RF voltage modulation at discrete frequencies, for application to proton extraction using crystal channeling

RF voltage modulation at a finite number of discrete frequencies is described in a Hamiltonian framework. The theory is applied to the problem of parasitic extraction of protons from a circulating beam in a high energy hadron collider, using a bent crystal as a thin “septum” extraction element. Three modes of employing discrete resonances are discussed: a strong, single drive resonance which may be used to excite protons to hit deep within the crystal; a single resonance ramped in such a manner that the island can carry trapped particles from low to high amplitudes; and overlapping resonances to create a chaotic band for separating the moving island and the large amplitude island. Simulations are used to confirm the expected dynamics, and finally a prototypical extraction scheme is described