Differential algebraic theory and calculation for arbitrary high order aberrations of electron lenses

Differential algebra is a new and promising mathematical method. Based on t he nonstandard analysis and formal power series theory, it presents a strai ghtforward method for computing arbitrary order nonlinearities of the nonli near dynamical system, with extremely high accuracy. The differential algeb raic theory and method for the calculation of arbitrary high order aberrati ons of electron lenses are studied. The compact and intuitionistic forms fo r the fifth order focusing aberrations from the differential algebraic coef ficients are given, considering the coupling relations between the DA coeff icients. Up to 7th order aberrations for the bell-shaped magnetic fields ar e calculated using the DA method. The patterns of all the 5th order aberrat ions and the 7th order aberrations are given. Some new types of the aberrat ions are observed.