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Phonon softening and anomalous mode near the $x_{c}=0.5$ quantum critical point in Ca$_{2-x}$Sr$_{x}$RuO$_4$

Inelastic neutron scattering is used to measure the temperature dependent phonon dispersion in Ca$_{2-x}$Sr$_{x}$RuO$_{4}$ ($x=0.4$, 0.6). The in-plane $Σ_{4}$ octahedral tilt mode softens significantly at the zone boundary of the high temperature tetragonal (HTT) \textit{I4}$_{\mathit{1}}$\textit{/acd} structure as the temperature approaches the transition to a low temperature orthorhombic (LTO) \textit{Pbca} phase. This behavior is similar to that in La$_2$CuO$_4$, but a new inelastic feature that is not found in the cuprate is present. An anomalous phonon mode is observed at energy transfers greater than the $Σ_{4}$ albeit with similar dispersion. This anomalous phonon mode never softens below $\sim 5$ meV, even for temperatures below the HTT-LTO transition. This mode is attributed to the presence of intrinsic structural disorder within the \textit{I4}$_{\mathit{1}}$\textit{/acd} tetragonal structure of the doped ruthenate.