Defects and Glassy Dynamics in Solid 4He: Perspectives and Current Status

We review the anomalous behavior of solid ⁴He at low temperatures with particular attention to the role of structural defects present in solid. The discussion centers around the possible role of two level systems and structural glassy components for inducing the observed anomalies. We propose that the origin of glassy behavior is due to the dynamics of defects like dislocations formed in ⁴He. Within the developed framework of glassy components in a solid, we give a summary of the results and predictions for the effects that cover the mechanical, thermodynamic, viscoelastic, and electro-elastic contributions of the glassy response of solid ⁴He. Our proposed glass model for solid ⁴He has several implications: (1) The anomalous properties of ⁴He can be accounted for by allowing defects to freeze out at lowest temperatures. The dynamics of solid ⁴He is governed by glasslike (glassy) relaxation processes and the distribution of relaxation times varies significantly between different torsional oscillator, shear modulus, and dielectric function experiments. (2) Any defect freeze-out will be accompanied by thermodynamic signatures consistent with entropy contributions from defects. It follows that such entropy contribution is much smaller than the required superfluid fraction, yet it is sufficient to account for excess entropy at lowest temperatures. (3) We predict a Cole-Cole type relation between the real and imaginary part of the response functions for rotational and planar shear that is occurring due to the dynamics of defects. Similar results apply for other response functions. (4) Using the framework of glassy dynamics, we predict low-frequency yet to be measured electro-elastic features in defect rich ⁴He crystals. These predictions allow one to directly test the ideas and very presence of glassy contributions in ⁴He.     

Shear Modulus in Viscoelastic Solid 4He

The complex shear modulus of solid ⁴He exhibits an anomaly in the same temperature region where torsion oscillators show a change in period. We propose that the observed stiffening of the shear modulus with decreasing temperature can be well described by the response of glassy components inside of solid ⁴He. Since glass is an anelastic material, we utilize the viscoelastic approach to describe its dynamics. The viscoelastic component possesses an increasing relaxation as temperature decreases. The response functions thus derived are identical to those obtained for a glassy, time-delayed restoring back-action. By generalizing the viscoelastic equations for stress and strain to a multiphase system of constituents, composed of patches with different damping and relaxation properties, we predict that the maximum change of the magnitude of the shear modulus and the maximum height of the dissipation peak are independent of an applied external frequency. The same response expressions allow us to calculate the temperature dependence of the shear modulus?? amplitude and dissipation. Finally, we demonstrate that a Vogel-Fulcher-Tammann (VFT) relaxation time is in agreement with available experimental data.     

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.      

In Pursuit of the Elusive Supersolid

The excitement following the initial report of supersolid behavior for ⁴He embedded in porous Vycor glass has been tempered by the realization that many of the early supersolid observations were contaminated by effects arising from an anomaly in the elastic properties of solid ⁴He. In an attempt to separate dynamic elastic effects from a true supersolid signal, we employed a torsional oscillator with two eigen-frequencies to study the ⁴He-Vycor system. We found that frequency-dependent elastic signals can entirely account for the observed period shift signals. Although, we conclude that supersolid does not exist for the ⁴He-Vycor case, the question of its presence in bulk samples remains open. In our current experiments we apply the two-frequency test to bulk samples of solid ⁴He. Again we find a frequency-dependent contribution arising from elastic effects; however, in some cases we also find a small frequency-independent contribution, which may indicate the existence of a remnant supersolid phase. Given the history of this subject such results must be treated with caution.     

Torsional oscillator for high magnetic field experiments

A new type of torsional oscillator for experiments in the high magnetic field of 12 Tesla is reported. A beryllium copper alloy (BeCu25) was chosen as a torsion rod for its reliable mechanical properties. Two non-metallic materials were tested for a torsion head except a small amount of silver paste for the electrode. For a quartz glass head, liquid³He was successfully cooled down below 0.5 mK at 12 Tesla. The quality factor was 2 × 10⁴ even at the highest field. On the other hand, a Stycast 1266 head in a field of 12 Tesla caused a large temperature difference between the liquid in the head and in the open space, in spite of a comparable quality factor with the quartz head.     

Simultaneous Measurement of Torsional Oscillator and?NMR in Solid 4He with 10?ppm of 3He

We have performed the simultaneous measurement of torsional oscillator and NMR in solid ⁴He with 10 ppm of ³He at 3.6 MPa. In this solid, NCRI response appears below about 400 mK. NMR measurement shows that there is the same kind of phase-separated ³He cluster which is found in our previous measurement in solid ⁴He with over a hundred ppm of ³He. When we warm the solid above the phase separation temperature, the cluster disappears gradually. Below and above the phase separation temperature, the distribution of ³He atoms changes significantly with long time constant, which is as long as a day. However, even in such a long time span, we do not observe any systematic changes in the torsional oscillator response. This result suggests that the phase separation and related changes of the distribution of ³He is not directly related to the impurity effect of the NCRI response.     

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     

Hysteretic Response of Torsionally Oscillated Solid 4He and Its Dependence on 3He Impurity Concentration

Measurements on hysteretic response of compound torsional oscillator containing annular-shaped solid ⁴He samples were carried out by varying the oscillator drive amplitude starting from high to low and then back up to the initial high value. Hysteresis in the oscillator frequency and amplitudes were observed only below an onset temperature. The hysteresis onset temperature (T _H ) did not depend on the oscillator frequency, width of the sample annulus, annealing and refreezing after melting. A systematic increase in T _H was observed as the ³He impurity concentration in solid ⁴He samples was increased. The dependence of T _H on ³He impurity concentration followed approximately that of the dissipation peak temperatures. Possible relationships of the observed hysteresis phenomena with models of solid ⁴He dynamics based on freezing of a vortex liquid and dislocation motion are discussed.     

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.     

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.     

Generalized Rotational Susceptibility Studies of Solid 4He

Using a novel SQUID-based torsional oscillator (TO) technique to achieve increased sensitivity and dynamic range, we studied TO’s containing solid ⁴He. Below ～250?mK, the TO resonance frequency f increases and its dissipation D passes through a maximum as first reported by Kim and Chan. To achieve unbiased analysis of such ⁴He rotational dynamics, we implemented a new approach based upon the generalized rotational susceptibility $\chi_{{}^{4}\mathrm{He}}^{ - 1}(\omega,T)$ . Upon cooling, we found that equilibration times within f(T) and D(T) exhibit a complex synchronized ultraslow evolution toward equilibrium indicative of glassy freezing of crystal disorder conformations which strongly influence the rotational dynamics. We explored a more specific $\chi_{{}^{4}\mathrm{He}}^{ -1}(\omega,\tau(T))$ with τ(T) representing a relaxation rate for inertially active microscopic excitations. In such models, the characteristic temperature T ^? at which df/dT and D pass simultaneously through a maximum occurs when the TO angular frequency ω and the relaxation rate are matched: ωτ(T ^?)=1. Then, by introducing the free inertial decay (FID) technique to solid ⁴He TO studies, we carried out a comprehensive map of f(T,V) and D(T,V) where V is the maximum TO rim velocity. These data indicated that the same microscopic excitations controlling the TO motions are generated independently by thermal and mechanical stimulation of the crystal. Moreover, a measure for their relaxation times τ(T,V) diverges smoothly everywhere without exhibiting a critical temperature or velocity, as expected in ωτ=1?models. Finally, following the observations of Day and Beamish, we showed that the combined temperature-velocity dependence of the TO response is indistinguishable from the combined temperature-strain dependence of the ⁴He shear modulus. Together, these observations imply that ultra-slow equilibration of crystal disorder conformations controls the rotational dynamics and, for any given disorder conformation, the anomalous rotational responses of solid ⁴He are associated with generation of the same microscopic excitations as those produced by direct shear strain.     

Observation of 4He Superfluidity in 1.8 nm-Pores

No Heading Superfluid properties of ⁴He adsorbed in uniform straight pore 1.8 nm in diameter were studied using a torsional oscillator. In the pore, the first one or two layers of adsorbed. ⁴He are solid, therefore the pore diameter is effectively reduced to about 1.1 or 0.4 nm. In order to investigate whether ⁴He becomes superfluid in such a narrow pore, we performed the oscillator experiments for two cases: ⁴He is adsorbed (1) on the bare substrate and (2) on the pore completely filled with N₂ atoms. In the latter case, only superfluid film coating the surface of the substrate grain can be detected. Compared with this case, an additional superfluid signal originating from ⁴He in the pore is observed for the bare substrate. This strongly suggests that ⁴He in the pore is superfluid.PACS numbers: 67.40.–w, 67.70.+n     

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.     

Torsional Oscillator Experiments with High Stability and Reproducibility

We report experiments of the torsional oscillator to observe the superfluid transition in ⁴ He films in porous glass (the pore diameter is 1m). Stability and reproducibility of the oscillator, which quite often is problematic in previous experiments, is essential for a quantitative analysis of observations in different conditions. It follows that the friction of the superfluid films and the energy dissipation of the solid films are derived from comparisons of measurements for different film thickness.     

Lattice Relaxation in Solid 4He: Effect on Dynamics of 3He Impurities

We present a new simplified derivation of the effect of lattice relaxation that accompanies the quantum tunneling of ³He impurities in solid ⁴He on the nuclear spin-lattice relaxation of the ³He impurities for very low impurity concentrations. As a result of the larger zero point motion of the ³He impurity compared to the ⁴He atoms, a significant lattice distortion accompanies the impurity as it moves through the lattice and the dynamics of the impurity depends on both the interaction energy between two ³He atoms and on the relaxation of the lattice for the tunneling impurity. Using a phenomenological model for the lattice relaxation we compare the nuclear spin-lattice relaxation rates observed at low temperatures with the dependence on temperature expected for a ⁴He lattice relaxation comparable to that observed by Beamish et al. (Phys. Rev. Lett. 96:195304, 2006).     

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.     

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.     

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.