In circular accelerators and storage rings, electrons emit synchrotron radiation in discrete photons, introducing quantum fluctuations into their motion. This discreteness causes the particles to undergo a random walk in energy and momentum space, leading to a diffusion process that shapes the energy spread of the beam and its emittance.
Mechanism An electron moving through a magnetic field emits radiation called synchrotron radiation. The expected amount of radiation can be calculated using the classical power. Considering quantum mechanics, however, this radiation is emitted in discrete packets of photons. For this description, the distribution of the number of emitted photons and also the energy spectrum for the electron should be determined instead. In particular, the normalized power spectrum emitted by a charged particle moving in a bending magnet is given by
S ( ξ ) = 9 3 8 π ξ ∫ ξ ∞ K 5 / 3 ( ξ ¯ ) d ξ ¯ . {\displaystyle S(\xi )={\frac {9{\sqrt {3}}}{8\pi }}\xi \int _{\xi }^{\infty }K_{5/3}({\bar {\xi }})d{\bar {\xi }}.}
This result was originally derived by Dmitri Ivanenko and Arseny Sokolov and independently by Julian Schwinger in 1949. Dividing each power of this power spectrum by the energy yields the photon flux:
F ( ξ ) = 1 ξ S ( ξ ) = 9 3 8 π ∫ ξ ∞ K 5 / 3 ( ξ ¯ ) d ξ ¯ . {\displaystyle F(\xi )={\frac {1}{\xi }}S(\xi )={\frac {9{\sqrt {3}}}{8\pi }}\int _{\xi }^{\infty }K_{5/3}({\bar {\xi }})d{\bar {\xi }}.}
The photon flux from this normalized power spectrum (of all energies) is then
N ˙ n o r m = 9 3 8 π ∫ ξ = 0 ∞ ∫ ξ ¯ = ξ ∞ K 5 / 3 ( ξ ¯ ) d ξ ¯ d ξ = Γ ( 11 / 6 ) Γ ( 1 / 6 ) 9 3 8 π = 15 3 8 . {\displaystyle {\dot {N}}_{norm}={\frac {9{\sqrt {3}}}{8\pi }}\int _{\xi =0}^{\infty }\int _{{\bar {\xi }}=\xi }^{\infty }K_{5/3}({\bar {\xi }})d{\bar {\xi }}d\xi =\Gamma (11/6)\Gamma (1/6){\frac {9{\sqrt {3}}}{8\pi }}={\frac {15{\sqrt {3}}}{8}}.}
The fact that the above photon flux integral is finite implies discrete photon emission. It is a Poisson process. The emission rate is
r γ = 5 3 6 r e e 0 β 4 γ ℏ | ρ | photons/sec . {\displaystyle r_{\gamma }={\frac {5{\sqrt {3}}}{6}}{\frac {r_{e}e_{0}\beta ^{4}\gamma }{\hbar |\rho |}}{\text{ photons/sec}}.}
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