Eur. Phys. J. D (2013) 67: 12DOI: 10.1140/epjd/e2012-30608-0

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Analysis of the quasicontinuum band emitted by highly ionisedtungsten atoms in the 4–7 nm range

Teresa Isabel Madeira1,a, Pedro Amorim1, Fernando Parente2, Paul Indelicato3, and Jose Pires Marques1

1 Centro de Fısica Atomica e Departamento de Fısica da Faculdade de Ciencias da Universidade de Lisboa, Campo Grande,1749-016 Lisboa, Portugal

2 Centro de Fısica Atomica e Departamento de Fısica da Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa,Monte da Caparica, 2829-516 Caparica, Portugal

3 Laboratoire Kastler Brossel, Ecole Normale Superieure, CNRS, Universite Paris 6, 4 place Jussieu, 75252 Paris Cedex 5,France

Received 2 October 2012 / Received in final form 16 November 2012Published online 31 January 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. Spectra emitted by highly ionized tungsten atoms from magnetically confined plasmas show acommon feature: a narrow structured quasi-continuum emission band most prominent in the range 4−7 nm,which accounts for 40−80% of the radiated power. This band has been fairly well explained by unresolvedtransitions from groups 4d-4p, 4f -4d (Δn = 0) and 5d-4f , 5g-4f and 5p-4d (Δn = 1). In this work we usea Multi-Configuration Dirac-Fock code in Breit self-consistent field mode to compute level energies andtransition probabilities for W27+ to W37+ ions contributing to this emission band. Intra-shell correlationwas introduced in the calculation for both initial and final states and all dipole and quadrupole radiativetransitions have been considered. The wavefunctions in the initial and final states are optimized separatelyand the resulting non-orthogonality effect is fully taken into account. The importance of some satellitelines was assessed. Together with the ionic distributions obtained by using the FLYCHK application andassuming that the initial states population depends statistically on the temperature we were able tosynthesize plasma emission spectrum profiles for several electron temperatures.

1 Introduction

In controlled thermonuclear reactions, atomic radiationis one of the primary energy loss mechanisms. Atomicphysics processes will thus play an important role in ITERperformances and in some of the diagnostics used to probethe plasma. This is especially true for the edge and diver-tor plasmas, where the knowledge of atomic physics pa-rameters is crucial to make the project succeed in dealingwith plasma-wall interactions and in taming problems liketritium retention or excessive heatloads. Atomic physics isalso essential for several (core) diagnostics because greatconfidence will be needed in the interpretation of theirsignals in order to gauge ITER performance.

The wavelength range of the radiation emitted bythe ions present in the plasma lies in the XR-IR rangeand therefore it may be considered as an atomic fin-gerprint for plasma diagnostics purposes. Although inatomic spectroscopy electric dipole lines are usually themost prominent, forbidden lines are not uncommon. In-deed, magnetic dipole (M1), magnetic quadrupole (M2)and electric quadrupole (E2) lines have been observed

a e-mail: tiamorim@fc.ul.pt

in the X-ray spectra of highly charged ions produced inhigh temperature, low density plasma sources such astokamaks [1–3], ECRIS [4,5], EBIT [6], and the Sun [7].Electric quadrupole lines are also found in astrophysicalplasmas and have been observed in high density laser pro-duced plasmas [8] and in beam foil interactions [9]. Two-photon decay of atomic levels in highly charged ions hasalso been observed [10] as well as magnetic octopole (M3)decay [11]. All these forbidden lines should also be ob-served in fusion plasmas as long as the electron tempera-ture is sufficiently high to guarantee the presence of highlyionised atoms [1,12]. The identification of such lines serveson one hand to provide fundamental tests of atomic struc-ture theory as well as checks of level population calcula-tions, and on the other hand, because the intensity of for-bidden transitions is sensitive to the electron density, thelines are used as a diagnostic tool in the spectroscopy ofsolar, astrophysical and laser produced plasmas.

For the next upcoming generation of fusion experi-ments, such as ITER, tungsten has been chosen as the ma-terial for plasma facing components, because it has a highmelting point, the necessary low erosion rates, and lowtritium retention, although the high radiative efficiency of

Page 2 of 6 Eur. Phys. J. D (2013) 67: 12

tungsten ions leads to stringent restrictions on the con-centration of tungsten ions in the burning plasma.

Tungsten plasma emission in the 4−7 nm range leadsto a quasicontinuum band and is believed to account for40−80% of the plasma radiated power [13,14]. Accordingto theoretical calculations using an average ion model, aconcentration of tungsten as small as 5×10−4 of the elec-tron concentration is sufficient to hinder breakeven con-ditions [15]. Which ions and radiative processes accountfor this narrow radiation band is an ongoing work of re-search. As suggested in reference [15], this band is formedby a large number of unresolved spectral lines, originatingmostly from Δn = 0 transitions within the n = 4 shell, al-though Δn = 1 transitions are also known to create quasi-continuum bands. These bands were observed in tokamakplasmas, in which tungsten occurred as an intrinsic impu-rity or was injected in controlled amounts by laser ablationtechniques [16].

The interpretation of the tungsten emission bands interms of atomic physics parameters has been the sub-ject of several papers. For these ions it was demonstratedthat the emission bands originate from transitions ofthe type 4p64dN−4p64dN−14f and 4p64dN−4p54dN+1,N = 4−9 [17,18]. More recently, it was also suggestedthat Δn = 1 transitions should account for the dis-crepancies still found between theory and experimentalresults around 5 nm [19]. Comparison of experimentaland calculated spectra has shown that this band is fairlywell explained by unresolved transitions from groups: (a)4d-4p, 4f -4d (Δn = 0) and (b) 5d-4f , 5g-4f and 5p-4d(Δn = 1) [17,19].

In the present work, attention has been focused onthe first group of transitions, in order to study featuresthat have not yet been explored or taken into account.We used a fully relativistic multiconfiguration Dirac-Fockmethod to evaluate the energies and transition proba-bilities of all levels considered. Excited levels that de-cay by dipole as well as by quadrupole radiative tran-sitions (ΔJ = 0,±1,±2) have been taken into accountbecause in laboratory tokamak plasmas and in variousastronomical objects, suitably chosen electric quadrupole(E2) forbidden lines serve as a basis for the determinationof reliable electron density [8] and temperature diagnos-tics [20], which are two rather important plasma parame-ters that enter the Lawson criteria. The effect of intra-shellelectronic correlation was evaluated and found to be im-portant as it introduced non-negligible deviations in thetransition energies and probabilities. The relative line in-tensities are calculated assuming a Boltzmann form forthe distribution of the ions over the different levels.

Eleven tungsten ion species have been studied, rang-ing from W27+ to W37+ because their presence has beenobserved in dedicated EBIT experiments [14] and becauseaccording to calculations using FLYCHK [21], assumingnon-LTE plasma conditions and an average electron den-sity of 1019 m−3, their abundances cannot be neglectedfor the electron temperature ranges under consideration.Finally, the effect of deviations from the electron dynam-ics theoretical model and experimental errors estimated

for plasma electron temperature measurements in fusiondevices, is taken into account. A list of initial and finalconfigurations of the transitions studied in this work arepresented in Table 1.

2 The MCDF method

The study of transition probabilities plays an importantrole in the determination of atomic abundances. Accu-rate estimates of radiative transition probabilities betweenmultiplet states are an important source for the success-ful experimental identification of astrophysical and labo-ratory plasmas spectra. Probabilities of magnetic dipoleand electric quadrupole transitions, in particular, are im-portant in plasma diagnostics, but the experimental de-termination of these quantities is difficult, and only anaccurate theoretical calculation can provide importantinformation [22].

We used the general relativistic MulticonfigurationDirac-Fock (MCDF) code of Desclaux and Indelicato[23–25] (mcdfgme) to calculate bound-state wavefunc-tions and energies. Details of the method, including theHamiltonian and the processes used to build the wave-functions can be found elsewhere [26,27].

The total wavefunction is calculated with the help ofthe variational principle. The total energy of the atomicsystem is the eigenvalue of the equation

Hno pairΨΠ,J,M (. . . , ri, . . .) = EΠ,J,MΨΠ,J,M (. . . , ri, . . .),(1)

where Π is the parity, J is the total angular momentumeigenvalue, and M is the eigenvalue of its projection on thez axis Jz . In this equation, the hamiltonian is given by:

Hno pair =N∑

i=1

HD(ri) +∑

i<j

Vij(|rij |), (2)

where HD is the one electron Dirac operator and Vij is anoperator representing the electron-electron interaction oforder one in α. The expression of Vij in Coulomb gauge,and in atomic units, is:

Vij =1rij

(3a)

− αi · αj

rij(3b)

− αi · αj

rij

[cos

(ωijrij

c

)− 1

](3c)

+ c2 (αi · ∇i) (αj · ∇j)cos

(ωijrij

c

) − 1ω2

ijrij, (3d)

where rij = |ri − rj | is the inter-electronic distance, ωij isthe energy of the exchanged photon between the two elec-trons, αi are the Dirac matrices and c is the speed of light.We use the Coulomb gauge as it has been demonstratedthat it provides energies free from spurious contributionsat the ladder approximation level and must be used inmany-body atomic structure calculations [28,29].

Eur. Phys. J. D (2013) 67: 12 Page 3 of 6

Table 1. List of initial and final configurations of the transitions studied in this work (Ar core implicit) for tungsten ions.

Ion Initial state Final state Lines ΔJ

37+ 3d10 4s2 4p5 4d2 + 3d10 4s2 4p6 4f 3d10 4s2 4p6 4d 91 0, ±1, ±236+ 3d10 4s2 4p5 4d3 + 3d10 4s2 4p6 4d 4f 3d10 4s2 4p6 4d2 465 0, ±1, ±236+ 3d10 4s2 4p5 4d2 5s + 3d10 4s2 4p6 4f5s 3d10 4s2 4p6 4d 5s 225 0, ±1, ±235+ 3d10 4s2 4p5 4d4 + 4p6 4d2 4f 3d10 4s2 4p6 4d3 2439 0, ±135+ 3d10 4s2 4p5 4d3 5s + 4p6 4d 4f5s 3d10 4s2 4p6 4d2 5s 2108 0, ±135+ 3d10 4s2 4p5 4d4 + 4p6 4d2 4f 3d10 4s2 4p6 4d3 1175 ±234+ 3d10 4s2 4p5 4d5 + 3d10 4s2 4p6 4d3 4f 3d10 4s2 4p6 4d4 9864 0, ±1, ±234+ 3d10 4s2 4p5 4d4 5s + 3d10 4s2 4p6 4d2 4f5s 3d10 4s2 4p6 4d3 5s 9304 0, ±133+ 3d10 4s2 4p5 4d6 + 3d10 4s2 4p6 4d4 4f 3d10 4s2 4p6 4d5 13166 0, ±1, ±233+ 3d10 4s2 4p5 4d5 5s + 3d10 4s2 4p6 4d3 4f5s 3d10 4s2 4p6 4d4 5s 23266 0, ±132+ 3d10 4s2 4p5 4d7 + 3d10 4s2 4p6 4d5 4f 3d10 4s2 4p6 4d6 10732 0, ±1, ±232+ 3d10 4s2 4p5 4d6 5s + 3d10 4s2 4p6 4d4 4f5s 3d10 4s2 4p6 4d5 5s 33481 0, ±131+ 3d10 4s2 4p5 4d8 + 3d10 4s2 4p6 4d6 4f 3d10 4s2 4p6 4d7 4697 0, ±1, ±231+ 3d10 4s2 4p5 4d7 5s + 3d10 4s2 4p6 4d5 4f5s 3d10 4s2 4p6 4d6 5s 27626 0, ±130+ 3d10 4s2 4p5 4d9 + 3d10 4s2 4p6 4d7 4f 3d10 4s2 4p6 4d8 977 0, ±1, ±230+ 3d10 4s2 4p5 4d8 5s + 3d10 4s2 4p6 4d6 4f5s 3d10 4s2 4p6 4d7 5s 12299 0, ±129+ 3d10 4s2 4p5 4d10 + 3d10 4s2 4p6 4d8 4f 3d10 4s2 4p6 4d9 44 0, ±1, ±229+ 3d10 4s2 4p5 4d9 5s + 3d10 4s2 4p6 4d7 4f5s 3d10 4s2 4p6 4d8 5s 2890 0, ±128+ 3d10 4s2 4p6 4d9 4f 3d10 4s2 4p6 4d10 6 0, ±127+ 3d10 4s2 4p6 4d9 4f2 3d10 4s2 4p6 4d10 4f 227 0, ±1

The term (3a) represents the Coulomb interaction,the term (3b) is the Gaunt (magnetic) interaction, theterm (3c) the Breit (ω2

ij) retardation and the (3d) standsfor higher-order terms in the retardation operator. In thisexpression the ∇ operators act only on rij and not on thefollowing wavefunctions.

The electron-electron interaction is described by thesum of the Coulomb and the Breit interaction. The mcd-fgme code can include as an option, the Breit operator inthe self-consistent field process. All calculations are donefor finite nuclei using a Fermi distribution with a thicknessparameter of 2.3 fm. The nuclear radii are taken from ref-erence [30]. All calculations are performed replacing theelectron mass by the reduced mass in the Dirac equation,using the latest atomic mass table [31,32].

The so-called Optimized Level (OL) method was usedto determine the wavefunction and energy for each stateinvolved. In this way, spin-orbitals in the initial and fi-nal states for the radiative transitions are not orthogo-nal, since they have been optimized separately. This non-orthogonality effect is fully taken into account [33,34],using the formalism proposed by Lowdin [35]. Thelength gauge has been used for all radiative transitionprobabilities.

3 Results and discussion

In relativistic atomic-structure calculations, the Breit in-teraction has traditionally been treated as a first or-der perturbation correction based on the no-pair DFHamiltonian. In contrast to the perturbative approach,incorporation of the Breit term in the SCF processhas the advantage that both the electrostatic and Breit

interactions are included to the same order in SCF poten-tials within the algebraic approximation [36].

Using the Desclaux and Indelicato MCDF code withthe Breit term included in the self-consistent process,we have first calculated the 4p54dN+1 + 4p64dN−14f →4p64dN electric dipole transition probabilities for theW27+ – W37+ charge state range. The initial state con-figurations are mixed due to electronic correlation. Thiseffect leads to a narrowing of the spectral interval con-taining the important transitions around the 5 nm re-gion, as had already been pointed out by Jonauskaset al. [17]. Intra-shell correlation was also included in thefinal state (4p64dN−24f2 for N ≥ 2). The result is plottedin Figure 1a.

The same procedure was then adopted for the satellitetransitions between configurations built from the previ-ous ones with a 4d electron excited to the 5s sub-shell:4p54dN5s + 4p6 4dN−24f5s → 4p64dN−15s, becausethese lines lie on the same energy range. To the best of ourknowledge, these transition probabilities had not been cal-culated before because it is in general assumed that theircontribution to the spectra is negligible due to the low rel-ative ion population in these excited levels. However, ourcalculations show that the influence of the presence of the5s spectator electron on the transition probabilities, whichcan be visualized in Figure 1b, may have to be taken intoaccount.

We also calculated the electric and magnetic quadru-pole transition probabilities for the same sets of initial andfinal configurations. The results are plotted in Figure 2.This figure shows a pattern that is different from the onesresulting from electric dipole transitions, the strongestcontributions being located in the 6−7 nm region.

To obtain relative line intensities, we assumed that thedistribution of ions over the different initial i-levels, for the

Page 4 of 6 Eur. Phys. J. D (2013) 67: 12

1

3

5

4 4.5 5 5.5 6 6.5 7 7.5wavelength (nm)

1

3

5

1

3

5

1

3

5

1

3

5

1

3

5

gA [1

013 s-1

] 1

3

5

1

3

5

1

3

5

1

3

5

1

3

5

(a)

1

3

5

4 4.5 5 5.5 6 6.5 7 7.5wavelength (nm)

W27+

1

3

5 W28+

1

3

5 W29+

1

3

5 W30+

1

3

5 W31+

1

3

5

gA [1

013 s-1

]

W32+

1

3

5 W33+

1

3

5 W34+

1

3

5 W35+

1

3

5 W36+

1

3

5

(b)

W37+

Fig. 1. (Color online) Calculated electric dipole transition probabilities for (a) 4p54dN+1 +4p64dN−14f → 4p64dN transitionsand (b) 4p54dN5s + 4p64dN−24f5s → 4p64dN−15s transitions.

temperature T , has a Boltzmann form [37]

N qi = N q

1

gi

g1e−

EikT . (4)

Here, N q1 and g1 are the population and the statistical

weight of the ground level, respectively, gi is the statisticalweight of level i, and Ei is the energy of this level relativeto the ground level. For each ion, the line intensity for atransition i → k is given by:

Iik = �ωikAikN qi , (5)

where Aik is the transition probability and N qi is the pop-

ulation of the level i.Plasma electron temperature measurements are usu-

ally provided by three complementary diagnostics:Thomson scattering (TS), electron cyclotron emission(ECE) and the X-ray pulse height analysis (PHA) [38].These procedures are based on the assumption that elec-trons have a Maxwellian behaviour. However, plasmasgenerated in fusion devices are not ideal systems and aresustained in conditions that can make electron dynamicsto depart from the ideal case: magnetic field anysotropyand gradients, auxiliary heating mechanisms, magnetohy-drodynamics phenomena and plasma discharges shorter

than the average time to attain electrodynamic equilib-rium. In most cases not even LTE conditions are met [39].Deviations from the Maxwellian behaviour are known tooccur and electron temperature measurements are fre-quently affected by errors that can go up to 30% [40–42].

To account for these deviations, we calculated, for sev-eral electron temperature values in the range used in toka-mak plasmas, the charge-state distributions in the plasmausing the FLYCHK code [21] and assuming non-LTE plas-ma conditions and a typical average electron density of1019 m−3. The FLYCHK code solves rate equations forlevel population distributions by considering collisionaland radiative (CR) atomic processes. The results are plot-ted in Figure 3 for electron temperatures ranging from0.9 keV to 1.8 keV.

Using the obtained charge state distribution for sev-eral electron temperatures and the calculated atomic rela-tive intensities, convoluted with a 0.01 nm wide Gaussianinstrumental function, we were able to simulate transi-tion spectra shown in Figure 4. A comparison with ex-periment must be performed with precaution, mainly forthree reasons:

1. Calculation of line intensities is crucial. One needs toknow the upper levels relative population distributionwithin each charge state and for a specific temperature.

Eur. Phys. J. D (2013) 67: 12 Page 5 of 6

2.06.01.01.41.8

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

wavelength (nm)

2.06.01.01.41.8

2.06.01.01.41.8

2.06.01.01.41.8

5.01.01.52.02.53.0

gA [1

05 s-1

]

5.01.01.52.02.53.0

5.01.01.52.02.5

2.0

6.0

1.0

1.4

1.53.04.56.07.59.0

(a)

2.06.01.01.41.8

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

wavelength (nm)

W29+

3.0

9.0

1.5

2.1

W30+

5.0

1.5

2.5

3.5

W31+

2.01.01.82.63.4

W32+

3.09.01.52.12.7

gA [1

05 s-1

]

W33+

5.01.01.52.02.53.0

W34+

2.04.06.08.01.01.2

W35+

2.04.06.08.01.0

W36+

2.04.06.08.01.0

(b)

W37+

Fig. 2. (Color online) Calculated electric quadrupole transition probabilities for (a) 4p54dN+1 + 4p64dN−14f → 4p64dN

transitions and (b) 4p54dN5 + 4p64dN−24f 5s → 4p64dN−15s transitions.

Fig. 3. Charge-state distribution, assuming non-LTE plasmaconditions and an average electron density of 1019 m−3, forelectron temperatures ranging from 0.9 keV to 2.4 keV, calcu-lated using the FLYCHK application.

Here, we assumed that this distribution depends onthe temperature in agreement with equation (4). Inaddition, the temperature and density dependence ofthe ionisation and recombination rates, and particletransport affect the charge state distribution;

2. Fusion-grade plasmas are usually contaminated by awide variety of impurities, depending mainly on theplasma wall facing materials, which can also radiatein this wavelength range [1,43,44]. This work, howeveris intended to evaluate the specific contribution fromtungsten ions. No effort was put into identifying andstudying the contribution from other set of impurities

1.0E+17

3.0E+17

5.0E+17

4.0 4.5 5.0 5.5 6.0 6.5 7.0wavelength (nm)

0.9 keV

1.0E+17

4.0E+17

7.0E+171.1 keV

2.0E+17

8.0E+17

1.4E+181.2 keV

2.0E+17

8.0E+17

1.4E+18

Inte

nsity

[arb

.uni

ts]

1.3 keV

2.0E+17

8.0E+17

1.4E+181.4 keV

2.0E+17

8.0E+17

1.4E+181.5 keV

2.0E+17

8.0E+17

1.4E+18

1.8 keV

Fig. 4. Simulated emission spectra profiles, from W27+ toW37+ ions, in 4−7 nm band for electron temperatures rangingfrom 0.9 keV to 1.8 keV.

present in any of the two fusion devices – the ASDEXUpgrade tokamak and the LHD stellarator;

3. There may be other tungsten ions configurations con-tributing to the spectra in this wavelength range.

Page 6 of 6 Eur. Phys. J. D (2013) 67: 12

4 Conclusions

In this work we use the Multi-Configuration Dirac-Fockcode of Desclaux and Indelicato [24,25], with the Breitterm included in the self-consistent process, to computethe energies and transition probabilities of more than twohundred and fifty thousand lines from W27+ to W37+

ions which are important for tokamak plasma diagnostics.Intra-shell correlation was introduced for both initialand final states in the calculation and all dipole andquadrupole radiative transitions have been considered.The effect of the presence of a 5s spectator electron on thetransition probabilities is also presented. Although this ef-fect had been mentioned before in some publications [17],the results had never been presented. Finally, by mixingcharge states using the FLYCHK code and assuming thatthe initial states population dependence on the temper-ature has a Boltzmann form, we were able to simulatetransition spectra for different temperatures.

This research was supported in part by FCT projectPest-OE/FIS/UI0303/2011, PTDC/FIS/117606/2010, FCTGRANT SFRH/BPD/69627/2010, Faculty of Sciences, LisbonUniversity, and Laboratoire Kastler Brossel (“UMR No. 8552”of the ENS, CNRS and UPMC). P.I. acknowledges supportfrom the Helmholtz Alliance HA216/EMMI and the Pro-gramme Hubert Curien PESSOA 20022VB.

References

1. M. Klapisch, J.L. Schwob, M. Finkenthal, B.S. Fraenkel,S. Egert, A. Bar-Shalom, C. Breton, C. DeMichelis, M.Mattioli, Phys. Rev. Lett. 41, 403 (1978)

2. M. Bitter, K.W. Hill, N.R. Sauthoff, P.C. Efthimion, E.Meservey, W. Roney, S.V. Goeler, R. Horton, M. Goldman,W. Stodiek, Phys. Rev. Lett. 43, 129 (1979)

3. E. Kallne, J. Kallne, J.E. Rice, Phys. Rev. Lett. 49, 330(1982)

4. M.C. Martins, J.P. Marques, A.M. Costa, J.P. Santos, F.Parente, S. Schlesser, E.-O.L. Bigot, P. Indelicato, Phys.Rev. A 80, 032501 (2009)

5. J.P. Santos, M.C. Martins, A.M. Costa, J.P. Marques, P.Indelicato, F. Parente, Phys. Scr. T144, 014005 (2010)

6. T. Putterich, R. Neu, C. Biedermann, R. Radtke, ASDEXUpgrade Team, J. Phys. B 38, 3071 (2005)

7. A.H. Gabriel, C. Jordan, Phys. Lett. A 32, 166 (1970)8. J.C. Gauthier, J.P. Geindre, P. Monier, E. Luckoenig, J.F.

Wyart, J. Phys. B 19, L385 (1986)9. D.D. Dietrich, G.A. Chandler, R.J. Fortner, C.J. Hailey,

R.E. Stewart, Phys. Rev. Lett. 54, 1008 (1985)10. P.H. Mokler, S. Reusch, A. Warczak, Z. Stachura, T.

Kambara, A. Muller, R. Schuch, M. Schulz, Phys. Rev.Lett. 65, 3108 (1990)

11. P. Beiersdorfer, A.L. Osterheld, J. Scofield, B. Wargelin,R.E. Marrs, Phys. Rev. Lett. 67, 2272 (1991)

12. K.B. Fournier, W.H. Goldstein, M. May, M. Finkenthal,Phys. Rev. A 53, 709 (1996)

13. T. Putterich, R. Neu, R. Dux, A.D. Whiteford, M.G.O’Mullane, the ASDEX Upgrade Team, Plasma Phys.Control. Fusion 50, 085016 (2008)

14. C. Biedermann, R. Radtke, AIP Conf. Proc. 1125, 107(2009)

15. R. Isler, R. Neidigh, R. Cowan, Phys. Lett. A 63, 295(1977)

16. R. Radtke, C. Biedermann, J.L. Schwob, P. Mandelbaum,R. Doron, Phys. Rev. A 64, 012720 (2001)

17. V. Jonauskas, S. Kucas, R. Karazija, J. Phys. B 40, 2179(2007)

18. Y.J. Rhee, D.H. Kwon, Int. J. Mass Spectrom. 271, 45(2008)

19. C.S. Harte, C. Suzuki, T. Kato, H.A. Sakaue, D. Kato, K.Sato, N. Tamura, S. Sudo, R. D’Arcy, E. Sokell, J. White,G. O’Sullivan, J. Phys. B 43, 205004 (2010)

20. E. Biemont, C.J. Zeippen, Comments At. Mol. Phys. 33,29 (1996)

21. H.-K Chung, M.H. Chen, W.L. Morgan, Yu. Ralchenko,R.W. Lee, High Energy Density Phys. 1, 3 (2005)

22. H. Ray, Astrophys. J. 579, 914 (2002)23. J.P. Desclaux, in Methods and Techniques in Computatio-

nal Chemistry : Small Systems of METTEC, edited by E.Clementi (STEF, Cagliary, 1993), Vol. A, p. 253

24. J.P. Desclaux, Comput. Phys. Commun. 9, 31 (1975)25. P. Indelicato, J.P. Desclaux, MCDFGME, a multi-

configuration Dirac-Fock and general matrix elements pro-gram (v.2011), http://dirac.spectro.jussieu.fr/mcdf

26. I.P. Grant, H.M. Quiney, Adv. At. Mol. Phys. 23, 37(1988)

27. P. Indelicato, Phys. Rev. A 51, 1132 (1995)28. O. Gorceix, P. Indelicato, Phys. Rev. A 37, 1087 (1988)29. E. Lindroth, A.M. Martensson-Pendrill, Phys. Rev. A 39,

3794 (1989)30. I. Angeli, At. Data Nucl. Data Tables 87, 185 (2004)31. G. Audi, A. Wapstra, C. Thibault, Nucl. Phys. A 729, 337

(2003)32. A. Wapstra, G. Audi, C. Thibault, Nucl. Phys. A 729, 129

(2003)33. P. Indelicato, Phys. Rev. Lett. 77, 3323 (1996)34. P. Indelicato, Hyp. Int. 108, 39 (1997)35. P.O. Lowdin, Phys. Rev. 97, 1474 (1955)36. Y. Ishikawa, H.M. Quiney, G.L. Malli, Phys. Rev. A 43,

3270 (1991)37. I.I. Sobel’man, Introduction to the Theory of Atomic, Spec-

tra (Pergamon Press, Oxford, 1972), p. 30038. Burning Plasma Diagnostics AIP Conference Proceedings,

edited by F.P. Orsitto, G. Gorini, E. Sindoni, M. Tardochi(American Institute of Physics, Melville, New York, 2008),Vol. 988

39. R.W.P. McWhirter, in Plasma Physics and Nuclear FusionResearch, Chapter 10: Plasma Radiation (Academic Press,London, 1981)

40. G. Zhuang, R. Behn, I. Klimanov, P. Nikkola, O. Sauter,Plasma Phys. Control. Fusion 47, 1539 (2005)

41. T.I. Madeira, Ph.D. thesis, IST-UTL, Lisbon, Portugal,2009

42. E. de la Luna, V. Krivenski, G. Giruzzi, C. Gowers, R.Prentice, J.M. Travere, M. Zerbini, Rev. Sci. Instrum. 74,1414 (2003)

43. M. Finkenthal, L.K. Huang, S. Lippmann, H.W. Moos, P.Mandelbaum, J.L. Schwob, M. Klapisch, Phys. Lett. 127,255 (1988)

44. K.B. Fournier, W.H. Goldstein, M. May, M. Finkenthal,Phys. Rev. A 53, 709 (1996)