Elastic Response in the Dilute non-Kramers System Y$_{1-x}$Pr$_x$Ir$_2$Zn$_{20}$

Ultrasonic investigations of the single-site quadrupolar Kondo effect in diluted Pr system Y$_{0.966}$Pr$_{0.034}$Ir$_2$Zn$_{20}$ are reported. The elastic constant $(C_{11}-C_{12})/2$ is measured down to ~40 mK using ultrasound for the dilute system Y$_{0.966}$Pr$_{0.034}$Ir$_2$Zn$_{20}$ and the pure compound YIr$_2$Zn$_{20}$. We found that the elastic constant $(C_{11}-C_{12})/2$ of the Pr-dilute system exhibits a logarithmic temperature dependence below $T_0$ ~0.3 K, where non-Fermi-liquid (NFL) behavior in the specific heat and electrical resistivity is observed. This logarithmic temperature variation manifested in the $\Gamma_3$-symmetry quadrupolar susceptibility is consistent with the theoretical prediction of the quadrupolar Kondo effect by D. L. Cox. On the other hand, the pure compound YIr$_2$Zn$_{20}$ without $4f$-electron contributions shows nearly no change in its elastic constants evidencing negligible phonon contributions. In addition, clear acoustic de Haas-van Alphen (dHvA) oscillations in the elastic constant were detected for both compounds on applying magnetic field. This is mainly interpreted as contribution from the Fermi surface of YIr$_2$Zn$_{20}$.