Model formalism of liquid 3He-B at equilibrium

The approximate formal treatment of the nuclear spin system of normal liquid ³He given some time ago is extended to the ordered ³He phase. The formalism leads to the prediction of normal thermal behavior of ³He-B at lower pressures and at temperatures approaching its phase-boundary temperatures. In contrast to the disordered normal liquid phase, which is thermally anomalous, the entropy of the ³He-B decreases on isothermal compression, or its isobaric volume expansion coefficient is positive. The equilibrium thermal behavior of ordered ³He-B is thus qualitatively different from that of disordered liquid ³He. Experimental control of these aspects of the liquid ³He phase transformation is lacking at the present time. Both early and new ³He-B paramagnetic susceptibility data, extended recently over a wide reduced-temperature range, disclose a fundamental competition between the spontaneous ordering mechanism responsible for the existence of ³He-B and the specific ordering process imposed upon this phase on application of an external constant and uniform magnetic field. As a consequence, magnetized ³He-B will be shown to increase its entropy on isothermal magnetization and to cool on adiabatic magnetization. The magnetocaloric effect is, however, only moderate. The competition of the ordering process leads to the delay or possibly even to the suppression of the formation of the ordered phase, a state of affairs foreseen in our earlier work. At low or moderate magnetic field strengths, the zero-field phase-boundary temperatures are shown to shift toward lower temperatures while, simultaneously, the order of the phase change decreases, from second order, in the absence of the field, to first order. Although of model-theoretic character, involving limitations of various types, the rich physical content of ³He-B at equilibrium clearly emerges in the present work.Work performed under the auspices of the U.S. Department of Energy.     

Surface tension of liquid3He at low temperature

The thermal contribution to the surface tension of liquid³He due to the single quasiparticle motion is estimated in the low-T regime using a local approximation for the entropy. The density and temperature dependence of the effective mass is shown to play a crucial role in determining the behavior of (T). The theoretical predictions explain the anomalous behavior of (T) recently observed at low temperature by Suzuki et al.³ Predictions for the temperature dependence of the interfacial tension of liquid³He-⁴He mixtures are also given.     

Measurements of the viscosity and the slip length on liquid3He at low temperatures

Measurements of the viscosity and the slip length are reported for both the normal liquid and superfluid³He within the slip approximation. We used the hollow torsional oscillator. For the normal liquid³He the slip length is shorter than the theoretical expectation and there is no anomaly near the superfluid transition, but the viscosity shows the anomalous deviation from the Fermi liquid behavior which is thought to be attributed to the superfluid fluctuation. For the superfluid³He, the temperature dependence of the slip length agrees with the theory qualitatively and the viscosity agrees well with the theory quantitatively as far as the slip approximation is valid.     

The Kapitza resistance to a variety of metallic surfaces below 0.3 K

The Kapitza resistance has been measured between liquid or solid ³ He or ⁴ He and surfaces of Mg, Cu, W, and Au in the temperature range 0.03–0.3 K. The experimental data are in relatively good agreement with the acoustic mismatch theory modified so as to include phonon attenuation in the metal. Below ～0.1 K the Kapitza resistance of liquid ³ He is often reduced relative to that of liquid ⁴ He, suggesting the presence of an enhanced thermal conduction between liquid ³ He and metals as has been observed in other laboratories.     

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.     

The Origin of the Magnetic Channel of the Thermal Boundary Resistance Between Liquid 3He and Ag Sinters at Very Low Temperatures

At temperatures below about 10mK, the temperature dependence of the thermal boundary resistance between liquid ³ He and metal sinters deviates significantly from the behaviour expected from the acoustic mismatch theory (R _K T ^–3). This behaviour is explained in terms of the existence of a magnetic channel for the coupling between the ³ He quasiparticles and the sinter; the origin of this coupling, however, is unknown so far. In our studies of the magnetic properties of Ag sinters made of powders of submicron grain size we have detected and identified a magnetically ordering subsystem which, although it contributes only a few ppms to the total mass, dominates the magnetic properties of the sinters. We present the results of our investigation of the magnetic properties of these Ag sinters which strongly deviate from the behaviour of bulk silver, and discuss a possible mechanism how a magnetically ordered subsystem may open up the magnetic channel of the thermal boundary resistance to liquid ³ He at very low temperatures. The main conclusion of our study is that our results should allow the tuning of the thermal boundary resistance between liquid ³ He and sintered metal heat exchangers at very low temperatures.     

Nuclear spin ordering of solid3He in applied magnetic fields

The effects of a magnetic field on nuclear spin ordering in solid³He have been observed through the influence of the field on the solid entropy. By making use of the Clausius-Clapeyron equation, the³He melting temperature as a function of pressure was determined in several magnetic fields from 0 to 1.2 T from measurements of the latent heat of solidification in a Pomeranchuk cell. The solid entropy as a function of temperature was then calculated from the melting curve slope in each field by again utilizing the Clausius-Clapeyron equation. The solid ordering manifested itself as a substantial reduction in entropy occurring over a narrow temperature interval. A portion of the magnetic phase diagram of solid³He was established by plotting the ordering temperature versus field. In low fields the reduction in entropy occurs over a very narrow temperature interval, and the ordering temperature is depressed by the field. Above about 0.41 T, however, the ordering region is broadened and moves to significantly higher temperatures with increasing fields. The behavior in low fields is tentatively interpreted as a magnetic phase transition from the paramagnetic to an ordered phase, while the effect in higher fields is believed to represent paramagnetic ordering of the nuclear spins by the applied field.Work supported in part by the National Science Foundation.     

The mechanical behavior of a vibrating wire in superfluid3He-B in the ballistic limit

An experimental procedure is described for the characterization of the response of very small vibrating wires to liquid³He at temperatures down to 120 µK. The relative scales of the mean free path in the liquid and the radius of the wire play a significant role in the interpretation of the results. It is shown that the same ideas concerning the transition from hydrodynamic to ballistic behavior as the temperature is reduced can be applied both to saturated³He-⁴He solutions and to superfluid³He-B.     

The Fermi Liquid theory of4He in3He

The Zharkov-Silin Fermi Liquid theory of solutions of⁴He in normal (non-superfluid) liquid³He is reviewed and slightly extended. The theory is expected to be valid only below 0.1 K, and it predicts that there should be a hundred-fold increase in the diffusion coefficient as the temperature is lowered into this region. The limited range of validity explains the apparent disagreement between the recent very low temperature measurements of the phase separation line by Nakamura et al. and extrapolations from higher temperatures. In the low temperature experiments the⁴He concentration X₄ is so small that there is no macroscopic phase separation, only a gradual thickening of the⁴He-rich film on the walls. We confirm that the phase separation temperature T_ps(X₄) estimated from the thickening is close to the values which would be observed in an ideal experiment with a macroscopic phase. Fits to T_ps(X₄) including the new data show that the⁴He effective mass m ₄ ^* is close to, and may be equal to, the bare mass m₄. The difference in binding at zero pressure between⁴He in liquid⁴He and in liquid³He is (E₄₄–EE₄₃)/k_B=(0.21+0.03/–0.01)) K. Using the volume measurements of Laheurte to calculate the pressure dependence of E₄₃ indicates that the difference in binding has a minimum of (0.0±0.2) K near 11 atm. This implies that the solubility of⁴He in³He is enhanced in this region of pressure. The behavior of the spinodal line at low temperature, and the possibility of observing Bose condensation in a metastable solution of⁴He in liquid³He are also discussed.     

Surface Magnetism of 3He in Grafoil

We report the magnetic susceptibility of ³He in Grafoil filled with pure liquid³He at 27.6 bar and at temperatures down to 0.1 mK with a cw NMR method. It is composed of two contributions: from the bulk liquid and from the adsorbed layer of ³He on the Grafoil surface. The latter shows a well-known strong ferromagnetic tendency and can be fitted to a Curie–Weiss law in the high temperature region. The obtained Weiss temperature is surprisingly large compared with the previous ones.     

Instabilities of the liquid solid interface

Bodensohn et al.¹ observed that a sudden temperature change of the liquid above 1.1 K caused the interface between solid and liquid⁴He to become corrugated; they proposed that the instability of the interface was caused by the introduction of a heat current induced by the temperature gradient. Subsequently Grinfeld² showed that a non hydrostatic stress of the crystal could cause an instability; this idea was put forward by Balibar, Edwards and Saam³ to explain the observation of Bodensohn et al. A more detailed analysis has been given recently by Bowley and Nozières⁴. The two causes of instability are analyzed in the present paper for both normal³He and superfluid⁴He. The phase diagram and critical heat current for normal³He are presented. The corrugations will appear most rapidly near the minimum of the melting curve at 0.32 K and need a temperature gradient of order a few µK/cm. For superfluid⁴He the inability of the liquid to support thermal conduction (heat is transported by second sound) changes the nature of the instability, with the result that dissipation at the interface becomes important. As a consequence only the Grinfeld instability is observable in practice so that corrugations appear with wavelength between 6 and 8 mm.     

Thermal conductivity of normal liquid3He at finite temperatures

Greywall's experimental data for the thermal conductivity of normal liquid³He are reanalyzed. The temperature dependence at various pressures seems consistent with a law of corresponding states in the sense that it can be described by a universal function of a reduced temperature. It is suggested that the appreciable dynamical screening of the scattering amplitude for the quasiparticles in³He is responsible for this behavior. For illustration the thermal conductivity is calculated at finite temperature for a class of screened interaction models to leading order in the screening parameter. Good agreement is found with experiment by fitting the single Landau parameterA _a ⁰ .     

Review paper: Heat transfer between liquid helium and solids below 100 mK

The many experimental determinations of heat transfer between liquid helium and solids are reviewed and compared with the existing theories. Generally heat transfer is a complex process at low temperatures, involving parallel heat paths with several resistances in series in each path. The standard theories for the individual resistances are reviewed and as far as possible a definite value of the expected resistance obtained. The experiments have been considered in five groups: cerium magnesium nitrate in liquid³He, cerium magnesium nitrate in dilute³He in liquid⁴He, metals in liquid³He, metals in dilute³He in liquid⁴He, and miscellaneous experiments. Where appropriate, the experimental results have been reevaluated in terms of standard models for heat transfer and presented as resistance versus temperature on diagrams showing the theoretical predictions. Between 20 and 100 mK many measurements show good agreement with acoustic mismatch theory for thermal boundary resistance. Below 20 mK most experiments have been made with finely divided solids (sintered metals or powdered paramagnetic salts); invariably the resistance has been anomalously low, with the metal/³He systems showingR T ^–1 and the metal/dilute³He in⁴He systems showingR T ^–3. The cerium magnesium nitrate/helium experiments have shown a temperature—independent resistance and sometimes anR T ^~1 dependence. These resistances have been attributed to the spin-lattice resistance in cerium magnesium nitrate. Evidence for heat transfer by magnetic coupling has been reviewed and it is concluded that the positive evidence has other explanations, while the lack of dependence upon helium pressure,³He phase, and large magnetic fields is strong negative evidence. If the disagreement of experimental results with standard theory is not to be attributed to magnetic coupling, then several theoretical questions remain to be answered; these questions are posed. Basically, at low temperatures the excitations in solids and liquid helium have wavelengths and mean free paths much larger than the size of the finely divided particles or pores between the particles. Thus theories for bulk solids and bulk liquid helium are not appropriate for describing excitations and their interactions at these low temperatures.     

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.     

Behavior of a vibrating wire near the liquid3He-vacuum interface and in a saturated3He-4He mixture

In this paper we describe the effect of the vicinity of a liquid³He-vacuum interface on the behavior of a vibrating wire viscometer. It was found that in the fluid near the liquid interface the quality factor is lower than in the bulk liquid. We further report on the observation of a doubling of the resonance peak of the wire in a saturated³He-⁴He mixture. The frequencies and amplitudes of the two peaks strongly depend on the distance between the phase boundary and the vibrating wire; the temperature and the velocity of the phase boundary have no significant influence on the peak frequencies. The observed peak doubling is attributed to the coupling of the vibrating wire with a standing second-sound wave in the dilute phase where the volume of the dilute phase can be regarded as a resonating cavity for second sound.     

Thermal resistance between a paramagnetic salt (CMN) and liquid helium at millikelvin temperatures

We report new measurements of the thermal resistance between cerium magnesium nitrate (CMN) and liquid helium. AtT=100 mK, the thermal resistanceR for powdered CMN in contact with liquid³He-⁴He or liquid³He is about 2000 times the Kapitza boundary resistance. From the field and temperature dependences ofR we conclude that it is dominated by the phonon-bottlenecked spin-lattice resistance. However, if high-purity liquid³He is in contact with CMN, we observe, below 20 mK, a magnetic boundary resistance in parallel to the spin-lattice resistance. This resistance is two orders of magnitude smaller than the spin-lattice resistance atT=10 mK, and is described byR T ^1.4. The absolute values of the magnetic boundary resistance, as well as the spatially limited phonon-bottlenecked resistance, are extremely sensitive to the surface treatment of the CMN crystals.     

Nanokelvin thermometry at temperatures near 2 K

We present a method of temperature measurement based on a liquid³He vapor pressure thermometer with a resolution of one part in 10⁹ of the absolute temperature over the range from 1.6 to 2.2 K. The thermometer, as well as apparatus suitable for the assessment of the resolution and stability of the device, are described in detail. A method for the determination of a fixed point on the temperature scale with a resolution of 2×10^?9 is presented. Two different procedures for monitoring the long-term stability of the thermometer are discussed. The present resolution and stability of the thermometer are an improvement by two orders of magnitude over conventional germanium resistance thermometry. Although this performance level is adequate for presently planned phase transition experiments using liquid⁴He, future improvements by yet another order of magnitude seem feasible and will bring the device within an order of magnitude of the thermal noise limit.     

Variational calculation of the binding energy of one3He impurity in liquid4He

We present a variational microscopic study of the liquid⁴He with one³He impurity at zero temperature, within the framework of the average correlation approximation. In this approach all calculations can be referred to quantities of liquid⁴He. The chemical potentials of liquid⁴He and of a³He impurity are studied in the liquid range of densities and the results show an overall good agreement with the experimental data. Detailed comparisons with previous calculations are also presented. Due to the presence of the pressure term, the calculation of chemical potentials is a severe test to the capability of the wave function and the interaction to obtain a good saturation density. The calculation of the chemical potential of the impurity using the Lennard-Jones and the Aziz potentials, points out the different balance of kinetic and potential energies obtained in the variational minimization of these two potentials.Supported by a Fulbright Fellowship.     

Thermal resistance between powdered cerium magnesium nitrate and liquid helium at very low temperatures

Experiments have been performed from magnetic temperatures of 2–20 mK on the effect of minute⁴He impurities and a magnetic field on the thermal resistance between powdered cerium magnesium nitrate (CMN) and liquid³He. The thermal resistance decreases with decreasingT and increasing small field but is increased dramatically at a givenT by the addition of roughly a monolayer of⁴He. The resistance is interpreted as resulting from a surface magnetic coupling between CMN and liquid³He.This work has been supported by the U.S. Atomic Energy Commission under Contract No. AT(04-3)-34, P.A. 143, and by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant No. AF-AFOSR-631-67A.