Field of inserted charges during Scanning Electron Microscopy of non-conducting samples

Three different approaches to calculating the electric potential in an inhomogeneous dielectric next to vacuum due to a charge distribution built up by the electron beam are investigated. An analytical solution for the electric potential cannot be found by means of the image charge method or Fourier analysis, both of which do work for a homogenous dielectric with a planar interface to vacuum. A Born approximation gives a good approach to the real electric potential in a homogenous dielectric up to a relative dielectric constant of 5. With this knowledge the electric potential of an inhomogenous dielectric is calculated. Also the electric field is calculated by means of a particle-mesh method. Some inhomogeneous dielectric configurations are calculated and their bound charges are studied. Such a method can yield accurate calculations of the electric potential and can give quantitative insight in the charging process. A capacitor model is described to estimate the potential due to the charge build up. It describes the potential build up in the first microseconds of the charging. Thereafter, it seems that more processes have to be taken into account to describe the potential well. This potential can further be used in a macroscopic approach to the collective motion of the electrons described by the Boltzmann transport equations or a derived density model, which can be a feasible alternative approximation to the more commonly used Monte-Carlo simulation of individual trajectories.