Experimental application of sum rules for electron energy loss magnetic chiral dichroism

We present a derivation of the orbital and spin sum rules for magnetic circular dichroic spectra measured by electron energy loss spectroscopy in a transmission electron microscope. These sum rules are obtained from the differential cross section calculated for symmetric positions in the diffraction pattern. Orbital and spin magnetic moments are expressed explicitly in terms of experimental spectra and dynamical diffraction coefficients. We estimate the ratio of spin to orbital magnetic moments and discuss first experimental results for the Fe $L_{2,3}$ edge.

Figure 1

Diffraction pattern for an EMCD experiment. (a) Experimental diffraction pattern for an iron sample oriented in (110) two-beam configuration. The momentum-transfer vectors for the two symmetrical positions pos1 and pos2 are represented by arrows. (b) Fourfold diffraction pattern which has been used to express the differential cross section. The two different positions of the spectrometer aperture ($\mathbf{OS}=\delta g{\mathbf{e}}_x\pm ϵg{\mathbf{e}}_y$, with $\delta$ and $ϵ$ real numbers) which have been considered are indicated by open circles. The Bragg spots are represented by solid circles. A pair of integers $(n,m)$ is associated with each Bragg spot, as shown for four of them.

Figure 2

(a) EELS signal measured at the two symmetric positions pos1 and pos2 in the diffraction pattern of an iron sample oriented in the (110) two-beam configuration. (b) Corresponding dichroic signal.