Expansion of a finite-size plasma in vacuum

Abstract

The expansion dynamics of a finite size plasma is examined from an analytical perspective. Results regarding the charge distribution as well as the electrostatic potential are presented. The acceleration of the ions and the associated cooling of the electrons that takes place during the plasma expansion is described. An extensive analysis of the transition between the semi infinite and the finite size plasma behaviour is carried out. Finally, a test of the analytical results, performed through numerical simulations, is presented. PACS numbers:The production of energetic particles through the interaction of an intense laser pulse with a solid target is a topic that has been widely investigated during the last three decades. For sufficiently long pulses, the emitted particles emerge from a coronal plasma formed by the laser on the target foil and the expansion of the plasma in vacuum plays a key role in such a phenomenon. Thus, despite in Refs. [1,2,3,4] the expansion of a semi infinite plasma has been widely investigated, present experiments often involve "thin foils" as targets [5,6,7,8,9,10,11] which are in some cases no thicker than a few tens of a Debye length, and we believe that, in these cases, the experimental results must be analyzed in terms of the expansion of a finite size plasma. The finite-size and the semi-infinite cases are very different from each other, as in the latter case an infinite amount of energy is available. As a consequence, even if the ions are accelerated, the energy of the electron plasma remains constant. On the contrary, in the case of a finite size plasma an exchange of kinetic energy between electrons and ions takes place as long as the two populations have different velocity distributions, i.e., as long as a charge separation is present. The aim of this work is to provide a detailed analytical description of the thermal expansion of a globally neutral, finite size, unidimensional plasma. Analytical predictions will be compared with numerical results obtained with a Particle in Cell (PIC) code. The configuration at the initial time t 0 , which formally can be defined as the hydrostatic equilibrium in the limit of infinitely massive ions, is specified by the ion density n i0 (x) ≡ n i (x, t 0 ) = n 0 θ(a − |x|), and by the Boltzmann-like electron profile n e0 (x) ≡ n e (x, t 0 ) = n exp (eΦ(x, t 0 )/T e0 ), where a is the half-thickness of the plasma, 2an 0 is the total number of positively charged particle, θ(x) = 0 for x < 0 and θ(x) = 1 for x > 0, Φ(x, t 0 ) is the electrostatic potential,n is the density at the position where Φ(x, t 0 ) = 0, T e0 the initial electron plasma temperature and ∞ −∞ n e0 (x)dx = 2an 0 because the system is globally neutral. For the sake of * Electronic address: betti@df.unipi.it notational simplicity we are considering a plasma where the ion charge is equal and opposite of that of the electrons. Measuring space in units of the initial electron Debye length λ d,0 = (T e0 /4πe 2 n 0 ) 1/2 and rescaling the potential Φ(x, t) as Φ(x, t) → φ(x, t) = eΦ...

Collisionless expansion of a Gaussian plasma into a vacuum

“…͑6͒ still holds, with a time-dependent, but not space-dependent, temperature. Vlasov 35 or particle-in-cell codes 36 are more exact descriptions which up to now are used on a limited scale in the present context.…”

Influence of a finite initial ion density gradient on plasma expansion into a vacuum

“…On the other hand, for a finite plasma slab, the electrons cool down in the expansion while they progressively give their thermal energy to the ions. 18,[22][23][24][25][26] If the ion density is a step function at the plasma surface, the resulting maximum electric field associated with the electron Debye sheath is given by E = ͱ 2k B T e / e D , where T e is the electron temperature, e is the elementary charge, and D is the local Debye length. On the other hand, if the ion density has an initial scale length l ss with l ss ӷ D , the electron density is equal to the ion density, with E = k B T e / el ss , down to the point where the local Debye length exceeds the ion scale length, and where the electron Debye sheath appears.…”

Laser Acceleration of Ultrashort Ion Bunches and Femtosecond Neutron Sources

“…In the rear sheath acceleration (RSA) model, ions on the rear side of the target (especially protons from surface impurities) are accelerated up to MeV energies by the intense electrostatic field generated by the high-energy, "fast" electrons escaping from the target. A recent experiment measured this field directly [8] and confirmed the validity of models of plasma expansion in vacuum to describe the RSA mechanism [9,10].…”