$	$^{3} {\rm P}_{2}$	$^{1} {\rm S}_{0}$	$^{1} {\rm P}_{1}$
7.0(+04)	2.86(-13)	8.42(-14)	2.53(-13)	4.21(-13)	9.55(-14)	2.53(-13)
	1.07(-12)	2.94(-13)	8.77(-13)	1.47(-12)	2.35(-14)	8.07(-13)
	0	0	0	0	0	0
1.4(+05)	2.02(-13)	5.73(-14)	1.72(-13)	2.86(-13)	6.72(-14)	1.72(-13)
	6.91(-13)	1.68(-13)	5.04(-13)	8.44(-13)	1.52(-14)	4.58(-13)
	0	0	0	0	0	0
2.8(+05)	1.41(-13)	3.78(-14)	1.13(-13)	1.89(-13)	4.71(-14)	1.13(-13)
	4.28(-13)	9.12(-14)	2.74(-13)	4.56(-13)	9.30(-15)	2.46(-13)
	8.56(-18)	3.67(-20)	1.34(-19)	2.38(-19)	2.90(-18)	1.17(-17)
7.0(+05)	8.68(-14)	2.02(-14)	6.05(-14)	1.01(-13)	2.89(-14)	6.05(-14)
	2.10(-13)	3.72(-14)	1.12(-13)	1.86(-13)	4.60(-15)	9.85(-14)
	6.58(-15)	5.98(-16)	1.50(-15)	1.75(-15)	2.00(-15)	9.76(-15)
1.4(+06)	5.86(-14)	1.15(-14)	3.46(-14)	5.76(-14)	1.95(-14)	3.46(-14)
	1.15(-13)	1.79(-14)	5.36(-14)	8.94(-14)	2.50(-15)	4.64(-14)
	4.88(-14)	1.03(-14)	2.54(-14)	2.82(-14)	1.26(-14)	8.57(-14)
2.8(+06)	3.82(-14)	6.02(-15)	1.81(-14)	3.01(-14)	1.27(-14)	1.81(-14)
	5.97(-14)	8.18(-15)	2.46(-14)	4.11(-14)	1.30(-15)	2.09(-14)
	9.17(-14)	2.56(-14)	6.27(-14)	6.85(-14)	2.16(-14)	1.76(-13)

 

 

Table 5: Same as Table 4 for O VII

$T_{\rm e}$	$^{3} {\rm S}_{1}$	$^{3} {\rm P}_{0}$	$^{3} {\rm P}_{1}$	$^{3} {\rm P}_{2}$	$^{1} {\rm S}_{0}$	$^{1} {\rm P}_{1}$
9.0(+04)	3.36(-13)	9.90(-14)	2.97(-13)	4.95(-13)	1.12(-13)	2.97(-13)
	1.24(-12)	3.51(-13)	1.05(-12)	1.75(-12)	2.50(-14)	9.63(-13)
	0	0	0	0	0	0
1.8(+05)	2.37(-13)	6.74(-14)	2.02(-13)	3.37(-13)	7.89(-14)	2.02(-13)
	8.03(-13)	2.02(-13)	6.05(-13)	1.01(-12)	1.63(-14)	5.50(-13)
	0	0	0	0	0	0
3.6(+05)	1.66(-13)	4.45(-14)	1.34(-13)	2.23(-13)	5.53(-14)	1.34(-13)
	4.95(-13)	1.10(-13)	3.29(-13)	5.50(-13)	1.01(-14)	2.96(-13)
	9.67(-19)	1.86(-20)	5.19(-20)	6.73(-20)	3.42(-19)	1.36(-18)
9.0(+05)	1.02(-13)	2.39(-14)	7.16(-14)	1.19(-13)	3.40(-14)	7.16(-14)
	2.44(-13)	4.52(-14)	1.35(-13)	2.27(-13)	4.90(-15)	1.19(-13)
	3.55(-15)	4.90(-16)	1.32(-15)	1.54(-15)	1.05(-15)	6.02(-15)
1.8(+06)	6.90(-14)	1.37(-14)	4.11(-14)	6.85(-14)	2.30(-14)	4.11(-14)
	1.34(-13)	2.18(-14)	6.49(-14)	1.09(-13)	2.70(-15)	5.65(-14)
	3.90(-14)	8.49(-15)	2.26(-14)	2.57(-14)	1.01(-14)	7.43(-14)
3.6(+06)	4.51(-14)	7.19(-15)	2.16(-14)	3.60(-14)	1.50(-14)	2.16(-14)
	6.99(-14)	1.00(-14)	3.01(-14)	5.02(-14)	1.40(-15)	2.56(-14)
	8.19(-14)	2.11(-14)	5.60(-14)	6.28(-14)	1.98(-14)	1.65(-13)

 

 

Table 6: Same as Table 4 for Ne IX

$T_{\rm e}$	$^{3} {\rm S}_{1}$	$^{3} {\rm P}_{0}$	$^{3} {\rm P}_{1}$	$^{3} {\rm P}_{2}$	$^{1} {\rm S}_{0}$	$^{1} {\rm P}_{1}$
1.4(+05)	4.33(-13)	1.27(-13)	3.82(-13)	6.37(-13)	1.44(-13)	3.82(-13)
	1.59(-12)	4.57(-13)	1.37(-12)	2.28(-12)	3.40(-14)	1.26(-12)
	0	0	0	0	0	0
2.8(+05)	3.05(-13)	8.69(-14)	2.61(-13)	4.35(-13)	1.02(-13)	2.61(-13)
	1.02(-12)	2.62(-13)	7.89(-13)	1.31(-12)	2.20(-14)	7.19(-13)
	0	0	0	0	0	0
5.6(+05)	2.14(-13)	5.75(-14)	1.73(-13)	2.88(-13)	7.12(-14)	1.73(-13)
	6.35(-13)	1.43(-13)	4.31(-13)	7.22(-13)	1.36(-14)	3.88(-13)
	1.22(-18)	1.19(-20)	3.74(-20)	5.73(-20)	5.29(-19)	2.01(-18)
1.4(+06)	1.31(-13)	3.09(-14)	9.28(-14)	1.55(-13)	4.38(-14)	9.28(-14)
	3.15(-13)	5.93(-14)	1.78(-13)	2.97(-13)	6.70(-15)	1.57(-13)
	3.65(-15)	2.68(-16)	8.31(-16)	1.19(-15)	1.38(-15)	6.78(-15)
2.8(+06)	8.90(-14)	1.78(-14)	5.35(-14)	8.91(-14)	2.97(-14)	5.35(-14)
	1.73(-13)	2.86(-14)	8.55(-14)	1.43(-13)	3.60(-15)	7.45(-14)
	3.66(-14)	4.47(-15)	1.38(-14)	1.96(-14)	1.23(-14)	7.45(-14)
5.6(+06)	5.83(-14)	9.39(-15)	2.82(-14)	4.70(-14)	1.94(-14)	2.82(-14)
	8.97(-14)	1.32(-14)	3.96(-14)	6.60(-14)	1.90(-15)	3.40(-14)
	7.37(-14)	1.09(-14)	3.37(-14)	4.75(-14)	2.32(-14)	1.57(-13)

 

 

Table 7: Same as Table 4 for Mg XI

$T_{\rm e}$	$^{3} {\rm S}_{1}$	$^{3} {\rm P}_{0}$	$^{3} {\rm P}_{1}$	$^{3} {\rm P}_{2}$	$^{1} {\rm S}_{0}$	$^{1} {\rm P}_{1}$
2.0(+05)	5.30(-13)	1.56(-13)	4.69(-13)	7.82(-13)	1.77(-13)	4.69(-13)
	1.94(-12)	5.65(-13)	1.70(-12)	2.83(-12)	4.30(-14)	1.56(-12)
	0	0	0	0	0	0
4.0(+05)	3.74(-13)	1.07(-13)	3.20(-13)	5.34(-13)	1.25(-13)	3.20(-13)
	1.25(-12)	3.25(-13)	9.80(-13)	1.63(-12)	2.80(-14)	8.90(-13)
	0	0	0	0	0	0
8.0(+05)	2.62(-13)	7.07(-14)	2.12(-13)	3.54(-13)	8.74(-14)	2.12(-13)
	7.78(-13)	1.78(-13)	5.36(-13)	8.96(-13)	1.76(-14)	4.85(-13)
	1.19(-18)	7.95(-21)	2.92(-20)	4.91(-20)	6.68(-19)	2.45(-18)
2.0(+06)	1.61(-13)	3.81(-14)	1.14(-13)	1.91(-13)	5.38(-14)	1.14(-13)
	3.85(-13)	7.39(-14)	2.22(-13)	3.70(-13)	8.60(-15)	1.98(-13)
	3.36(-15)	1.71(-16)	6.40(-16)	9.60(-16)	1.61(-15)	7.10(-15)
4.0(+06)	1.09(-13)	2.21(-14)	6.62(-14)	1.10(-13)	3.64(-14)	6.62(-14)
	2.13(-13)	3.56(-14)	1.07(-13)	1.79(-13)	4.80(-15)	9.38(-14)
	3.31(-14)	2.83(-15)	1.07(-14)	1.59(-14)	1.36(-14)	7.11(-14)
8.0(+06)	7.17(-14)	1.17(-14)	3.50(-14)	5.83(-14)	2.39(-14)	3.50(-14)
	1.11(-13)	1.64(-14)	4.95(-14)	8.27(-14)	2.50(-15)	4.27(-14)
	6.62(-14)	6.90(-15)	2.62(-14)	3.88(-14)	2.47(-14)	1.44(-13)

 

 

Table 8: Same as Table 4 for Si XIII

$T_{\rm e}$	$^{3} {\rm S}_{1}$	$^{3} {\rm P}_{0}$	$^{3} {\rm P}_{1}$	$^{3} {\rm P}_{2}$	$^{1} {\rm S}_{0}$	$^{1} {\rm P}_{1}$
2.8(+05)	6.23(-13)	1.84(-13)	5.52(-13)	9.19(-13)	2.08(-13)	5.52(-13)
	2.25(-12)	6.65(-13)	2.00(-12)	3.33(-12)	5.30(-14)	1.85(-12)
	0	0	0	0	0	0
5.5(+05)	4.39(-13)	1.25(-13)	3.76(-13)	6.27(-13)	1.46(-13)	3.76(-13)
	1.44(-12)	3.84(-13)	1.15(-12)	1.92(-12)	3.50(-14)	1.05(-12)
	0	0	0	0	0	0
1.1(+06)	3.08(-13)	8.32(-14)	2.49(-13)	4.16(-13)	1.03(-13)	2.49(-13)
	8.92(-13)	2.10(-13)	6.32(-13)	1.05(-12)	2.10(-14)	5.74(-13)
	1.29(-18)	8.47(-21)	3.32(-20)	6.78(-20)	9.70(-19)	3.46(-18)
2.8(+06)	1.90(-13)	4.49(-14)	1.35(-13)	2.24(-13)	6.32(-14)	1.35(-13)
	4.45(-13)	8.71(-14)	2.61(-13)	4.37(-13)	1.06(-14)	2.34(-13)
	3.43(-15)	1.77(-16)	6.55(-16)	9.75(-16)	2.09(-15)	8.56(-15)
5.5(+06)	1.29(-13)	2.59(-14)	7.78(-14)	1.30(-13)	4.28(-14)	7.78(-14)
	2.46(-13)	4.20(-14)	1.26(-13)	2.11(-13)	5.90(-15)	1.11(-13)
	2.97(-14)	2.50(-15)	9.28(-15)	1.35(-14)	1.48(-14)	6.93(-14)
1.1(+07)	8.43(-14)	1.37(-14)	4.12(-14)	6.86(-14)	2.81(-14)	4.12(-14)
	1.29(-13)	1.94(-14)	5.83(-14)	9.74(-14)	3.10(-15)	5.07(-14)
	5.84(-14)	5.93(-15)	2.21(-14)	3.20(-14)	2.55(-14)	1.30(-13)

Figure 4: Scaled effective collision strengths $\Upsilon ^{\rm s}=(Z-0.5)^{2}$$\Upsilon$versus $T^{\rm s}=T$(K)/(1000Z2) for He-like ions with Z=8,10,12,14 inside the n=2 level. The upper curves represent $\Upsilon ^{\rm s}$ with the resonance effect taken into account (plus, star, circle and cross respectively for Z=8,10,12and 14) and for comparison the lower curve (with plus) corresponds to $\Upsilon ^{\rm s}$ without resonance effect for Z=8. Note: the two Z=8 curves (with plus) are superposed

Figure 5: Simplified Gotrian diagram reporting the different contributions for the population of a given n=2 shell level. (1): direct contribution due to collisional excitation (CE) from the ground level (1s2) of He-like ions; (2)+(2'): CE upper level radiative cascade contribution; (3): direct RR or direct DR from H-like ions contribution; and (4)+(4'): RR or DR upper level radiative cascade contribution. Note: CE and DR are onlyeffective at high temperature

The 1s2l-1s2l' transitions (i.e. inside the n=2 shell) are very important for density diagnostic purpose. The data are from Zhang & Sampson ([1987]).

Below, we report scaled effective collision strength $\Upsilon^{\rm s}_{{ij}}=(Z-0.5)^{2}$ $\Upsilon_{{ij}}$. We also use a scaled electronic temperature $T^{\rm s}=T({\rm K})/(1000\,Z^{2}$). The ( Z-0.5)² coefficient has been chosen to obtain scaled $\Upsilon ^{\rm s}$ almost independent of Z (for 6 $\leq Z \leq$14). In Fig. 4, $\Upsilon^{\rm s}(T^{\rm s}$) is displayed for the four most important transitions (between 2³S₁ and 2³P_0,1,2 levels, and between 2¹S₀ and 2¹P₁ levels) including both direct and resonant contribution, and for comparison the direct contribution alone is shown for Z=8. We remark that the curves $\Upsilon ^{\rm s}$(T$^{\rm s}$) are nearly identical for different Z, and for these transitions the resonant contribution is quite negligible since the two curves for Z=8 are superposed. The rates for the transitions between 2³S₁ and 2³P_0,1,2 levels are proportional to their statistical weight. The curves for transitions 2³S₁-2³P₁ 2¹S₀-2¹P₁ are nearly identical.

These high values of $\Upsilon ^{\rm s}$ inside the n=2 shell and the small energy difference between these levels, favour transitions by excitation between the n=2 shell levels. Thus the excitation inside the n=2 shell should be taken into account even for low temperature plasmas.

Excitation from n=2 levels to higher shell levels can be neglected due to the weak population of the n=2 shell compared to the ground level (n=1) in a moderate density plasma and also due to the high $\Upsilon^{\rm s}(T^{\rm s}$) values inside the n=2 shell which favour transitions between the n=2 levels, as we see below. CE from the 1s² (ground) level to excited levels are only important for high temperature such as the hybrid case, due to the high energy difference between these levels. For 1s²-1s2l transitions, we have used the effective collision strength values from Zhang & Sampson ([1987]). These values include both non-resonant and resonant contributions.

CE rates for the 1s²-1snl ( $3\leq n\leq 5$) transitions are from Sampson et al. ([1983]). Their calculations do not include resonance effects but these are expected to be relatively small (Dubau [1994]). The rates converge as n^-3.

We have calculated the radiative cascade contribution from n>2 levels for each n=2 level. We have considered the first 49 levels, as fine-structure levels (LSJ); the contributions from the n=6 to $n=\infty$ levels are considered to converge as n^-3. The cascade contributions become more important for high temperatures. The cascade contribution (from n>2 levels) increases steadily with temperature and has an effect mostly on the 1s2s³S₁ level. The resonant contribution increases then decreases with temperature. For high temperature plasmas, cascade effects should be taken into account. For very low temperature plasmas only the direct non-resonant contribution is important, except for the 1s2s³S₁ level which also receives cascade from within the n=2 level, i.e. from 1s2p³P_0,1,2 levels, as long as the density does not redistribute the level population, i.e. the density is not above the critical density.

Tables 9, 10, 11, 12 and 13 report data which correspond respectively to the direct (b), the resonance (c), and the n>2 cascade (d) contributions.