Dh th 4e2 n0h the eld reaches its maximum at the

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Unformatted text preview: (21) which justifies the simple estimates used in Sec. II.C. However, the electrostatic potential (20) leads to an infinite acceleration of a test proton which is initially at zero energy in x ¼ 0. The reason is that the apparently reasonable choice of the Boltzmann relation poses several difficulties to the analysis (Passoni and Lontano, 2004) because, in order to have an electron density equal to zero at infinity, the selfconsistent electrostatic potential must diverge at large Rev. Mod. Phys., Vol. 85, No. 2, April–June 2013 765 distance from the target [mathematically,  ! À1 as x ! þ1, see Eq. (15)]. This is not a pathological consequence of the one-dimensional approximation but it is related instead to the fact that the Boltzmann relation implies the existence of particles with infinite kinetic energy, which is not physically meaningful [see also Sec. 38 of Landau and Lifshitz (1980)]. This unphysical behavior can be avoided by assuming an upper energy cutoff E c in the electron distribution function, so that e ! ÀE c as x ! þ1 and the electric field turns to zero at a finite distance. The cutoff assumption can be justified as a consequence of the laser-solid interaction producing electrons with a maximum kinetic energy and of the escape from the system of the most energetic ones (Lontano and Passoni, 2006; Passoni and Lontano, 2008). Experimental indications of target charging due to electron escape have been found by Kar et al. (2008b) and Quinn et al. (2009a). The finite range of the electric field driving TNSA is also apparent in direct measurements (Romagnani et al., 2005). Still using the Boltzmann relation, it can be assumed the 1D solution given by Eq. (20) to hold only up to a longitudinal distance roughly equal to the transverse size of the sheath, because at larger distances 3D effects should be taken into account (Nishiuchi et al., 2006). Alternatively, by assuming that the hot electron population occupies only a finite region of width h, the solution of Eq. (19) in the vacuum region 0 < x < h together with the corresponding electric field and electron density can be determined (Passoni and Lontano, 2004). Another possibility, explored by Schreiber et al. (2006), has been to heuristically assume that the hot electron expansion in vacuum creates a cylindrical quasistatic cloud in the vacuum, behind the target, and a circular positive surface charge on its rear face. The generated electrostatic potential is evaluated on the symmetry axis, along which the most energetic ions are accelerated. The total surface charge and the radius of the distribution are model parameters estimated from experiments (see also Sec. III.D). In order to consistently overcome the previously discussed limits, Lontano and Passoni (2006) proposed to solve the Poisson equation by assuming that a quasistationary state is established where only those electrons (trapped electrons) with negative total energy W ¼ mc2 ð À 1Þ À e are retain...
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