1. Collisional ionization in the beam body  Just behind the front, by continuity  →0 and the three body recombination  (T e,E) is negligible since T e ~ 1eV and n e <<n a. The return current heating is used for ionization:  The temperature is quasi-constant in this model and the drift velocity is not yet negligible: V E 2 ~ kT e /m  Numerically, we found that at a very small distance x’ (~ 1  m for n b =10 24 -10 26 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 Saha equilibrium quickly reached (~ 1  m) for an initial temperature in the body of about 2 eV or less Debayle A., Tikhonchuk V.T., Klimo O. UMR 5107- CELIA, CNRS - Université Bordeaux 1 - CEA Czech Technical University in Prague, FNSPE 112 1 2 Beam front description in the front reference frame  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 V E thus Keldysh formula for the tunnel ionization rate Low energy cross section approximation + single electron drift velocity ExEx X’X’ ni’ni’ nb’nb’ -x c ’ xf’xf’ EmEm n im ’ -v f t ’ 2n im ’ TeTe  n’ p pfpf p0p0 p pfpf p0p0 p pfpf p0p0 p pfpf p0p0 p pfpf p0p0 p pfpf p0p0 p pfpf p0p0 Electron coming back from the front f(p) Electrons slowed down in the beam body Eectrons moving faster than the front penetrate in The distribution function is even in the front reference frame The fastest electrons reach the beam front  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.  The main ionization processes, the beam energy losses and the target heating are analyzed analytically.  A 1D3V PIC simulations of a 8  m fast electron beam in a plastic target are presented (including electric- field ionization, collisional ionization of atoms by the plasma electrons, coulomb e-ion collisions and electron-atom collisions) Introduction hypotheses for Analytic resolution  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 (n i << n a ). The fast electron accumulation in the beam front produces the electrostatic field Fast ionization of a small part of the atoms by the electric field Quasi-constant electron temperature Weak contribution of the electric-field ionization in the beam body Fig4: Demonstration of the ionization process in the plastic target for the beam density 10 19 cm -3. Units normalized to maximum values (see legend), n b0 = 10 19 cm -3, V 2 = 0.9c, V 1 = 0.7c Strong contribution of the collisional ionization in the beam body PIC simulation with a step-like distribution function: Results of the beam front description  Results with two fast electron distribution functions:  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 Fig1 : the front velocity depending on the beam density with and without the energy loss S’ (dashed and solid lines) Fig2 : electric field maximum depending on the beam density with and without the energy loss S’ (dashed and solid lines).  b0 = 1 MeV 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’)V 1 = 0.7c, V 2 = 0.9c. Beam front velocity  Vlasov equation provides the density of fast electrons in function of the electric field potential,  min is the energy loss of the fastest electrons  Poisson equation provides the relaton of E x on . The electric field maximum is reached at x’=0:  The ion density conservation equation leads to: Quadratic approximation on E(  ) Linear approximation on n i (  ) 3 equations with 3 unknowns (V f, E m, n im ’) Beam body description  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 j b = cte Mono-energetic electrons Conclusion References 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 (< 10 24 m -3 ) The quasi-stationary approach is valid after a relaxation time: This time is short for distribution functions where the maximum is around V f : the electrons are quickly slowed to a velocity lower than V f In the beam front 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 T e The second contribution is the drift energy of electrons depending on E x. This contribution quickly disappears since the electric field decreases while the conductivity grows (~ 1  m) 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/V f ~ 3 - 4 fs) The energy loss in the beam body is slow compared to the relaxation time in the front t r. This confirms the quasi- stationary assumption. Energy loss in the beam body  In the quasi-constant temperature approximation the energy loss is: Fig7: front velocity V f dependence on time: n b0 = 10 26 m -3  b0 = 1MeV  Condition for the quasi-stationary solution: The relaxation time in the front must be shorter than the energy loss characteristic time in the beam body and Fig6: Simulation results for the front velocity V f depending on time Simulation results 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  m. 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 Qualitative description of the beam structure Transport of high current electron beams in dielectric targets Flat momentum distribution Current neutralization Fig5: Electron beam current density evolution Laser-electron conversion efficiencyFast electron temperature 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