Stark Broadening of Zn III Spectral Lines

Abstract

Stark widths for spectral lines within 24 multiplets of Zn III singlets and triplets have been calculated using modified semiempirical method for an electron density of 10¹⁷ cm⁻³ and temperatures from 5000 K up to 100,000 K. The obtained results have been used for the investigation of the influence of Stark broadening on Zn III spectral lines in stellar atmospheres and for the considerations of Stark width similarites within supermultiplets and transition arrays.

1. Introduction

Stark broadening parameters for numerous spectral lines of diferrent atoms and ions in various ionization stages may be very useful for consideration of various problems in astrophysics (for example [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].

Such data are particularly useful in astrophysics, for a number of problems like abundance determinations, stellar spectra analysis and synthesis, opacity and radiative transfer calculations, and other topics. Stark broadenig is usually the principal presure broadening mechanism in the case of white dwarfs, so that for the investigation of such celestial objects such data are particularly needed. The influence of Stark broadening has been considered in DO (D stands for Degenerate—The atmosphere of these stars is abundant with helium, which is indicated by the presence of ionized helium (He II) in their spectral lines), DB (they have atmosphere of helium, which is indicated by the presence of neutral helium (He I) in their spectral lines) (see e.g., [14]), DA (with atmospheres of hydrogen) dwarfs (see e.g., [15]) and in B subdwarfs (see e.g., [16]). It should be noted that such data may be also of interest in the case of A and late B type stars (see e.g., [15]).

Various atomic data for zinc, including Stark broadening parameters are of interest for abundance determination. Abundances for various galactic stars are needed, as it is underlined in [17], for the investigation of nucleosynthesis of zinc, which is not well understood [18]. Zinc is also significant for examination of the star formation rate, chemical enrichment of the Galactic bulge (see e.g., [18,19] and referenes therein) and the chemical evolution of the Universe at high redshifts, by studing abundances in damped Lyman-$\alpha$ systems (DLAs) [19,20,21,22].

The Zn III lines have been observed in spectra of various stars. As an example we can cite [23], where a Zn III line has been observed in ${\theta }^1$ Ori C, an O type star from multiple stellar system Theta Orionis. In ref. [24] zinc abundance in hot subdwarf stars of O type (sdO) HZ 44 and HD 127493 has been determined using spectral lines of Zn III an Zn IV and in [25], zinc abundances for intermediate He-sdOB star: the pulsators Feige 46 and LS IV–14${}^{\circ }$116 are based on 13 and 16 strong Zn III spectral lines, respectively.

There is no experimentally obtained data for Zn III spectral lines. Only one work with Stark widths for six Zn III multiplets [26] obtained by using the modified semiempirical method (MSE) [27]. In this paper, Stark full widths at half intensity maximum (FWHM) due to impacts with electrons for 24 Zn III multiplets will be presented. Calculations have been performed by using modified semiempirical method [27] and the obtained results are used for the investigation of the influence of Stark broadening on Zn III spectral lines in DB and DO white dwarfs as well as in A type stars. Additionally, similarities of Stark widths within supermultiplets and transition arrays wil be checked.

2. Theory

Modified semiempirical method (MSE) is particularly adequate for the calculation of Stark broadening parameters for isolated spectral lines emitted by non-hydrogenic ions due to their impacts with electrons, in the cases when there is not a set of atomic data needed for more sophisticated methods. The corresponding full width at half intensity maximum (FHWM) is [27]:

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

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

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

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

(1)

Here, i and f denote the initial and final atomic energy levels, respectively. The square of the matrix element $\{{\overrightarrow{R}}^2[n_k{\ell }_k{L}_k{J}_k,({\ell }_k\pm 1)L_{k^{\prime }}{J}_{k^{\prime }}],k=i,f\}$ is given by the expression:

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

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

(2)

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

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

(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 approximated as

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

(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 = 1 for neutrals and 3 for Zn III), $E_{ion}$ the appropriate spectral series limit, $Q(\ell L,{\ell }^{\prime }{L}^{\prime })$, multiplet factor and $Q(J,J^{\prime })$ line factor [28]. Needed Gaunt factors are $g(x)$ [29,30] and $\tilde{g}(x)$ [27]. Radial integrals $[R_{n_k^{*}{\ell }_k}^{n_k^{*}{\ell }_k\pm 1}]$ are obtained within the Coulomb approximation [31] and the tables from Ref. [32]. In the cases when the tables from ref. [32] are not applicable, for the higher principal quantum numbers, the method given in [33] can be used.

3. Results

In order to calculate Stark Full widths at half intensity maximum (FWHM), we applied the modified semiempirical method [27]. Needed atomic energy levels have been taken from [34,35]. The results for Stark FWHM due to collisions with electrons for 24 Zn III multiplets, for an electron density of 10${}^{17}$ cm${}^{-3}$ and plasma temperatures of 5000, 10,000, 20,000, 50,000 and 100,000 K are shown in

Table 1. This table gives electron-impact broadening (Stark broadening) Full Widths at Half Intensity Maximum (W) for Zn III multiplets, for an electron density of 10${}^{17}$ cm${}^{-3}$ and temperatures from 5000 to 100,000 K. Also, the $3kT/2\Delta E$, quantity is given, where $\Delta E$ is the energy difference between closest perturbing level and the closer of initial and final levels. This quantity shows the type of collisions contributing to line broadening. If this quantity is lower than one, collisions are elastic. Starting with the value of one, with its increase increases the influence of inelastic collisions.

Transition	T [K]	W [Å]	3kT/2$\mathsf{\Delta }$E
Zn III 4s${}^1$D–4p${}^1$Fo	5000	0.204 $\times {10}^{-1}$	0.855 $\times {10}^{-1}$
	10,000	0.144 $\times {10}^{-1}$	0.171
$\lambda$ = 1639.3 Å	20,000	0.102 $\times {10}^{-1}$	0.342
	50,000	0.644 $\times {10}^{-2}$	0.855
	100,000	0.456 $\times {10}^{-2}$	1.71
Zn III 4s${}^1$D–4p${}^1$Do	5000	0.200 $\times {10}^{-1}$	0.855 $\times {10}^{-1}$
	10,000	0.141 $\times {10}^{-1}$	0.171
$\lambda$ = 1619.6 Å	20,000	0.998 $\times {10}^{-2}$	0.342
	50,000	0.631 $\times {10}^{-2}$	0.855
	100,000	0.446 $\times {10}^{-2}$	1.71
Zn III 4s${}^1$D–4p${}^1$Po	5000	0.191 $\times {10}^{-1}$	0.855 $\times {10}^{-1}$
	10,000	0.135 $\times {10}^{-1}$	0.171
$\lambda$ = 1562.5 Å	20,000	0.954 $\times {10}^{-2}$	0.342
	50,000	0.603 $\times {10}^{-2}$	0.855
	100,000	0.426 $\times {10}^{-2}$	1.71
Zn III 4p${}^1$Fo–4d${}^1$G	5000	0.216 $\times {10}^{-1}$	0.136
	10,000	0.153 $\times {10}^{-1}$	0.271
$\lambda$ = 1335.8 Å	20,000	0.108 $\times {10}^{-1}$	0.543
	50,000	0.682 $\times {10}^{-2}$	1.36
	100,000	0.485 $\times {10}^{-2}$	2.71
Zn III 4p${}^1$Fo–4d${}^1$D	5000	0.229 $\times {10}^{-1}$	0.139
	10,000	0.162 $\times {10}^{-1}$	0.277
$\lambda$ = 1312.9 Å	20,000	0.114 $\times {10}^{-1}$	0.554
	50,000	0.723 $\times {10}^{-2}$	1.39
	100,000	0.515 $\times {10}^{-2}$	2.77
Zn III 4p${}^1$Fo–4d${}^1$F	5000	0.221 $\times {10}^{-1}$	0.141
	10,000	0.157 $\times {10}^{-1}$	0.281
$\lambda$ = 1304.8 Å	20,000	0.111 $\times {10}^{-1}$	0.562
	50,000	0.700 $\times {10}^{-2}$	1.41
	100,000	0.499 $\times {10}^{-2}$	2.81
Zn III 4p${}^1$Do–4d${}^1$P	5000	0.214 $\times {10}^{-1}$	0.125
	10,000	0.151 $\times {10}^{-1}$	0.250
$\lambda$ = 1357.5 Å	20,000	0.107 $\times {10}^{-1}$	0.500
	50,000	0.677 $\times {10}^{-2}$	1.25
	100,000	0.480 $\times {10}^{-2}$	2.50
Zn III 4p${}^1$Do–4d${}^1$D	5000	0.233 $\times {10}^{-1}$	0.139
	10,000	0.165 $\times {10}^{-1}$	0.277
$\lambda$ = 1325.8 Å	20,000	0.116 $\times {10}^{-1}$	0.554
	50,000	0.736 $\times {10}^{-2}$	1.39
	100,000	0.525 $\times {10}^{-2}$	2.77
Zn III 4p${}^1$Do–4d${}^1$F	5000	0.226 $\times {10}^{-1}$	0.141
	10,000	0.160 $\times {10}^{-1}$	0.281
$\lambda$ = 1317.5 Å	20,000	0.113 $\times {10}^{-1}$	0.562
	50,000	0.713 $\times {10}^{-2}$	1.41
	100,000	0.508 $\times {10}^{-2}$	2.81
Zn III 4p${}^1$Po–4d${}^1$P	5000	0.228 $\times {10}^{-1}$	0.125
	10,000	0.161 $\times {10}^{-1}$	0.250
$\lambda$ = 1400.3 Å	20,000	0.114 $\times {10}^{-1}$	0.500
	50,000	0.722 $\times {10}^{-2}$	1.25
	100,000	0.512 $\times {10}^{-2}$	2.50
Zn III 4p${}^1$Po–4d${}^1$D	5000	0.248 $\times {10}^{-1}$	0.139
	10,000	0.175 $\times {10}^{-1}$	0.277
$\lambda$ = 1366.7 Å	20,000	0.124 $\times {10}^{-1}$	0.554
	50,000	0.784 $\times {10}^{-2}$	1.39
	100,000	0.559 $\times {10}^{-2}$	2.77
Zn III 4p${}^1$Po–4d${}^1$S	5000	0.264 $\times {10}^{-1}$	0.174
	10,000	0.187 $\times {10}^{-1}$	0.348
$\lambda$ = 1203.3 Å	20,000	0.132 $\times {10}^{-1}$	0.696
	50,000	0.835 $\times {10}^{-2}$	1.74
	100,000	0.606 $\times {10}^{-2}$	3.48
Zn III 4s${}^3$D– 4p${}^3$Po	5000	0.195 $\times {10}^{-1}$	0.870 $\times {10}^{-1}$
	10,000	0.138 $\times {10}^{-1}$	0.174
$\lambda$ = 1668.0 Å	20,000	0.974 $\times {10}^{-2}$	0.348
	50,000	0.616 $\times {10}^{-2}$	0.870
	100,000	0.436 $\times {10}^{-2}$	1.74
Zn III 4s${}^3$D–4p${}^3$Fo	5000	0.186 $\times {10}^{-1}$	0.870 $\times {10}^{-1}$
	10,000	0.132 $\times {10}^{-1}$	0.174
$\lambda$ = 1604.1 Å	20,000	0.930 $\times {10}^{-2}$	0.348
	50,000	0.588 $\times {10}^{-2}$	0.870
	100,000	0.416 $\times {10}^{-2}$	1.74
Zn III 4s${}^3$D–4p${}^3$Do	5000	0.170 $\times {10}^{-1}$	0.870 $\times {10}^{-1}$
	10,000	0.120 $\times {10}^{-1}$	0.174
$\lambda$ = 1472.8 Å	20,000	0.848 $\times {10}^{-2}$	0.348
	50,000	0.536 $\times {10}^{-2}$	0.870
	100,000	0.379 $\times {10}^{-2}$	1.74
Zn III 4p${}^3$Po–4d${}^3$S	5000	0.168 $\times {10}^{-1}$	0.116
	10,000	0.119 $\times {10}^{-1}$	0.233
$\lambda$ = 1326.9 Å	20,000	0.841 $\times {10}^{-2}$	0.465
	50,000	0.532 $\times {10}^{-2}$	1.16
	100,000	0.376 $\times {10}^{-2}$	2.33
Zn III 4p${}^3$Po–4d${}^3$P	5000	0.177 $\times {10}^{-1}$	0.124
	10,000	0.125 $\times {10}^{-1}$	0.248
$\lambda$ = 1279.2 Å	20,000	0.883 $\times {10}^{-2}$	0.496
	50,000	0.558 $\times {10}^{-2}$	1.24
	100,000	0.396 $\times {10}^{-2}$	2.48
Zn III 4p${}^3$Po–4d${}^3$D	5000	0.177 $\times {10}^{-1}$	0.128
	10,000	0.125 $\times {10}^{-1}$	0.256
$\lambda$ = 1259.0 Å	20,000	0.885 $\times {10}^{-2}$	0.512
	50,000	0.559 $\times {10}^{-2}$	1.28
	100,000	0.397 $\times {10}^{-2}$	2.56
Zn III 4p${}^3$Fo–4d${}^3$G	5000	0.189 $\times {10}^{-1}$	0.125
	10,000	0.134 $\times {10}^{-1}$	0.249
$\lambda$ = 1316.1 Å	20,000	0.945 $\times {10}^{-2}$	0.499
	50,000	0.598 $\times {10}^{-2}$	1.25
	100,000	0.424 $\times {10}^{-2}$	2.49
Zn III 4p${}^3$Fo–4d${}^3$D	5000	0.193 $\times {10}^{-1}$	0.128
	10,000	0.136 $\times {10}^{-1}$	0.256
$\lambda$ = 1298.0 Å	20,000	0.964 $\times {10}^{-2}$	0.512
	50,000	0.609 $\times {10}^{-2}$	1.28
	100,000	0.432 $\times {10}^{-2}$	2.56
Zn III 4p${}^3$Fo–4d${}^3$F	5000	0.194 $\times {10}^{-1}$	0.129
	10,000	0.137 $\times {10}^{-1}$	0.258
$\lambda$ = 1291.2 Å	20,000	0.971 $\times {10}^{-2}$	0.517
	50,000	0.614 $\times {10}^{-2}$	1.29
	100,000	0.436 $\times {10}^{-2}$	2.58
Zn III 4p${}^3$Do–4d${}^3$P	5000	0.221 $\times {10}^{-1}$	0.124
	10,000	0.157 $\times {10}^{-1}$	0.248
$\lambda$ = 1424.0 Å	20,000	0.111 $\times {10}^{-1}$	0.496
	50,000	0.700 $\times {10}^{-2}$	1.24
	100,000	0.496 $\times {10}^{-2}$	2.48
Zn III 4p${}^3$Do–4d${}^3$D	5000	0.226 $\times {10}^{-1}$	0.128
	10,000	0.160 $\times {10}^{-1}$	0.256
$\lambda$ = 1399.0 Å	20,000	0.113 $\times {10}^{-1}$	0.512
	50,000	0.714 $\times {10}^{-2}$	1.28
	100,000	0.507 $\times {10}^{-2}$	2.56
Zn III 4p${}^3$Do–4d${}^3$F	5000	0.227 $\times {10}^{-1}$	0.129
	10,000	0.161 $\times {10}^{-1}$	0.258
$\lambda$ = 1391.0 Å	20,000	0.114 $\times {10}^{-1}$	0.517
	50,000	0.719 $\times {10}^{-2}$	1.29
	100,000	0.510 $\times {10}^{-2}$	2.58

. We estimate that the errors are within the limit of 40 per cent. We note that the dependence of Stark width on electron density remains linear until, for higher densities, the influence of Debye screening may be neglected. We note as well that the wavelengths given in Table 1 are calculated from the used energy levels so that they may be different from observed. The quantity $3kT/2\Delta E$, 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 ($\Delta E$), which indicates the type of collisions contributing to line broadening. If it is lower than one, the elastic collisions are dominant, since the energy of free electrons is lower than the threshold for excitaion of energy levels. With its increase above the value of one, increases and the influence of inelastic collisions.

There is no experimental data for Stark broadening of Zn III spectral lines. Stark widths for six Zn III s-p multiplets have been calculated in [26]. In Zn III there is a lot of energy level mixing treated in a non adequate way in [26]. Namely the contribution of a term is multiplied by its faction in the mixed energy level so that the obtained results are too small. So, we recalculated them assuming that the other fractions are mixed in the neighbouring levels so that the introduced error is not big.

Behaviour of Stark width with temperature in the case of multiplet with the largest (4p${}^1$P^o–4d${}^1$S, $\lambda$ = 1203.3 Å) and smallest (4p${}^3$P^o–4d${}^3$S, $\lambda$ = 1326.9 Å) width in Table 1, is shown in Figure 1. As usual, the decrease of Stark width values is faster for lower temperatures, where elastic collisions are dominant.

The results for Zn III Stark wdths are for multiplets belonging to two s-p supermultiplets: 4s${}^1$D–4p${}^1$L^o, 4s${}^3$D-4p${}^3$L^o (L = P, D, F), three singlet p-d supermultiplets 4p${}^1$L^o–4d${}^1$L’ (L = P, D, F; L’ = S, P, D, F, G), three triplet p-d, 4p${}^3$L^o–4d${}^3$L’ and two transition arrays: 4s–4p and 4p–4d. It is interesting to check how similar are Stark widths within the considered Zn III supermultiplets and transition arrays. Such similarities, if exist, may be used to check consistency of experimental or theoretical values and to estimate unknown line widths with the help of other values existing in literature.

In ref. [36] is concluded, on the basis of examination of existing experimental data, that the differences of Stark widths in angular frequency units, belonging to the same supermultiplet are usually within the limits of about 30 per cent and 40 per cent in the case of transition arrays. To check this for Zn III, in

Table 2. This table gives electron-impact broadening (Stark broadening) Full Widths at Half Intensity Maximum (W) in [Å] and in [${10}^{12}$ s${}^{-1}$] units for Zn III spectral lines, for an electron density of 10${}^{17}$ cm${}^{-3}$ and T = 10,000 K.

Transition	$\mathit{\lambda }$ [Å]	W [Å]	W [10${}^{12}$ s${}^{-1}$]
Zn III 4s${}^1$D–4p${}^1$Fo	1639.3	0.144 $\times {10}^{-1}$	0.101
Zn III 4s${}^1$D–4p${}^1$Do	1619.6	0.141 $\times {10}^{-1}$	0.101
Zn III 4s${}^1$D–4p${}^1$Po	1562.5	0.135 $\times {10}^{-1}$	0.104
Zn III 4p${}^1$Fo–4d${}^1$G	1335.8	0.153 $\times {10}^{-1}$	0.162
Zn III 4p${}^1$Fo–4d${}^1$D	1312.9	0.162 $\times {10}^{-1}$	0.177
Zn III 4p${}^1$Fo–4d${}^1$F	1304.8	0.157 $\times {10}^{-1}$	0.174
Zn III 4p${}^1$Do–4d${}^1$P	1357.5	0.151 $\times {10}^{-1}$	0.154
Zn III 4p${}^1$Do–4d${}^1$D	1325.8	0.165 $\times {10}^{-1}$	0.177
Zn III 4p${}^1$Do–4d${}^1$F	1317.5	0.160 $\times {10}^{-1}$	0.174
Zn III 4p${}^1$Po–4d${}^1$P	1400.3	0.161 $\times {10}^{-1}$	0.155
Zn III 4p${}^1$Po–4d${}^1$D	1366.7	0.175 $\times {10}^{-1}$	0.176
Zn III 4p${}^1$Po–4d${}^1$S	1203.3	0.187 $\times {10}^{-1}$	0.243
Zn III 4s${}^3$D– 4p${}^3$Po	1668.0	0.138 $\times {10}^{-1}$	0.093
Zn III 4s${}^3$D–4p${}^3$Fo	1604.1	0.132 $\times {10}^{-1}$	0.097
Zn III 4s${}^3$D–4p${}^3$Do	1472.8	0.120 $\times {10}^{-1}$	0.104
Zn III 4p${}^3$Po–4d${}^3$S	1326.9	0.119 $\times {10}^{-1}$	0.127
Zn III 4p${}^3$Po–4d${}^3$P	1279.2	0.125 $\times {10}^{-1}$	0.144
Zn III 4p${}^3$Po–4d${}^3$D	1259.0	0.125 $\times {10}^{-1}$	0.148
Zn III 4p${}^3$Fo–4d${}^3$G	1316.1	0.134 $\times {10}^{-1}$	0.146
Zn III 4p${}^3$Fo–4d${}^3$D	1298.0	0.136 $\times {10}^{-1}$	0.152
Zn III 4p${}^3$Fo–4d${}^3$F	1291.2	0.137 $\times {10}^{-1}$	0.155
Zn III 4p${}^3$Do–4d${}^3$P	1424.0	0.157 $\times {10}^{-1}$	0.146
Zn III 4p${}^3$Do–4d${}^3$D	1399.0	0.160 $\times {10}^{-1}$	0.154
Zn III 4p${}^3$Do–4d${}^3$F	1391.0	0.161 $\times {10}^{-1}$	0.157

are shown Stark widths in Ångströms and angular frequency units for T = 10,000 K.

To transform Stark widths in Å units into angular frequency units one can use the expression:

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

(5)

where c is the speed of light.

If we analyse data in Table 2 we can conclude that for the supermultiplet 4s${}^1$D–4p${}^1$L^o (L = P, D, F), the greatest width is 6.7% larger from the smallest one if expressed in Ångströms and 3% in angular frequency units. In the case of 4s${}^3$D-4p${}^3$L^o supermultiplet these values are 15% and 11.8% respectively, for 4p${}^1$F^o–4d${}^1$(D,F,G) 5.9% and 9.25%, for 4p${}^1$D^o–4d${}^1$(P,D,F) 9.3% and 14.9%, and for 4p${}^1$P^o–4d${}^1$(S,P,D) 16.1and 56.8%. In the case of triplet p-d transitions we have for 4p${}^3$P^o–4d${}^3$(S,P,D) 5% and 16.5%, for 4p${}^3$F^o–4d${}^3$(D,F,G) 2.2% and 6.2%, and for 4p${}^3$D^o–4d${}^3$(P,D,F) 2.5% and 7.5%, For the case of the transition array 4s–4p these differences are 20% and 11.8% respectively and for 4p–4d, 57.1% and 91.3%.

We can see that for s-p transitions all is in accordance with the conclusions in [36]. The differences are well within the limits predicted in [36] and the differences are smaller for Stark widths in angular frequency units since in such a case, there is no influence of wavelengths which are similar. This is the situation for regularly ordered atomic energy levels. However, for p-d transitions the situation is quite opposite to the usual one, since atomic energy levels are not ordered in a regular way. The distance between upper terms entering in supermultiplets is big, and in the case of 4p${}^1$P^o–4d${}^1$(S,P,D) supermultiplet very big. In this case the energy difference between S, P and D components is 11,696.6 cm${}^{-1}$ while the closest distance to the corresponding f levels is 28,271.7 cm${}^{-1}$, so that for components closer to f levels the corresponding Stark width is considerably higher. But, on the other hand, when d level is closer to f levels, it is further from p levels and the wavelength of the corresponding transition is smaller, compensating partially the increase in the width, so that the widths in Å units are smaller than in angular frequency units in the case of big distances between upper terms in a supermultiplet, contrary to the usual situation with regular positions of atomic energy levels, described in [36]. For example in the case of 4p${}^1$P^o–4d${}^1$(S,P,D) supermultiplet, P component has the wavelenth 1400.3 Å and Stark width in angular frequency units 0.155 Å and S component 1203.3 Å and 0.243 Å respectively.

4. On the Stark Broadening in Stellar Atmospheres

In order to demonstrate the influence of Stark broadening on Zn III spectral lines in stellar spectra we investigated the ratio of Stark and Doppler widths as a function of Rosseland optical depth, in the case of multiplet with the largest (4p${}^1$P^o–4d${}^1$S, $\lambda$ = 1203.3 Å) and smallest (4p${}^3$P^o–4d${}^3$S, $\lambda$ = 1326.9 Å) Stark width in Table 1. In Figure 2 are presented their Stark and Doppler widths for a model of atmosphere of an A type star with effective temperature T${}_{eff}$ = 8500 K and logarithm of surface gravity log g = 4.5. We can see that for all considered Rosseland optical depths Doppler width dominates. The same is presented in Figure 3, for a model of DB white dwarf atmosphere with T${}_{eff}$ = 25,000 K and log g = 4.5 [37]. One can see that in this case, for both multiplets, Stark width dominates for all investigated Rosseland optical depths. For DO white dwarf atmosphere model with T${}_{eff}$ = 60,000 K and log g = 4.5 [37] (see Figure 4), Stark broadening dominates for higher values of Rosseland optical depths, while for lower ones Doppler broadening is higher, but for majority values comparable to the Stark. So we can see that Stark broadening is very important for white dwarfs. For DO dwarfs it is dominant in the case of Zn III lines for the bigger analyzed values of Rosseland optical depth, but for lower ones it can influence in the line wings. In the case of considered spectral lines of Zn III it could be neglected for A-type stellar atmospheres.

5. Conclusions

We calculated Stark widths for 24 Zn III multiplets using the modified semiempirical method [27]. The obtained data have been used to consider the influence of Stark broadening on spectral lines in the Zn III multiplets in atmospheres of A-type stars and DB and DO white dwarfs. Additionally, the results have been used to check similarities between Zn III Stark broadening parameters within supermultiplets and transition arrays.

The data for Stark widths of Zn III multiplets, obtained in this study, will be implemented in STARK-B database (http://stark-b.obspm.fr/ (accessed on 15 July 2022)) [39,40], which enters in Virtual Atomic and Molecular Data Center VAMDC (http://www.vamdc.org/ (accessed on 15 July 2022), [41,42]). The obtained Stark broadening parameters are of interest for modelling of stellar atmospheres, abundance determination of zinc, analysis and synthesis of Zn III lines in stellar spectra, opacity calculations, investigation, modelling and diagnostics of laboratory plasmas and for various technological applications in particular for optimisation of cutting, welding, melting and piercing of zinc by lasers.