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angle aperture axis beam emittance bending magnets beta function betatron motion betatron oscillations bunch cavity central orbit CERN CERN Accelerator School chromaticity circular accelerator closed orbit coefficients coherent collimator component constant coordinates coupling crossing damping defined defocusing density derived dipole dispersion function distribution doublet drift spaces effect electric field electron ellipse envelope-ellipse equation errors expression FODO FODO cell given gradient high energy horizontal and vertical impedance injection insertion instability integral kick lattice length linear Liouville's theorem longitudinal luminosity machine magnetic field matched Maxwell's equations measure multipole non-linear normal normalised obtained optics parameters Particle Accelerators perturbation phase advance phase space phase trajectory plane position primary collimator proton quadrupole magnets radius resonance revolution frequency scattering secondary collimator sextupole shown in Fig skew quadrupole solution space-charge stopband storage rings structure synchrotron radiation transfer line transfer matrix transformation transverse tune shift values velocity zero