Anomalous temperature dependence ofD andT 2 for dilute solutions of3He in solid4He

Results of measurements of the spin diffusion coefficientD and NMR relaxation timesT ₁,T ₂, and T_1p are presented for a range of fractional³He concentrations 1 × 10^–4x ₃2.5 × 10^–3 in solid⁴He at molar volumes 19.85V _m21.0 cm³ and temperatures 0.4<T<2 K in both hcp and bcc phases. We observe a minimumD(T) atx ₃=5×10^–4, which is interpreted in terms of a transition from coherent impuriton motion to thermally activated diffusion. ForT<0.8 K, (lnD)/(lnV _m)=60±8. TheT ₂ measurements show a minimum as a function of temperature forx ₃10^–3. TheT ₂ (T) andT ₁ (T) results yield values for activation energy and tunneling frequency of vacancies in these dilute solutions. Forx ₃=5×10^–4 andT 0.5 K,T ₂ (V _m) is anomalous.T ₁ measurements at the same concentration indicate there is an important contribution to the spectral density of dipole field fluctuations in the kHz region.Financial support provided for apparatus, materials, and a research studentship (ARA) by the Science Research Council.     

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.     

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.     

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.     

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.     

Spin diffusion in dilute,polarized 3He-4He solutions

Spin dynamics for arbitrarily polarized and very dilute solutions of ³He in liquid ⁴He are described. We began at a very fundamental level by deriving a kinetic equation for arbitrarily polarized dilute quantum systems based on a method due to Boercker and Dufty. This approach allows more controlled approximations than our previous derivation based on the Kadanoff-Baym technique. Our previous work is here generalized to include T-matrix interactions rather than the Born approximation. Spin hydrodynamic equations are derived. The general equations are valid for both Fermi and Bose systems. By use of a well-known phenomenological potential to describe the ³He-³He T-matrix we calculate longitudinal and transverse spin diffusion coefficients D and D _¦ and the identical-particle spin-rotation parameter . We confirm that these two diffusion constants differ at low T with D approaching a constant as T 0, and D_¦~1/T². Estimates of errors made by our approximations are considered in detail. Good agreement is found in comparison with data from both Cornell University and the University of Massachusetts. We find that the s-wave approximation is inadequate and that mean-field corrections are important. Comparison is also made between theory and the recent UMass viscosity measurements.     

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.     

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.     

Dielectric model of roton interactions in dilute solutions of 3He in 4He

The analogy between the theory of superfluidity and the theory of dielectrics can be extended to account for some of the properties of rotons in dilute solutions of ³He in superfluid ⁴He. These include the normal fluid density ratio and the shifts in energy of roton excitations relative to those of pure ⁴He.This research is supported in part by National Science Foundation grant DMR 76-21814 and by the Air Force Office of Scientific Research under grant AFOSR 76-2880A.     

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.     

Surface tension and mass density of dilute solutions of4He in3He

We report the first precise measurement on the surface tension of dilute solutions of⁴He in³He from 2.6 K to the phase separation temperature as well as the mass density of the solutions. Compared to liquid³He, the surface tension of dilute⁴He solutions increases by 3.5 ± 0.1 mdyne/cm for 2.5%⁴He and 7.4 ± 0.2 mdyne/cm for 5.2%⁴He at 1 K. The increase of the surface tension means a negative adsorption, i.e.,⁴He atoms are excluded from the free surface. The surface adsorption, which is calculated on the basis of the⁴He quasiparticle model, was compared with the data. Assuming that the effective potential near the surface for the⁴He quasiparticle is long-range, the temperature dependence of the surface adsorption is well explained.     

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 attenuation of second sound in dilute3He-4He solutions below 1 K

The attenuation of second sound in dilute solutions of³He in⁴He has been measured from the width of standing wave resonances. Results are presented here for³He molar concentrations of 1.3% and 5% in the temperature range 50 mK to 1 K at pressures of 0 and 10 atm. The results are compared with the theory of Saam, Baym, and Ebner and suggest that theoretical values of the phonon thermal conductivity lifetime are too great. Comparison is also made with recent measurements of the second-sound damping constant in very dilute solutions (X0.5%); our result shows no sign of the anomalous peak in the damping constant observed for such concentrations. The shear viscosity of the dilute solutions is also computed and found to be in agreement with the capillary flow measurements of Kuenhold, Crum, and Sarwinski.This work was supported in part by a grant from the Research Corporation.     

Nonlinear effects on spin waves in3He-4He dilute solutions

Spin dynamics of³He-⁴He dilute solution have been studied in the collisionless regime by NMR methods. In cw NMR experiments, we observed spin waves in a magnetic field gradient. The frequency difference between the first mode and the others obeyed a two-thirds power law in the strength of the field gradient. By using a pulsed NMR method, nonlinear spin dynamics were studied. We observed the large amplitude spin waves after an rf pulse. The frequency of each spin-wave mode depended on the tipping angle. By using the parameter determined in the present experiment, the numerical calculated results based on Leggett-Rice theory are in good agreement with the results of the pulsed NMR experiments.     

The nuclear magnetic susceptibility of dilute solutions of3He in4He at low temperatures

A theory of the nuclear magnetic susceptibility of dilute solutions of ³ He in ⁴ He at low temperatures is presented. It is shown that ^–1 is a linear function of the corresponding inverse free fermion magnetic susceptibility. It is also shown that the experimental Curie constant differs from the theoretical one by an amount which is of the order of the ³ He concentration. It follows that the existing experimental results are not adequate for the calculation of the Fermi liquid magnetic susceptibility enhancement factor.     

Theory of heat transfer between phonons and quasiparticles in dilute solutions of3He in4He

A calculation of the heat transfer rate between phonons and quasiparticles in solutions of³He in⁴He is described. The phonon-quasiparticle interaction of Baym and Ebner is used, and two processes contributing to the heat transfer are considered. One process is phonon absorption and emission with the excess momentum distributed among the quasiparticles by quasiparticle-quasiparticle interactions. The other is quasielastic scattering of the phonons by quasiparticles. Numerical results are presented for 0.1% and 0.5% solutions. The calculated heat transfer rate for the 0.1% solution is in agreement with the one reported experiment atT40 mK, but is too small at higher temperatures (50T60 mK).     

A pumping experiment in dilute3He-4He solutions at millikelvin temperatures

Very dilute³He-⁴He solutions provide a model system for the study of the effects of magnetic polarisation. It was proposed by the Lancaster group¹ that a pumping effect of³He should be observed under appropriate conditions in these solutions. In a strong magnetic field gradient, the³He should be pumped from the low field region towards the high field region, in which the³He is polarised. We have now observed this effect in 0.1% and 0.06% solutions, using a vibrating wire resonator in the low field region as the concentration indicator. The concentration change of a few percent under our present experimental conditions (9T maximum field, 10 mK) is of the expected magnitude so that development of the technique can now be considered.     

Characterization of spin polarized dilute3He-4He solutions below 500 mK

We have prepared a large sample of spin polarized liquid ³ He- ⁴ He at 500 mK. Polarization is obtained by laser optical pumping, and studied by pulsed NMR. We have observed a nuclear magnetization lifetime T ₁ longer than 1 hour, and an anomalously long, non linear decay of tranverse magnetization.Unité de recherche de l'Ecole Normale Supérieure et de l'Université Pierre et Marie Curie, associé au CNRS (URA 18).