Chapter 4
Impedance and Collective Effects
E. Metral, G. Rumolo, and W. Herr
As the beam intensity increases, the beam can no longer be considered as a
collection of non-interacting single particles: in addition to the “single-particle
phenomena”, “collective effects” become significant. At low intensity a beam of
charged particles moves around an accelerator under the Lorentz force produced
by the “external” electromagnetic fields (from the guiding and focusing magnets,
RF cavities, etc.). However, the charged particles also interact with themselves
(leading to space charge effects) and with their environment, inducing charges and
currents in the surrounding structures, which create electromagnetic fields called
wake fields. In the ultra-relativistic limit, causality dictates that there can be no
electromagnetic field in front of the beam, which explains the term “wake”. It is
often useful to examine the frequency content of the wake field (a time domain
quantity) by performing a Fourier transformation on it. This leads to the concept
of impedance (a frequency domain quantity), which is a complex function of
frequency. The charged particles can also interact with other charged particles
present in the accelerator (leading to two-stream effects, and in particular to electron
cloud effects in positron/hadron machines) and with the counter-rotating beam in
a collider (leading to beam–beam effects). As the beam intensity increases, all
these “perturbations” should be properly quantified and the motion of the charged
particles will eventually still be governed by the Lorentz force but using the total
electromagnetic fields, which are the sum of the external and perturbation fields.
Note that in some cases a perturbative treatment is not sufficient and the problem has
to be solved self consistently. These perturbations can lead to both incoherent (i.e. of
Coordinated by E. Metral
E. Metral () · G. Rumolo · W. Herr
CERN (European Organization for Nuclear Research), Geneva, Switzerland
e-mail: Elias.Metral@cern.ch; Giovanni.Rumolo@cern.ch; Werner.Herr@cern.ch
© The Author(s) 2020
S. Myers, H. Schopper (eds.), Particle Physics Reference Library,
https://doi.org/10.1007/978-3-030-34245-6_4
105
Impedance and Collective Effects
E. Metral, G. Rumolo, and W. Herr
As the beam intensity increases, the beam can no longer be considered as a
collection of non-interacting single particles: in addition to the “single-particle
phenomena”, “collective effects” become significant. At low intensity a beam of
charged particles moves around an accelerator under the Lorentz force produced
by the “external” electromagnetic fields (from the guiding and focusing magnets,
RF cavities, etc.). However, the charged particles also interact with themselves
(leading to space charge effects) and with their environment, inducing charges and
currents in the surrounding structures, which create electromagnetic fields called
wake fields. In the ultra-relativistic limit, causality dictates that there can be no
electromagnetic field in front of the beam, which explains the term “wake”. It is
often useful to examine the frequency content of the wake field (a time domain
quantity) by performing a Fourier transformation on it. This leads to the concept
of impedance (a frequency domain quantity), which is a complex function of
frequency. The charged particles can also interact with other charged particles
present in the accelerator (leading to two-stream effects, and in particular to electron
cloud effects in positron/hadron machines) and with the counter-rotating beam in
a collider (leading to beam–beam effects). As the beam intensity increases, all
these “perturbations” should be properly quantified and the motion of the charged
particles will eventually still be governed by the Lorentz force but using the total
electromagnetic fields, which are the sum of the external and perturbation fields.
Note that in some cases a perturbative treatment is not sufficient and the problem has
to be solved self consistently. These perturbations can lead to both incoherent (i.e. of
Coordinated by E. Metral
E. Metral () · G. Rumolo · W. Herr
CERN (European Organization for Nuclear Research), Geneva, Switzerland
e-mail: Elias.Metral@cern.ch; Giovanni.Rumolo@cern.ch; Werner.Herr@cern.ch
© The Author(s) 2020
S. Myers, H. Schopper (eds.), Particle Physics Reference Library,
https://doi.org/10.1007/978-3-030-34245-6_4
105