ABSTRACT
This chapter discusses the effects of quadrupole field error. Since quadrupole magnets determine the linear optics of the accelerators, deviations of the quadrupole fields from the design will distort the linear optics. In a circular accelerator, since the optics functions are derived from the parametrization of the one-turn transfer matrix and a quadrupole error anywhere in the ring will change the one-turn transfer matrix, a quadrupole error changes the optics functions everywhere. Dipole fields, including the fields in dipole magnets and the feed-down dipole components in quadrupoles and sextupoles, determine the beam orbit in an accelerator. The Hamiltonian dynamics approach gives a continuous description of the beam motion. In an ideal lattice that consists of drift spaces, dipoles, and quadrupoles, the beam motion in the horizontal plane is independent of the motion in the vertical plane, and vice versa. The nonlinear beam motion in circular accelerator lattices can be analyzed with the Hamiltonian dynamics approach or the Lie map approach.
