Lattice Distortion and Resonant X-Ray Scattering in DyB2C2

We study the resonant x-ray scattering (RXS) spectra at the Dy $L_{\rm III}$ absorption edge in the quadrupole ordering phase of DyB$_2$C$_2$. Analyzing the buckling of sheets of B and C atoms, we construct an effective model that the crystal field is acting on the $5d$ and $4f$ states with the principal axes different for different sublattices. Treating the $5d$ states as a band and the $4f$ states as localized states, we calculate the spectra within the dipole transition. We take account of processes that (1) the lattice distortion directly modulates the $5d$ states and (2) the charge anisotropy of the quadrupole ordering $4f$ states modulates the $5d$ states through the $5d$-$4f$ Coulomb interaction. Both processes give rise to the RXS intensities on $(00\frac{\ell}{2})$ and $(h0\frac{\ell}{2})$ spots. Both give similar photon-energy dependences and the same azimuthal-angle dependences for the main peak, in agreement with the experiment. The first process is found to give the intensities much larger than the second one in a wide parameter range of crystal field. This suggests that the main-peak of the RXS spectra is not a direct reflection of the quadrupole order but mainly controlled by the lattice distortion.