DERIVATION OF THE TRAJECTORY EQUATIONS OF ACHROMATIC QUADRUPOLES WITHHAMILTONIAN THEORY

Hamiltonian theory is used to develop a general method to derive the t rajectory equations for charged particles in magnetic, electrostatic a nd combined multipoles with a straight ion optical axis. The effect of the fringing fields of the multipoles, the influence of the momentum spread of the particles and parasitic aberrations are easily incorpora ted into the trajectory equations. With this method both the relativis tic and non-relativistic trajectory equations for a combined quadrupol e are derived. A combined quadrupole is the combination of a magnetic quadrupole and an electrostatic quadrupole. The correct ratio of the m agnetic and electrostatic field in a combined quadrupole results in an achromatic quadrupole. This is proved by the incorporation of the mom entum spread into the trajectory equations.