) ensuring that summing over all Brillouin zone points yields real-valued displacements. These atomic displacements vary in time (due to the oscillatory nature of phonon vibrations) and with the lattice position l. The oscillations of a specific atom in all unit cells are identical in amplitude and direction, although oscillations in different unit cells may be out of phase with each other at any given time, due to the phase factor e iq•R 0 lk . A quantity that encodes only the relative amplitudes and directions of atomic displacements is therefore used instead in the following analysis. The displacements of a single atom range from -|s qνlκα | to +|s qνlκα | (for any l). This observation motivates the definition of the following quantity that does not depend on l: The extent to which phonon eigenvectors are aligned with ion migration pathways can be analyzed by constructing an atomic displacement field associated with vacancy migration in each material, following ideas presented in Gordiz et al., which were developed in earlier publications. An atomic displacement field is comprised of a displacement vector for each of the atoms in a structure. Its components are here denoted as d κα , following the subscript notation in eq 3. The displacement field used in Gordiz et al., which is referred to here as the "MEP displacement field," was constructed as the tangent vector to the minimum energy path, calculated with DFT using the nudged elastic band method. Such calculations are also used to directly calculate ion migration barriers. Since the aim of this work is to avoid the high computational expense associated with these calculations, an alternative displacement field intended to be more suitable for high-throughput screening applications was constructed instead. Because the vacancy migration mechanisms for all materials investigated here involve only a single cation, the components of the field associated with non-migrating atoms were approximated to be 0. (In reality, these components are nonzero in the MEP displacement field, as they represent the displacements of other atoms that "move aside" to accommodate the migrating cation.) Consequently, any analysis of phonon mode alignments with this displacement field considers only contributions from migrating cations. In the present approximation, the 3 displacement components associated with the migrating cation were set to those of a normalized vector connecting the migrating ion to a neighboring cation of the same type. This approximate field is referred to here as the "neighboring cation displacement field." A comparison of these two displacement fields (Figure ) suggests that the neighboring cation displacement field approximates the MEP displacement field well for the materials considered here. For MgSc 2 S 4 and Mg 3 NAs, the field components associated with the migrating cation contribute 67% and 59%, respectively, to the total magnitude of the field (i.e. the sum of squares of its components). The displacement vector for the migrating cation in the neighboring cation displacement field for MgSc 2 S 4 is parallel to the corresponding vector in the MEP displacement field. For Mg 3 NAs, these two vectors differ in alignment by 21 • . A third field, termed the "channel-opening displacement field," approximates just the "moving aside" of other atoms to accommodate the migrating cation. Precise definitions of these three displacement fields are provided in the Supporting Information.