Discussion is begun with phonon's transporting character and Boltzman equation and based on "tunnelling level" theory developed in 70's. A system in which phonon state couples with two level system is considered. The interaction of phonons and that between phonon and boundary are neglected. Under extremely low temperature, for the amorsphous state solid, the interaction between phonon and tunnelling state is regarded as the main part to hinter the heat current and to make the heat conductivity of an amorphous state solid be a limitting value. The transition matrix element of interaction between phonon state and tunnelling state is calculated. Assuming that the particle in the tunnelling state obeys Boltzman Distribution and the medium is isotropic, for the heat conductivity of an amorphous solid, the T~2 relation is obtain. In the pioneer's work, relaxation time approximation was used to solve such a problem and the relaxation time was calculated with a quasi-classical method like that Enstein used to discuss the interaction between atom and radiation field. It should be noted that, in this thesis, our calculation is based on the quantum theory and the relaxation time approximation is not used. We only use Boltzman equation and quantum transition theory to consider this problem, and the physical image is more basic and distinct.
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