Spectroscopy of f-f transitions, crystal-field calculations, and magnetic and quadrupole helix chirality in DyFe₃(BO₃)₄

Recently, quadrupole helix chirality and its domain structure was observed in resonant x-ray diffraction experiments on DyFe$_3$(BO$_3$)$_4$ using circularly polarized x rays [T. Usui et al., Nature Materials 13, 611 (2014)]. We show that this effect can be explained quantitatively by calculating the quadrupole moments of the Dy$^{3+}$ ions induced by the low-symmetry ($C_2$) crystal-field (CF) component. In this work, the CF parameters for the Dy$^{3+}$ ions in the $P3_121$ ($P3_221$) phase of DyFe$_3$(BO$_3$)$_4$ are obtained from CF calculations based on the analysis of high-resolution temperature-dependent optical spectroscopy data. We also consider the helix chirality of the single-site magnetic susceptibility tensors of the Dy$^{3+}$ ions in the paramagnetic $P3_121$ ($P3_221$) phase and suggest a neutron diffraction experiment to reveal it.