4He in Nano-porous Media: Superfluidity and Quantum Phase Transition

This paper reviews recent findings of novel phenomena in ⁴He confined to a nano-porous glass. We examined pressure–temperature (P-T) phase diagram of ⁴He confined in a porous Gelsil glass that had nanopores 2.5 nm in diameter, by torsional oscillator and pressure studies. The obtained phase diagram is fairly unprecedented the superfluid transition temperature approaches zero at 3.4 MPa, and a novel nonsuperfluid phase exists between the superfluid and solid phase. These observations indicate that the confined ⁴He undergoes a superfluid-nonsuperfluid-solid quantum phase transition at zero temperature. We propose that the nonsuperfluid phase may be a localized Bose-condensed state in which global phase coherence is destroyed by a strong correlation between the ⁴He atoms or by a random potential. ⁴He in nanospace is an excellent model system for studying a strongly correlated Bose liquid and solid in a confinement potential.     

Phase separation in solid mixtures of helium-3 and helium-4

Phase separation temperatures have been determined in bcc³He-⁴He mixtures as a function of³He concentration and melting pressure from measurements of changes in the X-ray lattice parameter and Bragg peak shape. A new rigid tail dilution refrigerator cryostat was used to study³He-⁴He crystals with³He concentrations of 0.10, 0.20, 0.30, 0.45, 0.60, and 0.70 and melting pressures between 3.0 and 4.3 MPa. The phase separation temperatures determined are in good agreement with regular solution theory and give little support for an asymmetry in the coexistence curve expected from a Nosanow-type model and reported from previous experiments using other signatures of phase separation. At a given concentration, differences in phase separation temperatures determined from slow cooling and warming data, respectively, are as much as 25 mdeg, but this is less than half the differences reported from previous experiments. A bcc-hcp transformation was seen in a crystal with 10%³He at aboutT=0.3 K for a melting pressure of3.7 MPa.     

Magnetic Phase Transitions in bcc 3He

We study magnetic phase transitions in the bcc solid ³He by means of Monte Carlo simulations. We employ a classical spin model on the bcc lattice with multiple ring-exchange interactions. In the present study, we take into account up to 4-spin ring exchanges. In order to clarify the character of phase transitions we examine energy histograms generated by Monte Carlo simulations. It is shown that the phase transition between the cnaf and the paramagnetic phases is discontinuous at low magnetic fields while it is continuous at high magnetic fields. This result agrees qualitatively with experimental observations.     

A solid state Pomeranchuk refrigerator

Recently we have shown that YbInCu₄ and related compounds present a solid state Pomeranchuk effect. At their first order volume transition, a local moment phase coexist with a renormalized Fermi liquid in analogy with ³He at its melting curve. This structural transition can be affected by pressure and magnetic field. We show here that as in ³He the Pomeranchuk effect can be used to produce cooling. We discuss the efficiency and different ways of implementing the solid state Pomeranchuk refrigerator.     

NMR study of the solidification and melting of3He in porous glasses

TheP-T phase diagrams of the liquid-solid phase transition of³He in three porous glasses with different pore sizes have been determined from spin-lattice relaxation measurements in the temperature range 0.5–4.2 K. The onset of solidification of³He in the pores occurs at excess pressure over the bulk phase transition. The excess pressure depends on the pore size. A model of the phase transition in small pores which takes into account the contribution of the surface energy to the free energy is described and compared with experimental results. TheT ₁ relaxation mechanism of³He in the pores is found to be due to the surface relaxation when³He is in the liquid phase and due to the relaxation of bulk solid³He when it is in the solid phase.     

Phase transitions in the growth of4He films

Using a microscopic, variational approach we examine the growth of⁴He absorbed to graphite and alkali substrates. We find that superfluid layers are formed and their behavior as a function of coverage is closely related to the one of a purely two-dimensional superfluid. The growth of a new layer undergoes a phase transition from a cluster formation into the connected superfluid when the coverage is increased. Based on the important connection to the two-dimensional fluid we propose a microscopic theory of quantum vortices in⁴He films at zero temperature, in which single vortices are treated as quasiparticles. We calculate the energy needed to create the single vortex, vortex inertial mass, microscopic interaction between vortices and binding energy of the vortex-antivortex pair as a function of density. We predict that at the⁴He superfluid density less than about 0.037 Å^–2 the binding energy of the pair becomes negative, indicating a phase transition into a new state where vortex-antivortex pairs are spontaneously created.     

Self Diffusion in Solid 4He and 3He-4He Mixtures Near the bcc-hcp Phase Transition

We report experiments on the plastic flow of solid ⁴ He and ³ He- ⁴ He mixtures of 1.4% and 2.8% near the bcc-hcp transition. Plastic flow was generated by moving a wire through a macroscopic single crystal. We found that the plastic flow rate both in pure ⁴ He and in mixture helium crystals is enhanced in vicinity of the bcc-hcp phase transition. The results are interpreted in terms of self diffusion in the solid. Values of the self diffusion coefficient D_s at the respective transition temperatures of pure ⁴ He and of the mixtures are very close, and reach that found in normal liquids. The activation energy for self diffusion in the mixtures is lower by up to 3 K than in pure ⁴ He. We suggest that similar to what is observed in bcc metals, self-diffusion in solid He takes place through phonon assisted atom-vacancy exchange. The enhancement of the diffusion near the bcc-hcp transition is a result of the softening of a short wavelength transverse phonon. The temperature dependence of the energy of the phonon calculated using our data is in accord with the Landau theory of a phase transition driven by a soft mode. Work hardening was observed in mixture crystals, but not in pure ⁴ He. This implies that ³ He impurities pin dislocation lines.     

Evolution of the Concentrated Phase in Separated Solid 3He–4He Mixtures

Experimental and theoretical investigations of the growth kinetics of BCC ³ He precipitates in isotopic phase separation of HCP ³ He– ⁴ He are carried out. Low frequency pulsed NMR is used to study the mixture with an initial ³ He concentration of 3.18% at a pressure 3.7 MPa on stepped cooling down into the separation region. The separation time constant is shown to decrease monotonically with cooling. The evolution of heterophase structure resulted from separation is investigated by solving an equation for spatially non-homogeneous order parameter of decomposing mixture. The obtained temperature dependence of separation time constant is found to be in good agreement with the experiment.     

Inelastic Neutron Scattering Studies of First Order Phase Transitions in bcc Solid 4He

We report inelastic neutron scattering studies of the [110] transverse phonon branches of bcc ⁴He near the bcc-hcp transitions on the solid-liquid coexistence line and near the melting transition. The question behind the experiment was whether these transitions in a quantum solid are in any way different from what one observes in usual materials. In contrast to large softening of the T₁ branch seen near the bcc-hcp transition in group IV metals, we found that the transverse phonons in bcc ⁴He do not soften at all. Altough visual studies of the crystals near the transition are consistent with a martensitic transformation, neutron scattering indicates that the transition in solid ⁴He is different than in metals. Thus, the mechanism of the bcc-hcp transition remains an open question. Similar study done near the melting transition indicates that none of the phonons measured in the present experiment is affected by melting, which rules out a mechanical instability of the bulk as a mechanism of melting. Finally, in addition to the phonons, we observed a new feature at q=0 and at an energy transfer of 1.23meV which we attribute to neutron scattering by point defects. Similarly to the phonons, this feature did not change near any of the phase transitions.     

Phase Separation Kinetics of Solid 3He-4He Mixtures

The kinetics of isotopic phase separation in solid mixture of ³ He in ⁴ He with the initial concentration 2.05 % at various molar volumes has been investigated by precise pressure measurements. It has been shown that during both stepped and fast cooldown into the metastable region the equilibrium of coexisting phases is described by the exponential law with a characteristic time constant , The value of is found to decrease as the molar volume increases and the temperature lowers. It confirms that the growth of the ³ He-rich phase is connected with nonthermally activated (quantum) diffusion in the gas of delocalized ³ He quasiparticles. The obtained experimental results can be described only qualitatively by current kinetic theory of binary quantum solid mixtures. The conditions permitting the realization of the isotopic phase separation during the time observed in the experiment are analyzed. The effective quantum diffusion coefficient providing required ³ He atoms transport is about an order of magnitude higher than the corresponding value measured in NMR experiments. These conditions are probably fulfilled at the big concentration gradient which takes place at isotopic phase separation. The corresponding kinetic theory should be developed.     

Kinetics of Liquid 3He Droplets in Solid 4He Matrix

Precise measurements of pressure in the crystal at constant volume were used to obtain the data on growth and dissolution kinetics of liquid ³ He droplets formed as a result of isotopic phase separation of solid ³ He- ⁴ He Mixtures. We studied several crystals with an initial ³ He concentration of 2.05% in the pressure range of 26–27 bar. It is shown that the growth of the liquid droplets during the stepwise cooling of the two-phase crystal is correctly described by the superposition of two exponential processes: diffusion decomposition with a small time constant and strain relaxation with a big time constant. The strain layer near the droplet boundaries is due to a great difference in molar volume between the droplets and the matrix, and leads to a plastic deformation of the matrix and to a non-equilibrium ³ He concentration in the matrix. Under such conditions quantum diffusion is significantly suppressed and ³ He atom transport occurs only as the strain is relaxed.     

Phase boundaries and domain boundaries in crystals

Domain boundaries, i.e. interfaces between different orientation variants of the same crystal species, and phase boundaries, i.e. interfaces between two different modifications of the same compound, exhibit rather similar features. This has been investigated by means of polarized light and X-ray topography for a series of structurally related sulphates which were grown as large single crystals from aqueous solution. The major results are as follows: (i) Domain interfaces frequently adopted only a few orientations which are low-energy boundaries of best structural fit. These preferred orientations may be parallel to low-indexed lattice planes or to non-crystallographic planes (W and W^t walls according to the classification of Sapriel,Phys. Rev. B12, 1975, 5128). Illustrations of such (transition-induced) domain boundaries in KLiSO₄, NH₄LiSO₄, RbLiSO₄, CsLiSO₄,(NH₄)₃H(SO₄)₂ will be presented. (ii) For many first-order transitions the phase boundaries prefer planes of minimum strains, i.e. low energy, which again may be low-index lattice planes or non-crystallographic planes. These preferred orientations can be calculated from the strain tensor of the transition with the relative lattice-parameter changes as tensor components). If the transition isotherm deviates from the minimum strain orientation, characteristic zigzag boundaries with segments parallel to the (symmetrically equivalent) preferred planes may result. Zigzag phase boundaries have been observed in RbLiSO₄ and {N(CH_{3 4}}₂ZnCl₄. (iii) The shape and the density of transition-induced domains is influenced by the orientation of the phase boundaries and its velocity of motion. For the 708 K. transition of KLiSO₄ and the 413 K transition of (NH_{4 3}H(SO_{4 2} (in both cases loss of the trigonal axis), among the minimum-strain domain boundaries those normal to the phase boundary are preferred. In N(CH₃)_{4 2}ZnCl₄, the domain density increases with the phase boundary velocity.     

Phase Separation Line for Solid Mixtures of 3He in 4He

The excess pressure due to the phase separation of solid mixtures of ³He in ⁴He held at a constant volume was measured and used for constructing the phase separation diagram of this system. We obtained high-quality homogeneous samples of the solid mixtures after several cycles of cooling down and heating up the two-phase crystal. This gave reliable and reproducible experimental data without hysteresis efects. We compared the phase diagram line obtained with various theoretical approaches, which describe the phase separation of the helium isotope mixtures. The regular solution model can not describe the experimental data well and neither can the asymmetrical Mullin's model. Good agreement is observed only with the theory of Edwards and Balibar which takes into account the difference between the crystal symmetry (hcp and bcc) of the coexisting phases.     

The Effect of 3He Impurities on Anomalous Oscillations in the Expansion of Solid 4He

The regular periodic intensity bursts recently observed in the expansion of solid ⁴He into vacuum have anomalies that have been attributed to a transition to some new solid phase induced by excess vacancies. Here it is shown that a small concentration of ³He, from 1% down to 0.1%, added to the ⁴He solid is sufficient to remove all the anomalies. The origin of the observed anomalies is discussed in the light of these new experiments.      

Superfluid Phases of 3He in a Periodic Confined Geometry

Predictions and discoveries of new phases of superfluid ³He in confined geometries, as well as novel topological excitations confined to surfaces and edges of near a bounding surface of ³He, are driving the fields of superfluid ³He infused into porous media, as well as the fabrication of sub-micron to nano-scale devices for controlled studies of quantum fluids. In this report we consider superfluid ³He confined in a periodic geometry, specifically a two-dimensional lattice of square, sub-micron-scale boundaries (“posts”) with translational invariance in the third dimension. The equilibrium phase(s) are inhomogeneous and depend on the microscopic boundary conditions imposed by a periodic array of posts. We present results for the order parameter and phase diagram based on strong pair breaking at the boundaries. The ordered phases are obtained by numerically minimizing the Ginzburg-Landau free energy functional. We report results for the weak-coupling limit, appropriate at ambient pressure, as a function of temperature T, lattice spacing L, and post edge dimension, d. For all d in which a superfluid transition occurs, we find a transition from the normal state to a periodic, inhomogeneous “polar” phase with $T_{c_{1}} < T_{c}$ for bulk superfluid ³He. For fixed lattice spacing, L, there is a critical post dimension, d _c , above which only the periodic polar phase is stable. For d<d _c we find a second, low-temperature phase onsetting at $T_{c_{2}} < T_{c_{1}}$ from the polar phase to a periodic “B-like” phase. The low temperature phase is inhomogeneous, anisotropic and preserves time-reversal symmetry, but unlike the bulk B-phase has only $\mathtt{D}_{\text{4h}}^{\text{L}+\text{S}}$ point symmetry.     

Effect of Crystal Quality on HCP-BCC Phase Transition in Solid 4He

The kinetics of HCP-BCC structure phase transition is studied by precise pressure measurement technique in ⁴He crystals of different quality. An anomalous pressure behavior in bad quality crystals under constant volume conditions is detected just after HCP-BCC structure phase transition. A sharp pressure drop of 0.2 bar is observed at constant temperature. The effect observed can be explained if we suppose that microscopic liquid droplets appear on the HCP-BCC interphase region in bad quality crystals. After the interphase region disappearance, these droplets are crystallized with pressure reduction. It is shown that this effect is absent in high quality thermal-treated crystals.     

Phase Diagrams of 4He on Flat and Curved Environments

By means of diffusion Monte Carlo calculations, we obtained the phase diagrams of a first and second layer of ⁴He on graphene and on the outside of different isolated armchair carbon nanotubes with radii in the range 3.42 to 10.85 Å. That corresponds to tubes between the (5, 5) and (16, 16) in standard nomenclature. In both cases, the ground state is either a liquid (second layer on graphene and on nanotubes whose radii is greater than ～7 Å) or an incommensurate solid (for thinner tubes). In the former case, upon a density increase, the system undergoes a first-order phase transition to another incommensurate solid. A study of the influence of the C–He potential (isotropic or anisotropic) on the phase diagrams is also presented.     

Phase equilibrium in a polarized saturated3He—4He mixture

We present experimental results on the phase equilibrium of a saturated³He−⁴He mixture, which has been cooled to a temperature of 10–15 mK and polarized in a⁴He circulating dilution refrigerator to a stationary polarization of 15%, 7 times higher than the equilibrium polarization in the external field of 7 T. The pressure dependence of the polarization enhancement in the refrigerator shows that the molar susceptibilities of the concentrated and dilute phase of a saturared³He-⁴He mixture are equal atp=2.60±0.04 bar. This result affects the Fermi liquid parameters of the dilute phase. The osmotic pressure in the dilute phase has been measured as a function of the polarization of the coexisting concentrated phase up to 15%. We find that the osmotic pressure at low polarization (<7%) agrees well with thermodynamics using the new Fermi liquid parameters of the dilute phase.     

Measurements of Helium Density in Aerogel near the Liquid-Vapor Critical Point

We investigate the behavior of ⁴ He confined in silica aerogels near the bulk liquid-vapor critical point. Using a new mechanical technique to measure the density of the confined ⁴ He along isotherms, we find that the density continuously increases from a low density phase up to a dense phase as the pressure is increased up to slightly below the bulk saturated pressure. An hysteretic behavior is observed between emptying and filling, which is not uniquely due to thermal problems. We argue that, our observations are more in favor of some kind of capillary condensation than of a genuine first order phase transition.     

Supersolidity and Superfluidity of Grain Boundaries

We have looked for dc mass transport through solid ⁴He in a simple experiment with two communicating vessels filled with solid ⁴He in equilibrium with liquid ⁴He. Through good quality crystals, we have observed no mass transport, in contradiction with the hypothesis of a Bose–Einstein condensation of vacancies. Through crystals containing grain boundaries, we have found superfluid flow along these grain boundaries. We discuss these results in the context of other experiments on supersolidity.