TRANSVERSE-MODE COUPLING INSTABILITY IN A DOUBLE RF SYSTEM

The equations for transverse mode coupling in a storage ring with a double rf system are derived from a Hamiltonian formalism. The resulting integral equation is expanded into a set of orthogonal polynomials, and the expansion coefficients are then given by the solution of an infinite determinant. Truncation of this this determinant permits solution of the problem on a computer, and a code has been written which rinds the complex mode frequencies. The stability limits are determined by equating the imaginary part of the solution to the radiation damping rate. The theory is applied to LEP at injection with third higher-harmonic cavities, and the results were compared with those obtained by the multi-particle simulation program. They show excellent agreement, and predict that the threshold current will be reduced rather than improved.