Resonant x-ray scattering from chiral materials, $\alpha$-quartz and $\alpha$-berlinite

We study the resonant x-ray scattering at Si and Al K-edges from chiral materials, $\alpha$-quartz and $\alpha$-berlinite. We derive the general form of the scattering matrix for the dipole transition by summing up the local scattering matrices which satisfy the symmetry requirement. The oscillation term is obtained in the spectral intensity as a function of azimuthal angle with an expression of possible phase shift. We evaluate the parameters undetermined by the symmetry argument alone on the basis of underlying electronic structures given by the bond-orbital model. The spectra are calculated on forbidden spots $(001)$, $(00\bar{1})$, $(002)$, and $(00\bar{2})$ in circular polarizations without adjustable parameter, reproducing well the experimental curves depending on polarization, chirality, and scattering vector. Some discrepancies remain in the phase shift in $\alpha$-quartz.