Anomalous frequency dependence of the internal friction of vitreous silica

The internal friction Q(-1) and the sound velocity delta nu/nu of vitreous silica were measured at very low temperatures using mechanical double paddl e resonators operated at frequencies ranging from 0.33 to 14 kHz. Below sim ilar to 40 mK the internal friction showed an unexpected temperature and fr equency defendence, with absolute values of Q(-1) clearly exceeding those p redicted by the standard tunneling model. Even though the most plausible or igin of the observed excess internal friction appears to be the mutual inte raction between tunneling states, the results are difficult to reconcile qu antitatively with present theories taking into account this interaction.