Superfluidity of Pure 3He and Mixtures of 3He and 4He in Aerogel

We describe new torsional oscillator experiments on ³ He confined in 98.2% open aerogel. In one, we monitored the superfluid fraction of pure ³ He at T << T _c while we gradually changed the sample pressure. The resulting change in density alters ₀ of the superfluid ³ He relative to the distribution of the length scales (correlations) of silica in the aerogel. We observed a T = 0 normal-to-super fluid transition at a pressure of about 6.5 bar, in marked contrast to the bulk where liquid ³ He is superfluid at all pressures. In the second experiment, we measured the temperature dependence of the ³ He _s at a pressure of 21.6 bar with different amounts of ⁴ He present in the cell. Adding 2-3% ⁴ He slightly increases both T _c and _s . We found that for ⁴ He concentrations between 2% and 34%, the ³ He T _c increases by a very small amount. However, _s , which for pure ³ He in aerogel at 0.5T _c is no more than 11%, falls by another factor of 7. This behavior (constant T _c , reduced _s ) is similar to that observed in granular superconducting films where the long-range order is controlled by phase coherence between adjacent grains.     

Finite element method for thermomechanical response of near-incompressible elastomers

Summary The present study addresses finite element analysis of the coupled thermomechanical response of near-incompressible elastomers such as natural rubber. Of interest are applications such as seals, which often involve large deformations, nonlinear material behavior, confinement, and thermal gradients. Most published finite element analyses of elastomeric components have been limited to isothermal conditions. A basic quantity appearing in the finite element equation for elastomers is thetangent stiffness matrix. A compact expression for theisothermal tangent stiffness matrix has recently been reported by the first author, including compressible, incompressible, and near-incompressible elastomers. In the present study a compact expression is reported for the tangent stiffness matrix under coupled thermal and mechanical behavior, including pressure interpolation to accommodate near-incompressibility. The matrix is seen to have a computationally convenient structure and to serve as a Jacobian matrix in a Newton iteration scheme. The formulation makes use of a thermoelastic constitutive model recently introduced by the authors for near-incompressible elastomers. The resulting relations are illustrated using a near-incompressible thermohyperelastic counterpart of the conventional Mooney-Rivlin model. As an application, an element is formulated to model the response of a rubber rod subjected to force and heat.Notation A _i n _i /c - C Cauchy strain tensor - c VEC (C) - C ₁,C ₂ constants in Mooney-Rivlin model for elastomer - c ₂ VEC (C ²) - c _i eigenvalues ofC - c _e ,c _e , _e specific heat at constant strain - D _nl stiffness matrix due to the geometric nonlinearity - D _T ,D _T isothermal tangent modulus matrices - e VEC () - e _d VEC ( _d ) - f, f(T) thermal expansion function, =1/[1+(T-T ₀)/3] - f combined vector of nodal forces and heat fluxes - f _M consistent nodal force vector - f _T ,f _1T ,f _2T ,f _3T ,f _4T consistent heat flux vector - F deformation gradient tensor - g related tof _T - h time step - I _i invariants ofC - I 9×9 identity tensor - I identity tensor - i vectorial counterpart ofI:VEC(I) - J the Jacobian matrix in Newton iteration scheme - J determinant ofF - J _i invariants ofI ₁ ^–1/3 C - k, k(T) thermal conductivity - K tangent stiffness matrix - K _MM ,K _MT ,K _MP tangent stiffness submatrices - L,L _M ,L _P ,L _S lower triangular matrices related toLU decomposition ofK - M ₁,M ₂ strain-displacement matrices - N interpolation matrix - n surface normal vector - n _i (I _i /c) ^T - P matrix arising in theLU decomposition ofK - P the tension applied to the rubber rod - p (true) pressure - Q heat flux - q heat flux vector - r,r _M ,r _T ,r residual vectors - R,R ₁,R ₂,R ₃ matrices from thermal boundary conditions - R _s 1/2(R+R ^T ) - R _a 1/2(R–R ^T ) - R –R ₁+R ₂+R - R _s 1/2(R+R ^T ) - R _a 1/2(R-R ^T ) - s VEC() - S surface in undeformed configuration - t time - t traction referred to undeformed configuration - T, T ₀ temperature, reference temperature - T upper-shelf temperature in the surface convective relation - U upper triangular matrix inLU decompositionK - u displacement vector - v combined vector of nodal parameters - v _n value ofv at thenth time step - V volume in undeformed configuration - w strain energy density per unit undeformed volume - x position vector in deformed configuration - X position vector in undeformed configuration - volumetric thermal expansion coefficient - _c coefficient in the surface convective relation - ₁ strain-displacement matrix - _T interpolation matrix for thermal gradient: T - vector of nodal displacements - Lagrangian strain tensor - _d deviatoric Lagrangian strain tensor - interpolation function for - entropy per unit mass in the undeformed configuration - vector of nodal temperatures - þ isothermal bulk modulus - interpolation function forT - temperature-adjusted pressure,p/f ³(T) - mass density in the undeformed configuration - matrix arising inLU decomposition ofK - 2nd Piola-Kirchhoff stress - Cauchy stress tensor - , _M , _T , _c , ₀ Helmholtz free energy density function per unit mass - _i - _ij - vector of nodal values of - matrix arising in theLU decomposition ofK - near-incompressibility constraint function - internal energy density per unit mass - (·) variational operator - VEC(·) vectorization operator - symbol for Kronecker product of two tensors - tr(·) trace of a tensor - det(·) determinant of a tensor     

Nuclear Magnetism of Normal 3He and 3He-4He Mixtures in Aerogel

We report NMR experiments at 8 T on ³ He and ³ He- ⁴ He mixtures filling the pores of 95% porous aerogel, for temperatures T 6 mK. Magnetization measurements of pure ³ He reveal a localized layer approximately one monolayer thick. The longitudinal relaxation includes a component logarithmic in time, which is apparently associated with a fraction of the localized ³ He atoms. When the localized ³ He is displaced by adding ⁴ He the logarithmic relaxation disappears and T ₁ for the dominant exponential relaxation increases. Measurements of the spin diffusion coefficient with the aerogel filled with dilute solution in equilibrium with bulk phase-separated mixture provide an unambiguous determination of the spin mean free path,_s = 58 nm     

Anomalous temperature dependence ofD andT 2 for dilute solutions of3He in solid4He

Results of measurements of the spin diffusion coefficientD and NMR relaxation timesT ₁,T ₂, and T_1p are presented for a range of fractional³He concentrations 1 × 10^–4x ₃2.5 × 10^–3 in solid⁴He at molar volumes 19.85V _m21.0 cm³ and temperatures 0.4<T<2 K in both hcp and bcc phases. We observe a minimumD(T) atx ₃=5×10^–4, which is interpreted in terms of a transition from coherent impuriton motion to thermally activated diffusion. ForT<0.8 K, (lnD)/(lnV _m)=60±8. TheT ₂ measurements show a minimum as a function of temperature forx ₃10^–3. TheT ₂ (T) andT ₁ (T) results yield values for activation energy and tunneling frequency of vacancies in these dilute solutions. Forx ₃=5×10^–4 andT 0.5 K,T ₂ (V _m) is anomalous.T ₁ measurements at the same concentration indicate there is an important contribution to the spectral density of dipole field fluctuations in the kHz region.Financial support provided for apparatus, materials, and a research studentship (ARA) by the Science Research Council.     

3He diffusion in liquid isotopic mixtures

Spin echo techniques have been used to study the diffusion of³He in liquid isotopic mixtures with molar concentrations in the range fromX ₃=0.137toX ₃=0.850 over a temperature range from 3.0 to 0.4 K. AboveT the diffusive behavior is very similar to that of pure³He, although there is an increased scattering associated with⁴He. BelowT the diffusion coefficient increases with decreasing temperature and shows a dependence on the³He number density much like that of a classical gas. The excitations of⁴He have little influence on the³He diffusion coefficient, even at temperatures well above 1 K.The Rhodes Trust and the National Research Council of Canada supported one of us (R.B.H.) during the course of this research.     

Sound Modes of Superfluid 3He in Aerogel

We present the results of experiments on sound propagation at audio frequencies in ³ He-filled aerogel. Sound modes were observed at temperatures of 0.8–100 mK in an aerogel sample of 98% porosity. We find that below T _c for superfluid ³ He in the aerogel matrix the speed of sound in the composite system increases by as much as 1.5%. Also below the aerogel T _c new modes appear which correspond to propagation speeds of up to 10 m/s.     

Diffusive relaxation processes in liquid3He-4He mixtures. I. Normal phase

When a heat flux is switched on across a fluid binary mixture, steady state conditions for the temperature and mass concentration gradients T and c are reached via a diffusive transient process described by a series of terms modes involving characteristic times _n . These are determined by static and transport properties of the mixture, and by the boundary conditions. We present a complete mathematical solution for the relaxation process in a binary normal liquid layer of heightd and infinite diameter, and discuss in particular the role of the parameterA=k _T ² (/c) _T,P /TC _P,c coupling the mass and thermal diffusion. Herek _T is the thermal diffusion ratio, (/c) _T,P ^–1 is the concentration susceptibility, is the chemical potential difference between the components, andC _P,c is the specific heat. We present examples of special situations found in relaxation experiments. WhenA is small, the observable times (T) and (c) for temperature and concentration equilibration are different, but they tend to the same value asA increases. We present experimental results on four examples of liquid helium of different³He mole fractionX, and discuss these results on the basis of the preceding analysis. In the simple case for pure³He (i.e., in the absence of mass diffusion) we find the observed (T) to be in good agreement with that calculated from the thermal diffusivity. For all the investigated³He-⁴He mixtures, we observe (c) and (T) to be different whenA is small, a situation occurring at high enough temperatures. AsA increases with decreasingT, they become equal, as predicted. For the mixtures with mole fractionsX(³He)=0.510 and 0.603, we derive the mass diffusionD from the analysis of (c) and demonstrate that it diverges strongly with an exponent of about 1/3 in the critical region near the superfluid transition. As the tricritical point (T _t,X _t) is approached for the mixtureX=X _t0.675,D tends to zero with an exponent of roughly 0.4. These results are consistent with predictions and also with theD derived from sound attenuation data. We discuss the difficulties of the analysis in the regime close toT andT _t, with special emphasis on the situation created by the onset of a superfluid film along the wall of the cell forX=0.603 and 0.675.Work supported by grants from the National Science Foundation and the Research Corporation and by an A. P. Sloan fellowship to one of the authors (RPB).     

Thermal transport properties in the normal phase of dilute3He-4He mixtures

We present steady-state measurements of the thermal diffusion ratiok _T and of the heat conductivity for three dilute mixtures of³He in⁴He with concentrations 9×10^–3X(³He)5×10^–2 at saturated vapor pressure in the normal phase close to the superfluid transition. The data are compared with predictions by Dohm and Folk from the renormalization group (RG) theory. From auxiliary determinations of thermodynamic derivatives for these mixtures, we obtain the separation factor =–(k _T /T)×(/X) _T,P /(/T) _X,P above T over the range wherek _T is positive. Here is the mass density. From the transients of X(t) as a function of time, we obtain an estimation for the mass diffusion coefficientD and compare the results with predictions by Dohm and Folk and with results from other experiments.     

Spin diffusion in dilute,polarized 3He-4He solutions

Spin dynamics for arbitrarily polarized and very dilute solutions of ³He in liquid ⁴He are described. We began at a very fundamental level by deriving a kinetic equation for arbitrarily polarized dilute quantum systems based on a method due to Boercker and Dufty. This approach allows more controlled approximations than our previous derivation based on the Kadanoff-Baym technique. Our previous work is here generalized to include T-matrix interactions rather than the Born approximation. Spin hydrodynamic equations are derived. The general equations are valid for both Fermi and Bose systems. By use of a well-known phenomenological potential to describe the ³He-³He T-matrix we calculate longitudinal and transverse spin diffusion coefficients D and D _¦ and the identical-particle spin-rotation parameter . We confirm that these two diffusion constants differ at low T with D approaching a constant as T 0, and D_¦~1/T². Estimates of errors made by our approximations are considered in detail. Good agreement is found in comparison with data from both Cornell University and the University of Massachusetts. We find that the s-wave approximation is inadequate and that mean-field corrections are important. Comparison is also made between theory and the recent UMass viscosity measurements.     

Coherent diffusion of isotopic impurities in solid helium

A theory of coherent diffusion (impuriton propagation) of dilute isotopic impurities in solid He is described. An expression for the mean square distance R ² (t) traveled by the impurity is derived. For short times pure band propagation occurs and so R ² (t) grows as t ². For times longer than a characteristic impurity-phonon scattering time this time dependence becomes linear, and a diffusion constant can be defined. This characteristic scattering time is estimated to be ~7 × 10^–5 T ^–9 sec (T in K) for a ³He impurity in ⁴He at V _m = 21 cm³/mole. The corresponding diffusion constant is consistent with that measured in the NMR experiment of Allen and Richards if the ³He-⁴He tunneling frequency J ₃₄ is taken as 7 × 10⁵ sec^–1.     

3He in 99.5% Porous Aerogel at the Normal-Superfluid Transition

We have used a torsional oscillator to measure the superfluid density and dissipation near the superfluid transition of ³ He in aerogel of 99.5% porosity. We used a new cell (constructed at Penn State) for which the aerogel was grown in the pores of a 100 m silver sinter. The cell was tested with ⁴ He and showed no signs of the second-sound resonances that have interfered with previous torsional oscillator measurements. The measurements with ³ He, presented here, were taken at pressures of 1.34 and 4.13 bars. We observed values of _s / in the T 0 limit of 0.05 and 0.14 respectively. Our measurements show an increase in the dissipation on warming through T _c . This series of measurements is ongoing and temperature sweeps at various pressures are planned.     

Transport properties of helium near the liquid-vapor critical point. III. Thermal diffusion ratio,conductivity, and relaxation times of3He-4He mixtures

Measurements of the density and temperature gradient induced by a heat flux in two³He-⁴He mixtures along their critical isochore are reported. From these measurements, the thermal diffusion ratiok _T and the thermal conductivity are derived. In addition, similar measurements were made for³He. It is found that for the mixturesk _T and diverge with respective exponents of =1.23 and 0.6, which differs from the asymptotic predictions for binary mixtures. It therefore appears that the crossover to the expected mixture behavior has not been reached within the investigated range 10^–1>(T-T _c )/T _c >7×10^–4, where the limit close toT _c is set by the very long observed relaxation times. The characteristic times for the establishment of steady state conditions in the density and temperature gradients () and (T) are reported. Within experimental uncertainty they are found to be the same and nearly independent of isotopic composition. They are compared with the predictions from the hydrodynamic diffusion equations that lead to the characteristic times of the diffusion modes. The slowest calculated times are found to be in fair agreement with the observed ones.     

Susceptibility and Pressure Study of 3He nano-Clusters

Simultaneous measurements of pressure and magnetic susceptibility have been made in ³ He nano-clusters embedded in a ⁴ He matrix, following phase separation of the mixture. Susceptibility measurements extend from 0.5 mK to 10 mK for three different samples, which either undergo partial melting upon further cooling, or separate with liquid already present. The magnetic behavior of the clusters indicates solid fractions of 77, 54, and 19%, respectively, for pressures of 3.36 MPa, 3.06 MPa, and 2.88 MPa. The susceptibility of the 3.36 MPa sample follows a Curie law to the lowest temperature. For 3.06 MPa, we observe a kink in at 1.1 mK, which is approximately the ordering temperature T_N of the pure bulk ³ He if it existed at this pressure. However is almost constant down to 0.6 mK, with no drop at 1.1 mK, and no frequency shift, within our resolution of 10 Hz. Thus if there is magnetic ordering at 1.1 mK it is quite different than for bulk ³ He, not the U2D2 phase. For 2.88 MPa, follows a Curie-Weiss law with a positive Weiss =140 K, indicative of a ferromagnetic tendency, similar to that seen in 2D films.     

Nuclear magnetic relaxation of3He gas. II.3He-4He mixtures

Longitudinal relaxation timesT ₁ have been measured in³He-⁴He gas mixtures, using pulsed NMR, in the temperature range 0.6–15 K. Helium-3 number densities of the order of 10²⁴ atoms m^–3 were used. Relaxation takes place on or near the walls of the Pyrex sample cells and measurements ofT ₁ give information about the surface phases. A cryogenic precoating of solid molecular hydrogen was used to reduce the helium-substrate binding energy from 100 K on Pyrex to 13 K for³He and 15 K for⁴He. TheT ₁ data at high temperatures were similar to those observed previously in the pure³He-H₂ system. The presence of⁴He generally causedT ₁ to rise on cooling below 2 K due to the preferential adsorption of⁴He over³He at the surface. However,³He atoms that go into quasiparticle states in the superfluid helium film can be an extra source of relaxation. In uncleaned cells, relaxation probably takes place in quasiparticle states at the free surface of the superfluid film, which are bound with an energy of 5.1±0.3 K. Baking the Pyrex cells under vacuum and rf discharge cleaning the walls before sealing in the sample gas were found to increase the bulk gasT ₁ by two or three orders of magnitude. In a cleaned, sealed cell aT ₁ of 8 h was measured at 7.7 MHz and 0.8 K. In this case relaxation is probably occurring two or three helium layers away from the helium-hydrogen interface. It may be possible to observe a predicted minimum in the intrinsic dipolarT ₁ of the bulk gas by using a⁴He wall coating to suppress wall relaxation effects (which usually dominate the nuclear relaxation of the bulk gas).     

The viscosity and some related properties of liquid3He at the melting curve between 1 and 100 mK

The viscosity of liquid ³ He has been measured along the melting curve from 1 to 100 mK by means of a vibrating wire viscometer. In the normal Fermiliquid region we find 1/T² = 1.17–3.10T, where is in P and T in K. At the transition temperature T _A = 2.6 mK a rapid decrease occurs in _n , the viscosity of the normal component. Within 0.3 mK below T _A , _n decreases to about 25% of _A, but then becomes essentially constant. In the B phase _n first decreases to 20% of _A and then seems to increase below 1.4 mK. Data on _n , the density of the normal component, are also presented in the A and B phases. The results show that viscous flow is accompanied by a flow of zero dissipation, thus proving superfluidity in the A and B phases. The viscosity data at magnetic fields up to 0.9T have been related to theoretical calculations of the energy gap of superfluid ³ He near T _A . The splitting of the A transition and the suppression of the B phase in an external field were also measured.     

Shear viscosity measurements of liquid 3He-4He mixtures above 0.5 K

Simultaneous measurements of () and of the molar volume are reported for liquid mixtures of ³He in ⁴He over the temperature range between 0.5 and 2.5 K. Here is the shear viscosity and is the mass density. In the superfluid phase, the product of the normal components, _n and _n , is measured. The mixtures with ³He molefractions 0.30 < X < 0.80 are studied with emphasis on the region near the superfluid transition T and near the phase-separation curve. Along the latter, they are compared with data by Lai and Kitchens. For X > 0.5, the viscosity singularity near T becomes a faint peak, which however fades into the temperature-dependent background viscosity as X tends to the tricritical concentration X _t. Likewise, no singularity in is apparent when T _t is approached along the phase separation branches _– and ₊. Furthermore, viscosity data are reported for ³He and compared with previous work. Finally, for dilute mixtures with 0.01 X 0.05, the results for are compared with previous data and with predictions.     

Conductivity of the Wigner Solid on the Free Surface of Superfluid 3He-B

We have measured the conductivity of the 2D electron crystal (the Wigner solid) trapped on the free surface of superfluid ³ He-B down to 230 K. The resistance R(T) greatly decreases as temperature T decreases, obeying Arrhenius' law, R(T) exp (–(T)/k _B T), with (T) which we assign to the superfluid energy gap. This reveals that the scattering of ³ He quasiparticles by the WS determines the transport. As the input voltage increases, an anomalous behavior dominates the resistance just below the superfluid transition. The WS provides a new experimental method for probing both the bulk and free surface of superfluid ³ He.     

Novel Oscillating Aerogel Experiments in Superfluid 3He at Ultralow Temperatures

We present novel experiments on a disk of 98% aerogel oscillating in superfluid ³ He at ultralow temperatures. The aerogel dik is attached to a goal post shaped vibrating wire resonator and immersed in liquid ³ He cooled by a Lancaster style nuclear cooling stage. At low pressures we see no evidence for superfluidity within the aerogel down to our base temperature of below <0.11T_c. At higher pressures we observe large temperature dependent frequency shifts, reminiscent of torsional oscillator experiments. We find the transition temperature at 5 bar to be around 600K. The response of the resonator is highly non linear when the helium in the aerogel is superfluid. The resonant frequency decreases strongly with increasing wire amplitude. This offers an exciting new technique for measuring the superfluid properties of ³ He in aerogel in the ultralow temperature regime.     

NMR Studies on 3He-3He and 3He-4He tunneling motions in solid 3He-4He mixtures

Helium-3 nuclear spin relaxation times T ₁, T ₂, and T ₁have been measured for ³He-⁴He solid mixtures at the exchange plateau region (~0.5K). The ³He concentrations X ₃of the samples were 7.2, 2.9, 1.8, 1.4, 0.67, 0.65, and 0.22%, and their molar volumes varied between 19.9 and 20.9cm³/mole in hcp phase. The spectral density function J() for dipolar field fluctuations was determined in the low-frequency branch from T ₁measurements and in the high-frequency branch from conventional T ₁measurements. It was found that J() is given by J() = cJ()|_3–4 + (1–c)J()|_3–3, where J()|_3–4 is the spectral density function due to the ³He-⁴He tunneling motions, and J()|_3–3 is that due to the ³He-³He tunneling motions. Using the Torrey theory, the correlation frequency of the ³He-⁴He tunneling motions was evaluated from T ₁data, and was found to be in good agreement with Landesman 's theory.Supported in part by the Japan Society for the Promotion of Science through a grant to Y.H.     

Effects of temperature and magnetization on the maximum solubility of 3He in 4He

The laws x^m=x^m(T=0)(1+ _T T²) and x^m=x^m(B=0)(1+ _M B²), giving the effects of temperature and magnetization on the maximum solubility of ³He in ⁴He, are derived using the theory of normal Fermi systems. The parameters _T and _M are expressed in terms of measurable quantities, such as the specific heat and the magnetic susceptibility of the pure and dilute phases. Our relations are consistent with the experimental measurements of _T at SVP and predict a significant dependence of _T on pressure. Estimates of the parameter _M are given. The effects of temperature and magnetization on the osmotic pressure are also discussed.INFN sezione di Padova, gruppo collegato di Trento.