Thermodynamics of the Isentropic Sound Velocity Near the Superfluid Transition inHe4

Abstract

A thermodynamic relation between the isentropic velocity of sound $u$ in the zero frequency limit and the heat capacity for ${\mathrm{He}}^4$ near the superfluid transition temperature $T_{\lambda }$ is presented. The validity of the calculation depends upon the logarithmic temperature dependence of the heat capacity. The effect of the gravitational field on $u$ in samples of finite height is calculated. The minimum measurable velocity in real samples exceeds the velocity at $T_{\lambda }$ by at least 48 cm/sec. The major contribution to the velocity arises from a term whose temperature dependence is ${C_p}^{-1\mathrm{}}$. However, there are additional appreciable contributions whose temperature dependence is $T_{\lambda }-T$ and $(T_{\lambda }-T)\ln |T_{\lambda }-T|$. Comparison with the measurements of Barmatz and Rudnick shows that the present treatment quantitatively explains all the features of the experimental data. The calculation can be used to determine the contribution from dispersion to measured velocities at nonzero frequencies.