Stark Broadening of Al IV Spectral Lines

1. Introduction

Data on spectral line profiles, broadened by interactions with surrounding charged particles (Stark broadening) are of interest in many research fields as astrophysics (e.g. [1]), laboratory plasma ([2,3,4,5]) inertial fusion experiments ([6,7]), laser design and development ([8]), laser produced plasma research ([9,10,11]) and plasmas in technology ([12,13]). Particular interest for such data exists in astrophysics,where they are needed e.g. for stellar abundances investigations, stellar atmospheres modelling, spectral lines analysis and synthesis, radiative transfer etc.

The most interesting celestial objects for Stark brodening data applications are white dwarfs, where this is the principal presure broadening mechanism. The importance of Stark broadening has been investigated recently in DO, DB (see e.g., [14]), DA dwarfs (see e.g., [15]) and also in B subdwarfs (see e.g., [16]) and A and late B type stars (see e.g., [15]).

Astrophysical significance of aluminium is due to its high cosmic abundance, since it is the twelfth most common element in the Universe, and its spectral lines are commonly present in stellar spectra. For example, Smiljanic et al. [17] investigated aluminium abundances in giants and dwarfs and their implications on stellar and galactic chemical evolution using two samples: i) more than 600 dwarfs of the solar neighborhood and of open clusters and ii) low- and intermediate-mass clump giants in six open clusters. Carretta et al. [18] determined aluminium abundances for 90 red giant branch (RGB) stars in NGC 2808, and Smith [19] derrived atmospheric abundance of aluminium for a sample of forty normal, superficially normal, and HgMn type main-sequence late-B stars

Recently, Elabidi [21] used quantum mechanical theory [22,23] to calculate 20 Al IV lines belonging to 2p⁶–3s, 3s–3p and 3p–3d transitions. All these lines are in UV but in astronomy are particularly convenient spectral lines in the visible. In the case of Al IV such lines are from 4s–4p and 4p–4d transitions which are in Jℓ coupling. Here we calculated, using the modified semiempirical method [20], Stark widths for 23 lines and multiplets which are all in the visible part of the spectrum, so very convenient for astronomical observations and plasma diagnostics. The obtained results enable also to examine the influence of Stark broadening on Al IV spectral lines in DB and DO white dwarfs and A type stars. Moreover, the similarities of Stark widths within multiplets and supermultiplets may be checked for the cas of Jℓ coupling.

2. Theory

The modified semiempirical method (MSE) has been developed for the theoretical determination of line widths and shifts of isolated spectral lines which are emitted or absorbed by non-hydrogenic ions perturbed by their interaction with surrounding electrons. It may be particularly useful if we have not enough atomic data for the adequate application of a more precise theoretical method. In this formalism, the full width at half intensity maximum (FHWM) is given as [20]:

$$W_{MSE}=N\frac{8\pi }{3}\frac{{\hslash }^2}{m^2}{\left(\frac{2m}{\pi kT}\right)}^{1/2}\frac{\pi }{\sqrt{3}}\frac{{\lambda }^2}{2\pi c}\times$$

$$\times \{\sum _{{\ell }_i\pm 1}\sum _{L_{i^{\prime }}{J}_{i^{\prime }}}{\overrightarrow{R}}^2[n_i{\ell }_i{L}_i{J}_i,n_i({\ell }_i\pm 1)L_{i^{\prime }}{J}_{i^{\prime }}]\tilde{g}\left(x_{{\ell }_i,{\ell }_i\pm 1}\right)+$$

$$+\sum _{{\ell }_f\pm 1}\sum _{L_{f^{\prime }}{J}_{f^{\prime }}}{\overrightarrow{R}}^2[n_f{\ell }_f{L}_f{J}_f,n_f({\ell }_f\pm 1)L_{f^{\prime }}{J}_{f^{\prime }}]\tilde{g}\left(x_{{\ell }_f,{\ell }_f\pm 1}\right)+$$

$$+(\sum _{i^{\prime }}{\overrightarrow{R}}_{i{i}^{\prime }}^2{)}_{\Delta n\ne 0}g\left(x_{n_i,n_i+1}\right)+(\sum _{f^{\prime }}{\overrightarrow{R}}_{f{f}^{\prime }}^2{)}_{\Delta n\ne 0}g\left(x_{n_f,n_f+1}\right)\}.$$

(1)

Here, the index i is for the initial and index f for the final atomic energy level. The square of the matrix element $\{{\overrightarrow{R}}^2[n_k{\ell }_k{L}_k{J}_k,({\ell }_k\pm 1)L_{k^{\prime }}{J}_{k^{\prime }}],k=i,f\}$ may be expressed in the following maner:

$${\overrightarrow{R}}^2[n_k{\ell }_k{L}_k{J}_k,n_k({\ell }_k\pm 1)L_k^{\prime }{J}_k^{\prime }]=$$

$$\frac{{\ell }_{>}}{2J_k+1}Q[{\ell }_k{L}_k,({\ell }_k\pm 1)L_k^{\prime }]Q(J_k,J_k^{\prime }){\left[R_{n_k^{*}{\ell }_k}^{n_k^{*}({\ell }_k\pm 1)}\right]}^2.$$

(2)

Here, ${\ell }_{>}=\max ({\ell }_k,{\ell }_k\pm 1)$ and

$${\left(\sum _{k^{\prime }}{\overrightarrow{R}}_{k{k}^{\prime }}^2\right)}_{\Delta n\ne 0}={\left(\frac{3n_k^{*}}{2Z}\right)}^2\frac{1}{9}({n_k^{*}}^2+3{\ell }_k^2+3{\ell }_k+11).$$

(3)

In Equation (1)

$$x_{{\ell }_k,{\ell }_{k^{\prime }}}=\frac{E}{\Delta E_{{\ell }_k,{\ell }_{k^{\prime }}}},k=i,f$$

$E=\frac{3}{2}kT$ is the electron kinetic energy and $\Delta E_{{\ell }_k,{\ell }_{k^{\prime }}}=|E_{{\ell }_k}-E_{{\ell }_{k^{\prime }}}|$ is the energy difference between levels ${\ell }_k$ and ${\ell }_k\pm$1 ($k=i,f$),

$$x_{n_k,n_k+1}\approx \frac{E}{\Delta E_{n_k,n_k+1}},$$

where for $\Delta n\ne 0$, the energy difference between energy levels with $n_k$ and $n_k$+1, $\Delta E_{n_k,n_k+1}$ is represented by the following equation:

$$\Delta E_{n_k,n_k+1}=2Z^2{E}_H/n_k^{*3},$$

(4)

where $n_k^{*}={[E_H{Z}^2/(E_{ion}-E_k)]}^{1/2}$ is the effective principal quantum number, N electron density, T temperature, Z the residual ionic charge (e.g. Z=4 for Al IV), $E_{ion}$ the appropriate spectral series limit, $Q(\ell L,{\ell }^{\prime }{L}^{\prime })$, multiplet factor and $Q(J,J^{\prime })$ line factor [24]. Gaunt factors which are needed for calculation are denoted as $g\left(x\right)$ [25,26] and $\tilde{g}\left(x\right)$ [20]. Radial integrals $\left[R_{n_k^{*}{\ell }_k}^{n_k^{*}{\ell }_k\pm 1}\right]$ are calculated by using the Coulomb approximation [27] with the help of tables in [28].

In some cases, these tables are not applicable, because the values are out of them, in particular for higher principal quantum numbers, in such a case one can use the procedure described in Ref. [29].

3. Results

We calculated Stark Full widths at half intensity maximum (FWHM), by employing the modified semiempirical method [20]. The set of atomic energy levels, which need to be included as input data have been taken from [30,31]. In Table 1 are given the obtained results for electron-impact (Stark) FWHM for 23 Al IV transitions, calculated for an electron density of 10¹⁷ cm⁻³ and electron temperatures of 10,000, 20,000, 40,000, 80,000 and 160,000 K. Theoretical errors of our calculations are within the limit of 40 per cent. Concerning the dependence of results on electron density, it is linear if the influence of Debye screening is negligible. This influence should be checked only for high densities. We draw attention as well that the wavelengths given in Table 1 are not observed ones but they are calculated using atomic energy levels, so that they are not identical with observed ones. In Table 1 is also provided the value of $3kT/2\Delta E$. This is the ratio of mean energy of free electrons and the energy difference between closest perturbing level and the closer of initial and final levels ($\Delta E$), and it shows which type of collisions contribute to the line width. For values which are lower than one, dominate the elastic collisions. Namely in such a case the energy of colliding electrons is below the threshold for excitation of atomic energy levels. When it increases above this value, the contribution of inelastic collisions increases.

There is no experimental or other theoretical data for Stark broadening of Al IV transitions investigated here.

The presented here results for Al IV Stark wdths are from six multiplet, three belonging to the spectral series with parent term ²${\mathrm{P}}_{1/2}^o$: 4s ²[1/2]^o–4p ²[1/2], 4s ²[1/2]^o–4p ²[3/2] and 4p ²[1/2]–4d ²[3/2]^o, and three belonging to the spectral series with parent term ²${\mathrm{P}}_{3/2}^o$: 4s ²[3/2]^o–4p ²[1/2], 4p ²[1/2]–4d ²[1/2]^o, and 4p ²[1/2]–4d ²[3/2]^o. We have also four supermultiplets: 4s ²[K]^o–4p ²[K’] and 4p ²[K]–4d ²[K’]^o, in spectral series with parent terms ²${\mathrm{P}}_{1/2}^o$ and ²${\mathrm{P}}_{3/2}^o$ This gives us opportunity to check similarities of Stark widths within the present Al IV multiplets and supermultiplets.Namely, when such similarities are present, this may be used for example to check are the existing experimental or theoretical values consistent, and also to use existing data to estimate needed line widths which are unknown.

Wiese and Konjević [32] found, analysing experimental data from literature, that if we express Stark widths in angular frequency units, differences within the same multiplet should usually not exceede several per cent and about 30 per cent in the case of a supermultiplet. It is interesting to check this for Al IV transitions presnted here, which are in Jℓ coupling scheme. In oder to do this, we present in Table 2 Stark widths expressed in Ångströms and angular frequency units for T=80,000 K.

For the transformation of Stark widths between Å and angular frequency units one can use the following relation:

$$W\left(\mathring{A}\right)=\frac{{\lambda }^2}{2\pi c}W\left(s^{-1}\right),$$

(5)

where c is the speed of light.

If we analyse data in Table 2 we can see that in the spectral series with parent term ²${\mathrm{P}}_{1/2}^o$, in the case of multiplet 4s ²[1/2]^o–4p ²[1/2], the greatest width is 64% larger from the smallest one if expressed in Ångströms and 17% if the widths are in angular frequency units. For 4s ²[1/2]^o–4p ²[3/2] multiplet these values are 94% and 12%, and for 4p ²[1/2]–4d ²[3/2]^o 1.7% and 11%, respectively. For the spectral series with parent term ²${\mathrm{P}}_{3/2}^o$, these values are 62.5% and 9.5% in the case of 4s ²[3/2]^o–4p ²[1/2] multiplet, 65% and 0.4% for 4p ²[1/2]–4d ²[1/2]^o multiplet, and 60.5% and 0.5% for 4p ²[1/2]–4d ²[3/2]^o multiplet.

We can see that for considered Al IV transition in Jℓ coupling, these differences may be of the order of 10-20 per cent if expressed in angular frequency units which enables a rough check of consistency of existing experimental and theoretical data or an approximate check during experiment or calculation.

In the case of supermultiplet 4s ²[K]^o–4p ²[K’], belonging to the spectral series with parent term ²${\mathrm{P}}_{1/2}^o$, the maximal difference when line width is expressed in Ångströms is 210% and 89% if the widths are in angular frequency units. For the 4p ²[K]–4d ²[K’]^o supermultiplet, which belongs to the same spectral series, these values are 166% and 61%. The similar situation is and for supermultiplets in spectral series with the parent term ²${\mathrm{P}}_{3/2}^o$. For supermultiplet 4s ²[K]^o–4p ²[K’] these values are 124% and 85% respectively, while for 4p ²[K]–4d ²[K’]^o they are169% and 93%. We can see that the prediction [32] that the differences of Stark widths in angular frequency units, within the same supermultiplet usually not exceede about 30 per cent, in the case of Al IV transition in Jℓ coupling, considered here, is not satisfied.

4. On the Stark Broadening in Stellar Atmospheres

The obtained results are used here to show the influence of electron impacts on Al IV lines (Stark broadening) in stellar spectra. In order to demonstrate this influence we compared Stark and Doppler widths for different Rosseland optical depths, taking into account the transition with the largest (2s²2p⁵(²${\mathrm{P}}_{1/2}^o$)4s ²[1/2]^o–2s²2p⁵(²${\mathrm{P}}_{1/2}^o$)4p ²[3/2], $\lambda$ = 4250.2 Å) and the smallest (2s²2p⁵(²${\mathrm{P}}_{3/2}^o$)4p ²[5/2]–2s²2p⁵(²${\mathrm{P}}_{3/2}^o$)4d ²[3/2]^o, $\lambda$ = 3411.8 Å) Stark width values from Table 1. In Figure 1 are presented ratios of Stark and Doppler widths for an A type star atmospheric model with effective temperature ${\mathrm{T}}_{eff}$ = 8500 K and logarithm of surface gravity log g = 4.5 [34]. Recently, we performed a similar investigation for Zn III specrral lines in UV part of the spectrum [33].

As a difference, Al IV spectral lines considered here are all in the visible part of the spectrum. For UV Zn III lines we found that Doppler width is dominant for all considered Rosseland optical depths and that the Stark broadening could be neglected. In the case of Al IV lines in the visible, for higher values of Rosseland opacity or deeper atmospheric layers, Stark broadening may be a correction which increases the accuracy. In Figure 2 , such analysis is performed for a model of DB white dwarf atmosphere with ${\mathrm{T}}_{eff}$ = 25,000 K and log g = 4.5 [35].

One can see that for such atmospheric model Stark width is dominant for both transitions and for all investigated Rosseland optical depths, even more than in the case of previously investigated Zn III UV multiplets. For DO white dwarf atmosphere model with ${\mathrm{T}}_{eff}$ = 60,000 K and log g = 4.5 [35] (see Figure 3), we can see the difference in the influence of Stark broadening on UV and optical spectral lines. Namely, in the case of Al IV Stark broadening dominates for both transitions and for all Rosseland optical depths of interest, while for Zn III UV lines, Stark broadening is dominant for higher values of Rosseland optical depth, while for lower ones Doppler broadening is higher, but in most cases comparable to the Stark.

This is a demonstration of the influence of wavelength values. Namely in Stark broadening theories the wavelength enters as a square and the expression for Doppler width is linear with wavelength. So, with the increase of wavelength, Stark widths increase faster than Doppler ones. So, in visible part of the spectrum the influence of Stark broadening will be higher in comparison with Doppler broadening, than in UV part.

5. Conclusions

We calculated Stark widths for 23 Al IV transitions in Jℓ coupling, with the help of the modified semiempirical method [20]. For the considered transitions, which are all in the visible part of the spectrum, there are no other experimental or theoretical data. With the help of the obtained data we investigated the influence of Stark broadening on Al IV spectral lines in the visible part of the spectrum, in atmospheres of A-type stars and DB and DO white dwarfs. In comparison with lines in UV, the influence of Stark broadening in the visible part of the spectrum is higher due to the influence of higher wavelengths. We also used the obtained results to check regularities and similarities among Al IV Stark widths within multiplets and supermultiplets and found that in the case of Jℓ coupling (investigated here), differences of Stark widths within multiplets of Al IV transitions considered here are 10-20 per cent, while within investigated supermultiplets there is no similarities which would be useful for the check of consistency of experimental or theoretical results.

The Al IV Stark widths presented in this article will also enter in STARK-B database (http://stark-b.obspm.fr/ (accessed on 27 January 2023), [36,37], which enters in Virtual Atomic and Molecular Data Center VAMDC (http://www.vamdc.org/ (accessed on 27 January 2023), [38,39]).

The presented spectral line widths of Al IV, broadened by collisions with surrouning electrons (Stark broadening) may be used for many topics in astrophysics like for modelling of stellar atmospheres, abundance determination of zinc, analysis and synthesis of Al IV lines in stellar spectra, opacity calculations, investigation, modelling and diagnostics of laboratory plasmas as well as for various technological applications in particular for optimisation of cutting, welding, melting and piercing of aluminium by lasers. They also may be used for the diagnostic and modelling of an electrodynamic macro-particle accelerator arc plasma created by the evaporation of an Al-foil [40,41] and for diagnostics of plasmas generated during incipient laser ablation of aluminum in air [42].