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Nuclear Excitation in Plasmas - ECT

VIEWS: 25 PAGES: 59

									Nuclear Excitation in Plasmas

 G. Gosselin, P. Morel, V. Méot, N. Pillet
              (CEA, France)



      Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
       Nuclear Excitation Processes in Plasmas




                                                              Electron                Neutron
 Photon          NEEC                    NEET
Absorption                                                    Inelastic               Inelastic
                                                             Scattering              Scattering
      G. Gosselin                     P. Morel               G. Gosselin
         Now                          Next Talk                 Now
             Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                Multi-Scale Modeling

• Microscopic model : quantum tools
  – Fermi Golden rule
  – Perturbation methods
  – Partial wave scattering (PWBA, DWBA, …)
• Macroscopic model
  – Relativistic average atom
  – Statistical Thermodynamics
  – Plasma Physics
       Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
           Thermodynamic Equilibrium
• Ions, electrons and photons are at thermodynamic equilibrium

                    Ti  Te  Tr  T
• Nuclear populations are not
    Boltzmann law does not apply

                                                        Ei
                                                    

                    Ni 
                                 2J i  1 e           kT

                                                             Ej

                                2J       j  1e
                                n                       
                                                            kT

                                j1



           Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
           Thermodynamic Equilibrium

• Equilibration time (under stationary conditions)
                            ln 2
                        
                           e d
• Populations at equilibrium

                N f t     e
                            
                N i t     d

          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Internal Conversion / Electronic Capture




   Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                Internal conversion (IC)
• Fermi Golden rule
               Log 2 2
                                                                            r E r 
                                                                        2
      nuc
       IC      IC         i r H  f b
               TJ f J i 
• Final state density
               r E r   n (E r ) 2 J i  1
• Internal conversion rate for one electron
                                          nuc
                                ele
                                            IC
                                         2J b  1
                                 IC




          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
       Internal conversion coefficient
• Ratio : electron emission / photon emission
                          IC  J i       TJf  J i
                                    
                            Jf
                            
                           Jf  Ji       TJIC J i
                                             f



                                                                            2
         e 2 E 2J i  12J f  1  i
                                       J L Jf 
                                  1     1  R if
                                                                                     2

         c m e c 2   LL  1          0  
                                     2     2

              The internal conversion coefficient
           does not depend on the nuclear parameters

        Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
             Electronic capture (NEEC)
• Electron capture cross section
   – Fermi Golden rule
                         2
          NEEC E                                                        b E 
                                                                       2
                                           f b H  i r
                          ve
   – Final state density
                     b E   2 J f  1 gE  E r 




           Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                     NEEC cross section
                     2 2J f  1 Log2                              
      NEEC E  
                   2m e E 2J i  1 TJf J i                               
                                                                                  2

                                                        E  E r     2
                                                                           
                                                                           2
• Laboratory conditions
   – Non excited atom
   – Natural level widths
   – Conversion coefficient 




          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
              Relativistic Average Atom
• Thomas-Fermi-Dirac model
  – The atom lies in a spherical box whose radius is
    dictated by density
  – Average atom : mean value of binding energies
    and occupation numbers
  – Relativistic : necessary for heavy nuclei internal
    conversion calculations

                                           B. F. Rozsnyai, Phys. Rev. A5 (1972) 1137

          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                     Electronic Transition
• Average atom deficiency
   – Unique binding energy for each atomic shell



• Numerous electronic configurations
   Distribution of the binding energy over a statistical width e

                                                                         
                                                                            E  E r 2
       Gaussian envelope : gE  E r  
                                                               1              2e 2
                                                                   e
                                                               2e

           Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
           Macroscopic transition rates
• Energy constraint :  E  E b  0
• Excitation rate
                E
              
         e    NEEC E  v e f E  PE  E  dE
              0
• De-excitation rate
                        0
                  
             d   ele f E  PE  E  dE
                        IC
                   E

                                             G. D. Doolen, Phys. Rev. C18 (1978) 2547

          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                              Plasma transition rate
   • NEEC excitation rate
       2 J f  1 Te  Log 2                                        E            E 
e                           gE b  f FD E r 1  f FD E b  erf  r   erf  b 
       2 J i 1    TJf J i                                        e 2           e 2 


   • Internal conversion de-excitation rate
          Te  Log 2                                        E            E 
  d                  gE b  f FD E b 1  f FD E r  erf  r   erf  b 
            TJf J i                                        e 2           e 2 

          Principle of detailed balance fulfilled (Boltzmann)
                                        G. Gosselin, P. Morel, Phys. Rev. C70, 064603 (2004)

                    Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
      Enhanced nuclear decay in plasmas




• Decay of isomeric level through excitation of a level
  lying above the isomeric level
• Intuition : should become important when E ~ kT
• Isomeric lifetime changes in astrophysical plasmas

          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
          Electron Inelastic Scattering
• Free electrons in plasma
  Very low energy (~keV)
  Very huge numbers (> 1024 e/cm3)
• Inelastic scattering
  Coulomb excitation process
  Low cross section
  Attractive potential


         Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                      Coulomb Excitation
• Semi-classical approach
   – Classical trajectory (hyperbole)
   – Excitation probability deduced from time dependant
     perturbation method
   – Diverges for attractive potentials
• Quantum mechanics approach
   – Perturbation theory [Alder]
      • Multipole development
      • Cross section
   – How to choose a good wave function ?

           Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
         Quantum Coulomb Excitation
• PWBA Approximation
  – Plane wave
  – Works well at high energy
• DWBA Approximation
  – Distorted wave function
  – Works well over the whole energy domain
  – Radial equation can be solved by two methods
     • Exact solution (only with an unscreened potential)
     • WKB method



          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                          Radial Equation
                   d 2    2m                  1 
                            2 E  2 V r  
                                   2m
                    dr  2
                                              r 2       0
                                                        

• Exact solution
  – Hypergeometric functions
  – Numerical evaluation very difficult for low energies
• WKB Method
  – Approximate radial equation resolution
  – Solution by Langer
     • Asymptotic solutions  Bessel functions
         Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                                  WKB Method
• Radial equation
          2
              ur
           u                                                
                                                            1
                                                        mZe 
                                                           2
                                                              
                r 
                 
              Q 0
          dr 2                                         2
                                                     ( 
                                                     Q 2E  2
                                                      )
                                                      r2
                                                          
                                                          r 
            2
           dr                                              r


                                         Turning Point
Exponential                                                                     Oscillating
  Region                                                                         Region

u  r  
           1         r0 Qt dt   
                 exp                                      u  r  
                                                                      1         r Qt dt  
                                                                            cos 
           Qr     r
                                    
                                                                     Qr      r0
                                                                                             
                                                                                              




               Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                         WKB theory by Langer
• Radial equation
    – Variable and function change
• Oscillating region
              r
             
           8 Qt dt
             r0             
                                             r                           r       
u  (r )                   cos    J1/ 3  Qt dt   cos    J 1/ 3  Qt dt  
              3Qr          3             r0
                                                        
                                                               3             r0
                                                                                          
                                                                                           
                            

• Exponential region
                    r0
                
               8 Qt dt       
                                               r0                          r0           
                                                                                                  
    u  (r )  r                sin  I1/ 3  Qt dt     cos    K1/ 3  Qt dt      
                   Qr                      r
                                                           
                                                                  3           r
                                                                                                
                                                                                                 
                                
                  Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
G. Gosselin, N. Pillet, V. Méot, P. Morel, A. Dzyublik, Phys. Rev. C79, 014604 (2009)
          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                         Threshold artifact
• Non Zero cross section at threshold
   – Systematic effect for any nuclear transition
   – Same artifact in atom excitation in atomic physics
• Classical image
   – Electron acceleration in Coulomb field
• In plasma, no global acceleration as the total charge
  inside the ion sphere is zero
• For isolated ion, a pseudo-continuity could be
  reached with NEEC cross section ?

           Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                        Present and Future
• Self consistent excitation and de-excitation model
  –   Photon Absorption/Emission
  –   NEEC/Internal Conversion
  –   NEET/BIC
  –   Electron inelastic scattering
       • Principle of detailed balance at thermodynamic equilibrium
• Prospective
  – Screened ions for electron inelastic scattering
  – Multi-configuration effects in NEET

           Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Photonic excitation and de-excitation




  Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                          Photon absorption
• Breit and Wigner cross section
                   2J f  1                    
         h                c
                                2 2

                  22J i  1            2            2 
                                    h  h  E   
                                                     2

                                                                               4 
• If Doppler effect can be neglected
                                   
                                                  
                  e           g ,  h cd d
                            0

• If photons are at thermodynamic equilibrium
                             2J f  1 ln 2                     1
                        e                                 E
                             2J i  1 TJf J i
                                                       e   k B Tr
                                                                    1
            Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                               Photon emission
• Spontaneous emission                        • Mode occupation number
            ln 2
       d                                                        c2 I
                                                              n 
        sp

           TJ f Ji                                                2h3
• Total emission                              • Induced emission rate
                            E
                                                                                           2
       ln 2            e   k BTr                              ln 2 c I 
  d                                          ind
                                                       n   sp
                                                              d
                                                             TJ f J i 2h
                        E                        d                        3
      TJ f J i
                   e   k BTr
                               1

           Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
       Electronic configurations
                                  <  : no overlap
                                         Global treatment for NEEC
                                         Detailed treatment for NEET


                                  >  : heavy overlap
                                         Global treatment for NEEC
                                          and NEET

                                                                 
                                                                    E  E r 2
Gaussian envelope : gE  E r  
                                                       1              2e 2
                                                           e
                                                       2e
   Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
 GANIL : NEEC experiment
            Performed in September 2004




Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
               Blackbody photon distribution
• Thermodynamic equilibrium
• Photon distribution
                 8E 2
         f E   3 3
                               1
                              E
                 h c
                         e   k BT
                                    1
• Radiative intensity
                      8 3
   I   hc f h   2
                                      1
                                     h
                       c
                                e   k BT
                                           1

• Maximum : E=1.594 kBT

  The density of photons does not
   depend on the plasma density

                 Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                         Photon absorption
• Breit and Wigner cross section
             2J f  1                                              
   h                2 c 2
            22J i  1                                           
                                                                   2
                                               h  h  E   
                                                    2           2

                                                                 4
• Number of excited nuclei per unit of time

                                   
                                
 dNif    Ni f EdE g ,  d d hcdt  e Ni dt
      ' : photon frequency in ion local frame



           Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
     Excitation phenomena description
• Radiative processes
  – Photon absorption
  – Photon emission

• Bound/free electron processes
  – Internal conversion
  – Electron capture (NEEC)
• Enhanced de-excitation process

        Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
                       Photon Absorption
• With a smooth photon spectrum
                 2J f  1 2 3c3 ln 2
            e                    
                                       gE 
                 2J i  1 E TJ f Ji
                               2


• Example : 181Ta
                                                e  2.78 10 19 gE  s 1


                                                         N *   e N at t



          Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Asymptotic Enhancement Factor




 Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France
Gilbert GOSSELIN, Vincent MEOT, Pascal MOREL, Nathalie PILLET, CEA/DAM/DIF, France

								
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