Delocalization in inelastic scattering

How delocalized is an EELS signal? For instance, how far from a silicon atom must the electron probe be before a Si edge can be detected? A crude estimate of a localization length (λθE) suggests a maximum impact parameter of 50 Å for a typical plasmon loss. Yet with care, subnanometer resolution plasmon maps can be achieved. The improved resolution cannot be accounted for by dielectric screening as the dielectric screening length diverges for energy losses at and above the plasmon frequency. Niels Bohr [1] offered a classical explanation in 1913 leading to his adiabatic criterion for a cutoff impact parameter bmax = vω, for a fast electron, velocity v, and an energy loss of frequency ω. By starting with a quantum mechanical expression for the energy loss, both the semiclassical limit and surprisingly Bohrʹs criterion can be recovered and are found to be in excellent quantitative agreement with experiment. We show bmax can be thought of as a “dynamic” screening length and plays the same role as the screening length does in the elastic scattering from a Thomas-Fermi atom. In fact the parallel component of the inelastic scattering has the same form as elastic scattering from a Thomas-Fermi atom and much of our understanding of the role of the detector in annular dark-field imaging can be applied here. However, in inelastic scattering a dramatic improvement in resolution can be obtained with an off-axis detector as suggested by Ritchie and Howie [2]. Here we present experimental evidence of such an effect.