  Self-trapping and acceleration of ions in laser-driven relativistically
transparent plasma are investigated with the help of particle-in-cell
simulations. A theoretical model based on ion wave breaking is established in
describing ion evolution and ion trapping. The threshold for ion trapping is
identified. Near the threshold ion trapping is self-regulating and stops when
the number of trapped ions is large enough. The model is applied to ion
trapping in three-dimensional geometry. Longitudinal distributions of ions and
the electric field near the wave breaking point are derived analytically in
terms of power-law scalings. The areal density of trapped charge is obtained as
a function of the strength of ion wave breaking, which scales with target
density for fixed laser intensity. The results of the model are confirmed by
the simulations.
