Orbital angular momentum in electron diffraction and its use to determine chiral crystal symmetries

Abstract

In this work we present an alternative way to look at electron diffraction in a transmission electron microscope. Instead of writing the scattering amplitude in Fourier space as a set of plane waves, we use the cylindrical Fourier transform to describe the scattering amplitude in a basis of orbital angular momentum (OAM) eigenstates. We show how working in this framework can be very convenient when investigating, e.g., rotation and screw-axis symmetries. For the latter we find selection rules on the OAM coefficients that unambiguously reveal the handedness of the screw-axis. Detecting the OAM coefficients of the scattering amplitude thus offers the possibility to detect the handedness of crystals without the need for dynamical simulations, the thickness of the sample nor the exact crystal structure. We propose an experimental setup to measure the OAM components where an image of the crystal is taken after inserting a spiral phase plate in the diffraction plane and perform multislice simulations on α-quartz to demonstrate how the method indeed reveals the chirality. The experimental feasibility of the technique is discussed together with its main advantages with respect to chirality determination of screw axes. The method shows how the use of a spiral phase plate can be extended from a simple phase imaging technique to a tool to measure the local OAM decomposition of an electron wave, widening the field of interest well beyond chiral space group determination.

Quantum mechanical formulation of the Busch theorem

“…The generation of vortex beams as twisted photons [1], vortex neutrons [2], or vortex electrons has inspired versatile theoretical studies and interesting experiments or proposals to unveil the basic properties of such beams and of effects of quantum interference and coherence in particle collisions, inaccessible with ordinary beams [3][4][5][6][7][8][9]. Quantized vortex electrons-i.e., electron beams carrying a quantized orbital angular momentum (OAM)-generated in electron microscopes [10][11][12] can be applied as probes for the study of chiral [13] or magnetic structures [14] and enable magnetic mapping with atomic resolution [15].…”

Ein Whirlpool aus Elektronen

“…Man kann damit aber nicht rechts‐ von linksdrehenden Strukturen unterscheiden, da die Beugungsbilder identisch sind. Die helikale Phase der Sonde bricht diese Symmetrie; dadurch kann die Chiralität eines Kristalls unmittelbar aus dem Beugungsbild bestimmt werden .…”

Symmetry in the diffraction of beams carrying orbital angular momentum

“…Of course, the aperture, in general has a lower symmetry than the diffraction field. In inverse problems, the relation between the symmetry of the diffracted field and structure of the scatterer is of interest in all branches of diffraction physics [4][5][6][7]. Plane wave diffraction by regular polygonal apertures has been treated in a number of classic papers, which among other things, show that the diffraction pattern has the same rotational and mirror symmetries as the aperture transmission function [8][9][10][11].…”