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**Unformatted text preview: **(21) which justiﬁes the simple estimates used in Sec. II.C.
However, the electrostatic potential (20) leads to an inﬁnite
acceleration of a test proton which is initially at zero energy
in x ¼ 0. The reason is that the apparently reasonable choice
of the Boltzmann relation poses several difﬁculties to the
analysis (Passoni and Lontano, 2004) because, in order to
have an electron density equal to zero at inﬁnity, the selfconsistent electrostatic potential must diverge at large
Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013 765 distance from the target [mathematically, ! À1 as x !
þ1, see Eq. (15)]. This is not a pathological consequence of
the one-dimensional approximation but it is related instead to
the fact that the Boltzmann relation implies the existence of
particles with inﬁnite kinetic energy, which is not physically
meaningful [see also Sec. 38 of Landau and Lifshitz (1980)].
This unphysical behavior can be avoided by assuming an
upper energy cutoff E c in the electron distribution function, so
that e ! ÀE c as x ! þ1 and the electric ﬁeld turns to
zero at a ﬁnite distance. The cutoff assumption can be
justiﬁed as a consequence of the laser-solid interaction
producing electrons with a maximum kinetic energy and of
the escape from the system of the most energetic ones
(Lontano and Passoni, 2006; Passoni and Lontano, 2008).
Experimental indications of target charging due to electron
escape have been found by Kar et al. (2008b) and Quinn
et al. (2009a). The ﬁnite range of the electric ﬁeld driving
TNSA is also apparent in direct measurements (Romagnani
et al., 2005).
Still using the Boltzmann relation, it can be assumed the
1D solution given by Eq. (20) to hold only up to a longitudinal distance roughly equal to the transverse size of the
sheath, because at larger distances 3D effects should be taken
into account (Nishiuchi et al., 2006). Alternatively, by assuming that the hot electron population occupies only a ﬁnite
region of width h, the solution of Eq. (19) in the vacuum
region 0 < x < h together with the corresponding electric
ﬁeld and electron density can be determined (Passoni and
Lontano, 2004).
Another possibility, explored by Schreiber et al. (2006),
has been to heuristically assume that the hot electron expansion in vacuum creates a cylindrical quasistatic cloud in the
vacuum, behind the target, and a circular positive surface
charge on its rear face. The generated electrostatic potential is
evaluated on the symmetry axis, along which the most energetic ions are accelerated. The total surface charge and the
radius of the distribution are model parameters estimated
from experiments (see also Sec. III.D).
In order to consistently overcome the previously discussed
limits, Lontano and Passoni (2006) proposed to solve the
Poisson equation by assuming that a quasistationary state is
established where only those electrons (trapped electrons)
with negative total energy W ¼ mc2 ð À 1Þ À e are retain...

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