The quantitative measurement of magnetic moments from phase images of nanoparticles and nanostructures—I. Fundamentals

An approach that can be used to measure the magnetic moment of a magnetized nanoparticle or nanostructure from an electron-optical phase image is introduced. The measurement scheme is based on integration of the gradient of the measured phase image within a circular boundary that contains the structure of interest. The quantity obtained is found to be directly proportional to the magnetic moment of the particle, with a constant of proportionality that does not depend on the particle's shape or magnetization state. The measurement of magnetic moments from both simulated and experimental phase images is demonstrated, and strategies are presented that can be utilized to overcome sources of error associated with, for example, the presence of neighboring magnetic particles and the perturbation of the holographic reference wave.