Betatron Motion
For a particular particle, the deviation from an idea orbit will undergo “pseudo-harmonic” oscillation as a function of the path along the orbit:
The “betatron function” b(s) is effectively the local wavelength and  also defines the beam envelope.
Phase advance
b(s) is has the fundamental cell periodicity of the lattice
length of one, e.g., FODO cell
However, in general the phase (and therefore particle motion) does not, and indeed must not, follow the periodicity of the ring…
Lateral deviation in one plane
Closely spaced strong quads -> small b -> small aperture
Sparsely spaced weak quads -> large b -> large aperture
s
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