On the possibility to detect multipolar order in URu${}_2$Si${}_2$ by the electric quadrupolar transition of resonant elastic X-ray scattering

Resonant elastic X-ray scattering is a powerful technique for measuring multipolar order parameters. In this paper, we theoretically and experimentally study the possibility of using this technique to detect the proposed multipolar order parameters in URu${}_2$Si${}_2$ at the U-$L_3$ edge with the electric quadrupolar transition. Based on an atomic model, we calculate the azimuthal dependence of the quadrupolar transition at the U-$L_3$ edge. The results illustrate the potential of this technique for distinguishing different multipolar order parameters. We then perform experiments on ultra-clean single crystals of URu${}_2$Si${}_2$ at the U-$L_3$ edge to search for the predicted signal, but do not detect any indications of multipolar moments within the experimental uncertainty. We theoretically estimate the orders of magnitude of the cross-section and the expected count rate of the quadrupolar transition and compare them to the dipolar transitions at the U-$M_4$ and U-$L_3$ edges, clarifying the difficulty in detecting higher order multipolar order parameters in URu${}_2$Si${}_2$ in the current experimental setup.