**Unformatted text preview: **9 W cmÀ2 . Bottom: The effect of electron removal
by magnetic ﬁelds, showing that the proton beam emittance is not
signiﬁcantly affected. The images are for 6.5 MeV protons and the
target thickness is 40 m. From Cowan et al., 2004. Andrea Macchi, Marco Borghesi, and Matteo Passoni: Ion acceleration by superintense laser-plasma . . . 764 emittance, as shown in Fig. 14 (Cowan et al., 2004). This last
observation is important since, in order to take advantage of
the exceptionally small proton beam emittance in future
applications, e.g., to capture them into a postaccelerator,
removal of the comoving electrons without signiﬁcantly perturbing the protons is crucial.
The ultralow emittance stems from the extremely strong,
transient acceleration that takes place from a cold, initially
unperturbed surface and from the fact that during much of the
acceleration the proton space charge is neutralized by the
comoving hot electrons. Using the ion beam as a projection
source, having a low-emittance beam is equivalent to projecting from a virtual pointlike source located in front of the
target, with much smaller transverse extent than the ionemitting region on the target surface (Borghesi et al., 2004;
Nuernberg et al., 2009). As discussed in Sec. V.A, this property of laser-driven ion beams allows one to implement pointprojection radiography with high spatial resolution.
C. TNSA modeling The experimental observations and the considerations
summarized in Sec. III.A suggest the following assumptions,
leading to the formulation of a relatively simple system of
equations which can be investigated analytically and numerically (Passoni et al., 2004). First, we assume an electrostatic
approximation, so that the electric ﬁeld E ¼ Àr where the
potential satisﬁes Poisson’s equation
X
(14)
r2 ¼ 4e ne À Zj nj ;
j with the sum running over each species of ions, having
density nj and charge Zj . As a consequence of the laser-solid
interaction, the electron density ne may be described as
composed of at least two qualitatively distinct populations,
which will be labeled cold and hot in the following, having
densities nc and nh such that ne ¼ nc þ nh . In the simplest
approach, thermal effects are neglected for the cold population, while nh is given by a one-temperature Boltzmann
distribution (notice that in this section ‘‘e’’ indicates the
mathematical constant e ¼ expð1Þ ¼ 2:718 28 Á Á Á while e
indicates as usual the elementary charge),
nh ¼ n0h ee=Th : (15) This expression can be a reasonable ﬁrst approximation to
account for the presence of the self-consistent sheath ﬁeld and
has been used in many works on TNSA9 but, as discussed
below, it can lead to serious problems when the main goal is
the estimation of the maximum energy of the accelerated
ions. Alternatively, the electron dynamics can be included via
either ﬂuid or kinetic equations. It is mostly appropriate to
consider two different ion species, a light (L) and a...

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