Theory and Practice of Electron Diffraction from Single Atoms and Extended Objects using an Electron Microscope Pixel Array Detector

摘要

What does the diffraction pattern from a single atom look like? How does itdiffer from the scattering from long range potential? With the development ofnew high-dynamic range pixel array detectors to measure the complete momentumdistribution, these questions have immediate relevance for designing andunderstanding momentum-resolved imaging modes. We explore the asymptotic limitsof long range and short range potentials. We use a simple quantum mechanicalmodel to explain the general and asymptotic limits for the probabilitydistribution in both and real and reciprocal space. Features in the scatteringpotential much larger than the probe size cause the bright-field disk todeflect uniformly, while features much smaller than the probe size, instead ofa deflection cause a redistribution of intensity within the bright-field disk.Because long range and short range features are encoded differently in thediffraction pattern, it is possible to separate their contributions indifferential phase contrast (DPC) or Center-of-Mass (CoM) imaging. The shapeprofiles for atomic resolution CoM imaging are dominated by the shape of theprobe gradient and not the highly-singular atomic potentials or their localfields. Instead, only the peak height shows an atomic-number sensitivity, whoseprecise dependence is determined by the convergence angle. At lower convergenceangles, the contrast oscillates with increasing atomic number, similar tobright field imaging. The range of collection angles impacts DPC and CoMimaging differently, with CoM being more sensitive to the upper cutoff limit,while DPC is more sensitive to the lower cutoff.