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Radiation losses often determine the maximum
achievable energy in a charged particle accelerator.
Let us investigate radiation losses in various different types
of accelerator device using the relativistic Larmor formula.
For a linear accelerator, the motion is onedimensional. In this case,
it is easily demonstrated that

(1650) 
where use has been made of Eq. (1648), and
is
the particle momentum in the direction of acceleration
(the direction, say). Here, is the particle rest mass.
Thus, Eq. (1649) yields

(1651) 
The rate of change of momentum is equal to the force exerted on the
particle in the direction, which, in turn, equals the change in the
energy, , of the particle per unit distance. Consequently,

(1652) 
Thus, in a linear accelerator, the radiated power depends on the
external force acting on the particle, and not on the actual
energy or momentum of the particle.
It is obvious, from the above
formula, that light particles, such as electrons,
are going to radiate a lot more than heavier particles, such as protons.
The ratio of the power radiated to the power
supplied by the external sources is

(1653) 
since for a highly relativistic particle. It is
clear, from the above expression, that the radiation losses in an electron
linear accelerator are negligible unless the gain in energy is
of order
MeV in a distance of
meters. That is
MeV/meter.
Typical energy gains are less that 10 MeV/meter. It follows, therefore,
that
radiation losses are completely negligible in linear accelerators, whether
for electrons, or for other heavier particles.
The situation is quite different in circular accelerator devices, such
as the synchrotron and the betatron. In such machines, the momentum
changes rapidly in direction as the particle rotates, but the change in energy
per revolution is small. Furthermore, the direction of acceleration is
always perpendicular to the direction of motion. It follows from
Eq. (1649) that

(1654) 
where is the orbit radius. Here, use has been made of the standard
result
for circular motion. The radiative
energy loss per revolution is given by

(1655) 
For highly relativistic () electrons, this expression yields

(1656) 
In the first electron synchrotrons, meter,
GeV. Hence,
keV per revolution. This
was less than, but not negligible compared to, the energy gain of
a few keV per turn. For modern electron synchrotrons, the limitation
on the available radiofrequency power needed to overcome radiation
losses becomes a major consideration, as is clear from the
dependence of the radiated power per turn.
Next: Angular distribution of radiation
Up: Relativity and electromagnetism
Previous: The Larmor formula
Richard Fitzpatrick
20060202