Shear viscosity measurements of liquid 4He and 3He-4He mixtures near the tricritical point

A torsional oscillator cell is described, by means of which simultaneous precision measurements of () and of the molar volume can be made in liquid ⁴He-⁴He mixtures over the temperature range between 0.5 and 3 K. Here is the mass density, the shear viscosity and in the superfluid phase they become the contributions _n and _n of the normal component. The results of for ⁴He near the superfluid transition are compared with the predictions by Schloms, Pankert and Dohm, and by Ferrell. Measurements of () are reported for mixtures with 0.64X0.74, where X is the ³He mole fraction. Those for X = 0.67 and 0.70 are compared with data by Lai and Kitchens. The viscosity experiments show no evidence of a weak singularity at the tricritical point.     

Concentration susceptibility of 3He-4He mixtures near the superfluid transition

Measurements of the concentration susceptibility % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabiqaaiaacaGaaeqabaWaaeaaeaaakeaacuaHhpWygaacai% abg2da9iabgkGi2kaadIfacaGGVaGaeyOaIyRaeuiLdqKaaiykamaa% BaaaleaacaWGubaabeaaaaa!3C99!\[\tilde \chi = \partial X/\partial \Delta )_T \] are reported for ³He-⁴He mixtures at saturated vapor pressure and at constant mole fraction X(³He) as a function of temperature. Here is the isotopic difference in chemical potentials. The mixtures cover the range from X = 0.60 to X = 0.677. Particular emphasis is given to the region near the lambda line for three mixtures and to the region close to the tricritical point. The method makes use of the vertical concentration gradient induced in the mixture by gravity. This difference X is measured via the dielectric constant over a height of 2 mm using a capacitance technique. The predicted peak of the susceptibility at the superfluid transition is observed and as the tricritical point is approached, this peak is progressively broadened by gravity effects. The data are compared with those from other methods and also with estimations based on calorimetric data for more dilute mixtures. The susceptibility data are transformed into results along a path at constant chemical potential . Sufficiently close to T , these can be fitted to an expression with a weakly singular term, which is consistent with the predictions on the grounds of universality. These predictions are based on the calorimetric data of more dilute mixtures. The width of this critical region is found to be consistent with theoretical estimations by Riedel, Meyer, and Behringer. The tricritical scaling scheme by these authors is tested by the new data and the resulting scaling curve is found to be in reasonable agreement with that obtained from earlier data by Goellner, Behringer, and Meyer, except in the region closest to the tricritical point. There the new data appear to be more consistent with measurements from light scattering. In addition, the concentration susceptibility for more dilute mixtures (0.05 < X < 0.4) is calculated both from calorimetric data and from saturated vapor pressure measurements and the results are found to be internally consistent.Supported by grants from the AFOSR and from the NSF.     

Transport properties in3He-4He mixtures near the superfluid transition

We report measurements of the temperature, density, and concentration gradients in³He-⁴He mixtures, induced by a vertical heat flux. The flat horizontal cell included two superposed capacitors and the density was determined by means of the dielectric constant method. The experiments were carried out on mixtures with mole fractionsX ₃=0.37, 0.15, and 0.05 at saturated vapor pressure, with special emphasis on the region near the superfluid transition. Our measurements under steady-state conditions give the conductivity , the thermal diffusion ratiok _T , and the coefficient of thermal expansion. We describe the singular behavior of these quantities in the neighborhood ofT (X). In the superfluid phase, we test with fair success a relation by Khalatnikov between gradX/ gradT and several static properties. From the relaxation times needed to attain steady-state conditions, and in combination with measured static and transport properties, we obtain in the normal phase the mass diffusionD, which diverges strongly asT is approached. In the superfluid phase, we test successfully a scaled relation that results from the solution of Khalatnikov's hydrodynamic equations. From our data the dispersion relations for scattered light are calculated: _o/q ² in both the normal and the superfluid phases and ₂/q ² in the normal phase.Research supported by NSF grant DMR 8024056.     

Sound propagation in 3He-4He mixtures near the superfluid transition

Measurements of the acoustic attenuation and dispersion in liquid ³He-⁴He mixtures near the superfluid transition T (x) are reported. The frequency range is /2gp=1–45 MHz and the ³He mole fraction X of the mixtures is 0.007, 0.05, 0.15, and 0.36. Comparisons are made with the measurements of Buchal and Pobell for similar mixtures obtained in the kHz region, and on the whole, the consistency between the two experiments is very satisfactory. An analysis is then performed using both the kHz and MHz data. In the normal phase, where the energy dissipation is caused by order parameter fluctuations having a lifetime _F , the attenuation data can all be scaled according to the expression = (T )f(_F. Here (T )^1+y, with y being a function of the mole fraction X and _F(T–T )^–x, with x increasing weakly with X. In the superfluid phase, we attempt a similar scaling representation, which is found to be fairly successful, but where x(T\s-T ) is roughly 15% larger than x(T>T ). In the superfluid phase we also analyze the attenuation data, assuming the additivity of relaxation and fluctuation-dissipation mechanism, and discuss the relaxation times so derived. In contrast to the attenuation, the dispersion data cannot be brought satisfactorily into a scaling representation. However, at T , we find U()-U(0)^y as predicted by Kawasaki, where y is in good agreement with the values from attenuation experiments.Supported by a grant from the National Science Foundation.     

Velocity of second sound and superfluid density near the superfluid transition in3He-4He mixtures

Using superleak condenser transducers, the velocity of second soundU ₂ has been measured near the superfluid transition temperature T in³He-⁴He mixtures with molar concentrationsX of³He of 0.0, 0.038, 0.122, 0.297, and 0.440. We have obtained the superfluid density _s/ fromU ₂ on the basis of linearized two-fluid hydrodynamics. The results for _s/ are consistent with those obtained from the oscillating disk method, as expected from two-fluid hydrodynamics. The value of _s/ at eachX could be expressed by a single power law, _s/=k, where =1-T/R, with the experimental uncertainty. It is found that the exponent is independent of concentration forX0.44 within the experimental uncertainty. This concentration independence of is in agreement with the universality concept. From the conclusion that the values of are universal forX0.44, the concentration dependence of the superfluid component _s is expressed by an empirical equation _s(X, )=²_s(0, ). It is found that corresponds to the volume fraction of⁴He in the superfluid³He-⁴He mixture. The value of is in agreement with that obtained from the measurement of the molar volume by others.This paper is based on a thesis submitted to Tokyo University of Education in partial fulfillment of the requirements for the Ph.D. degree.     

Shear viscosity measurements of liquid 3He-4He mixtures above 0.5 K

Simultaneous measurements of () and of the molar volume are reported for liquid mixtures of ³He in ⁴He over the temperature range between 0.5 and 2.5 K. Here is the shear viscosity and is the mass density. In the superfluid phase, the product of the normal components, _n and _n , is measured. The mixtures with ³He molefractions 0.30 < X < 0.80 are studied with emphasis on the region near the superfluid transition T and near the phase-separation curve. Along the latter, they are compared with data by Lai and Kitchens. For X > 0.5, the viscosity singularity near T becomes a faint peak, which however fades into the temperature-dependent background viscosity as X tends to the tricritical concentration X _t. Likewise, no singularity in is apparent when T _t is approached along the phase separation branches _– and ₊. Furthermore, viscosity data are reported for ³He and compared with previous work. Finally, for dilute mixtures with 0.01 X 0.05, the results for are compared with previous data and with predictions.     

Fermi liquid behaviour of the viscosity of3He-4He mixtures

We have investigated³He-⁴He mixtures at³He-concentrations 0.98%x9.5% by the vibrating wire technique in the temperature range 1 mKT 100 mK and at pressures 0 bar p 20 bar. In the degenerate regime of the mixtures the Landau theory of Fermi liquids predicts a temperature dependence of the viscosity proportionalT ^–2. We report on the first observation of this behaviour at 3 mKT 10 mK for all investigated concentrations and pressures. At temperatures below about 20 mK slip corrections had to be taken into account due to the increase of the quasiparticle mean free path at very low temperatures. The low-temperature cut-off in T ² = constant indicates the transition into the ballistic regime of the mixtures, where the mean free path of the quasiparticles exceeds the radius of the vibrating wire. Our results for the pressure dependence of the viscosity as well as for its magnitude show substantial differences from predictions based on pseudopotential theory. However, a calculation of with the quasiparticle interaction potential of recent solubility measurements in mixtures agrees well with our experimental data, in particular the pressure independence of .     

Anomalous nuclear spin relaxation in liquid 3He-4He mixtures near the lambda transition

The longitudinal nuclear spin relaxation time T ₁ was measured for liquid ³He-⁴He mixtures under saturated vapor pressure in an epoxy sample chamber in which the wall effect is emphasized. The molar concentration of ³He was 0.12, 0.20, 0.35, 0.45, and 1.00. The temperature was regulated to better than 5 K in the range 1.2–2.1 K. A decrease of T ₁ was clearly observed several millikelvins below T (the superfluid transition temperature). T ₁ had a cusp like temperature dependence as a whole where the temperature is far from. T . This anomalous behavior of T ₁ is discussed.Work based on part of a dissertation submitted to the University of Tsukuba in partial fulfillment of the degree of Doctor of Science by one of the authors (T.K.).     

Thermal transport properties of helium near the superfluid transition. II. Dilute3He-4He mixtures in the superfluid phase

Measurements of the average thermal conductivity _exp hQ/T and of the thermal relaxation time to reach steady-state equilibrium conditions are reported in the superfluid phase for dilute mixtures of³He in⁴He. Hereh is the cell height,Q is the heat flux, andT is the temperature difference across the fluid layer. The measurements were made over the impurity range 2×10^–9<X(³He)<3×10^–2 and with heat fluxes 0.3<Q<160 µW/cm². Assuming the boundary resistanceR _b , measured forX<10^–5, to be independent ofX over the whole range ofX, a calculation is given for _exp. ForQ smaller than a well-defined critical heat fluxQ _c (X) X ^0.9, _exp is independent of Q and can be identified with the local conductivity _eff, which is found to be independent of the reduced temperature = (T–T)/T for –10^–2. Its extrapolated value at T is found to depart forX10^–3 from the prediction X ^–1 , tending instead to a weaker divergence X ^–a witha0.08. A finite conductivity asX tends to zero is not excluded by the data, however. ForQ >Q _c (X), a nonlinear regime is entered. ForX10^–6, the measurements with the available temperature resolution are limited to the nonlinear conditions, but can be extrapolated into the linear regime forX2×10^–7. The results for _exp(Q),Q _c (X), and _eff(XX) are found to be internally consistent, as shown by comparison with a theory by Behringer based on Khalatnikov's transport equations. Furthermore, the observed relaxation times (X) in the linear regime are found to be consistent forX>10^–5 with the hydrodynamic calculations using the measured _eff(X). ForX<10^–5, a faster relaxation mechanism than predicted seems to dominate. The transport properties in the nonlinear regimes are presented and unexplained observations are discussed.     

Second-sound velocity and superfluid density in3He-4He mixtures near superfluid transition points—Applicability of universality and scaling laws

From second-sound velocity measurements, superfluid densities near superfluid transition points were determined in ³ He- ⁴ He mixtures under saturated vapor pressure. The critical exponent of the superfluid density thus obtained increases about 15% with increasing ³ He concentration up to 40 mole percent, which contradicts the universality concept. Furthermore, the critical exponents obtained here and the exponents of the specific heat do not satisfy the scaling laws. A new concept of universality is introduced and discussed.Submitted in partial fulfillment of the requirements for the Ph.D. degree at The University of Tokyo.     

The behavior of the4He viscosity near the superfluid transition

Measurements of the shear viscosity at saturated vapor pressure through the lambda transition indicate a singular behavior of the form |1 – (/)|=A ^x , (where =|1–(T/T )|, with equal values for the critical exponent on both sides of the transition.Work sponsored by Consiglio Nazionale delle Ricerche, Rome (Italy).     

Universality and the scaling laws in the superfluid phase transition of3He-4He mixtures

From the second-sound velocityU ₂ near the superfluid transition point, the superfluid densities in³He-⁴He mixtures, _s (X) and _s (), were deduced along the paths of constant³He concentrationX and of constant chemical potential difference of³He and⁴He. The following critical exponents of _s are determined: (a) =XX for _s (X) in the(X, T) plane,(b) X for _s (X) in the(, T) plane, and(c) for _s () in the(, T) plane. It is found that and _X change by about 4–6% relative to with increasing³He concentration up toX=0.4 and by 8–10% up toX=0.53. It seems that, belowX=0.53, universality hold for . Values of have been found to be in good agreement with the critical exponent of _s in pure⁴He under constant pressure. The values of and _X forX0.53 are also found to be consistent with the scaling relations in the (,T) plane of³He-⁴He mixture.Work performed in part while at the Electrotechnical Laboratory.     

Thermodynamic derivatives in3He-4He mixtures in the critical region of the superfluid transition

By assuming a phenomenological Ginzburg-Landau-Wilson free energy for³He-⁴He mixtures near the critical surface of the superfluid transition we can express thermodynamic derivatives of the form (?a/?b) _c,d in terms of a correlation function of ¦ψ¦², where Gy is the order parameter. Herea, b, c, and d are any four of the quantities entropys, number densityn, molar concentrationX, temperatureT, pressurep, and chemical potential difference Δ=μ₃?μ₄. Relationships among these thermodynamic derivatives are those of the Pippard-Buckingham-Fairbank approximation in the³He-⁴He case. Some combinations of the coefficients in the free energy are equal to derivatives on the critical surface and remain invariant with respect to renormalization group transformations. A general scheme is also developed to give correlation function expressions for thermodynamic derivatives.     

Vibrating-wire measurements in vacuum,liquid3He and liquid3He-4He mixtures

In order to overcome the 200µK - barrier in the refrigeration of liquid ³ He- ⁴ He mixtures we have constructed an experimental cell using only pure materials to minimize possible origins for heat leaks into the liquid. With this arrangement we were able to cool a saturated6.8%- mixture to a temperature of 150µK. A vibrating wire which was immersed in pure ³ He floating on top of the phase-separated mixture was used as a thermometer. This wire was calibrated in a second experiment with pure ³ He only in the cell. In superfluid ³ He-B at T0.15 mK the damping of the wire due to the quasiparticles becomes very small, and we observe typical characteristics of the vacuum damping of the wire which was extensively examined before filling any liquid into the cell.     

Thermal transport properties in helium near the superfluid transition. III. Dilute3He-4He mixtures in the normal phase

An experimental study is presented for the thermal conductivity and the thermal relaxation for dilute mixtures of³He in⁴He with concentration 9×10^–4X(³He)5×10^–2 at saturated vapor pressure and in the normal phase near the superfluid transition. The conductivity results for are compared with predictions by Dohm and Folk from field-theoretic renormalization group(RG) theory. The conductivity _s =[^–1(T)–^–1(T)]^–1, is compared with Ahlers' phenomenological arguments, and also with predictions by Dohm and Folk and by Onuki. The temperature difference transient T(t) across the fluid, measured as a function of timet after switching on and off the heat current, is analyzed. The thermal diffusion ratiok _T and the mass diffusion coefficientD are obtained by fitting the calculated transient to the experimental one. The results are compared with the predictions that follow from the RG approach by Dohm and Folk. Very good agreement is obtained fork _T. The transient is not very sensitive toD, and hence the determination is not accurate. Yet within the uncertainty, the deducedD also agrees with predictions. Appendices give (1) the corrections to from finite heat effects, (2) the calculation of the concentration susceptibility (X/) _T,P , and (3) the calculation procedure for ,k _T, andD using the RG approach of Dohm and Folk.     

Temperature-, concentration- and pressure-dependence of the viscosity of liquid3He-4He mixtures at low temperatures

We have investigated the viscosity of liquid³He-⁴He mixtures at various³He - concentrations (0.98%x9.5%) in the temperature range1 mK T 100 mK and at pressures 0 bar P 20 bar. At T10 mK the Fermi-liquid behaviour T² = const. as well as x^4/3 could be confirmed. However, there are significant deviations from theoretical predictions for the magnitude of the viscosity as well as for its pressure dependence.