MNRAS 432, 2247–2251 (2013)
doi:10.1093/mnras/stt634
Advance Access publication 2013 May 9
Stark broadening of resonant Cr II 3d5 –3d4 4p spectral lines
in hot stellar atmospheres
Z. Simić,1,2‹ M. S. Dimitrijević1,2,3 and S. Sahal-Bréchot3
1 Astronomical
Observatory, Volgina 7, 11060 Belgrade, Serbia
Newton Institute of Chile, Yugoslavia Branch, 11060 Belgrade, Serbia
3 Observatoire de Paris-Meudon, F-92195 Meudon, France
2 Isaac
Accepted 2013 April 3. Received 2013 March 18; in original form 2013 February 13
ABSTRACT
New Stark broadening parameters of interest for the astrophysical, laboratory and technological plasma modelling, investigations and analysis for nine resonant Cr II multiplets have
been determined within the semiclassical perturbation approach. In order to demonstrate one
possibility for their usage in astrophysical plasma research, obtained results have been applied
to the analysis of the Stark broadening influence on stellar spectral line shapes.
Key words: atomic data – line: profiles – stars: atmospheres – white dwarfs.
1 I N T RO D U C T I O N
Chromium lines are interesting for astrophysics due to their presence in stellar atmospheres, where they are identified in a large
number, so that data on their profiles are obviously of interest for example to determine chromium abundance and investigate chromium
stratification in stellar atmospheres (Dimitrijević et al. 2005, 2007)
as well as for the diagnostics and modelling of stellar plasma and for
analysis and synthesis of stellar spectra. They have been identified
in A-type star spectra, such as e.g. in o Peg (Adelman 1991), 7 Sex
(Adelman & Philip 1996), φ Aqu (Caliskan & Adelman 1997) and
Przybylski’s star (Cowley et al. 2000), which are also chemically
peculiar stars. Namely the majority of chemically peculiar stars
are of A spectral type. As an example, in the spectrum of φ Aqu,
Caliskan & Adelman (1997) identified 28 Cr II spectral lines and
noted an overabundance of log[Cr/H] = −5.82 ± 0.27, in comparison with the Solar value of −6.26. Also for Przybylski’s star,
Cowley et al. (2000) identified 15 Cr II lines and noted an overabundance with the value log [Cr/H] = −5.92 ± 0.26. Cr II spectral lines
are before Fe II and Ti II in number and intensity in the Ae/Be Herbig
star V380 Ori, where 25 Cr II lines were found (Shevchenko 1994).
Since in stellar atmospheres, layers, where the Stark broadening
contribution to the line profiles is important, exist (Lanz et al. 1988;
Popović, Dimitrijević & Tankosić 1999a; Popović, Dimitrijević &
Ryabchikova 1999b; Popović, Milovanović & Dimitrijević 2001;
Tankosić, Popović & Dimitrijević 2003; Dimitrijević et al. 2003a,
2004, 2007; Dimitrijević, Jovanović & Simić 2003b; Simić et al.
2005, 2006; Hamdi et al. 2008; Simić, Dimitrijević & Kovačević
2009b), Stark broadening parameters for singly ionized chromium
spectral lines are obviously of astrophysical interest.
E-mail: [email protected]
Dimitrijević et al. (2007) calculated Stark broadening parameters
for Cr II spectral lines of seven multiplets belonging to 4s–4p transitions and applied the obtained results to the analysis of Cr II line
profiles observed in the spectrum of the Cr-rich star HD 133792.
There is only one experimental result on the Stark broadening of
Cr II spectral lines, obtained by Rathore et al (1984). The Stark
widths and shifts of Cr II 3120.36, 3124.94 and 3132.05 Å of the
multiplet 5 (4s 4 D−4p 4 F o ) have been measured in a T-tube plasma.
These results have been compared with values predicted from established systematic trends and regularities. Using regularities and
systematic trends, Lakićević (1983) also made an attempt to estimate Stark broadening parameters of the Cr II 2065.65 Å line. In
order to enlarge and complete Stark broadening data for astrophysically interesting Cr II lines, we have determined Stark broadening
parameters for nine resonant Cr II 3d5 –3d4 4p multiplets in the visible and UV wavelength range (2060–6073 Å). There are no other
available theoretical or experimental data for considered multiplets.
The obtained results have also been used to analyse the influence of
Stark broadening on Cr II spectral lines in a hot A-type star and DB
white dwarf atmospheres.
2 THEORETICAL REMARKS
For the determination of Stark broadening parameters the semiclassical perturbation (SCP) formalism (Sahal-Bréchot 1969a,b) has
been used. For updates see Fleurier, Sahal-Bréchot & Chapelle
(1977), Sahal-Bréchot (1974, 1991), Dimitrijević, Sahal-Bréchot
& Bomier (1991), Dimitrijević & Sahal-Bréchot (1996) and the review article of Dimitrijević (1996). The full width at half-maximum
intensity (FWHM) (W = 2w) and shift (d) of an isolated spectral
line broadened by electron impacts can be expressed (in angular
frequency units), for an ionized emitter, in terms of cross-sections
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
2248
Z. Simić, M. S. Dimitrijević and S. Sahal-Bréchot
for elastic and inelastic processes as
⎞
⎛
σii (v) +
σff (v) + σel ⎠ ,
W = N vf (v) dv ⎝
d =N
i =i
f =f
RD
2πρ dρ sin(2ϕp ).
vf (v) dv
(1)
R3
In the above equations, N is the electron density, f (υ) the
Maxwellian velocity distribution function for electrons, ρ the impact parameter of the incoming electron, i (respectively, f ) denotes the perturbing levels of the initial state i (respectively, final state f), σii (υ) [respectively, σff (υ)] is the inelastic crosssection, and here it is expressed by an integral over the impact
parameter ρ of the transition probability Pii (ρ, υ) [respectively,
Pff (ρ, υ) ] as
RD
1
σii (υ) = πR12 +
2πρ dρ
Pii (ρ, υ).
(2)
2
R1
i =i
i =i
The elastic cross-section is
RD
σel = 2πR22 +
2πρ dρ sin2 δ + σr ,
R2
δ=
(ϕp2
+
1
ϕq2 ) 2 .
(3)
The phase shifts ϕp and ϕq due, respectively, to the polarization
potential (r−4 ) and to the quadrupolar potential (r−3 ) are given in
section 3 of chapter 2 in Sahal-Bréchot (1969a) and RD is the Debye
radius. The cut-offs R1 , R2 and R3 are described in section 1 of chapter 3 in Sahal-Bréchot (1969b). We denote by σr the contribution of
the Feshbach resonances (Fleurier et al. 1977).
For the determination of the ion-impact widths and shifts, the
corresponding equations are analogous to equations (1)–(3), but
without the Feshbach resonance contribution to the width. Also, in
comparison with electrons different are and hyperbolic paths, since
for electrons the Coulomb force is attractive and for perturbing ions
repulsive.
3 R E S U LT S A N D D I S C U S S I O N
The atomic energy levels which are needed for the determination of
Stark broadening parameters are from Wiese & Musgrove (1989).
The needed oscillator strengths have been determined by using
the method of Bates & Damgaard (1949) and the tables of Oertel & Shomo (1968), while for higher levels, the calculations have
been performed as in van Regemorter, Hoang Binh & Prud’homme
(1979). Within the method of Bates and Damgaard, only oscillator strengths for transitions allowed in the LS coupling are different
from zero. For simpler spectra it is usually enough to take all perturbing levels with n = 0, ±1and ± 2, but for a particular transition
we add the perturbing levels until the corresponding sum rules are
satisfied.
Concerning the use of a set of Bates and Damgaard’s oscillator
strengths, which is complete according to the corresponding sum
rules, Ben Nessib, Dimitrijević & Sahal-Bréchot (2004) and Hamdi
et al. (2007), who calculated line widths and shifts of Si V and Ne V
ions, compared SCP ab initio Stark widths obtained with Bates and
Damgaard oscillator strengths and with SUPERSTRUCTURE (Thomas–
Fermi–Dirac interaction potential model with relativistic corrections; Eissner, Jones & Nussbaumer 1974) oscillator strengths.
They obtained that the difference between the two sets of calculations did not exceed 30 per cent. Since the accuracy of the SCP
method is about 20 per cent, such a difference, due to different sets
of oscillator strengths used in Stark broadening calculation, is not
crucial.
Electron-, proton- and helium ion-impact broadening parameters – W (FWHM) and d (shift for five resonant Cr II 3d5 –3d4 4p
multiplets) – are shown in Table 1, together with the results of
present calculations of helium ion-impact broadening parameters
for additional four resonant lines. For these lines in Table 1 are
included electron- and proton-impact broadening parameters, previously communicated on a conference and published in the corresponding proceedings (Simić et al. 2009a). This table shows Cr II
Stark broadening parameters, for a perturber density of 1017 cm−3
and temperatures from 5000 to 100 000 K. The quantity C (given
in Å cm−3 ), when divided by the corresponding full width at halfmaximum, gives an estimate for the maximum perturber density for
which tabulated data may be used. The obtained data could not be
compared with experimental results, but in Dimitrijević et al. (2007)
the Stark broadening parameters for other Cr II lines, obtained with
the same method, have been compared with the experimental results of Rathore et al (1984) and an acceptable agreement has been
obtained. Also, they are successfully used in the same article for
the synthesis of stellar Cr II lines and their comparison with the observed ones, which indirectly indicates that the present results are
also correct.
The largest Stark widths are for resonant Cr II 6073.4, 5279.6 and
4588.2 Å spectral lines, corresponding to transitions with the lower
term 3d5 4 F. Fig. 1 shows the full width at half-maximum intensity
as a function of the temperature for electron-impact broadening.
In order to show how the obtained results may be used for the
investigation of the influence of the Stark broadening mechanism
for Cr II spectral lines in stellar plasma conditions, we have compared Doppler and Stark widths for Cr II multiplets 3d5 4 F–3d4 4p
4 o
F (λ = 5279.6 Å) and 3d5 4 F–3d4 4p 4 Po (λ = 6073.4 Å) for a Kurucz (1979) A-type star atmosphere model with Teff = 8500 K and
log g = 4.0 (Fig. 2). Due to the characteristics of the Lorentz distribution function describing the Stark broadening contribution and
the Gauss distribution function describing the Doppler contribution
even when the Stark width is smaller than the Doppler one, Stark
broadening will contribute to the line wings, so that it is obvious
that layers where the Stark broadening influence is not negligible
exist.
To show it more clearly, we synthesized the Cr II 4588.2 Å line
profile using the SYNTH code (Piskunov 1992) (Fig. 3) and DIPSO
program package for the corresponding equivalent width (EW), for
Teff = 8 750 K and log g = 4.0 as a function of the chromium
abundance. One can see in Fig. 3 that the influence of Stark broadening increases in line wings with the increase of the chromium
abundance as expected. Namely with the increase of the abundance
of chromium, an element with much lower ionization potential than
hydrogen and helium, the usual main constituents of stellar plasma,
the electron density increases due to the increase of the degree
of plasma ionization and the corresponding Stark line width also
increases.
It is also important to consider the influence of Stark broadening
on the EW of chromium, since it is important to see the possible
errors in the chromium abundance determination, if this broadening mechanism is neglected. For this reason, Fig. 4 shows (denoted
with a dotted line) the ratio of the EW for the Cr II spectral line
4588.2 Å with (EW2 ) and without (EW1 ) Stark broadening contribution, as a function of the chromium abundance expressed as
log[Cr/H]. The full line denotes the ratio of the EW without Stark
broadening: with variable chromium abundance (EW2 ) and with the
Stark broadening of Cr II spectral lines
Table 1. Electron-, proton- and He II-impact broadening parameters for Cr II, 3d5 –3d4 4p spectral lines, for a perturber density
of 1017 cm−3 and temperatures from 5 000 to 100 000 K. The calculated wavelength of the transitions (in Å) and parameter
C are also given. This parameter when divided by the corresponding Stark width gives an estimate for the maximal perturber
density for which the line may be treated as isolated. WIDTH is FWHM. We note that for the first four lines, electronand proton-impact broadening parameters are previously communicated on a conference and published in the corresponding
proceedings (Simić et al. 2009a).
Perturber density is 1.E+17 cm−3
Perturbers are:
Electrons
Protons
Helium ions
Transition
T (K)
Width (Å)
Shift (Å)
Width (Å)
Shift (Å)
Width (Å)
Shift (Å)
Cr II
6 S−6 Po
2060.4 Å
C = 0.15E+21
5000
10 000
20 000
30 000
50 000
100 000
0.514E−01
0.382E−01
0.282E−01
0.238E−01
0.196E−01
0.157E−01
−0.334E−03
−0.379E−03
−0.438E−03
−0.425E−03
−0.460E−03
−0.515E−03
0.148E−02
0.268E−02
0.382E−02
0.431E−02
0.473E−02
0.528E−02
−0.542E−04
−0.120E−03
−0.232E−03
−0.311E−03
−0.405E−03
−0.547E−03
0.219E−02
0.335E−02
0.433E−02
0.468E−02
0.508E−02
0.553E−02
−0.541E−04
−0.118E−03
−0.214E−03
−0.278E−03
−0.355E−03
−0.446E−03
Cr II
4 F−4 Do
4588.2 Å
C = 0.40E+21
5000
10 000
20 000
30 000
50 000
100 000
0.382
0.284
0.212
0.182
0.155
0.133
0.718E−01
0.491E−01
0.378E−01
0.319E−01
0.265E−01
0.219E−01
0.102E−01
0.175E−01
0.244E−01
0.268E−01
0.295E−01
0.329E−01
0.117E−02
0.244E−02
0.416E−02
0.505E−02
0.639E−02
0.770E−02
0.147E−01
0.218E−01
0.269E−01
0.290E−01
0.315E−01
0.339E−01
0.115E−02
0.227E−02
0.359E−02
0.439E−02
0.522E−02
0.630E−02
Cr II
4 F−4 Fo
5279.6 Å
C = 0.53E+21
5000
10 000
20 000
30 000
50 000
100 000
0.480
0.358
0.268
0.229
0.194
0.165
0.743E−01
0.514E−01
0.399E−01
0.338E−01
0.274E−01
0.229E−01
0.120E−01
0.209E−01
0.293E−01
0.325E−01
0.357E−01
0.398E−01
0.874E−03
0.188E−02
0.337E−02
0.425E−02
0.546E−02
0.679E−02
0.174E−01
0.261E−01
0.326E−01
0.352E-01
0.383E−01
0.414E−01
0.868E−03
0.180E−02
0.304E−02
0.370E−02
0.464E−02
0.556E−02
Cr II
4 F−4 Po
6073.4 Å
C = 0.70E+21
5000
10 000
20 000
30 000
50 000
100 000
0.793
0.577
0.425
0.357
0.294
0.255
0.264
0.197
0.155
0.134
0.110
0.920E−01
0.144E−01
0.258E−01
0.368E−01
0.414E−01
0.459E−01
0.521E−01
0.411E−02
0.806E−02
0.127E−01
0.156E−01
0.184E−01
0.221E−01
0.210E−01
0.320E−01
0.408E−01
0.442E−01
0.483E−01
0.528E−01
0.397E−02
0.742E−02
0.111E−01
0.132E−01
0.152E−01
0.182E−01
Cr II
4 D−4 Do
3378.0 Å
C = 0.25E+21
5000
10 000
20 000
30 000
50 000
100 000
0.159
0.120
0.903E−01
0.774E−01
0.654E−01
0.547E−01
0.646E−02
0.399E−02
0.383E−02
0.355E−02
0.352E−02
0.356E−02
0.543E−02
0.929E−02
0.130E−01
0.142E−01
0.156E−01
0.174E−01
0.322E−03
0.695E−03
0.125E−02
0.161E−02
0.205E−02
0.257E−02
0.781E−02
0.116E−01
0.143E−01
0.154E−01
0.168E−01
0.181E−01
0.320E−03
0.669E−03
0.114E−02
0.138E−02
0.176E−02
0.212E−02
Cr II
4 D−4 Fo
3738.4 Å
C = 0.26E+21
5000
10 000
20 000
30 000
50 000
100 000
0.218
0.168
0.129
0.111
0.951E−01
0.822E−01
0.344E−01
0.240E−01
0.183E−01
0.155E−01
0.126E−01
0.106E−01
0.585E−02
0.103E−01
0.144E−01
0.160E−01
0.176E−01
0.196E−01
0.431E−03
0.929E−03
0.167E−02
0.211E−02
0.270E−02
0.336E−02
0.851E−02
0.128E−01
0.161E−01
0.174E−01
0.189E−01
0.204E−01
0.429E−03
0.891E−03
0.151E−02
0.183E−02
0.230E−02
0.276E−02
5000
10 000
20 000
30 000
50 000
100 000
5000
10 000
20 000
30 000
50 000
100 000
0.309
0.229
0.168
0.141
0.115
0.994E−01
0.152
0.114
0.854E−01
0.731E−01
0.625E−01
0.539E−01
0.916E−01
0.799E−01
0.730E−01
0.654E−01
0.592E−01
0.498E−01
0.363E−01
0.253E−01
0.195E−01
0.164E−01
0.138E−01
0.114E−01
0.514E−02
0.932E−02
0.136E−01
0.154E−01
0.174E−01
0.201E−01
0.440E−02
0.756E−02
0.106E−01
0.116E−01
0.128E−01
0.144E−01
0.245E−02
0.463E−02
0.701E−02
0.836E−02
0.972E−02
0.116E−01
0.669E−03
0.137E−02
0.227E−02
0.277E−02
0.341E−02
0.409E−02
0.742E−02
0.114E−01
0.147E−01
0.161E−01
0.177E−01
0.197E−01
0.633E−02
0.942E−02
0.116E−01
0.126E−01
0.137E−01
0.147E−01
0.233E−02
0.411E−02
0.603E−02
0.683E−02
0.794E−02
0.945E−02
0.655E−03
0.126E−02
0.195E−02
0.238E−02
0.278E−02
0.334E−02
5000
10 000
20 000
30 000
50 000
100 000
0.160
0.120
0.904E−01
0.775E−01
0.657E−01
0.563E−01
0.315E−01
0.216E−01
0.168E−01
0.143E−01
0.116E−01
0.968E−02
0.424E−02
0.745E−02
0.105E−01
0.117E−01
0.128E−01
0.143E−01
0.444E−03
0.939E−03
0.165E−02
0.200E−02
0.258E−02
0.313E−02
0.618E−02
0.930E−02
0.117E−01
0.126E−01
0.137E−01
0.149E−01
0.439E−03
0.886E−03
0.144E−02
0.176E−02
0.214E−02
0.257E−02
Cr II
4 P−4 Po
3637.7 Å
C = 0.32E+21
Cr II
4 P−4 Do
3046.9 Å
C = 0.20E+21
Cr II
4 G−4 Fo
3197.6 Å
C = 0.19E+21
2249
2250
Z. Simić, M. S. Dimitrijević and S. Sahal-Bréchot
Figure 1. Stark widths for resonant Cr II 4588.2, 5279.6 and 6073.4 Å
spectral lines as a function of temperature.
Figure 4. Ratio of EW for Cr II 4588.2 Å as a function of log[Cr/H].
Dotted line – ratio of EW2 (the EW with Stark broadening included) and
EW1 (without it). Full line – ratio of the EW without Stark broadening
contribution: EW2 denotes the EW with variable chromium abundance and
EW1 the EW with the Solar value of chromium abundance.
Figure 2. Thermal Doppler and Stark widths for Cr II multiplets 3d5 4 F–
3d4 4p 4 Fo (λ = 5279.6 Å) and 3d5 4 F–3d4 4p 4 Po (λ = 6073.4 Å), for an
A-type star atmosphere model with Teff = 8500 K and log g = 4.0, as a
function of the Rosseland optical depth.
Figure 5. Thermal Doppler and Stark widths for Cr II multiplets 3d5 4 F–
3d4 4p 4 Fo (λ = 5279.6 Å) and 3d5 4 F–3d4 4p 4 Po (λ = 6073.4 Å), for a
DB white dwarfs atmosphere model (Wickramasinghe 1972) with Teff =
15 000 K and log g = 8.0, as a function of the optical depth.
Figure 3. Comparison of the Cr II 4588.2 Å line profile (‘a’) without the
Stark broadening contribution and with this contribution for different Cr
abundances log[Cr/H]: ‘b’ – Solar one; ‘c’ – (−3.75); ‘d’ – (−3.25); ‘e’ –
(−2.75); Teff = 8750 K, log g = 4. [This figure was preliminary reported in a
conference and published in the corresponding proceedings (Simić 2010).]
Solar value (EW1 ). We can see in Fig. 4 that, in the considered case,
the neglecting of Stark broadening will introduce non-negligible
errors in abundance determination which increase with the increase
of chromium abundance.
The influence of the Stark broadening on Cr II spectral lines for
DB white dwarf plasma conditions is investigated here for the multiplets 3d5 4 F–3d4 4p 4 Fo (λ = 5279.6 Å) and 3d5 4 F–3d4 4p 4 Po
(λ = 6073.4 Å) by using the corresponding model with Teff =
15 000 K and log g = 8 (Wickramasinghe 1972). For the considered model atmosphere of the DB white dwarfs, the pre-chosen
optical depth points at the standard wavelength λs = 5150 Å(τ 5150 )
are used in Wickramasinghe (1972) and here as the difference to
the A-type star model (Kurucz 1979), where the Rosseland optical
depth scale (τ Ross ) was taken. As one can see in Fig. 5, for the DB
white dwarf atmosphere plasma conditions, thermal Doppler broadening is much less important than in A-type stars, in comparison
with the Stark broadening mechanism, which is the most important
one here.
Stark broadening of Cr II spectral lines
We have demonstrated that the Stark broadening of the considered Cr II lines may be non-negligible in A-type star atmospheres,
especially in the line wings and when chromium is overabundant.
We also note that it is even more important for the modelling of subphotospheric layers. Its neglection, as the present analysis shows,
may contribute to errors in chromium abundance determination.
The obtained results also show that it is the principal broadening
mechanism in DB white dwarf atmospheres. Also, Stark broadening
parameters for nine resonant Cr II multiplets are determined.
Cr II Stark broadening parameters given in Table 1 will also be
included in the STARK-B data base (Sahal-Bréchot, Dimitrijević
& Moreau 2012), a part of Virtual Atomic and Molecular Data
Center (Dubernet et al. 2010; Rixon et al. 2011), cf. also SahalBréchot (2010). This data base is devoted to diagnostics modelling
and investigations of stellar atmospheres, but also for laboratory,
fusion and laser produced plasmas research and modelling.
AC K N OW L E D G E M E N T S
This work is a part of the project 176002 ‘Influence of collisional
processes on astrophysical plasma spectra’ supported by Ministry
of Education and Science of Republic Serbia.
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