** Up:** Laboratory observation and modeling

**Subsections**

#

3 Atomic data and collisional-radiative model

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
for the transitions
from *n* = 3 levels and
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 (), which confirms
applicability of the steady state
equilibrium.
Total ionization rates from the *n* = 3, 4resonant levels were found to be a factor of
10^{1} - 10^{2} 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
1s^{2}2*l* (*l* = s, p), 1s^{2}3*l* (*l* = s, p, d),
1s^{2}4*l* (*l* = s, p, d, f),
1s2s2*l* (*l* = s, p), 1s2s3*l* (*l* = s, p, f),
1s2s4*l* (*l* = s, p, d, f).
The extended model includes levels up to *n* = 5,
including configurations of the type 1s2s*nl*.
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 1s^{2}2s^{2}, 1s^{2}2s*nl*,
and 1s
^{2}2*l*'*nl*'', where *n* = 2, 3, 4.
The extended model comprises
configurations of the type 1s2s^{2}*nl* (*n* = 2, 3, 4)
and 1s2s2p*nl* (*n* = 2, 3, 4).

**Ca** **XVI-**
The configurations 2s^{2}*nl*,
2s2p*nl* (*n* = 2, *l* = s, p; *n* = 3, *l* = s, p, d; *n* = 4; *l* = s, p, d, f)
and 2s2p^{3} are included in the basic model of
147 levels.
The extended model, in addition to the configurations
listed above, includes the 5*l* (*l* = s, p, d, f) and
the 1s2s^{2}2p*nl* (*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
2s^{2}2p^{2}, 2s^{2}2p*nl*2s2p^{2}*nl*,
2s
^{2}*nl*^{2} and 1s^{2}2p^{4} (*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 1s^{2}2s^{2}2p5*l*(*l* = s, p, d, f), 1s^{2}2s^{1}2p^{2}5*l* (*l* = s, p, d, f),
and 1s2s^{2}2p^{2}*nl* (*n* = 2, *l* = p; *n* = 3,
*l* = s, p, d; *n* = 4, 5, *l* = s, p, d, f).

**Ca** **XIV-**
We use the configurations 2s^{2}2p^{2}*nl*,
2s2p^{3}*nl* and 2s
^{2}*nln*'*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:
2s^{2}2p^{4},
2s^{2}2p^{3}*nl* (*n* = 3; *l* = s, p, d and *n* = 4; *l* = s, p, d, f),
2s 2p^{4}*nl* (*n* = 3; *l* = s, p, d and *n* = 4; *l* = s, p, d, f),
2s 2p^{6},
2s^{2}2p
^{2}*nln*'*l*' (*n* = 3; *l* = s, p and *n*' = 3; *l* = s, p) and
2s^{2}3s^{2}3p^{2}.

**Ca** **XII-**
The 267 levels of the configurations
2s^{2}2p^{5},
2s^{2}2p^{4}*nl* (*n* = 3; *l* = s, p, d and *n* = 4; *l* = s, p, d, f),
2s 2p^{6} and
2s 2p^{5}*nl* (*n* = 3; *l* = s, p, d and *n* = 4; *l* = s, p, d, f)
are used in the basic model ion.

** Up:** Laboratory observation and modeling

Copyright The European Southern Observatory (ESO)