3 Atomic data and collisional-radiative model

3.1 Details of computations

The atomic data for CR modeling was generated by the HULLAC suite of computer codes. HULLAC computes the ab initio intermediate coupled wave functions, level energies and transition probabilities, using the fully relativistic parametric potential code RELAC (Klapisch [1971]; Klapisch et al. [1977]). All double excited states with energies less than or equal to the energy of the highest included resonant level were also included in the model for each ion. The electron collision strengths were calculated in distorted wave (DW) approximation and averaged over Maxwellian distribution to produce collisional excitation rates by the CROSS code (Bar-Shalom et al. [1988]). Effects of proton-ion collisions were not included in our model. Proton collisions can change populations of closely spaced fine structure levels, such as Ca XVI -Ca XIII ground levels. Stratton et al. ([1985]) demonstrated that for the Ti, Cr, Fe and Ni ions this effect is weak, with the exception of carbon-like ions. Since electron impact excitation is the main source of excited level population at the typical tokamak temperature and density, the accuracy of the electron excitation rate coefficients is critical for calculating accurate spectral line intensities. This especially concerns the ions which have metastable levels with relatively large populations. The DW approximation does not take into account scattering resonances and coupling between scattering channels. Finkenthal et al. ([1987]) pointed out the importance of these effects for CR modeling of lower Z elements. In the described tokamak experiments electron energies are typically much greater than collisional excitation thresholds of the XUV lines, and the DW approximation, therefore, proves adequate. There has been a number of theoretical studies which compare R-matrix and DW calculated electron collision strengths for Ca XVIII - Ca XIII ions (for example, Huang et al. [1987]; Bhatia et al. [1986]; Dufton et al. [1983]; Zhang & Pradhan [1994]; Bhatia & Doschek [1993]; Aggarwal [1992]; Baliyan & Bhatia [1994]). The collision strengths, calculated by the two methods, agree to better than 30% for the temperature range considered (100 - 1000 eV). Therefore, the DW excitation rate coefficients, used in this work, are of adequate accuracy.

Using the ab initio level energies, transition probabilities and collisional excitation rate coefficients generated by HULLAC, quasi-steady state (QSS) level population calculations were performed for the temperatures and densities of interest. All E1, M1 and M2 radiative transitions and all collisional excitation and de-excitation transitions were included. The quadrupole (E2) radiative transitions were found to be negligible for the L-shell calcium ions and were not included in the models. For the extended detailed non-LTE calculations, which involved the adjacent ion species, ionization and recombination rates were generated as follows: ionization rates, including inner-shell, were calculated according to the Lotz formula (Lotz [1968, 1970]) using the ab initio level energies. Autoionization probabilities were calculated by RELAC in the DW approximation. Recombination rates were calculated based on detailed balance principle. In particular, dielectronic recombination was taken into account by calculating radiationless capture rates from the ab initio autoionization rates.

Atomic rates from CHIANTI database were also used for CR calculations of line intensities. To the extent of our measurements, a large set of CHIANTI calcium data has been benchmarked in the present work. CHIANTI includes the best available electron impact excitation and radiative decay rates for E1 and M1 transitions of n = 2 configurations of Ca XV, Ca XIV, Ca XII, n = 2, 3 configurations of Ca XVII, Ca XVI, Ca XIII, and n=2, 3, 4, 5 configurations of Ca XVIII (Dere et al. [1997]; Landi et al. [1999]).

   
3.2 Model ions

Two models were generated for each ion: a basic model and an extended model. The basic models included a reduced number of levels and basic atomic processes (collisional excitation and de-excitation and radiative decay), and were found to be sufficient for most cases. The extended models were generated for particular test cases and therefore included greater number of levels and additional atomic processes (such as K-shell excitation, autoionization, collisional ionization from metastable and excited levels). The effect of radiative cascades on the populations of n = 2 levels is found to be $\le 15 \%$ for the transitions from n = 3 levels and $\le 5 \%$ from n = 4, 5 levels. For the range of plasma parameters of both experiments, we have checked and concluded that inner-shell ionization does not change the relative level populations significantly ($\le 15 \%$), which confirms applicability of the steady state equilibrium. Total ionization rates from the n = 3, 4resonant levels were found to be a factor of 10¹ - 10² less than total radiative decay rates from these levels.
Ca XVIII- The basic model for lithium-like calcium includes 67 energetically lowest levels from the configurations 1s²2l (l = s, p), 1s²3l (l = s, p, d), 1s²4l (l = s, p, d, f), 1s2s2l (l = s, p), 1s2s3l (l = s, p, f), 1s2s4l (l = s, p, d, f). The extended model includes levels up to n = 5, including configurations of the type 1s2snl. A He-like calcium model with all levels up to n = 4 was also used.
Ca XVII- Our basic model for beryllium-like calcium ion included 125 levels of the configurations 1s²2s², 1s²2snl, and 1s ²2l'nl'', where n = 2, 3, 4. The extended model comprises configurations of the type 1s2s²nl (n = 2, 3, 4) and 1s2s2pnl (n = 2, 3, 4).
Ca XVI- The configurations 2s²nl, 2s2pnl (n = 2, l = s, p; n = 3, l = s, p, d; n = 4; l = s, p, d, f) and 2s2p³ are included in the basic model of 147 levels. The extended model, in addition to the configurations listed above, includes the 5l (l = s, p, d, f) and the 1s2s²2pnl (n = 2, l = p; n = 3, l = s, p, d, and n = 4, l = s, p).
Ca XV- The basic model ion includes 377 levels of the configurations 2s²2p², 2s²2pnl2s2p²nl, 2s ²nl² and 1s²2p⁴ (n = 2, l = s, p; n = 3, l= s, p, d; n = 4; l = s, p, d, f). The extended model ion comprised a total of 1056 levels: the configurations mentioned above and the configurations 1s²2s²2p5l(l = s, p, d, f), 1s²2s¹2p²5l (l = s, p, d, f), and 1s2s²2p²nl (n = 2, l = p; n = 3, l = s, p, d; n = 4, 5, l = s, p, d, f).
Ca XIV- We use the configurations 2s²2p²nl, 2s2p³nl and 2s ²nln'l', where n = 2, l = s, p; n = 3, l = s, p, d; n = 4, l = s, p, d, f and n' = 3, l' = s, p, d. The total number of levels in the basic model is 546.
Ca XIII- The model ion contains 542 levels of the following configurations: 2s²2p⁴, 2s²2p³nl (n = 3; l = s, p, d and n = 4; l = s, p, d, f), 2s 2p⁴nl (n = 3; l = s, p, d and n = 4; l = s, p, d, f), 2s 2p⁶, 2s²2p ²nln'l' (n = 3; l = s, p and n' = 3; l = s, p) and 2s²3s²3p².
Ca XII- The 267 levels of the configurations 2s²2p⁵, 2s²2p⁴nl (n = 3; l = s, p, d and n = 4; l = s, p, d, f), 2s 2p⁶ and 2s 2p⁵nl (n = 3; l = s, p, d and n = 4; l = s, p, d, f) are used in the basic model ion.