144
E. Metral et al.
systematically observed at injection [144]. The reason why the instability appears in
both planes is that the integrated central density of electrons causing the instability
comes mainly from the electron cloud in quadrupole magnets and therefore affects
equally both planes. This instability is controlled by means of high chromaticity
and high octupole strength. Besides, electron cloud instabilities have been observed
also at high energy (6.5 TeV) but mainly in the vertical plane and for values of
bunch currents lower than nominal. This has been explained as due to the onset of a
central stripe in all the dipoles (appearing when the bunch intensity decays), which
leads to an integrated electron density capable of making a 6.5 TeV beam unstable
[145]. Concerning long bunches, a great deal of evidence indicates that the primary
instability limiting the performance of the LANL-PSR is an e-p instability [146].
Growth of the electron cloud results from multipacting of the electrons on the walls
of the vacuum chamber during passage of the trailing edge of the proton beam,
when the electrons can receive a net acceleration toward the wall. The instability
was controlled by various measures to enhance Landau damping and transverse
feedback. In coasting beams, an e-p instability was first observed in the LBNLBevatron [147] and CERN-ISR [139]. While in the Bevatron this instability was
combated with active feedback and beam bunching, in the ISR additional pumping
was installed to improve the vacuum from 0.1 to 0.01 nTorr and the number of
clearing electrodes was increased to sweep away the electrons.
Several analytical approaches have been used to study the electron cloud
instability, including few-particles models and an attempt to apply the TMCI theory
to the electron cloud wake field, modeled as a broadband resonator. However
the most widespread and comprehensive approach makes use of particle tracking
simulations with localized electron cloud kicks. The numerical modeling of the
interaction between an electron cloud and a particle bunch is discussed in Sect. 4.7.
Simulation codes have had the merit to reveal interesting features of the electron
cloud instability, which distinguish it from other types of conventional instabilities.
For example, the electron cloud wake field is not only a function of the distance
between source and probe particles, but it depends on the locations of the two
separately. This translates into an impedance with a double frequency dependence
[148]. Another interesting finding was that, for constant beam emittances (transverse
and longitudinal) and constant bunch length, the electron cloud instability threshold
decreases with the beam energy [149]. The reason of this anomalous behaviour
is that, although the beam becomes stiffer at higher energies, its transverse sizes
become smaller and the pinching effect on the electron cloud is amplified.
Concerning the multi-bunch instability, the usual approach is to calculate the
bunch-to-bunch wake field with an electron cloud build-up code (which correctly
models the electron cloud dynamics in the space between two bunches) and then
apply the multi-bunch analytical formula to assess the threshold for the beam
stability. Simulation codes with bunches modeled as single macroparticles, valid
for machines operating with short bunches, have been also developed to study
numerically the multi-bunch instabilities due to electron cloud in a more selfconsistent manner.
E. Metral et al.
systematically observed at injection [144]. The reason why the instability appears in
both planes is that the integrated central density of electrons causing the instability
comes mainly from the electron cloud in quadrupole magnets and therefore affects
equally both planes. This instability is controlled by means of high chromaticity
and high octupole strength. Besides, electron cloud instabilities have been observed
also at high energy (6.5 TeV) but mainly in the vertical plane and for values of
bunch currents lower than nominal. This has been explained as due to the onset of a
central stripe in all the dipoles (appearing when the bunch intensity decays), which
leads to an integrated electron density capable of making a 6.5 TeV beam unstable
[145]. Concerning long bunches, a great deal of evidence indicates that the primary
instability limiting the performance of the LANL-PSR is an e-p instability [146].
Growth of the electron cloud results from multipacting of the electrons on the walls
of the vacuum chamber during passage of the trailing edge of the proton beam,
when the electrons can receive a net acceleration toward the wall. The instability
was controlled by various measures to enhance Landau damping and transverse
feedback. In coasting beams, an e-p instability was first observed in the LBNLBevatron [147] and CERN-ISR [139]. While in the Bevatron this instability was
combated with active feedback and beam bunching, in the ISR additional pumping
was installed to improve the vacuum from 0.1 to 0.01 nTorr and the number of
clearing electrodes was increased to sweep away the electrons.
Several analytical approaches have been used to study the electron cloud
instability, including few-particles models and an attempt to apply the TMCI theory
to the electron cloud wake field, modeled as a broadband resonator. However
the most widespread and comprehensive approach makes use of particle tracking
simulations with localized electron cloud kicks. The numerical modeling of the
interaction between an electron cloud and a particle bunch is discussed in Sect. 4.7.
Simulation codes have had the merit to reveal interesting features of the electron
cloud instability, which distinguish it from other types of conventional instabilities.
For example, the electron cloud wake field is not only a function of the distance
between source and probe particles, but it depends on the locations of the two
separately. This translates into an impedance with a double frequency dependence
[148]. Another interesting finding was that, for constant beam emittances (transverse
and longitudinal) and constant bunch length, the electron cloud instability threshold
decreases with the beam energy [149]. The reason of this anomalous behaviour
is that, although the beam becomes stiffer at higher energies, its transverse sizes
become smaller and the pinching effect on the electron cloud is amplified.
Concerning the multi-bunch instability, the usual approach is to calculate the
bunch-to-bunch wake field with an electron cloud build-up code (which correctly
models the electron cloud dynamics in the space between two bunches) and then
apply the multi-bunch analytical formula to assess the threshold for the beam
stability. Simulation codes with bunches modeled as single macroparticles, valid
for machines operating with short bunches, have been also developed to study
numerically the multi-bunch instabilities due to electron cloud in a more selfconsistent manner.