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Checking the dependence on the upper level ionization potential of electron impact widths using quantum calculations

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Using their dependence on the upper level ionization potential and the rest core charge of the emitter, electron impact line widths of several highly charged ions have been recently predicted. For most of them, neither theoretical nor experimental
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  Eur. Phys. J. D 61 , 285–290 (2011)DOI:10.1140/epjd/e2010-10298-4 Regular Article T HE E UROPEAN P HYSICAL J OURNAL D Checking the dependence on the upper level ionization potentialof electron impact widths using quantum calculations H. Elabidi 1 , a and S. Sahal-Br´echot 2 1 Groupe de Recherche en Physique Atomique et Astrophysique, Facult´e des Sciences de Bizerte,Universit´e du 7 Novembre `a Carthage, 7021 Zarzouna, Tunisia 2 LERMA, Observatoire de Paris, CNRS UMR 8112, ENS, UPMC, UCP, 5 Place Jules Janssen,92190 Meudon, FranceReceived 18 May 2010 / Received in final form 22 October 2010Published online 3rd December 2010 –c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2010 Abstract. Using their dependence on the upper level ionization potential and the rest core charge of theemitter, electron impact line widths of several highly charged ions have been recently predicted. For mostof them, neither theoretical nor experimental data are available (Mg VII, Na VII, Al VIII, Na VIII, Mg IX,Al X, Mg X, Al XI, Si XI, Ti XI, Cr XIII, Cr XIV, Fe XV, Fe XVI, Ni XVIII and Fe XXIII). The aimof this work is to check the above dependence and systematic trends of line widths by using quantumcalculations. The comparison between the quantum results and the predicted ones shows that for the sameelement, the agreement is better for lower ionization degrees. 1 Introduction Stark broadening of multiply charged ion lines is of a greatinterest for modelling and investigating subphotosphericlayers[1]. Such data are alsoused to analyse element abun- dance and to estimate radiative transfer through stellarplasmas for some classes of hot stars, and to investigateother astrophysicalproblems[2]. Throughout recent years, the theory and calculations of Stark broadening have beenimproved and experiments have been used to check avail-able theoretical results and to provide new Stark broad-ening data. But despite the numerous available data (cal-culated and measured), the results for heavy and highlycharged ions are still few and insufficient. This lack of datais especially inconvenient for opacity modeling of stellaratmospheres.It is well known[1–3] that a great number of data of  Stark broadening parameters of spectral lines of differentelements are needed for opacity calculations. Many exper-iments and methods (semiclassical, quantum-mechanical,semi-empirical) of Stark broadening calculation have beendeveloped for several years. In parallel, many efforts havebeen made to perform simple methods and approximateapproaches that can provide these data with enough reli-ability and more computational simplicity when accuratedata are not available. The study of possible regularitiesand systematic trends of Stark broadening parameters isone of these methods[4–11], and references therein. This method can be interesting for the acquisition of new data a e-mail: haykel.elabidi@fsb.rnu.tn and for the evaluation of available theoretical and experi-mental results.Many regularities of Stark broadening have been in-vestigated[4,5,9]: the dependence on the rest core charge Z  c of the emitter as seen by the optical electron, or on thenuclear charge number or on the upper level ionizationpotential χ . According to[11], the last approach is more advantageous (achievement of better fittings, the Starkwidth dependence on χ is theoretically expected ... ).In[11], the established regularities were extended to higher ionization degrees and used for the prediction of Stark impact widths for the line when they are not avail-able (theoretical or experimental) and srcinating fromthe same type of transition (3 s –3  p transition arrays). It isexpected in that paper that the predicted results by thismethod are as accurate as the data used for the establish-ment of Stark width dependencies. So, it would be of agreat interest for the check of these predicted results andfor the accuracy of this method to provide Stark broad-ening data calculated or measured at the same plasmaconditions [11].This is the aim of the present work. We will use ourquantum formalism for electron impact broadening in in-termediate coupling as developed in[12]. It is adapted for highly charged ions where LS  coupling breaks down. Wewill apply this formalism to perform calculations of elec-tron impact broadening parameters for all the ions stud-ied in[11] and to compare them with the predicted val- ues based on the regularities approach in order to checkits validity. Since these quantum calculations have beenchecked for Be-like[13,14] and for Li-like ions[15,16],  286 The European Physical Journal D where good agreements have been found with experimen-tal results, they can be especially adapted to performthis task. The ions investigated here are Mg VII, Na VII,Al VIII, Na VIII, Mg IX, Al X, Mg X, Al XI, Si XI, Ti XI,Cr XIII, Cr XIV, Fe XV, Fe XVI, Ni XVIII and Fe XXIII.These ions are of astrophysical interest[17,18] and some of  them are important for diagnostics of fusion plasmas [19]. 2 Theory 2.1 Outline of the quantum formalism of electronimpact widths and computational method The quantum formalism and the computational procedureare described elsewhere[12,14,16] and only a few details are given here for the sake of completeness. The impactapproximation is at the basis of the formalism. It meansthat the time interval between collisions is much longerthan the duration of a collision. The expression of the full width at half maximum  W  obtained in[14] is: W  = 2 N  e   m  2  2 mπk B T   12 × ∞   0 Γ  w ( ε )exp  − εk B T   d  εk B T   , (1)where k B is the Boltzmann constant, N  e is the electrondensity, T  is the electron temperature and Γ  w ( ε ) =  J  T i J  T f lK i K f  K  i ,K  f  ,J  T f  ,J  T i  2  J  i K  i lK  f  J  f  1  2 ×  K  i J  T i sJ  T f  K  f  1  2 [1 − (Re( S  i )Re( S  f  ) + Im( S  i )Im( S  f  ))] , (2)where −→ L i + −→ S  i = −→ J  i , −→ J  i + −→ l = −→ K  i and −→ K  i + −→ s = −→ J  T i . L and S  represent the atomic orbital angular momentumand spin of the target, l is the electron orbital momen-tum, the superscript T  denotes the quantum numbers of the total electron + ion system. S  i ( S  f  ) are the scatter-ing matrix elements for the initial (final) levels, expressedin the intermediate coupling approximation, Re(S) andIm(S) are respectively the real and the imaginary partsof the S-matrix element,  abcdef   represent 6–  j symbolsand we adopt the notation [ x,y,... ] = (2 x +1)(2 y +1) ... Both S  i and S  f  are calculated for the same incident elec-tron energy ε = mv 2 / 2. Equation (1) takes into accountfine structure effects and relativistic corrections resultingfrom the breakdown of the LS  coupling approximation forthe target.The calculation will be done through the SUPER-STRUCTURE code (SST)[20] for the atomic structure in intermediate coupling and DISTORTED WAVE (DW)code[21] for the scattering problem as in[14]. This weak coupling approximation for the collision part assumed inDW is adequate for highly charged ions colliding with elec-trons since the close collisions are of small importance.The JAJOM code[22] is used for the scattering problem in intermediate coupling. R -matrices in intermediate cou-pling and real (Re S ) and imaginary part (Im S ) of thescattering matrix S have been calculated using the trans-formed version of  JAJOM [23] and the program RtoS[24] respectively. The evaluation of Re S and Im S is done ac-cording to:Re S =  1 − R 2  1 + R 2  − 1 , Im S = 2 R  1 + R 2  − 1 . (3)The relation S = ( 1 + iR )( 1 − iR ) − 1 guarantee the uni-tarity of the S -matrix. 2.2 Electron impact width regularities The investigation of electron impact parameter regulari-ties needs an accurate set of theoretical and experimen-tal data at the same plasma conditions (electron densityand temperature). These data are given for different tem-peratures and densities, so we have to normalize them toparticular ones at which the other regularities will be in-vestigated. The dependence of electron impact widths islinear with electron density for non hydrogenic emittersexcept for high densities when the Debye cutoff plays anon negligible role. However, the dependence on the elec-tron temperature is different from line to line for all spec-tra and is not well established, so a great care has to betaken in performing this dependence. Several works dealwith this subject [16,25–27]. At low temperatures (low energies), the electron im-pact width for ion emitters is commonly approximated bythe T  − 1 / 2 dependence. This is due to the fact that at lowenergies the collision strength varies very slowly with en-ergy and it is almost constant [28]. This a consequenceof the Coulomb attraction between the radiating ions andthe colliding electron. As a consequence, the cross sectionis inversely proportional to the square of the electron ve-locity. After integration over the Maxwellian electron ve-locities distribution given in formula (1), we readily obtainthat W  ∝ T  − 1 / 2 .At high temperatures, we can use the Bethe approxi-mation of cross sections [29]. This approximation is usu-ally considered as a simplification of the Born approxima-tion valid at high energies and especially good when thedifference of energy E  j − E  i = hν  between the atomic lev-els of the radiator is small. It is shown in [30] that whenthe initial kinetic energy ε = 12 mv 2 i is very much greaterthan hν, the expression of the cross section for the Betheapproximation is: σ i → j = f  i → j 3 πe 4 hν  1 ε ln4 εhν  ∝ ln εε, (4)where f  i → j is the oscillator strength of the transition i → j . At high temperatures, we can write the mean ki-netic energy as  ε  = 12 m  v 2  = 32 kT  . Writing that the  H. Elabidi and S. Sahal-Br´echot: Dependence on the upper level ionization potential of electron impact widths 287 0.00 0.02 0.04 0.06 0.080.00.10.20.30.4     W    (    A    ˚    ) ln(T)/T 1/2 Fig. 1. Quantum electron impact width of the Mg X 3 s 2 S 12 –3  p 2 P o 32 transition as a function of ln( T  ) / √ T  for the temperaturerange (1 . 6 × 10 4 –5 × 10 6 K) at density N  e = 10 18 cm − 3 . 0.00 0.01 0.02 0.03 0.04 0.050.00000.00040.00080.00120.00160.0020 ln(T)/T 1/2     W    (    A    ˚    ) Fig. 2. Same as in Figure1but for the Fe XVI 2  p 6 3 s 2 S 12 –2  p 6 3  p 2 P o 32 transition for the temperature range (5 × 10 4 –5 × 10 6 K). 0.00 0.01 0.02 0.03 0.04 0.050.00000.00040.00080.00120.0016     W    (    A    ˚    ) ln(T)/T 1/2 Fig. 3. Same as in Figure1but for the Fe XXIII 2 s 3 s 3 S 1 –2 s 3  p 3 P o2 transition for the temperature range (5 × 10 4 –5 × 10 6 K). width is proportional to [ vσ ] av ≈ [ v ] av [ σ ] av , where [ ... ] av denotes the average over the Maxwellian electron veloci-ties distribution, we find that W  ∝ ln T  √  T  . This asymptoticform is valid at high temperatures as it is shown in Fig-ures1–3. In these figures, the behavior of the electron im- pact width at high temperature for the ions Mg X, Fe XVIand Fe XXIII is in agreement with the ln T  √  T  dependence. W  is almost linear with ln T  √  T  at high temperatures whichis in accordance with the Bethe approximation.In [11], instead of the T  − 1 / 2 temperature dependencefor ion lines, the authors used, for each line, the wholespectrum of functions of the form[9]: f  ( T  ) = A + BT  − C  , (5)where A , B and C  are determined using an accurate set of data for a particular line. The above temperature depen-dence (Eq. (5)) given by[11] is adequate for intermediate temperatures.In severalworks[4,5,8], Puri´cet al. have suggestedthat a useful form for the electron impact width is W  = A k χ − k where χ is the upper level ionization potential and the co-efficient A k is independent of  χ but is a function of therest core charge Z  c , electron density N  e and temperature T  . The dependence form of  W  on χ combined with equa-tion (5) yields to the following formula for the electronimpact width [9]: W  P  = N  e f  ( T  ) a 1 Z  c 1 c χ − b 1 , (6)the coefficients a 1 , b 1 and c 1 are independent of electrontemperature and density, ionization potential for partic-ular transition and of the rest core charge of the emit-ter. Using a set of known Stark broadening parameters todetermine these coefficients, the expression (6) would besuitable for the calculation of new data by interpolationor extrapolation. The authors in[11] indicated that the dependence of electron impact width on N  e , T  , Z  c and χ is universal and applicable for all types of transitions. Butthey expected a better accuracy of results when obtainedusing regularities within the lines srcinating from partic-ular type of array, e.g. from 3 s –3  p transition arrays for alarge range of electron temperatures and densities. In thefollowing, we discuss their conclusions on the basis of thestudied ions. 3 Results and discussion Our quantum calculations of electron impact widths(FWHM) have been performed at an electron density N  e = 10 19 cm − 3 and electron temperature T  = 5 × 10 5 Kallowing us to compare them with the predictions from theelectron impact width regularities of [11]. All the stud- ied lines are srcinating from the same transition array(3 s –3  p ) and for the ions Mg VII, Na VII, Al VIII, Na VIII,Mg IX, Al X, Mg X, Al XI, Si XI, Ti XI, Cr XIII, Cr XIV,Fe XV, Fe XVI, Ni XVIII and Fe XXIII. We note also thatthe transitions in all these ions, except Mg X and Al XI  288 The European Physical Journal D Table 1. Present quantum electron impact widths compared to the results W  P  from[11] at N  e = 10 19 cm − 3 and T  = 5 × 10 5 K.Emitter Sequence Transition λ (˚A) W  W W  P  Mg X Li 3 s 2 S 12 –3  p 2 P o 32 2215 . 0 0 . 88 1 . 69Al XI Li 3 s 2 S 12 –3  p 2 P o 32 1997 . 6 0 . 59 1 . 69Na VIII Be 2 s 3 s 1 S 0 –2 s ( 2 S)3  p 1 P o1 3179 . 0 1 . 99 1 . 47Mg IX Be 2 s 3 s 1 S 0 –2 s ( 2 S)3  p 1 P o1 2814 . 2 1 . 31 1 . 58Al X Be 2 s 3 s 1 S 0 –2 s ( 2 S)3  p 1 P o1 2535 . 0 0 . 84 1 . 50Si XI Be 2 s 3 s 1 S 0 –2 s ( 2 S)3  p 1 P o1 2312 . 5 0 . 62 1 . 51Fe XXIII Be 2 s 3 s 3 S 1 –2 s 3  p 3 P o1 549 . 51 0 . 009 1 . 73Na VII B 3 s 2 S 12 –( 1 S)3  p 2 P o 32 1752 . 2 0 . 79 1 . 80Mg VII C 2  p 3 s 3 P o2 –2  p ( 2 P o )3  p 3 P o1 1350 . 4 0 . 31 1 . 35Al VIII C 2  p 3 s 3 P o1 –2  p ( 2 P o )3  p 3 S 1 1223 . 7 0 . 22 1 . 47Cr XIV Na 2  p 6 3 s 2 S 12 –2  p 6 3  p 2 P o 32 389 . 86 0 . 0078 1 . 70Fe XVI Na 2  p 6 3 s 2 S 12 –2  p 6 3  p 2 P o 32 335 . 41 0 . 0049 1 . 88Ni XVIII Na 2  p 6 3 s 2 S 12 –2  p 6 3  p 2 P o 32 320 . 56 0 . 0037 1 . 85Ti XI Mg 2  p 6 3 s 2 1 S 0 –3 s 3  p 3 P o1 568 . 98 0 . 034 3 . 43Cr XIII Mg 2  p 6 3 s 2 1 S 0 –3 s 3  p 3 P o1 328 . 27 0 . 011 3 . 79Fe XV Mg 2  p 6 3 s 2 1 S 0 –3 s 3  p 3 P o1 417 . 26 0 . 012 4 . 00 studied in [16], have not been studied theoretically or ex-perimentally so far. In Table1,we report our quantum electron impact widths W  compared to those estimatedin[11] using the formula(6) of the dependence on the upper level ionization potential for the above mentionedelectron density and temperature values.The averaged ratio ( W/W  P  ) between our results andthose from[11] is about 1 . 63, and we can see from Table1that, for the same element, this ratio increases with theionization degree (we have excluded from these remarksthe case of Na VII and Na VIII and the results of theMg-like ions: Ti XI, Cr XIII and Fe XV that will be dis-cussed separately in the next paragraphs). At these highionization degrees, the two approximations used in[11] breakdown: the LS  coupling and the Coulomb approxima-tions. Consequently, regarding the fact that our quantummethod is supposed to be especially adapted for highlycharged ions, we can conclude that, for the same element,the formula (6) of [11] based on the upper level ionization potential dependence is more adapted to the lower ioniza-tion degrees than for the higher ones and to lines arisingfrom highly excited levels. Moreover, if we look at Figure2of [11], we can see that the number of experimental and theoretical results used to scale electron impact widthswith temperature is different from an ionization stage toan other, especially, from low to high ionization degrees.This number decreases with the increase of  Z  c . For ex-ample, for the highest ionization degrees there is only oneresult for Z  c = 12 and four results for Z  c = 11. But, for lowionization degrees, the number of the experimental resultsis enough to carry out the temperature scaling (about 28results for Z  c = 2). Consequently, we think that it will bevery difficult for high ionization stage to evaluate – witha sufficient accuracy – the coefficients A , B and C  used inthe temperature scaling procedure, and that the discrep-ancy between our results and the Puri´c ones is expectedfor highly charged ions.There is an unexpected disagreement in the case of Na VII and Na VIII, where the ratio ( W/W  P  ) is 1.76for Na VII but it is about 1.47 for Na VIII. Regardingto the results of all other ions presented here and theirbehavior with Z  c , the agreement between our results andthose from[11] is expected to be better for Na VII than for Na VIII. The srcin of this problem is still unexplained.The other important point that can be noted fromTable1is the significant discrepancy in the case of all Mg-like ions presented here (Ti XI, Cr XIII and Fe XV). Theaveraged ratio ( W/W  P  ) for the three cases is about 3.74.To study this discrepancy, we have performed quantumand semiclassical perturbation (SCP) calculations for twodifferent transitions of four other Mg-like ions. The SCPcalculations have been done using our atomic data fromthe SST code. The ions are K VIII, Ca IX, Sc X and Ti XI,and one of the two transitions studied here is the same asstudied in Table1(2  p 6 3 s 2 –3 s 3  p ). The comparison pre-sented in Table2shows that the agreement between ourresults and the semiclassical ones is good for temperatureand density used in the present paper (the averaged ratio( W/W  SCP  is about 1.14). Then, our calculations are ex-pected to give good results even for the Mg-like ions andmay be the approximated formula of [11] – in its presentform – is less adapted to such isoelectronic sequences thatcontain many electrons.This problem may be caused by the fact that the elec-tron undergoing the transition is one of two electrons in3 s (3 s 2 –3 s 3  p transition), but for all other cases the transi-tion is 3 s –3  p . Another reason that can be the srcin of theabove discrepancy is the number of electrons in the ions  H. Elabidi and S. Sahal-Br´echot: Dependence on the upper level ionization potential of electron impact widths 289 Table 2. Present quantum electron impact widths compared to present semiclassical perturbation calculations W  SCP  at N  e =10 19 cm − 3 (Mg-like ions).Emitter Transition λ (˚A) T  (10 5 K) W  W W  SCP  K˜VIII 2  p 6 3 s 2 1 S 0 –3 s 3  p 1 P o1 519 . 40 2 0 . 056 1 . 125 0 . 049 1 . 443 s 3  p 1 P o1 –3 s 3 d 1 D 2 441 . 40 2 0 . 041 0 . 955 0 . 032 1 . 10Ca IX 2  p 6 3 s 2 1 S 0 –3 s 3  p 1 P o1 466 . 20 2 0 . 037 0 . 975 0 . 034 1 . 143 s 3  p 1 P o1 –3 s 3 d 1 D 2 359 . 02 2 0 . 028 0 . 905 0 . 022 1 . 10Sc X 2  p 6 3 s 2 1 S 0 –3 s 3  p 1 P o1 422 . 63 2 0 . 025 0 . 865 0 . 024 1 . 263 s 3  p 1 P o1 –3 s 3 d 1 D 2 357 . 79 2 0 . 021 0 . 915 0 . 016 1 . 07Ti XI 2  p 6 3 s 2 1 S 0 –3 s 3  p 1 P o1 386 . 14 2 0 . 018 0 . 805 0 . 017 1 . 143 s 3  p 1 P o1 –3 s 3 d 1 D 2 327 . 18 2 0 . 017 0 . 965 0 . 013 1 . 13 of this isoelectronic sequence (12 electrons) compared tothe other sequences studied here (Li, Be, B, C). For theNa-like ions, that have a comparable number of electronswith the Mg-like ones, we have not detected the same dif-ference, because there is only one 3 s -electron undergoingthe transition instead of two equivalent electrons in thelower level as in Mg-like ions. 4 Conclusions We have presented in this paper quantum mechanical cal-culations of electron impact widths for (3 s –3  p ) transitionarray for the following ions (Mg VII, Na VII, Al VIII,Na VIII, Mg IX, Al X, Mg X, Al XI, Si XI, Ti XI, Cr XIII,Cr XIV, Fe XV, Fe XVI, Ni XVIII and Fe XXIII). Theyhave been compared to the results based on the depen-dence of electron impact widths on the upper level ioniza-tion potential. An acceptable agreement has been foundfor lower ionization degrees for the same element. The dif-ference found for high ionization degrees may be relatedto the LS  coupling and the Coulomb approximations usedin[11]. We can relate also this disagreement to the num- ber of the experimental and theoretical results used in[11] to achieve the temperature scaling of widths. This numberdecreases significantly from the lower ionization degrees tothe higher ones where it becomes incapable to accomplishthe temperature scaling procedure with sufficient accu-racy.We have detected a significant discrepancy betweenour results and those of [11] for the cases of all Mg-likeions. This problem may be caused by the fact that theelectron undergoing the corresponding transition is one of the two 3 s -electrons (2  p 6 3 s 2 –3 s 3  p ), contrary to the othertransitions studied here for which this electron is onlyone 3 s -electron (example 2  p 3 s –2  p 3  p ). And to see whetherour formalism is adapted to the magnesium isoelectronicsequence, we have calculated the electron impact widthssrcinating from the same transition array (2  p 6 3 s 2 –3 s 3  p )but for four other Mg-like ions (K VIII, Ca IX, Sc X andTi XI). The comparison of our results to those of the semi-classical perturbation theory shows a good agreement.Our calculations are interesting for the test of severalregularities of electron impact broadening and especiallythe dependence on the upper level ionization potential.They can be used also for the establishment of new Starkbroadening regularities. Reciprocally, the comparison withthe present results and maybe with other future resultscan be very useful for checking the validity of our formal-ism. The authors would like to thank J. Dubau and M. Cornille fortheir help in the use of the SST/DW/JAJPOLARI/RtoS com-puter codes and their adaptation to line broadening. This workhas been supported by the Tunisian research unit 05/UR/12-04, the French one UMR 8112 and the bilateral cooperationagreement between the Tunisian DGRS and the French CNRS(project code 09/R 13-03, project No. 22637). References 1. M.J. Seaton, J. Phys. B 20 , 6363 (1987)2. M.S. Dimitrijevi´c, Astron. Astrophys. Trans. 22 , 389(2003)3. M.S. Dimitrijevi´c, S. Sahal-Br´echot, IAU Colloquium 137  ,ASP Conference Series, edited by W.W. Werner, A. Baglin(1993), Vol. 404. J. Puri´c, I.S. Laki´cevi´c, V. Glavonji´c, Phys. Lett. A 76 ,128 (1980)
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