\frac{C_V}{Nk} \sim 9 \left({T\over T_D}\right)^3\int_0^{\infty} {x^4 e^x\over \left(e^x-1\right)^2}\, dx .
In the low temperature limit, the limitations of the Debye model mentioned above do not apply, and it gives a correct relationship between (phononic) heat capacity, temperature, the elastic coefficients, and the volume per atom (the latter quantities being contained in the Debye temperature).