Spectroscopy of f-f transitions, crystal-field calculations, and magnetic and quadrupole helix chirality in DyFe$_3$(BO$_3$)$_4$

Recently, quadrupole helix chirality and its domain structure was observed in resonant x-ray diffraction experiments on DyFe3(BO3)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 Dy3+ ions induced by the low-symmetry (C2) crystal-field (CF) component. In this work, the CF parameters for the Dy3+ ions in the P3121 (P3221) phase of DyFe3(BO3)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 Dy3+ ions in the paramagnetic P3121 (P3221) phase and suggest a neutron diffraction experiment to reveal it.