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

**Unformatted text preview: **ed, while those with positive total energy are lost from
the system. The corresponding trapped electron density, given
R
by nh ¼ W<0 fe ðx; pÞdp, is included in the Poisson equation
and the analytical solutions are determined (Lontano and
Passoni, 2006; Passoni and Lontano, 2008; Passoni,
Bertagna, and Zani, 2010b). As a general feature, the potential, the electrostatic ﬁeld, and the electron density distributions go to zero at a ﬁnite position xf of the order of several
hot Debye lengths.
If both electron populations, hot and cold, are considered,
it is possible to ﬁnd an implicit analytical solution of Eq. (19)
both inside the target and in the vacuum region. Using
a two-temperature Boltzmann relation to describe the electron density, that is, ne ¼ n0h expðe=Th Þ þ n0c expðe=Tc Þ,
the electric ﬁeld proﬁle turns out to be governed by the
parameters a n0c =n0h and b Tc =Th , as shown in
Fig. 15 (Passoni et al., 2004). The presence of the cold 766 Andrea Macchi, Marco Borghesi, and Matteo Passoni: Ion acceleration by superintense laser-plasma . . . 2. Plasma expansion into vacuum FIG. 15. Electric ﬁeld proﬁle in a sheath with two electron
temperatures. The ﬁeld is normalized to Th =eDh and is shown
for cold-to-hot electron temperature ratio b ¼ Tc =Th ¼ 0:01 and
for different values of the pressure ratio ab ¼ p0c =p0h ¼ 1
(dotted line), ab ¼ 10 (dashed line), and ab ¼ 100 (solid line).
The x coordinate is normalized to the cold electron Debye length
Dc corresponding to ab ¼ 10. From Passoni et al., 2004. electron population strongly affects the spatial proﬁles of the
ﬁeld, which drops almost exponentially inside the target over
a few cold electron Debye lengths. An estimate of Tc , as
determined by the Ohmic heating produced by the return
current (see Sec. II.B.3), is required. A simple analytical
model of the process has been proposed (Davies, 2003;
Passoni et al., 2004), to which we refer for further details
and results.
The quasistatic approach allows one to draw several general properties of the accelerating TNSA ﬁeld. The spatial
proﬁle is characterized by very steep gradients, with the ﬁeld
peaking at the target surface and decaying typically over a
few m distance. The most energetic ions, accelerated in the
region of maximum ﬁeld, cross the sheath in a time shorter
than the typical time scale for plasma expansion, electron
cooling, and sheath evolution. As a consequence the static
approximation will be more accurate for the faster ions.
Assuming a time-independent ﬁeld also requires the electron
cloud to not be affected by the ions ﬂowing through it, which
implies a number of accelerated ions much smaller than that
of the hot electrons, Ni ( Ne . A quasistatic model not
requiring this assumption was proposed by Albright et al.
(2006) who included effects of the accelerated ion charge
on the electric ﬁeld by modeling the layer of light ions
(having areal charge density QL ) as a surfac...

View
Full
Document