Ultrasonic investigation of quadrupole ordering in HoB 2 C 2

The elastic properties of ${\mathrm{HoB}}_{2}{\mathrm{C}}_{2}$ undergoing an antiferroquadrupole ordering with a tetragonal structure have been investigated by means of ultrasonic measurements. The transverse ${(C}_{11}\ensuremath{-}{C}_{12})/2,$ ${C}_{44},$ and ${C}_{66}$ modes exhibit characteristic softenings with decreasing temperature in a paramagnetic phase I down to an ordered phase IV at ${T}_{C1}=5.9\mathrm{K}.$ The softening of ${(C}_{11}\ensuremath{-}{C}_{12})/2,$ ${C}_{44}$ and ${C}_{66}$ is described by a formula ${C}_{\ensuremath{\Gamma}}{=C}_{\ensuremath{\Gamma}}^{0}(T\ensuremath{-}{T}_{C}^{0})/(T\ensuremath{-}\ensuremath{\Theta})$ due to the quadrupole-strain interaction for the $4f$ state of ${\mathrm{HoB}}_{2}{\mathrm{C}}_{2}$ consisting of a ground E doublet with orbital degeneracy and an excited singlet with a small-energy splitting $\ensuremath{\Delta}.$ The significant softening of the elastic constant and the ultrasonic attenuation in phase IV between ${T}_{C1}=5.9\mathrm{K}$ and ${T}_{C2}=5.0\mathrm{K}$ have been found. The ultrasonic dispersion in phase IV indicates the slowing down of the quadrupole fluctuation time, $\ensuremath{\tau}=7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}\mathrm{sec}.$

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