Session L17: Solid Helium

Shear Modulus Anomaly in Solid Helium

The search for supersolidity was given impetus by recent experiments in which solid helium appeared to decouple from a torsional oscillator, but other phenomena which characterize superflow have not yet been observed. Both experiments and theory indicate that defects are involved in supersolidity and these should also affect the solid's mechanical behavior. We have measured the shear modulus of solid helium at extremely low frequencies and strains, using a new method, and observe anomalous stiffening at temperatures below 200 mK. It has the same dependence on temperature, measurement amplitude, $^{3}$He impurity concentration and annealing as the torsional oscillator decoupling. This elastic behavior is explained in terms of a dislocation network which is pinned by $^{3}$He at the lowest temperatures but becomes mobile above 100 mK. Moving dislocations appear eliminate the decoupling and disrupt possible supersolidity.   

Effect of Annealing and Stresses on the Shear Modulus of Solid Helium

Recent measurements suggest that defects are crucial to ``supersolid'' behavior. Dislocations produced during crystal growth or by plastic deformation can have dramatic effects on a solid's mechanical properties. We have made pressure/flow and elastic/acoustic measurements on solid helium and have studied the effects of annealing near melting and of applying large stresses at low temperatures. Pressure gradients are greatly reduced by annealing, but only at temperatures quite close to melting. In our elastic measurements, we observed a large stiffening of the shear modulus in the same temperature range where decoupling was observed in torsional oscillators, behavior which can be understood in terms of the response of mobile dislocations above 100 mK. Annealing usually raises the high temperature shear modulus but leaves the low temperature modulus unchanged, as expected if annealing eliminates some dislocations. Applying large stresses further increases the high temperature modulus, but these changes are reversed by heating above 0.5 K, suggesting that the defects introduced by stressing the crystal are much easier to anneal than the dislocations produced during growth.   

Amplitude Dependence of the Shear Modulus Anomaly in Solid Helium

The shear modulus of solid $^{4}$He increases substantially in the temperature range below 200 mK where torsional oscillator measurements showed mass decoupling apparently associated with supersolidity. The amount of helium which decoupled depended on the oscillator amplitude, which was interpreted in terms of a supersolid critical velocity of order 10 microns/second. We observed a similar amplitude dependence in our shear modulus anomaly - the stiffening at low temperatures decreased above a critical drive level. By varying the measurement frequency (from 20 to 2000 Hz) and by changing the sample's dimensions, we conclude that the amplitude dependence we see is a function of the stress or displacement in the solid helium rather than the velocity. This contrasts with recent torsional oscillator measurements in which the amplitude dependence scaled with sample velocity. However, the amplitude dependence in our modulus measurements begins at stresses comparable to those in torsional oscillators and at low temperatures it shows hysteretic behavior similar to that seen in torsional oscillators.   

Finite-Element Modeling of Acoustic Modes in Solid $^4$He

Using a finite-element method we have calculated the low-frequency eigenmodes of various torsional oscillators (TO) in the literature. To elucidate the relationship between the elasticity [1] and apparent nonclassical rotatation inertia (NCRI) of solid $^4$He, we have investigated the dependence of the torsion mode on the shear modulus of $^4$He. If we exclude supersolidity, we find the inferred increase in the shear modulus that is necessary to account for typical frequency shifts in TO studies is significantly larger than that reported in Ref. [1] and nearly unphysical. Experiments are in progress to understand the connection between NCRI and the increased shear modulus. [1] J. Day and J. R. Beamish, arXiv:0709.4666v1 (2007).   

Decoupling NCRI from shear modulus changes in solid $^4$He

Day and Beamish report a significant increase in the shear modulus of solid $^4$He [1] below 250 mK with temperature dependence similar to the non-classical rotational inertia (NCRI) response seen in torsional oscillator measurements [2]. Finite element calculations show that stiffening of the solid $^4$He could mimic very small NCRI signals [3]. We have constructed a one-piece, welded oscillator which is designed to minimize the effect from stiffening of the solid helium. Preliminary data will be presented. \newline [1] J. Day and J. R. Beamish, arXiv:0709.4666v1 (2007). \newline [2] E. Kim and M. H. W. Chan, Science \textbf{305}, 1941 (2004). \newline [3] A. C. Clark and M. H. W. arXiv:0711.3619v1 (2007).   

Studies of Non-Classical Rotational Inertia of Solid $^4$He

We have constructed a compound torsional oscillator which could be operated at two resonant modes (the first at 496 and the second at 1172 Hz). This device allowed us to study the non-classical rotation inertia of the \underline{identical} solid $^4$He at the two oscillator modes driven separately. We present here recent studies of NCRI when the two modes are simultaneously excited. The idea was to drive the first mode at variable high amplitude and to detect its effect on NCRI fraction by the second mode. We expected that when the solid $^4$He was driven at high amplitudes with the first mode to produce significant reduction in NCRI fraction, the same reduction would be measured with the second mode driven simultaneously at very low amplitude. On the contrary, the observed reduction in NCRI fraction by the second mode was much \textit{smaller} than that expected from the first mode. If the driver/detector roles of the first and second modes were reversed, the amount of reduction of NCRI fraction detected and induced by a high drive amplitude of the second mode became \textit{greater} in the first mode driven at a low amplitude. The critical drive amplitude effects of NCRI induced in one mode are not entirely ``seen'' by the other mode in our oscillator.   

Torsional oscillator and synchrotron x-ray experiments on solid helium in aerogel

A number of Torsional Oscillator experiments have shown that the Non Classical Rotational Inertia fraction in solid $^{4}$He is strongly dependent on sample preparation. Samples presumed to be the most pristine show the smallest fraction, rapidly quenched samples a much larger one. On the assumption that samples grown in a strongly disordered environment would similarly show a large NCRIf, we have made T.O. measurements on solid samples grown in 95{\%} porous silica aerogel. Contrary to expectation, these samples show a behavior that is very similar to high purity samples grown from the superfluid phase. Subsequent x-ray diffraction experiments show that the solid grown in aerogel is highly polycrystalline, with a hcp crystal structure (as in bulk) and a crystallite size of approximately 100 nm. X-ray diffraction experiments were performed at the Advanced Photon Source, Argonne national laboratory. This work was supported through NSF DMR-0706339 (MHWC) and DE-FG01-05ER05-02 (CAB).   

Non-classical response of solid helium confined in Vycor glass

The non-classical rotational inertia of solid $^{4}$He was observed by a series of torsional oscillator experiments[1]. Probable heat capacity signature of the supersolid transition which coincides with the non-classical mass decoupling is also observed[2]. Recent torsional oscillator experiments suggest that disorder and the orientation of a helium crystal may play an important role in the appearance of non-classical rotational inertia. Accordingly, the investigation on the role of defects and crystal orientation may provide crucial clues to understand supersolidity. Solid helium confined in porous media cannot possess well-defined crystal orientation and is likely heavily populated with defects. Here we will present the pressure dependence of the non-classical rotational inertia in solid helium confined in Vycor glass by torsional oscillator techniques. [1] E. Kim and M. H. W. Chan, \textit{Science} \textbf{305}, 1941 (2004); \textit{Nature} \textbf{425}, 227 (2004); \textit{J. Low Temp. Phys.} \textbf{138}, 859 (2005); \textit{Phys. Rev. Lett. }\textbf{97}, 115302 (2006). [2] X. Lin, A. C. Clark, M. H. W. Chan, \textit{Nature} \textbf{449,}1025(2007).   

Torsional oscillators and the entropy dilemma of solid $^{4}$He

Solid $^{4}$He is viewed as a nearly perfect Debye solid. Yet, recent calorimetry measurements by Chan's group (JLTP \textbf{138} (2005) 853 and Nature \textbf{449} (2007) 1025) indicate that at low temperatures the specific heat has both cubic and linear contributions. These features appear in the same temperature range where measurements of the torsional oscillator period suggest a supersolid transition. We analyze (Phys. Rev. B \textbf{75} (2007) 094201) the specific heat and compare the measured with the estimated entropy for a proposed supersolid transition with 1{\%} superfluid fraction and find that the observed entropy is too small. We suggest that the low-temperature linear term in the specific heat is due to a glassy state that develops at low temperatures and is caused by a distribution of tunneling systems in the crystal. We propose that dislocation related defects produce those tunneling systems. Further, we argue (Phys. Rev. B \textbf{76} (2007) 014530) that the reported mass decoupling is consistent with an increase in the oscillator frequency as expected for a glass-like transition. The glass model offers an alternate interpretation of the torsional oscillator experiments in contrast to the supersolid nonclassical rotational inertia (NCRI) scenario.   

the excitation spectrum of solid 4He

Speculation about a possible Bose condensation in solid $^{4}$He has existed for decades and has recently been further stimulated by the discovery of an acoustic anomaly and a decrease in the moment of inertia at temperatures below 200 mK. The excitation spectrum played an important role in understanding the properties of superfluid liquid helium so that, if there is such a condensation in solid $^{4}$He,$^{ }$the spectrum might also confirm it and aid in understanding it. We have measured the excitation spectrum of solid $^{4}$He by neutron scattering using the Disc Chopper Spectrometer at NIST. We have identified a sharp line in the spectrum as vacancy wave excitations. The dispersion relation for these excitations has a minimum energy of 1 meV and is quadratic. This vacancy mode intersects the longitudinal phonon mode at 1.1 meV and the two become degenerate at higher energies. The degenerate mode has a linear dispersion law with smaller slope than the longitudinal acoustic mode at lower energies. No change in the spectrum was observed below 200 mK. The spectrum has strong similarities to the spectrum in superfluid liquid helium.   

Torsion Oscillator Studies of Solid Helium-4

We will present results of torsion oscillator experiments on solid helium-4 below 300 mK.   

Pairing states for the ring exchange t-J model

Multiple spin interactions, introduced by ring exchange process, are important for many systems, including solid $^{3}$He and High Tc cuprates. In high Tc cuprates, the dominant term is the 2-spin antiferromagnetic interaction, which leads to d-wave singlet pairing upon doping. In solid $^{3}$He, on the other hand, the 4-spin interaction plays an important role, and it's interesting to determine how the Cooper pairing state may differs from that of cuprates. In this work, we apply the renormalized mean-field theory [1] to a modified t-J model, where the J term includes the 4-spin interaction introduced by the ring exchange. Our result shows that a mixed state of singlet and triplet pairing optimizes the energy. At half filling, the pairing state is unphysical, due to the fact that there is no double occupancy. Upon doping or with intrinsic vacancies, the paring state becomes physical. Such a mechanism may introduce supersolidity in bulk solid $^{3}$He and solid $^{3}$He absorbed on a substrate at very low temperature. [1] F.C. Zhang, C. Gros, T.M. Rice and H. Shiba, Supercond. Sci. Technol. 1 36 (1988)