Specific heat of normal and superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ on the melting curve

A method has been developed for determining the specific heat of liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ on the melting curve as a function of temperature and magnetic field. This approach depends on the accurate measurement of pressure and volume responses to heat pulses applied to the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in a Pomeranchuk cell. Analysis of a number of different experiments at a particular melting pressure yields both the specific heat of the liquid and its temperature. The thermodynamic determination of the temperature has been separately discussed in another publication. Measurements were performed between 1.1 and 23 mK in magnetic fields up to 8.8 kOe. From the normal-fluid specific-heat data the low-temperature value of the effective mass at the melting curve was found to be $\frac{m^{*}}{m}=5.5\mathrm{}\pm 0.2$. This is substantially smaller than that reported by Wheatley. Specific-heat discontinuities at the $A$, $A_1$, $A_2$, and $B$ superfluid transitions have been measured. These give values for certain combinations of the coefficients of the fourth-order invariants in a Ginzburg-Landau expansion. Comparison was made with the predictions of spin-fluctuation theories. It was found that these alone cannot account for the behavior of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ at melting pressures. The entropy difference between the $A$ and $B$ phases was calculated from the specific-heat data and compared with that calculated from (i) measurement of the latent heat at the $B\to A$ transition, and (ii) measurement of the suppression of the $B$ transition by magnetic field, $B$ phase susceptibility data, and a magnetic Clausius-Clapeyron equation. The different methods give a consistent picture in which the thermal differences between $A$ and $B$ phases are quite small. The $A$-phase specific heat at $\frac{T}{T_c}\sim 0.5$ appears to have a weaker dependence on temperature than that expected for the limiting low-temperature behavior of the Anderson-Brinkman-Morel state.