Fig. 3. Lattice dynamics and atomic tunneling splitting.

a Phonon dispersion measured using inelastic X-ray scattering for longitudinal (circles) and transverse (triangles) polarization, along with first-principles simulations (peach lines). The low-energy transverse acoustic phonon (TA₁) is polarized along the c axis and is responsible for large-amplitude thermal displacements along this direction. b High-energy resolution inelastic neutron scattering spectra measured on BaTiS₃ powder at 2.4 K. An excitation is observed at 0.46 meV. The top right panel shows the spectrum integrated over momentum, Q. The bottom left panel shows the tunneling peak intensity versus Q, and the bottom right panel shows the momentum-integrated spectrum at different temperatures. c Tunneling splitting energy, ΔE, versus oscillator energy (ℏω) from an analytical model for three double-well separation distances. The inset shows the double-well parameters and the symmetric and antisymmetric wave functions. The well separation predicted from the tunneling splitting (intersection of the red curve and pink horizontal bar) agrees with that inferred from the TA₁ phonon energy (intersection of the red curve and blue vertical bar).