Third sound in3He-4He mixture films

We report third-sound studies of³He-⁴He mixture films for 0.3T1.8 K carried out in two different experimental situations. In one experiment detailed measurements of the amplitude of both the temperature and thickness variation of the film were made for pure⁴He and the results for |T/d| are in good agreement with the predictions of Bergman. Mixture studies in this apparatus were made difficult due to the presence of capillary condensation. In spite of this a number of interesting results are reported. Subsequent measurements were made in a new apparatus where capillary condensation effects could be demonstrably avoided. Helium films of 5.7 layers of⁴He were studied as a function of added amounts of³He. BelowT0.7 K the third-sound measurements are consistent with model calculations based on a bilayer film. We conclude that for 0.3T0.6 K films of the type we have studied are isotopically layered in the van der Waals field provided by the substrate.     

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.     

Transport properties of helium near the liquid-vapor critical point. II. Thermal conductivity of a3He-4He mixture

Measurements of the thermal conductivity are reported for an 80%³He-20%⁴He mixture above the critical point along several isotherms and near-critical isochores, using the same techniques and apparatus described for a study of³He. Using again the assumption that the observed conductivity can be decomposed into a sum of a regular and a singular contribution _reg and _sing, it is shown that along two near-critical isochores, _sing diverges. In particular, along the isochore showing the largest at the phase transition, the divergence is nearly the same as for³He and can be roughly characterized by a simple power law (T-T_c)^– with 0.58. This observation is contrary to predictions that foresee _sing0 asT _c is approached. The relaxation times characterizing the attainment of steady state conditions after switching the heat flux on and off show a similar behaviour as a function of reduced temperature as do those for pure³He. This result might indicate a substantial coupling between concentration and entropy diffusion. In the Appendix, the correlation length for³He nearT _c is calculated from heat conductivity, viscosity, and specific heat data.Work supported by Grant DMR 8024056 of the National Science Foundation.     

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.     

Critical behavior of liquid4He at negative pressures

We examine the equation of state of liquid⁴He at negative pressures close to the spinodal density _s where the hydrodynamic speed of sound vanishes. The non-analytic behavior of the equation of state and the speed of sound in the vicinity of the spinodal density are calculated in two and in three dimensions; we find for the speed of sound the non-analytic behavior mc _s ² ( — _s)^2/5 in three dimensions and mc _s ² [(-_s)/¦ln(-_s)¦]^1/2 in two dimensions.We then examine the low density regime numerically, using a semi-analytic microscopic theory. It is found that non-analytic exponents are visible only in a negligible density regime around the spinodal point. Estimates for the spinodal densities, and the range of critical fluctuations are provided.     

Sound Propagation in 3He and 4He Above the Liquid-Vapor Critical Point

A scaled plot is presented of published sound attenuation and dispersion data of ³ He and ⁴ He along the critical isochore above T_c as a function of the reduced frequency ^* . Here ^* / with the frequency and the order parameter relaxation rate, where the latter is determined from experimental transport and thermodynamic data. For a given isotope, the scaled data obtained at different frequencies lie on a single curve. These curves are different, however, for ³ He and ⁴ He. The resulting scaled plot is compared with recent predictions by Onuki, and by Folk and Moser. The crossover-term contribution in from the background thermal conductivity data is compared with the form derived by Bhattacharjee and Ferrell. In the Appendix, the calculation of for both ³ He and ⁴ He from experiments is described and discussed.     

Equation of state of3He near its liquid-vapor critical point

We report high-resolution measurements of the pressure coefficient (P/T) for³He in both the one-phase and two-phase regions close to the critical point. These include data on 40 isochores over the intervals–0.1t+0.1 and–0.2+0.2, wheret=(T–T _c )/T _c and =(– _c )/ _c . We have determined the discontinuity (P/T) of (P/T) between the one-phase and the two-phase regions along the coexistence curve as a function of . The asymptotic behavior of (1/) (P/T) versus near the critical point gives a power law with an exponent (+–1)^–1=1.39±0.02 for0.010.2 or–1×10 ^–2t–10 ^–6 , from which we deduce =1.14±0.01, using =0.361 determined from the shape of the coexistence curve. An analysis of the discontinuity (P/T) with a correction-to-scaling term gives =1.17±0.02. The quoted errors are fromstatistics alone. Furthermore, we combine our data with heat capacity results by Brown and Meyer to calculate (/T) _c as a function oft. In the two-phase region the slope (²/T ²)_c is different from that in the one-phase region. These findings are discussed in the light of the predictions from simple scaling and more refined theories and model calculations. For the isochores 0 we form a scaling plot to test whether the data follow simple scaling, which assumes antisymmetry of – ( _c ,t) as a function of on both sides of the critical isochore. We find that indeed this plot shows that the assumption of simple scaling holds reasonably well for our data over the ranget0.1. A fit of our data to the linear model approximation is obtained for0.10 andt0.02, giving a value of =1.16±0.02. Beyond this range, deviations between the fit and the data are greater than the experimental scatter. Finally we discuss the (P/T) data analysis for ⁴ He by Kierstead. A power law plot of (1/) P/T) versus belowT _c leads to =1.13±0.10. An analysis with a correction-to-scaling term gives =1.06±0.02. In contrast to ³ He, the slopes (²/T ²)_c above and belowT _c are only marginally different.Work supported by a grant from the National Science Foundation.     

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.     

Shear viscosity of liquid4He and3He-4He mixtures,especially near the superfluid transition

High-resolution measurements of are reported for liquid⁴He and³He-⁴He mixtures at saturated vapor pressures between 1.2 and 4.2 K with particular emphasis on the superfluid transition. Here is the mass density, the shear viscosity, and in the superfluid phase both and are the contributions from the normal component of the fluid ( _n and _n ). The experiments were performed with a torsional oscillator operating at 151 Hz. The mole fraction X of³He in the mixtures ranged from 0.03 to 0.65. New data for the total density and data for _n by various authors led to the calculation of . For⁴He, the results for are compared with published ones, both in the normal and superfluid phases, and also with predictions in the normal phase both over a broad range and close to T. The behavior of and of in mixtures if presented. The sloped/dT near T and its change at the superfluid transition are found to decrease with increasing³He concentration. Measurements at one temperature of versus pressure indicate a decreasing dependence of on molar volume asX(³He) increases. Comparison of at T, the minimum of _n in the superfluid phase and the temperature of this minimum is made with previous measurements. Thermal conductivity measurements in the mixtures, carried out simultaneously with those of , revealed no difference in the recorded superfluid transition, contrary to earlier work. In the appendices, we present data from new measurements of the total density for the same mixtures used in viscosity experiments. Furthermore, we discuss the data for _n determined for⁴He and for³He-⁴He mixtures, and which are used in the analysis of the data.     

NMR Studies on 3He-3He and 3He-4He tunneling motions in solid 3He-4He mixtures

Helium-3 nuclear spin relaxation times T ₁, T ₂, and T ₁have been measured for ³He-⁴He solid mixtures at the exchange plateau region (~0.5K). The ³He concentrations X ₃of the samples were 7.2, 2.9, 1.8, 1.4, 0.67, 0.65, and 0.22%, and their molar volumes varied between 19.9 and 20.9cm³/mole in hcp phase. The spectral density function J() for dipolar field fluctuations was determined in the low-frequency branch from T ₁measurements and in the high-frequency branch from conventional T ₁measurements. It was found that J() is given by J() = cJ()|_3–4 + (1–c)J()|_3–3, where J()|_3–4 is the spectral density function due to the ³He-⁴He tunneling motions, and J()|_3–3 is that due to the ³He-³He tunneling motions. Using the Torrey theory, the correlation frequency of the ³He-⁴He tunneling motions was evaluated from T ₁data, and was found to be in good agreement with Landesman 's theory.Supported in part by the Japan Society for the Promotion of Science through a grant to Y.H.     

Continuing Specific Heat Measurements of 3He in 3He - 4He Mixture Films

Additional data from our on-going experiments for the heat capacity of ³ He in ³ He- ⁴ He mixture films on a Nuclepore substrate are reported over the temperature range 90T165 mK, for ³ He coverages between 0.05 and 1.4 bulk-density atomic lagers, on a ⁴ He film of thickness 4.33 bulk-density atomic lagers. This is a two-dimensional Fermi liquid system, in which we can change the ³ He coverage and thus tune the Fermi temperature.     

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.     

Heat capacity of3He/4He mixtures

We measured the specific heat capacity of three dilute mixtures (2.6, 1.07, and 0.44%) of³He in⁴He and of pure⁴He. The⁴He contribution to the specific heat of the mixture is subtracted, leaving only the³He part. This is fitted to a theoretical expression over the whole temperature range from 10 to 700 mK. Assuming a dispersion relation of the form =² k ²(1+k ²)/2m*, the fits yield the value of and the effective mass m* of each mixture. The average value of is –0.076±0.01 A² and the effective mass in the limit of zero concentration is (2.23±0.02)m ₃. These are compared to values deduced from other measurements.     

Alkali Atoms attached to 3He Nanodroplets

No Heading We have experimentally studied the electronic 3p 3s excitation of Na atoms attached to ³ He droplets by means of laser-induced fluorescence as well as beam depletion spectroscopy. From the similarities of the spectra (width/shift of absorption lines) with these of Na on ⁴ He droplets, we conclude that sodium atoms reside in a dimple on the droplet surface and that superfluid-related effects are negligible. The experimental results are supported by Density Functional calculations at zero temperature, which confirm the surface location of Na, K and Rb atoms on ³ He droplets. In the case of Na, the calculated shift of the excitation spectra for the two isotopes is in good agreement with the experimental data.PACS 68.10.–m, 68.45.–v, 68.45.Gd     

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.     

The transverse acoustic impedance of dilute solutions of3He in superfluid4He

The transverse acoustic impedanceZ=R–iX of dilute solutions of³He in superfluid⁴He has been measured at a frequency (/2) of 20.5 MHz at temperaturesT from 30 mK to the transition at T. The³He concentrations studied werec=0.014, 0.031, 0.053, 0.060, and 0.092 below 1 K, thoughc decreased slightly near the point. The impedance was found from the temperature dependence of the quality factor and the resonant frequency of anAT-cut quartz crystal resonator immersed in the liquid. Below 1 K,Z is due to the Fermi gas of³He quasiparticles, and in the collisionless limit, 1 ( is a relaxation time),R remains constant whileX goes to zero. Measurements ofR(c, T) andX(c, T) were analyzed to determine the momentum accommodation coefficient (c, T) and (c, T). The relaxation times were in good agreement with previous work, while (c, T) was independent ofc, but increased from 0.29±0.03 below 0.1 K to 1.0±0.1 above 0.8 K. Various mechanisms are suggested to explain this. Between 1.0 and 1.5 K the³He quasiparticles and the thermally excited rotons are in the hydrodynamic region, 1. Values of the total viscosity (c, T) were obtained and analyzed to give the³He gas viscosity and the³He-³He and roton-³He scattering rates, both of which were energy-dependent. The superfluid healing length a was also measured. Near the point we founda=(0.1±0.03)^–2/3 nm, where =1–T/T, proportional to the phase coherence length . Our data are consistent with the hypothesis that _s/T is a universal constant for superfluid dilute solutions, where _s is the superfluid density. Between 1.0 and 1.8 K we found thata(c, T) was comparable to measurements in³He-⁴He films.     

The thermomechanical effect in normal liquid3He

A thermomechanical effect has been found in normal liquid³He. The effect appears only when the temperature gradient is established across liquid³He confined in small pores so that the³He quasiparticle scattering is boundary limited. This was achieved with a porous plug of packed 70 nm copper-oxide powder. The magnitude of P/gDT varies from 4 C at 2 mK to C at 20 mK, where P is pressure, T is temperature and C is the heat capacity per unit volume. The sign is positive, meaning that the³He moves from cold to hot. If viewed as an analogue of thermoelectricity, the magnitude of this thermomechanical effect is unsurprising. However, as shown in the following paper, it is an order of magnitude larger than a theoretical prediction for liquid³He.     

Diffusive relaxation processes in liquid3He-4He mixtures. I. Normal phase

When a heat flux is switched on across a fluid binary mixture, steady state conditions for the temperature and mass concentration gradients T and c are reached via a diffusive transient process described by a series of terms modes involving characteristic times _n . These are determined by static and transport properties of the mixture, and by the boundary conditions. We present a complete mathematical solution for the relaxation process in a binary normal liquid layer of heightd and infinite diameter, and discuss in particular the role of the parameterA=k _T ² (/c) _T,P /TC _P,c coupling the mass and thermal diffusion. Herek _T is the thermal diffusion ratio, (/c) _T,P ^–1 is the concentration susceptibility, is the chemical potential difference between the components, andC _P,c is the specific heat. We present examples of special situations found in relaxation experiments. WhenA is small, the observable times (T) and (c) for temperature and concentration equilibration are different, but they tend to the same value asA increases. We present experimental results on four examples of liquid helium of different³He mole fractionX, and discuss these results on the basis of the preceding analysis. In the simple case for pure³He (i.e., in the absence of mass diffusion) we find the observed (T) to be in good agreement with that calculated from the thermal diffusivity. For all the investigated³He-⁴He mixtures, we observe (c) and (T) to be different whenA is small, a situation occurring at high enough temperatures. AsA increases with decreasingT, they become equal, as predicted. For the mixtures with mole fractionsX(³He)=0.510 and 0.603, we derive the mass diffusionD from the analysis of (c) and demonstrate that it diverges strongly with an exponent of about 1/3 in the critical region near the superfluid transition. As the tricritical point (T _t,X _t) is approached for the mixtureX=X _t0.675,D tends to zero with an exponent of roughly 0.4. These results are consistent with predictions and also with theD derived from sound attenuation data. We discuss the difficulties of the analysis in the regime close toT andT _t, with special emphasis on the situation created by the onset of a superfluid film along the wall of the cell forX=0.603 and 0.675.Work supported by grants from the National Science Foundation and the Research Corporation and by an A. P. Sloan fellowship to one of the authors (RPB).     

Vortex-Loops and Phase Nucleation in Superfluid 4He and 3He

We propose a new model for the nature of the nucleation of solid from the superfluid phases of ⁴ He and ³ He. A fast solidification event in the superfluid results in a local release of pressure and a velocity field in the superfluid. This in turn facilitates the nucleation of vortex-loops. The kinetic energy gain of this process balances the surface tension, as the solid surface is quickly covered by many vortex-loops (hairy snow-ball). We show that this scenario gives good agreement with experiments on heterogeneous nucleation.     

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.