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

The quoted proton energies are explicitly kinetic. For mean ring radius $\bar{R}=215$ m,Using these expressions rather than setting $\beta =1$ at injection givesThe frequency approaches $c/C$ as energy increases; it is the velocity increase, not a change in the prescribed mean ring circumference, that produces the frequency sweep.

The RF harmonic number is the number of RF cycles per revolution, so $f_{\mathrm{RF}}=h{f}_{\mathrm{rev}}$. For $h=32$,The synchronous phase fixes which phase of that voltage is sampled, not the frequency ratio. For a sine voltage, ${\phi }_s=0$ gives zero net accelerating voltage and describes a stationary ideal-energy condition; an actual energy ramp requires a phase producing the required energy gain. A different phase-origin convention changes that statement about the zero of phase, but not the boxed frequencies.

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.

Both machines are proposed electron–positron linear colliders, using sources, damping rings, bunch compressors, two opposing main linacs, and final-focus and beam-delivery sections that bring very small bunches to an interaction point. The spent beams go to extraction lines and beam dumps rather than making repeated collision turns.

In the exam-era International Linear Collider reference design, polarized electrons and an undulator-based positron source feed damping rings and two superconducting main linacs; the full $500$ GeV centre-of-mass facility is about $31$ km long. The Compact Linear Collider adds high-current drive-beam generation and deceleration lines parallel to its main linacs; its $3$ TeV concept is roughly $48$ km long. The linear layout avoids the repeated synchrotron-radiation losses that would be severe for multi-hundred-GeV electrons in a storage ring. ^{[ref] [ref]}

For a comparison appropriate to this 2012 examination, use the reference-design and conceptual-design targets rather than later revised starting stages:The ILC reference programme included scanning approximately $200$–$500$ GeV and a possible $1$ TeV upgrade. The CLIC 2012 studies included lower-energy construction stages as well as the ultimate multi-TeV machine. These are design goals, not achieved operating luminosities.

Collider luminosity sets the event rate for a process of cross-section $\sigma$ through $dN/dt=\mathcal{L}\sigma$; it is not radiated optical power. ^{[ref] [ref]}

The International Linear Collider uses externally powered superconducting niobium RF cavities at about $1.3$ GHz, with a reference average accelerating gradient of $31.5$ MV/m. Superconducting walls reduce RF dissipation, allowing relatively long RF pulses and bunch trains, at the cost of cryogenic infrastructure.

The Compact Linear Collider uses normal-conducting copper structures at about $12$ GHz and a main-linac gradient near $100$ MV/m in its multi-TeV design. A separate high-current, lower-energy electron drive beam is decelerated in power-extraction structures; the resulting RF power accelerates the high-energy, low-current main beams. This two-beam concept supplies short, high-power pulses without requiring a high-power klystron at every main-beam structure.

ILC: superconducting low-loss RF; CLIC: high-gradient normal-conducting two-beam RF. Both rely on damping and final focusing to obtain very small collision spots, so energy alone does not determine their luminosity. ^{[ref] [ref]}