Single Electron Self-coherence and Its Wave/Particle Duality in the Electron Microscope

Intensities in high-resolution phase-contrast images from electron microscopes build up discretely in time by detecting single electrons. A wave description of pulse-like coherent-inelastic interaction of an electron with matter implies a time-dependent coexistence of coherent partial waves. Their superposition forms a wave package by phase decoherence of 0.5 - 1 radian with Heisenbergs energy uncertainty ΔEH = ħ/2 Δt-1 matching the energy loss ΔE of a coherent-inelastic interaction and sets the interaction time Δt. In these circumstances, the product of Planck's constant and the speed of light hc is given by the product of the expression for temporal coherence λ2/Δλ and the energy loss ΔE. Experimentally, the self-coherence length was measured by detecting the energy-dependent localization of scattered, plane matter waves in surface proximity exploiting the Goos-Hänchen shift. Chromatic-aberration Cc-corrected electron microscopy on boron nitride (BN) proves that the coherent crystal illumination and phase contrast are lost if the self-coherence length shrinks below the size of the crystal unit cell at ΔE > 200 eV. In perspective, the interaction time of any matter wave compares with the lifetime of a virtual particle of any elemental interaction, suggesting the present concept of coherent-inelastic interactions of matter waves might be generalizable.