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Eur. Phys. J. D 54, 51–64 (2009) DOI: 10.1140/epjd/e2009-00167-8 Regular Article T HE EUROPEAN P HYSICAL JOURNAL D Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory H. Elabidi 1, a , S. Sahal-Br´ echot 2 , and N. Ben Nessib 3 1 Groupe de Recherche en Physique Atomique et Astrophysique, Facult´ e des Sciences de Bizerte, Zarzouna 7021, Tunisia 2 LERMA, Observatoire de Paris, CNRS, Universit´ e Pierre et Marie Curie, Place Jules Janssen, 92190 Meudon, France 3 Groupe de Recherche en Physique Atomique et Astrophysique, INSAT, Centre Urbain Nord B.P. 676, 1080 Tunis, Tunisia Received 13 December 2008 / Received in final form 11 April 2009 Published online 6 June 2009 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2009 Abstract. Quantum mechanical results for the electron impact Stark widths of the 3s–3p transitions in ten Li-like ions from C IV to P XIII are carried out. The atomic structure is obtained through a scaled Thomas-Fermi-Dirac-Amaldi potential (SST numerical code) with relativistic corrections. The distorted wave method is used for the calculation of the S-Matrix, and Feshbach resonances are included by means of the Gailitis method. A comparison with other theoretical and available experimental results is done. Except for Ne VIII, we find that the agreement between our quantum results and the experiments gets better when Z increases, which is not the case for the available close-coupling quantum ones. The behavior of the Stark width with the charge Z and the electron temperature Te is also studied and in contrast to previous studies, an improved agreement with experimental Z-scaling is obtained. We show that the relative difference between widths of the two fine structure lines of the same multiplet increases with Z from 0.5% for C IV to about 12% for P XIII, proving the increasing importance of fine structure effects. The importance of the Feshbach resonances is discussed and a comparison with available semi-classical perturbation results is given. PACS. 32.70.Jz Line shapes, widths, and shifts – 34.80.Dp Atomic excitation and ionization – 95.30.Dr Atomic processes and interactions 1 Introduction Stark broadening of spectral lines plays an important role in the diagnostics and modelling of various cosmic and laboratory plasmas, in particular they can be used for the determination of electron densities N e of laboratory plasmas, for radiative transport calculations of astrophys- ical plasmas or to test various theoretical approximations. Especially, the study of lithium like transitions of ion- ized emitters is of interest in a number of fields including extreme ultraviolet laser development, the study of radia- tion transport in stellar atmospheres and for the analysis of the conditions in experiments on inertial confinement plasmas [1]. The scaling of Stark widths with the effective charge Z = z + 1 “seen” by the optical electron, where z is the charge of the ion provides an useful test of the evalua- tion for different Z values. It enables also measured Stark widths to be extrapolated to higher emitter charges along an isoelectronic sequence. The greatest difficulty in this type of study is the necessity to compare theoretical re- a e-mail: [email protected] sults to experimental ones measured at different electron temperature T e . Namely, experimental data for low-charge ions usually determined at relatively low temperatures have to be compared with the results for higher charge ions measured at much higher temperatures. Since Stark broadening parameters depend upon both, electron den- sity N e (linear dependence: excluding many particles ef- fects beyond Debye approximation), and T e one has to per- form temperature scaling whenever comparison of results measured at different temperatures is performed. Since widths W (T e ) dependencies are not yet firmly established one has to be cautious and try to determine these depen- dencies [2,3]. Since the pioneering and basic papers by Baranger [46] and by Griem et al. [7], many meth- ods have been developed for the evaluation of broadening parameters: we can cite semi-classical perturbation ones (SCP) [811], semi-empirical [12], modified semi-empirical (MSE) [13,14] and close-coupling quantum mechanical calculations [1519]. These quantum results are in contra- diction with experimental and the majority of the other theoretical ones. The first use of the convergent close coupling (CCC) method for line broadening calculations

Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

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Page 1: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

Eur. Phys. J. D 54, 51–64 (2009)DOI: 10.1140/epjd/e2009-00167-8

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Quantum Stark broadening of 3s–3p spectral lines in Li-like ions;Z-scaling and comparison with semi-classical perturbation theory

H. Elabidi1,a, S. Sahal-Brechot2, and N. Ben Nessib3

1 Groupe de Recherche en Physique Atomique et Astrophysique, Faculte des Sciences de Bizerte, Zarzouna 7021, Tunisia2 LERMA, Observatoire de Paris, CNRS, Universite Pierre et Marie Curie, Place Jules Janssen, 92190 Meudon, France3 Groupe de Recherche en Physique Atomique et Astrophysique, INSAT, Centre Urbain Nord B.P. 676, 1080 Tunis, Tunisia

Received 13 December 2008 / Received in final form 11 April 2009Published online 6 June 2009 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2009

Abstract. Quantum mechanical results for the electron impact Stark widths of the 3s–3p transitions inten Li-like ions from C IV to P XIII are carried out. The atomic structure is obtained through a scaledThomas-Fermi-Dirac-Amaldi potential (SST numerical code) with relativistic corrections. The distortedwave method is used for the calculation of the S-Matrix, and Feshbach resonances are included by meansof the Gailitis method. A comparison with other theoretical and available experimental results is done.Except for Ne VIII, we find that the agreement between our quantum results and the experiments getsbetter when Z increases, which is not the case for the available close-coupling quantum ones. The behaviorof the Stark width with the charge Z and the electron temperature Te is also studied and in contrastto previous studies, an improved agreement with experimental Z-scaling is obtained. We show that therelative difference between widths of the two fine structure lines of the same multiplet increases with Zfrom 0.5% for C IV to about 12% for P XIII, proving the increasing importance of fine structure effects.The importance of the Feshbach resonances is discussed and a comparison with available semi-classicalperturbation results is given.

PACS. 32.70.Jz Line shapes, widths, and shifts – 34.80.Dp Atomic excitation and ionization – 95.30.DrAtomic processes and interactions

1 Introduction

Stark broadening of spectral lines plays an important rolein the diagnostics and modelling of various cosmic andlaboratory plasmas, in particular they can be used forthe determination of electron densities Ne of laboratoryplasmas, for radiative transport calculations of astrophys-ical plasmas or to test various theoretical approximations.Especially, the study of lithium like transitions of ion-ized emitters is of interest in a number of fields includingextreme ultraviolet laser development, the study of radia-tion transport in stellar atmospheres and for the analysisof the conditions in experiments on inertial confinementplasmas [1].

The scaling of Stark widths with the effective chargeZ = z + 1 “seen” by the optical electron, where z is thecharge of the ion provides an useful test of the evalua-tion for different Z values. It enables also measured Starkwidths to be extrapolated to higher emitter charges alongan isoelectronic sequence. The greatest difficulty in thistype of study is the necessity to compare theoretical re-

a e-mail: [email protected]

sults to experimental ones measured at different electrontemperature Te. Namely, experimental data for low-chargeions usually determined at relatively low temperatureshave to be compared with the results for higher chargeions measured at much higher temperatures. Since Starkbroadening parameters depend upon both, electron den-sity Ne (linear dependence: excluding many particles ef-fects beyond Debye approximation), and Te one has to per-form temperature scaling whenever comparison of resultsmeasured at different temperatures is performed. Sincewidths W (Te) dependencies are not yet firmly establishedone has to be cautious and try to determine these depen-dencies [2,3].

Since the pioneering and basic papers byBaranger [4–6] and by Griem et al. [7], many meth-ods have been developed for the evaluation of broadeningparameters: we can cite semi-classical perturbation ones(SCP) [8–11], semi-empirical [12], modified semi-empirical(MSE) [13,14] and close-coupling quantum mechanicalcalculations [15–19]. These quantum results are in contra-diction with experimental and the majority of the othertheoretical ones. The first use of the convergent closecoupling (CCC) method for line broadening calculations

Page 2: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

52 The European Physical Journal D

was for the case of the B III 2s–2p resonance doublet [16]for which the experimental width [20] was found to beabout a factor of 2 larger than the CCC calculations.This method was also applied for the Be-like ions fromboron to neon and it was shown that experimental andtheoretical results only agree for low-charge ions and thedifference between experiment and theory increases withZ and can reach a factor of 2 for Ne VII [18]. The samedisagreement was found for Li-like results [19], in allcases, experimental results [2,21,22,24] are higher thanthe quantum ones, and the disagreement increases withZ. In [25], Griem and Ralchenko have suggested thatnew accurate calculations and detailed comparisons withexperiments are needed.

The present work is a continuation of a two firstones [26] and [27–29]. In [26] a new quantum mechani-cal expression for electron impact broadening calculationsfor intermediate coupling was obtained. In [27–29], weperformed the first applications for the 2s3s–2s3p andthe 3s–3p transitions respectively in Be-like and Li-likeions from carbon to neon. We used the UCL atomiccode package SUPERSTRUCTURE/DW/JAJOM. Thispackage has been used for many years and provides finestructure wavefunctions, energy levels, wavelengths, ra-diative probability rates and electron impact collisionstrengths [30–32]. In [27–29], we have extended the useof these codes to electron impact linewidths calculations.

The agreement obtained between our quantum resultsand the experimental ones encourages us to perform cal-culations for the 3s–3p doublet lines of Li-like ions fromC IV to P XIII. This is the aim of this paper. Since ourresults present the best agreement with experiments forhighly charged ions, especially for Ne VIII (contrary toother theoretical treatments), it will be possible to deter-mine the temperature dependence of the linewidths withmore confidence and use it for the Z-scaling.

2 Theory and computational procedure

2.1 Theory

The basic formula for the electron impact broadening ofa spectral line corresponding to a transition i → f is thegeneral formula of Baranger [6]:

2w = Ne

∞∫

0

vfM (v)dv

×( ∑

i′ �=i

σii′ (v)+∑f ′ �=f

σff ′(v)+∫

|fi(θ, v) − ff (θ, v)|2 dΩ

)

(1)

where Ne is the electron density, v the velocity of thescattered electron and fM (v) the Maxwellian electron ve-locity distribution. σii′ (σff ′) are inelastic cross sectionsconnecting the initial (final) level with other perturbinglevels denoted by primes. The fi(θ, v) and ff (θ, v) are elas-tic scattering amplitudes for the target ion in the initial

and final levels respectively, and the integral is performedover the scattering angle θ, with dΩ being the element ofsolid angle. The formulation of the quantum-mechanicalexpression for electron impact broadening for intermedi-ate coupling is described in [26]. The expression obtainedin [26] for the width of an isolated line (Ji = Ji′ andJf = Jf ′) is:

w = π

(�

m

)2

Ne

×∑

JTi JT

f ll′KiKf Ki′Kf′

(−1)2Ji+Ki+Ki′+Kf+Kf′+2JTf +l+l′+1

× [Ki, Kf , Ki′ , Kf ′ ]12

[JT

f , JTi

]

2

{JiKilKfJf1

}

×{

JiKi′ l′

Kf ′Jf1

}{KiJ

Ti s

JTf Kf1

}{Ki′J

Ti s

JTf Kf ′1

}

×∫ ∞

0

fM (v)v

dv{δl′lδKi′KiδKf′Kf

−Re[S∗IC

F (Δf ′Jf l′Kf ′sJTf ; ΔfJf lKfsJT

f )

× SICI (Δi′Jil

′Ki′sJTi ; ΔiJilKisJ

Ti )

] }. (2)

This formula takes into account fine structure effects andrelativistic corrections resulting from the breakdown ofthe LS coupling approximation for the target. w is thehalf width at half maximum (HWHM) and:

−→Li +

−→Si =

−→Ji

−→Ji +

−→l =

−→Ki

−→Ki + −→s =

−→JT

i (3)

where L and S represent the atomic orbital angular mo-mentum and spin of the target, l and l′ are the electronorbital momentum before and after collision, the super-script T denotes the quantum numbers of the total elec-tron+ion system. SIC

I (Δi′Jil′Ki′sJ

Ti ; ΔiJilKisJ

Ti ) is the

scattering matrix element for the initial level, denotedas I, expressed in the intermediate coupling approxima-tion, the same definition as for the complex conjugateS∗IC

F (Δf ′Jf l′Kf ′sJTf ; ΔfJf lKfsJT

f ) of the scattering ma-trix S for the final level, denoted as F , Re (. . .) is the real

part of (. . . ),{

abcdef

}represent 6–j symbols and we adopt

the notation [x, y, . . .] = (2x + 1)(2y + 1). . . Both SI andSF are calculated for the same incident electron energyE = mv2/2.

We will use the distorted wave approximation withEissner’s [33] method and computer code. In fact a ba-sic approximation of Eissner’s code consists in neglectingl �= l′. Consequently the contribution of l �= l′ is zero whenusing this code. In addition, we will neglect Ki �= Ki′ andKf �= Kf ′ , which is valid for the high values of l whichplay the most important role in highly charged ions. Thus

Page 3: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

H. Elabidi et al.: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions 53

we obtain the following expression for the linewidth whichwill be used throughout our calculations:

w = Ne

(�

m

)2 (2mπ

kBTe

) 12

×∞∫

0

Γw (E) exp(− E

kBTe

)d

(E

kBTe

)(4)

where kB is the Boltzmann constant, Te is the electrontemperature and

Γw(E) =∑

JTi JT

f lKiKf

[Ki, Kf , JT

f , JTi

]

2

{JiKilKfJf1

}2{KiJ

Ti s

JTf Kf1

}2

× [1−(

Re (SICI )Re (SIC

F )+Im(SICI )Im (SIC

F ))]

.(5)

Note that the approximation l = l′ is also at the basis ofthe classical path approximation.

2.2 Computational procedure

The atomic data have been computed using a com-puter package developed at University College, London(UCL). The SUPERSTRUCTURE (SST) program [34]calculates energy levels and radiative data taking intoaccount configuration interaction. The radial functionsare calculated assuming a scaled Thomas-Fermi-Dirac-Amaldi (TFDA) potential. Relativistic corrections (spin-orbit, mass, Darwin and one-body) are introduced accord-ing to the Breit-Pauli approach [35] as a perturbation tothe non-relativistic Hamiltonian.

The scattering problem is treated in the non-relativistic distorted wave approximation using the UCLdistorted wave (DW) program of Eissner [33]. The reac-tance matrices R are calculated in LS coupling. The UCLprogram JAJOM [36] uses the term coupling coefficients(TCC) calculated by SST and these matrices to producecollision strengths in intermediate coupling. R-matrices inintermediate coupling and real (ReS) and imaginary part(ImS) of the scattering matrix S have been calculated us-ing the transformed version of JAJOM (JAJPOLARI [37])and the program RtoS [38] respectively. The evaluation ofReS and ImS is done according to:

ReS =(1 − R2

) (1 + R2

)−1, (6)

andImS = 2R

(1 + R2

)−1. (7)

Thanks to the relation S = (1 + iR)(1 − iR)−1, the S-matrix is unitary. The integral in equation (4) is evaluatednumerically using the trapezoid method with a variablestep of electron energy E.

The Feshbach resonances were not included in [27–29].They are included in the present paper by means of theGailitis formula [39]. We have extrapolated the factorΓw(E) from equation (5) that contains the scattering ma-trix S below the threshold for the corresponding inelasticprocess.

3 Comparison of the present resultswith available experiments

In the following subsections, we present electron impactfull widths at half maximum (FWHM) in A (W = 2w) forthe 3s–3p doublets lines of lithium like ions from C IVto P XIII. Our results are compared to the measuredwidths obtained by the two experimental groups fromBelgrade, Yugoslavia: [2] from the Institute of physicsand [24] from the Faculty of Physics and the “Bochumgroup” [21–23,40] from the Ruhr Universitat in Bochum,Germany. The two first groups used a low-pressure pulsedarc and the second used a gas-liner pinch. Note that themeasurements of the Bochum group were carried out onlyfor the 3s 2S1/2–3p 2Po

3/2 and for electron temperaturesand densities higher than those used by the two groups ofBelgrade. The results of Bottcher et al. [40] were carriedout only for the C IV, N V and O VI ions and only atone temperature value 14.51 × 104 K (12.5 eV) and onedensity value 1.8×1018 cm−3, but in the following we willnot consider these results since they were rectified fouryears later by the same group [21]. The accuracy of theexperimental widths is most often given by [41] except fora few cases where another reference is provided. For eachion, our calculations are made in the range of electrontemperatures corresponding to the available experimentalresults. For the cases where there are no comparison withexperiments, we perform calculations for electron temper-atures from 105 K to 20 × 105 K and for an electron den-sity 1018 cm−3.

3.1 Results for C IV

The linewidths of this ion were measured for lower tem-peratures by Sreckovic et al. [24], for intermediate temper-atures by Blagojevic et al. [2] and for much higher tem-peratures by Glenzer et al. [21]. In Table 1, we reportour quantum results W , the experimental Wm and othertheoretical linewidths: CCC for convergent close couplingcalculations of Ralchenko et al. [19], SCP: for the calcu-lations within the semi-classical perturbation formalism(Sahal-Brechot [8,9,42,43]), updated by Dimitrijevic andSahal-Brechot [44,45] performed in [2,41], G: for the sim-plified semi-classical equation (526) of [10] used for cal-culations in [2,21], MSE: for the modified semi-empiricalformula of Dimitrijevic and Konjevic [13,14] used for cal-culations in [2,21] ([21] is for the two last temperatures8.13 and 9.98 × 105 K), S: for the quantum mechani-cal calculations after Seaton [15], HB: for calculationsby Hey in [46] based on the semi-classical Gaunt factorapproximation developed in [47–50] and A: for the nonperturbative semi-classical method by Alexiou [51]. Semi-classical calculations for C IV have been performed in [52]but comparisons have been done with some older experi-ments ([53–55]. . . ), so we don’t include these data and weuse only the most recent semi-classical calculations. Theexperimental widths are determined with an error of about23−30% [41]. For all cases, experimental results are lower

Page 4: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

54 The European Physical Journal D

Table 1. Stark widths in A for the C IV 3s 2S−3p 2Po transitions. Present quantum calculations W and other theoreticalresults (CCC: quantum calculations of Ralchenko et al. [19], SCP: calculations within the semi-classical perturbation formalism(Sahal-Brechot [8,9,42,43]), updated by Dimitrijevic and Sahal-Brechot [44,45] performed in [2,41], G: simplified semi-classicalequation (526) of [10] used for calculations in [2,21], MSE: modified semi-empirical formula of Dimitrijevic and Konjevic [13,14]used for calculations in [2,21], S: quantum mechanical calculations after Seaton [15], HB: calculations by Hey in [46] and A: nonperturbative semi-classical method by Alexiou [51]) are compared with experimental widths Wm ([24]a, [2]b, [21]c). Temperaturevalues are given in 104 K and density values in 1017 cm−3.

Transition Te Ne Wm W WmW

WmWCCC

WmWSCP

WmWG

WmWMSE

WmWS

WmWHB

WmWA

2S1/2−2Po3/2 1.57 1.45 1.138a 1.467 0.78

5801.31 A 1.70 1.87 1.296a 1.819 0.71

1.78 1.96 1.317a 1.862 0.71

1.83 1.82 1.264a 1.706 0.74

1.90 1.66 1.196a 1.530 0.78 0.80

7.24 0.58 0.244b 0.348 0.70 1.16 0.84 0.80 1.06 1.22

7.83 0.76 0.313b 0.446 0.70 1.16 0.84 0.80 1.05 1.23

8.01 0.8 0.329b 0.467 0.70 1.19 0.85 0.80 1.06 1.24

8.13 15 6.70c 8.709 0.77 1.28 0.92 0.90 1.19 1.19 0.93

9.98 24 9.70c 13.127 0.74 1.24 0.86 0.88 1.15 1.14 0.92

2S1/2−2Po1/2 1.57 1.45 0.964a 1.476 0.65

5811.97 A 1.70 1.87 1.154a 1.829 0.63

1.78 1.96 1.084a 1.872 0.58

1.83 1.82 1.074a 1.716 0.63

1.90 1.66 1.042a 1.539 0.68 0.70

1.95 1.44 0.762a 1.319 0.58 0.59

1.98 1.37 0.735a 1.245 0.59 0.60

2.03 1.25 0.694a 1.123 0.62 0.62

7.24 0.58 0.229b 0.349 0.66 1.09 0.79 0.75 0.99 1.14

7.83 0.76 0.304b 0.448 0.68 1.11 0.82 0.77 1.02 1.19

than our calculated ones. The averaged ratio of the exper-imental results to our quantum ones (Wm/W ) is about0.68, this ratio is about 1.16 for the CCC quantum re-sults and 1.23 for the quantum calculations of Seaton [15].These quantum results are closer to the experiments thanours. The semi-classical and the modified semi-empiricalcalculations show a better agreement with experimentalresults.

Figures 1 and 2 display our quantum, experimentaland the other theoretical results for the C IV 3s 2S1/2–3p 2Po

3/2 and 3s 2S1/2–3p 2Po1/2 linewidths. Our quantum

results are higher than all other theoretical ones, espe-cially for low temperatures, and at higher temperaturesthe difference becomes lower. This is due to the effect ofresonances which is important at low temperatures.

3.2 Results for N V

The linewidths of N V 3s–3p transitions are measuredby all the experimental groups cited above [2,21,40]. Itis noted in [41] that for this ion, error in the experimentaldetermination of widths is 23−30%. We present in Table 2and in Figures 3 and 4 our quantum linewidths W com-pared to the same experimental [2,21] and other theoreti-cal results as those of Table 1. The averaged ratio Wm/W

0 2 4 6 8 10 12 14 16 18 20 22 24

0

1

2

3

4

5

6

7

8

9

10

11

Te(10

4K)

W(A

˚)

Fig. 1. Stark width W as a function of the electron temper-ature for the C IV 3s 2S1/2–3p 2Po

3/2 transition at an elec-

tron density Ne = 1018 cm−3. Present work: ◦, CCC [19]:•, SCP [2,41]: �, S [15]: �. Experimental values: [24]: ∗, [2]:×, [21]: +. Notations are the same as those of Table 1.

is 0.71 for our calculations and 1.5 for the CCC ones.The semi-classical results of Griem and of Alexiou arein good agreement and give an averaged ratio Wm/WA,G

Page 5: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

H. Elabidi et al.: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions 55

0 2 4 6 8 10 12 14 16 18 20 22 24

0

1

2

3

4

5

6

7

8

9

10

11

Te(10

4K)

W(A

˚)

Fig. 2. Same as in Figure 1 but for the C IV 3s 2S1/2–3p 2Po1/2

transition.

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0

1

2

3

4

5

Te(10

4K)

W(A

˚)

Fig. 3. Same as in Figure 1 but for the N V 3s 2S1/2−3p 2Po3/2

transition. SCP [2,41,56]: �, MSE [2,21]: �. Experimental val-ues: [2]: × and [21]: +.

2 4 6 8 10 12 14 16 18 20 22 24

0

1

2

3

4

5

Te(10

4K)

W(A

˚)

Fig. 4. Same as in Figure 3 but for the N V 3s 2S1/2−3p 2Po1/2

transition.

0 2 4 6 8 10 12 14 16 18 20 22 24 26

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

Te(10

4K)

W(A

˚)

Fig. 5. Same as in Figure 1 but for the O VI 3s 2S1/2−3p 2Po3/2

transition. SCP [2,41,57]: �.

0 2 4 6 8 10 12 14 16 18 20 22 24

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

Te(10

4K)

W(A

˚)

Fig. 6. Same as in Figure 5 but for the O VI 3s 2S1/2−3p 2Po1/2

transition.

about 1.06. Only our results and the semi-classical onesof Dimitrijevic and Sahal-Brechot [56] overestimate theN V linewidths. The modified semi-empirical method ofDimitrijevic and Konjevic and the semi-classical Gauntfactor approximation of Hey agree well with each other.

3.3 Results for O VI

The errors of measurements of this ion are about 23%.Comparison between our results and the measured onesshows that the ratio Wm/W is about 0.83 and we notethat only our values are larger than the experimental ones.Table 3 and Figures 5 and 6 show that our results arevery close to the experiments and that the modified semi-empirical calculations agree with the CCC ones. The ex-perimental results exceed the CCC calculations by about

Page 6: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

56 The European Physical Journal D

Table 2. Same as in Table 1 but for the N V 3s 2S−3p 2Po transitions. SCP calculations: [2,41,56] and experimental resultsWm: [2]b, [21]c.

Transition Te Ne Wm W WmW

WmWCCC

WmWSCP

WmWG

WmWMSE

WmWHB

WmWA

2S1/2−2Po3/2 7.24 0.86 0.186b 0.291 0.64 1.49 0.90 1.05 1.32

4603.74 A 7.83 0.99 0.210b 0.323 0.65 0.91 1.05 1.32

8.01 1.13 0.234b 0.364 0.64 0.90 1.03 1.30

8.07 1.21 0.243b 0.387 0.63 1.42 0.88 1.00 1.26

8.23 1.33 0.282b 0.422 0.67 0.94 1.06 1.34

17.3 12 2.2c 2.667 0.82 1.61 1.09 1.18 1.51 1.37 1.03

21.7 16 2.7c 3.182 0.85 1.54 1.10 1.12 1.40 1.29 1.03

25.3 20 3.4c 3.684 0.92 1.16 1.10 1.39 1.26 1.09

27.7 23 3.8c 4.040 0.94 1.16 1.12 1.42 1.28 1.10

2S1/2−2Po1/2 7.24 0.86 0.186b 0.294 0.63 1.49 0.90 1.05 1.32

4619.97 A 7.83 0.99 0.204b 0.325 0.63 0.89 1.02 1.29

8.01 1.13 0.231b 0.367 0.63 0.89 1.02 1.28

8.07 1.21 0.249b 0.391 0.64 1.42 0.90 1.03 1.29

8.23 1.33 0.282b 0.426 0.66 0.94 1.06 1.34

Table 3. Same as in Table 1 but for the O VI 3s 2S−3p 2Po transitions. SCP calculations: [2,41,57] and experimental resultsWm: [2]b, [21]c.

Transition Te Ne Wm W WmW

WmWCCC

WmWSCP

WmWG

WmWMSE

WmWHB

WmWA

2S1/2−2Po3/2 6.19 1.38 0.178b 0.213 0.84 1.66 1.01 1.27 1.55

3811.35 A 6.55 1.09 0.136b 0.165 0.82 1.62 1.00 1.25 1.53

7.97 1.42 0.171b 0.200 0.86 1.69 1.05 1.28 1.56

9.63 10 1.00c 1.313 0.76 1.49 0.92 1.21 1.42 1.28 0.97

13.34 13 1.40c 1.505 0.93 1.79 1.11 1.38 1.64 1.44 1.17

18.1 21 1.80c 2.151 0.84 1.58 1.01 1.16 1.37 1.21 1.03

20.31 24 2.10c 2.341 0.90 1.65 1.07 1.26 1.50 1.32 1.10

2S1/2−2Po1/2 6.19 1.38 0.176b 0.215 0.82 1.64 1.00 1.26 1.53

3834.24 A 6.55 1.09 0.132b 0.167 0.79 1.57 0.97 1.21 1.48

7.97 1.42 0.178b 0.203 0.88 1.76 1.09 1.33 1.64

65%. The semi-classical calculations of Dimitrijevic andSahal-Brechot and Alexiou agree well with experiments.

3.4 Results for F VII

Measurements of F VII linewidths were performed by theBochum group [22] with an error of about 23%. We pointout from Table 4 (same notation as in Tab. 1) and Fig-ures 7 and 8 that discrepancies between our results andthe experimental ones become very small and all theoret-ical calculations underestimate the linewidths except ourquantum results. We note that we obtain in this case thebest agreement with Wm/W = 0.96. The measured widthsexceed the CCC ones by about 87%. Good agreement withexperiments is also obtained for the semi-classical calcu-lations of Dimitrijevic and Sahal-Brechot (Wm/W = 1.1).

3.5 Results for Ne VIII

Experimental determinations of the Ne VIII linewidthshave been performed only for the 3s 2S1/2−3p 2Po

3/2

transition and only by the Bochum group: by Glenzeret al. [21] for the two temperatures (34.47 and 49.32) ×104 K and more recently by Hegazy et al. [23] for 46.40 ×104 K. We only discuss the recent results [23], becausethose in [21] have been discarded from discussion in [23].The errors in the measurements are about 30% [41]. Wepresent in Table 5 and Figure 9 our results comparedto other theoretical and to the above mentioned exper-imental ones (we have not plotted the 3s 2S1/2−3p 2Po

1/2

linewidths because there are no experimental results forit). We obtain also in this case the best agreement withexperimental results. It is shown in Table 5 and Figure 9

Page 7: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

H. Elabidi et al.: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions 57

Table 4. Stark widths in A for the F VII 3s 2S−3p 2Po transitions as a function of temperature at density Ne = 1018 cm−3.CCC: quantum calculations of Ralchenko et al. [19], SCP: semi-classical calculations [58]; G, MSE and HB calculations arefrom [22]; A: results from [51] and experimental results Wm are from [22]. Temperature values are given in 104 K and densityvalues in 1017 cm−3.

Transition Te Ne Wm W WmW

WmWCCC

WmWSCP

WmWG

WmWMSE

WmWHB

WmWA

2S1/2−2Po3/2 16.71 15.7 0.87 0.912 0.95 1.88 1.06 1.51 1.75 1.36 1.27

3246.56 A 19.26 21.0 1.11 1.147 0.97 1.87 1.09 1.48 1.71 1.34 1.28

21.47 29.2 1.49 1.519 0.98 1.86 1.11 1.51 1.69 1.35 1.27

2S1/2−2Po1/2 16.71 15.7 0.87 0.929 0.94 1.88 1.08

3276.99 A 19.26 21.0 1.11 1.168 0.95 1.87 1.11

21.47 29.2 1.49 1.548 0.96 1.86 1.14

0 2 4 6 8 10 12 14 16 18 20 22 24

0,0

0,2

0,4

0,6

0,8

1,0

1,2

Te(10

4K)

W(A

˚)

Fig. 7. Stark widths for the F VII 3s 2S1/2−3p 2Po3/2 transi-

tion as a function of temperature at density Ne = 1018 cm−3.Present work: ◦, CCC [19]: •, SCP [58]: �, G [22]: �, MSE [22]:�, HB [22]: �, A [51]: �. Experimental results [22]: +.

0 2 4 6 8 10 12 14 16 18 20 22 24

0,0

0,2

0,4

0,6

0,8

1,0

1,2

Te(10

4K)

W(A

˚)

Fig. 8. Same as in Figure 7 but for the F VII 3s 2S1/2−3p 2Po1/2

transition.

Table 5. Stark widths in A for the Ne VIII 3s 2S1/2−3p 2Po3/2

transition. Present quantum calculations W , CCC: quantumcalculations of Ralchenko et al. [19], SCP: semi-classical calcu-lations [59,60], EDK: modified semi-empirical calculations [61]are compared to the experimental widths Wm: [23] at tem-perature value Te = 46.40 × 104 K and density value Ne =1018 cm−3.

Transition Wm W WmW

WmWCCC

WmWSCP

WmWEDK

2820.70 A 0.29 0.212 1.37 2.3 1.38 1.84

0 5 10 15 20 25 30 35 40 45 50 55 60

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

Te(10

4K)

W(A

˚)

Fig. 9. Stark widths for the Ne VIII 3s 2S1/2−3p 2Po3/2 tran-

sition as a function of temperature at density Ne = 1018 cm−3.Present work: ◦, CCC [19]: •, SCP (present calculations andin [59,60]): �, EDK [61]: �. Experimental widths [23]: �.

that, only for this case, our quantum results are smallerthan the measured ones, and the ratio Wm/W is 1.37. TheCCC results of Ralchenko et al. are a factor 2.3 smallerthan those of Hegazy et al. [23]. The semi-classical widthscalculated in this paper agree well with our quantum ones.

The ratio of the calculations in [61] based on theinclusion of accurate collision strengths in the modifiedsemi-empirical method to the experimental results ofHegazy et al. [23] is 1.84.

At the end of our comparison with experience we findthat in all cases studied, our quantum results are higher

Page 8: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

58 The European Physical Journal D

0 2 4 6 8 10 12 14 16 18 20 22

0,00

0,04

0,08

0,12

0,16

0,20

0,24

0,28

0,32

Te(10

5K)

W(A

˚)

Fig. 10. Stark widths for the Na IX 3s 2S1/2−3p 2Po3/2

(solid line) and for the 3s 2S1/2−3p 2Po1/2 (dashed line) transi-

tions as a function of electron temperature at electron densityNe = 1018 cm−3. Present quantum results: ◦ and semi-classicalresults SCP: � (some SCP results are from [59,60]).

than the measured ones except the case of Ne VIII. Twoimportant remarks have to be noted: on the one hand, inthis isoelectronic sequence, the CCC quantum results ofRalchenko et al. are closer to the experiments than ourones only for the lowest Z element (C IV), and for theother cases our calculations give more satisfactory resultsthan the CCC ones. On the other hand, from C IV toF VII, the agreement between experiments and our re-sults is improved progressively with the increase of Z (seeSect. 4.3).

3.6 Results for Na IX to P XIII

For the Na IX results, there are the semi-classical calcula-tions (SCP) that have been performed by Dimitrijevic andSahal-Brechot in [59,60] and the present SCP results forelectron temperatures in the range 2 × 104 K−2 × 106 Kand electron densities 1016−1022 cm−3 (the present resultsare only at Ne = 1018 cm−3). In Table 6 and Figure 10, wereport our quantum widths compared to the semi-classicalones for the Na IX 3s 2S−3p 2Po transitions. The semi-classical results are higher than our ones especially at hightemperatures.

Several calculations and measurements have been car-ried out for the four other ions (Mg X, Al XI, Si XIIand P XIII) but not for the transitions considered in thepresent paper. The SCP linewidths of 7 Al XI and 9 Si XIImultiplets have been calculated by Dimitrijevic and Sahal-Brechot in [62]. The widths of the 2p−3d line of Si XII andof the 2s−3p, 2p−3d, 2s−4p, 2p−4d, 2s−5p, 2p−5d linesof Mg X have been calculated by Fill and Schoning in [63]using a quantum mechanical method for the calculationof complete Stark profiles for multielectron ions. A moredetailed description of this method can be found in [64].A Measurements of 3d−5f line profiles for Mg X, Al XI,P XIII and Cl XV have been reported in [65].

0 2 4 6 8 10 12 14 16 18 20 22

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0,20

Te(10

5K)

W(A

˚)

Fig. 11. Stark widths for the Mg X 3s 2S1/2−3p 2Po3/2 (solid

line) and for the 3s 2S1/2−3p 2Po1/2 (dashed line) transi-

tions as a function of electron temperature at electron den-sity Ne = 1018 cm−3. Present quantum results: ◦ and presentsemi-classical results SCP: �.

0 2 4 6 8 10 12 14 16 18 20 22

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

Te(10

5K)

W(A

˚)

Fig. 12. Same as in Figure 11 but for the Al XI 3s 2S−3p 2Po

transitions.

In addition, in the present work, by using the Sahal-Brechot’s computer code with our SST atomic structure,we have calculated the missing SCP line widths for theMg X, Al XI, Si XII and P XIII 3s−3p transitions. Com-parison with all these semi-classical results are reportedin Table 6 and Figures 11−14. All SCP calculations arehigher than our quantum results, the relative differencebetween the two approaches increases with Z from 12%for Mg X to 34% for P XIII.

4 Discussions

4.1 Comparison with SCP and CCC results

We can see from the previous figures (Figs. 1−9) thatat high temperatures, our calculations become closer tothe semi-classical and the CCC ones than at low tem-peratures. Since short range effects decrease at high tem-perature, quantum DW and CCC results become closer,

Page 9: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

H. Elabidi et al.: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions 59

Table 6. Present Stark widths W in A for the 3s 2S−3p 2Po transitions of ions from Na IX to P XIII compared to semi-classicalperturbation calculations WSCP (values designed by ∗ are from [59,60] and the other are calculated in this work). Temperaturesvalues are given in 105 K and electron density is 1018 cm−3.

Na IX Mg X Al XI Si XII P XIIITe W WSCP W WSCP W WSCP W WSCP W WSCP

2S1/2−2Po3/2

1 0.281 0.261 0.181 0.182 0.124 0.131 0.079 0.097 0.051 0.0732 0.200 0.195∗ 0.131 0.133 0.090 0.095 0.059 0.070 0.039 0.0535 0.129 0.134∗ 0.084 0.091 0.059 0.064 0.039 0.047 0.027 0.03510 0.090 0.104∗ 0.059 0.070 0.042 0.049 0.028 0.036 0.020 0.02620 0.061 0.082∗ 0.041 0.054 0.029 0.038 0.020 0.027 0.015 0.020

2S1/2−2Po1/2

1 0.292 0.271 0.193 0.191 0.129 0.140 0.086 0.106 0.057 0.0822 0.208 0.200 0.139 0.140 0.095 0.102 0.064 0.076 0.044 0.0595 0.135 0.138 0.089 0.095 0.063 0.069 0.042 0.051 0.030 0.03910 0.094 0.107 0.063 0.073 0.045 0.052 0.031 0.039 0.023 0.02920 0.063 0.084 0.043 0.057 0.031 0.041 0.021 0.030 0.016 0.022

0 2 4 6 8 10 12 14 16 18 20 220,00

0,02

0,04

0,06

0,08

0,10

0,12

Te(10

5K)

W(A

˚)

Fig. 13. Same as in Figure 11 but for the Si XII 3s 2S−3p 2Po

transitions.

0 2 4 6 8 10 12 14 16 18 20 22

0,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

Te(10

5K)

W(A

˚)

Fig. 14. Same as in Figure 11 but for the P XIII 3s 2S−3p 2Po

transitions.

such as SCP results. This confirms the fact that the semi-classical perturbation approximation is valid at high en-ergies.

4.2 Contribution of resonances

We have shown that the contribution of resonances de-creases when the electron temperature Te increases. Wenotice the effect of resonances at low Te by comparing thebehavior of our linewidths versus temperature in Be-likeions [28] where we did not include resonances and thepresent calculations. Thanks to the inclusion of Feshbachresonances, our linewidths now decrease with Te, as ex-pected by the basic theory of ion-electron collisions. Werecall that the contribution of resonances in our calcula-tions is probably overestimated for low-charged ions. Infact it is known that the Gailitis formula overestimatesthe average over Feshbach resonances for low-charged ions.This was already noticed by Fleurier et al. [42] who intro-duced the semi-classical limit of the Gailitis formula in theSCP expression of the ion electron impact width.

4.3 Comparison with experimental results versus Z

In Figure 15 we display the ratio (Wm/W ) of each experi-mental result to the correspondent calculated value by ourmethod versus the effective charge Z, we display experi-mental results for all temperatures of measurement. Thisratio increases with Z and get closer to unit for the caseof F VII. All the experimental results are lower than thepresent quantum ones (except the case of Ne VIII). Be-cause in [21] the experimental results are higher than thetheoretical ones, Glenzer et al. suggested that the increas-ing of this discrepancy with Z could indicate an increas-ing in the contribution of ion quadrupole broadening athigher temperature for Z > 4. In our case, the experimen-tal results become higher than the theoretical ones only for

Page 10: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

60 The European Physical Journal D

4 5 6 7 8

0

1

2

Z

Wm/W

Fig. 15. Ratio of the experimental to the present quantumresults for the 3s 2S1/2−3p 2Po

3/2 transition versus the effec-tive charge Z. Experimental results: +: Glenzer et al. [21,22],×: Blagojevic et al. [2], ∗: Sreckovic et al. [24], �: Hegazyet al. [23].

4 5 6 7 80

1

2

3

0

1

2

3

Ne VIIIN V O VI F VIIC IV

Z

Wm/W

th

Fig. 16. Averaged ratio of experimental [2,21–24] to availabletheoretical results for the 3s 2S1/2−3p 2P0

3/2 transition ver-sus the effective charge Z. ◦: present quantum calculations, •:quantum (CCC) calculations [19], �: calculations within thesemi-classical perturbation formalism [2,41,56–60], �: simpli-fied semi-classical calculations based on equation (526) of [10]performed in [2,21,22], �: modified semi-empirical calcula-tions [13,14] performed in [2,21,22], �: quantum calculationsafter Seaton [15], �: calculations of Hey and Breger [46] ([22]for the F VII), �: calculations by Alexiou [51].

Z = 8 (the case of Ne VIII) and unfortunately, there areno experimental results beyond Ne VIII. So Stark widthmeasurements for ions with Z > 8 would be very welcometo conclude about the limit of Z for which ion quadrupolecontribution becomes important.

We show in Figure 16 the averaged ratio of the ex-perimental results [2,21–24] to all theoretical ones. Weremark that for all theoretical calculations considered in

this work, the averaged ratio of experimental to theoreti-cal results Wm/Wth increases with the effective charge Zof the emitter and diverges at high Z except our calcula-tions that match the experimental data better than all theother theoretical predictions. For our calculations, the ra-tio Wm/W increases to be around unit (0.68 for C IV, 0.71for N V, 0.84 for O VI and 0.96 for F VII). For the case ofNe VIII, all the theoretical results are smaller than the ex-perimental ones, furthermore we have the best agreementwith the experiments, the ratio Wm/W is about 1.37. Butit increases from F VII to Ne VIII faster than from C IV toF VII. Maybe, the measurements of Hegazy et al. overesti-mate the Ne VIII linewidths. We notice that the measure-ment of Ne VIII linewidth was made by Glenzer et al. [21]and the same experience was re-made later (2003) withsame equipment by Hegazy et al. In this last experience,the new results found were lower. It seems that our cal-culations become more close to the experiments at highZ, but regarding the case of Ne VIII (that it shows thebiggest difference with experiments), it is difficult to reachconclusion about this discussion. So it would be very im-portant to perform measurements of the 3s−3p linewidthsstudied here for some Li-like ions with Z ≥ 9 (Na IX,Mg X, Al XI...). Such measurements will be very usefulfor testing our calculations and to better see the trend ofthe ratio Wm/W versus Z. With only the present data,and with the fact that the CCC calculations give betteragreement with experiments than our ones only for thecase of C IV, we can conclude with caution that our cal-culations are more adapted for highly charged ions. Thisis expected, since the accuracy of the SST and the DWapproximations increases when Z increases.

4.4 Fine structure effects

For multicharged ions, relativistic effects must be takeninto account in atomic structure calculations. In fact, forhigh ionization states of ions, LS coupling breaks downwhen calculating collision strengths in electron-ion scat-tering [66,67]. Therefore, the widths of the fine structurecomponents can be different. Jones [68] distinguished twodistinct types of relativistic corrections:

(a) Relativistic corrections resulting from the motionof the colliding electron and its interaction with the target.

(b) Relativistic corrections resulting from breakdownof the LS approximation for the target.

On the one hand, Walker [69] showed that the (a) cor-rections contributed less than 10% to the cross sections forthe target charge Z ≤ 25. Thus these corrections can beneglected for the ions which are of interest in the presentwork. On the other hand, Jones [66] and further papersshowed that the (b) corrections were essential. In fact,owing to the high electron velocity, the relativistic partof the Hamiltonian of the target can be neglected dur-ing the collision process. Therefore the reactance matri-ces can be calculated in LS coupling in the first stage.In the second stage, the collision strengths for intermedi-ate coupling are calculated using these reactance matricesand term coupling coefficients (TCC) obtained for atomic

Page 11: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

H. Elabidi et al.: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions 61

0 2 4 6 8 10 12 14 16 18 20 22 24

0

2

4

6

8

10

Te(10

4K)

W(A

˚)

Fig. 17. Present Stark width W as a function of the elec-tron temperature for the C IV 3s 2S1/2−3p 2Po

3/2 (solid line)

and 3s 2S1/2−3p 2Po1/2 transitions (dashed line) at an electron

density Ne = 1018 cm−3.

structure calculations including relativistic effects. A sim-ilar method has been extensively developed and used fornon-LTE calculations of level populations and line inten-sities for the multicharged ions that are observed in thesolar corona for instance, or in astrophysical objects in theX band of wavelengths; see for example CHIANTI [70,71]and references therein.

We check the assumption that for high ionizationstates of ions, the widths of the fine structure compo-nents can be different by giving in Figures 17 and 18 thewidths of the two fine structure lines 3s 2S1/2−3p 2Po

3/2

and 3s 2S1/2−3p 2Po1/2 of C IV and P XIII respectively. It

is clear that fine structure effects increase with the emit-ter charge Z. The relative difference between the widthsof the two fine structure components increases from 0.5%for C IV to about 12% for P XIII. This difference can bemore important for much higher ionization states of ions,this will be the aim of the future works in which, ions thathave high ionization states will be studied.

4.5 Behavior of the width with Te and Z-scaling

The variation of the linewidths with electron tempera-ture Te for the range of temperatures from 2 to 20 eVis given approximatively by: W ∝ T−α

e with α takes thefollowing values for each ion: α = 0.32 for C IV, 0.55for N V, 0.37 for O VI, 0.36 for F VII, 0.35 for Ne VIII,0.44 for Na IX, 0.41 for Mg X, 0.36 for Al XI, 0.33 forSi XII and 0.48 for P XIII which give an averaged value(α)av of about 0.40. The corresponding variation given byRalchenko et al. in [19] gave a values of α smaller thanour ones, and the CCC-averaged value (αCCC)av is about0.26. Our (α)av show a Te-dependence close to that oftenused in the literature.

0 2 4 6 8 10 12 14 16 18 20 22

0,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

Te(10

5K)

W(A

˚)

Fig. 18. Same as in Figure 17 but for the P XIII3s 2S1/2−3p 2Po

3/2 and 3s 2S1/2−3p 2Po1/2 transitions.

In several precedent works, Z-scaling have been stud-ied for Li-like ions: In [21], Glenzer et al. have used thetemperature dependence of the Stark widths as deter-mined by the choice of effective Gaunt factor to scale alltheir experimental results at the same electron tempera-ture (12.5 eV), the considered ions are C IV, N V, O VIand Ne VIII. They have calculated theoretical widths forthis value of temperature. They found that theoretical ap-proaches [10,13,15,46] show nearly a Z−2 dependence ofthe Stark width in contrast to the experimental resultswhich did not show this scaling. In [2], Blagojevic et al.studied the Li-like ions: B III, C IV, N V, and O VI. Theyscaled their experimental widths to the electron tempera-ture of 7.5 eV (8.7×104 K) using the Te-dependence fromtheir semi-classical calculations (Table I in Ref. [2]). Theyfound that: (i) the agreement between experiments andtheir semi-classical calculations is good except for ionswith the highest and lowest Z (Ne VIII and B III) and(ii) the inclusion of ion broadening does not influence thisagreement to a large extent. Recently, Hegazy et al. [23]studied the 3s 2S1/2−3p 2Po

3/2 transition in Ne VIII, to in-vestigate the electron density dependence of Stark widths,they have plotted widths against temperature in the range20−70 eV, but they found that no clear temperature de-pendence was obvious. So they scaled all widths to a tem-perature of 40 eV employing a temperature scaling derivedfrom the quantum-mechanical calculations of Ralchenkoet al. [19] (i.e. (αCCC)av = 0.26). To study the Z-scaling,Hegazy et al. [23] have applied the above temperature scal-ing to the original data of the other ions and have shownthat widths are related to the ions charge Z by a simplepower law: W ∝ Z−1.13.

Since our results are the closest to the experimentalones (especially for higher Z), we will adopt our averagedvalue of α (0.40) to scale the experimental linewidths atthe electron temperature used for the Z-scaling study.We display in Figure 19 the Stark widths (in angularfrequency units) as a function of log Z. The present

Page 12: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

62 The European Physical Journal D

0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95

11,8

12,0

12,2

12,4

12,6

Ne VIIIN V O VI F VIIC IV

log10

(Z)

log 10

[W(s

-1)]

Fig. 19. Stark width W (in angular frequency units) of thetransition 3s 2S1/2 −3p 2Po

3/2 for C IV, N V, O VI, F VII andNe VIII as a function of log Z. The present results (quantum: ◦and SCP: �) are calculated at Te = 14.51×104 K (12.5 eV) andNe = 1 × 1018 cm−3. The experimental results of the Bochumgroup [21–23]: + are taken directly from Figure 3 of [23], theresults of Blagojevic et al. [2]: × are scaled using the relationW ∝ T−α

e with our value of α (0.40). Solid line: linear fit ofthe quantum results, dashed line: linear fit of the SCP results,dotted line: linear fit of the experimental results (+).

quantum and semi-classical results are calculated at Te =12.5 eV = 14.51 × 104 K and Ne = 1 × 1018 cm−3, theexperimental widths of Blagojevic et al. [2] are normalizedto Te = 12.5 eV using our Te dependence and the results ofthe Bochum group [21–23] are taken from Figure 3 of [23](using their scaling method mentioned above). We show inFigure 19 only the ions for which experimental results areavailable. We find that our calculated widths are relatedto Z by a Z−1.70 power law and the SCP ones by a Z−1.56

power law. The widths of Blagojevic et al. [2] (only forC IV, N V and O VI) show a nearly Z−0.79 dependence.The results of the Bochum group a Z−1.13 dependence.

It appears from these results that there is a disagree-ment between the experimental Z-scaling of Stark widthsand the correspondent theoretical predictions. This dis-agreement, reported also in [2,21,40], may be due to thefact that the temperature dependence of Stark widthsis not yet satisfactorily established. A good explanationof the observed difference with the experimental resultswould probably need more precise evaluation of higher-multipole contributions to the broadening, whose relativeimportance increases with Z.

In Figure 20, we display our Stark widths and the SCPones (in angular frequency units) as a function of log Zfor all ions considered in the present work, we find thatour calculations give widths that are related to the chargeby W ∝ Z−1.84, but SCP widths are related to Z byW ∝ Z−1.43.

0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2

11,6

11,8

12,0

12,2

12,4

12,6

12,8

log10

(Z)

log 10

[W(s

-1)]

Fig. 20. Present quantum: ◦ and present SCP: � Stark widths(in angular frequency units) of the transition 3s 2S1/2−3p2Po

3/2 for ions from C IV to P XIII as a function of log Z. Re-

sults are given for Ne = 1.8×1018 cm−3 and Te = 14.51×104 K.Solid line: linear fit of the quantum results, dashed line: linearfit of the SCP results.

5 Summary of the results and conclusion

We have presented quantum mechanical calculations -including Feshbach resonances- of Stark widths for the3s−3p doublet transitions in ten Li-like ions (from C IVto P XIII). Contrary to previous quantum calculations,an improved agreement has been found with experimen-tal results especially for ions with high ionization de-gree. The difference between experiments and our calcu-lations increases with the effective charge Z to be closeto unit for high Z. The CCC results are closer to the ex-perimental ones only for the case of C IV. This showsthat our method is more adapted to the calculation oflinewidths especially for high Z through the use of theSST/DW/JAJPOLARI/RtoS codes and with the intro-duction of the relativistic effects. To conclude about thevalidity of our theoretical calculations for higher ioniza-tion degree, it may be very interesting to perform 3s−3plinewidth measurements for Li-like ions with Z > 8.

To perform a Z-scaling of linewidths, the experimentalresults have to be normalized to the same electron temper-ature value Te. This normalization needs a knowledge ofthe Te-dependencies that are not yet firmly established inthe literature. We find that our 3s−3p Li-like linewidthsare related to the electron temperature by W ∝ T−α

e

where α change from an ion to an other with an averagedvalue of 0.40. This value is different from that predictedby the CCC calculations of Ralchenko et al. [19] (0.26).We have used our Te-dependencies to normalize experi-mental results of Blagojevic et al. [2] to Te = 12.5 eV.The results of Blagojevic et al. [2] show nearly a Z−1 de-pendence. The results of the Bochum group [21–23] showa Z−1.13 dependence for ions from C IV to Ne VIII. OurStark widths for ions from C IV to Ne VIII calculated

Page 13: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions; Z-scaling and comparison with semi-classical perturbation theory

H. Elabidi et al.: Quantum Stark broadening of 3s–3p spectral lines in Li-like ions 63

at Te = 12.5 eV show a Z−1.70 dependence, but if weconsider all ions from C IV to P XIII, we find that our ap-proach shows a Z−1.84 dependence. The SCP results showa Z−1.43 dependence. The disagreement between experi-mental results and theoretical predictions of the Z-scalingof linewidths still remaining, it can be explained by theevaluation of higher-multipole contributions to the broad-ening.

This work has been supported by the Tunisian research unit05/UR/12-04 and a bilateral cooperation agreement betweenthe French CNRS and the Tunisian DGRSRT. The authors areindebted to J. Dubau and M. Cornille for their invaluable helpin the use of the SST/DW/JAJPOLARI/RtoS computer codesand their adaptation to line broadening. They are also gratefulto N. Feautrier, G. Peach and S. Alexiou for helpful commentsand valuable discussions.

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