142
E. Metral et al.
which move slowly and exhibit a dissipation rate depending on their probability
of being elastically reflected at the chamber surface. Due to the second part of
the decay evolution, the electron clearing time between bunch trains can become
painfully long. Three effects are suspected to be responsible for the long memory
and lifetime of the electron cloud. First, nonuniform fields, such as quadrupoles or
sextupoles, may act as magnetic bottles and trap electrons for an indefinite time
period [125, 137]. Second, if the probability of elastic reflection really approaches
one in the limit of zero electron energy, as is suggested by measurements [122], lowenergetic electrons could survive nearly forever, bouncing back and forth between
the chamber walls, independently of the magnetic field. Third, slow ions produced
by residual gas ionization have been also suspected to be long lived in the beam
chamber and, therefore, possibly cause (or help) electron trapping and long survival
time.
4.5.2 The Electron Cloud Instability
When a positron/hadron beam interacts electromagnetically with the electron cloud
that has formed in the beam chamber, a coherent oscillation of both electrons and
beam particles can grow from any small initial perturbation of the beam distribution,
e.g. from the statistical fluctuations due to the finite number of beam particles. This
instability can be considered as a two-stream instability of the same type as studied
in plasma physics. Such instabilities can be very fast, since in the new generation
of high intensity rings operating with many closely spaced bunches, the density of
electrons can become quickly very large. Even machines operating with bunches
spaced by hundreds of ns can actually suffer from electron cloud, because of the
long survival time of low energy electrons in the beam pipe. Electron clouds can
cause single-bunch instabilities as well as multi-bunch dipole mode instabilities. The
multi-bunch instability appears when the electron cloud can carry a sufficiently long
memory as to couple subsequent bunches. The single-bunch phenomenon, instead,
is driven by a pinched electron cloud, which, over one single passage of the bunch
through it, is able to transfer information from an offset bunch head to the bunch
tail. Obviously, although this second type of instability is caused by a single-bunch
mechanism, it can only occur in multi-bunch operation, since the electron cloud
requires a train of several bunches to build up. For single-bunch instabilities caused
by multi-bunch built electron clouds, electrons usually only perform a low number
of oscillations while the bunch is passing (typically between fractions of unit and
few units), and the bunch effectively interacts with a pre-existing cloud produced
by the preceding bunches and filling almost uniformly the beam pipe prior to the
bunch arrival. The number of electrons does not change appreciably during one
bunch passage. In reality, another possible head tail effect resulting into a different
type of two-stream instability was observed in some machines operating with long
bunches. In this case, the instability is intimately related to an electron cloud
from “trailing-edge multipacting”, described in the previous section. The electrons
E. Metral et al.
which move slowly and exhibit a dissipation rate depending on their probability
of being elastically reflected at the chamber surface. Due to the second part of
the decay evolution, the electron clearing time between bunch trains can become
painfully long. Three effects are suspected to be responsible for the long memory
and lifetime of the electron cloud. First, nonuniform fields, such as quadrupoles or
sextupoles, may act as magnetic bottles and trap electrons for an indefinite time
period [125, 137]. Second, if the probability of elastic reflection really approaches
one in the limit of zero electron energy, as is suggested by measurements [122], lowenergetic electrons could survive nearly forever, bouncing back and forth between
the chamber walls, independently of the magnetic field. Third, slow ions produced
by residual gas ionization have been also suspected to be long lived in the beam
chamber and, therefore, possibly cause (or help) electron trapping and long survival
time.
4.5.2 The Electron Cloud Instability
When a positron/hadron beam interacts electromagnetically with the electron cloud
that has formed in the beam chamber, a coherent oscillation of both electrons and
beam particles can grow from any small initial perturbation of the beam distribution,
e.g. from the statistical fluctuations due to the finite number of beam particles. This
instability can be considered as a two-stream instability of the same type as studied
in plasma physics. Such instabilities can be very fast, since in the new generation
of high intensity rings operating with many closely spaced bunches, the density of
electrons can become quickly very large. Even machines operating with bunches
spaced by hundreds of ns can actually suffer from electron cloud, because of the
long survival time of low energy electrons in the beam pipe. Electron clouds can
cause single-bunch instabilities as well as multi-bunch dipole mode instabilities. The
multi-bunch instability appears when the electron cloud can carry a sufficiently long
memory as to couple subsequent bunches. The single-bunch phenomenon, instead,
is driven by a pinched electron cloud, which, over one single passage of the bunch
through it, is able to transfer information from an offset bunch head to the bunch
tail. Obviously, although this second type of instability is caused by a single-bunch
mechanism, it can only occur in multi-bunch operation, since the electron cloud
requires a train of several bunches to build up. For single-bunch instabilities caused
by multi-bunch built electron clouds, electrons usually only perform a low number
of oscillations while the bunch is passing (typically between fractions of unit and
few units), and the bunch effectively interacts with a pre-existing cloud produced
by the preceding bunches and filling almost uniformly the beam pipe prior to the
bunch arrival. The number of electrons does not change appreciably during one
bunch passage. In reality, another possible head tail effect resulting into a different
type of two-stream instability was observed in some machines operating with long
bunches. In this case, the instability is intimately related to an electron cloud
from “trailing-edge multipacting”, described in the previous section. The electrons