A theory of the electric quadrupole contribution to resonant x-ray scattering: Application to multipole ordering phases in Ce_{1-x}La_{x}B_{6}

TL;DR

This paper develops a theoretical framework for analyzing electric quadrupole contributions to resonant x-ray scattering, applying it to Ce_{1-x}La_{x}B_{6} to interpret experimental spectra and symmetry behaviors in multipole ordered phases.

Contribution

The authors derive a comprehensive expression for RXS amplitude considering energy dependence, enabling detailed analysis of multipole order parameters in localized f electron systems.

Findings

The derived formula accurately reproduces experimental RXS spectra at Ce L_{2,3} edges.

Azimuthal angle dependence reveals sixfold symmetry in AFQ phase and threefold in AFO phase.

Hexadecapole order contributes to the sixfold symmetry observed in the AFQ phase.

Abstract

We study the electric quadrupole (E2) contribution to resonant x-ray scattering (RXS). Under the assumption that the rotational invariance is preserved in the Hamiltonian describing the intermediate state of scattering, we derive a useful expression for the RXS amplitude. One of the advantages the derived expression possesses is the full information of the energy dependence, lacking in all the previous studies using the fast collision approximation. The expression is also helpful to classify the spectra into multipole order parameters which are brought about. The expression is suitable to investigate the RXS spectra in the localized f electron systems. We demonstrate the usefulness of the formula by calculating the RXS spectra at the Ce L_{2,3} edges in Ce_{1-x}La_{x}B_{6} on the basis of the formula. We obtain the spectra as a function of energy in agreement with the experiment of Ce_{0.7}La_{0.3}B_{6}. Analyzing the azimuthal angle dependence, we find the sixfold symmetry in the \sigma-\sigma' channel and the threefold onein the \sigma-\pi' channel not only in the antiferrooctupole (AFO) ordering phase but also in the antiferroquadrupole (AFQ) ordering phase, which behavior depends strongly on the domain distribution. The sixfold symmetry in the AFQ phase arises from the simultaneously induced hexadecapole order. Although the AFO order is plausible for phase IV in Ce_{1-x}La_{x}B_{6}, the possibility of the AFQ order may not be ruled out on the basis of azimuthal angle dependence alone.