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E. Metral et al.
a single particle) and coherent (i.e. of the centre of mass) effects, in the longitudinal
and in one or both transverse directions, leading to beam quality degradation or
even partial or total beam losses. Fortunately, stabilising mechanisms exist, such
as Landau damping, electronic feedback systems and linear coupling between the
transverse planes (as in the case of a transverse coherent instability, one plane is
usually more critical than the other).
Beam instabilities cover a wide range of effects in particle accelerators and they
have been the subjects of intense research for several decades. As the machines
performance was pushed new mechanisms were revealed and nowadays the challenge consists in studying the interplays between all these intricate phenomena, as it
is very often not possible to treat the different effects separately [1, 2]. This field is
still very active as can be revealed by the recent (and future) international workshops
devoted to this subject [3–5].
This chapter is structured as follows: space charge is discussed in Sect. 4.1, wake
fields (and related impedances) in Sect. 4.2, the induced coherent instabilities in
Sect. 4.3 and the Landau damping mechanism in Sect. 4.4. The two-stream effects
are analyzed in Sect. 4.5, concentrating mainly on electron cloud, while beam–beam
effects are reviewed in Sect. 4.6, before concluding in Sect. 4.7 by the numerical
modelling of collective effects.
4.1 Space Charge
E. Metral
4.1.1 Direct Space Charge
Two space charge effects are distinguished: the direct space charge and the indirect
(or image) one ([6–9], and references therein). The direct space charge comes from
the interaction between the particles of a single beam, without interaction with the
surrounding vacuum chamber. Consider two particles with the same charge (for
instance protons) in vacuum. They will feel two forces: the Coulomb repulsion
(as they have the same charge) and the magnetic attraction (as they represent
currents moving in the same direction, leading to an azimuthal magnetic field).
Let’s assume that particle 1 is moving with speed v 1 = β 1 c with respect to the
laboratory frame, with β the relativistic velocity factor and c the speed of light. In its
rest frame, particle 1 produces only an electrostatic field, which can be computed,
and applying the relativistic transformation of the electromagnetic fields between
the rest and laboratory frames, the magnetic contribution can be obtained. Note
that there is no magnetic contribution in the longitudinal plane (B s = 0), which
leads to the longitudinal Lorentz force F s = eE s , where e is the elementary charge,
s the azimuthal coordinate and E s the longitudinal electric field. The transverse
E. Metral et al.
a single particle) and coherent (i.e. of the centre of mass) effects, in the longitudinal
and in one or both transverse directions, leading to beam quality degradation or
even partial or total beam losses. Fortunately, stabilising mechanisms exist, such
as Landau damping, electronic feedback systems and linear coupling between the
transverse planes (as in the case of a transverse coherent instability, one plane is
usually more critical than the other).
Beam instabilities cover a wide range of effects in particle accelerators and they
have been the subjects of intense research for several decades. As the machines
performance was pushed new mechanisms were revealed and nowadays the challenge consists in studying the interplays between all these intricate phenomena, as it
is very often not possible to treat the different effects separately [1, 2]. This field is
still very active as can be revealed by the recent (and future) international workshops
devoted to this subject [3–5].
This chapter is structured as follows: space charge is discussed in Sect. 4.1, wake
fields (and related impedances) in Sect. 4.2, the induced coherent instabilities in
Sect. 4.3 and the Landau damping mechanism in Sect. 4.4. The two-stream effects
are analyzed in Sect. 4.5, concentrating mainly on electron cloud, while beam–beam
effects are reviewed in Sect. 4.6, before concluding in Sect. 4.7 by the numerical
modelling of collective effects.
4.1 Space Charge
E. Metral
4.1.1 Direct Space Charge
Two space charge effects are distinguished: the direct space charge and the indirect
(or image) one ([6–9], and references therein). The direct space charge comes from
the interaction between the particles of a single beam, without interaction with the
surrounding vacuum chamber. Consider two particles with the same charge (for
instance protons) in vacuum. They will feel two forces: the Coulomb repulsion
(as they have the same charge) and the magnetic attraction (as they represent
currents moving in the same direction, leading to an azimuthal magnetic field).
Let’s assume that particle 1 is moving with speed v 1 = β 1 c with respect to the
laboratory frame, with β the relativistic velocity factor and c the speed of light. In its
rest frame, particle 1 produces only an electrostatic field, which can be computed,
and applying the relativistic transformation of the electromagnetic fields between
the rest and laboratory frames, the magnetic contribution can be obtained. Note
that there is no magnetic contribution in the longitudinal plane (B s = 0), which
leads to the longitudinal Lorentz force F s = eE s , where e is the elementary charge,
s the azimuthal coordinate and E s the longitudinal electric field. The transverse