*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **selfsimilar solution for isothermal plasma expansion; Eq. (22). The
front of charge separation at x ¼ xf ðtÞ and the rarefaction front at
x ¼ Àcs t are also indicated. The electric ﬁeld is uniform in the
Àcs t < x < xf ðtÞ region. Andrea Macchi, Marco Borghesi, and Matteo Passoni: Ion acceleration by superintense laser-plasma . . . 767 the inertia of ions is important and the assumption of quasineutrality must be abandoned. Ultimately, a self-consistent
analysis can be developed through numerical simulations (see
Sec. III.C.4). Still assuming, for simplicity, that only a single
ion population and a single-temperature Boltzmann electron
population are present, and ni ðt ¼ 0Þ ¼ n0 ÂðÀxÞ, Eq. (20)
can be used to deﬁne the initial conditions for the electric
ﬁeld at the time t ¼ 0 at which the ion acceleration process
begins. The following interpolation formulas for the electric
ﬁeld and ion velocity at the ion front:
sﬃﬃﬃ
2 E0
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ;
E ðtÞ ’
e 2 þ 1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
uf ðtÞ ’ 2cs lnð þ 2 þ 1Þ; (23) (24) pﬃﬃﬃﬃﬃ
where ¼ !pi t= 2e, give the correct behavior at t ¼ 0 for
both the electric ﬁeld [see Eq. (21)] and the front velocity, and
reduce to previous expressions for !pi t ) 1. These formulas
ﬁt well numerical calculations by Mora (2003) using a
Lagrangian ﬂuid code. Related results of similar studies using
ﬂuid and kinetic descriptions can be found in the literature.12
The major drawback of Eq. (24) is that the maximum
velocity of ions, and hence the cutoff energy, diverges logarithmically with time. This is not surprising, being an unavoidable consequence of the isothermal assumption and the
chosen boundary conditions: the system has an inﬁnite energy
reservoir in the electron ﬂuid and thus it is able to accelerate
ions indeﬁnitely. Nevertheless, the simplicity of Eq. (24) has
proven to be attractive; thus it has been suggested to insert a
phenomenological ‘‘maximum acceleration time’’ tacc at
which the acceleration should stop. Such a formula has been
used in attempts to ﬁt experimental data (Fuchs et al., 2006b).
We return to this point in Sec. III.D. There is no easier way to
remove this unphysical behavior from the 1D planar model
but to give a constraint of ﬁnite energy (per unit surface). In
this way, the electron temperature decays in time due to the
plasma expansion and to collisional and radiative losses. The
electron cooling cooperates with the effects of ﬁnite acceleration length and maximum electron energy in the determination
of a ﬁnite value for the maximum energy gain.
The expansion of plasma slabs (foils) of ﬁnite thickness,
and hence of ﬁnite energy, has been considered analytically
and numerically. In these models the electron temperature is
taken as a function of time Th ¼ Th ðtÞ, determined either
by the energy conservation equations (Betti et al., 2005;
Mora, 20...

View
Full
Document