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.     

Viscosity of saturated3He-4He mixture below 200 mK

The shear viscosity of saturated³He-⁴He mixture has been measured at temperatures between 7 mK and 200 mK using a vibrating-wire viscometer and a calibrated pressure cell. The reliability of the vibrating-wire technique was tested by measuring the viscosity of pure⁴He. The results are internally consistent. A phenomenological expression is given for the viscosity of saturated³He-⁴He mixture between 7 mK and 80 mK.     

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.     

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.     

NMR characterization of phase separation in highly polarized3He-4He solutions above 450 mK

Laser optical pumping techniques have been successfully applied to concentrated³He-⁴He gaseous mixtures. Using a³He cryostat, these mixtures were subsequently liquefied and studied down to 450 mK, using pulsed NMR techniques.Observation of highly nuclear polarized solutions and of phase separation (demixion) in the mixtures is reported. Measurements of magnetic relaxation times and the dynamics of coexisting phases in this system are discussed.Unité de Recherche de l'Ecole Normale Supérieure et de l'Université Pierre et Marie Curie, associée au C.N.R.S. (URA18)     

Transport properties of spin polarized3He-4He mixtures

We have calculated the viscosity, thermal conductivity, and longitudinal spin diffusion coefficient in³He-⁴He mixtures which are spin polarized. The calculation applies to all temperature regimes. We have also calculated the Onsager cross coefficient which arises because of the coupling between heat and longitudinal spin currents. The interaction between³He quasiparticles is taken to be a constant as a first approximation. We have also investigated the changes brought about by allowing the interaction to vary with the momentum of the quasiparticle.     

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.     

First viscosity of dilute3He-4He mixtures below 0.6 K

Starting with the Boltzmann transport equation, the first viscosity of dilute³He-⁴He mixtures for various³He concentrations x is evaluated up to around T 0.6 K by including the contribution from three-phonon processes (3PP) in the anomalous elementary excitation spectrum of liquid⁴He. Due to 3PP, the characteristic time for³He viscosity at high temperatures, i.e., T2T_F where T_F is the³He Fermi temperature, is evaluated as 5 × 10^–12/xT, which is smaller than the value estimated by Rosenbaum et al. This is interpolated with in the degenerate (quantum) region, TT_F. The obtained viscosities are in better agreement with experimental results than those of Baym and Saam, whose theory does not include 3PP. However, at very low concentrations there exists a discrepancy between the present theory and experiments, so that an alternate treatment should be considered.     

The velocity of second sound in dilute3He-4He solutions below 0.6°K

The velocity of second soundu ₂ in dilute solutions of³He in⁴He has been measured by a pulse, time-of-flight technique using a new type of mechanical transducer. Tables of measured values ofu ₂ as a function of temperature between 0.03 and 0.6° K are presented for³He concentrations in the range 0.14–6.3% at the saturated vapor pressure and under hydrostatic pressures of 10 and 20 atm. Estimates of the upper limit of the intrinsic attenuation of second sound have also been made and these are consistent with theory provided that the experimental values of the phonon thermal conductivity lifetime are used in place of the values calculated by Baym and Ebner.Work supported by a grant from the National Science Foundation.On leave of absence from University of Nottingham, U.K.     

The spin diffusion coefficient of3He in3He-4He solutions

The transport properties of³He in³He-⁴He solutions with molar concentrations of 5, 9, 14, and 24% have been studied for 0.9 KT2.5 K. The spin diffusion coefficientD _s and the longitudinal relaxation timeT ₁ were measured by the spin-echo method for temperatures both above and below the solution lambda temperatureT . The spin-echo method measures the diffusion coefficient for magnetizationD _s, which differs from the usual diffusion coefficient for particlesD belowT .D _s depends on the³He-³He scattering cross section _FF and the³He-roton/phonon cross section _FB, whileD depends only on _FB. The distinction betweenD _s andD is elaborated in terms of a simple mutual-friction model for diffusion. The two scattering mechanisms are clearly evident in the behavior ofD _s as a function of concentrationx and temperature. The contribution due to the³He-³He scattering is inversely proportional tox, indicating that the³He can be treated in first approximation as a classical gas (the Pomeranchuk model). The predictions of various theoretical models are compared with the results, where possible, but most of the previous theoretical work is not applicable to the concentration range and temperatures of these measurements.Supported in part by the National Science Foundation and the U.S. Office of Naval Research.     

Anisotropic spin diffusion and multiple spin echoes in3He-4He solutions

We propose a ₁-t₁-₂ pulse-NMR experiment to detect the spin-diffusion anisotropy, =D-D, in degenerate spin-polarized³He-⁴He mixtures, where D and dare the transverse and longitudinal spin diffusion coefficients. In such an experiment the nonlinearity of the dynamics produces multiple spin echoes (MSE). At the ³He concentration x₃ 4% the spinrotation parameter vanishes (M 0), so that the nonlinearity of the equations of motion is entirely due to the anisotropy. In this situation, detection of MSE amounts to observation of D. For slight anisotropy, i.e. D/D 0.25, we use a perturbation scheme similiar to that developed by Einzel et al. (in that case, for small M and small demagnetizing field) to calculate the second and third echo heights. For larger anisotropy we numerically calculate the echo heights. We find that for D/D = 0.5 the heights are 2 % of the first echo, and should be detectable. The (₁, ₂) tip-angle dependence of the D echoes is different from that of the M and demagnetization echoes, and furthermore, they occur at right angles to these echoes (in spin space). Thus, even when small spin-rotation and demagnetization effects are present, the ₁-t₁-₂ experiment provides a sensitive means of detecting the anisotropy.     

The effect of a strong magnetic field on the damping of second sound in dilute3He-4He mixtures below 100 mK

We have measured the increase in damping of second sound in 0.1% and 0.05%³He-⁴He solutions upon application of a magnetic field of 10 T. The increase is caused by changes in the thermal conductivity and viscosity of the mixture as the system becomes spin polarized. We show how to fit the data for the viscosity in zero-field to obtain an estimate of the interaction between the³He quasiparticles. It is then possible to predict the viscosity in a field of 10 T using the calculation of Hampson et al. The data agree with the theory to within 2%.     

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.     

Enhancedp-wave pairing in polarized3He-4He mixtures

The potential model of Bardeen, Baym, and Pines is generalized to describe polarized³He-⁴He-mixtures. The zero-field strength and range of the potential as a function of³He-density and pressure are determined by transport data and, for the first time, the magnetostriction. It is found that at higher pressures the magnetostriction leads to a³He-density-dependence in the strength of the potential, an effect which only negligibly appears when fitting the transport data. The model predicts ap-wave superfluid transition with a transition temperatureT _c ^p which is highest at the maximum solubility. Including the correction for the polarization-dependence of the maximum solubility, a large increase inT _c ^p with polarization is found. The size of the enhancement depends on the zero fieldT _c ^p which in turn depends on which set of normal state data we fit.     

Mass and spin diffusion near the λ line in dilute3He-4He mixtures

We examine the possibility of using NMR and other measurements on very weak solutions of ³ He in liquid ⁴ He to investigate the superfluid phase transition. It is found that even for these very weak solutions the mass (D _m) and spin (D _s) diffusion coefficients associated with the ³ He behave in radically different ways:D _m is predicted to diverge asT approaches T from above as (T–T) ^–1/3 , while the behavior ofD _s depends on the type of experiment, and for the ordinary spin-echo type has no particular singularity.Based on a D.Phil. thesis submitted to the University of Sussex by M. A. Eggington.     

NMR studies of highly polarized liquid3He-4He mixtures

An efficient technique for maintaining large³He nuclear spin polarizations in liquid³He-⁴He mixtures has been developed and used to polarize samples contained in narrow tubes. Results obtained in a U-shaped cell and a coil-shaped cell are compared. A systematic study of the effects of dipolar fields within the sample on the characteristics of the observed long-lived magnetostatic modes is in progress. NMR signals obtained in the coil-shaped cell reveal phase separation and the existence of a highly polarized³He-rich region in the sample.Laboratoire de L'Ecole Normale Supérieure et de L'Université Pierre et Marie Curie, associé au CNRS (URA 18)     

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.     

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.