Phonons in higher dimensions and the heat capacity of solids

4 Phonons in higher dimensions and the heat capacity of solids 

The analysis that leads to Eq. (4.39) can be repeated for three-dimensional systems and for solids with more than one atom per unit cell, however analytical results can be obtained only for simple models. Here we discuss two such models and their implications with regard to thermal properties of solids. We will focus on the heat capacity, Eq. (4.35), keeping in mind that the integral in this expression is actually bound by the maximum frequency. Additional infonnation on this maximum frequency is available via the obvious sum rule 

This model assumes that all the normal mode frequencies are the same. Taking Eq. (4.42) into account the density of modes then takes the fonn 

For T oo this gives Cy = iNks- This result is known as the Dulong-Petit law that is approximately obeyed for many solids at high temperature. This law reflects the thermodynamic result that in a system of classical oscillators, each vibrational degree of freedom contributes an amount ks (SM ks for each kinetic and each positional mode of motion) to the overall heat capacity. 

In the low temperature limit Eq. (4.44) predicts that the heat capacity vanishes like 

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