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22, 52.92 ), where γo and αl are the quartic and quadratic coefficients of the double-well potential, respectively. With these parameters, we found that ωl depends on T sensitively, especially at moderate temperatures. For T > 25 K, its square reasonably complies with the Barrett formula (J.H. Barrett, Phys. Rev. 86 (1952) 118). However, ωl does not saturate below 25 K. We believe this deviation reflects that, below 25 K, SPA no longer applies and a simple tunneling picture may be more suitable. This idea also explains why isotope substitution has a considerable effect only at low temperatures. We have adopted this idea in our calculations. Also, for BT, the temperature dependence of the S mode is neglected for simplicity, which however should be important for quantitative features, as suggested by the expression of ωs.
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When θ = 1, σi will be turned into static variables, namely, it shall not have its own dynamics. Nevertheless, thermal fluctuations will impose incoherent dynamics upon it [1]. Because it is in a deep double-well potential, this kind of dynamics will have pronounced physical consequences. On the contrary, no interesting physics can be forecasted in connection with the thermal fluctuations of ηi at θ = 0, since it is in a featureless single-well potential.
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