  New exact results are obtained for relativistic acceleration of test positive
ions in the non-Boltzmann laminar zone of a planar electron sheath evolving
from an initially mono-energetic electron distribution. The electron dynamics
is analyzed against the background of motionless foil ions. The limiting
gamma-factor of accelerated ions is shown to be determined primarily by the
values of the ion-electron charge-over-mass ratio and the initial gamma-factor
of the accelerated electrons: there exists a critical relationship between
these two quantities, which determines whether the ion can overtake the
electron front moving with initial electron velocity. It is proven that in
reality protons and heavier test ions can never catch up with the electron
front; for their maximum energy an appropriate intermediate asymptotic formula
is derived. The domain of applicability of the laminar-zone results is analyzed
in detail.
