Classification of Half-Integer Resonance Dynamics

Abstract

Slow extraction is accomplished at Fermilab by exciting a half integer resonance with quadrupoles and octupoles. The physics of this process can be idealized by a Hamiltonian of the form H = ( +cos(2 +/PHI/))J+(kappa+cos2 )JS = 1/4(pS+qS)(pS(kappa+1)+qS(kappa-1)) + 1/2( +cos/PHI/)pS+1/2( -cos/PHI/)qS - pq sin/PHI/, where q = 2J sin and p = 2J cos . The angle-action pair ( ,J) are canonically conjugate variables; q represents horizontal displacement from an equilibrium orbit, scaled by betatron functions and by the ratio of quadrupole to octupole harmonics. The three control parameters , kappa, and /PHI/, are related to the distributions and strengths of quadrupoles and octupoles around the accelerator: is the difference between the tune and the half integer n/2, scaled by the nth harmonic quadrupole driving term; kappa is the ratio of Oth to nth harmonic octupole driving terms; and /PHI/ is the relative phase between the octupole and quadrupole harmonics. The allowed dynamics of this system have been classified by mapping its transition boundary, the catastrophe and Maxwell surfaces in the ( , kappa, /PHI/) control space. We describe some results here.