Self-similar expansion of finite-size non-quasi-neutral plasmas into vacuum: Relation to the problem of ion acceleration

A new self-similar solution is presented which describes nonrelativistic expansion of a finite plasma mass into vacuum with a full account of charge separation effects. The solution exists only when the ratio Λ=R∕λD of the plasma scale length R to the Debye length λD is invariant, i.e., under the condition Te(t)∝[ne(t)]1−2∕ν, where ν=1, 2, and 3 corresponds, respectively, to the planar, cylindrical, and spherical geometries. For Λ⪢1 the position of the ion front and the maximum energy Ei,max of accelerated ions are calculated analytically: in particular, for ν=3 one finds Ei,max=2ZTe0W(Λ2∕2), where Te0 is the initial electron temperature, Z is the ion charge, and W is the Lambert W function. It is argued that, when properly formulated, the results for Ei,max can be applied more generally than the self-similar solution itself. Generalization to a two-temperature electron system reveals the conditions under which the high-energy tail of accelerated ions is determined solely by the hot-electron population.