Adiabatic theory of particle trapping due to Coulomb crossing of one-dimensional betatron resonance

In the presence of the space charge (s.c) forces the synchrotron oscillations result in periodical modulation of the s.c. tune shift, periodical crossing of betatron resonances and particle trapping in resonance islands. The trapping effect for one-dimensional resonance is considered using classical perturbation theory and 'frozen core' approach to calculation of the s.c. forces. The beam losses and the emittance growth are analyzed for the arbitrary order resonance; the numerical results are given for the third-order resonance