We measured the density of vibrational states (DOS) and the specific heat of various glassy and crystalline polymorphs of SiO 2 . The typical (ambient) glass shows a well-known excess of specific heat relative to the typical crystal (α-quartz). This, however, holds when comparing a lower-density glass to a higherdensity crystal. For glassy and crystalline polymorphs with matched densities, the DOS of the glass appears as the smoothed counterpart of the DOS of the corresponding crystal; it reveals the same number of the excess states relative to the Debye model, the same number of all states in the low-energy region, and it provides the same specific heat. This shows that glasses have higher specific heat than crystals not due to disorder, but because the typical glass has lower density than the typical crystal. DOI: 10.1103/PhysRevLett.112.025502 PACS numbers: 63.20.-e, 07.85.-m, 76.80.+y The low-temperature thermodynamic properties of glasses are accepted to be anomalously different from those of crystals due to the inherent disorder of the glass structure. At temperatures of ∼10 K, the specific heat of glasses shows an excess relativetothatofthecorrespondingcrystals.Theexcessspecific heat is related to a distinct feature in the spectrum of the atomic vibrations: At frequencies of ∼1 THz, glasses exhibit an excess of states above the Debye level of the acoustic waves, the socalled "boson peak." The excess of specific heat and the boson peak are universally observed for all glasses and by all relevant experimental techniques. However, the results still do not converge to a unified answer to how disorder causes these anomalies.Themajorityofthemodelsexplainthebosonpeakbyappealing tovarious glass-specific features. Theseincludelow-energy optical modes [1], onset of mechanical instability related to saddle points in the energy landscape [2] or to jamming [3][4][5], local vibrationalmodes of clusters [6] or locally favoured structures [7], librations [8] or other coherent motions [9] of molecular fragments, crossover of local and acoustic modes [10], quasilocal vibrations of atoms in an anharmonic potential [11], broadening of vibrational states in the Ioffe-Regel crossover regime [12], spatial variation of the elastic moduli [13], breakdown of the continuum approximation [14,15], and topologically diverse defects [16], to cite the most important ones.Alternatively, the boson peak is identified as the counterpart of the acoustic van Hove singularities of crystals, i.e., explained by the piling up of the vibrational states of the acousticlike branches near the boundary of the pseudoBrillouin zone [17][18][19][20].Diverging in explanations of the boson peak, all models agree that the excess states and the excess specific heat of
The lattice dynamics in Bi 2 Te 3 and Sb 2 Te 3 were investigated both microscopically and macroscopically using 121 Sb and 125 Te nuclear inelastic scattering, x-ray diffraction, and heat capacity measurements. In combination with earlier inelastic neutron scattering data, the element-specific density of phonon states was obtained for both compounds and phonon polarization analysis was carried out for Bi 2 Te 3 . A prominent peak in the Te specific density of phonon states at 13 meV, that involves mainly in-plane vibrations, is mostly unaffected upon substitution of Sb with Bi revealing vibrations with essentially Te character. A significant softening is observed for the density of vibrational states of Bi with respect to Sb, consistently with the mass homology relation in the long-wavelength limit. In order to explain the energy mismatch in the optical phonon region, a ∼20% force constant softening of the Sb-Te bond with respect to the Bi-Te bond is required. The reduced average speed of sound at 20 K in Bi 2 Te 3 , 1.75(1) km/s, compared to Sb 2 Te 3 , 1.85(4) km/s, is not only related to the larger mass density but also to a larger Debye level. The observed low lattice thermal conductivity at 295 K, 2.4 Wm −1 K −1 for Sb 2 Te 3 and 1.6 Wm −1 K −1 for Bi 2 Te 3 , cannot be explained by anharmonicity alone given the rather modest Grüneisen parameters, 1.7(1) for Sb 2 Te 3 and 1.5(1) for Bi 2 Te 3 , without accounting for the reduced speed of sound and more importantly the low acoustic cutoff energy.
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