Using electron vortex beams to determine chirality of crystals in transmission electron microscopy

We investigate electron vortex beams elastically scattered on chiral crystals. After deriving a general expression for the scattering amplitude of a vortex electron, we study its diffraction on point scatterers arranged on a helix. We derive a relation between the handedness of the helix and the topological charge of the electron vortex on one hand and the symmetry of the higher-order Laue zones in the diffraction pattern on the other for kinematically and dynamically scattered electrons. We then extend this to atoms arranged on a helix as found in crystals which belong to chiral space groups and propose a method to determine the handedness of such crystals by looking at the symmetry of the diffraction pattern. In contrast to alternative methods, our technique does not require multiple scattering, which makes it possible to also investigate extremely thin samples in which multiple scattering is suppressed. In order to verify the model, elastic scattering simulations are performed, and an experimental demonstration on ${\mathrm{Mn}}_2{\mathrm{Sb}}_2{\mathrm{O}}_7$ is given in which we find the sample to belong to the right-handed variant of its enantiomorphic pair. This demonstrates the usefulness of electron vortex beams to reveal the chirality of crystals in a transmission electron microscope and provides the required theoretical basis for further developments in this field.

Figure 1

Schematic diagram of the scattering paths of a focused vortex beam to scatter to points ${\mathit{k}}_1$ and ${\mathit{k}}_2$ in the FOLZ opposite each other. The black path represents the first-order Born approximation in which the electron scatters only once. The red path is a second-order Born approximation in which the electron first scatters to a point in the ZOLZ and then to the FOLZ. The green path represents an event in which the electron scatters to the FOLZ and then scatters within this zone through a scattering vector of the ZOLZ. When neglecting multiple scattering between different Laue zones, the symmetry of the diffraction pattern is also conserved in the dynamical approximation.

Figure 2

Schematic representation of the experimental setup. A vortex beam is focused directly on a screw axis in the sample further simplified assuming elastic scattering is dominated by the helically arranged heavy atoms. The radius of the vortex is chosen such that it matches the distance of the atoms to the screw axis.

Figure 3

(top) Multislice simulation of the first-order Laue zone of the diffraction pattern of (left) an $m=-1$ vortex and (right) an $m=+1$ vortex scattered on right-handed ${\mathrm{Mn}}_2{\mathrm{Sb}}_2{\mathrm{O}}_7$, with SG $P{3}_{1}21$. The vortex is centered along the screw axis with the size of the vortex equal to the distance of the Sb atoms from the screw axis, approximately 1.2 Å. The energy of the probe is 300 keV, the convergence angle is 8 mrad, and the spherical aberration is $1 \mu \mathrm{m}$. The thickness of the sample is 20 nm. The centrosymmetry of the first-order ring can be seen for the right-handed enantiomorph for $m=+1$. (bottom) Experimental CBED pattern of a 1.2-Å-sized vortex probe pointed at a threefold screw axis in ${\mathrm{Mn}}_2{\mathrm{Sb}}_2{\mathrm{O}}_7$ for $m=-1$ (left) and $m=+1$ (right). The contrast was adapted to show only the peaks in the FOLZ.

Figure 4

Comparison of the (a) and (b) simulated and (c) and (d) experimental $I_{FOLZ}\left(\theta \right)$ for $\theta \in \{0,\pi \}$ (blue) and $\theta \in \{\pi ,2\pi \}$ (red) for (a) and (c) $m=-1$ and (b) and (d) $m=+1$ on the $P{3}_{1}21$ enantiomorph of ${\mathrm{Mn}}_2{\mathrm{Sb}}_2{\mathrm{O}}_7$. From the position of the peaks it is clear that the $m=+1$ vortex scattered on the right-handed screw axis gives a centrosymmetric FOLZ, which is not the case for the $m=-1$ vortex, where the differences are indicated by arrows.

Figure 5

Multislice simulation of the first-order Laue zone of the diffraction pattern of (left) an $m=-1$ vortex and (right) an $m=+1$ vortex scattered on right-handed ${\mathrm{Mn}}_2{\mathrm{Sb}}_2{\mathrm{O}}_7$, with thicknesses of (top) 40 nm and (bottom) 60 nm. Although dynamical scattering is expected to be dominant, the centrosymmetry of the FOLZ still can be seen for $m=+1$, as was the case for the 20-nm-thick sample.

Figure 6

Comparison of the simulated $I_{FOLZ}\left(\theta \right)$ for $\theta \in \{0,\pi \}$ (blue) and $\theta \in \{\pi ,2\pi \}$ (red) for (a) and (c) $m=-1$ and (b) and (d) $m=+1$ on the $P{3}_{1}21$ enantiomorph of ${\mathrm{Mn}}_2{\mathrm{Sb}}_2{\mathrm{O}}_7$  for sample thicknesses of (a) and (b) 40 nm and (c) and (d) 60 nm. In (b) and (d) it can be seen that the positioning of the peaks is still centrosymmetric for samples up to 60 nm, although the shape of the peaks gets altered. This symmetry is not at all present in the $m=-1$ vortex case, for which the main differences are indicated by arrows.

Figure 7

Simulated diffraction pattern of a chiral set of point scatterers with fourfold symmetry for a plane wave and a vortex probe. When the right-handed enantiomorph is mirrored along the beam direction to get the left-handed enantiomorph, the diffraction of the plane wave stays identical, while that of the vortex is different. This shows that vortex beam scattering in the kinematical approximation will, in contrast to plane-wave scattering, depend on the chirality of a potential.