Tunneling and Rattling in the Intermetallic Cage Compound VAl$_{10+\delta}$

We have used a variety of thermodynamic and transport measurements to study the low-energy ``rattling mode'' in the so-called ``Einstein solid,'' VAl$_{10+\delta}$ [1]. The rattling is thought to arise from weakly bound Al atoms that occupy the large void at the center of Z$_{16}$ Friaf polyhedra, of which there are eight per unit cell in the VAl$_{10+\delta }$ structure. We find that the temperature dependence of our heat capacity and electrical resistivity data are indeed consistent with an Einstein oscillator-type localized mode. In contrast, we find that the $T$-dependence of our elastic constant and internal friction data deviate significantly from that expected for an Einstein oscillator. Indeed, the moduli data show a local minimum at $\sim $20 K, followed by a broad local maximum centered at $\sim $40 K, whereas in the same temperature range the internal friction data show a peak. Both of these observations strongly suggest that the atom inside the polyhedron undergoes quantum mechanical tunneling. We propose a model for the interatomic potential inside the polyhedron that accounts for both the rattling and tunneling behavior. \\[4pt] [1] A. D. Caplin, G. Gr\"{u}ner, and J. B. Dunlop, Phys. Rev. Lett. \textbf{30}, 1138 (1973).