Chapter 7.5, Problem 61P

The heat capacity of liquid ${}^{\text{4}}\text{He}$

below 0.6 K is proportional to $T^3$ with the measured value $C_{\text{V}}/Nk={\left(T/4.67\text{K}\right)}^3$. This behavior suggests that the dominant excitations at low temperature are long–wavelength phonons. The only important difference between phonons in a liquid and phonons in a solid is that a liquid cannot transmit transversely polarized waves—sound waves must be longitudinal. The speed of sound in liquid ${}^{\text{4}}\text{He}$ is 238 m/s, and the density is $0.145{\text{g/cm}}^{\text{3}}$. From these numbers, calculate the phonon contribution to the heat capacity of ${}^{\text{4}}\text{He}$ in the low–temperature limit, and compare to the measured value.