State is established where only those electrons

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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...
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This document was uploaded on 09/28/2013.

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