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Quantum mechanical calculations of the electron-impact broadening of spectral lines for intermediate coupling

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A new quantum mechanical expression for calculating electron-impact broadening is obtained for intermediate coupling. It would be especially useful for calculations of widths and shifts of fine structure spectral lines of multicharged ions where LS
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  I NSTITUTE OF P HYSICS P UBLISHING J OURNAL OF P HYSICS B: A TOMIC, M OLECULAR AND O PTICAL P HYSICS J. Phys. B: At. Mol. Opt. Phys. 37 (2004) 63–71 PII: S0953-4075(04)64419-1 Quantum mechanical calculations of theelectron-impact broadening of spectral lines forintermediate coupling Haykel Elabidi 1 , N´ebil Ben Nessib 1 and Sylvie Sahal-Br´echot 2 , 3 1 Groupe de Recherche en Physique Atomique et Astrophysique, Facult´e des Sciences de Bizerte,7021 Zarzouna, Tunisia 2 Laboratoire d’Etude du Rayonnement et de la Mati`ere en Astrophysique, Observatoire de Paris,Section de Meudon, UMR CNRS 8112, Bˆatiment 18, 5 Place Jules Janssen,F-92195 Meudon Cedex, FranceE-mail: Sylvie.Sahal-Brechot@obspm.fr Received 4 June 2003, in final form 15 September 2003Published 9 December 2003Online at stacks.iop.org/JPhysB/37/63( DOI: 10.1088/0953-4075/37/1/004 ) Abstract A new quantum mechanical expression for calculating electron-impactbroadening is obtained for intermediate coupling. It would be especiallyuseful for calculations of widths and shifts of fine structure spectral lines of multicharged ions where LS coupling breaks down. 1. Introduction Spectroscopicdiagnosticsofspectrallinesareveryimportantforplasmaandstellaratmospheremodelling. Fromthesestudies,onecandetermineelectrondensities,temperatures,abundancesand so on. For stellar atmospheres, collisional broadening parameters are required for a largenumber of lines of various elements. In addition, Stark broadenings of resonance lines of multicharged ions play a role in stellar interior modelling and are frequently unavailable. Infact, instellarinteriors,theelectrondensityis highenoughthatelectroncollisionalbroadeningcontributes significantly to the radiative accelerations needed for evolutionary models andopacity calculations. Alecian et al [1] have shown that the most important source of linebroadening for multicharged iron lines such as Fe XIV comes from pressure broadening.ManysemiclassicalcalculationsoftheimpactStarkbroadeningofspectrallineshavebeencarried out in the last few decades. However, there have been very few quantum calculations.WecanquoteBelyandGriem[2],whostudiedthewidthoftheMgIIresonanceline,andBarnesand Peach [3], who studied the shape and shifts of the resonance lines of Ca II perturbed byelectron collisions. These authors used the close coupling method. Dimitrijevic et al [4] gavea detailed comparison between close coupling quantum calculations and those made withinthe semiclassical perturbationtheory for the electron-impact broadeningof the Li I resonance 3 Author to whom any correspondence should be addressed.0953-4075/04/010063+09$30.00 © 2004 IOP Publishing Ltd Printed in the UK 63  64 H Elabidi et al lines. Seaton [5] studied the widths of the lines of C III with the R -matrix method. All thesecalculations have been performed within the LS coupling approximation; the fine structureeffects and other relativistic effects were not included in the atomic structure. In that case thewidths and shifts of the fine structurecomponentsare equalto the width andshift of the globalmultiplet.For multicharged ions, relativistic effects must be taken into account in atomic structurecalculations. Therefore the widths and shifts of the fine structure components can bedifferent. In addition, for multicharged ionic lines perturbed by electrons, the semiclassicalapproximation is poor for the calculation of widths and shifts, because the levels are farfrom each other, and thus close collisions can be important, especially for resonance lines.Consequently, quantum calculations of spectral line broadening including relativistic effectswouldbewelcomeformultichargedions,sincetheyremainunavailableasfarasweareaware.In fact, for high ionization states of ions, LS coupling breaks down when calculatingcollision strengths in electron–ion scattering [6, 7]. Relativistic effects must be included.Jones [8] distinguished two distinct types of relativistic correction:(a) Relativistic corrections resulting from the motion of the colliding electron and itsinteraction with the target.(b) Relativistic corrections resulting from breakdown of the LS coupling approximation forthe target.On the one hand, Walker [9] showed that the (a) corrections contributed less than 10% tothe cross sections for the target charge Z   25. Thus these corrections can be neglected forthe ions which are of interest in the present work.On the other hand, Jones [6] and further papers showed that the (b) corrections wereessential. Infact, owingtothe highelectronvelocity,therelativisticpartofthe Hamiltonianof the target can be neglected during the collision process. Therefore the reactance matricescan be calculated for LS coupling in the first stage. In the second stage, the collisionstrengths for intermediate coupling are calculated using these reactance matrices and termcoupling coefficients (TCC) obtained for atomic structure calculations including relativisticeffects. A similar method has been extensively developed and used for non-LTE calculationsof level populationsand line intensities for the multichargedions that are observedin the solarcorona for instance, or in astrophysical objects in the X band of wavelengths; see for exampleCHIANTI [10] and references therein. Computer codes calculating the atomic structure formulticharged ions including various relativistic effects are available. Computer codes givingthe collisional S -matrix elements within the ‘distorted wave approximation’, that is valid forelectron–multichargedion collisions, are also available.The collision S -matrix is also an essential part of the line broadening calculations. Byadapting the aforesaid method to line broadening theory, we provide in the present paper atheoretical formula giving the width and the shift of a spectral line perturbed by electroncollision for intermediate coupling, i.e. including the (b) relativistic corrections. In our nextpaper, we will extend the computercodes developedfor the computationof the non-LTE levelpopulationsandlineintensitiestocalculationsoflinewidthsandlineshiftsbasedontheformulafrom the present paper. Results will be provided for a line of astrophysical interest. 2. Theory 2.1. Generalities If the perturbers are moving rapidly, the broadening and shift of the line arise from a series of binarycollisionsbetweentheatomandoneoftheperturbers. Thisistheimpactapproximation  Electron-impact broadening for intermediate coupling 65 which is valid if  wτ   1 , (1)where w is the half-width at half-maximum and τ  is the average time of collision. For atransition between an upper set of levels i , i  with quantum numbers J  i  M  i , J  i   M  i  and a lowerset of levels f  , f   with quantum numbers J   f  M   f  , and J   f    M   f   , we can write the profile of theline as [3, 11–13]  I  (ω) = 1 π Re  ii   f f    i f  ∗ |  | i   f  ∗  i   f  ∗ | [ W  + iD − i (ω − ω if  )  I  ] − 1 | i f  ∗  . (2) W  and D represent respectively the width and the shift operator, I  is the unit operator and  is an operator corresponding to the dipole line strength defined by  i f  ∗ |  | i   f  ∗  = q 2  i | r  | f   i  | r  | f    , (3)where r  represents the emitter coordinates and q 2 = e 2 /( 4 πε 0 ) . In the case of isolated lines,the upper (respectivelylower) set of levels is limited to the degeneratemagnetic sublevels M  i ,  M  i  (respectively M   f  , M   f   ). Then we write (3) in terms of reduced matrix elements that areindependent of magnetic quantum numbers; they are defined by  i f  ∗ |  | i   f  ∗  = D i f i   f    i f  ∗ ||  || i   f  ∗  , (4)where  D i f i   f   =  µ ( − 1 )  J  i +  J  i  −  M  i −  M  i   J  i 1 J   f  −  M  i µ M   f   J  i  1 J   f   −  M  i  µ M   f    . (5)  a b ca  b  c   are the 3-  j symbols [14]. Hence we have w + id  =  i f  ∗ || W  + iD || i   f  ∗  =   M  i ,  M  i  ,  M   f  ,  M   f    D i f i   f    i f  ∗ | W  + iD | i   f  ∗  . (6)In the following, we give the expressions for the width and shift of a spectral line: firstly,for LS coupling; secondly, for intermediate coupling (i.e., including relativistic effects in theatomic structure of the radiating ion). 2.2. Calculations for LS coupling For LS coupling, the levels are defined as follows: | i  = |  i S i  L i lsL T i S T  M  T i M  T s  , (7)where S i and L i are the spin and the angular momentum of the target. s = 1 / 2 and l are thespin and the angular momentum of the perturber electron. S T and L T i are the spin and theangular momentum of the system ( emitter + electron ) and M  T s and M  T i are their projections.  i denotes a particular linear combination of the states denoted by | C  i α i S i  L i  . The symbol C  i stands for a particular configuration and α i is a degeneracy parameter which is used whenmore than one term S i  L i belongs to the configuration C  i . For an allowed dipolar transition,we have S i = S  f  = S [3] and the transformation for this coupling is |  i  L i  M  i SM  s lmsm s  =   L T i M  T i S T  M  T s C   L i l L T i  M  i m M  T i C  S s S T  M  s m s M  T s |  i SL i lsL T i S T  M  T i M  T s  , (8)where C  a b cd e f  are the Clebsch–Gordan coefficients; m and m s are, respectively the projectionsof   l and  s .  66 H Elabidi et al We replace | i  , | i   , | f   and | f    in (6) by the expression in (7) and we use (8) and thefollowing relationship: w + id  = π  ¯ hm  2  N    ∞ 0  f  (v)v d v  l  0 ( 2 l + 1 ) [1 − S ii ( l ,v) S ∗  f f  ( l ,v) ] , (9)where S ii ( l ,v) S ∗  f f  ( l ,v) =   M  i M  i  M   f  M   f   mm   D i f i   f    S  I  (χ i   M  i  lm  ; χ i  M  i lm ) × 12 l + 1 S ∗ F  (χ  f    M   f   lm  ; χ  f  M   f  lm  . (10) χ i and χ  f  represent all nonmagneticquantum numbers associated with the unperturbedstates i and f  of the emitter. The subscripts I  and F  are introduced to emphasize that the S matrixelements correspond to different total energies E   I  and E  F  defined by [15]  E   J  = E   j + ε  j ε  j = ¯ h 2 k  2  j 2 µ(  J  , j ) = (  I  , i ),( F  , f  ). (11)We findthatthehalf-half-width w andshift d  canbeevaluatedintermsofthescatteringmatrixelements S  I  and S F  ofthesystem ( ion + electron ) ,whenfinestructureeffectscanbeneglected,in the following manner [3, 16]: w + id  = π  ¯ hm  2  N    L T i L T  f  S T ll  ( − 1 )  Li +  Li  + l + l  [  L T i , L T  f  , S T ]2[ S ] ×  L i L T i l L T  f  L f  1  L i  L T i l   L T  f  L f   1   ∞ 0  f  (v)v d v × { δ l  l δ  L i  L i δ  L f    L f  − S  I  ( i  SL i  l  12  L T i S T ;  i SL i l 12  L T i S T ) × S ∗ F  (  f   SL f   l  12  L T  f  S T ;   f  SL f  l 12  L T  f  S T ) }; (12)we adopt the notation [  x , y ,  z ,... ] = ( 2  x + 1 )( 2  y + 1 )( 2  z + 1 )... .The transition here is from an initial atomic level i having quantum numbers L i , S i tothe final state f  having quantum numbers L f  , S  f  . The summation is taken over the totalangular momenta L T i and L T  f  of the system ( emitter + perturber ) in the initial and final statesof the transition, the total spin S T of the system and the perturber angular momenta l and l  respectively before and after the collision. N  is the electron density and f  (v) is the Maxwelldistribution of velocities for electrons at temperature T  defined by  f  (v) = 4 πv 2  m 2 π k  B T   3 / 2 exp  − m v 2 2 k  B T   ; (13)the coefficients of the type  a b ca  b  c   are 6-  j symbols [14]. k  B is the Boltzmann constant. 2.3. Calculations for JK coupling To find the expression of  w + id  for intermediate coupling (taking into account the finestructureoftheemitter),it is necessarytolookforitfor JK  coupling,definedbythefollowing  Electron-impact broadening for intermediate coupling 67 relationships:  S i +   L i =  J  i   J  i +  l = K  i  K  i +  s =  J  T i , (14)where all the quantumnumbers in (14) have the same definitions as those in (7). The level for  JK  coupling is defined by [7, 17] | i  J K   = |  i S i  L i J  i lK  i sJ  T i M  T i ; (15)the transformation (8) becomes for this coupling |  i S i  L i J  i  M  i lmsm s  =   J  T i M  T i K  i k  i C  K  i s J  T i k  i m s M  T i C   J  i l K  i  M  s m k  i |  i S i  L i J  i lK  i sJ  T i M  T i  (16)where k  i is the projection of the quantum number K  i defined in (14).We now apply (6) and write  i   f  ∗ | W  + iD | if  ∗  in the new basis |  i S i  L i J  i lK  i sJ  T i M  T i  ;we find w + id  = π  ¯ hm  2  N    J  T i M  T i J  T  f  M  T  f  K  i k  i K  f  k   f  K  i  k  i  K   f   k   f   M  i M   f  M  i  M   f   ll  mm  µ ( − 1 )  J  i +  J  i  −  M  i −  M  i  ×  J  i 1 J   f  −  M  i µ M   f   J  i  1 J   f   −  M  i  µ M   f    C  K  i  s J  T i k  i  m s M  T i C   J  i  l  K  i   M  i  m  k  i  × C  K  f   s J  T  f  k   f   m s M  T  f  C   J   f   l  K   f    M   f   m  k   f   C  K  i s J  T i k  i m s M  T i C   J  i l K  i  M  i m k  i C  K  f  s J  T  f  k   f  m s M  T  f  C   J   f  l K  f   M   f  m k   f  ×   ∞ 0  f  (v)v d v { δ l  l δ  J  i  J  i δ  J   f   J   f  δ K  i  K  i δ K  f   K  f  δ  L i  L i  δ  L f  L f   δ S i S i  δ S  f  S  f   − S  I  ( i  S i   L i  J  i  l  K  i  sJ  T i ;  i S i  L i J  i lK  i sJ  T i ) × S ∗ F  (  f   S  f    L f    J   f   l  K  f   sJ  T  f  ;   f  S  f  L f  J   f  lK  f  sJ  T  f  ) } , (17)where S  I  and S ∗ F  are now scattering matrices written in the form for JK  coupling. The ∗ signdenotes the complex conjugates of the matrix elements. Using standard angular momentumcoupling theory [18], we can simplify (17); it becomes w + id  = π  ¯ hm  2  N    J  T i M  T i J  T  f  M  T  f  K  i K   f  K  i  K  f   ll  µ ( − 1 )  J  i +  J  i  + K  i + K  i  + K  f  + K  f   +2  J  T  f  + l + l  +1 × [ K  i , K  f  , K  i  , K  f   ] 1 / 2 2[  J  T  f  ]  C   J  T  f  1 J  T i  M  T  f  µ M  T i  2  J  i K  i lK  f  J   f  1  ×  J  i  K  i  l  K  f   J   f   1  K  i J  T i s J  T  f  K  f  1  K  i  J  T i s J  T  f  K  f   1   ∞ 0  f  (v)v d v × { δ l  l δ  J  i  J  i δ  J   f   J   f  δ K  i  K  i δ K  f   K  f  δ  L i  L i  δ  L f  L f   δ S i S i  δ S  f  S  f   − S  I  ( i  S i   L i  J  i  l  K  i  sJ  T i ;  i S i  L i J  i lK  i sJ  T i ) × S ∗ F  (  f   S  f    L f    J   f   l  K  f   sJ  T  f  ;   f  S  f  L f  J   f  lK  f  sJ  T  f  ) } . (18)With the orthogonality formula   M  T  f  µ  C   J  T  f  1 J  T i  M  T  f  µ M  T i  2 = 1 and   M  T i 1 = [  J  T i ], we obtain
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