Figure (a) shows the calculated harmonic phonon dispersion of α-Li 3 N. We find an imaginary mode at the Γ-point of the vibrational Brillouin zone that corresponds to out-of-plane vibrations of Li ions in the Li 2 N plane. The associated phonon eigenvector and PES for this mode are shown in Figures and The PES shows a double-well potential that is characteristic of displacive phase transitions . The saddle point corresponds to the P 6/mmm space group of the initial crystal structure, while the two local minima have lower P 3m1 symmetry. If a displacive phase transition were to happen around T s , this could be the mechanism behind the superionic transition. Out-of-plane motion of lithium ions has preivously been identified as an essential part of the diffusion mechanism in α-Li 3 N . Our data, however, show the energy barrier between the two wells is only 1.20 meV, which corresponds to a thermal energy of 14 K. At temperatures above 14 K, therefore, atomic vibrations will behave as for a single harmonic (quadratic) potential . Even below 14 K, the P 3m1 phase may not be stable due to zero-point motion and quantum tunnelling; such behaviour has been observed in the quantum paraelectric SrTiO 3 . The effects of the double-well PES are thereofore expected to be largely negligible within the stability range of α-Li 3 N (> 300 K cf. Figure ), and to have litte effect at the superionic transition temperature T s ≈ 678 K. We conclude from this analysis that the displacive phase transition predicted for α-Li 3 N does not contribute towards the mechanism of the superionic transition observed in Li 3 N. 1 1 An imaginary mode was also observed in the phonon dispersion of β-Li3N at the K-point and is shown in Figure (a) (SI). In this case, however, the energy barrier between the two wells of the PES shown in Figure (b) (SI) is 40.95 meV. This corresponds to a thermal energy of 475.2 K, which is beyond the stability range of the β-phase according to the calculated phase diagram and neutron powder diffraction