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**Unformatted text preview: **e layer of density
nL ¼ ðQL =ZL eÞðx À xL Þ. Equation (19) is then solved as a
function of the instantaneous position xL . An extension of this
model, using an adiabatic description of the hot electrons,
was proposed by Andreev et al. (2008) to investigate the
variation of the maximum light ion energy as a function of the
heavy ion target thickness.
On the basis of the above discussion we expect static
models to be most reliable for estimating the cutoff in the
ion energy spectrum. This estimate requires a few parameters
as input, depending on the model. This issue will be discussed
in Sec. III.D.
Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013 A description of ion acceleration over relatively long times
and/or under conditions such that the quasistatic modeling of
Sec. III.C.1 is not valid anymore demands for the inclusion of
the ion dynamics. The description may be based either on a
ﬂuid model, using Eqs. (16) and (17), or on a kinetic one
using Eq. (18).
The simplest approach is obtained using a 1D ﬂuid approach, invoking quasineutrality, using Eq. (15) and assuming
a single ion and electron population expanding in the semiinﬁnite space x > 0. Equation (19) is substituted by the
simpler condition ne ¼ Zi ni , the index i denoting the single
ion component. The boundary conditions are that the electron
density should remain equal to the background value well
inside the plasma, so that ne ðÀ1Þ ¼ n0 , and should vanish in
vacuum far from the surface, i.e., ne ðþ1Þ ¼ 0. Together with
Eqs. (16) and (17), the resulting system admits the classical
self-similar solution ﬁrst found by Gurevich, Pariiskaya, and
Pitaevskii (1966),
x
x
(22)
À1 ;
ui ¼ cs þ ;
ni ¼ n0 exp À
cs t
t
where x=t is the self-similar variable, L ¼ ni =j@x ni j ¼ cs t is
the local density scale length, and the expressions are valid
for x > Àcs t. The proﬁles corresponding to Eqs. (22) are
sketched in Fig. 16.
As a consequence of the quasineutral approximation, the
physical quantities describing the plasma dynamics present
several diverging behaviors, such as the unlimited increase of
ui with x. This implies that the neutral solution must become
invalid at some point, which can be estimated by equating
the local density scale length L to the local Debye length D .
This provides xf ðtÞ ¼ cs t½2 lnð!pi tÞ À 1, the corresponding velocity uf ¼ dxf =dt ¼ 2cs lnð!pi tÞ, and the electric
ﬁeld at the ion front Ef ¼ Eðxf Þ ¼ 2E0 =!pi t, where E0 ¼
ð4n0 Th Þ1=2 . This estimate gives twice the self-similar ﬁeld
E ¼ Th =ecs t. The argument also deﬁnes the front of the
fastest ions moving at velocity uf and thus it gives also the
high-energy cutoff in the energy spectrum of the ions in this
description.
Equations (22) are also singular for t ! 0, i.e., at
the earliest instants of the expansion, when quasineutrality also breaks down. In general, in the sub-ps regime FIG. 16. Sketch of the density and velocity proﬁles from the...

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