Interpreting Torsional Oscillator Measurements: Effect of Shear Modulus and Supersolidity

The torsional oscillator is the chief instrument for investigating supersolidity in solid ⁴He. These oscillators can be sensitive to the elastic properties of the solid helium, which show anomalies over the same range of temperature in which the supersolid phenomenon appears. In this report we present a detailed study of the influence of the elastic properties of the solid on the periods of torsional oscillators for the various designs that have been commonly employed in supersolid measurements. We show how to design an oscillator which measures supersolidity, and how to design one which predominantly measures elasticity. We describe the use of multiple frequency TOs for the separation of the elastic and supersolid phenomena.     

Transverse Ultrasound Measurements in 4He Single Crystals

Recently, Kim and Chan (Science 305:1941, 2004; Phys. Rev. Lett. 97:115302, 2006) have reported an anomalous decoupling transition of solid ⁴He in a torsional oscillator measurement, and interpret their results as evidence for non-classical rotational inertia and a possible supersolid phase of ⁴He. The detailed nature and properties of such a “supersolid” state in ⁴He are still far from being clear, although there are clues from experiments involving ³He impurities, different sample cell geometries, annealing effects and grain boundary flow. Defects produced during crystal growth or deformation (e.g. dislocations) may affect supersolidity, or even produce it, and they are expected to have significant impact on the elastic properties of the solid. The supersolid fraction could also decouple from the lattice and produce a decrease in the transverse sound speed. We have begun the experiments in this laboratory to study such effects, measuring the velocity and attenuation of transverse ultrasound at 10 MHz in ⁴He single crystals grown at constant pressure.      

Observation of Non-Classical Rotational Inertia in Bulk Solid 4He

In recent torsional oscillator experiments by Kim and Chan (KC), a decrease of rotational inertia has been observed in solid ⁴He in porous materials (Kim, E., Chan, M.H.W. in Nature 427:225, 2004; J. Low Temp. Phys. 138:859, 2005) and in a bulk annular channel (Kim, E., Chan, M.H.W. in Science 305:1941, 2004). This observation strongly suggests the existence of “non-classical rotational inertia” (NCRI), i.e. superflow, in solid ⁴He. In order to study such a possible “supersolid” phase, we perform torsional oscillator experiments for cylindrical solid ⁴He samples. We have observed decreases in rotational inertia below 200 mK for two solid samples (pressures P=4.1 and 3.0 MPa). The observed NCRI fraction at 70 mK is 0.14%, which is about 1/3 of the fraction observed in the annulus by KC. Our observation is the first experimental confirmation of the possible supersolid finding by KC.     

Observation of Non-Classical Rotational Inertia in Solid 4He Confined in Porous Gold

No Heading Recent torsional oscillator measurements on solid ⁴He confined in Vycor glass at 62 bars show supersolid response, an abrupt drop in rotational moment of inertia, at 175 mK¹. We have investigated the pore-size dependence of the supersolid behavior by confining solid ⁴He in a different porous host, porous gold, of a considerably larger pore diameter. When solid ⁴He in porous gold is cooled below 0.2 K a sharp drop in the resonant period is found. The supersolid response exhibits a strong dependence on the amplitude of oscillation.PACS numbers: 67.80–s. 67.80 Mg., 0.5.70.Fh, 05.30.Jp     

Low Frequency Elastic Measurements on Solid ^{4}He in Vycor Using a Torsional Oscillator

Torsional oscillator (TO) experiments involving solid \(^{4}\) He confined in the nanoscale pores of Vycor glass showed anomalous frequency changes at temperatures below 200 mK. These were initially attributed to decoupling of some of the helium’s mass from the oscillator, the expected signature of a supersolid. However, these and similar anomalous effects seen with bulk \(^{4}\) He now appear to be artifacts arising from large shear modulus changes when mobile dislocations are pinned by \(^{3}\) He impurities. We have used a TO technique to directly measure the shear modulus of the solid \(^{4}\) He/Vycor system at a frequency (1.2 kHz) comparable to that used in previous TO experiments. The shear modulus increases gradually as the TO is cooled from 1 K to 20 mK. We attribute the gradual modulus change to the freezing out of thermally activated relaxation processes in the solid helium. The absence of rapid changes below 200 mK is expected since mobile dislocations could not exist in pores as small as those of Vycor. Our results support the interpretation of a recent TO experiment that showed no anomaly when elastic effects in bulk helium were eliminated by ensuring that there were no gaps around the Vycor sample.     

Influence of Deformation on the Formation of a Glassy Phase of 4He in the Supersolid Region

A new series of experiments has been performed to find out the conditions for the formation of a glass phase in solid ⁴He in the region where an anomalous behavior attributed to the supersolid effect was observed previously. A special two-chamber cell was used to deform the sample in situ. The contribution of the glass phase was identified by analyzing the measured temperature dependence of the sample pressure under different experimental conditions. It is found that the contribution of the glass phase increased sharply after deformation of the sample and practically disappears after its annealing.     

Absence of Slow Sound Modes in Supersolid 4He

Torsional oscillator experiments on solid ⁴He have been interpreted as showing mass decoupling similar to what one observes in a superfluid. Within the context of a two-component model for the supersolid one would expect the appearance of a second, slow acoustic mode. We have searched for this mode using an acoustic resonance technique. We have used porous membranes in bulk solid ⁴He analogous to a second sound experiment in the superfluid. We also investigated solid helium in Vycor using piezoelectrically driven titanium diaphragms (analogous to a fourth sound experiment in the liquid). Our measurements have shown no indication of an additional sound mode in the kHz range.     

The Quest for Bose-Einstein Condensation in Solid 4He

Ever since the seminal torsional oscillator (TO) measurements of Kim and Chan which suggested the existence of a phase transition in solid ⁴He, from normal to a ??supersolid?? state below a critical temperature T _c = 200 mK, there has been an unprecedented amount of excitement and research activity aimed at better understanding this phase. Despite much work, this remarkable phase has yet to be independently confirmed by conventional scattering techniques, such as neutron scattering. We have carried out a series of neutron scattering measurements, which we here review, aimed at observing Bose-Einstein condensation (BEC) in solid ⁴He at temperatures below T _c . In bulk liquid ⁴He, the appearance of BEC below T _?? signals the onset of superfluidity. The observation of a condensate fraction in the solid would provide an unambiguous confirmation for ??supersolidity??. Although, our measurements have not yet revealed a non-zero condensate fraction or algebraic off diagonal long-range order n ₀ in solid ⁴He down to 65 mK, i.e. n ₀=(0±0.3)%, our search for BEC and its corollaries continues with improved instrumentation.     

Effect of Impurities on Supersolid Condensate: A Ginzburg–Landau Approach

Basing our arguments on a wavefunction that contains both positional and superfluid order, we propose a Ginzburg–Landau functional for a supersolid with the two order parameters necessary to describe such a phase: density n _B (r) and supersolid order parameter ψ _V . We argue that adding lighter. ³He atoms to a ⁴He supersolid produces attractive regions for vacancies, leading to patches of higher T _c. On the other hand, the supersolid stiffness decreases in this granular state with increased ³He disorder. Both effects are linear in ³He concentration.     

Anomalous Response of 4He Confined in Nanoporous Media to Torsional Oscillation

The existence of “Non-Classical Rotational Inertia (NCRI)” in solid ⁴He below 0.2?K has been controversial and interpreted by a number of different theories. We report on torsional oscillator measurements for ⁴He in a nanoporous Gelsil glass, which has a network of nanopores with 3.5?nm in diameter. In addition to the usual “low-T NCRI” with an onset temperature 0.15?K, we find a larger decrease in rotational moment of inertia in a broad range of temperature from 0.2 to 1.9?K. This “high-T inertial anomaly” is accompanied with multiple dissipation peaks, but has no dependence on torsional oscillation velocity unlike the low-T NCRI. Since the high-T anomaly is observed also in confined liquid states, it originates in amorphous solid ⁴He layer near the pore wall. Our result shows that different types of supersolid—like phenomena, i.e. inertial anomalies, can coexist in a single ⁴He sample, even with genuine superfluidity of liquid ⁴He.     

The Glassy Response of Solid 4He to Torsional Oscillations

We calculated the glassy response of solid ⁴He to torsional oscillations assuming a phenomenological glass model. Making only a few assumptions about the distribution of glassy relaxation times in a small subsystem of otherwise rigid solid ⁴He, we can account for the magnitude of the observed period shift and concomitant dissipation peak in several torsion oscillator experiments. The implications of the glass model for solid ⁴He are threefold: (1) The dynamics of solid ⁴He is governed by glassy relaxation processes. (2) The distribution of relaxation times varies significantly between different torsion oscillator experiments. (3) The mechanical response of a torsion oscillator does not require a supersolid component to account for the observed anomaly at low temperatures, though we cannot rule out its existence.     

Dislocation Mobility and Anomalous Shear Modulus Effect in $$^$$He Crystals

We calculate the dislocation glide mobility in solid \(^4\)He within a model that assumes the existence of a superfluid field associated with dislocation lines. Prompted by the results of this mobility calculation, we study within this model the role that such a superfluid field may play in the motion of the dislocation line when a stress is applied to the crystal. To do this, we relate the damping of dislocation motion, calculated in the presence of the assumed superfluid field, to the shear modulus of the crystal. As the temperature increases, we find that a sharp drop in the shear modulus will occur at the temperature where the superfluid field disappears. We compare the drop in shear modulus of the crystal arising from the temperature dependence of the damping contribution due to the superfluid field, to the experimental observation of the same phenomena in solid \(^4\)He and find quantitative agreement. Our results indicate that such a superfluid field plays an important role in dislocation pinning in a clean solid \(^4\)He at low temperatures and in this regime may provide an alternative source for the unusual elastic phenomena observed in solid \(^4\)He.     

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.     

Quantum lattice gas and the existence of a supersolid

We discuss, within mean field theory, the possible phase diagrams of a quantum lattice gas model (as considered by Matsuda and Tsuneto atT=0) with particular reference to the existence of a supersolid phase, displaying both long-range diagonal (or crystalline) order and long-range off-diagonal order (as characteristic of a superfluid). For parameter ranges which provide a reasonable representation of the known phase diagram of ⁴ He, we show that a supersolid phase may appear between the usual solid and superfluid phases. Such a phase might not, however, extend down to absolute zero.     

Specific heat of amorphous3He films and confined liquid3He

We have measured the heat capacities of³He films and liquid³He in porous Vycor glass at 10 to 600 mK. With increasing the film thickness from 1 to 3 atomic layers, the specific heat evolves gradually from that typical to solid to that of liquid³He. At about 2 atomic layers, however, its low-temperature part is nearly temperature-independent; we interpret this as a result of gradual freezing of spins in an amorphous solid³He film with decreasing the temperature. The contribution of liquid³He in the center of the Vycor pores can be described as the specific heat of bulk liquid³He at corresponding pressures in the range 0 to 28 bar. The thickness of amorphous solid on the pore walls increases with external pressure roughly linearly. Preplating the walls with⁴He allows to determine the positions of³He atoms contributing to the surface specific heat at 10 to 50 mK. In addition, the contribution from the specific heat of³ He -⁴He mixing at 100 to 600 mK is discussed as a function of pressure and amount of⁴He.0n leave from ISSP Acad. Sci. of Russia, Chernogolovka, Russia     

Flow and Supersolidity in Helium

Recent torsional oscillator measurements showed evidence of non-classical rotational inertia in solid helium at temperatures below 200 mK and generated a great deal of interest in a possible supersolid state. The nature and properties of such a state are still unclear, although experiments involving ³He impurities and crystal annealing may provide clues. It would be very interesting to know whether supersolids share any of the other unusual properties of superfluids: superleaks, persistent currents, second sound and quantized vortices. We have studied the response of solid helium to pressure differences, in order to look for unusual flow properties that might be associated with supersolidity. Our measurements involved both helium confined in the nanometer pores of Vycor glass and bulk solid helium, at temperatures as low as 30 mK. Near melting, solid helium flows very easily but we did not see any evidence of superflow at low temperatures. If helium does become a supersolid at low temperatures, then its response to pressure gradients must be very different from that of liquid helium. We describe these and other experiments and discuss the role that defects may play in the low temperature behavior of solid helium.     

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.     

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.      

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.