Trichroism in energy-loss near-edge structure spectroscopy: Polarization dependence of near-edge fine structures

The goal of this paper is to relate the current of inelastically scattered electrons collected in a transmission electron microscope (TEM) to the double differential electron energy-loss cross section. Up to now, this relationship, which depends on the point symmetry around the probed atom site, has been essentially studied in a situation called dichroism. This situation can be encountered when a principal threefold, fourfold, or sixfold rotation axis through the probed atom site exists. The electron energy-loss cross section is then a linear combination of longitudinal and transversal cross sections, and the weights of these components are ${\cos }^2{\theta }_{\mathbf{q}}$ and ${\sin }^2{\theta }_{\mathbf{q}}$, where ${\theta }_{\mathbf{q}}$ is the angle between the scattering wave vector $\mathbf{q}$ and the principal rotation axis. The first aim of this paper is to find the dependence on $\mathbf{q}$ of the cross section in all other cases, that is to say, when the symmetry around the probed atom site is described with one of the eight low symmetry point groups $C_1$, $S_2$, $C_{1h}$, $C_2$, $C_{2h}$, $C_{2v}$, $D_2$, and $D_{2h}$. In these eight cases of low symmetry, three distinct situations called trichroism can be distinguished. In these situations, the cross section is expressed in terms of the cross sections obtained for six, four, or three particular orientations of the scattering wave vector. The second aim of this paper is to provide an expression of the inelastically scattered electron current collected in a TEM for these three situations of trichroism. This current is expressed in terms of experimental parameters, such as the incident beam convergence, the collector acceptance, the electron beam kinetic energy, and the sample orientation. As in the case of dichroism, magic conditions can be found, for which the collected current becomes independent of the single-crystal sample orientation. The case of the C $K$ edge in the nonstoichiometric ${\mathrm{V}}_6{\mathrm{C}}_5$ metallic carbide with a trigonal symmetry is given as an illustration.

Figure 1

Geometry of electron scattering. Coordinates $(x,y,z)$ and $(X,Y,Z)$ are linked to the sample to be studied and the incident electron probe, respectively. The $z$ axis coincides with the main rotation axis through the probed atom site, and the $Z$ axis is oriented along the incident beam. The angle between both axes ${\chi }_0$ and the azimuthal angle ${\delta }_0$ define the relative orientation of the sample with respect to the probe.

Figure 2

Curves $R_c=0$ for different values of the incident electron kinetic energy in keV: (a) $E_c\approx 0$, (b) $E_c=50$, (c) $E_c=100$, (d) $E_c=200$, (e) $E_c=300$, and (f) $E_c=400$.

Figure 3

(Color online) Magic collection angle (in ${\theta }_E$ unit) versus incident electron kinetic energy (in keV), in the case of parallel illumination $({\alpha }_m=0)$.

Figure 4

In the case of the stoichiometric phase VC, any carbon has 6 first nearest neighbor V atoms (large circles) and $12\text{second}$ nearest neighbor carbons (small circles). In the case of the nonstoichiometric ${\mathrm{V}}_6{\mathrm{C}}_5$ phase shown here, a carbon has either nine (dichroism) or ten (trichroism) second nearest neighbor carbons. In this figure, vacancies are represented by small gray circles.

Figure 5

Intrinsic components versus electron energy loss, obtained with ${\widehat{\mathbf{q}}}^{\star }$ parallel to (a) ${\mathbf{e}}_z$, (b) ${\mathbf{e}}_y$, (c) ${\mathbf{e}}_x$, (d) ${\mathbf{e}}_{xy}$, (e) ${\mathbf{e}}_{zx}$, and (f) ${\mathbf{e}}_{yz}$. The fine structures near the C $K$ edge of VC is represented by the thin line curve (g).

Figure 6

Plot of the ${\mathbf{e}}_{zx}$ intrinsic component (thick full line) and the average of ${\mathbf{e}}_x$ and ${\mathbf{e}}_z$ intrinsic components (thin full line) versus energy.

Figure 7

This figure shows the EELcs for ${\widehat{\mathbf{q}}}^{\star }={\mathbf{e}}_z$ (a). The collected current is calculated in the case of an incident beam parallel to the principal rotation axis at different conditions: ${\alpha }_m=0.3\mathrm{mrad}$ and (b) ${\beta }_m=0.3\mathrm{mrad}$; (c) ${\beta }_m=1.305\mathrm{mrad}$ (magic conditions) and (d) ${\beta }_m=5\mathrm{mrad}$.