Cahill-Pohl model

FIGURE 8,4 Thermal conductivity of quartz and glassy silica as a function of temperature [7]. The quartz thermal conductivity exhibits a T behavior at low temperature, a peak at about 10 K, then reduction at higher temperatures. This is typical of a crystalline solid. For amorphous glass the thermal conductivity increases as T2 plateaus between 1 to 10 K and then increases monotonically with temperature. Also plotted are the predictions of the Cahill-Pohl and Einstein models. The Cahill-Pohl model provides accurate predictions for temperatures higher than 50 K but cannot predict the low temperature behavior. The Einstein model predictions are much lower than the measured values. 

Cahill and Pohl [8,9] recently developed a hybrid model which has the essence of both the localized oscillators of the Einstein model and coherence of the Debye model. In the Cahill-Pohl model, it was assumed that a solid can be divided into localized regions of size A./2. These localized regions were assumed to vibrate at frequencies equal to to = 2kvsIX where v, is the speed of sound. Such an assumption is characteristic of the Debye model. The mean free time of each oscillator was assumed to be one-half the period of vibration or x = it/to. This implies that the mean free path is equal to the size of the region or XI2. Using these assumptions, they derived the thermal conductivity to be... 

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