# Introduction

STARK-B is a database of calculated widths and shifts of isolated lines of atoms and ions due to electron and ion collisions.

This database is devoted to modeling and spectroscopic diagnostics of stellar atmospheres and
envelopes. In addition, it is also devoted to laboratory plasmas, laser equipments and technological
plasmas. So, the domain of temperatures and densities covered by the tables is wide and depends on
the ionization degree of the considered ion. The temperature can vary from several thousands for
neutral atoms to several hundred thousands of Kelvin for highly charged ions. The electron or ion
density can vary from 10^{12} (case of stellar atmospheres) to several 10^{19}cm^{-3} (some white dwarfs and
some laboratory plasmas).

The impact approximation and the isolated line approximation are applied, so that the line profile is Lorentzian. The basis for calculations is the computer code which evaluates electron and ion impact broadening of isolated spectral lines of neutral atoms and ions, using the semiclassical-perturbation approach developed by Sahal-Bréchot (1969ab, 1974), and supplemented in Fleurier et al. (1977), see below. This computer code has been updated by Dimitrijević and Sahal-Bréchot in their series of papers, Dimitrijević and Sahal-Bréchot (1984) and following papers. The data are derived from this series of papers and are cited in the tables. The accuracy of the data varies from about 15-20 percent to 35 percent, depending on the degree of excitation of the upper level, and on the quality of the used atomic structure entering the calculation of scattering S-matrix leading to the widths and shifts. The more the upper level is excited, the more the accuracy is good. In the earlier papers, the used atomic structure was the so-called “Bates and Damgaard” one (Coulomb wavefunctions + quantum defect). More recent atomic structure data are introduced in the latest papers. Fore more details, the reader is invited to refer to the papers cited in the tables for the used atomic data and atomic levels.

## The impact approximation

The impact approximation is valid when the mean duration τ of a collision is much smaller than the mean interval ΔT between two collisions (Baranger 1958 abc).
ΔT is of the order of the inverse of the colisional line width γ expressed in angular frequency units.

τ can be written as :

**τ= <ρ> / <ν>** ,

where <ρ> is a typical impact parameter and <ν> the mean velocity of the collider :

**<ν> = (8kT / πμ) ^{1/2}**,

μ being the reduced mass, T the temperature, and k the Bolzmann constant.

An order of magnitude of <ρ> can be derived from the line width γ and is obtained by writing (N being the density of the perturbers)

**γ=N<ν>π<ρ>**.

^{2}The validity condition of the impact approximation can be written as :

**N V <<1**,

**V=π<ρ>**is the collision volume (Baranger 1958abc).

^{3}The impact values of the widths and shifts are given in the tables, except when N V > 0.5. Then the cells are empty and marked by an asterisk preceding the cell. Widths values for 0.1 <N V < 0.5 are marked by an asterisk in the cell preceding the value. See the Section "Data description" for more details.

In the far wings, Δω being the detuning in angular frequency units, the validity condition for the generalized impact approximation becomes

**τΔω<<1**.

When the impact approximation is not valid (especially for ion colliders), the quasistatic approximation can be used. As shown by Baranger (1962) for ion emitters and polarization interaction potenial, and by Sahal-Bréchot (1991) for the quadrupolar interaction which is in fact dominant due to the Coulomb repulsion, the quasistatic broadening is completely negligible in the wings. For neutrals emitters, the polarization part of the interaction is most often dominant and can be obtained by the A parameter of Griem (1974). This A parameter is provided in a few tables, where it is calculated with the method described by by Ben Nessib et al. (1996).

## The isolated line approximation

At high densities or for lines arising from high levels, the electron impact width becomes comparable to the separation ΔΕ(nl, nl+1) between the perturbing energy levels and the initial or final level : the corresponding levels become degenerate and the isolated line approximation is invalid (Griem 1974). In order to check the validity of this approximation, we have defined a parameter C in Dimitrijevic and Sahal-Bréchot (1984).

C is given in the tables of the database and is given by:

**C = Nλ ^{2}10^{-8} (E_{j} –E_{j’})**,
where N is the electron density of the medium in cm

^{-3}, λ the wavelength of the line in Å, and

**(E**, where i and f denote the initial and final levels of the studied line, and i’ and f’ the corresponding closest perturbing levels. The units are in cm-1.

_{j}–E_{j’}) = min[(E_{i}–E_{i’}), (E_{f}–E_{f’})]See also “Data description”.

## The semiclassical perturbation approximation (SCP)

When the impact approximation is valid, the collisional broadening becomes an application of the theory of collisions (Baranger 1958abc).

In the semiclassical approximation, rectilinear trajectories are used for neutral emitters (or absorbers), and hyperbolic trajectories for ionic emitters (or absorbers) colliding with charged particles.

Within the second order perturbation approximation, dipolar, polarization and quadrupolar interactions are taken into account (Sahal-Bréchot 1969ab and earlier papers), updated for complex atoms and ions (Sahal-Bréchot 1974). The details of calculations of the widths and shifts can be found on these papers. For ionic emitters, the original computer code has been updated by including Feshbach resonances in elastic and fine structure transitions by using the semiclassical limit of the Gailitis formula (Fleurier et al. 1977).

Debye shielding effect is also taken into account. It is negligible at low densities or for lines arising from low levels. Then widths and shifts are proportional to the density.

## The Modified Semiempirical Method (MSE)

The MSE (Modified Semi-Empirical) method may be used when we have not a sufficiently complete set of energy levels for an adequate semiclassical treatment. So STARK-B includes data obtained using the MSE method (Dimitrijević & Konjević 1980, Dimitrijević & Kršljanin 1986, supplemented by Popović & Dimitrijević (1996) for complex atoms).

The accuracy of the MSE method is about 40-50%.

In comparison with the semiclassical perturbation approach (Sahal-Bréchot, 1969ab), the modified semiempirical approach needs a considerably smaller number of atomic data. In fact, if there are no perturbing levels strongly violating the assumption that the closest perturbing levels are levels with Δn=0 for line width calculations, we only need energy levels nl with

**❘n-n _{i} ❘=Δn=0** and

**(l = l**(resp. n

_{i}± 1)_{f}, l

_{f}), where n is the principal quantum number, l the orbital momentum and i and f denote the initial and final levels of the transition.

In addition, the perturbing levels with Δn≠0 which are needed for a full semiclassical investigation are lumped together and approximately estimated in the MSE method.

- Baranger, M. 1958a, Phys. Rev. A, 111, 481-493
- Baranger, M. 1958b, Phys. Rev. A, 111, 494-504
- Baranger, M. 1958c, Phys. Rev. A, 112, 855-865
- Baranger M., 1962 "Spectral line broadening by plasmas", edited by D. R. Bates. Library of Congress Catalog Card Number 62-13122. in "Atomic and Molecular Processes", pp. 493-548, Acad. Press Inc., New-York
- Ben Nessib, N., Ben Lakhdar, Z. et Sahal-Bréchot, S., 1996,, Phys. Scr., 54, 608-613.
- Dimitrijevic, M.S., and Sahal-Bréchot, S.: 1984,JQSRT 31, 301-313
- Dimitrijevic M.S., & Konjevic J., 1980, JQSRT, 24, 451-459
- Dimitrijevic M.S. & Krsljanin, V. 1986, A&A, 165, 269-274
- Fleurier C., Sahal-Bréchot, S., and Chapelle, J.: 1977, JQSRT, 17, 595-604
- Griem, H. R. 1974, "Spectral line broadening by plasmas", Pure and Applied Physics, New York: Academic Press, USA
- Popovic L. C., & Dimitrijevic M.S., 1996, Phys. Scr., 53, 325-331
- Sahal-Bréchot, S.: 1969a, A&A 1, 91-123
- Sahal-Bréchot, S.: 1969b, A&A 2, 322-354
- Sahal-Bréchot, S.: 1974, A&A 35, 319-321
- Sahal-Bréchot, S.: 1991, Astron.Astrophys. 245, 322-330