Vibrational frequencies. The vibrational spectrum of a material has significant importance for the fields of ionic conduction and thermal conduction. For ionic transport, the attempt frequency of the mobile species is included in the Arrhenius pre-factor (Supplemental Note 4). Often the Debye frequency D is used to approximate the attempt frequency despite not probing the vibrations of the mobile species directly. In fact, especially in disordered and soft materials, the Debye-frequency often fails its original purpose to estimate the maximum vibrationalfrequency, but rather falls into the center of the vibrational density of states. With that, it provides a useful experimental insight to the average dynamics of the lattice, but it should not be expected to be an accurate descriptor for the vibrational properties of the mobile ion. A better descriptor for the attempt frequency would be a vibrational frequency E specific for the mobile ion that is conceptually similar to an Einstein oscillator. This concept is especially useful for materials with an underlying guest-host structural motif, like the mobile ion within a rigid sub-lattice in superionic conductors 2 or guest atoms and associated "rattler modes" in the skutterudites and clathrates, which have rather dispersion-less phonon branches (Supplemental Note 5). 31 Here, we find that the average Ag + vibrational frequency (1.2 ± 0.2 THz; range = 1.0 -1.5 THz) determined from X-ray diffraction measurements is consistent with the average Ag + frequency found computationally (2 THz), and perfectly aligns with the lowest frequency peak in the vibrational density of states (Figure ). Meanwhile, the average Debye frequency (3.1 ± 0.2 THz; range = 2.5 -3.4 THz) found from speed of sound measurements does capture the average vibrational frequency of the entire density of states (~3 THz), but does not correspond to any particular feature. A detailed discussion of the measurements and further comparison can be found in the Supporting Information (Supplemental Notes 2,3,5). It is important to note that the ability to directly probe vibrations of the mobile ion provides experimental insights to the frequency spectrum relevant for ionic transport. From a thermal transport point-of-view, the Debye frequency is frequently shown to provide a strong and simple approximation even for complex materials, while the determination of Einstein frequencies can provide a useful tool if atom-specific frequencies play a significant role. Thus, the characterization of both and provides complementary information in examining thermal and ionic transport together. that, the Ag + on this lattice site has an inherently higher displacement in the direction of the jump, typically believed to be favorable for ionic transport.