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E. Metral et al.
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 [48]. In most accelerators, the RF acceleration mechanism generates an azimuthal non-uniformity of the particle density and consequently
the work of Laslett, Neil and Sessler for continuous beams [35] 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 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 (this leads to a longrange 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 anti-damped
transverse motion. For M equi-populated equi-spaced bunches, M coupled-bunch
mode numbers exist (n = 0, 1, . . . , M − 1), 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 φ = 2πn/M.
Pellegrini [49] and, independently, Sands [50, 51] 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 slip 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 m = . . . , −1, 0, 1, . . . , also
called head–tail (or azimuthal) mode number, was introduced. This mode describes
the number of betatron wavelengths (with sign) per synchrotron period.
The work of Courant and Sessler, or Pellegrini and Sands, was done for particular
impedances and oscillation modes. Using the Vlasov formalism, Sacherer unified
E. Metral et al.
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 [48]. In most accelerators, the RF acceleration mechanism generates an azimuthal non-uniformity of the particle density and consequently
the work of Laslett, Neil and Sessler for continuous beams [35] 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 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 (this leads to a longrange 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 anti-damped
transverse motion. For M equi-populated equi-spaced bunches, M coupled-bunch
mode numbers exist (n = 0, 1, . . . , M − 1), 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 φ = 2πn/M.
Pellegrini [49] and, independently, Sands [50, 51] 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 slip 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 m = . . . , −1, 0, 1, . . . , also
called head–tail (or azimuthal) mode number, was introduced. This mode describes
the number of betatron wavelengths (with sign) per synchrotron period.
The work of Courant and Sessler, or Pellegrini and Sands, was done for particular
impedances and oscillation modes. Using the Vlasov formalism, Sacherer unified