Published on October 29, 2007
Slide1: Test ion acceleration in the field of expanding planar electron cloud M. M. Basko Institute for Theoretical and Experimental Physics, Moscow 33rd EPS Conference on Plasma Physics Rome, Italy, 19-23 June 2006 Slide2: Question: how to calculate the maximum energy of an accelerated test ion? The usual approach, based on using the Boltzmann relation Motivation turns out to be inadequate because it leads to a diverging answer the true value of the “diverging” logarithmic factor depends on how accurately the Botlzmann-Maxwell distribution is established in its far tail, for which there simply may be not enough time after the laser pulse. Conclusion: in general, the Boltzmann relation cannot be used to solve our problem. Slide3: Formulation of the problem At time t=0 all electrons ― treated as a collisionless charged fluid ― of a uniform planar foil are accelerated to the same velocity v0. The main foil ions are assumed to be infinitely heavy and staying at rest. A test ion of charge +eZp and mass mp begins to accelerate at t=0 from its initial position x=0 at the foil surface. Questions: Can a test ion reach the electron velocity v0 ? Earlier this problem was considered by Bulanov et al. (Pl. Phys. Rep., 30, p.18, 2004); here we correct and expand the results of their work. Can it overtake the electron front ? Slide4: Principal parameters and variables There are three independent dimensionless parameters which govern our problem Principal unknown variables: Slide5: Electron trajectories xe(t,) Analytical results are most readily obtained for =0 (l0 0, n0l0 = finite). Slide6: Electron cloud structure Asymptotically, the relaxation zone expands linearly in time, with 0 = 0.250.02 for =0, 0=1. In the quasi-equilibrium core of the relaxation zone one can use the Boltzmann relation. In the laminar zone the present solution applies. Slide7: Motion of test ions (general features) Motion of a test ion can be analyzed analytically only in two cases: the ion trajectory lies entirely in the laminar zone; the ion trajectory lies entirely in the quasi-equilibrium core, where the hot electrons obey the Boltzmann relation. Our new exact results are exclusively for test ions accelerated in the laminar zone ! They apply within a limited domain of the , , 0 parametric space. Slide8: Ion equations of motion Dimensionless equations of motion (=0): Transformation to new “hyperbolic” variables: Phase equation: Integral trajectories in the , phase plane Slide9: Maximum energy of a test ion only sufficiently light test ions with > cr can catch up with the electron front and overtake it; Main results: for each value of 0 there exists a critical value cr=mcr(g0) of parameter = Zpme/mp such that Alternatively, for each value of m there exists a critical value of g0 = gcr(m) such that a test ion with mass ratio m overtakes the electron front when g0 > gcr(m) . heavier test ions with < cr lag behind the electron front, and only asymptotically, on a time scale ~exp(0.5 -1), approach the electron velocity v0; Slide10: Acceleration of test ions by ultra-relativistic electrons Only for highly ultra-relativistic electrons with 0 >> 1000 does the critical charge-over-mass ratio cr begin to drop significantly below 1/8 ! Slide11: Practical implications Acceleration of protons: protons can overtake the electron front only when 0 > 0.5 exp(1827.5) = 2·10793 ― which is beyond any realistic value ! for 0 -1 << 1 the proton trajectory lies inside the equilibrium Boltzmann zone: Target example: 1μm of gold (ionization z=50, ne=3x1023 cm-3), 0=100 = 10. for 0 > 348 ( = 0) → 537 ( > 3) the proton trajectory lies inside the laminar zone; Slide12: Conclusions In practice, protons and heavier test ions can never catch up with the electron front. If hot electrons are non-relativistic or moderately relativistic, acceleration of test ions in the non-equilibrium laminar zone near the electron front is relatively inefficient; the situation is close to the Boltzmann equilibrium. For highly relativistic electrons (0 > 300) the non-equilibrium structure of the expanding electron cloud becomes important. In the planar geometry, the maximum ion energy depends logarithmically either on elapsed time, or on the traveled distance.