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**Unformatted text preview: **Williams (1981), and Kishimoto et al.
(1983). More recent works stimulated by the TNSA experiments
include Kovalev and Bychenkov (2003), Mora (2003, 2005), Betti
et al. (2005), Ceccherini et al. (2006), and Peano et al. (2007).
11
A list of papers describing TNSA models mostly based on a
static modeling includes Passoni and Lontano (2004, 2008), Passoni
et al. (2004), Albright et al. (2006), Lontano and Passoni (2006),
Robinson, Bell, and Kingham (2006), and Schreiber et al. (2006). Andrea Macchi, Marco Borghesi, and Matteo Passoni: Ion acceleration by superintense laser-plasma . . . the electron cloud to be spatially uniform in the plane normal
to the ion motion.
Note that all the models proposed to describe TNSA are, to
a large extent, phenomenological, i.e., they need as input
parameters physical quantities which are not precisely
known. Since these descriptions give a simpliﬁed picture
of the acceleration process, the best model in this context
may be considered the one which provides the best ﬁt of
experimental data with the lowest set of laser and target
parameters. This issue will be discussed in Sec. III.D. In
principle, these difﬁculties could be overcome performing
realistic numerical simulations, but these generally consider
‘‘model’’ problems due to intrinsic difﬁculties in the numerical study of these phenomena, such as, for example, the large
variations of density from the solid target to the strongly
rareﬁed expansion front. At present, a complementary use
of simple models, presented in Secs. III.C.1, III.C.2, and
III.C.3, and advanced simulations, discussed in Sec. III.C.4,
seems the most suitable option to theoretically approach
TNSA.
1. Quasistatic models Static models assume, on the time scale of interest (i.e., in
the sub-ps regime), immobile heavy ions, an isothermal that
laser-produced hot electron population, and a sufﬁciently low
number of light ions so that their effect on the evolution of the
potential can be neglected and they can be treated as test
particles. In this limit, if Eq. (15) is used to describe hot
electrons and one neglects thermal effects for cold electrons,
the potential in planar geometry is determined by
@2
¼ 4e½n0h ee=Th À ðZH n0H À n0c Þ
@x2
¼ 4en0h ½ee=Th À ÂðÀxÞ; (19) where we assumed the background charge to ﬁll the x < 0
region with uniform density. The corresponding electron
density and electric ﬁeld can be calculated, as well as the
energies of test ions moving in such potential. This can be
considered the simplest self-consistent approach to describe
the TNSA accelerating ﬁeld. The solution of Eq. (19) in the
semi-inﬁnite region x > 0 is (Crow, Auer, and Allen, 1975)
2T
x
À1 ;
(20)
ðxÞ ¼ À h ln 1 þ pﬃﬃﬃﬃﬃ
e
2eDh
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
where Dh ¼ Th =4e2 n0h . The ﬁeld reaches its maximum
at the surface and is given by
sﬃﬃﬃ
2
Eð0Þ ¼ E0 ;
e E0 ¼ Th
;
eDh...

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