Mean free path of phonon

The thermal conductivity of bulk silicon (148 W K m ) is dominated by phonons electronic contributions are negligible. Due to restrictions of the mean free path of phonons in the porous network the thermal conductivity of micro PS is reduced by two or three orders of magnitude at RT, compared to the bulk value. Because of the larger dimensions of its network, meso PS shows a thermal conductivity several times larger than that of micro PS, for the same value of porosity. Thermal oxidation at low temperatures (300°C) is found to decrease the thermal conductivity of meso PS by a factor of about 0.5 [Pe9]. In contrast to bulk Si the thermal conductivity of PS is found to decrease with decreasing temperature [Be21, La4, Ge9, Lyl]. 

Total thermal conductivity is a sum of the lattice and electronic parts, K = Ki + Ke- The lattice part of the thermal conductivity describes the scattering of phonons on the vibrations of atoms, whereas the electronic part describes thermal conductivity appearing due to conduction electrons and is related to the electrical conductivity Wiedemann-Franz equation, = a T Lo, where T is the absolute temperature and Lq is the ideal Lorenz number, 2.45 X 10 Wf2K [64]. The electronic part of the thermal conductivity is typically low for low-gap semiconductors. For the tin-based cationic clathrates it was calculated to contribute less than 1% to the total thermal conductivity. The lattice part of the thermal conductivity can be estimated based on the Debye equation /Cl = 1 /3(CvAvj), where C is the volumetric heat capacity, X is the mean free path of phonons and is the velocity of sound [64]. The latter is related to the Debye characteristic temperature 6 as Vs = [67t (7V/F)] . Extracting the... 

Debye temperature from either heat capacity or structural data and assuming that the mean free path of phonons is the average distance between the guest atoms, the lattice part of the thermal conductivity was calculated for several tin-based cationic clathrates to be in the range of 0.7-0.9 W m K , which is in good agreement with the experimental data [31, 32, 56, 58]. 

The mean free path of phonons is affected by a variety of processes. All of these processes therefore determine the thermal conductivity of ceramics. One major process that influences the mean free path is the Umklapp process. At low temperatures, the mean free path corresponding to this process becomes large. 

If there is a difference in temperature in a material, a flow of thermal energy occurs. The flow of thermal energy, Q, is proportional to the temperature gradient, dTtdx, and the coefficient is called the thermal conductivity, K. Thus, Q = KdTIdx. The thermal conductivity, K, is approximately proportional to the heat capacity (O, the phonon speed (v), and the mean free path of phonons (/) of the material K CyVl (175). At low temperatures, the mean free path (/) is almost constant and the heat capacity (C) is proportional to Tl Thus, K is proportional to Tl At high temperatures, C is almost constant and / is proportional to TThus, K is proportional to T". The thermal conductivity of cBN appears to have this general tendency as shown in Fig. 33 (147,176-178). 

Strongly rising resonant scattering of phonons due to HO (8) leads to that at some energy E i the mean free path of phonons becomes equal to their wave length [17,22]. The value of this energy is determined by expression [22]... 

Mean free path of phonons in Laj Caj Cr03 calculated from the thermal conductivity and heat capacity... 

The thermal conductivity of pure LaCrOs linearly depends on the inverse temperature, which indicates that the main carrier of thermal conduction are phonons. The phonon tends to be scattered in the solids by heavy metal ions such as lanthanum (La +) in the lattice. Moreover, the alkaline earth or rare-earth substitution causes the decrease of thermal conductivity at room temperatures, which indicates that the phonon is also scattered by the substituted cations which randomly occupied the lattice. As a result, the thermal conductivity of calcium substituted samples showed no dependence on temperature. The length of mean free path of phonon (/) can be calcnlated from thermal conductivity, molar volume (Tm), molar heat capacity (Cp) and velocity of the sound (m) as follows ... 

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