Quantum Effects in the Radiation from Accelerated Relativistic Electrons

The radiation associated with the operation of a betatron or synchrotron, due to the transverse acceleration of relativistic electrons by the magnetic field of the machine, is discussed qualitatively. Order of magnitude expressions for the angular opening of the cone within which the radiation is emitted, and for the wavelength of the maximum of the radiated spectrum, are obtained by means of qualitative arguments based on classical physics. It is then shown, from a consideration of the rate of spread of a wave packet that describes the electron, that quantum effects are negligible so long as the ratio $γ$ of the total energy to the rest energy of the electron is small in comparison with the square root of the ratio of the orbit radius $R$ to the reduced Compton wavelength $λ̄c$ of the electron. This criterion for the validity of classical theory is substantially equivalent to, but not identical with, the commonly accepted criterion $γ2≪R/λ̄c$⁠, which is derived from the assumption that the momentum of an emitted photon must be small in comparison with the momentum of the electron. Thus classical theory is expected to be valid for electron energies small in comparison with $1015$ ev if the magnetic field strength does not exceed $104$ oersted.