4.10 Einstein and Cahill Pohl Minimum Phonon Conductivities

In the Einstein solid thermal conductivity model, the atomic vibrational states are directly used (not related to phonon modes). The complete derivation of the Einstein thermal conductivity is given in Appendix C. He considers the interaction (through a spring constant) of the first, second, and third neighboring atoms with a central atom. He uses a quantum specific heat capacity and a single frequency and arrives at [24]

Einstein model from (C.25) in Appendix C,

where n is the number density of atoms, and T _E is the Einstein temperature.

Cahill and Pohl [51] extend the proceeding equation to include a range of frequencies (details are given in Appendix C.2), and arrive at

Cahill Pohl model from (C.36) in Appendix C,

where u _{p, ?} and T _{D , ?} are the phonon (acoustic) speed and the Debye temperature for polarization ? and are related through (4.27), by use of the Debye model.

It is generally accepted that k _{p, C- P} is the lowest thermal conductivity for a solid, provided the proper values of u _{p, ?} or T _{D , ?} [they can be related through (4.63)] are available. Based on this k _{p, C- P} is used for thermal conductivity of amorphous solids and polymers. Figure 4.17 shows the variation of k _{p, C- P} with respect to temperature. At low temperatures ( T < T _{D , ?}) it...