A single quadrupole magnet focuses a charged beam in one transverse plane while defocusing it in the other — a lone quad can never focus in both x and y at once. The fix, discovered independently by Christofilos and by Courant, Livingston & Snyder in the early 1950s, is alternating-gradient ("strong") focusing: chain a focusing quad (QF) and a defocusing quad (QD) back to back, each separated by a field-free drift. Net result — the beam is focused in both planes on average, the same principle behind every synchrotron and storage ring built since, including the LHC.
Each element is a linear map on the transverse phase-space vector (x, x′). A drift of length L and a thin quadrupole of focal length f = 1/k act as 2×2 matrices:
Drift(L) = [[1, L], [0, 1]]
Quad(f) = [[1, 0], [-1/f, 1]] (focusing: f > 0)
Multiplying the matrices for one FODO cell (QF·O·QD·O) gives the one-turn map. Its trace fixes the phase advance per cell μ via cos μ = ½·trace(M) — the betatron oscillation frequency. Summed over every cell in the ring this gives the tune Q, the number of transverse oscillations per revolution. If |½·trace(M)| exceeds 1 the motion is no longer a bounded rotation in phase space but a hyperbolic runaway: the amplitude grows every cell and the beam is lost on the vacuum-chamber wall — this is the stability condition that sets the practical limit on how strong a quad can be for a given cell length.
- Quadrupole strength k — sets the focal length f = 1/k of every magnet; stronger quads focus harder but shrink the stable window.
- Drift length L — the field-free spacing between magnets; a longer drift raises the phase advance for the same k.
- Injected beam offset — the initial displacement of the ring of test particles entering the lattice; the envelope curves scale with it exactly (this is linear optics).
- Envelope — the ±(amplitude × transported unit trajectory) curve every particle at that amplitude must stay within; toggle it to see just the individual particles instead.