FODO Synchrotron: Betatron Tune, Emittance and Stop Bands

A linear transverse-beam-dynamics playground for a FODO synchrotron. Each thin quadrupole and drift is a $2 \times 2$ symplectic transfer matrix; the one-turn map gives the Courant-Snyder (Twiss) parameters, the betatron tune from $\cos\mu = \mathrm{trace}(M)/2$, and the invariant emittance ellipse. Panel A shows the ring with alternating focusing and defocusing quads, a circulating bunch and the periodic $\beta$ function across one cell. Panel B is the transverse phase space: a tracked particle lands turn after turn on its invariant ellipse, with the single-particle emittance constant turn after turn, or, in a stop band, no ellipse exists and the amplitude diverges. Panel C is the stability diagram, $\mathrm{trace}(M_{\mathrm{cell}})/2$ and the tune versus quad focal length, with the $f < L/4$ stop band and the integer / half-integer resonance lines, plus the dipole magnetic rigidity $B\rho = p / (0.299792458 q)$ and $d p / d t = q v B$. Each element map is symplectic ($\det = 1$), the betatron tune follows from $\cos\mu = \mathrm{trace}(M)/2$ with $\beta\gamma - \alpha^2 = 1$, the Courant-Snyder invariant is conserved over many turns, and outside the stability band the amplitude diverges on resonance.