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Radiation damping in accelerator physics is a way of reducing the beam emittance of a high-velocity beam of charged particles.
There are two main ways of using radiation damping to reduce the emittance of a particle beam—damping rings and undulators—and both rely on the same principle. They induce synchrotron radiation to reduce the particles' momentum, then replace that with momentum in the desired direction of motion.
As particles are steered around a closed loop ( not necessarily circular), the lateral acceleration causes them to give off synchrotron radiation, thereby reducing the size of their momentum vectors without changing their orientation (ignoring quantum effects for the moment). Included around the ring are accelerating sections that replace the energy lost to the synchrotron radiation in such a way that the momentum vector of the particle will be restored to its original size, but will make a smaller angle with the desired trajectory.
Because tight corners enhance synchrotron radiation, damping rings are often small. When a long beam of particles is needed, to fill a large particle accelerator, the damping ring may be extended with long straight sections.
Undulators and Wigglers
When faster damping is required than can be provided by the turns inherent in a damping ring, it is common to add undulator or wiggler magnets to induce more synchrotron radiation. These are devices with periodic magnetic fields that cause the particles to oscillate transversely, equivalent to many small tight turns. These operate using the same principle as damping rings and this oscillation causes the charged particles to emit synchrotron radiation, which is then replaced in accelerating sections.
The many small turns in an undulator have the advantage that the cone of synchrotron radiation is all in one direction, forward. This is easier to shield than the broad fan produced by a large turn.
SLAC damping rings home page, including a non-technical description of the damping rings at SLAC.