Sum Rules in X-Ray Resonant Raman Scattering: Recovering the Co Ground State Information in ${\mathrm{C}\mathrm{o}\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_4$ as a Test Case

We present an approach to x-ray resonant Raman scattering suitable for sum-rule analysis. In magnetic systems it gives ground state information as represented by an expansion in coupled multipoles of orbital and spin moments. The experiment is based on the angular dependence of the integrated peak intensities. In $3d$ transition elements one obtains information up to order $4$ from the channel $2p^{6}3d^n\to 2p^{5}3d^{n+1}\to 2p^{6}3s^{1}3d^{n+1}$. By combining these results with $L_{2,3}$ absorption dichroism, we show quantitatively the atomiclike properties of cobalt in ${\mathrm{C}\mathrm{o}\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_4$ up to high-order multipoles.

Figure 1

Schematics of IRRS in perpendicular geometry ($\delta =0$). P is helicity vector of incident circularly polarized photon beam, $\mathbf{\epsilon }$ the direction of detected photons, and M magnetization. The IRRS detector moves along the cone with azimuthal angle $\beta$ and semiaperture $\alpha$.

Figure 2

Measured IRRS spectra in Co ferrite at $\beta =\delta =0$ (lower panel): thick and thin lines correspond to opposite circular polarizations, the experimental data points are given in the difference spectrum. Angular dependence of IRRS signals: $\delta$ scan of $J_{\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{r}}$ (inset of lower panel), $\beta$ scan of normalized $J_{\mathrm{s}\mathrm{u}\mathrm{m}}$ (central panel), and $\beta$ scan of $J_{\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{r}}$ (upper panel).