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**Unformatted text preview: **heavy (H)
population: in this way it is possible, e.g., to model the
acceleration of light species present on the surfaces of a solid
target made of heavy ions.
9
See, e.g., Mora (2003), Passoni et al. (2004), Albright et al.
(2006), Nishiuchi et al. (2006), and Robinson, Bell, and Kingham
(2006). Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013 Depending on the description of the ion populations, two
main categories of TNSA models, to be discussed in detail in
Secs. III.C.1 and III.C.2, respectively, may be identiﬁed as
follows. The ﬁrst includes static models in which it is assumed
that the light ions, or at least the most energetic ones, are
accelerated in the early stage of the formation of the sheath, so
that the latter may be assumed as stationary. With these
conditions, the effects of the light ions on the electrostatic
potential are usually neglected, while the heavy ion population
of the target is considered immobile. The aim is thus to
provide the most accurate description of the sheath depending
on assumptions on the hot electron distributions. The second
category includes dynamic models where the system is described as a neutral plasma in which the ions acquire kinetic
energy in the course of the sheath evolution. In several cases a
unique ion component is considered. This approach is therefore strongly connected to the classic problem of plasma
expansion in vacuum, ﬁrst considered by Gurevich,
Pariiskaya, and Pitaevskii (1966). In a cold ﬂuid description,
neglecting relativistic effects the ions are described by
@uj
Zj e
þ uj Á ruj ¼ À
r;
@t
mj (16) @nj
þ r Á ð nj u j Þ ¼ 0
@t (17) ðj ¼ L; H Þ; where uj ¼ uj ðr; tÞ is the ﬂuid velocity. If the ions are
described kinetically, their Vlasov equation for the phasespace distribution fj ¼ fj ðr; v; tÞ is
@fj
Zj e
@fj
þ v Á rfj À
¼ 0:
r Á
@t
@v
mj (18) Most of the general studies of plasma expansion and related
ion acceleration developed both before and after TNSA experiments10 as well as more speciﬁc models of TNSA11 so far
proposed in the literature can be considered as suitable simpliﬁcations of the previous equations, falling into one of the
two above mentioned categories (or suitable combinations of
them) and obtained adding further, physically motivated assumptions. Most of these models assume a 1D geometry,
consistently with the electrostatic approximation, and planar
in most cases. This latter assumption, when applied to TNSA
modeling, requires the rear surface to be sufﬁciently ﬂat and
10
General studies of plasma expansion in vacuum include
Gurevich, Pariiskaya, and Pitaevskii (1966), Allen and Andrews
(1970), Widner, Alexeff, and Jones (1971), Crow, Auer, and Allen
(1975), Pearlman and Morse (1978), Denavit (1979), Mora and
Pellat (1979), and Dorozhkina and Semenov (1998). Early papers
focused on modeling ion acceleration in laser-produced plasmas
include Pearlman and Morse (1978), Wickens, Allen, and Rumsby
(1978), True, Albritton, and...

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