Dielectronic  Recombina/on  Computa/ons  with  the    

HULLAC  Code  Ehud  Behar  

Department  of  Physics,  Technion,  Israel  Visi/ng  Department  of  Astronomy,  

University  of  Maryland,  USA    

Outline  

•  The  HULLAC  Code  •  Method:  DW  DR  rates  •  Results  for  W  •  Laboratory  Benchmarking  •  Advanced  Topics    

HULLAC  

•  Hebrew  University  Lawrence  Livermore    Atomic  Code  

•  Conceived  by  Marcel  Klapisch,  mostly  wriMen  by  Avi  Bar-­‐Shalom,  with  contribu/ons  by  Joseph  Oreg  (Racah  Algebra)  

•  Later  programming  contribu/ons  from  Bill  Goldstein,  Michel  Busquet  and  others  

•  New  Collisional-­‐Radia/ve  version  by    M.  Klapisch  and  M.  Busquet  

HULLAC  Basic  Approxima/ons  

•  Parametric  poten/al  for  energies  •  Configura/on  mixing  •  Distorted  Wave  •  Factoriza/on  Interpola/on  method  for  cross  sec/ons  /  rate  coefficients  

•  Isolated  resonance  approxima/on,  but  DR+RR  interference  possible  to  compute  

Rela/vis/c  Parametric  Poten/al  (PP)  (Klapisch  1977)  

•  Treat  e-­‐e  interac/ons  as  perturba/on  

•  PP  U(ri)  is  a  screened  hydrogenic  poten/al  

H = H0 hiDirac

elec.i! +UPP (!ri )

"

#$%

&'+ H1 (Ze / ri + e

2 / rijelec.i! (UPP (!ri )

"

#$%

&'+ H2 (Breit + Lamb)

UPP (r) = !1r

qs e!! sr f!(! s;r)

shell s" + z

#

$%

&

'(

f!(! s;r) = (1! j2! + 2

) (! sr)j

j!j=0

2!+1"

qss! + z = Z

Level  Energies  

•  PP  is  found  through  the  varia/onal  principle  by  itera/ng  αs  values  to  minimize  level  energies  ΣEl    

•  Used  to  produce  complete  set  of  single-­‐electron  orthogonal  wavefunc/ons,  and  then  asymmetric  mul/-­‐electron  states  

•  Perturba/on  theory  with  H1  and  H2  yield  final  energies  

•  Clean,  fast  method  that  always  converges  and  provides  orthogonal  wavefunc/ons  suitable  for  bound  and  con/nuum  rate  coefficients  

Factoriza/on-­‐Interpola/on    (Bar-­‐Shalom  et  al.  1998)  

 •  “Factoriza/on”  of  electrosta/c  matrix  element  

 to  “angular”  and  “radial”  parts  

!ij (! ) = 8 " i, ji (! )( )JT 1rklk<l

" ! j, j j( )JT2

ji j jJT"

! ! bound statej(nlj)! unbound elec

!ij (! ) = 8 " i ZT ji, j j( )! j

2

ji j jJT" QT ji, j j( )

Factoriza/on-­‐Interpola/on  (contd.)  

•  Separate  bound  electron  coupling  computed  from  Racah  algebra  

•  Con/nuum/radial  parts  -­‐  both  direct  and  exchange  -­‐  integrals  vary  slowly  with  energy  ε,  which  allows  for  efficient  interpola/on  

•  Extended  to  bound-­‐free  transi/ons,  i.e.,  Autoioniza/on  (Oreg  et  al.  1991)  

DR  Method  Recap  •  Include  all  radia/ve  and  Auger  channels  •  Obtain  Edi,  A,  and  Aa  rates  from  code  •  Test  for  and  include  configura/on  mixing  •  For  rate  coefficients  

–  Convolve  with  Maxwellian  (Te)  –  Add  up  thousands  of  doubly  excited  level  contribu/ons    –  Extrapolate  to  high-­‐n  high-­‐l  (plasma  dependent)  –  Fit  with  few-­‐component  exponen/al  func/on  (op/onal)  

•  For  beam  experiments  –  Compute  resonance  strength  –  Convolve  with  beam  profile  –  Iden/fy  resonances    

DR  Coefficients  -­‐  Capture  •  Cross  Sec/on  and  Resonance  Strength    (i  –  ini/al,    d  –  doubly  excited,  electron  ε  =  Ed  –  Ei  =  Edi)  –  Narrow  natural-­‐width  Loren/zan  profile  

•  Capture  Rate  Coefficient    

!idDC Te( ) =<!v >Maxwell =

gd2gi

2!!2

mkTe

!"#

$%&

3/2

Adia e'Edi /kTe

! idDC !( ) = gd

2gi

2!!( )38!m!

Adia Ld !( )

Ld !( )

SidDC = ! id

DC "( )d! =!gd2gi

2!!( )38!m Ed " Ei( ) Adi

a

DR  Coefficients  -­‐  Stabiliza/on  

•  Branching  ra/o  for  stabiliza/on  

•  With  low  lying  doubly  excited  levels,  decays  to  unbound  (d’)  and  stabiliza/on  to  bound  (d’’)  doubly  excited  levels  can  be  important  

•  Decays  to  unbound  levels  can  be  followed  by  autoioniza/on  or  by  further  decay  

BD (d) =Adj

j<d! + Add 'B

D (d ')i<d '<d! + Add ''

d ''<i<d!

Adi 'a

i '! + Adj

j<d! + Add '

i<d '<d! + Add ''

d ''<i<d!

Non-­‐Resonant  Decays  

Low  Lying  Resonances  

 •  Sensi/ve  to  code  accuracy  

•  Resonance  yields  maximum  contribu/on  at  kTe=  2/3  Edi  

•  Low/high  resonances  contribute  at  low/high  temperatures  

Low  Resonances  –  All  the  Trouble  

!idDC Te( )! e"Edi /kTe

Not  for  Fusion  Plasmas,  but  

•  Photoionized    plasmas  tend  to  be  highly-­‐ionized  but  cold  kTe  <<  EIon  

•  DR  resonances  of  the  lowest  lying  doubly  excited  levels  are  dominant  

Savin  et  al.  2002  

Savin  et  al.  2002  (contd.)  Resonance  Strengths  &  Rate  Coefficients  

 

For  high-­‐lying  levels  n-­‐3  Scaling  

! idfD Te( ) = !id

DC Te( )BD (d)!Adia Adf

Adia + Adf

! bound ! n"3/2

# for$n > 0 Adia ,Adf ! n

"3

#! idDC Te( )! n"3

for!n = 0 Adf " n0

andmixed cases

What  About  High-­‐l  ?  

High-­‐n  and  Low-­‐E  

Ar-­‐like  W  

Benchmarking  Computed  DR  Data  with  the  Electron  Beam  Ion  Trap  

N-­‐like  to  Si-­‐like  W  Biedermann  et  al.  2009  

Benchmarking  Computed  DR  Data  (contd.)  

Biedermann  et  al.  2009  LMM  (LMN)  resonances    

Advanced  Topics  based  on  elegant  methods  by  V.  Jacobs  

•  Plasma  density  effects   •  RR  +  DR  interference  (Fano  profile  cross-­‐sec/ons)  

•  Not  drama/c  for  rates  •  C.f.  R-­‐matric  methods  

Summary  &  Ques/ons  •  DR  calcula/ons  are  challenging  and  difficult  to    mass-­‐produce  

•  Need  to  carefully  inspect  several  effects  on  a  case  by  case  basis  –  High  -­‐  n  –  High  -­‐  l  –  Non-­‐resonance  stabiliza/on  –  Decays  to  autoionizing  levels  –  Density  effects  

•  Refinements  can  be  tedious  –  need  to  define  the  accuracy  required  

•  Benchmarking  with  experiments  is  invaluable