A&A 423, 397-400 (2004)

DOI: 10.1051/0004-6361:20047100

**N. Ben Nessib ^{1} - M. S.
Dimitrijevic^{2,3} - S. Sahal-Bréchot^{4}**

1 - Groupe de Recherche en Physique Atomique et
Astrophysique, Faculté des Sciences de Bizerte, 7021 Zarzouna,
Tunisia

2 - Astronomical Observatory, Volgina 7, 11160 Belgrade,
Serbia

3 - Institute Isaac Newton of Chile, Yugoslavia Branch, Yugoslavia

4 - Laboratoire d'étude du Rayonnement et de la Matière en
Astrophysique, UMR CNRS 8112, Observatoire de Paris-Meudon, 92195
Meudon, France

Received 19 January 2004 / Accepted 3 May 2004

**Abstract**

Using a semiclassical perturbation approach, we have
obtained Stark broadening parameters for 16 Si V multiplets using
atomic data calculated ab initio with the SUPERSTRUCTURE code. In
order to complete Stark broadening data for the most important
charged perturbers in stellar atmospheres, Stark broadening
parameters for proton-, He II-, and Si II-impact line
widths and shifts are also presented. Results have been obtained
for an electron density of 10^{17} cm^{-3} as a function of
temperature. Moreover, we have performed the same calculations
with oscillator strengths calculated within the Coulomb
approximation. The differences, which are within the error bars of
the semiclassical perturbation approach (
)
confirm
that the Bates & Damgaard approximation may be used to complete
the atomic data set needed for the Stark broadening calculations
for multicharged ions like Si V, where more reliable oscillator
strength values are not available.

**Key words: **atomic processes - line: profiles - atomic data

The semiclassical perturbation formalism (Sahal-Bréchot 1969a,b) has been used in a series of papers for large scale calculations of Stark broadening parameters for a number of spectral lines of various emitters from neutral atoms up to twelve times ionized atoms (see e.g., Dimitrijevic 1997, and references therein). The results of such calculations are of interest for a number of different problems in physics, astrophysics and plasma technology. In astrophysics, such atomic data are of interest e.g. for numerical modelling of stellar atmospheres or abundance determinations, problems correlated with nucleogenesis, mixing between atmosphere and interior, stellar structure and evolution (see e.g., Boesgaard 1988), and opacity calculations (Seaton 1988). Moreover, Stark broadening parameters are of interest for the investigation and diagnostics of laboratory and laser-produced plasmas, as well as for the theoretical studies of regularities and systematic trends.

In the case of four-times ionized atoms there exists a sufficiently complete set of the needed atomic data required for sophisticated semiclassical calculations for a number of astrophysically interesting lines. Therefore, large scale calculations have been performed with the semiclassical perturbation formalism (Sahal-Bréchot 1969a,b) for C V (Dimitrijevic & Sahal-Bréchot 1995a, 1996a), N V (Dimitrijevic & Sahal-Bréchot 1992), O V (Dimitrijevic & Sahal-Bréchot 1995b,c), P V (Dimitrijevic & Sahal-Bréchot 1995a, 1996a), S V (Dimitrijevic & Sahal-Bréchot 1997, 1998a) and V V (Dimitrijevic & Sahal-Bréchot 1998b) spectral lines. When we did not have a sufficiently complete set of reliable atomic data, the full semiclassical method could not be applied adequately, and simpler methods (Dimitrijevic & Konjevic 1980; Griem 1974, Eq. (526)) were used for the calculation of C V, O V, F V, Ne V and Al V Stark broadening parameters, including 6 multiplets of Si V (Dimitrijevic 1993a,b). Such data are of interest for the consideration of radiative transfer through subphotospheric layers (Seaton 1988), for modelling of some hot star atmospheres such as PG 1159 pre-white dwarfs with an effective temperature between 100 000 and 140 000 K (Werner & Heber 1991), as well as for fusion plasmas and laser-produced plasmas. The development of soft X-ray lasers, where Stark broadening data are needed to calculate gain values, to model radiation trapping and to consider photoresonant pumping schemes (see e.g., Fill & Schöning 1994; Griem & Moreno 1980), provided an additional interest in such results.

With the development of space-borne spectroscopy, together with the possibility of obtaining high resolution spectra for a wide wavelength range including X-rays not detectable from the Earth's surface, the interest in such results is increasing. For example the Low Energy Transmission Grating (LETG) on board the X-ray space observatory CHANDRA provides high quality spectra in the wavelength region between 5 and 140 Å, and contains a series of lines which can provide density and temperature diagnostics for relatively hot and dense stellar plasmas. An analysis of the X-ray high resolution spectrum of some stars shows spectra which may be rich in emission lines in the soft X-ray region, including Si V lines.

The aim of this paper is twofold. First we will compute Si V Stark broadening parameters within the semiclassical formalism by using oscillator strengths from the SUPERSTRUCTURE code to provide new Stark broadening data of astrophysical interest. Additionally, we perform the same calculations using the Coulomb approximation (Bates & Damgaard 1949; Oertel & Shomo 1968), in order to estimate the error introduced in the Stark broadening parameters due to uncertainties in the oscillator strengths obtained from the Coulomb approximation.

The energy levels of SiV are calculated using the general atomic
structure code SUPERSTRUCTURE developed at the University College,
London and described by Eissner et al. (1974). SUPERSTRUCTURE can
calculate electric dipole (E1), electric quadrupole (E2), magnetic
dipole (M1) and magnetic quadrupole (M2) transition probabilities.
The wave functions are determined by diagonalization of the
non-relativistic Hamiltonian using orbitals calculated in a scaled
Thomas-Fermi-Dirac-Amaldi (TFDA) potential. The scaling parameters
have been obtained by a self-consistent energy minimization
procedure on all term energies of the eleven configurations
,
,
and
(
). The relativistic corrections:
spin-orbit, mass, Darwin and one-body, are introduced according to
the Breit-Pauli approach of Bethe & Salpeter (1957) in intermediate coupling *LSJ*. The main input data of this code are the charge of the ion and the configurations to be used in the
model.

Stark broadening parameter calculations have been performed within
the semiclassical perturbation method (Sahal-Bréchot 1969a,b). A
detailed description of this formalism with all the innovations
and optimizations is given in Sahal-Bréchot (1969a,b, 1974, 1991),
Fleurier et al. (1977), Dimitrijevic et al. (1991),
Dimitrijevic & Sahal-Bréchot (1996b), and only the basic
details of calculations will be presented here. The Stark full
width (*W*) at the intensity half maximum (FWHM) and shift (*d*)
of an isolated spectral line, may be expressed as (Sahal-Bréchot 1969a,b)

(1) |

where

(2) |

and the elastic cross section is given by

(3) |

The phase shifts and due respectively to the polarisation potential (

For electrons hyperbolic paths due to the attractive Coulomb force were used, while for perturbing ions the hyperbolic paths are different since the force is repulsive.

The formulae for the ion-impact widths and shifts are analogous to Eqs. (1)-(3), without the resonance contribution to the width. The results obtained within the semiclassical perturbation approach (Sahal-Bréchot 1969a,b) have been compared with critically selected experimental data for 13 He I multiplets (Dimitrijevic & Sahal-Bréchot 1985) and it was found that the agreement between experimental data and semiclassical calculations is within 20%, which is within the limits of predicted accuracy of the semiclassical method (Griem 1974). In addition, recent measurements of line widths for F V (Uzelac et al. 1993; Blagojevic et al. 1996), N V (Glenzer et al. 1992; Blagojevic et al. 1999) and O V (Blagojevic et al. 1999) are in excellent agreement with the corresponding semiclassical perturbation calculations (Blagojevic et al. (1996) for F V, Dimitrijevic & Sahal-Bréchot (1992) for N V and Dimitrijevic & Sahal-Bréchot (1995b,c) for O V).

By combining the SUPERSTRUCTURE code for calculating energy levels and oscillator strengths and the code for the Stark broadening calculations we calculated Stark broadening parameters ab initio.

Using atomic energy levels obtained from the SUPERSTRUCTURE code, we have also calculated oscillator strengths using the Coulomb approximation. For higher levels, the method described by van Regemorter et al. (1979) has been used.

Electron-impact broadening widths (FWHM) and shifts for 16 Si V multiplets, have been calculated with atomic energy levels and oscillator strengths obtained using the SUPERSTRUCTURE Code.
Results have also been obtained using the Bates & Damgaard method
for the oscillator strengths. These data are shown in Table 1
(available only in electronic form) for a perturber density of
10^{17} cm^{-3} and temperatures from 50 000 up to 500 000 K.
Additionally, in order to provide Stark broadening data for the
most important ionic perturbers in stellar atmospheres, proton-,
ionized helium-, and Si II-impact broadening parameters are
calculated for the same perturber densities and temperatures.
Again, atomic energy levels and oscillator strengths are obtained
using the SUPERSTRUCTURE code, and the results are shown in Table 2 (available only in electronic form). We found results similar to
those obtained using the Bates & Damgaard approximation for the
set of oscillator strengths for transitions to the perturbing
levels. Such temperatures are of interest for the modelling and
analysis of X-ray spectra, such as the spectra obtained by
CHANDRA, modelling of some hot star atmospheres (e.g. PG 1195
type), subphotospheric layers, soft X-ray lasers and
laser-produced plasmas. Higher temperatures are of interest for
fusion plasmas as well as for stellar interiors. We also specify a
parameter C (Dimitrijevic & Sahal-Bréchot 1984), which gives
an estimate for the maximum perturber density for which the line
may be treated as isolated, when it is divided by the
corresponding full width at half maximum. For each value given
in Table 1 the collision volume (*V*) multiplied by the perturber
density (*N*) is much less than one and the impact approximation
is valid (Sahal-Bréchot 1969a,b). When the impact approximation
is not valid, the ion broadening contribution may be estimated by
using the quasistatic approach (Griem 1974; Sahal-Bréchot 1991).
In the region where neither approximation is valid, a unified-type
theory should be used. For example, in Barnard et al. (1974) a
simple analytical formula for such a case is given. The accuracy
of the results shown in Tables 1 and 2 decreases when broadening
by ion interactions becomes important.

If we compare results for Stark widths obtained with oscillator
strengths calculated with SUPERSTRUCTURE ()
and by using the
Bates & Damgaard approximation (
), the average ratio of
and
is 1.09 for *T* = 50 000 K and 1.10 for 500 000 K. Since in Stark broadening calculations we use a set of atomic data where a particular oscillator strength value is not always
critical, the result obtained confirms that the Bates & Daamgard
approximation may be useful for Stark broadening calculations in
the case of ions such as Si V, when more reliable data are not
available. For shifts the disagreements are larger, but the shifts
obtained here are typically 2-3 orders of magnitude smaller than
the widths. In fact, for the widths all the important
contributions have a positive sign. For the present shifts
important contributions have different signs, leading to mutual
cancellations. Therefore, since the usual final accuracy for the
shift is about 20-30 per cent of the width value, the accuracy of
the shifts determined here is bad. If we compare our present
results with Stark width results obtained in Dimitrijevic
(1993a) by using the modified semiempirical method (Dimitrijevic
& Konjevic 1980) and with the results obtained in
Dimitrijevic (1993b) by using Griem's simplified semiclassical
approach (Griem 1974, Eq. (526)) we can see that the average ratio
of the present widths to the simplified semiclassical ones is
1.30, and to the modified semiempirical method 1.77. As to the
difference between the present work and that of Dimitrijevic
(1993a,b), we have here a complete set of atomic energy levels
needed for an adequate application of the semiclassical
perturbation approach. One also has to take into account that
these energy levels are calculated, while in Dimitrijevic
(1993a,b), experimental atomic energy levels were used.

In conclusion, we see that using the SUPERSTRUCTURE code one obtains a set of energy levels and oscillators strengths, enabling an ab initio calculation of Stark broadening parameters. This is suitable especially for multicharged ions when other theoretical and experimental atomic data are scarce. This work also suggests that the set of oscillator strength values obtained with the Bates & Damgaard approximation may be useful for Stark broadening calculations when more reliable data are not available.

This work is a part of the project "Influence of collisional processes on astrophysical plasma line shapes'' (GA 1195), supported by the Ministry of Science and Environment Protection of Serbia.

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