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.     

Elementary Excitations in Solid and Liquid 4He at the Melting Pressure

Recent discovery of a nonclassical rotational inertia (NCRI) in solid ⁴He below 0.2 K by Kim and Chan has revived great interest in the problem of supersolidity and initiated intensive study on the properties of solid ⁴He. A direct proof that the onset of NCRI corresponds to the supersolid transition would be the observation of a corresponding drop of the entropy of solid ⁴He below the transition temperature. We have measured the melting pressure of ultrapure ⁴He in the temperature range from 0.01 to 0.45 K with several single crystals grown at different pressures and with the accuracy of 0.5 μbar. In addition, supplementary measurements of the pressure in liquid ⁴He at constant volume have been performed, which allowed us to eliminate the contribution of the temperature-dependent properties of the pressure gauge from the measured melting pressure data. With the correction to the temperature-dependent sensitivity of the pressure gauge, the variation of the melting pressure of ⁴He below 320 mK obeys the pure T ⁴ law due to phonons with the accuracy of 0.5 μbar, and no sign of the transition is seen (Todoshchenko et al. in JETP Lett. 85:454, 2007). This sets the upper limit of ∼5⋅10⁻⁸ R for a possible excess entropy in high-quality ⁴He crystals below 320 mK. At higher temperatures the contribution from rotons in the superfluid ⁴He has been observed. The thermal expansion coefficient of the superfluid ⁴He has been measured in the range from 0.01 to 0.7 K with the accuracy of ∼10⁻⁷ 1/K, or by two orders of magnitude better than in previous measurements. The roton contributions to the melting pressure and to the pressure in liquid at a constant volume are consistent and yield the value of 6.8 K for the roton gap, which is very close to the values obtained with other methods. As no contribution due to weakly interacting vacancies to the melting pressure of ⁴He has been observed, the lower limit of about 5.5 K for their activation energy can be set.      

Nucleation of 4He Crystal at Melting Pressure by Acoustic Waves

No Heading When an acoustic wave pulse of several msec was applied to superfluid ⁴He at melting pressure, ⁴He crystal was nucleated on the transducer and rapidly disappeared after the pulse. We measured thresholds of the acoustic wave power which induced a nucleation in the temperature range between 0.09 K and 1 K. The treshold of the nucleation exhibited no temperature dependence below 0.3 K. The nucleation was not induced above 0.6 K even if the injected acoustic pulse was strong. 0.6 K is close to the inversion temperature at which the direction of the force by acoustic wave was inverted on the solid-liquid interface of ⁴He [Phys. Rev. Lett. 90, 075301 (2003)]. Our observation indicates that the effect of acoustic radiation pressure contributes to the heterogeneous nucleation by acoustic waves.PACS numbers: 67.80.–s, 68.08.–p, 81.10.–h, 43.25.Qp.     

Melting and Crystallization of 3He Inclusions in Two-phase Solid 3He-4He Mixtures

The peculiar features of the phase diagram for the ³ He- ⁴ He system make it possible to melt the ³ He inclusions formed during phase separation of the mixture by further cooling and to crystallize them in subsequent heating. The kinetics of these processes is studied on a sample with a molar volume of 20.54 cm ³ /mole (P=31.7 bar) using pressure measurements. The time dependence of the crystal pressure P(t) is measured on cooling at a rate of 10 mK/h followed by heating. The dependence P(t) has two distinct rises in pressure, the first rise being associated with the phase separation of the mixture and the second one with the melting of the ³ He inclusions formed. It is shown that the melting of the ³ He inclusions is almost complete after the fast cooling and the observed pressure jump is in good agreement with the corresponding change in the molar volume. The repeated crystallization of the inclusions is found to give rise to a large pressure gradient near the boundary of the inclusions, suppressing quantum diffusion considerably. This may result in an incomplete crystallization of the inclusions. The experimentally observed difference between the initial and final pressure in the sample corresponds to the fact that approximately 20% of the ³ He remains in the liquid state.     

Simultaneous Measurement of Torsional Oscillation and Ultrasound Propagation in Solid 4He at Low Temperatures

Preliminary results of simultaneous measurements of 10 MHz longitudinal ultrasound propagation in solid ⁴He loaded onto torsional oscillator (TO) are reported. Temperature dependence of sound velocity and attenuation and that of amplitude and frequency of TO are measured. The properties of dislocation lines present in the solid samples are extracted from the ultrasound data and are compared with the shifts in TO frequency below 100 mK.     

The Formation and Melting of 3He Clusters in Phase-Separated Solid 3He-4He Mixtures

Melting pressure measurements have been made on the ³ He-rich phase formed by cooling a solid mixture of 0.6% ³ He in ⁴ He through phase separation, at pressures between 2.78 and 3.56 MPa. For comparison, simultaneous observations were made with the mixture confined in the pores of a silver sinter and in an open volume connected to the same fill-line. Samples in the sinter cell solidify at higher pressures than the open cell and equilibrate more quickly. Both cells exhibit hysteresis between melting and freezing temperatures, and the transitions are broader in the open-volume cell. Relative to pure ³ He, the melting pressure is elevated by as much as 60 kPa in the sinter cell and 20 kPa in the open cell. The size of the pressure change on melting indicates that a large fraction of the ³ He remains solid at pressures below the melting curve, and this effect is more pronounced in the sinter cell. Measurements are in progress to measure the magnetisation through the magnetic ordering transition.     

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.     

Superfluid Vorticity and 1/f Noise in Melting of Solid 4He

One of the ways to reach a zero-vorticity state in superfluid ⁴ He is to melt solid helium slowly at temperatures well below T . This method takes advantage of the small density difference between the liquid and solid phases and of the high critical velocities for vortex formation at low T. However, pinning of the liquid-solid interface on the walls, revealed by 1/f-like pressure fluctuations of up to 1-10 bar in our setup, has consistently led to remanent vortex line densities of 4 · 10³ lines per cm ².     

Layer by Layer Growth of Solid 4He on Graphite down to 0.1 K

The growth of solid ⁴He on graphite from the superfluid phase is known to occur at pressures well below the bulk solidification point. The number of adsorbed layers increases with pressure and the solid growth undergoes non-continuous layer-by-layer growth at low temperature. We have studied this growth using the torsional oscillator method for isotherms down to 0.1?K. In contrast with simple layer-by-layer growth scenarios, our evidence suggests that the growth of adsorbed solid ⁴He is more complex and less solid is present on the graphite substrate at low temperature.     

Crystal Growth and Melting of Nuclear-Ordered Solid 3He

We measured crystal growth and melting rate of nuclear-ordered solid ³ He in a rectangular sample cell. The melting rate was much faster than the growth rate at all temperatures in the low field phase (U2D2) and strongly depended on temperature. The melting occurred on a rough surface and the dissipation mechanism of the melting was compared with the surface magnon scattering mechanism. The crystal growth rate did not depend on temperature and was compared with the spiral growth model associated with screw-dislocations. We found that a large chemical potential difference started to develop when the crystal grew large enough so that the sample became almost single domain at the upper part of the sample cell. We attributed this saturation to the pinning of the screw dislocation on the sample wall. This chemical potential difference decreased rapidly when the crystal was in the high field phase and the crystal grew as it did before the saturation. The crystal continued to grow even after the crystal returned to the U2D2 phase from the high field phase.     

Spin Diffusion Processes in Solid 3He-4He Mixtures Near the BCC-HCP Phase Transition at the Melting Curve

The spin diffusion coefficient of 1% ³He in solid of ⁴He has been measured in the vicinity of the BCC-HCP phase transition at the melting curve by pulsed NMR. The applied spin echo technique does allow to distinguish the contributions from all of coexisting phases. In addition to well-known diffusion in BCC, HCP, and bulk liquid phases the new fast diffusion process is observed and the diffusion coefficient of this process is shown to be close to that in liquid mixture being dependent on the time between the NMR pulses (bounded diffusion). The possible reason for the effect may be connected with formation of liquid droplets during the BCC-HCP transition.     

Nucleation of Solid 4He by Acoustic Waves

When an acoustic wave pulse was applied to metastable superfluid ⁴He above the megting pressure, a bulk ⁴He crystal was nucleated in the superfluid at low temperatures. Overpressure in which the metastable liquid could exist was investigated by changing the acoustic wave power. A larger power pulse could nucleate the crystal in smaller overpressures, which decreased linearly in the acoustic wave power.     

Analysis of the Melting Process of Magnetized Solid 3He

We report on our analysis of the melting process of highly magnetized solid ³ He. Akimoto et al. found that the solid-liquid interface becomes unstable during melting in a magnetic field of 9 T. The liquid then penetrates into the solid in the form of cellular dendrites. The instability, which was predicted by Puech et al., was attributed to a Mullins-Sekerka type of instability due to the magnetization gradient on the solid side. The measurements on which we base our analysis clearly show gradients on both sides of the interface. We present an extension of the linear stability analysis for this situation, as well as a numerical calculation of the dispersion relation of interface deformations. Our results are in good agreement with the experiment and explain the initial suppression of the instability, caused by the magnetization gradient in the liquid.     

On the Mobile Behavior of Solid 4He at High Temperatures

We report studies of solid helium contained inside a torsional oscillator, at temperatures between 1.07 K and 1.87 K. We grew single crystals inside the oscillator using commercially pure ⁴He and ³He-⁴He mixtures containing 100 ppm ³He. Crystals were grown at constant temperature and pressure on the melting curve. At the end of the growth, the crystals were disordered, following which they partially decoupled from the oscillator. The fraction of the decoupled He mass was temperature and velocity dependent. Around 1 K, the decoupled mass fraction for crystals grown from the mixture reached a limiting value of around 35%. In the case of crystals grown using commercially pure ⁴He at temperatures below 1.3 K, this fraction was much smaller. This difference could possibly be associated with the roughening transition at the solid-liquid interface.     

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.     

Evidence of Homogeneous Nucleation at Phase Separation in Solid Mixtures of 4He in 3He

A generalized relation for the characteristic time constants of mixture separation was found in the framework of the homogeneous nucleation theory. It allows us to describe the data on the separation kinetics at various concentrations c, molar volumes, and supercooling levels. Using precise pressure measurements, the separation time constants were obtained for 5 samples with the ⁴He concentration of 2.2–3.34% in the pressure range 31.5–38.6 bar. The data obtained for all the samples are in good agreement with the generalized relation if the impurity diffusion coefficient is proportional to c ^?4/3. The value of surface tension on the inclusion boundary was estimated from comparison between experiment and theory. It is in adequate agreement with the data obtained in other experiments. The obtained resugts are considered as evidence of homogeneous nucleation at phase separation of solid³He-⁴He mixtures.     

Measurements of Non-classical Rotational Inertia of Solid 4He at Two Frequencies

We have developed a double resonance compound torsional oscillator for studying non-classical rotational inertia (NCRI) of solid ⁴He at two different frequencies. The torsional oscillator consists of two beryllium copper torsion members, and two masses. The two masses rotate in phase in the first mode and out of phase in the second mode at resonance frequencies, 496 Hz and 1173 Hz, respectively. Samples of solid ⁴He (commercial grade with ³He impurity level less than 1 ppm) are grown with the blocked capillary method at pressures between 27 and 42 bar. Temperature dependences of NCRI signals at two different frequencies are observed in identical solid ⁴He at 37 bar.      

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.