A phenomenological model is developed to explain a new set of detailed torsional oscillator data for hcp 4He. The model is based on Anderson's idea of the vortex fluid (vortex tangle) in solid 4He. Utilizing a well-studied treatment of dynamics of quantized vortices we describe how the "local superfluid component" is involved in rotation (torsional oscillation) via a polaried vortices tangle. The polarization in the tangle appears both due to aligning the remnant or thermal vortices and due to penetration of additional vortices into volume. Both are supposed to occur in a relaxation manner, and the inverse full relaxation time tau^-1 is the sum of them. One of them is found to change linearly with respect to the rim velocity Vac. The developed approach explains the behavior of both NLRS and Delta Q^-1 seen in the experiment. We reproduce not only the unique Vac dependence, but also obtain new information about the vortices tangle, for example, a divergence in tau at extrapolated T ~ 30 mK.
X-ray diffraction experiments show that solid 4He grown in aerogel is highly polycrystalline, with a hcp crystal structure (as in bulk) and a crystallite size of approximately 100 nm. In contrast to the expectation that the highly disordered solid will have a large supersolid fraction, torsional oscillator measurements show a behavior that is strikingly similar to high purity crystals grown from the superfluid phase. The low temperature supersolid fraction is only ~3x10-4 and the onset temperature is ~ 100 mK.
In these torsional oscillator experiments the samples of solid $^4$He were characterized by measuring their thermal conducitvity. Polycrystalline samples of helium of either high isotopic purity or natural concentration of $^3$He were grown in an annular container by the blocked-capillary method and investigated before and after annealing. No correlation has been found between the magnitude of the low-temperature shift of the torsional oscillator frequency and the amount of crystalline defects as measured by the thermal conductivity. In samples with the natural $^3$He concentration a substanti…
We investigate the origin of a resonant period drop of a torsional oscillator (TO) containing solid ${}^{4}$He by inspecting its relation to a change in elastic modulus. To understand this relationship directly, we measure both phenomena simultaneously using a TO with a pair of concentric piezoelectric transducers inserted in its annulus. Although the temperature, ${}^{3}$He concentration, and frequency dependence are essentially the same, a marked discrepancy in the drive amplitude dependence is observed. We find that this discrepancy originates from the anisotropic response of polycrystallin…
The rigid double-torus torsional oscillator (TO) is constructed to reduce any elastic effects in-herent to complicate TO structures, allowing explicit probing for a genuine supersolid signature. We investigated the frequency- and temperature-dependent response of the rigid double-torus TO containing solid 4He with 0.6 ppb 3He and 300 ppb 3He. We did not find evidence to support the frequency-independent contribution proposed to be a property of supersolid helium. The frequency-dependent contribution which comes from the simple elastic effect of solid helium coupled to TO is essentially respons…
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 predo…
We have investigated the NMR properties of dilute 3He impurities in solid 4He contained in a torsional oscillator (TO) by the simultaneous measurement of the NMR and the torsional oscillator response of the so-called supersolid 4He. From measurements on samples with one hundred to a few hundred ppm of 3He, we have found three different states of 3He. The first is the homogeneously distributed isolated 3He atom in a solid matrix of 4He. The second is the 3He cluster in a homogeneous 4He matrix, which appears below the phase separation temperature of a solid mixture. The third is the 3He cluster…
We have measured the response of a torsional oscillator containing polycrystalline hcp solid $^{4}$He to applied steady rotation in an attempt to verify the observations of several other groups that were initially interpreted as evidence for macroscopic quantum effects. The geometry of the cell was that of a simple annulus, with a fill line of relatively narrow diameter in the centre of the torsion rod. Varying the angular velocity of rotation up to 2\,rad\,s$^{-1}$ showed that there were no step-like features in the resonant frequency or dissipation of the oscillator and no history dependence…
A decrease in the rotational period observed in torsional oscillator measurements was recently taken as a possible indication of a supersolid state of helium. We reexamine this interpretation and note that the decrease in the rotation period is also consistent with a solidification of a small liquid-like component into a low-temperature glass. Such a solidification may occur by a low-temperature quench of topological defects (e.g., grain boundaries or dislocations) which we examined in an earlier work. The low-temperature glass can account for not only a monotonic decrease in the rotation peri…
The resonant period drop observed at low temperatures in torsional oscillators containing solid helium had been interpreted as a signature of supersolid. However, it was found that the shear modulus increase found in solid helium at the same low temperature could also decrease the resonant period of the torsional oscillator. We report the results of a study in two different torsional oscillators that were specially designed to minimize the shear modulus effect and maximize any possible supersolid response. We were able to place an upper limit on the nonclassical rotational inertia or supersoli…
Torsional oscillator 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 torsional oscillator (TO) technique to directly measure the shear modulus of the s…
The discovery by Kim and Chan (KC) of an anomalous decrease in the period of torsional oscillators (TO) containing samples of solid $^4$He at temperatures below 0.2 K was initially interpreted as a superfluid-like decoupling of a fraction of the solid moment of inertia from the TO. These experiments appeared to confirm the thirty-year-old theoretical prediction by Chester, Andreev, and Leggett of the existence of a low-temperature Bose-condensed supersolid phase in solid $^4$He. The initial results of KC lead to a flurry of experimental and theoretical activity and their results for bulk $^4$H…
Over the last five years several experimental groups have reported anomalies in the temperature dependence of the period and amplitude of a torsional oscillator containing solid $^4$He. We model these experiments by assuming that $^4$He is a viscoelastic solid--a solid with frequency dependent internal friction. We find that while our model can provide a quantitative account of the dissipation observed in the torsional oscillator experiments, it only accounts for about 10% of the observed period shift, leaving open the possibility that the remaining period shift is due to the onset of superflu…
The recent torsional oscillator results of Kim and Chan suggest a supersolid phase transition in solid 4He. We have used a piezoelectrically driven diaphragm to study the flow of solid helium through an array of capillaries. Our measurements showed no indication of low temperature flow, placing stringent restrictions on supersolid flow in response to a pressure difference. The average flow speed at low temperatures was less than 1.2x10-14 m/s, corresponding to a supersolid velocity at least 7 orders of magnitude smaller than the critical velocities inferred from the torsional oscillator measur…
Torsional oscillator experiments show evidence of mass decoupling in solid 4He. This decoupling is amplitude dependent, suggesting a critical velocity for supersolidity. We observe similar behavior in the elastic shear modulus. By measuring the shear modulus over a wide frequency range, we can distinguish between an amplitude dependence which depends on velocity and one which depends on some other parameter like displacement. In contrast to the torsional oscillator behavior, the modulus depends on the magnitude of stress, not velocity. We interpret our results in terms of the motion of disloca…
We have studied natural purity $^{4}$He single crystals and polycrystals between 10 and 600 mK using a torsional oscillator with a 2 cm$^{3}$ rigid cell made of sapphire with a smooth geometry. As the temperature was lowered, we observed sample dependent but reproducible resonant frequency shifts that could be attributed to a supersolid fraction of order 0.1%. However, these shifts were observed with single crystals, not with polycrystals. Our results indicate that, in our case, the rotational anomaly of solid helium is more likely due to a change in stiffness than to supersolidity. This inter…
In recent torsional oscillator experiments by Kim and Chan (KC), a decrease of rotational inertia has been observed in solid 4He in porous materials and in a bulk annular channel. 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 of 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 %…
Frequency shifts and dissipations of a compound torsional oscillator induced by solid $^4$He samples containing $^3$He impurity concentrations ($x_3$ = 0.3, 3, 6, 12 and 25 in units of 10$^{-6}$) have been measured at two resonant mode frequencies ($f_1$ = 493 and $f_2$ = 1164 Hz) at temperatures ($T$) between 0.02 and 1.1 K. The fractional frequency shifts of the $f_1$ mode were much smaller than those of the $f_2$ mode. The observed frequency shifts continued to decrease as $T$ was increased above 0.3 K, and the conventional non-classical rotation inertia fraction was not well defined in all…
The quantized vortex state appearing in the recently discovered new states in hcp 4He since their discovery is discussed. Special attention is given to evidence for the vortex state as the vortex fluid (VF) state and its transition into the supersolid (SS) state. Its features are described. The historical expla- nations 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 va- cancies, interstitials and other possible excitations in the quantum solids which are expected because of the large zero-point motion…
The torsional oscillator experiments described here examine the effect of disorder on the non-classical moment of inertia (NCRI) of a solid 4He sample. The NCRI increases with increasing disorder, but the period changes responsible for this increase occur primarily at higher temperatures, while at the lowest temperature the oscillator period remains relatively unaffected. This result is contrary to expectations based on a supersolid scenario and points to a non-superfluid origin for the NCRI.
Detailed studies of AC velocity Vac and temperature dependence of torsional oscillator responses of solid 4He are reported. A characteristic onset temperature T0 about 0.5 K is found, below which a significant Vac dependent change occurs in the energy dissipation for the sample at 32 bar. A Vac dependence of the non-classical rotational inertia fraction, NCRIF also appears below about T0. This value of T0 excludes the possible explanation of supersolid by liquid superfluidity in grain boundaries or other liquid related origins. The log(Vac) linear dependence was found in NCRIF. Furthermore, th…
We have performed a microscopic study of a straight quantized vortex line in three dimensions in condensed $^4$He at zero temperature using the Shadow Path Integral Ground State method and the fixed-phase approximation. We have characterized the energy and the local density profile around the vortex axis in superfluid $^4$He at several densities, ranging from below the equilibrium density up to the overpressurized regime. For the Onsager-Feynman (OF) phase our results are exact and represent a benchmark for other theories. The inclusion of backflow correlations in the phase improves the descri…
Torsional oscillator experiments on solid 4He show frequency changes which suggest mass decoupling, but the onset is broad and is accompanied by a dissipation peak. We have measured the elastic shear modulus over a broad frequency range, from 0.5 Hz to 8 kHz, and observe similar behavior - stiffening and a dissipation peak. These features are associated with a dynamical crossover in a thermally activated relaxation process in a disordered system rather than a true phase transition. If there is a transition in the DC response, e.g. to a supersolid state, it must occur below 55 mK.
We describe the first observations on the time-dependent dissipation when the drive level of a torsional oscillator containing solid He-4 is abruptly changed. The relaxation of dissipation in solid He-4 shows rich dynamical behavior including exponential and logarithmic time-dependent decays, hysteresis, and memory effects.
We apply our recently developed model of a Bose condensate of quantum kink wave in solid He4 to understand recent torsional oscillator experimental results of the citical velocities and the effect of the steady and oscillating rotations at around 0.1 degree K. When the D.C. rotation is present we find a decrease of the Q factor given by $Q^{-1} \propto f_{sf}\times Ω_{D}/ω_{TO}$ where $f_{sf}$ is the superfluid fraction; $Ω_{D}$, the D. C. angular rotation velocity, $ω_{TO}$, the torsional oscillator oscillating frequency. We estimate the AC critical velocity $Ω_A^{crit}$ as that required to g…
Flexural mode vibrations of miniature piezoelectric tuning forks (TF) are known to be highly sensitive to superfluid excitations and quantum turbulence in $\mathrm{^3He}$ and $\mathrm{^4He}$ quantum fluids, as well as to the elastic properties of solid $\mathrm{^4He}$, complementing studies by large scale torsional resonators. Here we explore the sensitivity of a TF, capable of simultaneously operating in both the flexural and torsional modes, to excitations in the normal and superfluid $\mathrm{^4He}$. The torsional mode is predominantly sensitive to shear forces at the sensor - fluid interfa…
We report measurements of the nuclear spin-lattice and spin-spin relaxation times of very dilute 3He in solid 4He in the temperature range 0.01 \leq T \leq 0.5 K for densities where anomalies have been observed in torsional oscillator and shear modulus measurements. We compare the results with the values of the relaxation times reported by other observers for higher concentrations and the theory of Landesman that takes into account the elastic properties of the 4He lattice. A sharp increase in the magnitude of the nuclear spin-lattice relaxation times compared to the the classical Landesman th…
Using a high-sensitivity torsional oscillator technique, we mapped the rotational and relaxational dynamics of solid helium-4 throughout the parameter range of the proposed supersolidity. We found evidence that the same microscopic excitations controlling the torsional oscillator motions are generated independently by thermal and mechanical stimulation. Moreover, a measure for the relaxation times of these excitations diverges smoothly without any indication for a critical temperature or critical velocity of a supersolid transition. Finally, we demonstrated that the combined temperature-veloci…
We have investigated the non-classical response of solid 4He confined in porous gold set to torsional oscillation. When solid helium is grown rapidly, nearly 7% of the solid helium appears to be decoupled from the oscillation below about 200 mK. Dissipation appears at temperatures where the decoupling shows maximum variation. In contrast, the decoupling is substantially reduced in slowly grown solid helium. The dynamic response of solid helium was also studied by imposing a sudden increase in the amplitude of oscillation. Extended relaxation in the resonant period shift, suggesting the emergen…
The quantum diffusion of ortho-H2 impurities in solid para-H2 has been investigated with a torsional oscillator technique. The unclustering dynamics of agglomerated impurities with increasing temperature are found to greatly affect the resonant period of oscillation. Based on the observed trends in both experimental data and complementary finite element calculations, we find that the oscillation period predominantly reflects changes in the solid's elastic response to the minute inertial stresses imposed on it. It is likely that the small but abrupt increase in the resonant period observed abov…
We study the dynamical process of the vortex tangle development under a spherically symmetric thermal counterflow around a heat source submerged into a bulk superfluid 4He. We reveal a peculiar vortex dynamics that is unique to this geometry, which is greatly diverse from the vortex dynamics in a homogeneous counterflow. Two types of heater are considered here, namely, a spherical heater with a solid wall and a point-like heater. In both cases, a spherical vortex tangle is formed surrounding the heater. The mechanism of vortex tangle development in the vicinity of a solid wall is strongly gove…
The paper investigates possibility of equilibrium solid-body rotation of a vortex bundle diverging at some height from a cylinder axis and terminating on a lateral wall of a container. Such a bundle arises when vorticity expands up from a container bottom eventually filling the whole container. The analysis starts from a single vortex, then goes to a vortex sheet, and finally addresses a multi-layered crystal vortex bundle. The equilibrium solid-body rotation of the vortex bundle requires that the thermodynamic potentials in the vortex-filled and in the vortex-free parts of the container are e…
A detailed torsional oscillator (TO) study on a stable solid 4He sample at 49 bar with To = 0.5K, is reported to T below the dissipation peak at Tp . We find hysteretic behavior starting below Tc = about 75 mK, in period shift, as well as in dissipation, with changes of AC excitation amplitude Vac . The derived difference of non-linear rotational susceptibility DeltaNLRS(T)hys across the hysteresis loop under a systematic condition is analyzed as a function of Vac and T . We propose that DeltaN LRS(T)hys is proportional to non- classical rortational inertia fraction, NCRIF, and is actually the…
We have studied the thermal history of the resonant frequency of a torsional oscillator containing solid 4He. We find that the magnitude of the frequency shift that occurs below 100 mK is multivalued in the low temperature limit, with the exact value depending on how the state is prepared. This result can be qualitatively explained in terms of the motion and pinning of quantized vortices within the sample. Several aspects of the data are also consistent with the response of dislocation lines to oscillating stress fields imposed on the solid.
The irrotational nature of superfluid helium was discovered through its decoupling from the container under rotation. Similarly, the resonant period drop of a torsional oscillator (TO) containing solid helium was first interpreted as the decoupling of solid from the TO and appearance of supersolid. However, the resonant period can be changed by mechanisms other than supersolid, such as the elastic stiffening of solid helium that is widely accepted as the reason for the TO response. To demonstrate the irrotational nature more directly, the previous experiments superimposed the dc rotation onto …
The non-classical rotational inertia fraction of the identical cylindrical solid $^4$He below 300 mK is studied at 496 and 1173 Hz by a double resonance torsional oscillator. Below 35 mK, the fraction is the same at sufficiently low rim velocities. Above 35 mK, the fraction is greater for the higher than the lower mode. The dissipation peak of the lower mode occurs at a temperature $\sim$ 4 mK lower than that of the higher mode. The drive dependence of the two modes shows that the reduction of the fraction is characterized by critical velocity, \textit{not} amplitude nor acceleration.
Diagonal and off-diagonal properties of 2D solid 4He systems doped with a quantized vortex have been investigated via the Shadow Path Integral Ground State method using the fixed-phase approach. The chosen approximate phase induces the standard Onsager-Feynman flow field. In this approximation the vortex acts as a static external potential and the resulting Hamiltonian can be treated exactly with Quantum Monte Carlo methods. The vortex core is found to sit in an interstitial site and a very weak relaxation of the lattice positions away from the vortex core position has been observed. Also othe…
A torsional oscillator study of solid para-hydrogen has been carried out down to 20 mK in search for evidence of superfluidity. This work was inspired by the observation of the supersolid phase in solid He-4. We found evidence of a possible phase transition, marked by an abrupt increase in the resonant period of oscillation and onset of extremely long relaxation times as the temperature was raised above 60 mK. The change in the period for para-hydrogen, in contrast to solid He-4, is not related to superfluidity as it fails a crucial test showing that it is not a consequence of irrotational flo…
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\cite{1,2}. We have also found a strong correlation between the NCRI signal and a high dissipation Q$^{-1}$ of 4 $\times$ 10$^{-6}$ of the oscillation above the transition temperature. Here, we present preliminary results of the annealing process in $^{4}$He 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 e…
Although solid helium-4 (4He) may be a supersolid it also exhibits many phenomena unexpected in that context. We studied relaxation dynamics in the resonance frequency f(T) and dissipation D(T) of a torsional oscillator containing solid 4He. With the appearance of the "supersolid" state, the relaxation times within f(T) and D(T) began to increase rapidly together. More importantly, the relaxation processes in both D(T) and a component of f(T) exhibited a complex synchronized ultraslow evolution towards equilibrium. Analysis using a generalized rotational susceptibility revealed that, while exh…
Kim and Chan have recently observed Non-Classical Rotational Inertia (NCRI) for solid $^4$He in Vycor glass, gold film, and bulk. Their low $T$ value of the superfluid fraction, $ρ_{s}/ρ\approx0.015$, is consistent with what is known of the atomic delocalization in this quantum solid. By including a lattice mass density $ρ_{L}$ distinct from the normal fluid density $ρ_{n}$, we argue that $ρ_{s}(T)\approxρ_{s}(0)-ρ_{n}(T)$, and we develop a model for the normal fluid density $ρ_{n}$ with contributions from longitudinal phonons and ``defectons'' (which dominate). The Bose-Einstein Condensation …
The lattice dynamics of solid 4He has been explored using pulsed NMR methods to study the motion of 3He impurities in the temperature range where experiments have revealed anomalies attributed to superflow or unexpected viscoelastic properties of the solid 4He lattice. We report the results of measurements of the nuclear spin-lattice and spin-spin relaxation times that measure the fluctuation spectrum at high and low frequencies, respectively, of the 3He motion that results from quantum tunneling in the 4He matrix. The measurements were made for 3He concentrations 16<x_3<2000 ppm. For 3He conc…
We perform numerical simulations of finite temperature quantum turbulence produced through thermal counterflow in superfluid $^4$He, using the vortex filament model. We investigate the effects of solid boundaries along one of the Cartesian directions, assuming a laminar normal fluid with a Poiseuille velocity profile, whilst varying the temperature and the normal fluid velocity. We analyze the distribution of the quantized vortices, reconnection rates and quantized-vorticity production as a function of the wall-normal direction. We find that the quantized vortex lines tend to concentrate close…
