4 Impedance and Collective Effects
117
Fig. 4.4 Longitudinal and transverse impedances and wake functions, in the case of resonator
impedances (f r = 1 GHz, Q = 100, R s = 20 , and R ⊥ = 20 M/m)
distributed kickers in the CERN SPS with the corresponding lumped impedance:
exactly the same result was obtained [47].
4.3 Coherent Instabilities
E. Metral
The wake fields can influence the motion of trailing particles, in the longitudinal
and in one or both transverse directions, leading to energy loss, beam instabilities,
or producing undesirable secondary effects such as excessive heating of sensitive
components at or near the chamber wall. Therefore, in practice the elements of the
vacuum chamber should be designed to minimise the self-generated electromagnetic
fields. For example, chambers with different cross-sections should be connected
with tapered transitions; bellows need to be separated from the beam by shielding;
plates should be grounded or terminated to avoid reflections; high-resistivity
materials should be coated with a thin layer of very good conductor (such as copper)
when possible; etc.
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
117
Fig. 4.4 Longitudinal and transverse impedances and wake functions, in the case of resonator
impedances (f r = 1 GHz, Q = 100, R s = 20 , and R ⊥ = 20 M/m)
distributed kickers in the CERN SPS with the corresponding lumped impedance:
exactly the same result was obtained [47].
4.3 Coherent Instabilities
E. Metral
The wake fields can influence the motion of trailing particles, in the longitudinal
and in one or both transverse directions, leading to energy loss, beam instabilities,
or producing undesirable secondary effects such as excessive heating of sensitive
components at or near the chamber wall. Therefore, in practice the elements of the
vacuum chamber should be designed to minimise the self-generated electromagnetic
fields. For example, chambers with different cross-sections should be connected
with tapered transitions; bellows need to be separated from the beam by shielding;
plates should be grounded or terminated to avoid reflections; high-resistivity
materials should be coated with a thin layer of very good conductor (such as copper)
when possible; etc.
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