A Matrix Formalism for Landau Damping

Existing methods of analysing the e ect of bunch to bunch tune shifts on coupled bunch instabilities are applicable to beams with a single unstable mode, or a few non-interacting unstable modes. We present a more general approach that involves computing the eigenvalues of a reduced state matrix. The method is applied to the analysis of PEP-II longitudinal coupled bunch modes, a large number of which are unstable in the absence of feedback. Talk given at the 14th Advanced ICFA Beam Dynamics Workshop: Beam Dynamics Issues for E+ EFactories Frascati, Italy October 20{26 1997 Work supported by Department of Energy contract DE{AC03{76SF00515. A MATRIX FORMALISM FOR LANDAU DAMPING S. Prabhakar, J. Fox, H. Hindi Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, U.S.A ABSTRACT Existing methods of analysing the e ect of bunch to bunch tune shifts on coupled bunch instabilities are applicable to beams with a single unstable mode, or a few non-interacting unstable modes. We present a more general approach that involves computing the eigenvalues of a reduced state matrix. The method is applied to the analysis of PEP-II longitudinal coupled bunch modes, a large number of which are unstable in the absence of feedback.Existing methods of analysing the e ect of bunch to bunch tune shifts on coupled bunch instabilities are applicable to beams with a single unstable mode, or a few non-interacting unstable modes. We present a more general approach that involves computing the eigenvalues of a reduced state matrix. The method is applied to the analysis of PEP-II longitudinal coupled bunch modes, a large number of which are unstable in the absence of feedback.