Toward a quantitative analysis of the mirror method for characterizing insulation

The electrostatic mirror method developed by Le Gressus can be used to characterize the charge trapping properties of insulators. It is based on the measurements of the dark spot observed on a scanning electron microscope (SEM) screen working in the secondary electron mode when a beam of low energy electrons is used to probe the area of the sample where a charge has been trapped previously at high energy. The dark spot is the image of the end section of the emitter tube. The radius of the dark spot for a probing beam of energy eV is nearly proportional to the curvature radius of the equipotential surface V. For a trapped charge q_t with spherical symmetry (e.g. a point charge), the radius of curvature R(V) varies as 1/V, so that the plot of 1/R(V) is a straight line of slope λ/q_t, with λ=2πε₀(ε_r+1). However, if the charge is anisotropic, the 1/R(V) curve deviates from the straight line for probing electrons approaching the charge at high enough V. This work presents analytical derivations of the 1/R(V) function for a charged segment perpendicular to the sample surface and for a charged disc parallel to that surface, and also a good approximation for a charged cylinder. It shows that an oblate trapped charge leads to curves deviating downward from the above straight line and displaying a well-characterized maximum which usually is observed. Conversely, a prolate charge leads to an upward deviation. The work also considers the influence of the electrostatic image of the trapped charge in the substrate. Whereas this is negligible for thick samples, it becomes relevant for thin ones. The theoretical predictions are confirmed by the experimental data obtained with 100 μm thick samples, and the effect of the image charge has allowed detection of the film lamination influence on the anisotropy of the trapped charge, It is likely that this work might contribute significantly to the understanding of the trapping and detrapping processes in aging and breakdown mechanisms and, more practically, to the standardization of industrial insulators