Transcript Slide 1

Transport of high current electron beams in dielectric targets
1
1
Debayle A., Tikhonchuk V.T., Klimo O.
2
1
UMR 5107- CELIA, CNRS - Université Bordeaux 1 - CEA
2
Czech Technical University in Prague, FNSPE
Introduction

Beam front description in the
front reference frame
Efficient transformation of high intensity laser pulse into relativistic electron beam with high current density in
metal and insulator targets demonstrated by recent experiments.
Laser-electron conversion efficiency
Fast electron temperature
 Evolution of the electron distribution function:
Beam body description
 ee
xf

The energy loss in the beam body is supposed to be small: the
propagation is stationary.

The ionization by the electric field is higher than the collisionnal
ionization in the beam front since there is no free electrons in a
dielectric.
Vf
 They are heated by the elastic collisions electron-ion and
electron-atom:
T
n
C e  je E  gJ z i
t
t

2
e ne
je  E 
E
m e

Beam propagation through metals allowed by the current neutralization thanks to the target free electrons.

jb   je

The main ionization processes, the beam energy losses and the target heating are analyzed analytically.

A 1D3V PIC simulations of a 8 mm fast electron beam in a plastic target are presented (including electricfield ionization, collisional ionization of atoms by the plasma electrons, coulomb e-ion collisions and
electron-atom collisions)
 The ionization is due to the inelastic electron-ion collisions:
ni
n '
2
 V f i   (Te , E )ne (na  ni )  b (Te , E )ne ni
t
x'
 The electric field ionization
 The collisional ionization by the return current electrons moving
with the drift velocity VE
ni’
dni '
d
  ( E x )na   coll ni 
(ne ' (VE "V f ))
dx'
dx'
E
aE
 ( E x )   a a exp( a )
Keldysh formula for
Ex
Ex
the tunnel ionization
2
rate
mV
 coll  C VE ( E  1)na
2J z
Vf
VE 
 eEx
2
 VE "  f
m e
jb

  gJzV f

dni '
dx'
nmi ' S ' ( (0))  0
nb ' ( )d
 ( 0)
nim '  
 min
( ( Ex )na  collni )dx'
Quadratic approximation
on E()
Linear approximation on
ni()
3 equations with 3 unknowns (Vf, Em, nim’)
Em
Results of the beam front description
 Results with two fast electron distribution functions:
→0 and the three
n’
body recombination b(Te,E) is negligible since Te ~ 1eV and
ne<<na. The return current heating is used for ionization:
2
Vf
The ion density conservation equation leads to:
Low energy cross section
approximation + single
electron drift velocity
Te
 Just behind the front, by continuity
nb ' ( (0)) 
 ( 0)
Collisional ionization in the beam body
Te
t
VE m "
 0 Em2  2
Qualitative description of the beam structure
 The quasi-stationary state implies jb = cte
V f pb 0
)
c mc
 Atom ionization is described by:
 min
 The beam charge is neutralized behind the beam front thanks
to the newborn plasma electrons:
 b 0 '   f ( b 0 
Poisson equation provides the relaton of Ex on
. The electric field maximum is reached at x’=0:
 A part of the beam energy is lost for the ionization:
J dni '
S' z
eEx dx'
The ionization of atoms by the electric field in the beam front is weak
(ni << na).
Necessity of a strong ionization of the dielectric target by the fast electron beam during the propagation
time.
1
and
nb ' ( ) 
 0 1   / min
e min  mc 2 ( b 0 '1)
nb 0
E
 0 x  e(n'nb ' S ' )
x'
 The plasma electrons are thermalized in the e-e collision time:

Vlasov equation provides the density of fast
electrons in function of the electric field potential,
min is the energy loss of the fastest electrons
f ' (mc2  e )
f '
f '
thus
v'
 eEx
0
d
x'
p'
Ex  
dx '
 Electric field created by the fast electron charge accumulation:
hypotheses for Analytic resolution
1
Beam front velocity
ni ' ( x' )  nim '2 
2m e f jb
f ( p)  nb0 ( p  p0 )
f ( p)  nb0 H (V2  V ) H (V  V1 )
2nim’
nb’
2
ge2 J aV f
nim’
( x' xc ' )
Ex
X’
 The temperature is quasi-constant in this model and the drift
velocity is not yet negligible: VE2 ~ kTe/m

VE  kTe / m
x' 
ge2 J zV f
2 j m e f
2
b
(
m 2f jb2
2
e kTe
 4nim '2 )
 Numerically, we found that at a very small distance x’ (~ 1mm
for nb=1024 -1026 m-3): the drift velocity falls below the thermal
velocity  the collisional ionization is due to thermal electrons
jb2
2

2
Te0
ln( /(4nim
b )  1)
f(p)
Fig1 : the front velocity depending
on the beam density with and without
the energy loss S’ (dashed and solid lines)
Mono-energetic
electrons
pf p0
 the three body recombination rate equals to the collisional
ionization rate
 the heat rate is well above the collisional ionization rate
Saha equilibrium quickly reached
(~ 1 mm) for an initial temperature
in the body of about 2 eV or less
xf’
f(p)
f(p)
 The Saha equilibrium for electrons is reached if :
 (Te )ni  b (Te )ni3
-xc’
-vft’
p
f ( p)  nb0 ( p  p0 )
f(p)
Electron coming back
from the front
pf p0 p
Electrons slowed down
in the beam body
pf p0 p
f ( p)  nb0 ( p  p f )
f ( p)  nb0 ( p  p0c )  nb0 ( p  p0c,s )
f(p)
f(p)
The fastest electrons
reach the beam front
 The front velocity increases with the beam density and
tends to the maximum electron velocity
f(p)
Flat momentum
distribution
pf p0
Energy loss in the beam body
p
f ( p)  nb0 H ( p0  p) H ( p)
f ( p)  nb0 H ( p0  p) H ( p)
pf p0
Eectrons moving faster
than the front penetrate in
 In the quasi-constant temperature
approximation the energy loss is:

geJaV f 
2m e f jb2 '
'2
'
' 
(x )  
4nim 
( xc  V f t )  2nim

mc2 jb 
ge2 J aV f


p
pf p0
p
pf p0
Strong contribution of the collisional
ionization in the beam body
 Condition for the quasi-stationary
solution:
m v'
1 d c

'
c dt
e E
p
The distribution function is
even in the front reference
frame
PIC simulation with a step-like distribution function:
'
c
f ( p)  nb0 H (V2  V ) H (V  V1 )
The fast electron accumulation in the beam
front produces the electrostatic field
Fig7: front velocity Vf dependence on time: nb0= 1026 m-3
b0= 1MeV
Fast ionization of a small part of
the atoms by the electric field
 The electric field maximum weakly depends on the
beam density : the same amount of electrons penetrate
the beam front
Current neutralization
The electric field reaches values around 5 – 10% of the
atomic electric field and is therefore the main ionization
process in the beam front with the width of about 1 - 3 mm.
The front velocity depends on the beam density and
decreases slowly with time
The collisional ionization is the main process in the beam
body. It represents the main cause of atom ionization since
the electric field contribution is less than 15% of the matter
The plasma electron energy is quasi-constant in the
beam tail: this is consistent with the quasi-total conversion
of the ohmic heating into the collisional ionization process
Good current neutralization behind the beam front
Weak contribution of the electric-field
ionization in the beam body
Conclusion
In the beam front
The front velocity increases with the beam density: possible
filamentation instability
The electric field depends weakly on the beam density: the same
amount of electron penetrate the front
The energy loss caused by the ionization is not negligible for weak
beam densities (< 1024 m-3)
The quasi-stationary approach is valid after a relaxation time:
References
tr 
M. Manclossi et al, Phys. Rev. Lett., 96, 125002 (2006)
S.I. Krasheninnikov et al, Phys. Plasmas, 12, 1 (2005)
V.T. Tikhonchuk, Phys. Plasmas, 9, 1416 (2002)
O. Klimo et al, Phys. Rev. E, submitted
Fig6: Simulation results for the front velocity Vf
depending on time
Fig2 : electric field maximum depending
on the beam density with and without
the energy loss S’ (dashed and solid lines).
b0 = 1 MeV
Simulation results
Quasi-constant electron temperature
The relaxation time in the front must
be shorter than the energy loss
characteristic time in the beam body
Fig3 : front velocity, the electric field maximum,
the maximum ion density and the front thickness
depending on the beam density (without the energy
loss S’)V1 = 0.7c, V2 = 0.9c.
f0
e E
This time is short for distribution functions where the maximum is
around Vf: the electrons are quickly slowed to a velocity lower than
Vf
Fig4: Demonstration of the ionization process in the plastic target for the
beam density 1019 cm-3. Units normalized to maximum values (see legend),
nb0 = 1019 cm-3, V2 = 0.9c, V1 = 0.7c
m v'
Fig5: Electron beam current density evolution
In the beam body
The collisional ionization near the front is far from a
thermodynamic equilibrium. All the Ohmic heating is
converted into the collisional ionization.
This collisional ionization is split in two parts:
The first contribution is the thermal energy of
electrons depending on Te
The second contribution is the drift energy of
electrons depending on Ex. This contribution quickly
disappears since the electric field decreases while the
conductivity grows (~ 1mm)
The Saha equilibrium depends greatly on the temperature
just behind the ionization front. For a weak temperature
(~2eV), the thermodynamic equilibrium occurs after a time
around (t=x/Vf ~ 3 - 4 fs)
The energy loss in the beam body is slow compared to
the relaxation time in the front tr. This confirms the quasistationary assumption.