Microsoft Word - OverviewOfCoherentInstabilitiesForHHH04_v2.DOC

Single-bunch and coupled-bunch instability mechanisms in both longitudinal and transverse planes are reviewed. Stabilization by Landau damping, linear coupling, or feedbacks are also discussed. Benchmarking with some instability codes are shown as well as several experimental results. INTRODUCTION Two approaches are usually used to deal with collective instabilities. One starts from the single-particle equation while the other solves the Vlasov equation, which is nothing else but an expression for the Liouville conservation of phase-space density seen by a stationary observer. In the second approach, the motion of the beam is described by a superposition of modes, rather than a collection of individual particles. The detailed methods of analysis in the two approaches are different, the particle representation is usually conveniently treated in the time domain, while in the mode representation the frequency domain is more convenient, but in principle they necessarily give the same final results. The advantage of the mode representation is that it offers a formalism that can be used systematically to treat the instability problem. The first formalism was used by Courant and Sessler to describe the transverse coupled-bunch instabilities [1]. In most accelerators, the RF acceleration mechanism generates azimuthal non-uniformity of particle density and consequently the work of Laslett, Neil and Sessler for continuous beams [2] is not applicable in the case of bunched beams. Courant and Sessler studied the case of rigid (point-like) bunches, i.e. bunches oscillating as rigid units, and they showed that the transverse electromagnetic coupling of bunches of particles with each other can lead (due to the effect of imperfectly conducting vacuum chamber walls) to a coherent instability. The physical basis of the instability is that in a resistive vacuum tank, fields due to a particle decay only very slowly in time after the particle has left (long-range interaction). The decay can be so slow that when a bunch returns after one (or more) revolutions it is subject to its own residual wake field which, depending upon its phase relative to the wake field, can lead to damped or antidamped transverse motion. For M equi-populated equispaced bunches, M coupled-bunch mode numbers exist ( 1 ..., , 1 , 0 − = M n ), characterized by the integer number of waves of the coherent motion around the ring. Therefore the coupled-bunch mode number resembles the azimuthal mode number for coasting beams, except that for coasting beams there is an infinite number of modes. The bunch-to-bunch phase shift φ ∆ is related to the coupled-bunch mode number n by M n / 2π φ = ∆ . Pellegrini [3] and, independently, Sands [4,5] then showed that short-range wake fields (i.e. fields that provide an interaction between the particles of a bunch but have a negligible effect on subsequent passages of the bunch or of other bunches in the beam) together with the internal circulation of the particles in a bunch can cause internal coherent modes within the bunch to become unstable. The important point here is that the betatron phase advance per unit of time (or betatron frequency) of a particle depends on its instantaneous momentum deviation (from the ideal momentum) in first order through the chromaticity and the slippage factor. Considering a non-zero chromaticity couples the betatron and synchrotron motions, since the betatron frequency varies around a synchrotron orbit. The betatron phase varies linearly along the bunch (from the head) and attains its maximum value at the tail. The total betatron phase shift between head and tail is the physical origin of the head tail instability. The head and the tail of the bunch oscillate therefore with a phase difference, which reduces to rigid-bunch oscillations only in the limit of zero chromaticity. A new (within-bunch) mode number ... , 1 , 0 , 1 ..., − = m , also called head-tail mode number, was introduced. This mode describes the number of betatron wavelengths (with sign) per synchrotron period. It can be obtained by superimposing several traces of the directly observable average displacement along the bunch at a particular pick-up. The number of nodes is the mode number m . The work of Courant and Sessler, or Pellegrini and Sands, was done for particular impedances and oscillation modes. Using the Vlasov formalism, Sacherer unified the two previous approaches, introducing a third mode number ... , 1 , 0 , 1 ..., − = q , called radial mode number, which comes from the distribution of synchrotron oscillation amplitudes [6,7]. The advantage of this formalism is that it is valid for generic impedances and any high order head-tail modes. This approach starts from a distribution of particles (split into two different parts, a stationary distribution and a perturbation), on which Liouville theorem is applied. After linearization of the Vlasov equation, one ends up with Sacherer’s integral equation or Laclare’s eigenvalue problem to be solved [7]. Because there are two degrees of freedom (phase and amplitude), the general solution is a twofold infinity of coherent modes of oscillation ( ... , 1 , 0 , 1 ..., , − = q m ). At sufficiently low intensity, only the most coherent mode m q = (largest value for the coherent tune shift) is generally considered, leading to the classical Sacherer’s formulae in both transverse and longitudinal planes. For protons a parabolic density distribution is generally assumed, which is a reasonable approximation at relatively low energy, and the corresponding oscillation modes are sinusoidal. For electrons, the distribution is usually Gaussian, and the oscillation modes are described in this case by Hermite polynomials. In reality, the oscillation modes depend both on the distribution function and the impedance, and can only be found numerically by solving the (infinite) eigenvalue problem. However, the mode frequencies are not very sensitive to the accuracy of the eigenfunctions. Similar results are obtained for the longitudinal plane. TRANSVERSE Low Intensity At low intensity (i.e. below the intensity threshold given in the next section), the standing-wave patterns (head-tail modes) are treated independently. This leads to instabilities where the head and the tail of the bunch exchange their roles (due to synchrotron oscillation) several times during the rise-time of the instability. The complex transverse coherent betatron frequency shift of (sinusoidal) bunched-beam modes is given by Sacherer’s formula [6] ( ) ( ) , 2 1 , , 0 0 , 0 0 1 , , q m eff y x y x b y x q m Z L Q m I e j m Ω + = ∆ − γ β ω (1)