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When a charged particle is accelerated by a field, it would seem there are many possible photons that could be emitted to obey conservation of momentum. Is it random, or time-dependent, or based on the magnitude of the acceleration?

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Depends on the circumstances under which a particle is accelerated.

For instance, bremsstrahlung radiation occurs when a charged particle slows down on another charged particle (e.g., electrons driven into a target to generate X-rays). In this case, the radiation is a continuum and depends on the amount of energy a given particle loses to the interacting medium (there is a high-energy limit).

Synchrotron radiation, which occurs when charged particles undergo Larmor motion under the influence of a magnetic field, causes emission of photons at the angular frequency and its harmonics.

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    there are spectra for brehmsstrahlung http://en.wikipedia.org/wiki/Brehmstrahlung ands synchrotron in http://en.wikipedia.org/wiki/Synchrotron_radiation – anna v Jul 18 '14 at 05:13