Inhomogeneous level splitting in Pr$_{2-x}$Bi$_x$Ru$_2$O$_7$

We report that Bi doping drives Pr$_x$Bi$_{2-x}$Ru$_2$O$_7$ from an antiferromagnetic insulator ($x=2$) to a metallic paramagnet ($x\approx 1$) with a broad low $T$ maximum in $C/T$. Neutron scattering reveals local low energy spin excitations ($\hbar\omega\approx 1$ meV) with a spectrum that is unaffected by heating to $k_BT>>\hbar\omega$. We show that a continuous distribution of splittings of the non-Kramers Pr$^{3+}$ ground state doublet such as might result from various types of lattice strain can account for all the data.