Specific heat of He4 and He3-He4 mixtures at their λ transition

We have measured the specific heat near the λ transition of pure He4 and of five He3-He4 mixtures up to a mole fraction of 0.39 He3 in He4. Our data for He4 confirm the results of Ahlers revealing an asymmetry in the exponents above and below Tλ when the specific heat is represented by a simple-power-law temperature dependence. Our results for these exponents (α=0.012±0.002 and α′=-0.012±0.004) differ somewhat from Ahlers's. Our results can be reconciled with the requirement of scaling (α=α′) only by supposing substantial contributions to Cp are made by singular correction terms to a simple power law. The measured specific heat of the mixtures richest in He3 appears to be finite, continuous, and cusped at the λ line. These qualitative features have been termed "renormalization" by Fisher. An analysis of our mixture data with a power-law temperature dependence does not yield a fully renormalized exponent, but rather an effective exponent. We have calculated the following derivatives at the λ line: (sT)|p,λ, (φT)|p,λ, and (xT)|p,λ. We have used these derivatives to calculate the specific heat along paths of constant pressure and constant relative chemical potential Cpφ. This specific heat behaves very much like Cp of pure He4 supporting the idea of universality for the specific-heat exponents. It is also true that the same asymmetry in the branches above and below Tλ which is observed in pure He4 is retained in the mixtures. The persistence of the asymmetry of Cpφ as one moves along the λ line towards increasing He3 concentration (at the saturated vapor pressure of the mixtures) is analogous to the persistence of the asymmetry of Cp as one moves along the λ line towards increasing pressure in pure He4. In both cases a unique reconciliation of the asymmetry with scaling must await better predictions of the form of the corrections to scaling. We have used our calculated Cpφ to construct the derivative (xT)|p,φ hence paths of constant φ on the x-T plane, as well as the derivative (xφ)|p,T.