Heat capacity for real insulator

Question

In the Debye model, for temperatures $T \ll T_D \equiv \frac{\hbar\omega_D}{k_B}$, the molar heat capacity $c_V$ can be calculated as follows: $$c_V = \frac{12\pi^{4}}{5}R \left(\frac{T}{T_D}\right)^3$$ (where $12\pi^{4}/5\approx 234$).

Now if we take the limit $T\rightarrow 0$, we would expect from the Debye-model $c_V = 0$. However, this is apparently not the case for a real insulator, where I think we get $c_V = \alpha$, with $\alpha$ being a constant, in the limit $T\rightarrow 0$.

Can someone explain why?

Answer 1

The heat capacity does not go to a constant value at small temperatures.

The Third Law of Thermodynamics states that the heat capacity of any substance $c_{V}$ approaches $0$ as $T\rightarrow0$. The Debye model correctly predicts this. Moreover, it also gives the correct rate at which $c_{V}\rightarrow0$. Previously, the Einstein model had correctly predicted that $c_{V}\rightarrow0$ as $T\rightarrow0$, but $c_{V}$ vanished much more rapidly in the Einstein model than in real solids. Debye corrected that expression to get a numerically accurate expression for the low-temperature heat capacity of lattice solids.