Phonons reflect dynamic spin-state order in LaCoO$_3$

We investigate lattice dynamics in LaCoO₃ using inelastic neutron and x-ray scattering over T = 2–650 K, spanning the spin-state crossover at T₁ ≈ 100 K and the insulator–metal transition at T₂ ≈ 550 K. Comparison with quasi-harmonic ab-initio lattice-dynamical calculations helps reveal anomalous softening of a ≈10-meV oxygen phonon, confined to the temperature interval T₁ ≤ T ≤ T₂ and localized in momentum space at qSSO = (½,½,½)c. This wave vector corresponds to the spin-state ordering originally proposed by Goodenough [J. Phys. Chem. Solids 6, 287-297 (1958)]. Our results therefore provide momentum-resolved evidence for dynamic correlations of high-spin and low-spin Co³⁺ states in LaCoO₃, linking spin-state fluctuations to anomalous phonon renormalization.