4 Impedance and Collective Effects
173
Fig. 4.33 Example of numerical simulation with collective effects: schematic view of the
interaction of a single bunch with a longitudinal wake field at several locations in an accelerator
ring
the particles in the bunch feel the effect of the wake at the kick point, they are
subsequently transported to the next kick point in the accelerator optical model.
Since synchrotron motion plays a key role in most of the effects under study, it is
essential that particles are made to execute their synchrotron oscillations and move
across slices from turn to turn. This means that collective effects dealing with singlebunch problems need to have at least one model of synchrotron motion built in, and
that the bunch binning has to be regularly updated. Since the synchrotron motion
is slow enough, and in reality the RF cavities do really kick the beam particles
once or few times per turn, the longitudinal coordinates and the bunch slicing are
usually not updated more frequently than once per turn. However, the update of the
longitudinal coordinates from kick to kick point within one single turn, based on
the only drift from momentum spread, could become significant especially in space
charge simulations.
The simulation scheme with transverse wakes is basically the same as the one
displayed in Fig. 4.33, except that particles inside one slice do not feel the effect of
the same slice (as the transverse wakes are zero in the origin, for ultra-relativistic
particles) and dipolar and quadrupolar contributions can be separated, making the
wake kicks depending not only on the position of the source slice but also on the
position of the witness macroparticles. The simulation scheme with the electron
cloud is again similar to the one shown in Fig. 4.33, but the fundamental difference
is that there is a mutual action between beam and electron cloud, so that, while
macroparticles within a slice feel the effect of the electron cloud, the electron cloud
itself is also deformed by the action of the passing slice.
173
Fig. 4.33 Example of numerical simulation with collective effects: schematic view of the
interaction of a single bunch with a longitudinal wake field at several locations in an accelerator
ring
the particles in the bunch feel the effect of the wake at the kick point, they are
subsequently transported to the next kick point in the accelerator optical model.
Since synchrotron motion plays a key role in most of the effects under study, it is
essential that particles are made to execute their synchrotron oscillations and move
across slices from turn to turn. This means that collective effects dealing with singlebunch problems need to have at least one model of synchrotron motion built in, and
that the bunch binning has to be regularly updated. Since the synchrotron motion
is slow enough, and in reality the RF cavities do really kick the beam particles
once or few times per turn, the longitudinal coordinates and the bunch slicing are
usually not updated more frequently than once per turn. However, the update of the
longitudinal coordinates from kick to kick point within one single turn, based on
the only drift from momentum spread, could become significant especially in space
charge simulations.
The simulation scheme with transverse wakes is basically the same as the one
displayed in Fig. 4.33, except that particles inside one slice do not feel the effect of
the same slice (as the transverse wakes are zero in the origin, for ultra-relativistic
particles) and dipolar and quadrupolar contributions can be separated, making the
wake kicks depending not only on the position of the source slice but also on the
position of the witness macroparticles. The simulation scheme with the electron
cloud is again similar to the one shown in Fig. 4.33, but the fundamental difference
is that there is a mutual action between beam and electron cloud, so that, while
macroparticles within a slice feel the effect of the electron cloud, the electron cloud
itself is also deformed by the action of the passing slice.