Intrinsic third order aberrations in electrostatic and magnetic quadrupoles

Intrinsic aberrations are those which occur due to the finite length of the desired field configuration. They are often loosely ascribed to the fringing field. This is misleading as it implies that the effects can be minimized by shaping the fields. In fact, there is an irreducible component related to the broken symmetry. It is present even in the hard-edge limit, and moreover, the other (soft-edge) effects can be simply ascribed to the intrinsic aberration spread over a finite length. We rederive the aberration formulas for quadrupoles using a Hamiltonian formalism. This allows for an easy comparison of electrostatic and magnetic quadrupoles. For different combinations of large and small emittances in the two transverse planes, it is found that in some situations electrostatic quadrupoles have lower aberrations, while in others, magnetic quadrupoles are better. As well, we discuss the ways in which existing transport codes handle quadrupole fringe fields. Pitfalls are pointed out and improvements proposed.