Magnetic and natural optical activity of $f$–$f$ transitions in multiferroic Nd0.5Gd0.5Fe3(BO3)4

Spectra of absorption, magnetic circular dichroism, and natural circular dichroism of the $f$–$f$ transitions $^{4}I_{9/2}$ $\to$ $^{4}F_{3/2}$, $^{2}H_{9/2}$ + $^{4}F_{5/2}$, $^{4}S_{3/2}$ + $^{4}F_{7/2}$, $^{2}G_{7/2}$ + $^{4}G_{5/2}$, $^{2}K_{13/2}$ + $^{4}G_{7/2}$, and $^{4}G_{9/2}$ in the Nd$^{3+}$ ions in the Nd$_{0.5}$Gd$_{0.5}$Fe$_{3}$(BO$_{3}$)$_{4}$ crystal have been measured as a function of the temperature in the interval of 90–300 K. Temperature dependences of the magneto-optical activity (MOA) and natural optical activity (NOA) of the transitions have been obtained. It has been found that, in contrast to allowed transitions, the temperature dependence of the MOA of the $f$–$f$ transitions does not obey the Curie–Weiss law and the NOA depends on temperature. The NOA of some transitions changes the sign with variation in temperature. These phenomena have been explained by the presence of three contributions to the allowance of the $f$–$f$ transitions, which lead to three contributions of different signs to the MOA and NOA. The range of the MOA of the $f$–$f$ transitions in the Nd$^{3+}$ ion has been predicted theoretically and confirmed experimentally.