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used for this purpose. Some of them work in time domain, and can provide the wake
potentials for given source bunches (usually chosen to be short enough as to simulate
ideal pulse excitations and thus provide directly the wake functions). Other work in
frequency domain and output impedances, which need then to be back-transformed
into time domain to obtain the wake functions.
If the source of the perturbation is for example an electron cloud, then the
electron distribution of the cloud is usually calculated beforehand by means of an
electron cloud build-up code, and then its interaction with a coming bunch is calculated. Programs tracking simultaneously electrons and beam particles are presently
under development or test, due to the massive memory and CPU requirements to
solve this type of problems. On the other hand, generation and tracking of the ions
can be included in multi-bunch beam tracking programs to calculate the effect of
ions on bunched electron beams in a fully self-consistent manner. This is due to
the fact that, while ions do not move significantly during the passage of an electron
beam and allow modelling the bunches as charged disks, electrons can even perform
several oscillations during the passage of a bunch, which requires a much more
detailed modelling of the bunch.
4.7.2 Beam Dynamics
Beam dynamics tracking codes simulate the motion of beam particles inside an
accelerator by transporting them across a number of discrete points by means of
transformation matrices. In each of these points, additional kicks can be added,
modelling either nonlinear components and errors of the external fields or the
collective interactions. In the previous subsection, we have outlined the procedure
to calculate the excitation to be applied to the beam to compute its evolution
when it feels one or more collective interactions. To model the effects of space
charge, wake fields and electron clouds, it is certainly necessary to describe the
beam as an ensemble of macroparticles but beside that, its longitudinal structure
needs also to be detailed. In particular, to model coupled bunch instabilities the
relative positions of the macroparticles across the different bunches are necessary
to determine the total effect of the wake acting on each of them. For single-bunch
effects, a possible technique is to subdivide the bunch into several slices, so that the
macroparticles of each slice can feel the integrated effect of the wakes left behind by
the preceding slices (or the space charge from its own and the neighbouring slices,
or the electron cloud as was deformed by the previous slices). A possible scheme of
numerical simulation of a single bunch under the effect of a longitudinal wake field
is illustrated in Fig. 4.33.
The bunch is first divided into N slices and a kick must be applied to each
macroparticle within a given slice at a certain kick point. The kick depends on
the longitudinal wake function and the charge distribution of the preceding slices.
In the longitudinal plane, particles within a slice feel also the effect of the same
slice to which they belong, because the bunch suffers a net energy loss. After all
E. Metral et al.
used for this purpose. Some of them work in time domain, and can provide the wake
potentials for given source bunches (usually chosen to be short enough as to simulate
ideal pulse excitations and thus provide directly the wake functions). Other work in
frequency domain and output impedances, which need then to be back-transformed
into time domain to obtain the wake functions.
If the source of the perturbation is for example an electron cloud, then the
electron distribution of the cloud is usually calculated beforehand by means of an
electron cloud build-up code, and then its interaction with a coming bunch is calculated. Programs tracking simultaneously electrons and beam particles are presently
under development or test, due to the massive memory and CPU requirements to
solve this type of problems. On the other hand, generation and tracking of the ions
can be included in multi-bunch beam tracking programs to calculate the effect of
ions on bunched electron beams in a fully self-consistent manner. This is due to
the fact that, while ions do not move significantly during the passage of an electron
beam and allow modelling the bunches as charged disks, electrons can even perform
several oscillations during the passage of a bunch, which requires a much more
detailed modelling of the bunch.
4.7.2 Beam Dynamics
Beam dynamics tracking codes simulate the motion of beam particles inside an
accelerator by transporting them across a number of discrete points by means of
transformation matrices. In each of these points, additional kicks can be added,
modelling either nonlinear components and errors of the external fields or the
collective interactions. In the previous subsection, we have outlined the procedure
to calculate the excitation to be applied to the beam to compute its evolution
when it feels one or more collective interactions. To model the effects of space
charge, wake fields and electron clouds, it is certainly necessary to describe the
beam as an ensemble of macroparticles but beside that, its longitudinal structure
needs also to be detailed. In particular, to model coupled bunch instabilities the
relative positions of the macroparticles across the different bunches are necessary
to determine the total effect of the wake acting on each of them. For single-bunch
effects, a possible technique is to subdivide the bunch into several slices, so that the
macroparticles of each slice can feel the integrated effect of the wakes left behind by
the preceding slices (or the space charge from its own and the neighbouring slices,
or the electron cloud as was deformed by the previous slices). A possible scheme of
numerical simulation of a single bunch under the effect of a longitudinal wake field
is illustrated in Fig. 4.33.
The bunch is first divided into N slices and a kick must be applied to each
macroparticle within a given slice at a certain kick point. The kick depends on
the longitudinal wake function and the charge distribution of the preceding slices.
In the longitudinal plane, particles within a slice feel also the effect of the same
slice to which they belong, because the bunch suffers a net energy loss. After all