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Atomic-size vortex beams have great potential in probing materials' magneticmoment at atomic scales. However, the limited depth of field of vortex beamsconstrains the probing depth in which the helical phase front is preserved. Onthe other hand, electron channeling in crystals can counteract beam divergenceand extend the vortex beam without disrupting its topological charge.Specifically, in this paper, we report atomic vortex beams with topologicalcharge \pm1 can be coupled to the 2p columnar bound states and propagate formore 50 nm without being dispersed and losing its helical phase front. We gavenumerical solutions to the 2p columnar orbitals and tabulated thecharacteristic size of the 2p states of two typical elements, Co and Dy, forvarious incident beam energies and various atomic densities. The tabulatednumbers allow estimates of the optimal convergence angle for maximal couplingto 2p columnar orbital. We also have developed analytic formulae for beamenergy, convergence-angle, and hologram dependent scaling for variouscharacteristic sizes. These length scales are useful for the design ofpitch-fork apertures and operations of microscopes in the vortex-beam imagingmode.

microscopy and microanalysis

On-column 2p bound state with topological charge \pm1 excited by an  atomic-size vortex beam in an aberration-corrected scanning transmission  electron microscope