3He Correlation Energy in the Random Phase Approximation: the One-Ripplon-Exchange Potential in Thin 3He–4He Superfluid Films

A new approach to the calculation of correlation energies in the random phase approximation (RPA) is introduced. The method is applied to the system of adsorbed ³ He in thin ⁴ He superfluid films. The ³ He component interacts by means of the exchange of virtual surface interactions (ripplons) in addition to the usual atom-atom interaction. We report on a calculation of the RPA contribution to the ³ He ground-state energy of a previously introduced model for this effective interaction called the one-ripplon-exchange potential (OREP). V_OREP is manifestly frequency dependent and complex. In this paper we study the differences in the RPA contribution to the ground-state energy between the fully complex and frequency dependent V_OREP and V_OREP in the real, static limit.     

Mutual Friction in Superfluid 4He Near the λ-Line

We present experimental results for the thermal resistivity of superfluid ⁴He along several isobars between saturated vapor pressure and the melting pressure. The measurements are for the temperature range 1–T _c(q)/T <t<2×10^–5 and the heat-flux range 3<q<70 W/cm². Here t1–T/T , T is the transition temperature in the limit of zero q, and T _c is the transition temperature at finite q. The data suggest that the resistivity has an incipient singularity at T which can be described by the power law =(t/t ₀)^–(m+) where t ₀=(q/q ₀) ^x . However, the singularity is supplanted by the transition to a more highly dissipative phase at T _c(q)<T . The results suggest a mild dependence of m+ on the pressure P, but can be described quite well by m+=2.76, x=0.89, and q ₀=q _{0, 0}–q _{0, 1} P with q _{0, 0}=401Wcm^–2 and q _{0, 1}=–5.0Wcm^–2bars^–1. The results imply that the Gorter–Mellink mutual friction exponent m has a value close to 3.46 and is distinctly larger than the classical value m=3.     

On the possible superfluidity of 6He—Its phase diagram and those of 6He-4He and 6He-3He mixtures

The properties of pure ⁶He and ⁶He-⁴He and ⁶He-³He mixtures are studied within the context of the quantum theorem of corresponding states. For pure ⁶He, it is found that there is a small, but definite, region where it is expected to be superfluid. For the ⁶He-⁴He mixture, it is found that there is a -eutectic temperature, approximately 1.2 K, below which it is expected that both ⁶He and ⁴He will be part of the superfluid at all concentrations. Experiments to observe ⁶He superfluidity are discussed, and it is concluded that such experiments are feasible, although they may be quite difficult to carry out. Experiments are discussed to use a ⁶He generator (should it be possible to construct it) as a probe to study various properties of liquid and solid helium.Work supported in part grant from the National Science Foundation.     

Critical behavior of the free surface of liquid 4He near the λ point

The noncriticality of the free surface of liquid ⁴He near the point and the finite-size scaling postulate are combined to show that the surface tension can have two singularities, one due to rounding and another due to shifting. The rounding singularity can be reduced, via a further scaling assumption, to that previously suggested by Sobyanin and Hohenberg. Results from calculations based on continuous symmetry models and experiments on topologically 2D ⁴He films are used to argue that the shift singularity is ¦t¦^1–, which is consistent with the leading singularity observed by Magerlein and Sanders.     

Critical phenomena of liquid4He in the vicinity of the upper λ point

We determinedC _p along six isobars near T in the vicinity of the upper superfluid transition point (upper point) from measurements ofC _v and (P/T) _v along six isochores.C _p was analyzed with the functionC _p =(A/)^–(1+D^–)+B for T>T, and the same function with primed coefficients for T, whereD denotes the strength of the effect of the irrelevant variable. The present work clarified the effect of the pressure (irrelevant variable) on the critical behavior of ⁴ He near T, that is, the correction term due to the irrelevant variable increases with pressure even in the small range 3×10^–3. This indicates that the pressure depresses the true critical region. The universality of the amplitude ratioA/A was confirmed even in the vicinity of the upper point by specific heat measurements. With constraints ==–0.02, ==–0.5, andB=B the pressure-independent amplitude ratiosA/A=1.088±0.007 andD/D=0.85±0.2 were obtained.AD/AD=0.93±0.2 implies that the pressure has a similar effect onC _p in the normal fluid and superfluid regions, within experimental errors.     

Finite-size and gravity effects on the thermal conductivity of4He near the λ line

This paper reviews the opportunities for microgravity and Earth-based measurements of the thermal conductivityλ(t, L) of⁴He confined in cylindrical geometries of radiusL with axial heat flow at temperatures near the bulk superfluid-transition lineT _λ (P) (t is the reduced temperatureT/T _λ −1). It provides an evaluation of existing data forL=1 μm nearT _λ at saturated vapor pressure (SVP), and uses these to derive a scaling function for the resistivityR(t, L)=1/λ(t, L). The purpose of future measurements over a wide range ofL and of the pressureP will be to test the applicability of this function. In the present paper the scaling function is used to assess quantitatively the effect of gravity on potential Earth-based measurements. It is found that the gravity effect forR is particularly severe belowT _λ . For typical three-mm-high samples at SVP, values ofL significantly larger than 8 μm can only be investigated fully in micro-gravity. At higher pressures the gravity effect is larger. At 30 bar, three-mm samples withL≳4 μm require microgravity for measurements belowT _λ (P). Modern thermometry has sufficient resolution to permit quantitative measurements in micro-gravity of the anticipated finite-size effect for values ofL as large as 50 μm.     

Critical properties of4He in aerogel near the λ-line

We measured the superfluid fraction _s/ and the isobaric thermal expansion coefficient _P of⁴He confined in an aerogel. Data were obtained near the -line along several isobars. Powerlaw fits to the results for _s/ as a function of tT/T_c–1 (T_c is the transition temperature) give a pressure—independent exponent =0.755 when a confluent singular term is included. Fits to the _P data of power laws yield the specific-heat exponents –0.6 and –1.0 above and below T_c respectively. When an analytic background term a×t is included in the fit, the pressure-independent value =–0.59 is permitted, but the amplitude ratio A/A is found to be near zero and the coefficient of the analytic term is large. The measured values for and or are inconsistent with hyperscaling in three dimensions.     

Theory of helium under heat flow near the λ point. I. Interface of He I and He II

A He I-He II interface is shown to exist under heat flow and is studied near the point. The temperature in the superfluid region is found to be of the form T –T Q^3/4 if dynamic scaling is assumed, where T is the critical temperature andQ is the heat flow. In the normal region the temperature has a finite gradient. Here we are neglecting a small thermal resistance due to vortices in the superfluid region. In a finite system with size much greater than the correlation length a first-order phase transition occurs as an inverted bifurcation as the temperature at the cooler boundary is lowered slightly below T. Namely, the system will jump from the disordered state to a state in which the two phases are separated by an interface. The theory can be constructed analogously to the theory of superconductors in a magnetic field.     

Developed Capillary Turbulence on the Surface of Normal and Superfluid 4He

The capillary turbulence on the surface of normal and superfluid liquid ⁴He has been studied experimentally. It is observed for the first time that the value of the high-frequency boundary ω _b of the inertial interval increases significantly when liquid helium transits from normal to superfluid state, and that in superfluid He-II the correlation function of the surface deviation from equilibrium state in frequency representation can be described well by a power dependence with the index m close to −4.3.      

Experiments on 3He–4He Mixtures in Aerogel

We describe experiments on ³ He- ⁴ He mixtures in 98% open aerogel grown by cold-deposition (T80mK) of ⁴ He followed by the deposition of ³ He or alternatively by cooling down a homogenous mixture. The two approaches led to different ⁴ He film morphologies. We will also describe the observation of migration of ⁴ He from or toward the cell below 100 mK. This migration can lead to the exclusion of ⁴ He (other than the localized surface layers) for ⁴ He concentrations between 4% and 11% in the cell.     

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 is linear in temperature range studied whereas for the solution of 9.5% we observed the deviation from linear dependence above some critical value . 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 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.      

Magnetic Susceptibility of the Balian–Werthamer Phase of 3He in Aerogel

The equilibrium superfluid phase of ³He impregnated into high-porosity (98%) silica aerogels appears to be a non-equal-spin-pairing state in zero field at all pressures, which is generally assumed to be the Balian–Werthamer (BW) phase modified by the depairing effects of the aerogel structure. The nuclear magnetic susceptibility played a key role in identifying the B-phase of pure ³He with the BW state. We report theoretical calculations of the nuclear magnetic susceptibility for the BW model of superfluid ³He in aerogel within the framework of the Fermi-liquid theory of superfluid ³He. Scattering of quasiparticles by the aerogel, in addition to Fermi-liquid exchange corrections, leads to substantial changes in the susceptibility of the BW phase. The increase in the magnetic susceptibility of ³He-aerogel compared to pure³He-B is related to the polarizability of the gapless excitations and the impurity-induced local field. The limited data that is available is in rough agreement with theoretical predictions. Future measurements could prove important for a more definitive identification of the ordered phase, as well as for refining the theoretical model for the effects of disorder and scattering on the properties of superfluid ³He.     

??On the Superfluidity of Liquid He3??. Author??s Recollections

The history of the author??s paper (Pitaevskii in J. Exp. Theor. Phys. 37:1794, 1959; Sov. Phys. JETP 10:1267, 1960) on the superfluidity of liquid He³ is presented.     

Sound Velocity Measurements in the Low and the High Field Phases of the Nuclear-Ordered bcc Solid 3He in Magnetic Fields

We have measured the temperature and magnetic-field dependences of the sound velocity for one longitudinal and two transverse waves in the low field phase (LFP) and the high field phase (HFP) of nuclear spin ordered bcc solid ³He crystals with a single magnetic domain along the melting curve. From sound velocity measurements for various crystal orientations as a function of the sound propagation direction, we determined the elastic stiffness constants, c _ij (T,B). In the LFP with tetragonal symmetry for the nuclear spin structure, we extracted six nuclear spin elastic stiffness constants Δc _ij ^ℓ (T,0.06 T) from the temperature dependence of the sound velocity at 0.06 T and Δc _ij ^ℓ (0.5 mK,B) from the magnetic-field dependence of sound velocity at 0.5 mK. In the HFP with cubic symmetry for the nuclear spin structure, we extracted three Δc _ij ^h (T,0.50 T) at 0.50 T and Δc _ij ^h (0.5 mK,B) at 0.5 mK. At the first-order magnetic phase transition from the LFP to the HFP at the lower critical field B _c1, large jumps in sound velocities were observed for various crystal directions and we extracted three . Using the thermodynamic relation between Δc _ij and the change in the internal energy for the exchange interaction in this system, ΔU _ex(T,B), Δc _ij are related to the generalized second-order Grüneisen constants Γ _ij ^X ≡∂ ²ln X/∂ ε _i ∂ ε _j as Δc _ij (T,B)=Γ _ij ^X ΔU _ex(T,B), where X represents some physical quantity which depends on the molar volume and ε _j is the j-th component of a strain tensor. In the LFP, the Δc _ij ^ℓ (T,0.06 T) were proportional to T ⁴, and Δc _ij ^ℓ (0.5 mK,B) were proportional to B ². We extracted for the spin wave velocity in the LFP, s ^ℓ , from Δc _ij ^ℓ (T,0.06 T) and for the inverse susceptibility, 1/χ ^ℓ from Δc _ij ^ℓ (0.5 mK,B). In the HFP, Δc _ij ^h (T,0.50 T) were proportional to T ⁴ and Δc _ij ^h (0.5 mK,ΔB) were proportional to ΔB(≡B−B _c1). We obtained for the spin wave velocity in the HFP, s ^h , from Δc _ij ^h (T,0.50 T) and for B _c1 from Δc _ij ^h (0.5 mK,ΔB). The values obtained for and were compared with the Multiple Spin Exchange model (MSE) with three parameters by using analytic expressions for s ^ℓ and χ ^ℓ . The three-parameter MSE does not agree with the observed Δc _ij in the LFP.      

3He Effect on 2D Superfluidity in 3He–4He Mixture Films on Planar Gold

There have been a number of experiments exploring the nature of 2D superfluidity and the configuration of ³He–⁴He mixture films on various substrates. To date, a possible film-structure at T=0 is that of a simple layer model, ³He/superfluid ⁴He/solid-like ⁴He/substrate, in which the submonolayer superfluidity is strongly affected by the coverage of the ³He overlayer. Yet the mechanism is not been fully understood. In this paper, we report a QCM study at 60 MHz for the ³He effect on the superfluidity of mixture films on flat gold, mainly focusing on the anomalous depletion of the temperature dependence of the superfluid density σ _s. In the measurements, we kept the ³He coverage constant (n ₃= 0, 3.6, 7.2, 19.0, 57.2, or 92.8 μmol/m²) and then incrementally added ⁴He. We observed the evolution of the ³He effect on σ _s(T) with increasing ³He coverage; this depletion of σ _s(T) rapidly increases and then saturates near n ₃～1 layer. From the analysis of the linear-temperature region in the plot of the dissipation peak temperature T _p as a function of the superfluid ⁴He coverage n _4s and comparison with previous studies on Mylar and porous gold, we found a universal function for the strength of the ³He effect for all substrates.     

Transport in 3He–4He Mixtures in Restricted Geometry

The effect of wall scattering on transport in dilute degenerate ³He–⁴He mixtures in quasi-2D flow channels or films is discussed. The calculation of the quasiparticle mean free path combines particle–wall and particle–particle collisions including the interference between them. The spin polarization affects the wall-driven contribution by changing bulk mean free path and particle wavelength. The expressions for the wall contribution to transport coefficients are especially transparent in the limiting cases of large and small bulk mean free paths. The calculated temperature, concentration, and polarization dependences of the transport coefficients allow one to extract parameters of surface roughness from experimental data on transport.     

Mobility of 2D Electrons on Pure 4He and on 3He–4He Dilute Solution

The mobility of 2D electrons on pure ⁴He and on 0.5 % solution of ³He in ⁴He was investigated for different electron densities in the temperature range 0.12 to 1.3 K. The electrons in the same electron density show the same transition temperature from liquid state to Wigner crystal state in both pure ⁴He and in the solution. In the high temperature range where the gas-scattering is dominant, the electrons show a smaller mobility in the solution than in the pure ⁴He due to the electron collision with ³He gas atoms which have a higher vapor pressure. In the middle temperature range where the ripplon-scattering is dominant, the mobility in the solution is smaller than in ⁴He. This is explained by a smaller surface tension caused by ³He atoms collected at the surface. In the low temperature range where electrons are in the Wigner crystal state, the mobility gradually increases with decreasing temperature in the solution, while it stays almost constant in the pure ⁴He. The mobility increase is more pronounced in the low electron density. The results are qualitatively in agreement with the existing theory which includes the bulk ³He quasiparticle reflection from surface dimples and the effect of the surface layer of ³He atoms.     

Measurements of the A ⇄ B transition in a magnetic field for superfluid3He near the polycritical point

Measurements of both AB and BA transitions have been made as a function of magnetic field up to 100 G at temperatures and pressures close to the polycritical point (PCP). Forms for the specific heat differenceC _A–C _B=C ₀ (P)–(P)(T _c –T)/T _c and for the magnetization differenceM _A–M _B=H(T _c –T)/T _c fit the data satisfactorily except very nearT _c , where some other mechanism, thought to be a free energy difference due to residual heat flows, tends to stabilize the A phase. The pressure of the PCP, taken to be that at whichC ₀ =0, is found to be 21.22±0.02 bar.Work supported by the U.S. Energy Research and Development Administration under contract number E(04-3)-34, P.A. 143.     

Specific Heat Near the Superfluid Transition of a 0.9869 μm 4He Film

We report measurements of the specific heat of ⁴He near the superfluid transition while confined between silicon wafers at 0.9869 m separation. These data are analyzed to check on the behavior expected from correlation-length scaling. Comparison is also made with other data for planar confinement, as well as data for cylindrical confinement. These represent different lower-dimensional crossovers. We find that the present data scale very well above the bulk transition temperature, and in the region immediately below it. Near the specific heat maximum however, the data for planar confinement do not collapse on a universal curve. We compare these results with specific theoretical scaling functions. In particular we find that on the normal side, and for large enough values of the scaling variable, one can describe the data well using the concept of the surface specific heat. The locus of the data in this region agrees well with the most recent theoretical calculations.