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Recently, quadrupole helix chirality and its domain structure was observed inresonant x-ray diffraction experiments on DyFe$_3$(BO$_3$)$_4$ using circularlypolarized x rays [T. Usui et al., Nature Materials 13, 611 (2014)]. We showthat this effect can be explained quantitatively by calculating the quadrupolemoments 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 CFcalculations based on the analysis of high-resolution temperature-dependentoptical spectroscopy data. We also consider the helix chirality of thesingle-site magnetic susceptibility tensors of the Dy$^{3+}$ ions in theparamagnetic $P3_121$ ($P3_221$) phase and suggest a neutron diffractionexperiment to reveal it.

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