Localized Rattling of Al atoms in VAl$_{10+\delta}$

We have studied the localized rattling mode in the \lq{Einstein solid}\rq~VAl$_{10+\delta}$ using a suite of thermodynamic, transport, and neutron diffraction measurements. The rattling originates from Al atoms that occupy the large void within Z$_{16}$ Friaf polyhedra, of which there are eight per unit cell in the VAl$_{10+\delta}$ structure. Our heat capacity, thermal expansion, and electrical resistivity data are all qualitatively consistent with a low-frequency harmonic vibration of the atom. However, our neutron diffraction data show that the rattling atom potential is better described as sixth-order, rather than harmonic. Using a single-site, sixth-order effective potential for the rattling atom, we can explain our thermodynamic, transport, and structural data, including the unusual temperature dependence of the elastic constants.