Successive approximations for charged particle motion

Single-particle dynamics in electron microscopes, ion or electron lithographic instruments, particle accelerators, and particle spectrographs is described by weakly nonlinear ordinary differential equations. Therefore, the linear part of the equation of motion is usually solved and the nonlinear effects are then found in successive order by iteration methods. When synchrotron radiation is not important, the equation can be derived from a Hamiltonian or a Lagrangian. The Hamiltonian nature can lead to simplified computations of particle transport through an optical device when a suitable computational method is used. H. Rose and his school have contributed to these techniques by developing and intensively using the eikonal method [1–3]. Many ingenious microscopic and lithographic devices were found by Rose and his group due to the simple structure of this method [4–6]. The particle optical eikonal method is either derived by propagating the electron wave or by the principle of Maupertuis for time-independent fields. Maybe because of the time-dependent fields which are often required, in the area of accelerator physics the eikonal method has never become popular, although Lagrange methods had been used sometimes already in early days [7]. In this area classical Hamilitonian dynamics is usually used to compute nonlinear particle motion. Here the author will therefore derive the eikonal method from a Hamiltonian quite familiar to the accelerator physics community and reformulate it in a simplifying way. With the event of high-energy polarized electron beams [8] and plans for high-energy proton beams [9], nonlinear effects in spin motion have become important in high-energy accelerators. The author introduces a successive approximation for the nonlinear effects in the coupled spin and orbit motion of charged particles which resembles some of the simplifications resulting from the eikonal method for the pure orbit motion.