Self-similar plasma expansion of a limited mass into vacuum

A new self-similar solution is presented which describes non-relativistic expansion of a finite plasma mass into vacuum with a full account of charge separation effects. The solution exists only when the ratio ${\rm \Lambda} =$ R/${\lambda}_{\rm D}$ of the plasma scale length R to the Debye length ${\lambda}_{\rm D}$ is invariant, i.e. under the condition T$_{\rm e}$ $ \propto $ [ n$_{\rm e}$(t)] $^{1 - 2 /\nu}$, where ${\rm \nu} = 1$, 2, and 3 corresponds, respectively, to the planar, cylindrical, and spherical geometries. For ${\rm \Lambda} \gg 1$ the position of the ion front and the maximum energy E$_{\rm i,max }$ of accelerated ions are calculated analytically: in particular, for ${\rm \nu} =3$ one finds E$_{\rm i,max}=$ 2ZT$_{\rm e0}$W(${\rm \Lambda} ^{2}$/2), where T$_{\rm e0}$ 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 E$_{\rm i,max}$ can be applied more generally than the self-similar solution itself