Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S19

Use transverse coordinates $(x,x^{\prime })$ and define a positive signed focal length as focusing in that plane. A thin quadrupole magnet of length $\ell$ and signed gradient $G$ has integrated strength $1/f=|e|G\ell /p_0$; the electron charge and chosen field orientation are absorbed into the sign of $G$. Its beam transfer matrix isFor total electron energy $E_0$, $p_0=\sqrt{E_0^2-m_e^2{c}^4}/c$, so $f\simeq E_0/(|e|cG\ell )$ in the ultrarelativistic limit. The other transverse plane has the opposite signed focal length. A quadrupole magnet therefore focuses one plane while defocusing the other; the longitudinal magnetic field component is negligible in this paraxial idealization.

At fixed gradient and magnet length, $f\propto p$. If $E$ is total energy,Here ${\beta }_0=v_0/c$, not one of the Courant-Snyder parameters. For ultrarelativistic electrons this becomes $f\simeq f_0(1+\delta )$, or $1/f\simeq (1-\delta )/f_0$: higher-energy particles are bent and focused less strongly. If the symbol $E$ instead denotes kinetic energy $K$, the coefficient is $(K+m{c}^2)/(K+2m{c}^2)$, approaching $1/2$ in the nonrelativistic limit. Stating the energy convention avoids applying the ultrarelativistic result to a slow beam.

Let $D(d)=\left(\begin{array}{cc}1& d\\ 0& 1\end{array}\right)$ be the drift beam transfer matrix. Start immediately before the first lens and end immediately before its counterpart in the next FODO cell. Matrices act on column vectors from right to left. With $A=1/f_1$ and $B=1/f_2$, where both focal lengths are signed in the chosen plane,The two drifts account for both gaps. Direct multiplication gives $\det M=1$ andFor the equal-strength focusing-defocusing case, $f_1=f>0$ and $f_2=-f$:Choosing a different start within the cell gives a conjugate matrix, with the same trace and eigenvalues. If $f_2$ is specified as a positive magnitude of a defocusing lens, its signed value in these formulas is $-f_2$.

For a real uncoupled, lossless transverse map, $\det M=1$. Bounded nondegenerate oscillatory motion requires complex-conjugate eigenvalues $e^{\pm i\mu }$, equivalentlyIts Courant-Snyder parameters giveThe positive quadratic form ${\gamma }_T{x}^2+2{\alpha }_{T}x{x}^{\prime }+{\beta }_T{x}^{\prime 2}$ is invariant, so repeating the cell cannot produce secular growth. At trace $\pm 2$ a nontrivial Jordan block instead produces linear growth; only the exceptional maps $M=\pm I$ remain bounded at those exact boundaries. The criterion must hold in both transverse planes. For the symmetric FODO cell, both traces are $2-d^2/f^2$, giving

For ultrarelativistic particles with fixed magnet settings, $A(\delta )=A_0(1-\delta )$ and $B(\delta )=B_0(1-\delta )$. Hence the half-trace isOff-energy motion is stable when $|C(\delta )|<1$, separately in each transverse plane. Where $\sin {\mu }_0\ne 0$, its first-order phase shift isFor equal opposite strengths,Thus positive momentum error reduces the focusing and phase advance; negative error moves toward the overfocused unstable boundary. The unexpanded symmetric stability condition is $|f_0|(1+\delta )>d/2$, with $1+\delta >0$, within the ultrarelativistic energy scaling. Its natural chromaticity per cell is $d(\mu /2\pi )/d\delta =-\tan ({\mu }_0/2)/\pi$. Away from the ultrarelativistic limit replace the displayed first-order energy error by $\delta /{\beta }_0^2$. Linearizing at a stability boundary is singular and needs separate treatment.

For an ultrarelativistic electron of Lorentz factor $\gamma =E/(m_e{c}^2)=3000/0.511=5870.84$, the synchrotron critical frequency gives ${\epsilon }_c=\hslash {\omega }_c=3\hslash c{\gamma }^3/(2\rho )$. Thus, using the supplied constants,The angle-integrated spectral power per unit photon energy is proportional to $x{\int }_x^{\infty }{K}_{5/3}(u)du$, with $x=\epsilon /{\epsilon }_c$. It peaks at $x\simeq 0.286$, soThis is a power spectrum per linear energy interval; power per logarithmic interval and photon-count spectra have different peaks. The magnetic rigidity condition is $p=|e|B\rho$, with $pc\simeq E$, givingThe critical energy sets a broad spectral scale, not a maximum allowed photon energy.^[ref]

For a ring whose total bending angle $2\pi$ is supplied by dipoles with the computed radius, the single-electron energy loss per turn isA stored current $I$ corresponds to $I/|e|$ electron turns per second: the number of stored electrons times the revolution rate. ThereforeThis is the whole beam's radiated power, not the power accepted by one beamline. If the specified radius describes only one dipole and the other bends are different, the radiation integral is needed instead:The numerical result uses the standard equal-radius-bend assumption; the ring circumference, including straight sections, is not required. The RF system must replenish this loss to maintain stored beam energy.

Near the critical photon energy, the characteristic vertical divergence of synchrotron radiation is ${\sigma }_{\theta }\simeq 1/\gamma =1.70\times 10^{-4}\mathrm{rad}$ in the approximation requested. Ignoring electron source size and divergence, the rms vertical width $10\mathrm{m}$ downstream isThe full aperture width is $10\mathrm{mm}$, so its half-width is $5\mathrm{mm}=2.94{\sigma }_y$. It therefore passes almost all the central $8\mathrm{keV}$ photon beam. A Gaussian angular approximation would giveThe exact synchrotron angular distribution is not Gaussian, so this number is an illustrative containment estimate, not an exact throughput calculation. A finite aperture cannot transmit every photon in a distribution with tails. The assumed infinite horizontal aperture removes horizontal fan clipping.

Below the critical photon energy, radiation is emitted over a wider angular cone. The usual low-frequency characteristic-angle estimate isup to an order-one angular-width convention. At $\epsilon =0.012\mathrm{eV}$ this givesThe aperture should therefore be enlarged vertically. A full width of order $0.30\mathrm{m}$ covers $\pm 1$ characteristic width; approximately $0.90\mathrm{m}$ covers $\pm 3$ and gives containment comparable with the preceding X-ray estimate in the same Gaussian model. The original $10\mathrm{mm}$ width would pass only roughly $\mathrm{erf}[0.005/(\sqrt{2}0.149)]\simeq 2.7\%$ in that model. Equivalent optics could instead reduce the spot at a relocated aperture, at the expense of angular acceptance. These estimates concern geometric acceptance; THz coherence and bunch structure require further beam data.

An RF cavity stores an oscillating electromagnetic field. Its longitudinal electric field does work on a correctly phased bunch, with energy gain $\Delta E=q{V}_{\mathrm{RF}}\mathcal{T}\sin {\varphi }_s$ for a chosen phase convention and transit-time factor $\mathcal{T}$. The bunch arrival time is synchronized so successive gaps accelerate rather than decelerate it. The magnetic force guides and focuses the beam but does no work.

A linear accelerator uses new accelerating structures along its path; cell lengths and timing accommodate the particle speed, and each bunch normally traverses a given section once. A circular synchrotron reuses the same RF cavities each revolution. Its synchronous phase supplies both acceleration and radiation replacement; RF frequency or harmonic synchronization and bending-field strength follow the required orbit and particle speed. Longitudinal phase stability captures nearby particles into RF buckets.

For very high-energy electron and positron colliders, a linear geometry avoids repeated synchrotron losses: at fixed radius, loss per turn scales as $E^4/m^4$. This is not a statement that all high-energy colliders are preferably linear. The LHC was already a circular proton collider in the 2015 context; the proton's much larger mass greatly suppresses radiation at the same energy. Circular electron machines can also be competitive at lower energies. Magnet limits, luminosity, cost and beamstrahlung enter any real comparison.

Conventional RF cavities accelerate beams with controlled phase, energy spread and transverse beam emittance. Their material surfaces limit sustainable gradients to order $10$–$100\mathrm{MV}{\mathrm{m}}^{-1}$, depending on technology and operating conditions. Mature RF systems already reached multi-GeV and much higher beam energies in 2015. The 2013 ILC baseline proposed $250\mathrm{GeV}$ electrons colliding with $250\mathrm{GeV}$ positrons, approximately $31.5\mathrm{MV}{\mathrm{m}}^{-1}$ superconducting cavities and a roughly $31\mathrm{km}$ facility. These were design parameters, not an operating collider.^[ref]

In plasma wakefield acceleration, an intense laser displaces plasma electrons and excites a longitudinal charge-density wave. A suitably injected electron bunch gains energy from that wave, with fields of order tens to hundreds of $\mathrm{GV}{\mathrm{m}}^{-1}$. A concrete result available before this examination was $4.2\mathrm{GeV}$ from a $9\mathrm{cm}$ capillary, with $6\%$ rms energy spread, $6\mathrm{pC}$ bunch charge and $0.3\mathrm{mrad}$ rms divergence. These values characterize that experiment, not every laser-plasma beam; divergence alone is not an emittance measurement.^[ref]

A collider concept stages many separately driven plasma modules, with fresh laser pulses, interstage capture and refocusing, controlled injection, and a final collision focus. Dephasing and pump depletion limit a single stage. Although the accelerating sections could be much shorter than ILC linacs, a useful collider also requires high repetition rate and wall-plug efficiency, reproducible low energy spread, preserved normalized emittance, stable timing, high bunch charge and acceptable beam-beam effects. Positron acceleration is an additional challenge because an electron-optimized plasma wake need not focus positrons. Large gradient alone does not demonstrate the luminosity or beam quality of an RF collider.

In the cold unmagnetized linear plasma model, the refractive-index relation is $n^2=1-{\omega }_p^2/{\omega }_L^2$, withThe critical density occurs at ${\omega }_p={\omega }_L=2\pi c/{\lambda }_L$. Thusfor ${\lambda }_L=1\mu \mathrm{m}$. The corresponding frequency isA propagating conventional laser wake generally uses an underdense plasma, rather than this cutoff density. Strong relativistic laser motion, collisions or a magnetized plasma modify this simple critical-density condition.

A suitable accelerator-science example is strong focusing. The apparent technical contradiction is that increasing a quadrupole magnet's focusing strength improves confinement in one transverse plane while worsening it in the other. Abandoning that magnet entirely loses its strong focusing action.

An alternating-gradient lattice separates the two actions in space: a focusing quadrupole in one plane is followed, after a drift, by one with the opposite sign, and the pattern repeats. In the other plane their roles reverse. The symmetric thin-lens FODO cell has the same trace $2-d^2/f^2$ in both planes, so both are stable for $0<d/|f|<2$, despite each individual quadrupole defocusing one plane. The noncommuting sequence of drift and lens maps is what makes net confinement possible.

In a TRIZ interpretation, this resolves the contradiction by spatial separation and periodic alternation of opposing functions, rather than demanding that one static element satisfy both. The useful result is smaller beam envelopes and magnet apertures for a suitable lattice. This is a retrospective design analysis; it does not claim that the historical inventors actually followed TRIZ. The matrix calculation supplies a test of the proposed resolution rather than just attaching a slogan to an invention.