RF Voltage Modulation at Discrete Frequencies, for Application to Proton Extraction using Crystal Channeling W. E. Gabella”, J. Rosenzweiga, R. Kickb, and S. PeggsC “University of California Los Angeles, Physics Department 405 Hilgard Ave., Los Angeles, CA 90024 bIllinois Mathematics and Science Academy, Aurora, IL 60506 ‘Brookhaven National Laboratory, Upton, Long Island, NY 11973 Abstract RF voltage modulation at a finite number of discrete fre- quencies 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 sin- gle 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 sepa- rating the moving island and the large amplitude island. Simulations are used to confirm the expected dynamics, and finally a prototypical extraction scheme is described. I. INTRODUCTION We explore using RF voltage modulation to affect the flux of protons onto a bent crystal being used to extract the protons from a storage ring. A low flux, high energy pro- ton beam would be useful for both a test beam and for fixed target B-physics experiments at the next generation of hadron colliders [I]. RF noise has also been considered to enhance the flux on the crystal [Z]. RF modulation is used to affect the longitudinal phase space dynamics while keeping the beams relatively unaf- fected at the interaction regions, which are presumed to have zero dispersion [3]. At the crystal, the dispersion would be large relative to the betatron amplitude, so the longitudinal motion affect,s the beam distribution signifi- cantly. This leaves open the possibility of extracting beam (from the halo) while beam collisions occur. Channeling in a bent crystal could provide an econom- ical way to extract a small flux of protons from a stor- age ring. Extraction of circulating beam has recently been demonstrated at the CERN SPS [4]. For appropri- ate beam parameters, a significant fraction of impinging protons channel between the planes of symmetry in the crystal, executing “betatron” oscillations in the effective focusing force. If the crystal is adiabatically bent then channeled protons follow the bend [5,6] and are extracted from the storage ring. For the Tevatron experiment 853 [7], ‘7 meters of pulsed kicker magnets are replaced by a 3 centimeter long bent crystal to send 900 GeV protons down an abort beam line. Using voltage modulation islands can be placed and ma- nipulated in the RF bucket. The position, width and is- land tunes are well described by analytic theory. These islands will be used to affect the dynamics of single parti- cles. A large island near the RF separatrix gives particles a large step into the crystal, an island with a ramped mod- ulation frequency moves particles from smaller amplitudes to larger, and many overlapping islands form a stochastic layer that buffers between any ramped islands and large outer islands. II. ISLANDS IN THE RF BUCKET The longitudinal dynamics of a proton stored in a ring can be described by an effective Hamiltonian. This form for the Hamiltonian relies on the energy gain from a cavity being small relative to the particle energy, so the discrete system can be approximated by a continuous one. This is equivalent to requiring that the synchrotron tune, Qso, is small, then the longitudinal dynamics for a single proton is described by H(q,,,t) = 2~Qso($ + 1 - ,064) , (1) where p = 26,/6,,, g ives the relative momentum offset. Srep is the offset at the separatrix, 4 is the phase, or tirn- ing, of the proton at the RF cavity, and t is time mea- sured in turns around the ring. We do not discuss the above in much greater detail, but refer the reader to a standard treatment of RF phase stability [8]. The syn- chrotron tune given in terms of other RF parameters is Qz,, = h7pVo/(27rp2&), and the momentum offet at the separatrix is 6rep = 2Q,o/(hq), where h is the harmonic number and q is the phase slip factor. If the RF voltage is modulated at frequencies near twice the synchronous frequency, resonant islands appear in the RF bucket. Since the tune of a single particle in the RF 0-7803.1203-l/93%03.00 0 1993IEEE 233 © 1993 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. PAC 1993