Of the accelerated ion charge on the electric eld by

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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 1Š, 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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This document was uploaded on 09/28/2013.

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