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Subsections

Computing Fe  XIV line intensities

In the following sections the new atomic data presented above will be used to compute line emissivities and these will be compared both with previous theoretical models and observations. In particular Y98 identified significant discrepancies between theory and observation when comparing the CHIANTI/v1.0 Fexiv model (Sect. 4.14.2 of Dere et al. 1997) with data from the SERTS-89 instrument, Thomas & Neupert (1994) (hereafter TN94) and these issues will be directly addressed here.


  \begin{figure}\par\epsfxsize=8cm \epsfbox[69 52 528 440]{ds8798f1.eps} \par\end{figure} Figure 1: Collision strength for the 3s23p(2P $^{\rm o}_{1/2}$) - 3s3p2(2D5/2) transition. Solid line from present results averaged over 0.5 Ryd intervals. Open circles from Bhatia & Kastner (1993). Square from Dufton & Kingston (1991)


  \begin{figure}\par\epsfxsize=8cm \epsfbox[60 45 525 450]{ds8798f2.eps} \par\end{figure} Figure 2: Thermally averaged collision strength for the 3s23p(2P $^{\rm o}_{1/2})~-$ 3s3p2(2D5/2) transition. Solid line from present results, open circles from close-coupling calculation of Dufton & Kingston (1991), dashed line from distorted wave work of Bhatia & Kastner (1993)

  
5.1 Line emissivities

The line emissivity, $\epsilon_{\lambda}$, is defined as

\begin{displaymath}\epsilon_{\lambda} = \epsilon_{ij} = \Delta\kern-1ptE\,N_{j}\,A_{ji} \end{displaymath} (1)

for a transition between two levels of an ion with indices i and jthat give rise to a line at wavelength $\lambda$. $\Delta\kern-1ptE$ is the energy difference between the two levels, Nj is the number density of particles in the plasma that are in the emitting state of the ion, and Aji is the radiative probability for the transition. When dealing only with lines emitted by a single ion, it is more convenient to define the ion emissivity as

\begin{displaymath}\varepsilon_{\lambda} = \varepsilon_{ij} = \Delta\kern-1ptE\,n_{j}\,A_{ji} \end{displaymath} (2)

where nj is the fraction of the ions that are in the emitting state j.

For a plasma in steady state, in ionisation equilibrium and transparent to radiation, $\varepsilon_{\lambda} \propto I_{\lambda}$ where $I_{\lambda}$ is the observed intensity of a line, and so one can directly compare ratios of ion emissivities with ratios of the line intensities.

To compute the nj one needs to solve a set of linear equations which account for the atomic processes that populate and de-populate the levels of the ion. In the present work we will only consider electron excitation and de-excitation, and spontaneous radiative decay, to be consistent with the CHIANTI/v1.0 and B94 Fexiv models that we shall be comparing with. Other processes that are significant in some circumstances are photorecombination, photoexcitation and stimulated emission by a background radiation field, and proton excitation-de-excitation, although their effects are generally small for Fexiv emission from the solar atmosphere.

5.2 The Fe  XIV models

Data from three Fexiv models will be considered here and compared. The new model (to be referred to as SMY99) consists of the thermally averaged collision strengths from IP XIV for the ground transition, and those presented here for all other transitions. Radiative decay rates were calculated from the Basis 3 target referred to in Sect. 2. Level energies are the experimental values presented in Table 3.

This set of Fexiv data will be included in a fitted form in a future release of the CHIANTI database. Following the format of the rest of the CHIANTI database, the electron collision data has been assessed and spline fitted with a method based on that of Burgess & Tully (1992; see also Sect. 3.4 of Dere et al. 1997). Thermally averaged collision strengths ($\Upsilon$) have been computed over the temperature range $5.0\le \log\,T\le 10.0$, and it was found that for many transitions the variation of $\Upsilon$ with T was too complex to be fitted with the 5-point spline that is the basis of the Burgess & Tully (1992) method. In these cases a restricted set of temperatures had to be considered. The range over which the fits are most accurate is $5.4\le\log\,T\le 7.0$. Comparisons of $\Upsilon$'s derived from the spline fits with the original data generally give excellent agreement in this temperature range, with maximum differences of 5% in a few exceptional cases. The spline fit $\Upsilon$'s are those used by CHIANTI, and so the CHIANTI intensities should not be used outside the temperature range $5.4\le\log\,T\le 7.0$.

Although collision strengths have been computed for all possible transitions between the forty levels of the present Fexiv model, it is only necessary to consider the transitions that involve levels 1, 2 and 21 (see Table 3) as these are the only levels with significant population at typical coronal densities (Table 8). Thus only these transitions have been fitted.

The Fexiv model contained in version 1.0 of the CHIANTI database was described in Dere et al. (1997) and will be referred to as CH97. This consisted of the 12 levels of the 3s23p, 3s3p2 and 3s23d configurations, and thermally averaged collision strengths were taken from DK91 for the 3s-3p and 3p-3d transitions. For the ground transition, the IP XIV data were used. Radiative decay rates were from Froese Fischer & Liu (1986), and level energies were from the NIST database (Martin et al. 1995).

B94 presented an Fexiv model that consisted of the DK91 electron collisional data for the 12 3s23p, 3s3p2 and 3s23d levels, and the BK93 distorted wave collision data for all transitions involving the 28 levels of the 3p3 and 3s3p3d configurations. The radiative data and level energies are from BK93. The same 5 configuration target that was used for the collisional model was used by BK93 for the radiative data, and so these are less accurate than both the Froese Fischer & Liu (1986) data and that used in the SMY99 model.

    
Table 7: Thermally averaged collision strengths$^\dagger $ for the strongest EUV transitions
i j log (T[K])
    5.5 6.0 6.2 6.4 6.6 6.8 7.0 7.2
1 3 4.256(-2) 2.982(-2) 2.587(-2) 2.288(-2) 2.077(-2) 1.942(-2) 1.872(-2) 1.857(-2)
1 4 6.446(-2) 3.721(-2) 2.853(-2) 2.157(-2) 1.610(-2) 1.191(-2) 8.770(-3) 6.509(-3)
1 5 7.008(-2) 3.939(-2) 2.948(-2) 2.165(-2) 1.568(-2) 1.121(-2) 7.898(-3) 5.487(-3)
1 6 8.210(-1) 7.872(-1) 8.023(-1) 8.366(-1) 8.890(-1) 9.571(-1) 1.038( $\phantom{-}$0) 1.129( $\phantom{-}$0)
1 7 2.079(-1) 1.073(-1) 7.890(-2) 5.767(-2) 4.224(-2) 3.122(-2) 2.341(-2) 1.791(-2)
1 8 1.153( $\phantom{-}$0) 1.293( $\phantom{-}$0) 1.370( $\phantom{-}$0) 1.469( $\phantom{-}$0) 1.592( $\phantom{-}$0) 1.738( $\phantom{-}$0) 1.906( $\phantom{-}$0) 2.090( $\phantom{-}$0)
1 9 8.690(-1) 9.189(-1) 9.622(-1) 1.023( $\phantom{-}$0) 1.102( $\phantom{-}$0) 1.199( $\phantom{-}$0) 1.312( $\phantom{-}$0) 1.436( $\phantom{-}$0)
1 10 8.846(-1) 9.271(-1) 9.670(-1) 1.025( $\phantom{-}$0) 1.102( $\phantom{-}$0) 1.198( $\phantom{-}$0) 1.309( $\phantom{-}$0) 1.432( $\phantom{-}$0)
1 11 2.267( $\phantom{-}$0) 2.369( $\phantom{-}$0) 2.464( $\phantom{-}$0) 2.607( $\phantom{-}$0) 2.800( $\phantom{-}$0) 3.042( $\phantom{-}$0) 3.330( $\phantom{-}$0) 3.654( $\phantom{-}$0)
1 12 1.486(-1) 8.898(-2) 7.100(-2) 5.747(-2) 4.769(-2) 4.086(-2) 3.622(-2) 3.316(-2)
2 3 4.735(-2) 2.765(-2) 2.164(-2) 1.719(-2) 1.406(-2) 1.200(-2) 1.072(-2) 1.005(-2)
2 4 1.050(-1) 6.100(-2) 4.730(-2) 3.671(-2) 2.883(-2) 2.313(-2) 1.918(-2) 1.660(-2)
2 5 2.013(-1) 1.271(-1) 1.045(-1) 8.738(-2) 7.509(-2) 6.678(-2) 6.168(-2) 5.914(-2)
2 6 3.593(-1) 1.997(-1) 1.565(-1) 1.252(-1) 1.035(-1) 8.914(-2) 8.023(-2) 7.529(-2)
2 7 1.518( $\phantom{-}$0) 1.320( $\phantom{-}$0) 1.302( $\phantom{-}$0) 1.321( $\phantom{-}$0) 1.376( $\phantom{-}$0) 1.459( $\phantom{-}$0) 1.565( $\phantom{-}$0) 1.690( $\phantom{-}$0)
2 8 3.055(-1) 2.114(-1) 1.931(-1) 1.842(-1) 1.828(-1) 1.876(-1) 1.971(-1) 2.101(-1)
2 9 1.510( $\phantom{-}$0) 1.576( $\phantom{-}$0) 1.641( $\phantom{-}$0) 1.734( $\phantom{-}$0) 1.860( $\phantom{-}$0) 2.015( $\phantom{-}$0) 2.197( $\phantom{-}$0) 2.399( $\phantom{-}$0)
2 10 4.398( $\phantom{-}$0) 4.637( $\phantom{-}$0) 4.847( $\phantom{-}$0) 5.144( $\phantom{-}$0) 5.532( $\phantom{-}$0) 6.008( $\phantom{-}$0) 6.562( $\phantom{-}$0) 7.179( $\phantom{-}$0)
2 11 7.318(-1) 6.930(-1) 6.974(-1) 7.177(-1) 7.542(-1) 8.059(-1) 8.709(-1) 9.468(-1)
2 12 4.326( $\phantom{-}$0) 4.474( $\phantom{-}$0) 4.639( $\phantom{-}$0) 4.892( $\phantom{-}$0) 5.241( $\phantom{-}$0) 5.684( $\phantom{-}$0) 6.209( $\phantom{-}$0) 6.806( $\phantom{-}$0)
$^\dagger $ In this and subsequent tables, 4.256(-2) denotes 4.256 10-2.


 

 
Table 8: A comparison of fractional level populations predicted from the three theoretical models (SMY99, B94, CH97; see Sect. 5.2). The values have been calculated for a temperature of $\log\,T=6.2$, and electron numbers densities of 108, 1010 and 1012 cm-3
  log  $N_{\rm e} = 8$   log  $N_{\rm e} = 10$   log  $N_{\rm e} = 12$
Index SMY99 B94 CH97   SMY99 B94 CH97   SMY99 B94 CH97
1 9.85(-1) 9.85(-1) 9.86(-1)   5.44(-1) 5.49(-1) 5.53(-1)   3.52(-1) 3.56(-1) 3.53(-1)
2 1.51(-2) 1.45(-2) 1.40(-2)   4.54(-1) 4.46(-1) 4.47(-1)   6.33(-1) 6.31(-1) 6.47(-1)
                       
3 2.78(-10) 2.21(-10) 1.18(-10)   2.06(-8) 1.48(-8) 8.35(-9)   1.74(-6) 1.20(-6) 6.76(-7)
4 2.05(-9) 1.74(-9) 7.39(-10)   1.81(-7) 1.49(-7) 6.60(-8)   1.76(-5) 1.42(-5) 6.23(-6)
5 5.98(-10) 4.29(-10) 1.28(-10)   7.42(-8) 5.98(-8) 2.21(-8)   7.88(-6) 6.43(-6) 2.64(-6)
                       
6 1.02(-10) 7.82(-10) 6.63(-11)   6.27(-9) 4.85(-9) 3.99(-9)   4.55(-7) 3.53(-7) 2.76(-7)
7 2.58(-11) 1.73(-11) 5.34(-12)   6.75(-9) 4.84(-9) 3.36(-9)   8.40(-7) 6.17(-7) 4.67(-7)
                       
8 1.83(-11) 9.74(-12) 9.42(-12)   1.08(-9) 6.96(-10) 6.71(-10)   7.57(-8) 5.71(-8) 5.45(-8)
                       
9 6.94(-12) 7.09(-12) 9.59(-12)   6.56(-10) 5.27(-10) 7.16(-10)   6.32(-8) 4.45(-8) 6.03(-8)
10 6.09(-12) 4.23(-12) 5.30(-12)   1.00(-9) 6.88(-10) 8.72(-10)   1.16(-7) 7.99(-8) 1.03(-7)
                       
11 1.26(-11) 9.14(-12) 1.10(-11)   7.79(-10) 5.65(-10) 6.85(-10)   5.68(-8) 4.10(-8) 4.91(-8)
12 7.11(-13) 3.70(-13) 3.66(-13)   6.43(-10) 4.30(-10) 5.20(-10)   8.76(-8) 5.97(-8) 7.43(-8)
                       
21 6.51(-6) 4.07(-6) -   2.39(-3) 4.75(-3) -   1.51(-2) 1.30(-2) -



  \begin{figure}\par\epsfxsize=8cm \epsfbox[0 0 445 309]{ds8798f3.eps} \par\end{figure} Figure 3: A diagram illustrating the strongest EUV transitions. Wavelengths are given in Angströms

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