New theory of single bunch stability in a linac with quadrupole displacements

The analytical treatment previously described by Guingard and Hagel (1998) has been extended to include the important effect of magnetic quadrupole transverse displacements, the chromatic variation of the magnetic focusing, the energy spread along the bunch and possible microwave quadrupoles, the last two in relation to BNS damping The analytical treatment previously described has been extended to include the important effect of magnetic quadrupole transverse displacements, the chromatic variation of the magnetic focusing, the energy spread along the bunch and possible microwave quadrupoles, the last two in relation to BNS damping. Both, the longitudinal and transverse equations of motion are solved, the second by using the perturbation method with partial expansions developed for this theory. The localized nature of the quadrupole displacements is preserved by using thin lenses and the super-position principle for the kick effects. The causality principle applied to the downstream beam oscillations due to the kicks is introduced via Heaviside functions. The treatment presented provides formulae for the tuneshift in the bunch and first-order solutions for the transverse beam off-sets within the bunch. It presents a break-through in the recent efforts to solve the problem of the bunch stability theoretically, with realistic beam and linac models.