Passage through a half-integer resonance due to space charge for different initial distributions [particle beam]

We study a beam passing through a grid of intrinsic envelope resonances, where the half integer resonance is a maximum. At the initial stage a beam with any kind of distribution is very sensitive to the half integer and fourth-order resonances. During 60-100 turns the distribution becomes self-stabilised and very similar to a Gaussian. Maximum growth is observed for a maximum uniform distribution in real space. For intrinsic resonances self stabilization occurs when most particles remain in resonance