Notes on Synchrotron Radiation

Synchrotron radiation occurs when a charged particle, typically an electron, experiences an acceleration transverse to its velocity, as caused by a magnetic field. The following notes emphasize the case when the particle's velocity approaches the speed of light, the limit in which synchrotron radiation becomes prominent. 1 Total Rate The total (classical) radiation rate is obtained from the Larmor formula: P = dU dt = 2 3 e 2 a 2 c 3 = 2 3 e 2 γ 4 c ρ 2 , (1) where a is the acceleration in the electron's rest frame, and ρ is the radius of curvature. We have used a lab = v 2 /ρ ≈ c 2 /ρ for an electron with relativistic velocity. Also, a = γ 2 a lab with γ = 1/ 1 − v 2 /c 2 , noting that the acceleration is transverse, so the transformation involves two powers of the time dilation. The angular distribution of radiated power (for time t measured at the charge, i.e., for retarded time) is obtained via the invariant transformation of the Larmor angular distribution: dU dtdΩ = e 2 γ 4 c 4πρ 2 (1 − β cos θ) 2 − (1 − β 2) sin 2 θ cos 2 φ (1 − β cos θ) 5 , (2) where β = v/c and angles (θ, φ) are measured with respect to the direction of the electron's motion and with the x-axis towards the center of the electron's orbit. For highly relativistic motion, γ 1, the radiation is peaked forward with characteristic angle θ ≈ 1/γ. Then 1 − β cos θ ≈ (θ 2 + 1/γ 2)/2. In the plane of the orbit we have dU dtdΩ ≈ 2e 2 γ 10 c πρ 2