Transverse surface impedance of pair-correlated Fermi liquids. Application to normal and superfluid3He

Hydrodynamic experiments in quantum liquids at low temperatures are sensitively influenced by the interaction of the elementary excitations with the confining walls of the measuring cell. In the case of transverse oscillations of the walls such that the viscous penetration depth of the liquid is small compared to the container dimensions, the dynamical behavior of the quantum liquid is governed by the transverse surface of shear impedance. We calculate the shear impedance for an isotropic, pair-correlated Fermi liquid from a microscopic quasiparticle Landau-Boltzmann equation using variational methods. A general class of elastic quasiparticle-wall scattering laws is considered in deriving the impedance functional. We give exact results for the variational surface impedance in the low-frequency limit and discuss approximate results in context with special wall-scattering laws such as specular, backward and Andreev scattering. Finally we apply our theory to the experimentally accessible, pair-correlated Fermi liquid³He and particularly investigate the influence of pressure, temperature, frequency, and the parameters specifying the wall-scattering process on the shear impedance.     

Specific heat of normal and superfluid3He

Extensive measurements of the heat capacity of liquid ³ He in the normal and superfluid phases are reported. The experiments range from 0.8 to 10 mK and cover pressures from 0 to 32.5 bar in zero magnetic field. The phase diagram of ³ He, based on the platinum NMR temperature scale, is presented. In the normal liquid at low pressures and near the superfluid transitionT _c an excess specific heat is found. The effective massm* of³He is at all pressures about 30% smaller than the values reported earlier. The calculated Fermi liquid parameters F₀ and F₁ are reduced asm*/m, while the spin alignment factor (1 + Z₀/4)^–1 is enhanced from 3.1–3.8 to 4.3–5.3, depending on pressure. The specific heat discontinuity C/C atT _c is forP = 0 close to the BCS value 1.43, whereas at 32.5 bar C/C is 1.90±0.03 in the B phase and 2.04±0.03 in the A phase, revealing distinctly the pressure dependence of strong coupling effects. The temperature dependence of the specific heat in the B phase agrees with a model calculation of Serene and Rainer. The latent heatL at the AB transition is 1.14±0.02 µJ/mole forP = 32.5 bar and decreases quickly as the polycritical point is approached; at 23.0 bar,L = 0.03 ± 0.02 µJ/mole.Work supported by the Academy of Finland.     

Collisionless spin waves in normal and superfluid3He

Standing spin-wave modes in liquid³He have been studied by cw NMR at Larmor frequencies of 1, 2, and 4 MHz and pressures of 0, 6.3, and 12.3 bar. The spin waves, which produce peaks in the NMR line, are visible at temperatures below 5 mK at zero pressure. With the assumption of a slightly simplified sample shape and no transverse spin relaxation at the walls, the theory of Leggett fits the spin-wave frequencies in the normal liquid very well, giving a value of the Fermi liquid parameterF ₁ ^a =–0.6±0.2 at zero pressure. The width of some of the peaks is larger than expected from other determinations of the quasiparticle diffusion time _D . This could be due to wall relaxation or to deviations from the assumed sample geometry. In the superfluid A₁ and A phases, where the data cannot be fitted to existing theories, the spin-wave modes are shifted in frequency and suffer additional damping as the temperature is decreased. At still lower temperatures in the B phase an inversion of the spin-wave spectrum from one side of the NMR line to the other is observed, agreeing quantitatively with the predictions of the 1975 theory of Combescot.     

Negative ion motion in normal and superfluid 3He

We report the first measurements of negative ion motion in the superfluid phases of ³He and in the normal phase below 17 mK. Refrigeration was achieved with nuclear demagnetization of copper and we used a pulsed NMR platinum powder thermometer immersed in the liquid. In the A phase the longitudinal resonance frequency provided an additional high-resolution thermometer. In the normal phase we observed a strictly temperature-independent mobility. In the superfluid phases we found two velocity regimes. For small applied electric fields the velocity is a linear function of the field and the corresponding mobility increases monotonically toward lower temperatures. At high electric fields the velocity is a nonlinear function of the field as a result of the pair-breaking effect of the moving ion. Available theoretical calculations are only in partial agreement with our results.This work was supported financially by the Academy of Finland.Guggenheim Fellow on leave from Cornell University, Ithaca, New York.On leave from Regensburg University, Regensburg, West Germany, supported by Deutsche Forschungsgemeinschaft and Finnish Ministry of Education.     

Positive ion mobility in normal and superfluid 3He

The mobility of positive ions has been measured in the normal and superfluid phases of ³He at several pressures. Below 100 mK the normal phase mobility increases logarithmically with decreasing temperature down to the superfluid transition temperature T _c; it shows an anomalous jump near 100 mK. At low temperatures the drift velocity is nonlinear for electric fields exceeding 30 V/cm. In the superfluid the mobility, normalized to its value at T _c, is much less than for negative ions. We have also observed the anisotropic mobility in the A phase and the Landau critical velocity for pair-breaking in both superfluid phases.Work supported by the Academy of Finland.On leave of absence from Regensburg University, Regensburg, West Germany, supported by Deutsche Forschungsgemeinschaft.     

Zero and first sound velocity and Fermi liquid parameterF 2 s in normal3He

A precise velocity measurement of zero and first sound in normal³He was performed over the pressure range of 0.5 bar and 10 bar at a frequency of 389.1 MHz using a phase detection technique. From these measurements, the Fermi liquid parameter F ₂ ^s was determined as a function of pressure. The new value of F ₂ ^s was positive and less than 0.25 at all pressures below 10 bar. It allows propagation of transverse zero sound mode for all pressures within the current theoretical picture.     

Fluctuations above the superfluid transition in liquid3He

It is shown that fluctuations above the superfluid transition in liquid³He depend strongly upon the relative angular momentum1 for which Cooper pairing occurs. The effects may be calculated for any value of1 and they should be observable in the static magnetization, viscosity, and spin diffusion coefficient, giving a means of determining1. Conclusions to be drawn from existing experiments are discussed.Work supported by Energy Research and Development Administration.     

The spin diffusion in normal and superfluid Fermi liquids

Spin diffusion in paramagnetic spin systems is a dissipative process which acts so as to remove all spatial variation of the magnetization. In normal and superfluid Fermi liquids its physical origin lies in the nonconservation property of the macroscopic magnetization current associated with the thermal excitations, the Landau and Bogoliubov quasiparticles, respectively. In the hydrodynamic limit this dissipative process manifests itself in a constitutive relation connecting the decaying magnetization current with gradients in the magnetization density via a coefficient of spin diffusion. Exchange contributions to the quasiparticle interaction introduce, in addition, reactive processes, which can be associated with a rotation of the quasiparticle spin current about the direction of the spin polarization. This so-called spin current rotation—or Leggett-Rice effect—leads to nonhydrodynamic behavior of the spin diffusion whenever the exchange frequency becomes comparable to the inverse spin current relaxation time. In this article I would like to review our current understanding of diffusional spin transport, as influenced by nonhydrodynamic effects, in normal and superfluid Fermi systems.Dedicated to Ludwig Tewordt on the occasion of his 65th birthday.     

Propagation of solitary waves in a mixture of liquid3He and superfluid4He

The effect of an admixture of liquid³He on the propagation of solitary waves in superfluid⁴He is investigated. In the local density approximation, the mean field equations are reduced to the two-fluid equations describing the dynamics of the mixture of two liquids. The nonlinear perturbation approach is used to obtain a set of equations describing the density and velocity fluctuations of the homogeneous mixture at rest. The self-consistent solution of these equations is shown to be the solitary waves. Using a pseudopotential method, we perform numerical calculations to study the changes in the general properties of the waves caused by the³He impurity.     

Spin-dependent correlations in normal liquid 3He

Using a generalized Fermi hypernetted chain method on a Jastrow trial ground-state wave function, which includes a dependence on the z component of the nuclear spin, it is shown that spin correlations make a significant contribution to the ground-state energy of liquid ³He, accounting for much of the energy necessary to stabilize unpolarized liquid ³He relative to completely polarized ³He.Work supported in part by NSF grant DMR 7926447 and by the Deutsche Forschungsgemeinschaft.     

Heat flow in superfluid 3He

The thermal resistance in both superfluid phases of ³He has been measured at 20.0 and 29.6 bar in zero magnetic field. Heat conduction in ³He-B is shown to be primarily hydrodynamic, and a regime of reproducible heat flow behavior in the A phase is reported. The viscosity of each phase as a function of temperature is calculated using an equation of the two-fluid model, and critical velocity effects are discussed.Work supported by the U.S. Atomic Energy Commission under Contract No. AT(04-3)-34 P.A. 143.     

Magnetic excitations in superfluid 3He

Localized magnetic excitations (solitons) in superfluid ³He are studied theoretically. In the A phase, we determine the dispersion of solitons in both the longitudinal and the transversal configurations, while in the B phase we limit ourselves to the longitudinal soliton in the Leggett configuration. In the wall pinned configuration of the B phase we show that the magnetic perturbation propagates as spin wave. The effects of the spin-diffusion term on solitons as well as spin waves are considered. The soliton velocity decreases exponentially in time owing to the spin-diffusion term with a characteristic time % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% qedmvETj2BSbacfaGae83KdCKae8hiaaIae83waSLaeyyyIORaaiik% aiaaiodacaGGPaGaaiikaiaadogadaahaaWcbeqaamaaCaaameqaba% GaaGOmaaaaaaGcciGGVaGaamiraiabfM6axnaaCaaaleqabaWaaWba% aWqabeaacaaIYaaaaaaakiaacMcacaGGDbaaaa!4DB4!\[\Gamma [ \equiv (3)(c^{^2 } /D\Omega ^{^2 } )]\] where D is the spin-diffusion constant, is the Leggett frequency, and c is the spin-wave velocity.Supported by the National Science Foundation.     

Orbital modes in superfluid3He

The equation describing the rotation of the orbital coordinate in superfluid³He is obtained. It is shown that the orbital modes obey a diffusion-like equation with a finite energy gap. The transient behavior of the vector in the A phase after thed vector is suddenly rotated is considered.Supported by the National Science Foundation.On leave from Department of Applied Science, Tohoku University, Sendai, Japan.     

Persistent currents in superfluid3He

Measurements are reported of persistent currents in superfluid³He-B and³He-A. An ac gyroscope filled with 20 µm powder and mounted into a rotating nuclear refrigerator was employed. In³He-B, undiminished circulation was observed for 48 h; this implies an effective viscosity at least 12 orders of magnitude lower than in the normal fluid at the same temperature. AtP<15 bar, the observed critical velocity is independent of temperature but it is a weak function of pressure;v _c varies between 4 and 6 mm/sec. The response to rotation is hysteretic, with elastic potential flow at slow rotation and irreversible vortex flow at higher angular velocities. The persistent angular momentumL is reversible when thermally cycled in the B phase, and proportional to the superfluid fraction _s /. Above 15 bar the B phase splits into separate regions with different critical velocities. The measuredv _c in the phase existing only at high pressures is dependent on magnetic field; for example, at 23.0 bar,v _c (H=0) =5 mm/sec, butv _c (H=40 G) =15 mm/sec. In the low pressure phase,v _c is insensitive to a change in the magnetic field. The phase transition is of first order; the latent heatQ _G (1 µJ/mole) depends on the maximum angular velocity at which the cryostat was rotated. The transition is proposed to occur in the core structure of pinned quantized vortices sustaining persistent currents. In³He-A, currents could not be found to persist on an observable level. Direct measurements ofL atH=0 and atH=40 G, and repeated thermal cycling, showed that either the current decays rapidly orv _c <0.5 mm/sec.     

Zero-sound studies in superfluid 3He

The propagation of 5-, 15-, and 25-MHz sound in superfluid ³He has been studied in zero magnetic field over a wide pressure range. Measurements of the structure and location of the attenuation peaks, velocity changes, and the low-temperature attenuation are described. The degree of strong coupling in the B phase is discussed. Observations of superheating and supercooling near the polycritical point have been extended and comparison is made with theory.Work supported by the U.S. Energy Research and Development Administration under Contract No. E(04-3)-34, P.A. 143.     

NMR in flowing superfluid 3He

The effects of flow on the NMR absorption in the A and B phases of ³He have been studied. While no flow-induced changes in the absorption were seen in the B phase, a variety of phenomena were observed in the A phase. The phenomena include the development of time-dependent satellite peaks, height reductions of the signal, and a splitting of the signal, with a fraction of the signal shifting to lower frequencies. These effects are described and discussed in light of the existing theoretical models. In addition, the NMR shift in the static B-phase liquid confined between 135-m-spaced parallel plates has been measured as a function of temperature and the angle between the magnetic field and the plates. These results are compared with previous measurements and are found to be in good agreement. Finally, two effects observed in the static liquid, the disappearance of the B-phase signal under certain conditions and the splitting of the A-phase signal, are described.Research supported by the National Science Foundation.     

2D liquid3He near solidification: a highly correlated Fermi liquid

The magnetic susceptibility of two-dimensional liquid³He adsorbed on graphite has been measured by NMR techniques for submonolayer and second layer coverages in a large temperature range around the Fermi temperature. The susceptibility enhancement factor determined in the vicinity of second layer solidification is larger than that found in the bulk liquid at high pressures. The results are discussed in the framework of the quasi-localized and the paramagnon models of liquid³He.     

The thermomechanical effect in normal liquid3He

A thermomechanical effect has been found in normal liquid³He. The effect appears only when the temperature gradient is established across liquid³He confined in small pores so that the³He quasiparticle scattering is boundary limited. This was achieved with a porous plug of packed 70 nm copper-oxide powder. The magnitude of P/gDT varies from 4 C at 2 mK to C at 20 mK, where P is pressure, T is temperature and C is the heat capacity per unit volume. The sign is positive, meaning that the³He moves from cold to hot. If viewed as an analogue of thermoelectricity, the magnitude of this thermomechanical effect is unsurprising. However, as shown in the following paper, it is an order of magnitude larger than a theoretical prediction for liquid³He.