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.     

Spin diffusion in 3He-4He solutions under pressure

We have measured the spin diffusion coefficient D for ³He-⁴He solutions of nominal ³He concentration x = 1.3% and 5.0% at pressures of ~0, 10 and 20 atm, and an x=8.5% solution at 4, 10 and 20 atm. The results are interpreted in terms of the Bardeen-Baym-Pines theory for ³He-⁴He solutions, and curves for the effective ³He-³He interaction V(k) are presented. We find no evidence for a minimum in V(k) at nonzero k. Weak coupling BCS pairing transition temperatures calculated from the V(k) lie in the range 10^–6–10^–7 K.Research supported in part by Research Corporation.     

Longitudinal and transverse spin diffusion in3He-4He solutions in a strong magnetic field

Using pulsed NMR techniques, we have measured spin diffusion in³He-³He solutions with³He concentrations of 0.05%, 0.1%, 0.46%, 1.0%, 3.8% and 6.4% in a magnetic field of 8.8 Tesla for a temperature range 11 mK T 200 mK. We observe that the temperature dependence of the transverse spin diffusion coefficient D₁ deviates from that expected for an unpolarized Fermi liquid in the degenerate region in the 1.0%, 3.8% and 6.4% solutions. Moreover, by measuring both longitudinal and transverse spin diffusion coefficients in the 6.4%-mixture, we have verified experimentally the difference between them, and provided direct evidence for a field-induced anisotropy in spin diffusion. The results from the 0.05% and 0.1% solutions show agreement with the theory of Jeon and Mullin; however, no deviation of D₁ from that expected in an unpolarized mixture was observed because the³He is not in the degenerate regime for these very dilute systems for the temperatures we could achieve. The analysis of our measurements in terms of the Leggett-Rice equations also yields values for the spin rotation parameter M₀. Using our results along with previous measurements at various³He concentrations, we deduce a value for the s-wave quasiparticle scattering length of a=–0.88 ± 0.05 Å.     

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.     

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.     

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).     

The coupled phonon spectrum in3He-4He solutions

A simple extension of a model previously used for³He-⁴ He solutions is presented, in which the effective vertex functions between³ He and⁴ He density fluctuations are assumed to depend only on the momentum transfer. The possible zero-concentration³ He quasiparticle spectra are divided into two classes, those that do and those that do not intersect the⁴ He phonon spectrum. The present model calculations of the coupled phonon spectrum demonstrate that as the temperature is decreased the behavior of the shifts in the phonon spectrum can be considered as a signature of the class of³ He quasiparticle spectrum used. Comparison with recent scattering experiments, however, indicates that the existing data are not sufficient to distinguish unambiguously between the two classes of³ He spectrum. It is suggested that a low-temperature neutron scattering experiment be performed to settle this question.     

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.     

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.     

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.     

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.     

NMR Studies on 3He-3He and 3He-4He tunneling motions in solid 3He-4He mixtures

Helium-3 nuclear spin relaxation times T ₁, T ₂, and T ₁have been measured for ³He-⁴He solid mixtures at the exchange plateau region (~0.5K). The ³He concentrations X ₃of the samples were 7.2, 2.9, 1.8, 1.4, 0.67, 0.65, and 0.22%, and their molar volumes varied between 19.9 and 20.9cm³/mole in hcp phase. The spectral density function J() for dipolar field fluctuations was determined in the low-frequency branch from T ₁measurements and in the high-frequency branch from conventional T ₁measurements. It was found that J() is given by J() = cJ()|_3–4 + (1–c)J()|_3–3, where J()|_3–4 is the spectral density function due to the ³He-⁴He tunneling motions, and J()|_3–3 is that due to the ³He-³He tunneling motions. Using the Torrey theory, the correlation frequency of the ³He-⁴He tunneling motions was evaluated from T ₁data, and was found to be in good agreement with Landesman 's theory.Supported in part by the Japan Society for the Promotion of Science through a grant to Y.H.     

Spin diffusion and magnetization studies of3He in3He-4He mixture films

The longitudinal spin diffusion coefficient and magnetization of a submonolayer coverage of ³ He, n ₃ =0.0066 Å^–2, in a thin adsorbed ⁴ He film has been measured as a function of temperature0.024 K T0.5 K and ⁴ He coverage0.21 Å ^–2 n ₄ 0.32 Å ^–2.     

The influence of spin on second sound in superfluid3He-4He mixtures

The second-sound velocity of a superfluid mixture increases when a static and homogeneous magnetic field is applied. In particular, for dilute mixtures (molar ³ He concentration <0.1) and low temperatures, (0.1<T<0.6 K) the relative velocity increase is independent of concentration and is given by u ₂/u ₂5×10^–16 B _e ² T ^–2. It should be easily observed wheneverB _e T ^–1is larger than 10 ⁵ G K^–1 and can lead to a novel measurement of the magnetic susceptibility.     

Magnetic susceptibility and collisionless spin waves in liquid3He and3He-4He mixtures

The transverse spin susceptibility for a Fermi liquid in a uniform magnetic field is calculated in the collisionless regime using the Landau theory of Fermi liquids. The singularities corresponding to the l=0 and l=1 spin waves are investigated in pure and dilute³He as a function of the wave vector q up to the threshold of the Landau damping.     

A pulsed NMR study of 3He-4He solutions

We report a study of the Leggett-Rice effect in dilute ³He-⁴He solutions at a Larmor frequency of 71 MHz. We have measured the height and phase of the spin echoes following 2-pulse NMR sequences of three solutions with ³He concentrations of 0.56%, 1.4%, and 4.0%. The theory of Leggett fits the data extremely well and yields values of the parameter M ₀ and the transverse diffusion coefficient D. We are also able to evaluate M ₀ and D by analyzing multiple spin echoes and find good consistency between the two methods. From values of /(1+F ₀ ^a ) at various concentrations, we are able to determine the scattering length for ³He quasiparticles, a, quite accurately. We find a = –0.97±0.03 Å.     

Splitting of the phonon spectrum in3He-4He solutions

The dynamic structure function for ⁴ He- ⁴ He density correlationsS ⁴⁴(k, ) is calculated as a function of the wave vectork and frequency for a simple model of a ³ He- ⁴ He solution at various temperatures and small ³ He molar concentrationx. The inputs to the model include the measured ⁴ He phonon spectrum and the zero-concentration ³ He quasiparticle energy (above thek=0 value), which is taken to be quadratic ink and to intersect the phonon spectrum atk _c near the roton minimum. Taking into account the decay of a phonon into a quasiparticle-hole pair, which is effected in the model by a quantum hydrodynamic interaction, we find that the phonon spectrum is split into two distinct branches. Atk _c the splitting of the phonon branches depends on a coupling constant and the crossover parameter (k _c )/k _c . If in ³ He- ⁴ He solutions is large enough and (k _c )/k _c is small enough, then the phonon spectrum is split into two branches.Work performed in part under the auspices of the U.S. Atomic Energy Commission.Resident Student Associate, Summer 1972, at Argonne National Laboratory, Argonne, Illinois.Supported in part by a Faculty Research Grant and Fellowship from the Horace H. Rackham School of Graduate Studies at the University of Michigan.     

3He-3He quasiparticle interaction in3He-4He mixtures under pressure

Based on our recent phase separation curve of³He-⁴He solution at elevated pressures, we propose new³He-³He quasiparticle interaction potentials, which reproduce the existing experimental results pretty well except for³He effective mass under pressure.