where is the frequency of the phonon mode and corresponds to the activation barrier of this particular jump (for a mode-specific description see Supplemental Information 6). The ceiling operator is used because phonons are quantized. Intuitively, larger barriers require larger phonon fluctuations. For a Ag + vibrational mode where ℎ ≈ 5 meV, then Y ≈ 10 to 100 phonons using macroscopic values of for in Eq. 3.