Equilibrium beam distribution in an electron storage ring near linear synchrobetatron coupling resonances

Abstract

Linear dynamics in a storage ring can be described by the one-turn map matrix. In the case of a resonance where two of the eigenvalues of this matrix are degenerate, a coupling perturbation causes a mixing of the uncoupled eigenvectors. A perturbation formalism is developed to find eigenvalues and eigenvectors of the one-turn map near such a linear resonance. Damping and diffusion due to synchrotron radiation can be obtained by integrating their effects over one turn, and the coupled eigenvectors can be used to find the coupled damping and diffusion coefficients. Expressions for the coupled equilibrium emittances and beam distribution moments are then derived. In addition to the conventional instabilities at the sum, integer, and half-integer resonances, it is found that the coupling can cause an instability through antidamping near a sum resonance even when the symplectic dynamics are stable. As one application of this formalism, the case of linear synchrobetatron coupling is analyzed where the coupling is caused by dispersion in the rf cavity, or by a crab cavity. Explicit closed-form expressions for the sum/difference resonances are given along with the integer/half-integer resonances. The integer and half-integer resonances caused by coupling require particular care. We find an example of this with the case of a crab cavity for the integer resonance of the synchrotron tune. Whether or not there is an instability is determined by the value of the horizontal betatron tune, a unique feature of these coupling-caused integer or half-integer resonances. Finally, the coupled damping and diffusion coefficients along with the equilibrium invariants and projected emittances are plotted as a function of the betatron and synchrotron tunes for an example storage ring based on PEP-II.