Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S19 / 3 / b

The instantaneous beam lifetime iswhere losses make $dN/dt<0$. If the fractional loss rate is constant, $N(t)=N_0{e}^{-t/\tau }$ and $\tau$ is the time to fall to $1/e$ of the starting population. At fixed revolution frequency, the circulating current has the same exponential decay. Density-dependent loss mechanisms can make the instantaneous lifetime vary, so a single exponential need not fit every fill.

Particles are lost when a process drives their transverse amplitude or energy outside the machine's accepted region. Principal causes include scattering from residual gas; Touschek scattering within a bunch, which can convert transverse momentum into excessive longitudinal momentum; collision losses at an interaction point; diffusion from nonlinear resonances, field or RF noise; and collective instabilities or insufficient RF acceptance. Mechanical aperture restrictions turn large-amplitude motion into a loss.

Synchrotron-radiation energy loss is normally replenished by RF, so it does not by itself remove one electron for each emitted photon. Rare large energy deviations can nevertheless contribute to losses if they exceed the accepted energy range. Continuous top-up injection can hide the underlying decay and must be accounted for when measuring a lifetime.

Use a calibrated, nonintercepting DC beam current transformer to measure the circulating current repeatedly during a fill, with top-up injection absent or separately recorded. At fixed revolution frequency, $I=eN{f}_{\mathrm{rev}}$, so a fit to $\log I$ versus time gives slope $-1/\tau$ for exponential decay. The transformer's zero offset and long-term drift must be controlled, especially at low current. An AC current transformer or a calibrated bunch pickup can additionally resolve bunch charge, whereas a DC transformer provides the total circulating intensity needed here.