Statistical Physics

5.2 Heat Capacity of a Solid II – Debye Model 75that T D of diamond is exceptionally high. This is the reason that diamonddoes not obey the Dulong–Petit law at room temperature.This high Debye temperature can be understood from the well-knownproperties of diamond. The average frequency of the normal modes is thesame as the frequency in the Einstein model, and is given by ω = √ k/m,where k is the spring constant and m is the mass of an atom. Diamond isthe hardest solid known, which means that the bonds (covalent) between theatoms are strong, and so k must be large. Furthermore, the carbon atom isone of the lighter atoms. Therefore, it is natural that diamond has a highDebye temperature. Lead has properties opposite to those of diamond. It issoft, and the lead atom is one of the heaviest atoms. Therefore, the Debyetemperature of lead is low. We can estimate the “spring constant” k from theDebye temperature and the mass of an atom, and can compare the strengthsof various kinds of bonds. This is left as an exercise for readers at the end ofthis chapter.Fig. 5.6. Heat capacity calculated from the Debye model. The horizontal axis showsT/T D, where the Debye temperature T D is defined by k BT D ≡ ω max5.2.5 Physical Explanation for the Temperature DependenceThe Dulong–Petit law, obeyed at high temperature, is easy to understand.In this case k B T ≫ ω max , and so the equipartition law is obeyed for allnormal modes, as in the Einstein model at high temperature. The behavior atlow temperature can be understood as follows. From our experience with thediatomic molecule and the Einstein model, we know that the contribution tothe heat capacity from modes with frequencies ω that satisfy ω >k B T will benegligibly small. For these modes, 〈n〉 ∼0; these modes are dead, inert modes.Now, we can divide the modes roughly into two parts at a frequency ω T ≡