Rayleigh linewidth in4He near the gas-liquid critical point

Spectra of Rayleigh-scattered light in⁴He near the gas-liquid critical point have been measured, using a photon correlation method. Fitting the obtained relaxation times to Kawasaki's expression with background modification, we have obtained along the critical isochore the correlation length =(3.6±0.78)^{–0.534±0.046} Å and the high-frequency shear viscosity *=21.5±3.6 µP. * has been revealed to be in good agreement with the regular part of the viscosity _r=21±2µP. The singular part of the thermal diffusivity has been determined to beD _Ts =(4.9±2.7)×10^{–5–0.543±0.046} cm²/sec along the critical isochore.     

The decay rate of critical fluctuations in3He-4He mixtures near the gas-liquid critical point

We have measured the critical light scattering intensity and the Rayleigh line shape for³He and for³He-⁴He mixtures with compositionX(³He)=0.95, 0.79, and 0.63 along their respective critical isochores near the plait point. The experimental linewidth of³He is compared with the calculated one from heat conductivity and equation of state measurements, and satisfactory agreement is obtained. For mixtures, gravity effects in our cell of finite height prevent us from reaching the critical point along a path at strictly constant composition and density. HenceT _c cannot be determined directly. Using the prediction that the scattered light intensity in the mixtures has the same diverging behavior as for the pure fluid, we determine the reduced temperaturet[T – T _c(X)]/T _c from the intensity. The measured Rayleigh line shape can be expressed by a single decay rate as a function oft for a given scattering angle of the light beam. Our experiments show that in the mixtures is only weakly dependent on composition. Our analysis leads to the determination of the mass diffusion coefficientD, which is found to be nearly independent of composition and nearly equal to the thermal diffusivityD _T measured for³He. The results are discussed in the light of the predictions from mode coupling theory.     

Surface tension of liquid3He and4He near the liquid-gas critical point

The surface tension of liquid³He and⁴He was measured near the gas-liquid critical points in the reduced temperature range 3×10^–4–1, where t (T _c –T)/T _c . The critical exponents were found to be ₃=1.289±0.015 for³He and ₄=1.306±0.017 for⁴He. These values are very close to those for classical liquids, and are consistent with the value of 1.28 predicted by Widom, but are apparently different from the exponents previously obtained for liquid helium isotopes, which are near unity. The critical coefficients show good agreement with the quantum-corrected corresponding states theory for the Lennard-Jones 6–12 potential discussed by Young. The interface thickness is deduced from Widom's theory to bed=d ₀t^–v withd ₃₀=0.14±0.03nm and v₃=0.57±0.04 for³He, andd ₄₀=0.37±0.07 nm and v₄=0.58±0.01 for⁴He.     

Ultrasonic dispersion and attenuation near the liquid-gas critical point of4He

We describe sound velocity and acoustic attenuation measurements at several frequencies between 0.5 and 5 MHz along the critical isochore of ⁴ He immediately above the critical pointT _c. The dispersion is obtained with respect to the velocity measurements by Barmatz taken at 1.8 kHz along the critical isochore. The dispersion and attenuation results are analyzed following the theory of Kawasaki and using the more recent modifications resulting from the Fixman-Mistura approach as described by Garland and co-workers. We have determined the critical exponent for the correlation length divergence to be v=0.66±0.04 and found good internal consistency between certain constants appearing in the theoretical expressions for both sound dispersion and attenuation. These experimentally determined constants are compared with those calculated from other experiments. Deviations in the observed acoustic properties from the predicted ones are discussed. They are similar to those found in acoustic experiments and in light scattering experiments near the critical point of Xe.     

Diffusion in3He near its critical point

Pulsed NMR techniques were used to measure the spin diffusion coefficient in ³ He along eight isotherms between 3.0 and 4.2 K. Densities were also obtained simultaneously from the amplitude of the signal, assuming the magnetization obeys the Curie law. Absolute values of the density were obtained by normalizing to existing data. Although values of|T–T _c |/T _c and|– _c |/ _c of less than 10^–3 were achieved, no anomaly was observed. This is consistent with similar measurements on other substances. Our low-density results agree well with previous measurements and theory.Work supported in part by a National Science Foundation Department Science Development Grant, No. GU-2603, and the Ohio University Research Committee.     

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.     

Equation of state of a3He-4He mixture near its liquid-vapor critical point

Measurements of the pressure coefficient (P/T)_,x are reported for a ³ He- ⁴ He mixture with a mole fractionX=0.805 of ³ He in the neighborhood of the liquid-vapor critical point. These include data on 16 isochores taken over the density interval–0.50.5 and over the temperature range–0.1 t0.1, where =(– _c )/ _c andt=(T-T _c )/T _c ,with _c andT _c ,respectively, the critical density and temperature of the mixture. From the discontinuity of (P/T)_,x at the boundary between the two-phase and the one-phase regions we determine the dew-bubble curve nearT _c with better precision than was done in recentPVT experiments. From the extrapolation of data not approachingT _c closer than1 mK, (P/T)_,x along the critical isochore appears to be discontinuous atT _c ,while for the isochore / _c 0.92, (P/T)_,x is continuous across the dew curve. It is found that this latter isochore cuts the dew curve at its highest temperature. These observations are discussed in terms of general thermodynamic arguments and theoretical predictions of the asymptotic behavior. We calculate (P/T)_,x from the scaling equation of state proposed by Leung and Griffiths for ³ He- ⁴ He mixtures, using their numerical parameters. In spite of some systematic deviations, especially in the two-phase region, there is in general good agreement with experimental results. In particular, the shape of the measured dew-bubble curve and the apparent discontinuity of (P/T)_,x along the critical isochore show excellent agreement with theory.Work supported by a grant from the National Science Foundation. A report of this work has been presented at the Washington Meeting of the APS [Bull. Am. Phys. Soc. 20, 618 (1975)].     

Ultrasonic dispersion and attenuation near the liquid-gas critical point of 3He

We report ultrasonic dispersion and attenuation measurements near the liquid-gas critical point of ³He at frequencies from 0.5 to 5.0 MHz and densities from 0.89 _c to 1.15 _c . The singular part of the sound attenuation and the dispersion on the critical isochore _c = 0.0414 g/cm³ are analyzed in terms of the Kawasaki-Mistura theory. If the Ornstein-Zernike order parameter correlation function is assumed in the analysis, good agreement with our data is achieved, except close to the critical temperature T _cin the high-frequency region, where ^* = /_D 1. Here _D is the characteristic relaxation rate of the critical fluctuations. From a fit of the theory to our data, and assuming the inverse correlation length is expressed by = ₀, where = (T–T_c)/T_c with = 0.63, we obtain ₀ = (3.9 ± 0.4) × 10⁹ m^–1. It is found that a more realistic form of the correlation function, as proposed by Fisher and Langer and calculated by Bray, yields even poorer agreement with out data than does the classical Ornstein-Zernike form for ^* > 10. The same difficulties appear in the analysis of the available data for xenon. Thus, the present mode coupling theory is unable to satisfactorily describe the acoustic experiments on fluids near the liquid-vapor critical point over a large range of reduced frequencies ^*. In the appendix, we reanalyze previously reported ultrasonic data in ⁴He, taking into account the nonsingular term of the thermal conductivity. Using = 0.63, we obtain a good fit of the experiment to the theory in the hydrodynamic region with ₀ = (5.5 ± 1) × 10⁹ m^–1.Supported by a grant from the National Science Foundation.     

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.     

Transport properties of helium near the liquid-vapor critical point. IV. The shear viscosity of3He and4He

Shear viscosity measurements with a precision of 0.05% are reported for³He and⁴He along near-critical isochores 0.85 c <1.12, where _c is the critical density. The temperature range was –10^–4<<1, where =(T – T _c)/T _c is the reduced temperature. The experiments were carried out with a torsional oscillator operating at 158 Hz, driven at resonance in a phase-locked loop. The absolute value of the viscosity was obtained by calibration at the superfluid transition of⁴He, based on published values and from direct calculations using the free decay time constant of the oscillations. The data are analyzed in terms of a model using the recent mode-coupling (MC) expressions by Olchowy and Sengers, and where account is taken of the earth's gravity effects. The theory could be fitted very well to the experiment with a single free parameter, the cutoff wave numberq _D, which was found to be 3.0×10⁶ and 7.0×10⁶ cm^–1 for³He and⁴He, respectively. We have used for the critical exponent the MC predicted value of z=0.054, which permits a fit superior to that using z=0.064 predicted by dynamic renormalization group (DRG) theories. Detailed comparisons are made between the model calculations and data for various isochores and isotherms and good agreement is obtained. The effects of gravity are described in some detail. The predicted frequency effect in viscosity measurements is calculated for³He and is shown to be obscured by gravity effects. Using the Olchowy-Sengers formulas, we have also fitted the MC theory to the critical thermal conductivity data of³He, again withq _D as the only free parameter. This fit gaveq _D=6 × 10⁷ cm^–1, which in the ideal situation should have been the same asq _D from viscosity. We also discuss a representation of the³He viscosity data along the critical isochore by a power law and first correction-to-scaling erm. Using the viscosity and the critical conductivity data for³He, we have calculated the dynamic amplitude ratio and obtained =1.05±0.10, in agreement with predictions from MC and DRG theories. Also, agrees with data of classical fluids. Finally, a comparison is made of recent shear viscosity data for CO₂ by Bruschi and Torzo with those on He. The CO₂ data are also analyzed in terms of the MC theory, and the discrepancies are discussed. In the Appendices, we present the results of new compressibility measurements on³He along the critical isochore, as used in the MC analysis. We also present a brief analysis of the fluid hydrodynamics in the torsional oscillator leading to relations for the viscosity as a function of the measured quantities. Finally, we give a short outline of the vertical density profile calculations from the earth's gravity field for the calculations of the viscosity nearT _c.     

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.     

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.     

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).     

Superfluid density near the lambda point of4He under pressure

The superfluid density in ⁴ He was determined near T from the second-sound velocity as a function of T–T and pressure. The critical exponent of the superfluid density was found to depend, even slightly, on the pressure. Furthermore, the fundamental length ₀ in the coherence length = ₀ [1–(T/T)]^–' seemed to be proportional to the mean interatomic distance. The implications of the results are also discussed.This work was partly supported by The Ito Science Foundation and by The Nishina Memorial Foundation.     

On critical first-sound absorption in liquid4He near the λ-transition

The mode-mode coupling expression for the critical first-sound damping parameterD ₁ is renormalized using scaling and consistency arguments so as to avoid factorization of fluctuations and the uncertainty about the static susceptibility of the order parameter. Our functionD(,T –T ) shows the high-frequency behavior obtained by Kawasaki, supporting the idea of frequency scaling, and it provides a quantitative estimate in reasonable agreement with experiment.     

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.     

Dynamic scaling and ultrasonic attenuation in 4He near the superfluid transition point

Attenuation of first sound has been measured in ⁴He under saturated vapor pressure near the lambda temperature T at frequencies /2 ranging from 10.2 to 271 MHz. The frequency dependence of the critical part of the attenuation is determined and the dynamic scaling hypothesis is examined. Above the lambda point, it is found that the critical attenuation is described by a scaling function (, ) = ^1+y F(), where = ₀^–x and = T/T – 1, with the results x = 1.02±0.05 and y = 0.33±0.03. The characteristic frequency of the order-parameter fluctuation with the wave number k equal to the inverse correlation length is then proportional to ^x , which is in an excellent agreement with the prediction of dynamic scaling. Below the lambda point, a characteristic relaxation time or times shorter than previously expected at lower frequencies appears to exist in the present frequency range.Based on a Ph.D dissertation submitted by K. Tozaki to the University of Tokyo (1977).     

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.