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**Unformatted text preview: **an (1978), Wickens,
Allen, and Rumsby (1978), Gurevich, Anderson, and Wilhelmsson
(1979), True, Albritton, and Williams (1981), Kovalev, Bychenkov,
and Tikhonchuk (2002), and Diaw and Mora (2011). 768 Andrea Macchi, Marco Borghesi, and Matteo Passoni: Ion acceleration by superintense laser-plasma . . . electric ﬁeld (Tikhonchuk et al., 2005), obtaining for the velocpﬃﬃﬃ
ity proﬁle vL ’ cL 2ð1 þ x=cH tÞ1=2 . Noticeably, the L ions
velocity and density proﬁles vary slowly in space compared to
the H ion ones, and the L ion ﬂux is almost constant. Beyond the
H ion front, only L ions are present and they can be described
again by a single species expansion with vL ’ cL þ x=t [see
Eq. (22)]. However, matching of the velocity proﬁles in the
region behind the H ion front implies the existence of a transition region where the velocity is approximately constant. This
corresponds to a plateau region in the phase space and to a peak
in the L ion energy spectrum. The heuristic reason for plateau
formation is that the L ions are accelerated more efﬁciently
behind the H ion front than ahead of it. Figure 17 shows the
velocity spectrum from numerical results (Tikhonchuk et al.,
2005) using a Boltzmann-Vlasov-Poisson model (Bychenkov
et al., 2004) based on Eqs. (14), (15), and (18), compared with
analytical estimates from the self-similar solution.
According to the above model the peak energy of L
ions is
pﬃﬃﬃﬃﬃﬃﬃ
(26)
E L ’ ZL Th lnð4 2N=eÞ:
As an important indication from this model, the mass ratio
and the relative concentration of the two species might be
engineered to optimize the L ion spectrum. Several simulation studies14 have been devoted to this issue and to the
modeling of observations of multispecies spectra in both
planar and spherical (droplet) targets (see Sec. III.E).
4. Numerical simulations Even in their simplest formulation TNSA models are
highly nonlinear and the set of available analytical solutions
is limited. A numerical approach can be used to overcome
these limitations and to address additional effects.
Referring to the 1D problem of plasma expansion, an hydrodynamic two-ﬂuid approach may be used to take charge separation effects into account as reported by Mora (2003). The
hydrodynamic model, however, cannot take into account kinetic effects such as non-Maxwell distribution and breakdown
of equilibrium conditions. To address these effects a numerical
solution of the Vlasov equation for the distribution function of
electron and ions in phase space is needed. To this aim the PIC
approach (see Sec. II.D) may be used, with the drawback of
much larger computational requirements compared to hydrodynamics simulations. The reason is that, in order to obtain full
numerical convergence and accurate, low-noise results, a very
large number of particles should be used to resolve the strong
density variations in the plasma expansion.
In its simplest formulation the 1D simulation of collisionless
plasma exp...

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