Transverse divergence in target normal sheath acceleration of a thick contamination layer

The divergence of the ion beams accelerated via Target Normal Sheath Acceleration (TNSA) is investigated using a particle-in-cell plasma simulation code. A semi-analytical model is derived and validated by 2D simulations. The proton divergence is determined analytically at the end of the well-understood isothermal phase of the plasma expansion. In order to find the final velocity distribution the adiabatic phase of the acceleration is modeled as well. An analytical expression is given for the energy-dependent half envelope angle of a proton beam, which can be extended into three dimensions by assuming an axial symmetric distribution of the hot electrons. The model is also compared with experimental data, a good agreement in the energy dependent divergence is observed. The validity range of the model is discussed in time and space, which corresponds to typical TNSA parameters.