Multibunch Instabilities
The theory for transverse and longitudinal multibunch instabilities is reviewed. The coherent beam modes are classified, and the various mode numbers defining the coherent modes are explained. Sacherer’s longitudinal and transverse growth rate formulae ar
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lntroduction and Summary
The theory for transverse and longitudinal multibunch instabilities is reviewed. The coherent beam modes are classified, and the various mode numbers defining the coherent modes are explained. Sacherer's longitudinal and transverse growth rate formulae are discussed and compared with the commonly used short-bunch approximation and Robinson's characteristic equation. Coupling impedances with long-range wakes are particular troublesome for the large high-current colliders planned for the next decade. Theseare the higher-order modes of the RF cavities, the fundamental mode of the RF cavities, and the transverse resistive wall impedance.
Table 1. Classification of coherent beam modes
Longitudinal
Transverse
Coasting Beams Bunched Beams n = azimuthat mode number n = coupled bunch mode number = 1,2,3, ... 00 = 0, 1, 2, ... (M-1) m = phase plane periodicity = 1 (dipole), 2 (quadrupole), 3 (sextupole), ... (q =radial mode number) Mode coupling => Single-bunch "microwave" instability (turbulence). n = azimuthat mode number n = coupled bunch mode number = 0, 1, 2, ... (M-1) = -oo, ... , -1, 0, 1, 2, ... +oo m = head-tail mode number k = phase plane periodicity = 1 (dipole), 2 (quadrupole), = ... , -2, -1,0 1, 2, ... k = phase plane periodicity 3 (sextupole), ... = 1 (dipole), 2 (quadrupole), 3 (sextupole), .. Mode coupling => Single-bunch, fast, head-tail instability (turbulence).
M. Dienes et al. (eds.), Frontiers of Particle Beams: Factories with e+e - Rings © Springer-Verlag Berlin Heidelberg 1994
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Classification of Coherent Beam Modes
Coherent beam modes are classified according to whether the coherent beam motion is longitudinal or transverse, and according to whether the beam is bunched or debunched (coasting beam), Table 1. In this way four main classes of coherent beam modes are defined. Bearns are of course always bunched in e+/e- rings due to synchrotron radiation, but it is useful to classify the bearn modes in this general way. The complexity of the bearn motion increases from longitudinal to transverse, and from coasting to bunched, such that the mode description requires an increasing number of mode numbers.
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Longitudinal Bunched-Beam Modes
The generat theory for coherent bunched bearn modes and their interaction with the environment is due to Sacherer [1][2][3]. Basically two mode numbers describe the motion (see Fig. 1). The coup/ed bunch mode number n is defined as the number of waves of coherent motion per revolution, and resembles therefore the azimuthat mode number for coasting beams. For bunched bearns with M equidistant bunches, the bunch-to-bunch phase shift ~ is related to the coupled bunch mode number n by ~ =27tn!M. There are M coupled bunch modes numbered from 0 to (M-1). This is in cantrast to the azimuthat mode number for coasting bearns, where there is an infinite number of müdes. The within-bunch mode number m is the number
_&,$~
f~ tt~:·f~~l /
\ t
::Jk/(1) m-3
of periods of phase space density modulation per synchrotron period in the
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