Thermodynamics and experimental tests of static scaling and universality near the superfluid transition in He4 under pressure

The heat capacity at constant volume $C_{\nu }$ and the pressure coefficient ${\left(\frac{\partial P}{\partial T}\right)}_{\nu }$ were measured near the superfluid transition on six isochores. This work also yielded new results for the derivatives ${\left(\frac{\partial V}{\partial T}\right)}_{\lambda }$ and ${\left(\frac{\partial S}{\partial T}\right)}_{\lambda }$ along the λ line (V is the molar volume and S the entropy). An essentially complete and detailed description of the thermodynamics of the superfluid transition at all pressures is provided by the data. In particular, the results were used to derive the heat capacity at constant pressure $C_p$, the compressibility $\kappa$, the thermal expansion coefficient α, the ratio $\gamma \equiv \frac{C_p}{C_{\nu }}$, and the isentropic sound velocity u, along isobars. The heat capacity $C_p$ was examined carefully for its asymptotic behavior near $T_{\lambda }$. Although several interpretations of the data are possible with different assumptions about singular correction terms, it is clear that the exponents α and α′ are near zero, and that the ratio $\frac{A}{A^{{}^{\prime }}}$ of the amplitude above $T_{\lambda }$ to the amplitude below $T_{\lambda }$ is greater than unity and pressure dependent. The results are compared with the predictions of scaling and universality. The assumption of a pure-power-law singularity in $C_p$ results in α≠α′ at some pressures, and thus yields disagreement with scaling. The inclusion of singular higher-order contributions to $C_p$ in the analysis increases the uncertainty in the exponents derived from the data, and the prediction α=α′ of scaling falls within these larger uncertainties. The amplitude ratio remains greater than unity, and thus only nonzero exponents are consistent with scaling. The pressure dependence of $\frac{A}{A^{{}^{\prime }}}$ is not removed by the type of correction terms considered in the analysis, and is contrary to universality. When recent calculations for the exponent of the correction terms are assumed valid, then only a negative leading exponent is permitted by the data. This implies a finite $C_p$ at $T_{\lambda }$. If α=α′<0, then the data permit a continuous $C_p$ at $T_{\lambda }$ and at all pressures only if, contrary to universality, α and α′ depend upon the pressure.