MNRAS 431, 1039–1047 (2013)
doi:10.1093/mnras/stt228
Advance Access publication 2013 March 15
Stark broadening of Pb IV spectral lines
Rafik Hamdi,1‹ Nabil Ben Nessib,2,3 Milan S. Dimitrijević4,5,6
and Sylvie Sahal-Bréchot5
1 Groupe
de Recherche en Physique Atomique et Astrophysique, Faculté des Sciences de Bizerte, Université de Carthage, Tunisia
of Physics and Astronomy, College of Science, King Saud University. PO Box 2455, Riyadh 11451, Saudi Arabia
3 Groupe de Recherche en Physique Atomique et Astrophysique, INSAT, Université de Carthage, Tunisia
4 Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia
5 Laboratoire d’Étude du Rayonnement et de la Matière en Astrophysique, Observatoire de Paris, UMR CNRS 8112, UPMC, 5 Place Jules Janssen,
92195 Meudon Cedex, France
6 Institute Isaac Newton of Chile, Yugoslavia Branch, 11060 Belgrade, Serbia
2 Department
Accepted 2013 February 5. Received 2013 February 5; in original form 2012 December 12
ABSTRACT
Stark-broadening parameters have been calculated for 114 spectral lines of triply charged
lead ion (Pb IV) using semiclassical perturbation approach in the impact approximation. The
provided widths and shifts have been obtained for a set of temperatures from 20 000 to
300 000 K and an electron density of 1017 cm−3 . The studied lines correspond to transitions
between the configurations 5d10 nl−5d10 n l and 5d9 6s2 −5d10 nl. Energy levels and oscillator
strengths needed for this calculation have been calculated using a Hartree–Fock relativistic
(HFR) approach. Comparison has also been made with available theoretical and experimental
results. In addition, the regularity in the 5d10 6s 2 S1/2 −5d10 np2 Po1/2 spectral series has been
studied.
Key words: atomic data – atomic processes – line: profiles.
1 I N T RO D U C T I O N
Triply charged lead ion (Pb IV) belongs to the gold isoelectronic
sequence, its ground-state configuration is 4f 14 5d10 6s. This is an
interesting isoelectronic sequence with filled 4f and 5d subshells
and a single electron in the outer shell. This ion is characterized
by a strong resonance line, which is a candidate for spectroscopic
detection in hot DA white dwarfs (Vennes, Chayer & Dupuis 2005).
O’Toole (2004) reported the discovery of strong photospheric resonance lines of several heavy elements in the ultraviolet (UV) spectra
of more than two dozen sdB and sdOB stars at temperatures ranging
from 22 000 to 40 000 K. Among these lines, several correspond to
Pb IV ones. Pb IV 1313.1 Å resonance line was detected by Proffitt,
Sansonetti & Reader (2001) in the main-sequence B star AV 304.
Stark broadening of spectral lines is very important in DA and DB
white dwarf atmospheres (Simić, Dimitrijević & Kovačević 2009;
Dimitrijević et al. 2011; Dufour et al. 2011; Larbi-Terzi et al. 2012).
Hamdi et al. (2008) studied the influence of Stark broadening on
Si VI lines in DO white dwarf atmospheres and found that this mechanism is dominant in broad regions. Besides white dwarfs, Stark
broadening is the most important pressure-broadening mechanism
for A and B stars and this effect must be taken into account for investigation, analysis and modelling of their atmospheres. In Popović
et al. (2001), it was shown that Stark broadening can change the
E-mail: [email protected]
spectral line equivalent widths by 10–45 per cent. Hence neglecting this mechanism, significant errors in abundance determinations
may be introduced.
Alonso-Medina et al. (2010) carried out semi-empirical (SE)
calculations of Stark widths and shifts in the impact approximation
for 58 spectral lines of Pb IV using Griem’s (1968) formula with
a Gaunt factor suggested by Niemann et al. (2003). Atomic data
were determined using Hartree–Fock relativistic (HFR) approach
(Cowan 1981). They found that their values are for a factor of 2
lower than Dimitrijević & Sahal-Bréchot (1999). The latter ones
calculated widths and shifts of the lines within 6s 2 S−6p 2 Po and
6s 2 S−7p 2 Po multiplets with the impact semiclassical perturbation
(SCP) method (Sahal-Bréchot 1969a,b) with Bates and Damgaard
(Bates & Damgaard 1949) oscillator strengths. Alonso-Medina et al.
(2010) concluded that the Bates and Damgaard oscillator strengths
used by Dimitrijević & Sahal-Bréchot (1999) were overestimated
because the core-polarization effects were not included.
Due to the interest of SCP and SE methods for Stark-broadening
line profiles determination, and since Pb IV lines have been observed
in stellar spectra, it is important to clarify the reasons of this discrepancy.
Therefore, we have performed in this paper several SCP calculations of Stark-broadening parameters using different sets of
oscillator strengths. First, we have used the oscillator strengths
of Alonso-Medina, Colón & Porcher (2011), i.e. the same oscillator strengths as used in the SE work of Alonso-Medina
et al. (2010). Secondly, we have calculated oscillator strengths by
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
1040
R. Hamdi et al.
using the Cowan code [HFR correction approach (Cowan (1981)].
Thirdly, we have used the oscillator strengths of Safronova & Johnson (2004) obtained using third-order many-body perturbation theory. We have also calculated Stark-broadening parameters using the
modified semi-empirical method (MSE; Dimitrijević & Konjević
1980), with the atomic data taken from Alonso-Medina A., Colón
& Porcher (2011). Then our obtained results have been compared
to the SCP values of Dimitrijević & Sahal-Bréchot (1999) and to
the SE results of Alonso-Medina et al. (2010). They have also been
compared to the experimental ones of Bukvić et al. (2011). The
obtained results are used to clarify the reason for the discrepancy
between SE results of Alonso-Medina et al. (2010) and SCP results
of Dimitrijević & Sahal-Bréchot (1999).
In addition, we have provided in this work SCP impact Stark
widths and shifts for 114 spectral lines between the 5d10 nl−5d10 n l and 5d9 6s2 −5d10 nl configurations of Pb IV. The colliding particles
are electrons, protons and ionized helium. The energy levels and
oscillator strengths have been obtained with the Cowan code with
43 configurations.
Finally, the regularity of behaviour of Stark widths within the
5d10 6s 2 S1/2 −5d10 np2 Po1/2 spectral series has been studied.
2 T H E I M PAC T S E M I C L A S S I C A L
P E RT U R BAT I O N M E T H O D
A detailed description of this formalism with all the innovations is given in Sahal-Bréchot (1969a,b, 1974, 1991); Fleurier,
Sahal-Bréchot & Chapelle (1977); Dimitrijević, Sahal-Bréchot &
Bommier (1991); Dimitrijević & Sahal-Bréchot (1996). The profile
F(ω) is Lorentzian for isolated lines:
F (ω) =
w/π
,
(ω − ωif − d)2 + w2
(1)
where
Ei − Ef
,
i and f denote the initial and final states and Ei and Ef their corresponding energies.
The total width at half-maximum (W = 2w) and shift (d) (in
angular frequency units) of an electron-impact-broadened spectral
line can be expressed as
⎞
⎛
σii (v) +
σff (v) + σel ⎠
W = N vf (v)dv ⎝
ωif =
i =i
d =N
f =f
RD
2πρdρ sin(2ϕp ),
vf (v)dv
(2)
R3
where N is the electron density, f (υ) the Maxwellian velocity
distribution function for electrons, ρ denotes the impact parameter
of the incoming electron, i (resp. f ) denotes the perturbing levels
of the initial state i (resp. final state f ). The inelastic cross-section
σii (υ) (resp. σff (υ)) can be expressed by an integral over the
impact parameter ρ of the transition probability Pii (ρ, υ) (resp.
Pff (ρ, υ) ) as
RD
1
σii (υ) = πR12 +
2πρdρ
Pii (ρ, υ),
(3)
2
R1
i =i
i =i
and the elastic cross-section is given by
RD
2πρdρ sin2 δ + σr ,
σel = 2πR22 +
R2
1
δ = (ϕp2 + ϕq2 ) 2 .
(4)
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. All the cut-offs R1 , R2 and R3 are described in section 1 of
chapter 3 in Sahal-Bréchot (1969b). σr is the contribution of the
Feshbach resonances (Fleurier et al. 1977)
The formulae for the ion-impact widths and shifts are analogous
to equations (2)– (4), without the Feshbach resonances contribution
to the width. For electrons, hyperbolic paths due to the attractive
Coulomb force are used, while for perturbing ions the hyperbolic
paths are different since the force is repulsive.
The calculations need a relatively complete set of oscillator
strengths for transitions starting or ending on energy levels forming
the considered line, so that the corresponding oscillator strength
sum rules can be satisfied. In our present calculations, energy levels
and oscillator strengths have been carried out with the HFR approach using the Cowan code (Cowan 1981). We have adopted an
atomic model including 43 configurations: 5d9 6s2 , 5d9 6p2 , 5d10 ns
(6 ≤ n ≤ 11), 5d10 nd (6 ≤ n ≤ 11), 5d10 ng (5 ≤ n ≤ 11), 5d9 6s7s,
5d9 6s6d (even parity) and 5d10 np (6 ≤ n ≤ 11), 5d10 nf (5 ≤ n ≤
11), 5d10 nh (6 ≤ n ≤ 11), 5d9 6s6p (odd parity).
3 C O M PA R I S O N B E T W E E N D I F F E R E N T
C A L C U L AT I O N S
In order to explain the reason of the disagreement found between SE results of Alonso-Medina et al. (2010) and SCP ones of
Dimitrijević & Sahal-Bréchot (1999), the present section is devoted
to a comparison between different calculations for several lines of
Pb IV, using the impact approximation.
The first SCP calculations of Stark-broadening parameters of
Pb IV were performed by Dimitrijević & Sahal-Bréchot (1999). Energy levels were taken from Gutmann & Crooker (1973) and oscillator strengths were calculated with the Bates and Damgaard
method (Bates & Damgaard 1949). Stark widths were calculated
for 6s 2 S−6p 2 Po and 6s 2 S −7p 2 Po multiplets. The widths of Dimitrijević & Sahal-Bréchot (1999) are denoted here as WDS .
Alonso-Medina et al. (2010) used the SE Griem’s formula (Griem
1968) with a Gaunt factor suggested by Niemann et al. (2003).
Atomic data, obtained by using HFR approach of Cowan (1981)
with 14 configurations, were taken from Alonso-Medina et al.
(2011). Yet, it can be noticed that Griem (1974, p. 256) wrote that
not much is known about the accuracy of his SE formula (Griem
1968) for multiply charged ions. In fact, this formula was based on
the effective Gaunt factor proposed by Van Regemorter (1962) for
ions. Furthermore, Dimitrijević & Konjević (1980) showed that the
accuracy of SE line widths decreases with the increase of the charge
of emitter, and that these SE widths were considerably lower than
the results of experiments. Therefore, Alonso-Medina et al. (2010)
replaced the Gaunt factor of Van Regemorter (1962) by the Gaunt
factor suggested by Niemann et al. (2003). This is another reason
to investigate the accuracy of such adaptation of the SE method.
Regarding this work, we have performed three different sets of
calculations for the aim of comparison and discussion. SCP calculations using oscillator strengths and energy levels of Alonso-Medina
et al. (2011) obtained by Cowan code with 14 configurations (the
width is denoted by WSC1 ) and SCP calculations using oscillator strengths taken from Safronova & Johnson (2004) and energy
(the width is denoted by WSC2 ). Oscillator strengths of Safronova
& Johnson (2004) are obtained using third-order many-body
Stark broadening of spectral lines of Pb IV
1041
Table 1. Comparison between our electron impact Stark widths (WSC1 , WSC2 , WSC3 ), values
from Alonso-Medina et al. (2010) (WAM ) and values from Dimitrijević & Sahal-Bréchot (1999)
(WDS ). Present results: WSC1 – semiclassical Stark widths obtained using oscillator strengths
from Alonso-Medina et al. (2011); WSC2 – semiclassical Stark widths obtained using oscillator
strengths from Safronova & Johnson (2004); WSC3 – semiclassical Stark widths obtained using
oscillator strengths calculated with HFR method (Cowan 1981) and the atomic model given in
Section 2. WSC1 , WSC2 and WSC3 are obtained using energy levels from Alonso-Medina et al.
(2011). Results are given for an electron density of Ne = 1017 cm−3 .
Transition
T (K)
WSC1 (Å)
WSC2 (Å)
WSC3 (Å)
WDS (Å)
WAM (Å)
6s 2 S1/2 −6p 2 Po1/2
λ = 1313.1 Å
50 000
200 000
0.008 25
0.004 46
0.008 80
0.004 76
0.010 70
0.005 87
0.010 34
0.005 67
0.0051
0.0022
6s 2 S1/2 −6p 2 Po3/2
λ = 1028.6 Å
50 000
200 000
0.005 98
0.003 28
0.006 04
0.003 30
0.006 95
0.003 82
0.006 34
0.003 47
0.0051
0.0022
6s 2 S1/2 −7p2 Po1/2
λ = 476.7 Å
50 000
200 000
0.002 80
0.001 70
0.003 53
0.002 29
0.003 65
0.002 36
0.003 65
0.002 39
0.0017
0.0008
6s 2 S1/2 −7p 2 Po3/2
λ = 459.0 Å
50 000
200 000
0.003 10
0.002 01
0.003 36
0.002 22
0.003 36
0.002 19
0.003 38
0.002 22
0.0030
0.0014
perturbation theory with the Brueckner-orbital corrections which
account for core-polarization effects (Chou & Johnson 1997). In addition, we have performed SCP calculations with oscillator strengths
that we have calculated using the Cowan code with 43 configurations (the width is denoted by WSC3 ).
Furthermore, we have performed another calculation using the
MSE approach of Dimitrijević & Konjević (1980) and atomic data
of Alonso-Medina et al. (2011). MSE method is valid for singly and
multiply charged ions, it uses an effective Gaunt factor and has the
advantage that it does not involve a large number of atomic data. The
corresponding width is denoted by WMSE . For the details of MSE
calculations see Mahmoudi, Ben Nessib & Dimitrijević (2005). We
can notice that as a difference from MSE, the complete version of
Griem’s SE method (Griem 1968) needs the same set of atomic
data as the more sophisticated SCP method, so the same effort is
needed for simpler (SE) and more advanced (SCP) calculations, if
both codes are available.
tions; Eissner, Jones & Nussbaumer 1974) oscillator strengths for
Si V and Ne V ions. They found that the difference between the two
sets of calculations did not exceed 30 per cent. Thus, the difference
in oscillator strengths used in Stark-broadening calculation is not of
crucial importance, since the accuracy of the SCP method is about
20 per cent.
One additional reason which could explain the difference between SCP Dimitrijević & Sahal-Bréchot (1999) calculations and
SE Alonso-Medina et al. (2010) results, is the eventual existence
of a perturbing level close to the initial or final level of the studied
transitions, the transition to which is forbidden in Coulomb approximation but becomes allowed if configuration mixing is taken
into account. So the corresponding Bates and Damgaard oscillator
strength is zero, whereas the Hartree–Fock one can be large. The
existence of such levels can increase the width. The influence of a
possible close perturbing level will be discussed in Sections 4 and 5.
3.1 Role of oscillator strengths
3.2 Comparison between the three methods: SCP, SE
and MSE
Table 1 presents the different results for the lines of the 6s 2 S−6p 2 Po
and 6s 2 S−7p2 Po multiplets of Pb IV, for two temperatures (50 000
and 200 000 K), and for an electron density of 1017 cm−3 . Consequently, the use of the set of Bates and Damgaard’s oscillator
strengths in WDS calculations, which satisfies the corresponding
sum rules (Shore & Menzel 1965), cannot explain the large difference between WAM and WDS , for example for 476.7 Å line.
As we can see in Table 1, WDS is always larger than WSC1 . The
DS
is equal to 1.19 and cannot explain the difference
average ratio WWSC1
of the factor of 2 (found by Alonso-Medina et al. 2010 in some
cases). WDS is also larger than WSC2 but the difference is smaller
DS
is equal to 1.07 only. WSC3 are closer
since the average ratio WWSC2
to WDS and the average difference is 3 per cent.
A similar conclusion, concerning the use of a set of Bates and
Damgaard’s oscillator strengths, complete from the point of view
of the corresponding sum rules, was obtained by Ben Nessib,
Dimitrijević & Sahal-Bréchot (2004) and Hamdi et al. (2007)
who calculated line widths and shifts of Si V and Ne V ions. They
compared SCP ab initio Stark widths obtained with Bates and
Damgaard oscillator strengths and with SUPERSTRUCTURE (Thomas–
Fermi–Dirac interaction potential model with relativistic correc-
By using the same set of atomic levels and oscillator strengths, we
compare in Figs 1, 2 and 3 the three methods used in the calculations
for the widths: SCP (WSC1 ), SE (WAM ) and MSE (WMSE ). The three
theoretical widths are also compared with the experimental ones of
Bukvić et al. (2011).
Fig. 1 displays electron-impact Stark widths for the
6d 2 D5/2 −7p 2 Po3/2 line as a function of electron temperature for a
1017 cm−3 electron density. Our SCP and MSE calculations have
been obtained with energy levels and oscillator strengths from
Alonso-Medina et al. (2011).
As we can see in Fig. 1, WSC1 are slightly higher than WMSE . At
low temperatures, SE widths of Alonso-Medina et al. (2010) are
higher than SCP and MSE ones. At T = 200 000 K, the three methods give the same results. SE results overestimate the experimental
line width, our SCP and MSE results underestimate the experimental width, but the SCP value is in the lower limit of experimental
error.
Fig. 2 is the same as Fig. 1 but for the 7s 2 S1/2 −7p 2 Po3/2 transition.
At low temperatures WAM is higher than WSC1 and WMSE and at
high temperatures WAM is lower than our SCP and MSE results.
Our SCP width is close to the experimental ones, MSE results
1042
R. Hamdi et al.
Figure 1. Electron impact Stark widths FWHM for the 6d 2 D5/2 –7p 2 Po3/2
(λ = 3221.22 Å) line as a function of the electron temperature (T) at an
electron density of 1017 cm−3 . Solid line: our Stark widths obtained using SCP approach (Sahal-Bréchot 1969a,b) (WSC1 ) and oscillator strengths
from Alonso-Medina et al. (2011), Dotted line: our Stark widths obtained
using modified SE approach (Dimitrijević & Konjević 1980) and oscillator
strengths from Alonso-Medina et al. (2011) (WMSE ), Dashed line: Stark
widths of Alonso-Medina et al. (2010) obtained using SE formula (Griem
1968), with Gaunt factor suggested by Niemann et al. (2003) (WAM ), Full
circle: experimental Stark width (Bukvić et al. 2011).
Figure 2. Same as in Fig. 1 but for the 7s 2 S1/2 −7p 2 Po3/2 (λ = 3052.66 Å)
transition.
underestimate the experimental width and SE results overestimate
the experimental width, but the SCP value is on the lower limit of
experimental error.
Fig. 3 is the same as Fig. 1 but for the 5f 2 Fo7/2 −5g 2 G9/2 transition.
WAM is in the higher limit of the experimental error. WSC1 and
WMSE underestimate the experimental value but WMSE is closer.
WSC1 is lower than the experimental width by a factor of 3. At low
temperatures WMSE is lower than WAM but at high temperatures
WMSE is higher than WAM . At T = 100 000 K, SE and MSE approach
gives the same width.
One can see that in Figs 1 and 2, the results of three theoretical
methods are in much better agreement than in Fig. 3, where there
is a large difference. If we look at the partial energy level diagram
in Fig. 7, one can see that the structure of perturbing levels is not
regular. In particular, MSE theory assumes that the important con-
Figure 3. Same as in Fig. 1 but for the 5f 2 Fo7/2 −5g 2 G9/2 (λ = 2049.37 Å)
transition.
tribution to the line width proceeds from perturbing levels with the
same principal quantum number and that all other perturbing levels
are lumped together. But for the 5g 2 G9/2 level there is no 5h 2 Ho
perturbing term, and the energies of 5f 2 Fo levels are much lower
than the 6f 2 Fo and 7f 2 Fo ones. This introduces additional uncertainties in the MSE theoretical approach. Additional experimental
results will be of interest for checking and improving theory for
such a specific case which is more complicated than the previous
two.
By comparing our new large-scale SCP results for widths, given
in Section 5, with those of Alonso-Medina et al. (2010), we see that
the SE widths are not always lower than ours (as all values given
in Table 1). For many transitions, there is an acceptable agreement
between the two calculations. However, for a number of transitions
the SE results are greater than ours even by a factor of 2.
In Alonso-Medina & Colón (2011), the overestimation of theoretical widths obtained using SE formula for Sn III lines in some
cases when their values are above the existing experimental values is explained by the large number of perturbing levels used
for initial and final levels for each transition. From our point of
view, the set of atomic data used in the SE and SCP calculation of the width is relatively large but the number of significant perturbing levels involved in the calculation of the width is
not large. Adding other perturbing levels will have a negligible
effect on the final results for the line width. However, we note
that the shift calculations are more sensitive to the number of
perturbing levels. This is due to mutual cancellations of contributions with different signs. This is not the case for the widths,
where the contributions of different perturbing levels have positive
values.
Indeed, in the case of Pb IV the atomic model adopted by AlonsoMedina et al. (2010) is not so large. For example, the 8p and 9p
configurations are not included in their atomic model. In fact, these
configurations provide levels which significantly perturb the level
8s 2 S1/2 . For example, for the 7p 2 Po3/2 −8s 2 S1/2 transition, our width
at T = 20 000 K is equal to 0.374 Å. If we do not take into account
the 8p and 9p configurations, the width becomes 0.220 Å.
For the 6s 2 S−6p 2 Po multiplet, the widths of the fine structure
components of Alonso-Medina et al. (2010) are exactly the same.
The difference between our widths of the two fine structure components is 46 per cent. It must be noted that the difference between
the wavelengths of the two fine structure components is 27 per cent.
Stark broadening of spectral lines of Pb IV
1043
4 C O M PA R I S O N W I T H E X P E R I M E N T
Figure 4. Partial energy level diagram showing the principal perturbing
levels for 6s−6p transitions.
This large difference in the widths expressed in Å is only due to the
difference in the wavelengths, since in angular frequency units the
widths differ only by 6 per cent. The difference between the widths
of the two components found by Alonso-Medina et al. (2010) expressed in angular frequency units is 62 per cent. Partial energy level
diagram showing the principal perturbing levels for 6s 2 S−6p 2 Po
multiplet is presented in Fig. 4. This diagram shows that 6s2 2 D3/2 is
the nearest level to 6p 2 Po3/2 but its contribution is very small. In fact,
the value of the oscillator strength given in Alonso-Medina et al.
(2011) of the 6p 2 Po3/2 −6s2 2 D3/2 transition is only 7 × 10−5 . The
distances to the perturbing levels 6d 2 D, 6s 2 S and 7s 2 S are much
larger than the energy differences between 6p 2 Po3/2 and 6p 2 Po1/2 , so
that one expects that the widths of the two fine structure components are close. The large difference found by Alonso-Medina et al.
(2010) cannot be explained by the existence of a perturbing level
much closer to the upper level of one of the two neighbouring fine
structure transitions, which should be consequently more perturbed
in this case.
Recently, Bukvić et al. (2011) investigated Pb IV and Pb V spectral
line shapes in the laboratory helium plasma at electron temperatures
around 22 000 K and electron density between 5.1 × 1016 cm−3 and
9.1 × 1016 cm−3 . In Table 2, our results are compared to these
experimental results (Wm ) and also to the SE values of AlonsoMedina et al. (2010).
WSC = We +Wi , where We is electron-impact Stark width and
Wi is ionic-impact Stark width. Taking into account the experimental conditions, we have taken as ionic perturbers singly charged
helium ions. For each value given in Table 2, the collision volume (V) multiplied by perturber density (N) is much less than one
and the impact approximation is valid. The greatest value of N×V,
equal to 0.25, has been found for the 5g 2 G9/2 −6h 2 Ho transition
for collisions with ions. For the study of the 5g 2 G9/2 −6h 2 Ho transition, the atomic model given in Section 2 has been enriched by
the configurations 7i and 8i which give important perturbing levels
for 6h 2 Ho . In addition, since we have found that magnetic dipole
(M1) and electric quadrupole (E2) transitions have no influence on
the width, only electric dipole transitions (E1) are considered in our
calculations.
A number of levels in Alonso-Medina et al. (2011), used also
by Bukvić et al. (2011), belonging to the configuration 5d9 6s6p
are a mixture without a leading term. They are denoted as [1o ],
[2o ], . . . [27o ]. For some of them, there is a correspondence with
energy levels within LS coupling (Moore 1958; Alonso-Medina
et al. 2011). Namely 6d10 7p 2 Po1/2 , 7p 2 Po3/2 , 5f 2 Fo5/2 and 5f 2 Fo7/2
correspond to 5d9 6s6p [16o ]1/2 , [22o ]3/2 , [23o ]5/2 and [24o ]7/2 , respectively. In Table 2, experimental results of Bukvić et al. (2011)
are included only for transitions which could be described in LS
coupling, since our calculations have been performed only for such
Pb IV transitions.
The lines labelled as 4a and 4b in Table 2 correspond to the superposition of two close fine structure components: 6d 2 D3/2− 5f 2 Fo5/2
(2864.31 Å) and 6d 2 D5/2 −5f 2 Fo7/2 (2864.55 Å). By assuming local
thermodynamical equilibrium and that the lines are optically thin at
the temperature T = 2.38 104 K and the density Ne = 1017 cm−3 ,
Table 2. Comparison between our Stark widths (FWHM) (WSC ) obtained using atomic data calculated
using Cowan code (Cowan 1981), experimental values (Wm ) of Bukvić et al. (2011) and theoretical
values of Alonso-Medina et al. (2010) (WAM ). Results are given for an electron density of Ne =
1017 cm−3 .
Label
1
2a
2b
3
4a
4b
5
6
7
8
9
10
11
12
13
14
Transition
5f 2 Fo7/2 −5g 2 G
5g 2 G9/2 −6h 2 Ho
5g 2 G7/2 −6h 2 Ho
6p 2 Po1/2 −6s 2 2 D3/2
6d 2 D3/2 −5f 2 Fo5/2
6d 2 D5/2 −5f 2 Fo7/2
6d 2 D5/2 −5f 2 Fo5/2
6d 2 D3/2 −7p 2 Po3/2
6d 2 D5/2 −7p 2 Po3/2
6d 2 D3/2 −7p 2 Po1/2
7s 2 S1/2 −7p 2 Po3/2
7s 2 S1/2 −7p 2 Po1/2
7p 2 Po1/2 −7d 2 D3/2
7p 2 Po3/2 −7d 2 D5/2
7p 2 Po3/2 −7d 2 D3/2
7p 2 Po3/2 −8s 2 S1/2
λ (Å)
T (104 K)
Wm (pm)
WSC (pm)
WAM (pm)
2049.37
4534.46
4534.93
2154.01
2864.31
2864.55
3062.43
3002.76
3221.22
3962.49
3052.66
4049.84
2461.51
2978.20
3071.33
3145.47
2.00 ± 0.28
2.33 ± 0.33
2.33 ± 0.33
2.20 ± 0.30
2.38 ± 0.33
2.38 ± 0.33
2.22 ± 0.31
2.30 ± 0.32
2.30 ± 0.32
2.26 ± 0.32
2.22 ± 0.31
2.38 ± 0.33
2.32 ± 0.32
2.30 ± 0.32
2.26 ± 0.32
2.10 ± 0.30
49.3 ± 7.4
301 ± 24
19.0
260
54.2
300
5.0 ± 1.2
49.4 ± 7.4
4.08
40.4
37.3
22.5 ± 4.0
18.5 ± 3.8
31.2 ± 4.7
47.0 ± 7.0
25.5 ± 3.8
61.5 ± 9.2
24.9 ± 3.7
36.9 ± 5.5
43.3 ± 6.5
39.2 ± 6.0
21.7
26.6
31.2
48.1
29.4
51.2
31.0
40.0
46.7
42.8
43.3
31.1
22.0
34.2
28.4
1044
Figure
R. Hamdi et al.
5. Superposition
6d 2 D5/2 −5f 2 Fo7/2
of
6d 2 D3/2 −5f 2 Fo5/2
(2864.31 Å)
and
(2864.55 Å) line profiles.
Figure 6. Superposition of 5g 2 G9/2 −6h 2 Ho
5g 2 G7/2 −6h2 Ho (4534.93 Å) line profiles.
(4534.46 Å)
and
we have determined the global profile of this line. Under these conditions the intensities of the lines are additive and the total intensity
profile Itotal (λ) is given by the following formula:
Itotal (λ) = g1 A1 I1 (λ) + g2 A2 I2 (λ),
(5)
where g1 (resp. g2 ) is the statistical weight of the upper level of the
first component of the line (resp. the second component of the line).
A1 (resp. A2 ) is the transition probability of spontaneous emission
of the first component of the line (resp. the second component of
the line). I1 (λ) (resp. I2 (λ)) is the normalized profile of the first
component of the line (resp. the second component of the line) with
a half-width w1 and shift d1 (resp. half-width w2 and shift d2 ). A
normalized Lorentzian is given by
I (λ) =
1
w
,
π (λ − λif − d)2 + w2
(6)
where w = W/2 is the half width at half-maximum and d is the
shift. The resulting profile is plotted in Fig. 5. Using this profile,
we have found that the full width at half intensity maximum of
this global transition is 40.4 pm. The experimental width of this
global transition is 49.4 ± 7.4 pm (Bukvić et al. 2011). They found
that the composed profile is close to a Lorentz profile and that
the width of this composite distribution is less than the width of
the broader component. We have found that the composite profile
is not Lorentzian (see Fig. 5) and that the width of the composite
distribution is much larger than the width of a particular component.
The lines labelled as 2a and 2b in Table 2, are also a superposition of two close transitions: 5g 2 G9/2 −6h2 Ho (4534.46 Å) and
5g 2 G7/2 −6h 2 Ho (4534.93 Å). We have calculated the width of the
global line using the same method described above. Our calculated
width is 260 pm and the measured one is 301 ± 24 pm. Global
profile for this line is presented in Fig. 6.
The line labelled as 11 in Table 2, is also a superposition of
two close transitions: 7p 2 Po 1/2 −7d 2 D3/2 and 7p 2 Po 3/2 −7d 2 D3/2
(2461.51 Å). We have also calculated the width of the global line
using the same method described above. Our calculated width is
31.0 pm and the measured one is 24.9 ± 3.7 pm.
We have found a tolerable agrement with measured widths except
for one line: 5f 2 Fo7/2 −5g 2 G (2049.37 Å) labelled as 1 in Table 2
for which the experimental width is 49.3 ± 7.4 pm. In Fig. 7, we
show a partial energy level diagram for 5f −5g transitions. This
diagram shows that 5f 2 Fo7/2 −5g 2 G7/2 and 5f 2 Fo7/2 −5g 2 G9/2 lines
have close wavelengths and in fact these are also two superposed
Figure 7. Partial energy level diagram showing the principal perturbing
levels for 5f−5g transitions. Wavelengths 2049.37 Å and 1959.32 Å are
from Crawford, McLay & Crooker (1937) and λ = 2049.29 Å has been
calculated from Pb IV terms in table 1 of Crawford et al. (1937) and scaled
to the two previous observed wavelengths.
lines like in the three previous cases, but Alonso-Medina et al.
(2010) provided a width only for one component, so that Bukvić
et al. (2011) could not calculate the composite distribution. We have
also determined the global profile with the same preceding method,
and our width of 19.0 pm is still lower than the experimental one by
a factor greater than 2. The agreement with experiment for this line
can not be improved even we use the atomic data of Alonso-Medina
et al. (2011) (see Fig. 3).
5 L A R G E - S C A L E C A L C U L AT I O N S
5.1 SCP calculations for 114 transitions of Pb IV
Our SCP method allows us to study a large number of lines. In
fact, a large number of collisional data are needed for deriving
precise atmospheric parameters of hot star atmospheres as white
dwarfs (Dufour et al. 2011). Accurate and large Stark-broadening
tables are of crucial importance for sophisticated spectral analysis
by means of non-local thermodynamic equilibrium (NLTE) model
atmospheres (Rauch et al. 2007).
Thus, using our SCP code, we have calculated widths and
shifts for 114 transitions of Pb IV between the 5d 10 nl−5d10 n l and 5d 9 6s 2 −5d 10 nl configurations. The results are provided in
electronic form in the online journal as additional data (Table S).
Stark broadening of spectral lines of Pb IV
1045
Table 3. This table gives electron-, proton- and singly charged helium-impact broadening parameters for Pb IV lines calculated using
Cowan code (Cowan 1981) oscillator strengths, for a perturber density of 1017 cm−3 and temperature of 20 000 to 300 000 K. Calculated
wavelength of the transitions (in Å) and parameter C are also given. This parameter when divided with the corresponding Stark width
gives an estimate for the maximal pertuber density for which the line may be treated as isolated. We : electron-impact full Stark width
at half-maximum, de : electron-impact Stark shift, WH+ : proton-impact full Stark width at half-maximum, dH+ : proton-impact Stark
shift, WHe+ : singly charged helium-impact full Stark width at half-maximum, dHe+ : singly charged helium-impact Stark shift. This
table is available in its entirety for 114 Pb IV spectral lines in machine-readable form in the online journal as additional data. A portion
is shown here for guidance regarding its form and content.
Transition
T (K)
We (Å)
de (Å)
WH+ (Å)
dH+ (Å)
WHe+ (Å)
dHe+ (Å)
5f 2 Fo5/2 −7d 2 D3/2
3237.7 Å
C = 0.50E+20
20 000
30 000
50 000
100 000
200 000
300 000
0.462
0.399
0.338
0.281
0.239
0.219
0.389E−01
0.356E−01
0.336E−01
0.367E−01
0.323E−01
0.306E−01
0.184E−01
0.233E−01
0.312E−01
0.380E−01
0.449E−01
0.496E−01
0.126E−01
0.160E−01
0.210E−01
0.261E−01
0.310E−01
0.344E−01
0.223E−01
0.277E−01
0.320E−01
0.374E−01
0.428E−01
0.455E−01
0.110E−01
0.138E−01
0.174E−01
0.211E−01
0.252E−01
0.274E−01
5f 2 Fo5/2 −7g 2 G7/2
1174.1 Å
C = 0.34E+19
20 000
30 000
50 000
100 000
200 000
300 000
0.198
0.178
0.161
0.140
0.123
0.113
−0.146E−01
−0.167E−01
−0.141E−01
−0.131E−01
−0.120E−01
−0.984E−02
*0.191E−01
*0.216E−01
*0.246E−01
0.289E−01
0.329E−01
0.363E−01
*−0.117E−01
*−0.136E−01
*−0.166E−01
−0.203E−01
−0.231E−01
−0.253E−01
*0.192E−01
*0.214E−01
*0.237E−01
*0.266E−01
*0.289E−01
*0.301E−01
*−0.918E−02
*−0.110E−01
*−0.133E−01
*−0.162E−01
*−0.185E−01
*−0.198E−01
A sample of the results is only shown in Table 3 for guidance
regarding its form and content. The calculations have been made for
a perturber density of 1017 cm−3 and for a set of temperatures from
20 000 to 300 000 K. Stark widths (FWHM) and shifts are given for
electron-, proton- and singly ionized helium impact broadening. Energy levels and oscillator strengths needed for this calculation have
been determined using Cowan code (Cowan 1981) and the atomic
model described above in the end of Section 2 (43 configurations).
All wavelengths given in Tables 3 and S are calculated wavelengths. They have been determined from energy levels obtained
with the first three Cowan codes and the fourth part devoted for
scaling with experimental energy levels has not been used. So the
calculated wavelengths given in Table 3 are not good but the Starkbroadening parameters, which depend on relative and not absolute positions of energy levels are correct in angular frequency
units. The relationship between the width expressed in Å and the
width expressed in angular frequency units is given by the following
formula:
λ2
W (s −1 ),
(7)
2πc
where c is the speed of light. If we want to introduce a correction to
the width due to the difference between calculated and experimental
wavelength, one should use the following formula:
λexp 2
W.
(8)
Wcor =
λ
W (Å) =
In the above expression, Wcor is the corrected width, λexp is the
experimental wavelength, λ is the calculated wavelength and W is
the calculated width of Tables 3 and S. A similar formula can be
used for the shifts.
We also specify a parameter C (Dimitrijević & Sahal-Bréchot
1984), which gives an estimate for the maximal perturber density
for which the line may be treated as isolated, when it is divided by
the corresponding full width at half-maximum (FWHM). For each
value given in Tables 3 and S, the collision volume V multiplied
by the perturber density N is much less than one and the impact
approximation is valid (Sahal-Bréchot 1969a,b). For NV > 0.5, the
impact approximation breaks down and thus the values are not given.
For 0.1 < NV ≤ 0.5, the impact approximation reaches his limit of
validity and values are preceded by an asterisk. When the impact
approximation is not valid, the ion broadening contribution may be
estimated by using the quasi-static approach (Griem 1974; SahalBréchot 1991). In the region where none approximation is valid,
a unified-type theory should be used. For example, in Barnard,
Cooper & Smith (1974) a simple analytical formula is given for
such a case.
The difference between the widths of different fine structure
components of a multiplet is small except for 5f 2 Fo −7g 2 G and
6f 2 Fo −7g2 G multiplets for which the ratio of the widths of two
components attains 2. This large difference is due to the fact that the
upper level 7g 2 G9/2 of the considered transition and the closest perturbing level 7h 2 Ho11/2 have very close energies (
E = 240 cm−1 ).
This is not the case for the 7g 2 G7/2 , where the closest perturbing
level is more distant.
All the data given in the online Table S will be also inserted in
the STARK-B data base (Sahal-Bréchot, Dimitrijević & Moreau
2013), which is a part of Virtual Atomic and Molecular Data Center (Dubernet et al. 2010; Rixon et al. 2011), cf. also Sahal-Bréchot
(2010). This data base is devoted to diagnostics modelling and investigations of stellar atmospheres and also for fusion and laboratory
plasmas. As for diagnostics of stellar plasmas, Stark-broadening parameters are used in the determination of temperature and density
of laboratory plasmas. For example, in Hanif, Salik & Baig (2011),
spectroscopic emission of laser produced lead plasma was studied
and electron number density was determined from Stark-broadened
lines.
5.2 Systematic trends: behaviour with the principal quantum
number n along the 6s 2 S1/2 −np 2 Po1/2 series
In Fig. 8, electron-impact widths (in angular frequency units) for the
series 6s2 S1/2 −np2 Po1/2 of Pb IV are shown as a function of the principal quantum number of the upper state n, for T = 50 000 K and Ne =
1017 cm−3 . We can see a gradual increase of Stark width within the
considered spectral series. Such regular behaviour of Stark width is
the consequence of the gradual change of the energy separation between the initial (upper) level and the principal perturbing levels. As
1046
R. Hamdi et al.
No. 22637). This work is also a part of the project 176002 ‘Influence of collisional processes on astrophysical plasma line shapes’
supported by the Ministry of Education, Science and Technological
Development of Serbia.
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Figure 8. Electron-impact widths (in angular frequency units) for the series
6s 2 S1/2 −np 2 Po1/2 of Pb IV as a function of the principal quantum number of
the upper state. Dashed line: least square polynomial fitting (fourth order).
The correlation factor R2 = 1.
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S U P P O RT I N G I N F O R M AT I O N
Additional Supporting Information may be found in the online
version of this article:
Table 3. (http://mnras.oxfordjournals.org/lookup/suppl/doi:10.1093/
mnras/stt228/-/DC1).
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