Data on Stark Broadening of Sn II Spectral Lines

Abstract

Data on spectral line widths and shifts broadened by interactions with charged particles, for 44 lines in the spectrum of ionized tin, for collisions with electrons and H II and HeII ions, are presented as online available tables. We obtained them by employing the semiclassical perturbation theory for temperatures, T, within the 5000–100,000 K range, and for a grid of perturber densities from 10¹⁴ cm⁻³ to 10²⁰ cm⁻³. The presented Stark broadening data are of interest for the analysis and synthesis of ionized tin lines in the spectra of hot and dense stars, such as, for example, for white dwarfs and hot subwarfs, and for the modelling of their atmospheres. They are also useful for the diagnostics of laser-induced plasmas for high-order harmonics generation in ablated materials.

Data Set: The dataset is submitted as a Supplementary File.

Data Set License: CC-BY 4.0

1. Introduction

Data for the Stark broadening of spectral lines are needed for different scientific issues, such as, e.g., for stellar [1,2], laboratory [3,4], fusion [5,6], and laser-produced plasma research; modelling and diagnostics [7,8,9]; the development of lasers [2,10,11]; and the investigation and optimization of various plasmas in technology [12], such as, e.g., for the welding, melting, and piercing of metals by lasers, and for plasma light sources [2,13].

Recently, Ganeev [14] underlined that the diagnostics of laser-induced plasma is important for high-order harmonics generation (HHG) in ablated materials, accentuating also that the targets that are good for the creation of laser-induced plasma with optimal characteristics are the metals belonging to the period V of the periodic system of elements. Since Sn II is in the fifth period, it is useful for providing the corresponding Stark broadening parameters for spectral lines of Sn and its ions.

Spectral lines of Sn and its corresponding ions are observed in the spectra of different stars, and the data on Stark broadening of different spectral lines within the spectrum of tin in various ionization stages are useful for the analysis, synthesis, abundance determination, and modelling of stellar atmospheres. For example, observations of Sn I lines and the determination of tin abundance have been reported in Refs. [15,16,17], and Sn II spectral lines have been found, e.g., in the UV spectra of the metal-poor star HD 222925 [18]. Moreover, in the abundance analysis of CS 31082-001 [19] and 2MASS J00512646-1053170 [20], the tin has also been included. Additionally, it should be noted that, for the abundance determination of tin in HZ 44 and HD 127493, which are hot subdwarfs of sdO type, spectral lines of Sn III and Sn IV have been used [21]. Sn IV lines have also been found in the spectra of He-sdOB stars Feige 46 and LS IV-14°116 [22].

There is currently great interest in the Stark broadening parameters for Sn II, and a relatively large number of experiments have been performed [23,24,25,26,27,28,29,30,31]. The critical comments of some of them may be found in Refs. [32,33]. We note that several calculations also exist. The authors of Ref. [34] have applied the semiclassical method, in Refs. [35,36,37] various forms of semiempirical method were used, and in Ref. [38] regularities and systematic trends have been analysed to estimate Sn II Stark shifts.

Within the frame of the semiclassical perturbation theory [39,40,41], Stark widths and shifts for 44 Sn II spectral lines [42], broadened by collisions with electrons, protons, and He II ions, have been calculated for a grid of temperatures and electron densities. A detailed explanation of calculations, the analysis of obtained results and their comparison with existing experimental and theoretical data, and an investigation of the influence of Stark broadening in stellar atmospheres are given in Ref. [42]. In [42] are also data for a perturber density of 10¹⁵ cm⁻³ and temperatures from 5000 K up to 100,000 K, in tabular form. Here, in the Supplementary Materials, data for perturber densities from 10¹⁴ cm⁻³ up to 10²⁰ cm⁻³ are given online, in computer readable form.

2. The Semiclassical Perturbation Method

The semiclassical perturbation theory [39,40,41] has previously been used for the calculation of Stark widths and shifts. It is described in Ref. [42], and here will be given the basic equations needed to better understand how the results are obtained. In order to obtain the full width at half intensity maximum (FWHM-W) and shift (d) of an isolated spectral line of a non-hydrogenic ion, we employed the following expressions:

$\begin{array}{c}\hfill W=N\int vf\left(v\right)dv\left(\sum _{i^{\prime }\ne i}{\sigma }_{i{i}^{\prime }}\left(v\right)+\sum _{f^{\prime }\ne f}{\sigma }_{f{f}^{\prime }}\left(v\right)+{\sigma }_{el}\right)\end{array}$

(1)$\begin{array}{c}\hfill d=N\int vf\left(v\right)dv{\int }_{R_3}^{R_D}2\pi \rho d\rho sin\left(2{\phi }_p\right).\end{array}$

In the above equations, i and f represent the initial and final level, while $i^{\prime }$ and $f^{\prime }$ are their perturbing levels, $N$ is the perturber density of perturbers, $\upsilon$ is their velocity, $f\left(\upsilon \right)$ is the Maxwellian velocity distribution, and $\rho$ is the impact parameter for the considered perturbing particle.

The inelastic cross sections ${\sigma }_{k{k}^{\prime }}\left(\upsilon \right)$, $k=i,f$, is here presented as an integral of the transition probability $P_{k{k}^{\prime }}(\rho ,\upsilon )$ over $\rho$:

(2)$\sum _{k^{\prime }\ne k}{\sigma }_{k{k}^{\prime }}\left(\upsilon \right)={\displaystyle \frac{1}{2}}\pi R_1^2+{\int }_{R_1}^{R_D}2\pi \rho d\rho \sum _{k^{\prime }\ne k}{P}_{k{k}^{\prime }}(\rho ,\upsilon ).$

The elastic collisions and resonances are here presented as:

$\begin{array}{c}\hfill {\sigma }_{el}=2\pi R_2^2+{\int }_{R_2}^{R_D}2\pi \rho d\rho {sin}^2\delta +{\sigma }_r,\end{array}$

(3)$\delta ={({\phi }_p^2+{\phi }_q^2)}^{{\displaystyle \frac{1}{2}}}.$

Here, ${\sigma }_{el}$ is the elastic cross section and ${\phi }_p$ ($r^{-4}$) and ${\phi }_q$ ($r^{-3}$) are phase shifts due to the polarization and quadrupolar potential [39]. The symmetrization and cut-offs $R_1$, $R_2$, $R_3$, and $R_D$ are explained in [40]. The influence of Feshbach resonances, ${\sigma }_r$, is explained in [43].

3. Data Description

We present FWHM (W) and shift (d) of 44 Sn II spectral lines, broadened by collisions with e, H II, and He II ions, calculated using the semiclassical perturbation theory [39,40,41]. The atomic energy levels for the Sn II ion needed for the calculations have been taken from Refs. [44,45]. We performed calculations for temperatures of 5000 K, 10,000 K, 20,000 K, 30,000 K, 50,000 K, and 100,000 K, while the perturber densities are from 10¹⁴ cm⁻³ up to 10²⁰ cm⁻³. The data are contained in the Supplementary Materials, in computer readable form, as Tables S1–S14, for a perturber density of 10¹⁴ cm⁻³, and Tables S7–S20, for a perturber density of 10²⁰ cm⁻³. In the tables is present the parameter C [46], representing the condition of validity of the isolated line approximation. It enables one to determine up to which maximal density is valid for this approximation, when we divide it by the corresponding FWHM (W). Details of all calculations and the corresponding analyses and comparison with existing experimental and theoretical data can be found in Dimitrijević et al. [42]. In Ref. [42], the accuracy of the obtained results is checked by comparison with existing experimental and theoretical data. The theoretical error of the semiclassical perturbation method is estimated to be around 20% in the case of simple spectra. For a more complex spectrum such as Sn II, we assume that it is around 30%.

4. User Notes

We draw attention to the fact that the $\lambda$ in the tables in the Supplementary Materials is obtained from the energies used for the Stark width and shift calculations, so it is often different from experimentally obtained results.

If we want a different wavelength, for example, the experimental one, or for a line within a multiplet, we can do this by using, for the width for example, the expression:

(4)$W_{cor}={\left({\displaystyle \frac{{\lambda }_{new}}{\lambda }}\right)}^{2}W.$

and the analogous for the shift. In Equation (4), W_cor is the corrected width; ${\lambda }_{new}$ is the experimental, observed, or value of another line within the same multiplet; $\lambda$ is the wavelength calculated from atomic energy levels, or the averaged value for a multiplet; and W is the width from the tables in the Supplementary Materials.

We can calculate the line profile $F\left(\omega \right)$ (here, $\omega$ is in angular frequency units), using:

(5)$\begin{array}{c}\hfill F\left(\omega \right)={\displaystyle \frac{W/\left(2\pi \right)}{{(\omega -{\omega }_{if}-d)}^2+{(W/2)}^2}}.\end{array}$

Here

$\begin{array}{c}\hfill {\omega }_{if}={\displaystyle \frac{E_i-E_f}{\hslash }}\end{array}$

and $E_i$ and $E_f$ are the energies of the initial and final atomic energy level.

5. Conclusions

The computer readable data set available online as the Supplementary Materials is described in this article. It contains Stark full widths at half intensity maximum and shifts for 44 Sn II spectral lines broadened by collisions with electrons and H II and He II ions, obtained using the semiclassical perturbation theory [39,40,41]. The obtained results are presented for a grid of temperatures and perturber densities.

The Stark broadening data presented in this article are of particular significance for hot and dense stars like white dwarfs and hot subdwarfs, but also for A-type and late B-type stars. They are, for example, needed for the abundance determination of tin, the analysis and synthesis of stellar Sn II spectral lines, and the modelling of stellar atmospheres. This data set is also useful for the diagnostics of laser-produced plasmas, such as, for example, those needed for techniques for high-order harmonic generation in ablated materials. We also underline that excellent targets for optimal laser-induced plasma formation are the metal elements from the fifth period of the periodic table where tin belongs.

Author Contributions

Conceptualization, M.S.D.; methodology, S.S.-B. and M.S.D.; software, S.S.-B. and M.S.D.; validation, M.S.D., M.D.C., C.Y., and S.S.-B.; formal analysis, M.D.C., C.Y., and M.S.D.; investigation, M.S.D., M.D.C., and C.Y.; data curation, M.S.D., M.D.C., and C.Y.; writing—original draft preparation, M.S.D. and M.D.C.; writing—review and editing, M.S.D., M.D.C., and C.Y.; supervision, S.S.-B. and M.S.D. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

All data are contained in the Supplementary Materials.

Acknowledgments

This work has been supported with a STSM visit grant for M.S.D. within the framework of COST Action CA 18222 “Attosecond Chemistry—ATTOCHEM”. It is also based upon the work from COST Action CA22148-NEXT “An international network for Non-linear Extreme Ultraviolet to hard X-ray techniques”. Both actions are supported by COST (European Cooperation in Science and Technology—www.cost.eu, accessed on 15 November 2024).

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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