Electron motion in solenoidal magnetic fields using a first order symplectic integration algorithm

The use of nonsymplectic procedures in particle tracing codes for relativistic electrons leads to errors that can be reduced only at the expense of using very small integration steps. More accurate results are obtained with symplectic transformations for position and momentum. A first-order symplectic integration procedure requires an iterative calculation of the new position coordinates using the old momenta, but the process usually converges in three or four steps. A first-order symplectic algorithm has been coded for cylindrical as well as Cartesian coordinates using the relativistic equations of motion with Hamiltonian variables. The procedure is applied to the steering of a beam of 80-keV electrons by a weak transverse magnetic field superposed on a strong magnetic field in the axial direction. The steering motion is shown to be parallel to the transverse field rather than perpendicular as would be the case without the strong axial field