Dynamic measurements of thermal transport coefficients and boundary resistance I. Normal4He

The thermal conductivity,K, and the thermal diffusivity,D _T , of normal liquid⁴He have been obtained from the temperature response, ΔT (ω), across a fluid layer of thickness,d, to an ac heat flux,Q(t)=Q ₀ exp(iωt). Previous transient heat flux experiments measured the thermal relaxation of the fluid towards equilibrium and assumed the dominance of a single slowest mode. The present ac technique allows measurements under steady-state conditions while driving the system at a single frequency, ω. The response curve for ΔT(ω)/Q ₀ yields data forK,D _T and the boundary resistance,R _b . Boundary effects appear at frequencies higher than τ⁻¹ ≡ D_T/d² where the fluid is unresponsive to bulk heat transport. We use this fact to obtainR _b with high accuracy in the normal phase from the high frequency response. In addition, the apparatus permits the fluid thickness,d, to be varied continuously andin situ from zero to 3 mm, allowing for further consistency in the fluid measurements. This work also includes data for the onset of convection whereQ ₀>Q _c, andQ _c corresponds to the heat amplitude at convective onset.     

Experiments on the Self-Organized Critical State of 4He

When a heat flux Q is applied downward through a sample of ⁴He near the lambda transition, the helium self organizes such that the gradient in temperature matches the gravity-induced gradient in T _λ . All the helium in the sample is then at the same reduced temperature and the helium is said to be in the Self-Organized Critical (SOC) state. We have made the first measurements of the ⁴He SOC state specific heat, C _{∇ T} (T(Q)). There is no measurable difference between C _{∇ T} and the static zero-gravity ⁴He specific heat for temperatures between 650 and 250 nK below T _λ . Closer to T _λ , the specific heat is depressed and reaches a maximum value at 50 nK below T _λ . This depression is similar to that predicted theoretically as reported by R. Haussmann (Phys. Rev. B 60, 12349, 1999). Contrary to the expectations of theory, however, we see another depression far below T _λ . In addition, over the heat flux range of 30 nW/cm² to 13 μW/cm², we have made improved measurements of the speed of a recently discovered propagating thermal mode, which travels only upstream against the nominal heat flux of the SOC state. We are able to accurately predict the speed of this wave by treating the helium of SOC state as a traditional fluid with a temperature dependent thermal conductivity.     

Relaxation times and mass diffusion in superfluid dilute3He-4He mixtures

Relaxation times are reported from the transients observed during thermal conductivity _eff and thermal diffusionk _T ^* measurements in superfluid mixtures of³He in⁴He with a layer thickness of 1.81 mm. The experiments extend from 1.7 K toT and over a³He concentration range 10^–6X<5×10^–2. The agreement between the measured and the predicted from the two-fluid thermohydrodynamic equations is satisfactory forX>10^–3 but deteriorates for smaller³He concentrations. This situation is similar to that for _eff andk _T ^* results and indicates that the transport properties in very dilute mixtures with layers of finite thickness are not well understood. ForX>10^–3, the mass diffusion coefficientD _iso for isolated³He in⁴He has been determined from and from _eff measurements. There is an inconsistency by a constant numerical factor between these determinations. This problem might be related to the observations that in the superfluid phase, the relaxation times for different cell heightsh do not scale withh ². FromD _iso derived via the _eff data, the³He impurity-roton scattering cross section is determined. Comparisons with previous work are made.     

Thermal resistance between liquid3He and copper potassium tutton salt

The thermal resistance between liquid³He and copper potassium tutton salt (CPS) has been measured through its magnetic ordering temperature (T _c=29.6 mK). The thermal resistanceR for pure³He has a broad minimum near 60 mK and increases continuously throughT _cwith decreasing temperature, except for a dip atT _c. BelowT _c,R is proportional toT ^–1.5. Effects of⁴He coating have been studied by stepwise addition of⁴He into liquid³He. The thermal resistance increased drastically for the liquid containing 150 ppm⁴He and more for 95%⁴He. By sudden depressurization of the liquid³He containing 480 ppm⁴He, a considerable decrease ofR was observed. SinceR for pure³He was much smaller than the calculated Kapitza resistance, the present experimental results indicate the existence of surface magnetic coupling between liquid³He and CPS.     

Adiabatic Melting of 4He Crystal in Superfluid 3He at Sub-millikelvin Temperatures

Adiabatic melting of ⁴He crystal to phase separated ³He–⁴He solution (at T< 2 mK) is probably the most promising method to cool the dilute phase down to temperatures substantially below 0.1 mK. When started well below the superfluid transition temperature T _c of pure ³He, this process allows, in principle, to get the final temperature (T _f ) several orders of magnitude less than the initial one (T _i ). This work is the first practical implementation of the method below the T _c of ³He. The observed cooling factor was T _i /T _f =1.4 at 0.9 mK, being mainly limited by the bad performance of the superleak filling line, by incomplete solidification of ⁴He in the cell, and by the improper thermal contact between the cell wall and the liquid.     

Thermal Response of Superfluid 4He near T λ to an AC Heat Flux

We have measured the thermal response of a superfluid⁴He layer nearT _λ to a time-varying heat flux. In the low-frequency range studied, 1×10⁻³<f<0.1 Hz, we find the response to be independent of the applied frequencyf, a result consistent with the expectation that the measured response is due to the boundary resistanceR _b alone. These findings differ from those by Olafsen and Beharinger. The thermal conductivity cell used by these authors had extraneous surfaces in contact with the superfluid which were eliminated in the cell used for our experiments; we believe these surfaces were responsible for the frequency dependence observed in previous work. Furthermore, we show that AC and DC measurements ofR _b differ by no more than 1%, and that both exhibit a weak singularity nearT _λ.     

Transport properties for a dilute solution of3He in two-dimensional liquid4He

The temperature variations of the diffusion coefficientD(T), thermal diffusion ratio k _T (T) and thermal conductivity (T) in a dilute solution of³He atom in two-dimensional liquid helium are evaluated explicitly by solving the kinetic equations via phonon-phonon, phonon-roton, roton-roton, impurityelementary excitation and impurity-impurity scatterings. In the low-temperature region, the main contributions toD(T) and (T) come from the interactions between phonons and impurities, while in the high-temperature region the interactions between impurities and whole elementary excitations contribute more strongly toD(T) and (T) than those of only elementary excitations. For a dilute solution, the thermal diffusion ratio k _T (T), neglecting the internal mass counterflow, is much smaller than the effective thermal diffusion ratio k _T ^* (T), which is a function of thermostatic properties. The effective thermal conductivity _eff is much larger than the thermal conductivity and has different temperature dependence from the thermal conductivity. The behaviors of the two-dimensional diffusion coefficient and thermal conductivity are much like the bulk case, where they exhibit exponential decay with increasing temperature, although they are much smaller than those of the bulk case.     

Influence of Supercooling Level on Kinetics of Phase Separation in Solid Mixtures of 4He in 3He

We studied the kinetics of phase separation in the solid mixture of ⁴ He in ³ He at various supercooling in the temperature range of 100–200 mK through precise pressure measurements. The time dependences of the pressure change during phase separation were exponential. At small supercooling the characteristic time constant is almost independent of the final temperature T_f and is about 10 hours, which is considered to result from heterogeneous nucleation. In a narrow range of T_f decreases more than an order of magnitude. At larger supercooling is independent of T_f again. This behaviour agrees qualitatively with the theoretical consideration of the phase separation kinetics at homogeneous nucleation taking into account the finiteness of a cooling rate. The value of interphase surface tension has been obtained from comparison of the theory with the experimental results.     

Concentration dependence ofD,T 1, andT 2 for dilute solutions of3He in solid4He

Measurements of the spin diffusion coefficientD and NMR relaxation timesT ₁ andT ₂ are reported for dilute solutions of³He in solid⁴He at two molar volumes, 20.95 and 20.7 cm³. The weakly interacting impuriton model, for whichD ^–1 is proportional to impurity concentration, is observed only at fractional impurity concentrationx ₃ below 3 × 10^–4. Forx ₃ around 10^–3,T ₁ andT ₂ are controlled by the formation and breakup of³He₂ molecules.     

Heat transfer to He I and He II in pulsed thermal loading

Results are given on the effects of a pulsed thermal load on a copper specimen immersed in He I and He II. Estimates are made of the mean heat-transfer coefficient _p for He I, He II, and He I with T _l s.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 389–393, March, 1981.     

Scaling Results for Superfluid 3He in 98% Open Aerogel

We present experimental observations of the suppressed superfluid transition temperature, T _ca , superfluid fraction, ρ _s /ρ and Leggett frequency of ³He-B in aerogel, Ω _Ba . We determine T _ca from mass decoupling and the vanishing of the frequency shift away from the Larmor frequency in our different samples and different laboratories. We find that the suppressed transition temperature for ³He in aerogel occurs at a sample dependent, but approximately pressure independent, length, , where T _c and ξ ₀(P), are the transition temperature and the pressure dependent zero temperature coherence length for bulk ³He. T _ca also occurs at a pressure independent value of the Leggett frequency of bulk ³He-B. Further, we find that when the superfluid fraction and square of the Leggett frequency are plotted against T _ca −T (and not (T _ca −T)/T _ca ), the results of each measurement nearly collapse on to a pressure independent but sample dependent plot, with no further scaling. When plotted on a log–log scale, both measurements exhibit power laws in the range 1.33–1.45.      

Superfluidity of 4He Adsorbed on Nanoporous Activated Carbon Fibers

⁴He confined in nanoporous media is one of the most interesting nano-sized systems in the context of an interacting Bose system. In the present work, we study superfluidity of ⁴He adsorbed in nanoporous activated carbon fibers (ACFs). Up to a ⁴He coverage of n=22.6 μmol m⁻², no superfluidity is observed. Over 23.7 μmol m⁻², superfluid transition is observed at T _c ∼550 mK. With an increase in n the superfluid density enhances, but T _c is independent of n. These observations indicate that the thickness of the superfluid ⁴He films on the pore wall is restricted by a slit type pore shape of ACFs.     

3He Anomalously High Spin Diffusion in Solid Para-Hydrogen

NMR measurements of ³He spin diffusion coefficient in solid para-H₂ are carried out at the temperatures 0.45–1.5 K. The crystals have been grown under constant pressure 20–130 bar. The ³He concentrations in the initial para-H₂–³He gas mixtures were 0.1% and 0.3%. It is found out that the decay of echo amplitude vs both magnetic field gradient G and time interval τ between RF pulses is of non-exponential character, typical of one-dimensional diffusion in restricted geometry. The values of true spin-diffusion coefficient D _S measured are found to be ∼10⁻⁴ cm²/s at 20 bar. At 108 bar D _S value is one order of magnitude less. D _S does not depend on temperature. Such spin diffusion coefficient values seem to be anomalously high in comparison with well-known values of D _S =10⁻⁵ cm²/s for bulk liquid ³He at 27 bar and D _S =10⁻⁸ cm²/s for bulk solid ³He at 108 bar. The special experiments with the crystal annealing make it clear that the high spin diffusion here is connected with fast diffusion along dislocation lines.     

NMR study of 3He bubbles in phase-separated solid 3He –4He mixtures

³ He droplets embedded in a solid ⁴ He matrix have been studied by NMR and pressure measurements. One feature of the experiment is that the mixture crystals, of ³ He concentration 1%, are grown under constant pressure conditions to minimise the formation of defects. A number of sample pressures below 34 bar have been studied. Isotopic phase separation and the melting of the bubbles are clearly observed. Measurements of T₁ , T₂ and magnetisation give detailed information on the structure of the droplets. At an initial sample pressure of 28.3 bar preliminary measurements of the T₁ of the liquid bubbles show a temperature dependence of the form (A+ B/T²)^–1. This indicates that the expected relaxation in the liquid is augmented by a constant contribution, probably from the surface of the droplets.     

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 (/2) of 20.5 MHz at temperaturesT from 30 mK to the transition at T. The³He concentrations studied werec=0.014, 0.031, 0.053, 0.060, and 0.092 below 1 K, thoughc decreased slightly near the 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, 1 ( 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 (c, T) and (c, T). The relaxation times were in good agreement with previous work, while (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, 1. Values of the total viscosity (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 point we founda=(0.1±0.03)^–2/3 nm, where =1–T/T, proportional to the phase coherence length . Our data are consistent with the hypothesis that _s/T is a universal constant for superfluid dilute solutions, where _s is the superfluid density. Between 1.0 and 1.8 K we found thata(c, T) was comparable to measurements in³He-⁴He films.     

Properties of 3He in thin 4He films

Substantial progress has been made in our understanding of the properties of ³He atoms in thin films of ⁴He. For such films the ³He occupies discrete quantum states in the film and the system is both rich and complex. Here we discuss progress in this field from several points of view; we briefly discuss early heat capacity and third sound results and concentrate on more recent NMR measurements of the magnetization and relaxation times T ₁ and T ₂. Further experimental work and theory for systems of finite ³He coverage is needed to fully understand this fascinating system.Presenter at the Elba Summer School; authors are listed alphabetically.     

Kapitza resistance and thermal conductivity of Kapton in superfluid helium

We have determined simultaneously the Kapitza resistance, R_K, and the thermal conductivity, κ, of Kapton HN sheets at superfluid helium temperature in the range of 1.4–2.0 K. Five sheets of Kapton with varying thickness from 14 to 130 μm, have been tested. Steady-state measurement of the temperature difference across each sheet as a function of heat flux is achieved. For small temperature difference (10–30 mK) and heat flux density smaller than 30 W m⁻², the total thermal resistance of the sheet is determined as a function of sheet thickness and bath temperature. Our method determines with good accuracy the Kapitza resistance, R_K=(10540±444)T⁻³×10⁻⁶ K m² W⁻¹, and the thermal conductivity, κ=[(2.28±0.54)+(2.40±0.32)×T]×10⁻³ W m⁻¹ K⁻¹. Result obtained for the thermal conductivity is in good agreement with data found in literature and the Kapitza resistance’s evolution with temperature follows the theoretical cubic law.     

Thermal response of a fluid near its critical point:3He atT>T c

The local density response is studied in a simple fluid near the liquid-vapor critical point, subjected to temperature oscillations of its container. This investigation provides a new approach in the study of the adiabatic energy transfer (“piston effect”) in the fluid. The density response functionZ _F (ω, ε,z) is calculated for³He in the absence of stratification, where ω is the angular frequency, ε=(T−T _c )/T _c the reduced temperature,T _c =3.316 K the critical temperature, andz the vertical position in the container. Experiments are described where the density is measured by two superposed capacitive sensors in a cell of 3.5 mm height, and where the temperature oscillation frequencyf=ω/2π is varied between 10⁻⁴ and 2 Hz. Over the experimental range 5×10⁻⁴<ε<5×10⁻² there is in general reasonable agreement between predictions and experiments. The systematic departures might be accounted for by deviations from 1D geometry, which were not included in the calculations. Over the frequency and reduced temperature ranges, the damping effect from the critical bulk viscosity is predicted to be too small to be detectable. The observed effect of the stratification and its frequency dependence inZ _F are briefly discussed. In the appendix, the predicted critical acoustic attenuation from the bulk viscosity is compared with published data, the effect from finite thermal conductivity of the fluid container plates and also the corrections toZ _F for the effects of the cell sidewalls are calculated. F. Pobell     

Measurement of fountain pressure and other thermal properties in superfluid3He

The fountain pressure, heat capacity, and thermal conductivity of superfluid ³ He have been measured simultaneously at pressures of 29, 23.2, 10, and 2.5 atm and at the saturated vapor pressure. If the fountain equation, based on the two-fluid model, is used to compute the entropy from the observed fountain pressure, it is consistent with the entropy obtained by integrating the specific heat. The measured thermal conductivities are dominated by a diffusive conductivity which is inversely proportional to the temperatureT _c .Work supported by NSF, DMR-72-02926.     

Simulations of a Thermal Conductivity Cell Filled with Normal 4He and Dilute Mixtures of 3He in Superfluid 4He

We have simulated the thermal response of a cylindrical thermal conductivity cell filled with liquid helium to AC and DC heat fluxes. The conductivity cell in these simulations is realistic in that it includes sidewalls and gaps, which cannot be treated analytically or in a one-dimensional simulation. Our simulations are to able to account quantitatively for the apparent departure of the effective thermal conductivity, _eff , of dilute mixtures of ³ He in superfluid ⁴ He from theoretical predictions. We have recently demonstrated experimentally that this departure is due to the presence of gaps in previous thermal conductivity cells. These simulations also show that the additional phase lag in the response of normal ⁴ He to an AC heat flux, measured by Olafsen and Behringer, is due to gaps in the heated plate.