Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S19 / 1

Take a reference point immediately before the focusing quadrupole and let matrices act on $(x,x^{\prime })$ from the right. With $q=1/f$, define $Q_F=\left(\begin{array}{cc}1& 0\\ -q& 1\end{array}\right)$, $Q_D=\left(\begin{array}{cc}1& 0\\ q& 1\end{array}\right)$ and $D(L)=\left(\begin{array}{cc}1& L\\ 0& 1\end{array}\right)$. One cell isIts determinant is one. A different starting point gives a conjugate matrix with the same trace and stability; the orthogonal transverse plane replaces $q$ by $-q$.

For a periodic uncoupled lattice, $\mathrm{tr}M=2\cos \mu$ in the Twiss representation. Here $\cos \mu =1-L_1{L}_2{q}^2/2$, so bounded nondegenerate cell motion requiresThe zero-strength endpoint is only marginal and usually gives a drifting Jordan block. The phase advance increases from zero to $\pi$ on the positive-strength stable branch. Repeating the cell $N$ times gives ring eigenvalues $e^{\pm iN\mu }$ and tune $N\mu /(2\pi )$; it does not enlarge the stable strength interval. Integer ring phase advances can be resonance-sensitive in the presence of errors even when the ideal repeated matrix is bounded.

For fixed quadrupole fields and fractional momentum offset $\delta$, $q(\delta )=q_0/(1+\delta )$. Differentiate $\cos \mu =1-L_1{L}_2{q}^2/2$ at zero to obtain $d\mu /d\delta =-L_1{L}_2{q}_0^2/\sin \mu =-2\tan (\mu /2)$. ThusBoth transverse planes have the same natural result; the ring value is $N$ times this. It vanishes only in the zero-focusing limit, outside robust stable operation. Additional correcting elements, such as sextupoles where dispersion is nonzero, are required for zero corrected chromaticity.