Phase shift of charged metallic nanoparticles

We calculate the electrostatic potential generated by a charged conducting sphere located at some distance over a conducting substrate by means of two complementary approaches. The potential is then projected along the electron beam direction yielding the electron-optical phase shift. The scenario is compared with a uniformly charged sphere over the same substrate, a model that has been widely employed to interpret phase images of charged particles. We illustrate the implications of our findings in the context of transmission electron microscopy experiments performed on metallic nanoparticles, where this classical analysis can be considered as a useful and insightful starting point towards more accurate, yet more complicate, quantum mechanical approaches.