Collisional Model in LSP.
Simulation of Collisional Slowing Down of
Relativistic Electrons in Plasma.
E0  1 MeV, DT:   300 g/cm3 , Te  Ti  5 keV
A. Solodov, J. Myatt
University of Rochester
Laboratory for Laser Energetics
3d Meeting of the Fusion Science
Center for Extreme States of Matter
and Fast Ignition Physics
University of Rochester
26–27 January 2006
Summary
The LSP collisional model has been modified to include
relativistic effects and tested to reproduce correctly the collisional
slowing down of electrons in plasma predicted by the theory
• The non-relativistic LSP collisional model has been found correct
(except for the value of Coulomb logarithm which we modified) and
consistent with other theoretical models suggested for multi-fluid and
hybrid PIC simulations
• Relativistic electron scatter and drag rates have been calculated and
implemented in the LSP code
• The slowing-down of relativistic electron beams have been simulated
and compared with the theory
• With the improved collision model, the LSP code is capable of
reproducing the blooming, straggling and penetration predicted by the
theory
LSP1 can model larger, more dense plasmas for longer simulation
times than explicit PIC codes
LSP uses:
• an implicit solution of the electromagnetic fields;
• an implicit particle push;
• hybrid fluid-kinetic descriptions for electrons;
• inter- and intra-species collisions based on Spitzer collision rate:
- kinetic particle scattering off its own distribution and a pressure
gradient force for fluid species to model intra-species collisions,
- a frictional momentum push, dissipative heating and temperature
equilibration in collisions between species (both kinetic and fluid).
1D.
R. Welch, D. V. Rose, B. V. Oliver, R. E. Clark, Nucl. Instrum. Methods Phys.
Res. A 464, 134 (2001).
The inter-species collisions are based on the model of Rambo &
Procassini2 suggested, originally, for multi-fluid simulations
For a species  thermalizing with species  ,
 u 
     u  u ,

 t  c

2
3  T 
u

u

m



        T  T ,


2  t  c
where
  
4 2 Z2 Z 2 e 4  n
2 3/2
3m m T / m  T / m   2 / 9  u  u 


3m 
, m  m m /  m  m  .
 
m
1/ 3
A. Decoster 3 proposed to modify   and  , in particular,
   exp  -z 2 / 2  , z 
2P.
u  u
T / m  T / m 
1/ 2
.
W. Rambo and R. J. Procassini, Phys. Plasmas 2, 3130 (1995).
referred by others as private communication with A. Decoster.
3Unpublished,
,
Kinetic particle velocities are rotated in the weighted scattering
reference frame and the random velocity components are
advanced according to the temperature change
Drifting Maxwellian distributions are constructed at each grid cell:
 v  u 2 
in 1-D, f (v )  T / m  exp  
.
 T / m 


Particles are scattered in the weighted reference frame of
1/ 2
 
scattering spacies: u   u 
,       ,

 
 

sc
b
the total scattering angle   (2t  )1/ 2 .
Random velocity advance: v n 1/ 2  u   v n 1/ 2  u 1  T / T 
1/ 2
,
2
3  T 
sc 2

   m   u  u     T  T  m   u  u  .
2  t  c


physical temperature change
adjusment for a nonphysical
temperature change
We modify the scattering rates in the equations of Rambo &
Procassini to account for the relativistic effects
Relativistic electron temperature: Te 
pe2
3m  e

m  e2ue2
3 e
.
   u 
(1)

       u    u ,
 t  c

2
3  T 
(2)
(3)


m

u


u




   
   /      T  T ,
2  t  c



3m     1 
 m m   (2)
(1)
(3)
where  
    1    min   ,    ,  
   ,  

,
2m
 m m  

4 Z2 Z 2 e 4  n
  
,
2 2
3
m   v
2
 

1
2
1  v 
/ c2
 1     u    u  / c 2   9 / 3 
2
1/ 3

2
u 2   2u 2
 
/c .
2
The new scattering rates are obtained using the relativistic
Rutherford electron scattering cross-section4

4  e e 
2
d
 2 2 4 , for 
d
p v 
1
b
For a beam of relativistic electrons with momentum p  mv    , scattered by particles 
(1)
 
 n v  

p || d
m 
d   n v   t  1      ,
p d 
m 

(2)
 
 n v 
p  d
 d
d


2
n
v

,


  t
t
 2 d d  
p2 d 
2
2
4  e e  
2
me2 2v 4
.
Temperature equilibration of two species which do not drift with respect to ea ch other:
 
(3)
E d
m  2
3
3
 n v 
d   n v   t
.
2
E d 
2
m     1
v ee 
D 
D
2
ee
ei
Coulomb Log: ee  ln ee   ln
, dB

.
 , ei  ln ei , dB 
dB 
Dp 
dB
2me c  ee  1
2 ei mev ei
4E.
M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Pergamon Pres, Oxford (1981).
With the improved collisional model, the LSP code reproduces the
blooming, straggling and penetration of monoenergetic relativistic
electron beams predicted by the theory5
Theory5
Simulation
E0  1 MeV
DT:
  300 g/cm3
ne  7  1025 cm-3
Te  Ti  5 keV
5C.
K. Li and R. D. Petrasso, Phys. Rev. E 73, 016402 (2006)
With the improved collisional model, the LSP code reproduces the
blooming, straggling and penetration of monoenergetic relativistic
electron beams predicted by the theory5
Theory5
Simulation
E0  3 MeV
DT:
  300 g/cm3
ne  7  1025 cm-3
Te  Ti  5 keV
5C.
K. Li and R. D. Petrasso, Phys. Rev. E 73, 016402 (2006)
Conclusion
The LSP collisional model has been modified to include
relativistic effects and tested to reproduce correctly the collisional
slowing down of electrons predicted by the theory
• The non-relativistic LSP collisional model has been found correct
(except for the value of Coulomb logarithm which we modified) and
consistent with other theoretical models suggested for multi-fluid and
hybrid PIC simulations
• Relativistic electron scatter and drag rates have been calculated and
implemented in the LSP code
• The slowing-down of relativistic electron beams have been simulated
and compared with the theory
• With the improved collision model, the LSP code is capable of
reproducing the blooming, straggling and penetration predicted by the
theory