Temperature dependent phonon spectra are derived for an anharmonic lattice model using the selfconsistent harmonic approximation. The lattice consists of cage and guest ions with anharmonic guest- cage interaction. By using a quasi-isotropic harmonic interaction that leads to polarization-independent dynamical matrix, renormalized phonon spectra are derived explicitly for the whole Brillouin zone and for all branches. The largest softening with decreasing temperature occurs in an Einstein-like mode corresponding to oscillation of guest ions. Cage oscillations also show slight softening since the cage- cage force is partly mediated via guest ions. In the high-temperature region, frequency of the Einstein mode is proportional to T 1/4 in agreement with the behavior derived previously for an impurity model. As temperature decreases below the Einstein frequency, the renormalized phonon frequency decreases through an inflection point, and finally tends to T 2 variation. The behavior compares favorably with recent neutron scattering results in PrOs4Sb12. The temperature dependence of the sound velocity is also calculated, which agrees qualitatively with that observed in Sr 8Ga 16Ge 30.
ASJC Scopus subject areas
- Physics and Astronomy(all)