The recent increased interest in Lorentz Microscopy methods is due to two important factors: (1) the advent of digital detection systems (CCD cameras) and advanced computer controlled transmission electron microscopes, and (2) the continually decreasing length scale of magnetic recording media and related magnetic materials. Along with the increased experimental resolution and detection sensitivity one should ask the question: how small a magnetic moment can one detect with conventional Lorentz or electron holography techniques? And, perhaps more importantly, which technique should one use to obtain the best spatial resolution? To address these questions we identified a simple model system :the uniformly magnetized sphere.
The phase of the electron wave after passing through a region with a non-zero magnetic vector potential and an internal electrostatic potential Vscan be computed from the Aharonov-Bohm trajectory integral. For a spherical particle with a uniform magnetization and a constant mean inner potential (see geometry in Fig. la), this phase shift can be analytically computed.