## Introduction

STARK-B is a database of calculated widths and shifts of isolated lines of atoms and ions due to electron and ion collisions in the impact approximation. They are all published in refereed journals.

This database is devoted to modeling and spectroscopic diagnostics of stellar atmospheres and
envelopes. In addition, it is also devoted to laboratory plasmas, laser equipment 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 thousand for
neutral atoms to several hundred thousand 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 and is characterized by its width and its shift. Three methods and approximations are used in the tables: the Semi-Classical-Perturbation approximation (SCP) for electron and positive ion impacts, the Modified Semiempirical method (MSE) for electron impacts, and the quantum mechanical electron impact broadening for intermediate coupling, using the Superstructure code for the atomic data and the distorted-wave approximation for the scattering matrix (Q-SST-DW).

## 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 W 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 W and is obtained by writing (N being the density of the perturbers)

**W=N<ν>π<ρ>**.

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

**N V <<1**,

**V=π<ρ> ^{3}** is the collision volume (Baranger 1958abc).

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).

In conclusion, when the impact approximation is valid, the calculation of the broadening and shifting of the spectral lines becomes an application of the theory of collisions; the atom is described by its wave-functions, its energy levels and its atomic data, and the scattering matrix S for the atom colliding with one perturber only must be obtained. Then a summation (average) over all colliding perturbers has to be calculated. The Maxwell distribution of the velocities of the colliding perturbers is used. The interaction between the radiating neutral or ionized atom is the Coulomb interaction.

## The complete collision approximation

Within this approximation, atom–radiation and atom–perturber interactions are decoupled. This implies that the collision must be considered as instantaneous in comparison with the time, characteristic of the evolution of the excited state under the effect of the interaction with the radiation. In other words, the interaction process has time to be completed before the emission of a photon.

The complete collision approximation can become invalid in line wings, even if it remains valid in the line center.

Its condition of validity is:

τ << 1/ Δω , where Δω is the detuning.

## 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**

_{j}–E_{j’}) = min[(E_{i}–E_{i’}), (E_{f}–E_{f’})]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.

See the Section “Data Description” for details.

## The semiclassical perturbation approximation (SCP)

The calculations provide widths and shifts due to collisions with electrons and ions.. The impact approximation being valid, the collisional broadening becomes an application of the theory of collisions (Baranger 1958abc). The semi-classical scattering matrix must be obtained.

In the SCP approximation, the trajectory of the colliding electron (or ion) is classical and unperturbed by the atom-perturber interaction. Thus rectilinear trajectories for neutral emitters (or absorbers), and hyperbolic trajectories for ionic emitters (or absorbers) are used. Penetrating orbits are outside the scope of SCP method and code. Within the second order perturbation approximation for the atom-collider Coulomb interaction, dipolar, polarization and quadrupolar interactions are taken into account (Sahal-Bréchot 1969ab and earlier papers), updated for complex atoms and ions (Sahal-Bréchot 1974, Mahmoudi et al 2008). The details of calculations of the widths and shifts can be found on these papers. A review of the theory, of the approximations and details of calculations are given in Sahal-Bréchot et al. (2014).

For ionic emitters, the method 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, Sahal-Bréchot 2021).

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 original computer code, created by Sahal-Bréchot has been continuously updated by Dimitrijević and Sahal-Bréchot: Dimitrijević and Sahal-Bréchot (1984) and following papers. The data are derived from this series of papers and the references to the publications are cited in the tables. The accuracy of the data varies from about 15-20 percent to 35 percent for the width, depending on the degree of excitation of the upper level, and on the quality of the used atomic structure entering the calculation of the scattering S-matrix leading to the widths and shifts. The more the upper level is excited, the more the accuracy is good. For the shifts, the results can sometimes be less accurate, due to negative interference effects between the upper and lower levels of the studied line (Sahal-Bréchot et al. 2014).

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. For more details, the reader is invited to refer to the papers cited in the tables for the used atomic levels and atomic data.

In addition, we notice that the fine structure (and, *a fortiori*, hyperfine structure) can generally
be ignored, and consequently, the fine structure components (or hyperfine components) have
the same width and the same shift, which are equal to those of the multiplet. This is due to the
fact that the electronic spin S (and also the nuclear spin I), has no time to rotate during the
collision time τ (of the order of <ρ> / <ν>, the mean duration of the collision). This is only
true in LS coupling. Thus, widths and shifts are often only given for multiplets in the SCP
tables.

## 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 calculations provide widths due to electron collisions.

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.

## The Quantum mechanical method of electron-impact broadening of spectral lines in intermediate coupling: Distorted Wave approximation for the scattering S-matrix and SUPERSTRUCTURE code for the atomic data (Q-SST-DW)

The method is described in Elabidi et al. (2004), completed in Elabidi et al. (2008). It uses the Distorted Wave DW approximation for the scattering S-matrix and the SUPERSTRUCTURE code SST for the atomic data in intermediate coupling. Feshbach resonances have been included in Elabidi et al. (2009), using the Gailitis method (Gailitis 1963).

The UCL atomic code package SUPERSTRUCTURE/DW/JAJOM developed for the calculations of atomic data and collision cross-sections is used (Eissner 1972, Eissner et al. 1974, Saraph 1972). These codes have been extended to electron impact line widths calculations by Elabidi et al. (2004, 2008). For that, JAJOM has been modified into JAJPOLARI (Elabidi and Dubau, unpublished results) to produce the reactance matrices R and the collision strengths in intermediate coupling, Then the code RtoS (Dubau, unpublished results) has been developed and used to evaluate the real and the imaginary parts of the scattering matrix which are necessary for obtaining the line widths. Since widths and shifts are calculated in intermediate coupling, fine structure widths are given in the Q-SST-DW tables

The accuracy of the Q-SST-DW method is better than the SCP one especially at low temperatures, because the contributions of strong and close collisions to Stark broadening are important. The difference with the SCP calculations can become high, and the use of DW quantum mechanical calculations, even if they are weak-coupling approximation, improves the results when compared to experiments.

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