Theory of dilute solutions of3He in superfluid4He. I. Perturbation theory in a nonorthogonal basis

A new procedure is presented which enables the development of perturbative expansions for the energies and transition amplitudes in systems described by nonorthogonal basis functions. The procedure is based upon defining a new metric matrix, that is, by redefining the quantum-mechanical scalar product. The results justify certain previous applications of the Schmidt orthogonalization method. It is shown that the same results can be obtained using Löwdin orthogonalization. While the procedure was developed for application with the correlated basis function (CBF) approach to dilute³He-⁴He solutions, it can be applied in any nonorthogonality situation.Supported in part by the Office of Naval Research and in part by a Quinnipiac College Faculty Research Grant-in-Aid.A portion of this work has been submitted in a thesis by R.H.K. in partial fulfillment of the requirements of the Ph.D. degree to the Graduate School, University of Massachusetts, Amherst.     

Second-viscosity phenomena in dilute solutions of3He in superfluid4He

A theory of second-viscosity phenomena in dilute solutions of³He in superfluid⁴He is presented. The theory considers only phonon and³He quasiparticle excitations and is therefore valid at temperatures below about 0.6 K. It is shown, by an exact calculation, that within the framework of the Landau-Pomeranchuck model for the³He quasiparticle excitation energy, the four second-viscosity coefficients are related to one another and that only one of them is actually an independent kinetic coefficient. The relations between the second-viscosity coefficients are applied to analyze the expressions for the dissipative function and the first- and the second-sound attenuation coefficients. It is shown that the second-viscosity contribution to the second-sound attenuation is smaller by an order of magnitude than its contribution to the first-sound attenuation.     

An equilibrium theory of the dilute solutions of3He in superfluid4He at very low temperatures

An equilibrium theory of the dilute solutions of ³ He in superfluid ⁴ He is derived systematically. The theory is based on a model which (a) goes beyond the parabolic Landau-Pomeranchuk approximation for the ³ He quasiparticle energy by taking into account the fourth-order term in the momentum expansion of this quantity, (b) disregards contributions to the ³ He quasiparticle effective interaction whose order in the momentum is higher than two, and (c) allows the effective interaction to be nonlocal. The simplicity of the model enables the development of a unified parametrization of the various equilibrium properties of the solutions. The expressions obtained for these properties are both easy to apply and highly accurate over a wide temperature range spanning from T=0 to temperatures of the order of the ³ He quasiparticle degeneracy temperature. It is shown that the parameters appearing in the expression for the ³ He quasiparticle effective interaction at fixed ⁴ He number density are replaced in the fixed-pressure, low-temperature expansions of the equilibrium properties by other parameters whose appearance in the theory seems to be due to the renormalization of this interaction by the interactions between the ³ He quasiparticles and the virtual fluctuations of the ⁴ He number density Finally, a comparison is made between theory and experiment. Three quantities are considered in detail : the ³ He osmotic pressure and the ³ He quasiparticle inertial and specific heat effective masses. The analysis of the experimental data makes it possible to determine the parameters associated with the effective interaction at several pressures. It is found that the theory is, in general, in a very good accord with the experimental situation and that, within its framework, the experimental values of the osmotic pressure and the two effective masses are indeed consistent with one another.     

Time-dependent thermal convection in dilute solutions of 3He in superfluid 4He

Time-dependent thermal convection can occur in a unity-aspect-ratio Rayleigh-Beard convection cell containing a dilute solution of ³He in superfluid ⁴He when the fluid is heated from above. Results are presented primarily for a 0.24 mole % He solution at 0.925 K. Means is provided for introducing heat at the top and separately for a central plug and an outer ring such that both are at a constant temperature gDT above the bottom. A critical temperature difference T _cfor convection can be defined above which both steady and time-dependent convection occur. The time-dependent effects include a region of T. near T _cand characterized only by excessive noise, a region of somewhat higher T where there are intermittent major changes in the plug heating rate with a time distribution like that for random events, and a region at still higher T where periodic but nonsinusoidal variation of the heat flow is observed. When a long enough time, several months, has elapsed after cooling down the apparatus, time-dependent states no longer occur, and the heat flow above T _cis limited to steady convection. Briefly raising the temperature of the apparatus to 77 K is sufficient to restore the possibility of time-dependent states.     

Thermodynamics of the liquid-solid interface in dilute solutions of3He in4He

We show that the (p, T, x) phase diagram of⁴He close to melting pressure (25.3 bars) with small concentrations of³He and at very low temperatures has several unanticipated and novel features. For pressures between 25.3 bars and a triple point pressurep*, estimated to be 25.8 bars, we find a dilute liquid solution of³He in equilibrium with solid⁴He. The concentration of the liquidx _c increases from zero to the dilute liquid solubility limitx ₀ as the pressure increases from the pure⁴He melting pressure atT=0 top*. We explore the possibility of self-cooling by lowering the pressure throughp*. We also consider the effects of a strong magnetic field, and show that it lowersx _c. An estimate of the kinetic growth coefficient is given. Finally, we discuss the possible adsorption of³He on the melting front and the consequent faceting enhancement of the solid⁴He.     

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.     

Heat transport in dilute mixtures of3He in superfluid4He

We report new measurements of the effective thermal conductivity K_eff and relaxation time τ in dilute mixtures of³He in superfluid⁴He, with molar concentrationsX≤10⁻³. The temperature range extended fromT≈1.4 K toT _λ. Both K_cff and τ are found to agree with theoretical predictions, in contrast to previous experiments where significant differences were observed. A new thermal conductivity cell design was used which almost completely eliminates extraneous volumes and surfaces, and the earlier results are explained in relation to these design changes.     

λ Transition of dilute3He in4He solutions

Based on an empirically extended Landau roton formula and on the fact that³He atoms participate in the movement of the normal component of liquid⁴He, the variation of the temperature with³He concentration is obtained at the saturated vapor pressure. The results are in good agreement with experiment. The present approach can also be applied to higher pressures.     

Positive-ion mobility in dilute solutions of 4He in 3He

The mobility of positive ions in essentially pure ³He and in solutions with 100, 300, 1000, 3000, and 10,000 ppm of ⁴He has been studied at pressures of 60 Torr, 10, 20, and 25.6 bar. For temperatures below about 100 mK the pureliquid mobility data are consistent with a ln (T ₀/T) behavior except at the very lowest temperatures, where field-dependent effects may cause departures; both phenomena are in agreement with recent calculations of Bowley based on the Josephson-Lekner formalism. In the more concentrated solutions an increase in the ion radius (corresponding to a decrease in the mobility) is observed starting below ~200mK; this is interpreted as a polarization-potentialinduced phase separation resulting in a liquid ⁴He-rich halo about the solid ion core. At still lower temperatures a vanishing ⁴He content causes the solution data to rather abruptly rejoin the pure-liquid data. At still higher ⁴He concentrations additional very low-mobility species are observed which are likely ions with large metastable halos produced by a large ⁴He concentration at the field-emission tip where the ions are created.Work supported by the U.S. Energy Research and Development Administration and the National Science Foundation under Grants DMR-74-12186 and DMR-76-21370.     

The velocities of first and second sound in dilute solutions of3He in superfluid4He at very low temperatures

A unified theory of first and second sound in dilute solutions of ³ He in superfluid ⁴ He at very low temperatures is presented. The theory is based on the detailed semimicroscopic model for the ³ He quasiparticle excitations described by Disatnik and Brucker a few years ago. In contrast with Khalatnikov's macroscopic theory, the application of this model enables the derivation of relatively simple expressions for the sound velocities in which no omissions of terms representing contributions due to the thermal expansion are made. The sound velocities are given in the final expressions in terms of various parameters of the ³ He quasiparticle spectrum and effective interaction. These expressions are both highly accurate and easy to use over a wide temperature regime spanning from the quantum limit to temperatures of the order of the ³ He quasiparticle degeneracy temperature. The actual application of the theory to measurements of the sound velocities is described in detail. Numerical values or estimates for various characteristic parameters of the ³ He quasiparticle system, including in particular the ³ He quasiparticle effective mass, are obtained from the comparison between the theory and the experiment. The example of the second sound velocity is used to illustrate a procedure for analyzing data from very low-temperature measurements of the equilibrium properties of the solutions, which is expected to produce meaningful information regarding the parameters of the basic model. In practice, the theory is found to be in a very good accord with the measurements of the sound velocities. The result obtained for the zero concentration limit of the ³ He quasiparticle effective mass (m ₀ =2.19m ₃ ) is somewhat lower than the empirical estimates reported in the past. On the other hand, this result is in very good agreement with variational calculations based on the detailed microscopic theory of the solutions. A discussion of this and other results obtained from the comparison between theory and experiment is included.Work supported in part by the United States-Israel Binational Science Foundation.Killed in action, 16 October 1973.     

Temperature,pressure, and concentration dependence of the thermal conductivity of very dilute solutions of3He in superfluid4He

The thermal conductivity of³He–⁴He solutions at saturated vapor pressure has been measured for the concentration range 0.011–1.3 mole %³He and the temperature range 84–650 mK. Measurements were made at 10 and 24 atm for several of the concentrations. The thermal conductivity of solutions at 24 atm does not differ greatly from the thermal conductivity of pure⁴He at this pressure. Qualitative agreement with the Baym and Ebner theory is achieved only if the boundary scattering of phonons is treated in a different manner than suggested by them.     

Density of dilute 3He-4He solutions

The number density of dilute ³He-⁴He solutions under saturated vapor pressure is calculated within the framework of quantum hydrodynamics. For temperatures T small compared to the ³He Fermi temperature T _F, results are given for ³He concentrations large enough to manifest ³He quasiparticle interactions. The nondegenerate regime (TT _F) is also considered in the limit of small ³He concentration. Relevance to experimental determination of quasiparticle energies and interactions is emphasized.Supported by the National Science Foundation under Grant No. GP-40831.     

Relaxation times and mass diffusion in superfluid dilute3He-4He mixtures

Relaxation times are reported from the transients observed during thermal conductivity _eff and thermal diffusionk _T ^* measurements in superfluid mixtures of³He in⁴He with a layer thickness of 1.81 mm. The experiments extend from 1.7 K toT and over a³He concentration range 10^–6X<5×10^–2. The agreement between the measured and the predicted from the two-fluid thermohydrodynamic equations is satisfactory forX>10^–3 but deteriorates for smaller³He concentrations. This situation is similar to that for _eff andk _T ^* results and indicates that the transport properties in very dilute mixtures with layers of finite thickness are not well understood. ForX>10^–3, the mass diffusion coefficientD _iso for isolated³He in⁴He has been determined from and from _eff measurements. There is an inconsistency by a constant numerical factor between these determinations. This problem might be related to the observations that in the superfluid phase, the relaxation times for different cell heightsh do not scale withh ². FromD _iso derived via the _eff data, the³He impurity-roton scattering cross section is determined. Comparisons with previous work are made.     

Hydrodynamic heat flow in very dilute superfluid3He-4He mixtures

The superfluid hydrodynamics of heat flow is examined for very small mass concentrationsc of³He in⁴He in an effort to better understand recent results for the effective heat conductivity _eff, which appear to be in conflict with predictions. The full hydrodynamics contains a thermal boundary layer; within this layer the temperature and concentration gradients differ from those in the bulk fluid. An examination of finite heating effects based on the ansatz _eff c ^–p for smallc shows distinctly different behavior for experimental determinations of _eff whenp<1,p=1, andp>1. Thus, finite heating can be used as a probe to evaluate the exponentp.     

NMR measurements on dilute solutions of3He in solid4He

Measurements of the spin diffusion coefficientD and relaxation timesT ₁ andT ₂ are reported for³ He impurity in solid⁴ He of fractional concentrationx ₃ down to 10^–4. Forx ₃10^–3,T0.5 K, and molar volume 21.0 cm³,Dx ₃=1.2±0.4×10^–11 cm² sec^–2,T ₂ x ₃=1.7×10^–4 sec, andT ₁ x ₃=6.6×10^–2 sec at 2 MHz. The spectral densityJ() for dipole field fluctuations has been studied as a function of frequency from 10⁴ to about 10⁷ Hz. Two branches of approximately equal area have been found with scaling frequencies of about 10⁴ and 10⁶ Hz. It is argued that the high-frequency branch, which has some unusual structure, relates to ³ He- ⁴ He tunnelling, uninhibited by other ³ He impurities, and occurs at a rate of about 0.6 MHz for molar volume 21 cm³. The molar volume dependence of this tunneling appears to be similar to that found in pure, solid ³ He.     

Nucleation and phase separation in superfluid3He-4He solutions

The kinetics of the phase separation in³He-⁴He superfluid solutions is studied from anomaly in the first sound velocity and the dielectric constant. The histogram of experiments on solution supersaturation is obtained and used to study of phase separation probability. It was shown that above 50 mK the obtained temperature dependence of the supersaturation is qualitatively in accord with the thermal activation mechanism of nucleation. Below 50 mK the data may point towards a transition from classical to quantum mechanism of nucleation.     

Quasithermodynamic theory of He3 and He4 solutions

The conventional theory of nonideal solutions is modified on the basis of Gorter's hypothesis, and various properties of He³ and He⁴ solutions are investigated. The variation of -temperature with He³ concentration is explained adequately right up to the phase separation region. A comparison between the theoretical and experimental results for excess chemical potentials, excess entropy, and excess enthalpy is then carried out. In view of the recognized complex behavior of the solution in the region 1 to 2° K, the observed satisfactory agreement between theory and experiment may be regarded as a strong support for the assumptions made in our work.