Internal friction and sound velocity

We assume the simplified relation for a thermally activated relaxation rate given by Eq. 16 and add it to the tunneling rate (Eq. 17). For two well-defined harmonic potentials and within the approximations given by Eq. 10 and with A = V/Eo, it can be shown that the internal friction increases linear with temperature just above the temperature-independent region plateau) [11]  

This increase is observed for PS, (Fig. 4.13) applying Eq. 22 to the results below 20 K we obtain a zero point energy Eq= 13 2 K. In the same temperature region and due to the influence of the thermally activated relaxation the relative change of the sound velocity can be written as [11]  

A nearly linear temperature dependence of 0d/d is observed in both polymers at temperatures about below 50 K (Figs. 4.14 and 4.15). At T 1 K a crossover, due to tunneling and due to thermally activated relaxation, from the linear T dependence to the logarithmic 

Figvire 4.14 Relative change of sound velocity at v = 535 Hz for PMMA. The solid line represents the linear temperature dependence of the sound velocity at temperatures below 20 K used for the numerical calculations. Inset the same data but below 10 K in a semilogarithmic scale. 

T dependence can be observed for both polymers (insets in Figs. 4.14 and 4.15). Although the tunneling model with the assumption of thermally activated relaxation of the TS provides a reasonable fit for the linear temperature dependence, its origin is still controversial. We note that a linear temperature dependence of the sound velocity above a few Kelvin is a rather general behaviour observed in several amorphous [26, 27], disordered [28], and polycrystalline metals [29]. Nava [28] argued recently against an interpretation in terms of thermally activated relaxation of the TS for the linear dependence of the sound velocity. However, new acoustical results in polycrystalline materials indicate a hnear dependence of the sound velocity comparable with those found in amorphous materials [29]. 

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