Ultrasonic propagation in3He-4He mixtures near the tricritical point

We present measurements of the singular sound attenuation _sing in liquid mixtures of³He and⁴He near the superfluid transition and at temperatures above the phase separation curve. The mole fractionX of³He ranged from 0.55 to 0.73 and the frequency /2 varied between 1 and 45 MHz. The temperature range was 0.75–1.4 K, with the greatest emphasis on the tricritical region nearX _t=0.67 andT _t=0.87 K. From the change in slopedU/dT of the sound velocityU, we present a new determination of the phase separation curve, which is compared with previous measurements. The sound attenuation peak along the superfluid transition becomes broader in temperature asT is decreased. In addition, there is also an increase in sound attenuation as the phase separation temperatureT is approached. ForX<X _t these two peaks merge into one asX _t is approached. For a given frequency, the attenuation has a maximum value at the tricritical point. Estimates of the contribution _D of mass diffusion to the attenuation for³He-⁴He mixtures with 0<X<0.55 and comparison with experimental values show that _D becomes relatively more important asX increases, and that atX=0.55 it effectively accounts for all of the observed singular attenuation, at least at megahertz frequencies. Hence we assume that for mixtures withX>0.55 the observed attenuation can be analyzed solely in terms of the diffusive relaxation mechanism. The mass diffusion parameterD is then determined from the data. AtX=0.55,D diverges asT is approached, which is consistent with theoretical expectations and experimental results. NearT _t, there is a crossover to a tricritical regime, and it is found that approximatelyD(T–T _t) ^Z withZ=0.32±0.1. Mode coupling predictions are thatZ=1/2 while recent renormalization group calculations giveZ=1/3. The attenuation curves in the tricritical region at the various frequencies can be represented satisfactorily but not perfectly by a scaling function with a characteristic relaxation time (T–T _t) ^–x withx=1.7±0.15. This time corresponds to order-parameter fluctuations. Its temperature dependence is in excellent agreement with renormalization-group calculations that givex5/3, while expectations from dynamic scaling are thatx=3/2. Our analysis also gives the variation of the amplitudes of andD with the direction of approach toT _t. A comprehensive theory for interpreting all the data, in the normal as well as in the superfluid phase, is lacking at this time.Supported by a grant from the National Science Foundation. A preliminary account of this work was presented inBull. Am. Phys. Soc. 21, 229 (1976).     

Interfacial tension near the tricritical point in3He-4He mixtures

The interfacial tension ₁ of phase-separated³He-⁴He mixtures has been measured between 0.5 K and the tricritical point. The observed decrease of _i on approaching the tricritical point is much stronger than for ordinary critical points. Our data are consistent with a critical exponent =2.     

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.     

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.     

Phase diagram and concentration susceptibility of 3He-4He mixtures near the tricritical point

The inverse concentration susceptibility (/\x) _T of ³He-⁴He mixtures has been calculated from high-resolution vapor-pressure measurements and in situ measurements of the dielectric constant very close to the tricritical point. The measurements have been fitted to a scaling-law equation of state. In the normal fluid a logarithmic correction term was necessary to obtain a good fit. In the superfluid a regular correction term, proportional to (x – x _t)², was evaluated. New measurements of the phase-separation curve confirm our earlier measurements on the ³He-rich side and extend the measurements much closer to the tricritical point on the ⁴He-rich side. The tricritical point is at temperature T = 0.8669 ± 0.0005 K and mole fraction x = 0.6716 ± 0.0014.Work performed under the auspices of the U. S. Department of Energy.     

Thermodynamic properties of liquid3He-4He mixtures near the tricritical point. I. Vapor pressure measurements and their thermodynamic analysis

Sensitive vapor pressure (P _sat) measurements of ³ He- ⁴ He mixtures by means of a low-temperature strain gauge are described over the temperature range 0.5–1.5 K and the range 0.4<X<0.85, whereX is the ³ He mole fraction in the liquid phase. The vapor pressure cell is flat, with a height of only 2 mm, in order to reduce concentration gradients near the tricritical point. The pressure-sensitive device, which resolves changes of about 5×10 ^–8 atm, is described, and its advantages over a conventional manometer system are discussed. Data taken successively on mixtures of small mole fraction difference are used to locate the phase separation boundary in theT-X plane and also the lambda line from a change in (P _sat/T) _x at these transitions. The limiting slopes (dT/dX) and (dT/dX) of the phase separation curve and the lambda line in the vicinity of their junction point, the tricritical point, are presented and compared with previous work. From the vapor pressure data, the concentration susceptibility (X/) _T,P was obtained. Here = ₃ – ₄ is the chemical potential difference of the respective isotopic components ³ He and ⁴ He. It is shown that (X/) _t diverges as the tricritical point is approached along various paths in theT-X plane, and the relevant tricritical exponents are presented. The weak divergence of (X/) _T along the lambda line predicted from the postulates of Griffiths and Wheeler could not be detected and it is believed that such divergence has to occur in a temperature interval that is far too small to be resolved with present-day techniques. Furthermore, gravity effects might well prevent observation of the weak divergence. The lambda transition is well evidenced by a distinct shoulder in a plot of (X/) _T at constantX as a function ofT. This shoulder becomes smaller and gradually gets topped by a peak asX decreases from the tricritical mole fractionX _t . From a combination of vapor pressure and calorimetric data the chemical potential difference [(X, T)- _t ] is calculated between 0.78 and 1.22 K. Here _t is the value at the tricritical point. From this tabulation the critical line and its slope (d/dT) are obtained and compared with previous values based on calorimetric experiments only and with calculations based on the excess chemical potentials ₃ ^E and ₄ ^E derived from saturated vapor pressure data.Work supported by a Grant from the National Science Foundation.     

Dynamics near the tricritical point of a3He-4He mixture

By means of mode-mode coupling we compute damping constants near the tricritical point of a ³ He- ⁴ He mixture. Both on the coexistence curve and in the tricritical region we find : mass diffusion constant≈|?T _t | ^+1/2 , thermal diffusion constantDK _T ～|T?T _t | ^?1/2 , and third viscosity～|T?T _t | ^?3/2 , whereT is the temperature andT _t is the tricritical temperature. These results imply that damping constant of second sound～|T?T _t |S| ^?1/2 in the ordered phase.     

Interfacial tension near the tricritical point of3He-4He solutions

In a recent paper we made a calculation of the interfacial tension in a binary liquid solution. This involved consideration of the effects of composition gradient, and involved a characteristic length z giving the distance up or down the gradient over which a molecule was influenced by its neighbors. An equation was derived which related to z, the coexistence curve, properties of the critical point, and the thickness z of the interface. This equation has now been applied to³He-⁴He solutions, and its relation to scaling laws for the tricritical exponents has been discussed. This requires estimates of effective values of the tricritical exponents in the temperature range actually used from measurements in the literature of and other quantities, and from an estimate of z from the de Broglie wavelength, z has been estimated to be about 80Å at0.022 deg below the critical temperature. This has been compared to an estimate of the correlation length.Supported by the National Science Foundation.     

Thermal concentration gradients and the thermal diffusion ratio in3He-4He mixtures near the tricritical point

We report a study of transient effects in sound velocity measurements in several³He-⁴He mixtures near the tricritical point, from which by extrapolation we derive the equilibrium thermal concentration gradient and the thermal diffusion ratiok _T . In the superfluid phase, the thermal concentration gradient is found to be proportional to the concentration susceptibility.Research supported by a grant from the National Science Foundation.     

Thermodynamic properties of liquid3He-4He mixtures near the tricritical point. II. Data analysis by the scaling-field method

The scaling theory for tricritical phenomena by Riedel is applied to the analysis of thermodynamic properties of liquid³He-⁴He mixtures near the tricritical point. Within this theory experimental data for the phase diagram, the³He molar concentrationX, and the concentration susceptibility (X/) _T are discussed in terms of two scaling fields that are functions of the temperatureT and the difference = ₃ – ₄ of the chemical potentials of the two helium isotopes. The quantitiesX and (X/) _T in terms of thefields T and as independent variables are obtained for the intervals –0.1<T – T _t<0.53 K and –9< – _t <0.8 J/mole, from vapor pressure and calorimetric data described in a previous paper by Goellner, Behringer, and Meyer. The transformed data are analyzed to yield the tricritical exponents, amplitudes, scaling fields, and scaling functions. The values of the tricritical exponents are found to agree with those predicted by the renormalization-group theory of Riedel and Wegner. (Logarithmic corrections are beyond the precision of the present experiment.) Relations between amplitudes are derived and tested experimentally. The (linear) scaling fields are determined by using their relationship to geometrical features of the phase diagram. The data forX and (X/) _T are found to scale in terms of these generalized scaling variables. The sizes of the tricritical scaling regions in the normal and superfluid phases are estimated; the range of apparent tricritical scaling is found to be appreciably larger in the normal-fluid phase than in the superfluid phase. The tricritical scaling function for the concentration susceptibility is compared with the analogous scaling function for the compressibility of pure³He near thecritical gas—liquid phase transition. Finally, when the critical line near the tricritical point is approached along a path of constant < _t , the experimental data are found to exhibit the onset of the crossover from tricritical to critical behavior in qualitative agreement with crossover scaling.Work supported in part by the National Science Foundation through Grant No. GH-36882 and Grant No. GH-32007, and by a grant of the Army Research Office (Durham).     

A new method for measuring the osmotic response function of3He-4He mixtures near the tricritical point

A new method of measuring the osmotic response function in³He-⁴He mixtures near the tricritical point is described. The method has been used to measure the tricritical divergence of this quantity in the normal phase. The values of the tricritical exponent and amplitude are in fair agreement with those obtained by other methods.Boursier du CNRS Libanais     

Transport phenomena in He3-He4 mixtures near the tricritical point

We have made extensive measurements on transport phenomena in He³-He⁴ mixtures near the tricritical point and along the superfluid transition line at saturated vapor pressure. The He³ mole fraction X ranged from 0.51 to 0.72 and the temperature from 0.8 to 1.5 K. Our measurements were made under steady state conditions using a cell where we measured the vertical He³ concentration gradient X induced by a temperature gradient T produced by a vertical heat flux. The cell included two superposed capacitors and X was determined by means of the dielectric constant method. In this paper, we present a comprehensive report on our results for the thermal diffusion ratio k _T and the thermal conductivity both in the normal fluid and in the superfluid. In the tricritical region, k _T was found to diverge strongly as the tricritical point was approached; no singularity in was found. This behavior is consistent with theoretical predictions. In the region near the lambda line, remains finite, as expected, but k _T appears to have a stronger singularity than predicted by theory. The analysis of our experiment in the normal fluid for mixtures with X>0.51 was complicated by superfluid film flow along the walls of the sample cell. We describe this effect and analyze it with Khalatnikov's theory of superfluidity. However, for the mixture X = 0.51, where there is no such film, the behavior of k _T is consistent with predictions. The k _T data for the mixtures 0.6 < X < 0.7 could be cast into a tricritical scaling representation, similar to that for the concentration susceptibility. In the superfluid phase we test, for the first time and with fair success, a relation by Khalatnikov between X/T and static properties, measured in different experiments. Finally, we discuss the relaxation times that characterize the establishment of steady state conditions. From these data it is possible, under favorable circumstances, to obtain the mass diffusivity D.Supported by grant DMR77-20827 from the National Science Foundation.     

Critical and tricritical dynamics in3He-4He mixtures

A new formalism is presented to study the critical and tricritical dynamics of³He-⁴He mixtures near the superfluid transition for arbitrary concentration. In this fluid two conserved variables, the molar concentrationX and the entropys, are twofold coupled to a complex order parameter ψ first in the dynamic equations reversibly and second in the free energy dissipatively. However, at an intermediate concentrationX=X _D (which is 0.37 at the saturated vapor pressure) a linear combination ofX ands is found to be asymptotically decoupled from ψ both reversibly and dissipatively. There, dynamic renormalization group equations reduce to those of pure⁴He (or those of the F model) and some dynamic properties are common to those of pure⁴He. For example, atX≈X _D , the gradient ?(T?T _λ ) under heat flow goes to zero without³He mass flow, whereT _λ is the critical temperature, dependent onX, and the thermodiffusion ratio loses the singularity with the critical exponent α. Our dynamic renormalization group equations take into account the above two nonlinear couplings and can be used for any concentrations. Furthermore, using a linear response scheme, general relations are obtained among the kinetic coefficients. As a result the thermal conductivity on the λ line is found to be exactly proportional toX ^?1 at smallX. The coefficient in front ofX ^?1 can be expressed in terms of the diffusion constant of an isolated³He molecule in⁴He.     

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 .     

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 equilibration of fluids near the liquid-vapor critical point:3He and3He-4He mixtures

We study the temperature-equilibration process of fluids at constant volume in a thermal conductivity cell, where an initial temperature gradient relaxes to zero. The calculation is performed in the linear approximation for a pure fluid and a binary mixture. Near the critical point of the pure fluid, the adiabatic heating process, which takes place at constant volumeV, causes equilibration to proceed four times faster whenC _P /C _V 1 than for the process at constant pressureP. For the mixtures, the relaxation rate enhancement at constantV compared with constantP is restricted to a temperature region where the coupling between temperature and mass diffusion is small. The predictions are compared with experimental results for³He and for two³He-⁴He mixtures along their critical isochores. Finally, we discuss the thermal relaxation in the two-phase (liquid-gas) and one-phase (gas) regimes at the critical density, as measured with a conductivity and a calorimetry cell. The contrasting behavior for³He and a³He-⁴He mixture in these two regimes and under these different constraints is pointed out and discussed.     

Transport properties of helium near the liquid-vapor critical point. V. The shear viscosity of3He-4He mixtures

A torsional oscillator operating at 158 Hz has been used to measure the shear viscosity for two mixtures,³He_0.65-⁴He_0.35 and³He_0.80–⁴He_0.20. Data for each mixture are reported along near-critical isochores 0.85</ _c <1.18, where _c is the critical density. Just as in a previous paper on pure³He and⁴He, the observed viscosity includes the sum of background and critical contributions and the effect from the earth's gravity nearT _c . The analysis of the data provides (1) the background viscosity versus density and temperatureT, (2) the viscosity ratio / along the various isochores and also along selected isotherms obtained by interpolation, showing the critical contribution, and (3) the shape of the dew-bubble curve for each mixture, as obtained from the maxima in or from the discontinuities in the sloped/dT at the transition from the two-phase to the single-phase regime. A comparison between the data of these mixtures and those for pure fluids is presented. The background viscosity data along isotherms and along isochores are found to vary smoothly from³He to⁴He. The critical viscosity ratio [ / ]( _c ) for mixtures along the critical isochore is comparable with that for the pure fluids, i.e., it diverges weakly asT _c is approached. Hence, just as for previously investigated critical transport properties in³He-⁴He mixtures, there is no significant difference in behavior from that in the pure fluids.     

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.).     

Nucleation of transitions in liquid 3He-4He mixtures

We present a complete set of data describing the nucleation both of superfluidity and phase separation at container walls in liquid ³He-⁴He mixtures. The appearance of superfluidity and the growth of phase-separated films are measured locally at the walls. A theoretical interpretation is given, important results of which are that the phase separation and superfluid transitions become uncoupled when the transitions are localized near the walls and that quantum fluctuations most probably strongly affect the nature of the superfluid nucleation transition at low temperature. Further, comparison of our results with data for transitions in pure ⁴He films demonstrates that a universality principle governs the transitions in both systems.Partly supported by NSF Grant No. DMR 75-21886.Laboratoire associé au CNRS.