Specific heat ofHe4andHe3-He4mixtures at theirλtransition

We have measured the specific heat near the $\lambda$ transition of pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and of five ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures up to a mole fraction of 0.39 ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. Our data for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ confirm the results of Ahlers revealing an asymmetry in the exponents above and below $T_{\lambda }$ when the specific heat is represented by a simple-power-law temperature dependence. Our results for these exponents ($\alpha =0.012\mathrm{}\pm 0.002$ and ${\alpha }^{\prime }=-0.012\mathrm{}\pm 0.004$) differ somewhat from Ahlers's. Our results can be reconciled with the requirement of scaling ($\alpha ={\alpha }^{\prime }$) only by supposing substantial contributions to $C_p$ are made by singular correction terms to a simple power law. The measured specific heat of the mixtures richest in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ appears to be finite, continuous, and cusped at the $\lambda$ 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 $\lambda$ line: ${\left(\frac{\partial s}{\partial T}\right)|}_{p,\lambda }$, ${\left(\frac{\partial \phi }{\partial T}\right)|}_{p,\lambda }$, and ${\left(\frac{\partial x}{\partial T}\right)|}_{p,\lambda }$. We have used these derivatives to calculate the specific heat along paths of constant pressure and constant relative chemical potential $C_{p\phi }$. This specific heat behaves very much like $C_p$ of pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ supporting the idea of universality for the specific-heat exponents. It is also true that the same asymmetry in the branches above and below $T_{\lambda }$ which is observed in pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ is retained in the mixtures. The persistence of the asymmetry of