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4 Results and discussion

4.1 Ionization balance


Table 4: Observed and calculated relative line intensities in FTU tokamak
    Relative Intensity a
    Measured Calculated d
Ion $\lambda$ (Å) c   HULLAC CHIANTI

154.9 14 17.8 17.7
  157.8 17 17.2 13.4
  164.2 100 100.0 100.0
  167.4 40 39.8 48.0
  168.9 51 48.2 67.6
Ca XV 161.0 100 100.0 100.0
  171.6 34 19.2 18.2
  176.0 18 12.4 12.1
  176.9 46 31.7 30.4
  177.3 35 23.8 24.2
  181.9 98 97.1 95.9
  182.9 28 20.1 18.3
  201.0 48 41.8 39.5
Ca XIV 153.2 13 9.0 13.5
  165.3 27 38.0 53.0
  167.0 46 51.6 67.4
  183.5 15 32.7 33.9
  186.6 62 65.4 66.0
  189.0 12 10.3 14.0
  193.9 100 100.0 100.0
Ca XIII 156.7 37 34.3 34.1
  159.8 17 23.6 24.5
  161.7 100 100.0 100.0
  162.9 14 18.0 18.0
  164.1 23 22.9 23.0
  168.4 53 30.3 30.5


\end{figure} Figure 6: Synthetic line-integrated time-averaged XUV spectrum of highly ionized calcium in the TEXT tokamak

The LOS-integrated TEXT spectrum was modeled using fractional abundances estimated as described in Sect. 2.3. Ionization and recombination rates, used for this calculation, were taken from the recent work of Mazzotta et al. ([1998]). Their ionization equilibrium curves for the calcium ions differ from those of Arnaud & Rothenflug [1985], both in MA temperature and ionization fractions. The difference is most pronounced, up to 200%, for the lower ionization stages (Ca XIV - Ca XII) due to improved dielectronic recombination rates. Our calculated calcium ion ground state distributions closely reproduced the measured ones (Lippmann et al. [1987]). Using QSS emissivities, we constructed a line-integrated time-averaged synthetic TEXT spectrum (Fig. 6). All lines predicted by HULLAC are shown. This includes not only the $\Delta n=0$ transitions of interest, but also a large number of $\Delta n = 1$ and $\Delta n = 2$ transitions. These lines are grouped in two domains: 14 Å $\le \lambda \le$ 45 Å and 50 Å $\le \lambda \le$ 100 Å. The calculated intensities of these lines are typically a factor of 102 - 103 less than the $\Delta n=0$ lines. The 50 - 100 Å domain is covered by GRITS, however, line intensities are below the detector sensitivity limit. As seen in Fig. 6, these lines contribute insignificantly to $\Delta n=0$ line or background intensities. All lines were given a Gaussian shape with the FWHM of GRITS. The agreement between wavelengths from Kelly ([1987]) and HULLAC is $\le 1.5 \%$ for most lines, which, however, resulted in different blend patterns (comp. to Fig. 3). Good agreement between synthetic and measured spectra can be considered as indirect validation of the new fractional abundance calculations for calcium.

4.2 Individual ions

The calculated relative line intensities for Li I-like to F I-like calcium ions agree with the measured ones within the stated experimental error in most cases. In comparison with Lippmann et al. ([1987]), improved atomic rates result in a better agreement between the measured and calculated relative line intensities. Results for individual ions are discussed below. The emphasis is on the beryllium- to nitrogen-like calcium ions, due to their plasma temperature and density diagnostic potential. Calculated density dependence of the level populations is shown in Fig. 7. Both ground and excited level populations are in corona equilibrium at $n_{\rm e} = 10^9$ cm-3. The ground configuration levels approach Boltzmann values at $n_{\rm e} = 10^{10} -$ 1014cm-3, and transfer population to the excited 2pk+1 levels at different, $n_{\rm e}$-dependent, rates. This enables utilization of the $\Delta n=0$ E1 line intensity ratios as density diagnostics. Forbidden (M2) transitions between ground levels of Ca XV, Ca XIV and Ca XIII can also be used as a density diagnostics. These far ultraviolet lines have been observed in a solar active region by Feldman et al. ([1998]) and used to study physical conditions of the solar corona.
\end{figure} Figure 7: Modeled relative populations of the n=2 levels as a function of electron density at $T_{\rm e}$ values given in Table 1 (basic model): a) - Ca XVIII, b) - Ca XVII, c) - Ca XVI, d) - Ca XV, e) - Ca XIV, f) - Ca XIII, g) - Ca XII. Level populations are normalized to the total population per ion. The level labels are shown in Tables 2 and 3. The levels not shown in Tables 2, 3 are labeled as follows: b) Ca XVII: 6 - 1s22p $^{2}\,^{3}$P0; c) Ca XVI: 3 - 1s22s2p $^{2}\,^{4}$P1/2; 4 - 1s22s2p $^{2}\,^{4}$P3/2; 5 - 1s22s2p $^{2}\,^{4}$P5/2; d) Ca XV: 5 - 2s22p $^{2}\,^{1}$S0; 6 - 2s2p $^{3}\,^{5}$S2

Because of the high edge tokamak plasma density ($n_{\rm e}$ $\simeq$ 1012 cm-3), collisional quenching of the upper levels of these transitions overtakes radiative decay: both FTU and TEXT plasma edge spectra, recorded by the SPRED and NITS instruments, respectively, indicated no evidence of the forbidden lines. The line intensities, calculated for the edge tokamak densities, are a factor of 102 - 103less than those of typical XUV lines, which is beyond the photometric sensitivity limit of the instruments used.
Ca XVIII - The only XUV lines identified in solar flare spectrum are $\lambda302.2$ and $\lambda344.8$ (see references in Lawson & Peacock [1984]). The lines are populated according to their statistical weights. The discrepancy between calculations and measurements in the TEXT tokamak is attributed to the GRITS calibration uncertainty above 300 Å.

Ca XVII - An example of a persisting disagreement between calculations and solar flare measurements is a long-standing problem with the line intensity ratios of the Be I-like isosequence R = I(2s $^{2}\,^{1}{\rm S}_{0} -$ 2s2p $^{1}{\rm P}_{1})\,\,/\,\,
I$(2s2p $^{3}{\rm P}_{J'} - 2{\rm p}^{2}\,^{3}{\rm P}_{J''})$, J'=1,2 and J''=1,2. These intensity ratios are density sensitive (Doschek et al. [1977]). The Be I-like neon, magnesium, sulfur, argon and calcium lines have been observed in the solar flares by the NRL Skylab-based S082A spectroheliograph. The calculated line intensity ratios have consistently implied an electron density of $\sim 10^{13}$ cm-3 and higher, in contrast to the density of $ \le 10^{10}- 10^{12}$ cm-3, derived from other line intensity ratios (Doschek et al. [1977]; Bhatia & Mason [1983]; Dufton et al. [1983]; McCann et al. [1989]; Harra et al. [1992]). All six lines have been observed in TEXT. Calculated and measured relative intensities agree within 40%, with the two exceptions of $\lambda228.7$ and $\lambda244.0$, which are blended with the F IV lines. Except for the $\lambda192.9$ and $\lambda232.8$ lines, all other lines are relatively weak and the inferred densities have larger uncertainties or unrealistic values. We therefore investigated various CR effects on the $R_{1} = I(2{\rm s}^{2}\,^{1}{\rm S}_{0} - 2{\rm s}2{\rm p}\,^{1}{\rm P}_{1})\,\...
...2{\rm p}\,^{3}{\rm P}_{1,2} - 2{\rm p}^{2}\,^{3}{\rm P}_{1,2}) =
I(\lambda192.8$) $~/~I(\lambda232.8)$ ratio. As the number of levels, included in the model, increase, the calculated $I(2{\rm s}^{2}\,^{1}{\rm S}_{0} - 2{\rm s}2{\rm p}\,^{1}{\rm P}_{1})\,\,/\,\,
I(2{\rm s}2{\rm p}\,\,^{3}{\rm P} - 2{\rm p}^{2}\,\,^{3}{\rm P})$ ratios decrease, due to cascades to the $2{\rm p}^{2}$ levels. This effect is $\le$ 20% as n = 3,4 levels are added to the model. In particular, the R1ratio was found to be 61 (n=2 only; 10 lowest levels), 49 ($n=2,\,3$; 30 levels) and 47 ( $n=2,\,3,\,4$; 125 levels). The ratio, measured in TEXT, is $\sim 51 \pm 5$. Both $\lambda192.8$ and $\lambda232.8$ lines are primarily populated by electron impact excitation. The $2{\rm s}2{\rm p}\,^{1}{\rm P}_{1}$ level is populated from the ground, whereas the $2{\rm p}^{2}\,^{3}{\rm P}$levels are populated both from the ground and from the $2{\rm s}2{\rm p}\,^{3}{\rm P}$levels. The DWA rates for these transitions, calculated by HULLAC, are within $\le$ 15% of the R-matrix rates calculated by Dufton et al. ([1983]), for the 400 - 900 eV temperature range (with one exception of $2{\rm s}^{2}\,^{1}{\rm S}_{0} - 2{\rm s}2{\rm p}\,^{1}{\rm P}_{1}$, for which the HULLAC rate is $\simeq$ 35% smaller than the rate from Dufton et al. [1983]). The detailed model was used for Ca XVIII, Ca XVII and Ca XVI, to account for possible non-steady state contributions. The model was constructed for both tokamak and ionization equilibrium temperatures. Direct ionization is found to be the dominating process, the autoionization flux originating from the $1{\rm s}2{\rm l}^{2}$ levels of Ca XVI is negligible, and the population flux due to inner-shell ionization from $2{\rm s}^{2}2{\rm p}$ and $2{\rm s}2{\rm p}^{2}$ levels of Ca XVI to $2{\rm s}2{\rm p}\,^{3}{\rm P}$ levels of Ca XVII is several orders of magnitude less than the flux to the ground state. Significant departure from ionization equilibrium is required to populate the $2{\rm s}2{\rm p}~^3{\rm P}$ levels through inner-shell ionization (Feldman et al. [1992]). Our calculations indicate that if $n_{{\rm Ca XVII}}/n_{{\rm Ca XVI}} =
1:10$, at $n_{\rm e} = 3~10^{11}$ cm-3 up to 50% of the total population of $2{\rm s}2{\rm p}~^3{\rm P}$ levels is due to inner-shell ionization, and the R1 ratio is practically the same as at $n_{\rm e} = 3~10^{13}$ cm-3at ionization equilibrium conditions. We conclude, therefore, that the R1 ratio should be a reliable density diagnostics in the $n_{\rm e}$ range of 1010 - 1014 cm-3.

Intensity ratio of two $\Delta n=0$ lines of the same ion can be used as a temperature diagnostic due to the temperature dependence of the excitation rate coefficients, if the separation of the upper levels of these lines $\Delta E$is much less than $k\,T_{\rm e}$, as in the case of the resonant and intercombination lines of Be I-like calcium, R2(Ca XVII $) = I(2{\rm s}2{\rm p}\,^{3}{\rm P}_{1} - 2{\rm s}^{2}\,^{1}{\rm S}_{0})$/ $I(2{\rm s}2{\rm p}\,^{1}{\rm P}_{1} - 2{\rm s}^{2}\,^{1}{\rm S}_{0}) = $ $I(\lambda 371.0) / I(\lambda 192.8)$ ( $\Delta E \simeq 32$ eV). Figure 8 presents the calculated R2 as a function of logarithmic electron temperature in K. The lines have been observed in the solar flare on 09 August 1973 by the Skylab SO82A. The ratios are 0.05 - 0.08 (measured by Doschek et al. [1977]) and 0.027 - 0.058 (re-measured by Dufton et al. [1983]). Our calculations yield the logarithmic temperature between 5.9 and 7.2, derived from the latter measurement. The Ca XVII ionization equilibrium temperature is $\log T_{{\rm e}} = 6.74$ (Arnaud & Rothenflug [1985]) and 6.78 (Mazzotta et al. [1998]). McCann et al. ([1989]), using the SO82A observational data for the same flare, measured the R2 ratio for the ions S XIII and Ar XV, and derived the logarithmic temperatures of $\sim 6.5$ (sulfur), 6.0 and 6.45 (argon). These temperatures are within 20% of the ionization equilibrium temperatures of sulfur and argon, respectively. The temperature, inferred from the $I(\lambda 371.0) / I(\lambda 192.8)$ ratio, is consistent with the latter measurements and is reasonable for solar flares, although clearly much higher accuracies in the intensity measurements are needed to utilize this type of line ratio techniques.

Relative line intensities, calculated using the CHIANTI database, are close to the HULLAC calculation and consistent with the experimentally inferred intensities.

Ca XVI- The blending problem is especially aggravated in the 150 - 170 Å domain, densely populated by Ca XIV, Ca XV and Ca XVI lines. We note that our calculations did not reconcile the discrepancy between measured and calculated intensities of the $\lambda208.6$ and $\lambda224.6$ lines (upper levels $2{\rm p}^2~^2{\rm D}$). Lippmann et al. ([1987]) attributed them to the fact that their model did not include collisional excitation between 2s2p2 2D levels and the $2{\rm p}^3~^2{\rm D}$ levels, which resulted in overestimation of the upper level populations. Recent calculations of Keenan et al. ([1998]), which used R-matrix collision rates and included the $2{\rm p}^3$ levels as well, demonstrate a similar trend. HULLAC excitation rates between these terms are comparable to the excitation rates from the ground. According to our calculations, however, these excitation processes become noticeable ($\ge 10$% in the $2{\rm s}2{\rm p}^2~^2{\rm D}$ level populations) only at $n_{\rm e} \ge 5$ 1014 cm-3. The $\lambda\,$ 150 - 170 Å lines and the $\lambda208.6$ and $\lambda224.6$ lines were recorded from different TEXT discharges, which could have explained the difference. The relative line intensities of the $\lambda\,$ 150 - 170 Å lines from both TEXT and FTU datasets are in good agreement with HULLAC calculations. The ratio $R = I(\lambda224.6)/I(\lambda208.6$) = 1.3, recorded in TEXT, also agrees well with HULLAC prediction of 1.25. Keenan et al. ([1998]) pointed out that CHIANTI database gives abnormally low intensity of the latter ratio. Whereas this was due to a missing piece of data and has been corrected in v. 2.0 (Landi et al. [1999]), the CHIANTI relative intensities still differ from our measurements and HULLAC calculations. The measured branching ratios of the lines originating from the $2{\rm s}2{\rm p}^2\,^2{\rm P}_{1/2,3/2}$ levels agree well with computations.

Several Ca XVI line pairs are electron density sensitive, and they have been used by Dere et al. ([1979]) and Keenan et al. ([1998]) in application to solar flare diagnostics. In addition to the mentioned R ratio, the intensity ratios of $\lambda154.9, \lambda157.8, \lambda164.2,
\lambda168.9$ to $\lambda208.6$ are also density sensitive. The R ratio, however, seems to be a more reliable diagnostics, since the lines are close and subject to blends to a lesser degree.

\resizebox{\hsize}{!}{\includegraphics[angle=90]{ms9307f8.eps}}\end{figure} Figure 8: Predicted Ca XVII line intensity ratio $R_{2} = I(2{\rm s}2{\rm p}\,^{3}{\rm P}_{1} - 2{\rm s}^{2}\,^{1}{\rm S}_{0})$/ $I(2{\rm s}2{\rm p}\,^{1}{\rm P}_{1} - 2{\rm s}^{2}\,^{1}{\rm S}_{0}) = $ $I(\lambda 371.0) / I(\lambda 192.8)$as a function of electron temperature

Ca XV- As in the case of boron-like calcium, many of the lines are blended, and in some cases (150 - 170 Å) it was difficult to measure the brightnesses accurately. The $\lambda208$ blend was separated to the three components $\lambda208.3$, $\lambda208.7$ (Ca XV) and $\lambda208.6$ (Ca XVI) using our model. Ten out of fourteen recorded lines in TEXT and most of the lines recorded at FTU agree within the experimental error with the measurements. Both HULLAC and CHIANTI models include the important $2\,{\rm p}^4$ configuration. The radiative cascade effects are small: line intensities, calculated using HULLAC (n = 2, n = 2,3, and n=2,3,4 models) and CHIANTI data (n = 2 only), differ by $\le$ 5%. Several line ratios have been used to infer electron densities in solar flares (Dere et al. [1979]; Keenan et al. [1992], and references therein). Present HULLAC calculations do not improve upon density estimates previously published.

Ca XIV- HULLAC calculations agree within the error of measurements with both TEXT and FTU datasets for the majority of the lines. The effect of cascades from n = 3 and n = 4levels is weak, $\le 10$%. Inclusion of the $2\,{\rm p}^5$ levels affects the $2{\rm s}^{2}2{\rm p}^{3}-2{\rm s}\,2{\rm p}^{4}$ relative line intensities insignificantly ($\le 5$%). CHIANTI intensities are in overall good agreement with the measurements, however some line intensities disagree up to a factor of two. In the same time, there is a close agreement between several measured branching ratios and the branching ratios from HULLAC and CHIANTI. Diagnostics potential of Ca XIV lines includes the density-sensitive intensity ratios of the lines, originating from 2s2p$^4\;^4$P to the lines, originating from $2{\rm s}2{\rm p}^4\;^2{\rm S}, ^2{\rm P}, ^2{\rm D}$ (Feldman et al. [1980]). The line intensity ratios of $\lambda134$, $\lambda165$, $\lambda167$, $\lambda189$to $\lambda194$ are density sensitive up to $n_{\rm e} \simeq 10^{14}$ cm-3. Although $\lambda194$ is blended with a very strong $\lambda193$ Ca XVII line, it is possible to accurately separate them. The density predictions for TEXT, for example, based on the intensity ratio of $\lambda165$ and $\lambda189$ to $\lambda194$ are within 20% of the independently measured value. Three most intense XUV lines ( $\lambda183.5,
\lambda186.6$ and $\lambda193.9$) have been identified in the full Sun spectra (Behring et al. [1972]), and recently in the transition region (Brosius et al. [1998]).

Ca XIII- Both HULLAC and CHIANTI calculations agree well with the TEXT and FTU datasets. Several line intensity ratios, e.g. $\lambda131$, $\lambda160$ to $\lambda164$, can be used as density diagnostics in the range between 1010 and 1013 cm-3. The Ca XIII XUV lines have not been observed in solar plasmas.

Ca XII- The two XUV lines ( $\lambda141.0$ and $\lambda147.3$) have been identified in the full Sun spectra (Behring et al. [1972]). The lines share the same upper level $2{\rm s}2{\rm p}^6\,^2{\rm S}_{1/2}$ and their intensity ratio is simply a ratio of the transition probabilities, known with fairly high accuracy. As follows from Fig. 5, the $\lambda141$is a blend, which is the reason for a 20% difference between the measured and calculated relative intensities.
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