Elemental Dilution Effect on the Elastic Response due to a Quadrupolar Kondo Effect of the Non-Kramers Systems Y$_{1-x}$Pr$_x$Ir$_2$Zn$_{20}$

We measured the elastic constants $\left(C_{11}-C_{12}\right)/2$ and $C_{44}$ of the non-Kramers system $\rm{Y_{0.63}Pr_{0.37}Ir_{2}Zn_{20}}$ (Pr-37% system) by means of ultrasound to check how the single-site quadrupolar Kondo effect is modified by increasing the Pr concentration. The Curie-like softening of $\left(C_{11}-C_{12}\right)/2$ between 1 and 5 K of the present Pr-37% system can be reproduced by a multipolar susceptibility calculation based on the non-Kramers $\Gamma_3$ doublet crystalline-electric-field ground state. Further, on cooling below 0.15 K, a temperature dependence proportional to $\sqrt{T}$ was observed in $\left(C_{11}-C_{12}\right)/2$. This behavior rather corresponds to the theoretical prediction of the quadrupolar Kondo "lattice" model, unlike that of the Pr-3.4% system, which shows a logarithmic temperature dependence based on the "single-site" quadrupolar Kondo theory. In addition, we discuss the possibility to form a vibronic state by the coupling between the low-energy phonons and the electric quadrupoles of the non-Kramers doublet in the Pr-37% system, since we found a low-energy ultrasonic dispersion in the temperature region between 0.15 and 1 K.