Kinetic theory of periodic holes in debunched particle beams

A self-consistent theory of periodic hole structures in coasting beams in synchrotrons and storage rings is presented, extending the theory on localized holes. The analysis reveals new intrinsic nonlinear modes which owe their existence to a deficiency of particles trapped in the self-sustained potential well, showing up as notches in the thermal range of the distribution function. It is therefore the full set of Vlasov-Poisson equations which is invoked; linearized treatments as well their nonlinear extensions fundamentally fail to cope with this strongly nonthermodynamic phenomenon. Qualitative agreement with the holes recently found at the CERN proton synchrotron booster is shown.