Second-sound velocity, superfluid density, and universality near the superfluid transition inHe3-He4mixtures

A detailed analysis is presented for second-sound velocity ($u_2$) measurements in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures near $T_{\lambda }$ in terms of thermodynamic parameters and the superfluid fraction $\frac{{\rho }_s}{\rho }$. Several incorrect conclusions in the literature are corrected. The hydrodynamic expressions for the velocities of the propagating modes are obtained for the case where the thermal expansion coefficient does not vanish. The relevant thermodynamics near the $\lambda$ line is discussed, and expressions for the necessary parameters in terms of measured quantities are given. The importance of various contributions to $u_2$ is considered. The experimental data for $u_2$ are reviewed, and the corresponding results for the superfluid fraction are calculated. Special attention is given throughout to the propagation of errors from the uncertainty in thermodynamic parameters. The results for $\frac{{\rho }_s}{\rho }$ are found to be consistent with those obtained by the oscillating-disk method, as expected from two-fluid hydrodynamics. The difference in behavior of ${\rho }_s$ along paths of constant concentration and constant chemical potential is considered. The exponent of ${\rho }_s$ is found to be consistent with scaling and universality. The superfluid density is compared with the measured superfluid healing length $l$, and the parameter $\frac{l{\rho }_s}{T}$ is found to be independent of the concentration $X$. The data for ${\rho }_s$ are also found to be consistent with a concentration-independent free energy per correlation volume, as expected on the basis of universality arguments.