DERIVATION OF THE TRAJECTORY EQUATIONS OF ACHROMATIC QUADRUPOLES WITH HAMILTONIAN THEORY

Hamiltonian theory is used to develop a general method to derive the trajectory equations for charged particles in magnetic, electrostatic and 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 incorporated into the trajectory equations. With this method both the relativistic and non-relativistic trajectory equations for a combined quadrupole are derived. A combined quadrupole is the combination of a magnetic quadrupole and an electrostatic quadrupole. The correct ratio of the magnetic and electrostatic field in a combined quadrupole results in an achromatic quadrupole. This is proved by the incorporation of the momentum spread into the trajectory equations.