Critical dynamics near theλtransition inHe3-He4mixtures

Abstract

The critical dynamics of the $\ensuremath{\lambda}$ transition in $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures are studied by means of renormalized field theory applied to the model of Siggia and Nelson. A diagonal representation for the equations of motion is introduced, which greatly simplifies the computations in two-loop order. A universal connection is found with the asymptotic critical dynamics of pure $^{4}\mathrm{He}$ in all orders of perturbation theory. The observable critical dynamics of helium mixtures are dominated by nonuniversal crossover effects which can be properly described only within a nonlinear renormalization-group approach. The theory is applied to explain the observable critical and precritical temperature dependence of the mass diffusion $D$, of the thermal conductivity $\ensuremath{\kappa}$, of the thermal diffusion ratio ${k}_{T}$, and of the dynamic structure factor for $T\ensuremath{\ge}{T}_{\ensuremath{\lambda}}(X)$. Recent experimental data for the transport coefficients at the molar $^{3}\mathrm{He}$ concentration $X=0.05$ by Gestrich and Meyer are used to identify the nonuniversal parameters of the theory in the range $X\ensuremath{\ll}1$. Consistency with the dynamics of pure $^{4}\mathrm{He}$ ($X=0$) is verified. Predictions without adjustable parameters are made for the dynamic structure factor and the transport coefficients in very dilute mixtures. The Siggia-Kawasaki problem concerning the leading $X$ dependence of $\ensuremath{\kappa}({T}_{\ensuremath{\lambda}})$ in the $X\ensuremath{\rightarrow}0$ limit is resolved. It is demonstrated that Siggia's prediction $\ensuremath{\kappa}({T}_{\ensuremath{\lambda}})\ensuremath{\sim}{X}^{\ensuremath{-}1}$ is correct but not observable. Theoretical extrapolations to $X>0.05$ without adjustable parameters are presented and compared with measured transport coefficients at $X=0.11 \mathrm{and} 0.15$. The overall agreement is satisfactory. Deviations of order 15% exist with the thermal conductivity $\ensuremath{\kappa}$ at $X=0.15$. This may be attributed to dynamic effects arising from the singular specific heat and mass susceptibility, which are not included in the present analysis.