Stability of the higher order longitudinal modes for PEP

Technical Report · Sun Jul 01 04:00:00 EDT 1979

DOI:https://doi.org/10.2172/6401720· OSTI ID:6401720

Chao, A W

The theory of longitudinal instabilities of bunched beams is well known. Application to PEP has recently been considered by Pellegrini and Sands, who were mainly concerned with the rigid bunch oscillation of three bunches (i.e. the dipole mode with mode number m = 1). In this note, we will look at the stability of modes with arbitrary mode number m, first for a single-bunch beam and then or a beam consisting of three equally spaced bunches. The method of analysis is essentially that of Sacherer's, whose main result is an integral equation for the eigenmodes and eigenfrequencies of the longitudinal bunch oscillations. It turns out that the integral equation can be solved if the unperturbed beam has a uniform distribution inside an ellipse in the longitudinal phase space (the water-bag model). We study the impedance of a parallel resonator circuit representing the rf accelerating cavity and obtain expressions for the damping rates and frequency shifts for mode number m. In deriving these expressions, we have included contributions from the tails of the impedance and not only from its peak. The importance of these contributions has been pointed out by Zotter. For m = 1, our expression for the damping rate is the same as that for Robinson damping. We also find that the damping rate of a higher order mode (m />=/ 2) is small compared with that of the dipole mode (m = 1)--- this is expected since the rf wavelength is much longer than the bunch length. On the other hand, due to the contribution from the impedance tail, the mode frequency shift does not decrease with m rapidly. We have also included an estimate of the damping rates of the quadrupole modes using the PEP parameters. 4 refs., 1 fig.

• Research Organization: Stanford Linear Accelerator Center, Menlo Park, CA (USA)
• DOE Contract Number: AC03-76SF00515
• OSTI ID: 6401720
• Report Number(s): SLAC-PEP-NOTE-309; ON: DE89006345
• Country of Publication: United States
• Language: English

Related Subjects

43 PARTICLE ACCELERATORS
430200 -- Particle Accelerators-- Beam Dynamics
Field Calculations
& Ion Optics
430400* -- Particle Accelerators-- Storage Rings
BEAM BUNCHING
BEAM DYNAMICS
BOLTZMANN-VLASOV EQUATION
DIFFERENTIAL EQUATIONS
DISTRIBUTION FUNCTIONS
ELECTRIC IMPEDANCE
EQUATIONS
EQUATIONS OF MOTION
FUNCTIONS
IMPEDANCE
INTEGRAL EQUATIONS
MATHEMATICAL SPACE
PARTIAL DIFFERENTIAL EQUATIONS
PEP STORAGE RINGS
PHASE SPACE
SPACE
STABILITY
STORAGE RINGS