4 Impedance and Collective Effects
119
Fig. 4.5 Signal at the pick-up electrode for three different modes shown for several superimposed
turns (the red line corresponds to one particular turn), for the case of the parabolic amplitude
distribution and a constant inductive impedance (exhibiting therefore no growing oscillations!)
the two previous approaches, introducing a third mode number q ≡ m + 2k
(with 0 ≤ k < + ∞), called radial mode number, which comes from the distribution
of synchrotron oscillation amplitudes [52, 53]. 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 q (see Figs. 4.5 and
4.8). 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
[53]. Because there are two degrees of freedom (phase and amplitude), the general
solution is a twofold infinity of coherent modes of oscillation (m, q). At sufficiently
low intensity, only the most coherent mode (largest value for the coherent tune
shift) is generally considered, leading to the classical Sacherer’s formulae in both
transverse and longitudinal planes. Note that contrary to the space charge case,
these tune shifts are now complex, the imaginary part being linked to the instability
growth rate. For protons a parabolic density distribution is generally assumed and
the corresponding oscillation modes are sinusoidal (or close to it). 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
usually not very sensitive to the accuracy of the eigenfunctions. Similar results are
obtained for the longitudinal plane.
It is worth mentioning that the CERN ISR suffered from a beam instability
brought about by beams having different revolution frequencies. They could be
in the same vacuum chamber or coupled by the beam–beam effect. The name
of “overlap knock-out” [54] has been given to this phenomenon by which the
stack is subjected to transverse kicks from the bunches. This produces blow-up
of the stacked beam when the longitudinal frequency spectrum of the bunches
overlaps with the betatron frequency spectrum of the coasting stacked beam. Similar
problems limit the energy range of RHIC and proton lead in LHC [55].
119
Fig. 4.5 Signal at the pick-up electrode for three different modes shown for several superimposed
turns (the red line corresponds to one particular turn), for the case of the parabolic amplitude
distribution and a constant inductive impedance (exhibiting therefore no growing oscillations!)
the two previous approaches, introducing a third mode number q ≡ m + 2k
(with 0 ≤ k < + ∞), called radial mode number, which comes from the distribution
of synchrotron oscillation amplitudes [52, 53]. 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 q (see Figs. 4.5 and
4.8). 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
[53]. Because there are two degrees of freedom (phase and amplitude), the general
solution is a twofold infinity of coherent modes of oscillation (m, q). At sufficiently
low intensity, only the most coherent mode (largest value for the coherent tune
shift) is generally considered, leading to the classical Sacherer’s formulae in both
transverse and longitudinal planes. Note that contrary to the space charge case,
these tune shifts are now complex, the imaginary part being linked to the instability
growth rate. For protons a parabolic density distribution is generally assumed and
the corresponding oscillation modes are sinusoidal (or close to it). 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
usually not very sensitive to the accuracy of the eigenfunctions. Similar results are
obtained for the longitudinal plane.
It is worth mentioning that the CERN ISR suffered from a beam instability
brought about by beams having different revolution frequencies. They could be
in the same vacuum chamber or coupled by the beam–beam effect. The name
of “overlap knock-out” [54] has been given to this phenomenon by which the
stack is subjected to transverse kicks from the bunches. This produces blow-up
of the stacked beam when the longitudinal frequency spectrum of the bunches
overlaps with the betatron frequency spectrum of the coasting stacked beam. Similar
problems limit the energy range of RHIC and proton lead in LHC [55].