Effective ion acceleration of picosecond-duration well-collimated bunches in the strong relativistic interaction of a short laser pulse with a thin solid target has been experimentally demonstrated. In this work, with reference to the sharp rear solid–vacuum interface, where ion energization takes place, the one-dimensional Poisson–Boltzmann equation is analytically solved on a finite spatial interval whose extension is determined by requiring electron energy conservation, resulting in the consistent spatial distributions of the hot electrons created by the laser and of the corresponding electrostatic potential. Then, the equation of motions for an ensemble of test ions, initially distributed in a thin layer of the rear target surface, with different initial conditions, is solved and the energy spectrum corresponding to a given initial ion distribution is determined.