Phonons Griineisen parameter

The pressure-dependence of ZnO phonon-mode frequencies measured by Raman scattering was reported in [37, 130]. From that the Griineisen parameters 7j... 

Table 3.6. Griineisen parameters of the zone-center phonon modes of ZnO [37] ...

Elastic constants depend on pressure and temperature because of the anharmonicity of the interatomic potentials. From the dependence of bulk and shear moduli on hydrostatic and uniaxial pressure, third order elastic constants and Griineisen parameters may be determined. Griineisen parameter shows the effect of changing volume, V, on the phonon mode frequencies, co. 

The phenomenological Griineisen parameter electron-phonon coupling for the heavy-fermion compounds is very successful in describing the thermal expansion and temperature dependence of elastic constants. The parameters deduced from the different experiments agree with each other and they agree also with the parameters used in the microscopic description, sect. 4.3. 

The linear thermal expansion coefficient a can be determined from the derive of the temperature dependence of the cell dimension as a = d ln L(T) /dT. Here, L(T) is the cell dimension at the corresponding temperature T. The volumetric TEC can be expressed via fundamental parameters, such as the Griineisen parameter Yv, the bulk modulus K, the molar voliune Vm, the heat capacity C, phonon frequencies o), and the internal energy U ... 

We note that through the Griineisen parameters anharmonic corrections of the potential energy are involved in the thermal expansion coefficient. The approximation that the vibrational free-energy function depends on the volume of the crystal through the change of the phonon frequencies described by the first-order approximation... 

From the phonon nonlinerities, Griineisen [12] derived the expression for the volume TEC that is proportional to the product of the specific heat and Griineisen parameter,... 

Thermal Expansion and Griineisen Parameter. Specific-heat calculations usually neglect interactions of phonon states or different vibrational modes (harmonic approximation). However, thermal expansion arises from such interactions. The coefficient of thermal expansion a is calculated fiom oscillatory equations by assuming asymmetric binding potentials. At decreasing temperature, vibrations become more harmonic, and t 0 for T -> 0. Above T = 0, not only phonon states of the same mode interact, but also those of different modes. 

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