8 Accelerator Engineering and Technology: Accelerator Technology
369
Nuclotron [33] and FAIR [34] dipoles, achieve maximum field in the aperture of
the order of 2 T. Super-ferric correctors are a good niche application, because the
peak field at the pole tip remains modest. Recent examples of these magnets are
the super-ferric high-order correctors of the new interaction regions of the HighLuminosity upgrade of the LHC [35].
8.2 RF Cavities
E. Jensen
The acceleration of charged particles is possible with electromagnetic fields
thanks to the Lorentz force F = q(E + v × B). Since the increase of particle energy
is given by
dW = F · v dt = q (E· v + (v × B) · v) dt,
neither the magnetic field nor the transverse components of the electric field
contribute to energy exchange with the particle. Only the longitudinal electric field
component can be used to accelerate particles to higher energies; RF cavities for
acceleration thus must provide a longitudinal electric field.
In addition to the necessity that the electric field must have a longitudinal
component, a second condition is that the net electric field (or force) integrated
through the RF cavity on the path of the particle trajectory (taking its finite speed
into account) must not vanish. For substantial acceleration, this latter condition
naturally leads to the need for a time-varying field, as can be concluded from the
following line of thought: For field quantities constant in time, it follows from
Maxwell’s equations that the electric field is the gradient of a potential; to increase
the kinetic energy of a charged particle, it will thus have to pass through a potential
difference, which for technical (and safety) reasons will be limited to a few MV at
best, which in turn will limit the possible energy gain to a few MeV just once, i.e.
without the possibility to add stages. To reach larger energy gains, time-varying
fields are necessary. RF cavities provide time-varying fields at high frequencies
(radio frequency = RF, typically ranging from a few kHz to some 10 GHz).
However, when averaging over one period of the RF, the fields at any one location
inside an RF cavity average out to zero; so if a particle travels a large distance
through the time-varying fields in an RF cavity, it may experience both accelerating
and decelerating fields, which will lead to a reduction of the net acceleration. For
this reason, RF cavities are designed to concentrate the accelerating field over a
relatively short distance (the accelerating gap). Cavities may have more than one
gap, but in this case the distance between gaps must be adjusted to the particle
velocity (see Sect. 8.2.6 below).
Since the particle beam is normally travelling in a vacuum pipe, the RF cavity
must be compatible with this requirement as well. This can be done either by using
369
Nuclotron [33] and FAIR [34] dipoles, achieve maximum field in the aperture of
the order of 2 T. Super-ferric correctors are a good niche application, because the
peak field at the pole tip remains modest. Recent examples of these magnets are
the super-ferric high-order correctors of the new interaction regions of the HighLuminosity upgrade of the LHC [35].
8.2 RF Cavities
E. Jensen
The acceleration of charged particles is possible with electromagnetic fields
thanks to the Lorentz force F = q(E + v × B). Since the increase of particle energy
is given by
dW = F · v dt = q (E· v + (v × B) · v) dt,
neither the magnetic field nor the transverse components of the electric field
contribute to energy exchange with the particle. Only the longitudinal electric field
component can be used to accelerate particles to higher energies; RF cavities for
acceleration thus must provide a longitudinal electric field.
In addition to the necessity that the electric field must have a longitudinal
component, a second condition is that the net electric field (or force) integrated
through the RF cavity on the path of the particle trajectory (taking its finite speed
into account) must not vanish. For substantial acceleration, this latter condition
naturally leads to the need for a time-varying field, as can be concluded from the
following line of thought: For field quantities constant in time, it follows from
Maxwell’s equations that the electric field is the gradient of a potential; to increase
the kinetic energy of a charged particle, it will thus have to pass through a potential
difference, which for technical (and safety) reasons will be limited to a few MV at
best, which in turn will limit the possible energy gain to a few MeV just once, i.e.
without the possibility to add stages. To reach larger energy gains, time-varying
fields are necessary. RF cavities provide time-varying fields at high frequencies
(radio frequency = RF, typically ranging from a few kHz to some 10 GHz).
However, when averaging over one period of the RF, the fields at any one location
inside an RF cavity average out to zero; so if a particle travels a large distance
through the time-varying fields in an RF cavity, it may experience both accelerating
and decelerating fields, which will lead to a reduction of the net acceleration. For
this reason, RF cavities are designed to concentrate the accelerating field over a
relatively short distance (the accelerating gap). Cavities may have more than one
gap, but in this case the distance between gaps must be adjusted to the particle
velocity (see Sect. 8.2.6 below).
Since the particle beam is normally travelling in a vacuum pipe, the RF cavity
must be compatible with this requirement as well. This can be done either by using