The soft-potential model (an extension of the well-known tunneling model for two-level states in glasses) has been formulated in terms of soft-mode eigenvectors in order to characterize the localization of these modes. The interaction with high-frequency modes explains the absence of very small restoring-force constants. A quantitative comparison to specific-heat and neutron-scattering data from vitreous silica, amorphous selenium, and vitreous boron trioxide shows that both the two-level systems and the low-energy vibrational states can be explained by the same distribution of localized modes.

• Received 15 October 1990

DOI:https://doi.org/10.1103/PhysRevB.43.5039