4 Impedance and Collective Effects
171
appropriate transformation matrices. At each of these points, the interaction with
the desired collective effect will be applied (e.g. the beam’s own space charge field,
an electron cloud, a wake field). It is clear, therefore, that a numerical simulation
requires in the first place the knowledge of the driving term to be applied at each
interaction point. That is why in the following we separate the general simulation
into the solution of an electromagnetic problem, in which the collective interaction
is modelled and the resulting excitation on the beam is calculated (at least, its nonself-consistent part), and the beam tracking part, in which the evolution of a beam is
studied under the effect of this excitation. Note that most of numerical simulations
including collective effects are based on time domain models, as these are best suited
to describe the usually non-stationary beam evolution under the effect of collective
interactions.
4.7.1 The Electromagnetic Problem
The first step to set up a numerical simulation including a collective effect consists
of identifying the source of the self-induced perturbation acting on the beam and
modelling it in a way that can be subsequently used. We usually distinguish three
different types of collective interactions, which can take place with: (i) space charge
(see Sect. 4.1); (ii) wake fields from an accelerator component or part of the resistive
beam pipe (see Sect. 4.2); (iii) another “beam” of charged particles. This secondary
beam can be either a counter-rotating beam in a collider (see Sect. 4.6), or a static
cloud formed by the accumulation of particles, usually of opposite charge, around
the primary beam (see Sect. 4.5).
If the source of the perturbation is space charge, then two different approaches
are possible to compute its effect. Analytical formulae are available for the electromagnetic fields of coasting beams with ellipsoidal or Gaussian transverse sizes,
as well as of ellipsoidal or Gaussian bunches (in all dimensions). The additional
kicks given by these electromagnetic fields can be therefore calculated and applied
to the beam macroparticles in a finite number of locations along the ring (even if the
space charge interaction is in reality continuous). When doing that, self-consistency
requires that the sizes of the bunch are updated at every kick point. Another possible
approach consists of using the macroparticle distributions at each selected kick point
to calculate self-consistently the electric field with a Poisson solver and use it to
calculate the electromagnetic kicks on the macroparticles. It is worth noting that
the same approach can be used for beam–beam problems, because the shape of
the required electric field is the same, even if the coefficients need to be adapted
(electric and magnetic forces tend to cancel at ultra-relativistic energies for space
charge, while they add up for beam–beam).
If the source of the perturbation is a wake field from an accelerator component
or resistive wall, the shape of the relative wake function has to be calculated
beforehand. This is done analytically for some specific cases (e.g. resistive wall,
step or tapered transitions), but in general dedicated electromagnetic codes can be
171
appropriate transformation matrices. At each of these points, the interaction with
the desired collective effect will be applied (e.g. the beam’s own space charge field,
an electron cloud, a wake field). It is clear, therefore, that a numerical simulation
requires in the first place the knowledge of the driving term to be applied at each
interaction point. That is why in the following we separate the general simulation
into the solution of an electromagnetic problem, in which the collective interaction
is modelled and the resulting excitation on the beam is calculated (at least, its nonself-consistent part), and the beam tracking part, in which the evolution of a beam is
studied under the effect of this excitation. Note that most of numerical simulations
including collective effects are based on time domain models, as these are best suited
to describe the usually non-stationary beam evolution under the effect of collective
interactions.
4.7.1 The Electromagnetic Problem
The first step to set up a numerical simulation including a collective effect consists
of identifying the source of the self-induced perturbation acting on the beam and
modelling it in a way that can be subsequently used. We usually distinguish three
different types of collective interactions, which can take place with: (i) space charge
(see Sect. 4.1); (ii) wake fields from an accelerator component or part of the resistive
beam pipe (see Sect. 4.2); (iii) another “beam” of charged particles. This secondary
beam can be either a counter-rotating beam in a collider (see Sect. 4.6), or a static
cloud formed by the accumulation of particles, usually of opposite charge, around
the primary beam (see Sect. 4.5).
If the source of the perturbation is space charge, then two different approaches
are possible to compute its effect. Analytical formulae are available for the electromagnetic fields of coasting beams with ellipsoidal or Gaussian transverse sizes,
as well as of ellipsoidal or Gaussian bunches (in all dimensions). The additional
kicks given by these electromagnetic fields can be therefore calculated and applied
to the beam macroparticles in a finite number of locations along the ring (even if the
space charge interaction is in reality continuous). When doing that, self-consistency
requires that the sizes of the bunch are updated at every kick point. Another possible
approach consists of using the macroparticle distributions at each selected kick point
to calculate self-consistently the electric field with a Poisson solver and use it to
calculate the electromagnetic kicks on the macroparticles. It is worth noting that
the same approach can be used for beam–beam problems, because the shape of
the required electric field is the same, even if the coefficients need to be adapted
(electric and magnetic forces tend to cancel at ultra-relativistic energies for space
charge, while they add up for beam–beam).
If the source of the perturbation is a wake field from an accelerator component
or resistive wall, the shape of the relative wake function has to be calculated
beforehand. This is done analytically for some specific cases (e.g. resistive wall,
step or tapered transitions), but in general dedicated electromagnetic codes can be