Past exam of the particle physics course of the University of Oxford / 2019 / 2

Quadrupole magnets provide transverse restoring forces proportional to displacement from the reference orbit. Their magnetic field increases linearly away from the axis, so an off-axis particle receives a kick that can bring its trajectory back toward the design orbit. A magnetic quadrupole focuses in one transverse plane and defocuses in the other; a single quadrupole cannot confine both planes.

An alternating-gradient focusing lattice alternates quadrupole orientations with drift spaces. With suitable strengths and separations, a focusing-defocusing sequence gives stable betatron oscillations in both planes. The focusing acts on transverse position, while the phase advance through the lattice converts an initial angular deviation into position and then a corrective kick. The beam is thus transported within a bounded envelope rather than allowed to drift into the pipe. Dipoles provide the main bending; the quadrupole lattice controls the small deviations.

Write $z=\sqrt{ϵ}\omega \cos \psi$, with $\omega =\sqrt{\beta }>0$ and $\psi =\varphi -{\varphi }_0$. Differentiating twice givesThe coefficients must vanish for arbitrary initial phase ${\varphi }_0$. The sine coefficient givesThe conventional Courant-Snyder parameters normalize the phase so that $C=1$; any positive constant $C$ can be absorbed into the scale assigned to $\beta$ and $ϵ$. Explicitly, $\tilde{\beta }=\beta /C$ and $\widetilde{ϵ}=Cϵ$ preserve the orbit while making $\tilde{\beta }{\varphi }^{\prime }=1$. With this convention,The second expression is the Ermakov-Pinney equation for the beam envelope. It follows by substituting ${\varphi }^{\prime }=1/{\omega }^2$ into the cosine coefficient.

The printed equation omits the factor $\omega$ multiplying $k(s)$. For a concrete counterexample, take the constant periodic focusing strength $k=1/16$ and the constant envelope $\omega =2$. Then ${\varphi }^{\prime }=1/4$ and $z=2\sqrt{ϵ}\cos (s/4-{\varphi }_0)$ solve $z^{\prime \prime }+z/16=0$. The correct envelope equation is $0-1/8+(1/16)2=0$, whereas the printed equation gives $-1/8+1/16=-1/16$.

The associated invariant isThus $ϵ$ fixes the phase-space ellipse of a trajectory; a beam's rms geometric emittance uses the corresponding second-moment convention. Factors of $\pi$ depend on whether emittance means ellipse area or area divided by $\pi$ and do not alter the envelope equation.

For a beam with rms geometric emittance ${ϵ}_y$, its vertical size at the collision point is ${\sigma }_y^{\star }=\sqrt{{ϵ}_y{\beta }_y^{\star }}$. Reducing ${\beta }_y^{\star }$ therefore concentrates the beam, increasing the number of particles encountering the opposite beam per transverse area. For identical Gaussian beams colliding head-on,Keep bunch populations, horizontal size, collision frequency and emittances fixed. If both vertical beta values are decreased by a factor $f>1$, meaning ${\beta }_{y,\mathrm{new}}^{\star }={\beta }_{y,\mathrm{old}}^{\star }/f$, both vertical beam sizes decrease by $1/\sqrt{f}$, andThe luminosity is inversely proportional to the common vertical size, not its square, because only one transverse dimension has changed. The same ratio follows from the unequal-beam overlap denominator $\sqrt{{\sigma }_{y1}^2+{\sigma }_{y2}^2}$.

A final-focus quadrupole system on either side of the interaction point forms a waist there. Strong quadrupole doublets or triplets, with suitable matching sections, focus both planes. At a symmetric waist ${\beta }^{\prime }=0$ and the nearby drift gives $\beta (s)={\beta }^{\star }+s^2/{\beta }^{\star }$, so small ${\beta }^{\star }$ requires large beam envelopes and strong optical gradients in the adjacent magnets. Chromatic correction is needed for off-momentum particles. The formula assumes negligible crossing-angle and hourglass reductions; a bunch length comparable to ${\beta }^{\star }$ changes the effective overlap.

For a paraxial orbit $z^{\prime }\simeq p_z/p_s$, where $p_z$ is transverse momentum and $p_s$ longitudinal momentum. Acceleration mainly increases $p_s$, so a given transverse kick produces a smaller angular deviation. In an ideal matched accelerator without scattering or radiation, the conserved transverse canonical phase-space area becomes a smaller area in the coordinates $(z,z^{\prime })$. This is adiabatic damping of geometric emittance, not loss of canonical phase-space volume.

The invariant normalized emittance isHere ${\beta }_{\mathrm{rel}}$ is speed in units of $c$, distinct from the lattice beta function. Neglecting dilution mechanisms, each acceleration stage thus supplies a beam with smaller angular spread and geometric emittance for injection into the next stage.

A chain of increasingly energetic particle accelerators lets each ring operate over a manageable range of magnetic rigidity and radio-frequency settings. It also avoids injecting a large-emittance low-energy beam directly into the small-aperture high-field main ring. Higher injection energy reduces space-charge effects and improves capture and transmission. Emittance dilution, mismatching and collective instabilities can oppose the ideal damping; acceleration alone does not reduce normalized emittance.