Collisional ionization in the beam body

1
Transport of high current electron beams in
dielectric targets
1
1
2
Debayle A., Tikhonchuk V.T., Klimo O.
UMR 5107- CELIA, CNRS - Université Bordeaux 1 -
CEA
1
Czech Technical University in Prague, FNSPE
2
Introduction
Beam front velocity
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.
• Beam propagation through metals allowed by the
  current neutralization thanks to the target free
  electrons.
• Necessity of a strong ionization of the
  dielectric target by the fast electron beam
  during the propagation time.

Laser-electron conversion efficiency
Fast electron temperature
Beam body description

• Evolution of the electron distribution function
• Electric field created by the fast electron
  charge accumulation
• A part of the beam energy is lost for the
  ionization
• Atom ionization is described by
• The electric field ionization
• The collisional ionization by the return current
  electrons moving with the drift velocity VE

• Vlasov equation provides the density of fast
  electrons in function of the electric field
  potential, jmin is the energy loss of the fastest
  electrons

thus
hypotheses for Analytic resolution
and

• The plasma electrons are thermalized in the e-e
  collision time
• They are heated by the elastic collisions
  electron-ion and electron-atom
• The beam charge is neutralized behind the beam
  front thanks to the newborn plasma electrons
• The ionization is due to the inelastic
  electron-ion collisions
• The quasi-stationary state implies jb cte

• 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.
• The ionization of atoms by the electric field in
  the beam front is weak (ni ltlt na).

• Poisson equation provides the relaton of Ex on j.
  The electric field maximum is reached at x0

• The ion density conservation equation leads to

Keldysh formula for the tunnel ionization rate
Quadratic approximationon E(j)
Linear approximation onni(j)
Qualitative description of the beam structure
Low energy cross section approximation
single electron drift velocity
3 equations with 3 unknowns (Vf, Em, nim)
Collisional ionization in the beam body
Results of the beam front description
ni
Em
Te

• Results with two fast electron distribution
  functions

• Just behind the front, by continuity
  ?0 and the three body recombination b(Te,E) is
  negligible since Te 1eV and neltltna. The return
  current heating is used for ionization
• The temperature is quasi-constant in this model
  and the drift velocity is not yet negligible VE2
  kTe/m
• Numerically, we found that at a very small
  distance x ( 1mm for nb1024 -1026 m-3) the
  drift velocity falls below the thermal velocity ?
  the collisional ionization is due to thermal
  electrons
• The Saha equilibrium for electrons is reached if
  
• the three body recombination rate equals to the
  collisional ionization rate
• the heat rate is well above the collisional
  ionization rate

dn
2nim
nb
nim
Ex
X
-xc
xf
-vft
f(p)
f(p)
f(p)
Fig1 the front velocity dependingon the beam
density with and withoutthe energy loss S
(dashed and solid lines)
Mono-energetic electrons
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.
The fastest electronsreach the beam front
Electrons slowed down in the beam body
p
p
p
pf
p0
pf
p0
pf
p0
Electron coming back from the front

• The front velocity increases with the beam
  density and tends to the maximum electron
  velocity
• The electric field maximum weakly depends on the
  beam density the same amount of electrons
  penetrate the beam front

Saha equilibrium quickly reached ( 1 mm) for an
initial temperature in the body of about 2 eV or
less
f(p)
f(p)
f(p)
f(p)
Flat momentumdistribution
p
p
Fig2 electric field maximum depending on the
beam density with and without the energy loss S
(dashed and solid lines).eb0 1 MeV
p
p
pf
p0
pf
p0
pf
p0
pf
p0
Energy loss in the beam body
Eectrons moving faster than the front penetrate
in
Simulation results
The distribution function iseven in the front
referenceframe

• In the quasi-constant temperature approximation
  the energy loss is

• 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

PIC simulation with a step-like distribution
function
Strong contribution of the collisionalionization
in the beam body
The fast electron accumulation in the beamfront
produces the electrostatic field

• Condition for the quasi-stationary solution

Quasi-constant electron temperature
Fast ionization of a small part of the atoms by
the electric field
Current neutralization
Weak contribution of the electric-fieldionization
in the beam body
Fig7 front velocity Vf dependence on time nb0
1026 m-3eb0 1MeV
The relaxation time in the front must be shorter
than the energy loss characteristic time in the
beam body
Conclusion
In the beam body
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 sameamount of electron penetrate
  the front
• The energy loss caused by the ionization is not
  negligible for weak beam densities (lt 1024 m-3)
• The quasi-stationary approach is valid after a
  relaxation time
• This time is short for distribution functions
  where the maximum is around Vf the electrons are
  quickly slowed to a velocity lower than Vf

• 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 (tx/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 quasi-stationary assumption.

References
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
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
Fig6 Simulation results for the front velocity
Vf depending on time
Fig5 Electron beam current density evolution