Past exam of the symmetry and relativity course of the University of Oxford / 2015 / 1 / c

Interpret $3$ GeV as the electron's total energy; the alternative kinetic-energy reading changes the estimate only slightly at this scale. The Lorentz factor and orbital acceleration areUsing the supplied Larmor formula, ${\mu }_0=1/({ϵ}_0{c}^2)$ and the revolution time $2\pi R/v$,For $R=561$ m this gives $P=1.7405\times 10^{-10}$ W per electron and $T_{\mathrm{turn}}=1.17577\times 10^{-5}$ s. Their product isThis is the energy loss of one electron on a full circular orbit, not its energy loss per metre or the total loss of the whole bunch. The fractional loss is only about $4.26\times 10^{-6}$ of the $3$ GeV energy each revolution. Maintaining a steady stored energy requires replacing that energy through the accelerator's rf system.

Relativistic relativistic aberration concentrates emission into an angular scaleThis is a half-angle or characteristic divergence scale; a full cone opening is about twice it. The exact half-photon-count cone in a applies to isotropic rest-frame emission. A synchrotron emitter has its own rest-frame angular radiation pattern, so that count result should not be treated as an exact median-power formula for the synchrotron beam. It nevertheless gives the correct relativistic beaming scale. Electron-beam emittance, vertical angular spread and the detector's acceptance can broaden the experimentally observed divergence beyond this intrinsic estimate.

A fixed distant observer receives appreciable radiation while the velocity points within an angle of order $1/\gamma$ of the line of sight. The corresponding emission interval is $\Delta t_{\mathrm{em}}\sim R/(\gamma v)$. The source moves toward the detector during that interval, shortening the light-travel distance. In the far-field limit,where $\varphi$ is the angle between velocity and the detector direction. Near the tangent, $1-\beta \cos \varphi \sim {\gamma }^{-2}$, soThe emission interval itself is of order $3.2\times 10^{-10}$ s. Omitting the arrival-time compression would therefore miss many orders of magnitude.

For a definite angular window $-1/\gamma \le \varphi \le 1/\gamma$, integrating the arrival-time relation givesThe order-one coefficient depends on how the beam window or pulse width is defined. These are intrinsic single-electron radiation-flash scales. For bunch-length broadening of synchrotron radiation, a finite bunch moving through the same tangent point contributes an arrival-time spread of order its length divided by $v$; the longitudinal bunch profile and instrumental response must be convolved with the intrinsic flash. “Short bunch” alone does not specify an experimental bunch pulse duration at the attosecond scale, because no bunch length has been supplied.^[ref]

A shorter radiation flash has a broader Fourier transform. From iii, its characteristic angular-frequency bandwidth isAt fixed bending radius this scales as ${\gamma }^3$ as $v\to c$. The standard synchrotron critical angular frequency makes the same scaling explicit:A periodically circulating electron produces harmonics of its revolution frequency. The very large harmonic range, of order ${\gamma }^3$, forms the familiar effectively continuous broad synchrotron spectrum at ordinary detector resolution. The result concerns the intrinsic spectrum of the radiation, not merely the slow envelope imposed by a long incoherent bunch. Ordinary frequency is $\nu =\omega /(2\pi )$; mixing cycles per second with radians per second would introduce a spurious factor of $2\pi$ into photon-energy estimates.^[ref]

Using the conventional critical scale rather than guessing a precise Fourier-width coefficient,This is extreme ultraviolet and the very soft X-ray boundary, with a broad spectrum extending on either side; it is not a hard, multi-keV X-ray source at the stated large bending radius. The characteristic energy is not a sharp upper cutoff. For orientation, the standard angle-integrated spectral power per unit frequency peaks near $0.29E_c\simeq 31$ eV, also in the extreme ultraviolet; a particular angular acceptance weights that spectrum differently.

The crucial scale follows from both the $3$ GeV energy and the $561$ m radius. Using only the fact that synchrotron sources can make X-rays, or replacing the bending radius by a much smaller unstated value, would misidentify the characteristic photon energies here. The critical-energy convention and the broadband character are described by the primary radiation model.^[ref]