Nonlinear Ultrasound Propagation in Solid $$^4\hbox {He}$$ Compared with Shear Modulus Experiments

Ultrasound attenuation (\(\alpha \)) and velocity (V) at 9.6 MHz are measured in polycrystalline hcp \(^4\hbox {He}\). The ultrasound signal above 200 mK is linear and understood in terms of resonant vibration of dislocation segments pinned between network nodes with an average pinning length of 3.7 \(\mu \hbox {m}\), much shorter than 59 \(\mu \hbox {m}\) estimated from a shear modulus measurement. Dramatic changes in \(\alpha \) and V are observed below 200 mK. The changes are strongly dependent on temperature and are nonlinear and hysteretic. These effects result from pinning of dislocations by \(^3\hbox {He}\) impurities (nominal concentration of 0.3 ppm). The dislocation damping constant due to thermal phonons, the binding energy between dislocation and \(^3\hbox {He}\), and the average network pinning length obtained from the ultrasound data are compared with those from the shear modulus experiments.     

Charge Transport in Solid 4He

With the help of a new experimental technique strong anisotropy of the mobility of ions has been observed in hcp ⁴ He crystals. The mobility of positive ions in the direction of the six-fold axis is found to be 200 times higher than in the perpendicular direction. Activation energies of the mobility have been measured in both principal directions. They are equal to: 5.3K in the direction of the C₆-axis and 11K in the perpendicular direction. The behaviour of the ions' velocity in the strong electric field regime is also studied as a function of the orientation.     

Vortex Dynamics in hcp Solid 4He

The peculiar features noted in (Penzev et al. in Phys. Rev. Lett. 101: 065301, 2008) we conjecture are evidence of a vortex fluid state in solid He. We suggest to analyze this state by means of the dynamics of quantized vortices, as used for the tangle of vortices in superfluid turbulence. We introduce parameters of the vortex tangle dynamics, e.g., relaxation time for the drift of lines in parallel to the torsional oscillation axis. We briefly discuss the transition from the supposed vortex fluid state into the supposed supersolid state (Shimizu et al. in Phys. Rev. Lett., arXiv:0903.1326).     

A New Anomaly in Solid 4He

A new anomaly in the acoustic properties of solid⁴He has been observed when as little as a few parts per million of³He impurities are added. We show that the specific properties of the anomaly, can be explained by a continuous (second order) phase transition from a Bose condensed state above a critical temperature to a normal state below it. In this model the Bose particles which condense are long lived, thermally activated excitations.     

The Annealing Process in Solid 4He

We have used a torsional oscillator with square cross section and a resonance frequency of 185 Hz to confirm the nonclassical rotational inertia (NCRI) discovered by Kim and Chan (Nature 427:225, 2004; Science 305:1941, 2004). We have also found a strong correlation between the NCRI signal and a high dissipation Q ^?1 of 4×10^?6 of the oscillation above the transition temperature. Here, we present preliminary results of the annealing process in ⁴He at a pressure of 26 bar. When holding the temperature constant above 1 K we have observed a immediate rise in the period and a slow decay of the dissipation. The equilibrium value of Q ^?1 decreases with increasing temperature.     

Isochoric Compressibility of Solid 4He

We have performed experiments in which we inject atoms into hcp, solid ⁴He contained in a cell of fixed volume at pressures greater than the bulk melting pressure. We measure the change in pressure of the solid in response to the injection, which gives a measure of the isochoric compressibility. We show that at T??700?mK the solid undergoes very little growth and is incompressible. With decreasing temperature the compressibility rises, saturates near T??400 mK and may show weak evidence of a decrease near T??250 mK. Measurements at lower temperatures are necessary to fully test the predictions.     

Elastic Properties of Solid 4He

The shear modulus of solid ⁴He increases below 200 mK, with the same dependence on temperature, amplitude and ³He concentration as the frequency changes recently seen in torsional oscillator (TO) experiments. These have been interpreted as mass decoupling in a supersolid but the shear modulus behavior has a natural explanation in terms of dislocations. This paper summarizes early ultrasonic and elastic experiments which established the basic properties of dislocations in solid helium. It then describes the results of our experiments on the low temperature shear modulus of solid helium. The modulus changes can be explained in terms of dislocations which are mobile above 200 mK but are pinned by ³He impurities at low temperature. The changes we observe when we anneal or stress our crystals confirm that defects are involved. They also make it clear that the shear modulus measured at the lowest temperatures is the intrinsic value—it is the high temperature modulus which is reduced by defects. By measuring the shear modulus at different frequencies, we show that the amplitude dependence depends on stress in the crystal, rather than reflecting a superfluid-like critical velocity. The shear modulus changes shift to lower temperatures as the frequency decreases, showing that they arise from a crossover in a thermally activated relaxation process rather than from a true phase transition. The activation energy for this process is about 0.7 K but a wide distribution of energies is needed to fit the broad crossover. Although the shear modulus behavior can be explained in terms of dislocations, it is clearly related to the TO behavior. However, we made measurements on hcp ³He which show essentially the same modulus stiffening but there is no corresponding TO anomaly. This implies that the TO frequency changes are not simply due to mechanical stiffening of the oscillator—they only occur in the Bose solid. We conclude by pointing out some of the open questions involving the elastic and TO behavior of solid helium.     

Vortex-Loops and Solid Nucleation in Superfluid 4He and 3He

We propose a new model for the nature of the nucleation of solid from the superfluid phases of ⁴He and ³He. Unique to the superfluid phases the solid nucleation involves an extremely fast solidification front. We depart from the usual quasi-static treatment of solid nucleation by proposing that the nucleation of a solid seed is helped by the simultaneous nucleation of vortex-loops in the superfluid around it. It is the composite entity which is nucleated out of the over-pressurized liquid. This occurs when the local release of pressure creates a velocity field in the superfluid which in turn facilitates the nucleation of vortex-loops. The kinetic energy gain of this process balances the surface tension, as the solid surface is quickly covered by many vortex-loops (hairy snow-ball). We show that this scenario gives good agreement with many experiments on heterogeneous nucleation, where the energy barrier is found to differ with the classical theory of homogeneous nucleation by 8 orders of magnitude. We propose several experiments that could show the involvement of vortices with solid nucleation.     

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.     

Study of Solid 4He in Two Dimensions

Defects are believed to play a fundamental role in the supersolid state of ⁴He. We report on studies by exact Quantum Monte Carlo (QMC) simulations at zero temperature of the properties of solid ⁴He in presence of many vacancies, up to 30 in two dimensions (2D). In all studied cases the crystalline order is stable at least as long as the concentration of vacancies is below?2.5?%. In the 2D system for a small number, n _v , of vacancies such defects can be identified in the crystalline lattice and are strongly correlated with an attractive interaction. On the contrary when n _v ?10 vacancies in the relaxed system disappear and in their place one finds dislocations and a revival of the Bose-Einstein condensation. Thus, should zero-point motion defects be present in solid ⁴He, such defects would be dislocations and not vacancies, at least in 2D. In order to avoid using periodic boundary conditions we have studied the exact ground state of solid ⁴He confined in a circular region by an external potential. We find that defects tend to be localized in an interfacial region of width of about 15 ?. Our computation allows to put as upper bound limit to zero-point defects the concentration 3×10^?3 in the 2D system close to melting density.     

Quantized Vortex State in hcp Solid 4He

The quantized vortex state appearing in the recently discovered new states in hcp ⁴He since their discovery (Kim and Chan, Nature, 427:225–227, 2004; Science, 305:1941, 2004) is discussed. Special attention is given to evidence for the vortex state as the vortex fluid (VF) state (Anderson, Nat. Phys., 3:160–162, 2007; Phys. Rev. Lett., 100:215301, 2008; Penzev et al., Phys. Rev. Lett., 101:065301, 2008; Nemirovskii et al., arXiv:0907.0330, 2009) and its transition into the supersolid (SS) state (Shimizu et al., arXiv:0903.1326, 2009; Kubota et al., J. Low Temp. Phys., 158:572–577, 2010; J. Low Temp. Phys., 162:483–491, 2011). Its features are described. The historical explanations (Reatto and Chester, Phys. Rev., 155(1):88–100, 1967; Chester, Phys. Rev. A, 2(1):256–258, 1970; Andreev and Lifshitz, JETP Lett., 29:1107–1113, 1969; Leggett, Phys. Rev. Lett., 25(22), 1543–1546, 1970; Matsuda and Tsuneto, Prog. Theor. Phys., 46:411–436, 1970) for the SS state in quantum solids such as solid ⁴He were based on the idea of Bose Einstein Condensation (BEC) of the imperfections such as vacancies, interstitials and other possible excitations in the quantum solids which are expected because of the large zero-point motions. The SS state was proposed as a new state of matter in which real space ordering of the lattice structure of the solid coexists with the momentum space ordering of superfluidity. A new type of superconductors, since the discovery of the cuprate high T _c superconductors, HTSCs (Bednorz and Mueller, Z. Phys., 64:189, 1986), has been shown to share a feature with the vortex state, involving the VF and vortex solid states. The high T _c s of these materials are being discussed in connection to the large fluctuations associated with some other phase transitions like the antiferromagnetic transition in addition to that of the low dimensionality. The supersolidity in the hcp solid ⁴He, in contrast to the new superconductors which have multiple degrees of freedom of the Cooper pairs with spin as well as angular momentum freedom, has a unique feature of possessing possibly only the momentum fluctuations and vortex ring excitations associated with the possible low dimensional fluctuations of the subsystem(s). The high onset temperature of the VF state can be understood by considering thermally excited low D quantized vortices and it may be necessary to seek low dimensional sub-systems in hcp He which are hosts for vortices.     

Instability of Vacancy Clusters in Solid 4He

The behavior of pairs and clusters of vacancies in solid ⁴He crystals is studied with diffusion Monte Carlo method. We use a trial function suitable for describing solid ⁴He with long range order, arbitrary number of unoccupied sites and explicit Bose symmetry. It is found that vacancy clusters are unstable and collapse in all considered systems. We find no signature of stability of bound vacancy clusters of any size. Vacancies are found to exhibit weak attraction and a rough estimation of the binding energy is reported.     

Acoustic Resonances and Non-linearity in Solid 4He

Direct measurements of the low frequency shear modulus of solid ⁴He show a remarkable increase below 150?mK, a regime where torsional oscillator (TO) experiments show evidence of mass decoupling. Acoustic resonance measurements at higher frequencies confirm the unusual elastic behavior. A striking feature of both TO and shear modulus measurements is their amplitude dependence—the low temperature anomalies are reduced as the drive amplitudes increase. We have studied the amplitude dependence of acoustic resonances in both standard and isotopically pure ⁴He crystals. The resonance peaks shift to lower frequencies as the drive amplitude increases. At high amplitudes the peaks are asymmetric and exhibit both bistability and hysteresis—classic features of nonlinear oscillators. At the lowest amplitudes the peak frequency and shape are independent of drive amplitude and are non-hysteretic. The threshold for nonlinear behavior is lower for the isotopically pure crystal. We also studied the effects of annealing on the acoustic resonances. The nonlinear behavior was qualitatively unchanged but annealing often affected the shape of the resonance peaks, which often had a complicated structure at low temperatures.     

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.     

Dielectric Constant Measurements of Solid 4He

Careful measurements of the dielectric properties of solid ⁴He have been carried out down to 35 mK, considerably lower than the temperature range of previous studies. The sample was prepared from high purity gas with ³He concentrations of the order of 200 ppb and were formed by the blocked capillary method. The molar volume of the sample was 20.30 cm³. The dielectric constant of the samples was found to be independent of temperature down to 120 mK before showing a continuous increase with decreasing temperature and saturating below 50 mK. The total increase in ?? is 2 parts in 10^?5. The temperature dependence of ?? mimics the increase in the resonant frequency found in the torsional oscillator studies and also the increase found in the shear modulus measurements.     

Threshold Effect During Dissolution of 3He Inclusions in Solid 4He

A pressure jump has been found at the onset of the dissolution of bcc inclusions in separated solid³He - ⁴He mixture if the crystal is overheated above a certain critical value. This effect can be explained in the framework of a multistage dissolution process model.     

Flow of Solid 4He Near Melting

Torsional oscillator measurements on solid ⁴He have demonstrated non-classical rotational inertia (NCRI), indicative of a supersolid phase transition. Recent experiments indicate that the NCRI fraction depends on isotopic purity and perhaps on details of crystal growth and annealing, suggesting that defects may be involved. Our recent experiments have shown that solid helium does not flow in response to pressure gradients at low temperatures. Close to the melting temperature we do observe mass flow, but it decreases rapidly with temperature. For solid helium in the pores of Vycor the flow appears to be thermally activated and disappears below about half the melting temperature. Flow in bulk helium is restricted to a much narrower temperature range. Very close to melting (within 20 mK) the flow completely eliminates pressure differences in less than a minute. At slightly lower temperatures we saw flow, but significant pressure differences remained even after annealing.