Solutions of3He In superfluid4He Research Articles

Dissipation mechanisms of tuning fork vibrations in superfluid 3He–4He solutions

The amplitude–frequency characteristics of tuning forks immersed in superfluid 3He–4He solutions were measured in the temperature range of 0.1–2.5 K. The tuning fork resonance frequency and the resonance width were measured as functions of temperature for 5% and 15% concentrations of 3He and, for comparison, for pure 4He. The experimental results for the key dissipation mechanisms, that is, viscous friction and first and second-sound emission of a tuning fork, were analyzed. For separation and evaluation of the contributions of viscous damping and first and second-sound waves, “closed” (in a factory capsule) and “open” (without a capsule) tuning forks were used. The resonance width, which is a measure of dissipation of tuning fork vibrations, was found to be higher in solutions than in pure 4He, and to increase with increasing 3He concentration. It was shown that the existing analytical expression for the contribution of viscous friction provides a good description of the experimental data for 4He only in the hydrodynamic region. For the solutions, the viscosity contribution is consistent with the experiment only at high temperatures (above 1.4 K). For the “open” tuning fork, the contribution of the first-sound is consistent with the calculation results only for 4He, whereas for the solutions, the calculated values are underestimated compared to the experiment. The possible contribution of the second-sound to the dissipation of the tuning fork vibrations in solutions was estimated using experiments with the “closed” tuning fork. This contribution was found to vary non-monotonically with a maximum at temperatures of ≈ 0.6–0.8 K. The ratio of the tuning fork energy loss due to the thermal diffusion wave versus the loss due to the radiation flux of the second-sound wave in superfluid solutions was calculated using literature data with an accuracy of 10–3–10–4.

Convective Turbulence in Superfluid Solutions 3He–4He

In present work, we continue our experimental investigations of heat instability in superfluid 3He–4He solutions heated from below. We research two solutions with 3He concentrations 5.0% and 9.5% for temperature of 270 mK. It is found that for 5% solution the dependence \(\nabla T(\dot{Q})\) is linear in temperature range studied whereas for the solution of 9.5% we observed the deviation from linear dependence above some critical value \(\dot{Q}_{c}\) . This effect manifests the thermal instability which appears under start of phase separation in 9.5% solution if heat flow is switched on. For 5.0% solution where one does not observe the phase separation at the values of \(\dot{Q}\) applied, the instability was not observed. To identify the possible mechanism of a thermal instability in stratified solution, we estimated the dependence of the Nusselt number on relative Raileigh number Ra/Ra c . One observes that the dependence can be fitted as Nu=(Ra/Ra c ) b where b=0.31±0.04. Note that the dependence obtained agrees rather good with the empiric expression of (Busse in Rep. Prog. Phys. 41:1929, 1978) and connecting the numbers Nu and Ra for turbulent convection. This gives grounds to conclude the heat transfer in a stratified solution is realized by transition to the regime of turbulent convection.

Study of Bulk Longitudinal Spin Relaxation in 3He-4He Liquid Mixtures

Longitudinal spin relaxation has been investigated in spin polarised liquid 3He-4He mixtures in cesium-coated glass cells. The contribution of wall relaxation is found to be either negligible or easy to separate, and bulk relaxation times can be derived from systematic studies. We report on relaxation measurements at saturated vapour pressure, for temperatures ranging from 1.1 to 3K. 3He densities vary from 0.07 to 11 mmol.cm−3 (3He concentrations between x3≃0.3 and 34%), so that both normal and superfluid 3He-4He solutions are studied. At fixed temperature, the longitudinal relaxation time is inversely proportional to the 3He number density over the whole range of investigated concentrations. In contrast, it is almost independent of temperature (changes do not exceed 10%). Comparison with published data and expected behaviours is discussed.

Acoustic properties of supersaturated superfluid 3He 4He solutions

The first sound velocity in metastable long-lived superfluid 3He 4He mixtures are measured in the temperature range 30–200 mK and the pressure interval 0–20 atm. The metastable state is reached by continuously increasing the 3He concentration in situ due to changing the osmotic and fountain pressure. The experimental data on the relative sound velocity show a linear dependence for all pressures and can be described by scattering of acoustic phonons by 3He quasiparticles in the solution. The experimental data show no essential anomaly under transition from saturated to supersaturated state.

Optical generation of the first and second sounds in superfluid 3He–4He solutions

An optical method of generation of the first and second sounds in superfluid 3He–4He solutions is proposed. The method is based on illumination by amplitude-modulated laser beam with wavelength corresponding to a region of absorption. Numerical calculations of pressure and temperature as a function of time are carried out. It was found that the presence of 3He isotopes in the solution provides strong coupling between these parameters. As a result, the waves of the first sound involve oscillations of the temperature, while the waves of the second sound involve oscillations of the pressure.

Supersaturation and Nucleation in Superfluid 3He–4He Solutions under Elevated Pressures

Nucleation and phase separation kinetics of superfluid 3 He– 4 He solutions at various pressures and concentrations were investigated. The measurements of the first sound velocity and the attenuation as well as the liquid dielectric constant determination were used. The study was performed at a constant temperature and pressure and the phase transition was initiated by continuous increasing the 3 He concentration. It is shown that the pressure dependence of the attainable supersaturations is non-monotonic when passing through a maximum at pressure of about 0.1 Bar. The comparison has discovered great differences between the results obtained and theoretical predictions.

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.

Longitudinal spin relaxation in a dilute, degenerate Fermi gas at arbitrary polarization

We calculate the longitudinal magnetic relaxation timeT1−1 for a dilute, polarized Fermi gas with short range interactions at low temperatures. In the limit of small polarizationsT1−1 is proportional toT−2 in agreement with earlier predictions, while in the opposite caseT1−1 is strongly affected by polarization, and in the limit of large polarizations becomes proportional to the square of the energy separation between the Fermi surfaces for up and down spins. The possibility of measuringT1 in dilute solutions of3He in superfluid4He is investigated.

Nucleation and growth of the new phase in the supersaturated3He-4He superfluid solutions

Phase separation kinetics and growth of the new phase in supersaturated superfluid3He-4He solutions was studied using the method of continuous change of3He concentration in situ and developed theory. Simultaneous measurements of the first sound velocity and the dielectric constant were carried out during the change of concentration. The velocity and attenuation of the first sound and the dielectric constant in supersaturated metastable solutions were calculated on the assumption that the excess3He in the solution is in the form of both atoms and droplets of the new phase. It was shown that the sound velocity, unlike the dielectric constant, is very sensitive to the formation of droplets. Comparison of the theory and the experimental data allowed us to elucidate the growth dynamics of the nuclei of the new phase from the metastable supersaturated mixture.

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 (ω/2π) of 20.5 MHz at temperaturesT from 30 mK to the λ transition at Tλ. The3He concentrations studied werec=0.014, 0.031, 0.053, 0.060, and 0.092 below 1 K, thoughc decreased slightly near the λ 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, ωτ≫1 (τ 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 α(c, T) and τ(c, T). The relaxation times were in good agreement with previous work, while α(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, ωτ≪1. Values of the total viscosity η(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 λ point we founda=(0.1±0.03)e−2/3 nm, where e=1−T/Tλ, proportional to the phase coherence length ξ. Our data are consistent with the hypothesis that ξρs/T is a universal constant for superfluid dilute solutions, where ρ 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 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.

Brillouin scattering from superfluid 3He-4He solutions

We have carried out Brillouin scattering experiments in superfluid 3He-4He solutions at six molar concentrations (x = 0.050, 0.100, 0.135, 0.250, 0.398, and 0.606) for temperatures as low as 0.4 K. We have measured the relative scattered intensities from the three normal modes: first sound, second sound, and concentration fluctuations. We have obtained the velocity of high-frequency first sound (~600 MHz), the velocity of high-frequency second sound (~20–80 MHz), and, at low concentrations, rough indications of the damping of the modes. An exact calculation of the scattered intensities is presented and used together with our measured intensities to obtain values for three thermodynamic quantities: ϱ n /ϱ, (∂δ/∂x) P,T , and (∂δ/∂T) P,x .

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.

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 Lowdin 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.

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.

A kinetic theory of the dilute solutions of3He in superfluid4He. I. Thermodynamics and equations of motion

A semimicroscopic kinetic theory of the dilute solutions of3He in superfluid4He is presented. The theory considers only phonon and3He quasiparticle excitations and is therefore applicable at temperatures of up to about 0.6 K. The model underlying the theory utilizes a modified Landau-Pomeranchuck3He energy spectrum and a second-order momentum expansion for the effective3He quasiparticle interaction. In addition, it accounts for the effective phonon-3He quasiparticle interaction, to first order in the phonon momentum, by using a renormalized concentration-dependent sound velocity. The simplicity of the model enables the derivation of both a complete equilibrium theory and a complete set of equations of motion for the solutions. The resulting expressions for the thermodynamic properties and the macroscopic currents appearing in the equations of motion represent a useful parametrization of these quantities in terms of the parameters of the model. It is shown that the macroscopic currents can be written in a form which seems to have a simple physical interpretation. As expected, it is found that at local equilibrium the expressions for the currents reduce correctly to the respective phenomenological expressions.

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.