in3He-4He Mixtures Research Articles

Isaak Markovich Khalatnikov. 17 October 1919—9 January 2021

Isaak Khalatnikov made major contributions to two different branches of theoretical physics: quantum liquids, particularly liquid helium, and general relativity and cosmology. He made outstanding contributions to our theoretical understanding of superfluidity in liquid 4 He and in 3 He– 4 He mixtures. He developed the formulation of the two-fluid hydrodynamic equations, including the proper description of very complex dissipative and nonlinear effects. He also developed the idea, originally due to Landau, that the normal fluid can be described in terms of a gas of weakly-interacting excitations. His work on general relativity and cosmology began with the paper written with E. M. Lifshitz (ForMemRS 1982) in 1963. They extended Lifshitz's pioneering paper of 1946, showing how small departures from homogeneity and isotropy in the early Universe would have grown as the Universe expanded. They also tackled the problem of whether or not the equations of general relativity predict that space-time contains singularities, either in the past or in the future. They initially concluded that more general solutions would not contain singularities, but later work with Belinskiĭ not only showed that general solutions could contain singularities, but also indicated how the Universe would behave near a singularity. Later, Khalatnikov showed that nearly all cosmological solutions with a massive scalar field have a period of exponential or inflationary expansion. He became the founding director of the Landau Institute for Theoretical Physics of the USSR Academy of Sciences in 1964. He brought together a powerful group of theoreticians, creating a scientific centre that has played a major role in the development of theoretical physics.

PULSED NONLINEAR SOUND IN SUPERFLUID 4HE AND 4HE-3HE MIXTURES

Sound modes in 4 He and 4 He- 3 He mixtures which arise out of the two-fluid equations are made up up a vector convective flow and scalar temperature changes. A method for modeling nonlinear pulses of sound with geometric approximations to the vector and scalar components has recently been applied towards understanding nonlinear second sound near the lambda point. 6 The same method may be used, in general, for modeling linear and nonlinear sounds in Helium II. We demonstrate with a model for nonlinear second sound pulses in 3 He- 4 He mixtures and compare the results to experimental observations.

Wetting Phenomena in 3 He- 4 He Mixtures on Weak and Superweak Substrates

We explore wetting phenomena in3He-4He mixtures adsorbed on the newly discovered superweak oxidized alkali metal substrates. An approximate description of the relationship between the substrate work function and the substrate adsorbate potential well depth D is given. Within a slab model we derive the wetting phase diagram at liquid-vapor coexistence. On oxidized cesium at T = 0 we obtain the values of D for which triple point dewetting vanishes and for which pure3He ceases to wet. Reduction of D leads to the disappearance of wetting at the substrate-3He interface in an interesting scenario violating the Cahn wetting theorem.

High nuclear polarization in3He and3He-4He gas mixtures by optical pumping with a laser diode

New diode lasers delivering 50 mW output power at 1083 nm are shown to be efficient sources for optical pumping of helium. They can polarize nuclei in a3He gas up toM = 50% over the pressure range 0.4–1.6 torr. Larger nuclear polarizationsM of3He nuclei, of order 80%, can be obtained in3He-4He mixtures when the laser frequency is tuned to a4He line. A standard optical measurement of nuclear polarizationM has been extended to the case of3He-4He mixtures. The effect of various parameters on the steady-state polarizationM and on the pumping timeTp is discussed.

Universality versus nonuniversality of critical transport properties in liquid mixtures

The critical behavior near consolute points and plait points in mixtures and along lines connecting such points in the phase diagram belongs to the universality class of gas-liquid transitions in pure liquids. We give a survey of the results for the temperature dependence of transport coefficients, thermal conductivity, mass diffusion, and thermal-diffusion ratio, in mixtures within a non-asymptotic renormalization-group theory of critical dynamics. The observable critical behavior in some cases is nonuniversal and may be strongly concentration dependent. This is explained by different crossover temperatures in the singular Onsager coefficient of the order parameter and in the hydrodynamic transport coefficients. At the plait point the value of (ie1363-01) determines the crossover to the asymptotic behavior in the transport coefficients, and its smallness explains the situation in3He-4He mixtures. We also consider ionic solutions, where long-range forces may be present. The dynamical universality class in this case is different from that of mixtures with short-range interaction. As well as the “classical” static behavior for sufficient long-range interaction potentials, the dynamical critical behavior depends on the exponent of the power law for the spatial decrease in this interaction. This offers an additional possibility to determine this exponent by measuring the temperature dependence of the hydrodynamic transport coefficients.

Experiments to search for a substrate state for3he adjacent to surfaces in3He-4He mixtures

We describe three experiments designed to search for the bound state for3He adjacent to a substrate which has been predicted by Treiner and co-workers. In each case a bulk3He-4He solution is in contact with the substrate of interest. For a Nuclepore substrate, NMR techniques are used to establish that there is no 2D behavior for the3He in the cell; thus, no substrate state is present. For a cesium or hydrogen substrate the technique involves the measurement of the velocity of ordinary sound in the bulk solution as a function of temperature. The cesium and hydrogen experiments are incomplete.

Compression of3He by refluxing4He: A model for computing HEVAC effects in3He-4He mixtures

This article comprises the first part of a study concerning the effect that a flow of gaseous4He has on the concentration distribution of3He atoms in the presence of a super fluid film. We present a simple model in which hydrodynamic effects in the gas phase (diffusion and viscosity), and local thermodynamic equilibrium with a superfluid film are considered. Results are derived and discussed for the simple case in which a heat flow is sustained along a cylindrical tube lined with a helium film. This heat flux drives a superfluid flow in the film, and a corresponding counterflow of4He in the vapour. The pressure, temperature,3He concentration, and film thickness profiles along the tube are computed. Over a wide range of conditions, a dramatic reduction of the3He concentration, a large temperature increase, and a spectacular film thinning towards hotter regions are predicted to result from a heat flow. The results of a series of experiments supporting this model will be presented in a forthcoming article.

Zero-temperature attenuation and transverse spin dynamics in fermi liquids. III. Low spin polarizations

This is the third in a series of papers on microscopic theory of transverse dynamics in spin-polarized Fermi liquids. In the lowest order in polarization our exact general theory of Ref. 1 reproduces the conventional Landau-Silin-Leggett theory of Fermi liquids. The next term in polarization expansion contains a zero-temperature attenuation with the magnitude that does not depend on polarization. This attenuation results in a finite relaxation time in transverse spin dynamics at zero temperature and is responsible for anomalous temperature behavior of spin diffusion. The zero-temperature attenuation is determined by two angular harmonics of the derivatives of the mass operators and the irreducible vertex in off-shell directions, and cannot be expressed via standard Fermi liquid harmonics. At high polarizations, the parameters of transverse spin dynamics are calculated as an expansion in polarization. The expansion involves complex values of interaction function and energy on a set of isoenergetic surfaces with the radii between the Fermi momenta for up and down spins,p↓ andp↑. The results explain recent experimental data on spin diffusion in spin-polarized liquid3He↑ and3He↑-4He mixtures. The comparison with experimental data indicates that the superfluid transition temperature for3He in3He-4He mixtures may be much lower than the current estimates.

Multiple spin echoes in spin polarized Fermi liquids

Multiple spin echoes (MSEs) in3He and in3He-4He mixtures have been analyzed using a theoretical technique proposed by Einzel et al. In both systems MSEs are generated by Fermi liquid interactions, but for3He, MSEs are also generated at higher temperatures by the dipolar demagnetizing field. The theory is compared with the experimental results for3He and with our measurements for3He-4He mixtures. We have improved on the fit to the second echo heights for3He given by Einzel et al. by taking account of both the Leggett-Rice effect and the dipolar interaction in the calculation. Good fits were also found for the first and second echo heights for the3He-4He data. The third echo height is larger than expected if we assume that the total magnetization remains uniform in space. We have developed the theory for the case when this approximation is not made, and we find good agreement for the first five echoes.

Superleak transducers of second sound in3He-4He mixtures

In a recent second-sound experiment in a dilute {sup 3}H-{sup 4}He mixture, Church et al. were able to understand most of their data with the theory by d'Humieres, Launay, and Libchaber. However, significant discrepancies exist at lower frequencies. The authors show that diffusive currents across the superleak are responsible for this deviation. What is more, they are able to determine the diffusion coefficient D from the low-frequency data. The equations governing the performance of oscillating superleak transducers and Peshkov transducers are derived in a systematic manner. This pertains to pure He-II as well as to mixtures. Mistakes in earlier works on dissipative effects of superleak transducers are corrected. Explicit formulas are given for the amplitude and linewidth of second sound, and the experimentally relevant parameters are 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,3He0.65-4He0.35 and3He0.80−4He0.20. Data for each mixture are reported along near-critical isochores 0.85<ρ/ρ c <1.18, where ρ c is the critical density. Just as in a previous paper on pure3He and4He, the observed viscosity $$\tilde \eta$$ 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 $$\bar \eta$$ versus density ρ and temperatureT, (2) the viscosity ratio $$\tilde \eta$$ / $$\bar \eta$$ 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 η or from the discontinuities in the slopedη/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 [ $$\tilde \eta$$ / $$\bar \eta$$ ](ρ 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.

Analysis of spin wave resonances in3He-4He mixtures

Spin waves have been reported previously in3He-4He mixtures using cw-NMR. The spin waves form standing waves in the rf coil used to detect the NMR signal. Only one resonance was observed, corresponding to ak vector of 1 mm−1. This is in sharp contrast to similar experiments performed on pure3He, where several resonances were seen. An explanation is given of why this occurs and the resonant mode is identified. Furthermore, it is explained why there is an extra width to the resonance for temperatures below 2 mK. This is caused by radiation damping of the resonant mode. The theory is found to be in excellent agreement with the data. Further experiments are suggested, including details of an improved rf coil design, which should lead to information about the quasiparticle interaction.

Sound attenuation and3He-3He interaction in3He-4He liquid mixtures at saturated vapor pressure

The interaction potential between3He quasiparticles in3He-4He liquid mixtures is determined from the sound attenuation at saturated vapor pressure. Sound attenuation was measured in mixtures with3He mole fraction ranging from 0.0289 to 0.0573. The superfluid transition temperature of3He in mixtures and other properties were then estimated from the deduced interaction potential.

Thermal resistance between phonon and quasiparticle excitations in dilute mixtures of3He in4He

As part of a Kapitza resistance study, the thermal resistance between the phonon and3He quasiparticle excitations in three dilute3He-4He mixtures (0.03%, 0.1%, and 0.3%3He) has been measured over the temperature range 20–50 mK. The results are compared with a theory based upon the Baym-Ebner interaction between the excitations. The temperature dependences agree, but the measured resistance is smaller than the theoretical resistance by a factor of ∼2 that varies inversely with concentration. This concentration-dependent phonon-quasiparticle coupling discrepancy is consistent with the results of recent heat pulse absorption experiments by Husson et al. An indirect measurement was made of the same thermal resistance in3He-4He mixtures confined within the pores of a submicrometer sintered copper powder, with the results confirming a size-effect model for the resistance.

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. HenceTc 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 − Tc(X)]/Tc from the intensity. The measured Rayleigh line shape can be expressed by a single decay rate Γ as a function oft for a given scattering angle of the light beam. Our experiments show that Γ 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 diffusivityDT measured for3He. The results are discussed in the light of the predictions from mode coupling theory.

Transport properties in3He-4He mixtures near the superfluid transition

Transport properties in3He-4He mixtures near the superfluid transition

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.

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.

Interfacial tension near the tricritical point in3He-4He mixtures

The interfacial tension σ1 of phase-separated3He-4He mixtures has been measured between 0.5 K and the tricritical point. The observed decrease of σ i on approaching the tricritical point is much stronger than for ordinary critical points. Our data are consistent with a critical exponent μ=2.

Ultrasonic propagation in3He-4He mixtures near the tricritical point

We present measurements of the singular sound attenuation αsing 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 ω/2π 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 λ is decreased. In addition, there is also an increase in sound attenuation as the phase separation temperatureT σ 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 α D of mass diffusion to the attenuation for3He-4He mixtures with 0<X<0.55 and comparison with experimental values show that α 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 λ 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 τ∝(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 givex≈5/3, while expectations from dynamic scaling are thatx=3/2. Our analysis also gives the variation of the amplitudes of τ 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.