Transition to Turbulence for a Quartz Tuning Fork in Superfluid 4He

We have studied the resonance of a commercial quartz tuning fork immersed in superfluid ⁴He, at temperatures between 5 mK and 1 K, and at pressures between zero and 25 bar. The force-velocity curves for the tuning fork show a linear damping force at low velocities. On increasing velocity we see a transition corresponding to the appearance of extra drag due to quantized vortex lines in the superfluid. We loosely call this extra contribution “turbulent drag”. The turbulent drag force, obtained after subtracting a linear damping force, is independent of pressure and temperature below 1 K, and is easily fitted by an empirical formula. The transition from linear damping (laminar flow) occurs at a well-defined critical velocity that has the same value for the pressures and temperatures that we have measured. Later experiments using the same fork in a new cell revealed different behaviour, with the velocity stepping discontinuously at the transition, somewhat similar to previous observations on vibrating wire resonators and oscillating spheres. We compare and contrast the observed behaviour of the superfluid drag and inertial forces with that measured for vibrating wires.     

Vibrating Wire Measurements in Superfluid 3He at the Melting Curve Down to 0.53 mK

Measurements of the vibrating wire spectrum have been carried out in superfluid ³ He along the melting curve down to 0.53mK. We have observed that at temperatures below 0.3 T_c the width of the mechanical resonance of the wire decreases exponentially with 1/T, indicating the ballistic regime of collisions with quasiparticles. The value of the superfluid energy gap was found to be (1.99±0.05)T_c, in good agreement with the values obtained from heat capacity measurements. The vibrating wire was thereby calibrated for further experiments at temperatures below 0.5mK, where the sensitivity of the melting curve thermometry becomes rather poor.     

Transition to Quantum Turbulence Generated by Thin Vibrating Wires in Superfluid 4He

We report the turbulent transition in superfluid ⁴He generated by a vibrating wire as a function of its thickness. The response of a vibrating wire with a 3 μm diameter in superfluid ⁴He at 1.2 K reveals a hysteresis at the turbulent transition between an up sweep and a down sweep of driving force, while no hysteresis appears for wires with a thickness larger than 4.7 μm diameter. These results indicate that the 3 μm wire is efficient for reducing the number of vortex lines attached to it. A cover box and slow cooling also prevent vortex lines from attaching to a wire, resulting in a vortex-free vibrating wire. The effective mass of the vortex-free vibrating wire is almost constant in a wide range of velocities up to 400 mm/s; however, the wire density estimated from the resonance frequency is a half of the expected value of wire material, suggesting that a wire mass becomes lighter or a wire diameter becomes larger in the superfluid effectively.     

Instability of Quantized Vortices Attached to Oscillating Boundaries in Superfluid 4He

The motion of quantized vortices is studied using a vibrating wire in superfluid ⁴He. A vortex filtering method provides a superfluid practically free of remanent vortices in which the vibration of a wire cannot generate turbulence. Vortex lines are produced by cooling through the superfluid transition and remain forming bridges between a wire and a surrounding wall. Bridged remanent vortices increase the resonance frequency of a vibrating wire: the rate of an increase due to the remanent vortices is constant in a laminar flow regime and steeply increases in a turbulent flow regime with increasing wire velocity. These results suggest that oscillation of the bridged vortices provides a linear contribution to the wire vibration in the laminar flow regime, until instability occurs in the oscillation of the vortices, causing turbulence.      

Vibrating wire measurements in liquid3He II. The superfluid B phase

Measurements with a vibrating wire viscometer in superfluid³He-B at temperatures down to 0.6 mK are described. The need to consider compressibility of the superfluid component in any analysis of vibrating wire measurements is clearly demonstrated and a theoretical calculation of the force acting on a vibrating wire in a finite compressible superfluid is given. The experimental data are consistent with this calculation if theoretical values of the second viscosity ξ₃ are used in the analysis. The failure of the hydrodynamic theory when the quasiparticle mean free path1 is comparable to the wire radius a was observed, and an expression has been deduced for the force acting on the wire when1 is finite. Experimental and theoretical evidence is presented to show that this expression is valid for arbitrary1/a. Values of the viscosity obtained using this expression agree with those obtained in other experimental work and confirm the large discrepancy with theoretical calculations at low reduced temperatures.     

The Damping of a Quartz Tuning Fork in Superfluid 3He-B at Low Temperatures

We have measured the damping on a quartz tuning fork in the B-phase of superfluid ³He at low temperatures, below 0.3T _c. We present extensive measurements of the velocity dependence and temperature dependence of the damping force. At the lowest temperatures the damping is dominated by intrinsic dissipation at low velocities. Above some critical velocity an extra temperature independent damping mechanism quickly dominates. At higher temperatures there is additional damping from thermal quasiparticle excitations. The thermal damping mechanism is found to be the same as that for a vibrating wire resonator; Andreev scattering of thermal quasiparticles from the superfluid back-flow leads to a very large damping force. At low velocities the thermal damping force varies linearly with velocity, but tends towards a constant at higher velocities. The thermal damping fits very well to a simple model developed for vibrating wire resonators. This is somewhat surprising, since the quasiparticle trajectories through the superfluid flow around the fork prongs are more complicated due to the relatively high frequency of motion. We also discuss the damping mechanism above the critical velocity and compare the behaviour with other vibrating structures in superfluid ³He-B and in superfluid ⁴He at low temperatures. In superfluid ⁴He the high velocity response is usually dominated by vortex production (quantum turbulence), however in superfluid ³He the response may either be dominated by pair-breaking or by vortex production. In both cases the critical velocity in superfluid ³He-B is much smaller and the high velocity drag coefficient is much larger, compared to equivalent measurements in superfluid ⁴He.     

Observation of Laminar and Turbulent Flow in Superfluid 4He using a Vibrating Wire

No Heading We have investigated the laminar and the turbulent flow in superfluid ⁴He using a vibrating wire made of thin NbTi ( 2.5 m). The wire velocity as a function of applied force has shown a large hysteresis at the first cooling from normal fluid to the superfluid state. But after a couple of increasing and decreasing wire velocity we have found that the hysteresis vanished and the laminar and the turbulent flow are clearly separated at a critical velocity. The wire moving just after the first cooling must be influenced by remnant vortices nucleated through the superfluid transition. The appearance of the laminar flow below the critical velocity suggests that vortex strings on the wire seem to be selected as suitable sizes by a vibrating flow at higher velocities. We also measured the velocity dependence after immersing the wire directly into the superfluid and found that the laminar region expands up to a velocity much higher than the critical velocity observed above. This result indicates that remnant vortices are considerably reduced by the immersing method.PACS numbers: 67.40.Vs, 47.27.Cn     

Magnetic Distortion of the B-like Phase of Superfluid 3He Confined in Aerogel

We present measurements of the response of the B-like phase of superfluid ³He in aerogel to an applied flow. The measurements are made using a cylindrical piece of 98% silica aerogel attached to a vibrating wire resonator. The resonator is immersed in superfluid ³He at 16 bar pressure and at low temperatures. A variable magnetic field is applied such that the aerogel-confined superfluid may exist in the A-like or B-like phase, while the surrounding fluid is always in the bulk B-phase. The resonator response reveals a velocity dependence of the inferred aerogel-confined superfluid fraction. We discuss measurements of the temperature and magnetic field dependence of the response in the B-like phase. We find a significant field dependence indicating a strong magnetic distortion of the B-like phase order parameter.      

The specific heat capacity and thermal conductivity of normal liquid3He

By observing the diffusion of a heat pulse along a 10-cm column of normal liquid³He with the aid of two vibrating wire thermometers, it has been possible to measure the heat capacityC and thermal conductivityK of the liquid in the temperature range fromT _C to 10 mK and at pressures of 0.21, 4.39, 9.97, 20.01, and 29.32 bar. By using a Pt NMR thermometer, an LCMN thermometer, and a³He melting curve thermometer calibrated using the melting curve given by Greywall in 1983, a temperature scale has been established and (1) it has been shown that this melting curve is consistent in the temperature range 5–22 mK with the Korringa law for the Pt thermometer with a Korringa constant of 29.8±0.2 sec mK, (2) departures have been observed from the Curie-Weiss law for LCMN at low temperatures, and (3) values of the superfluid transition temperature have been obtained that are about 4% lower than the Helsinki values. The measured heat capacities agree well with those of Greywall, but values ofKT are higher than those of Greywall and show more temperature dependence below 10 mK. The implications for the present results of the very different melting curve given by Greywall in 1985 are discussed in an Appendix.     

Kapton Capacitance Thermometry at Low Temperatures and in High Magnetic Fields

A sensitive Kapton foil capacitance sensor, with size of 9.5 mm×4.5 mm, has been developed and used as a thermometer at ultra-low temperatures down to 1.2 mK and in high magnetic fields. There is no visible magnetic field dependence up to 15 T. The sensor was calibrated with ³He melting pressure thermometer (MPT) and vibrating wire (VW) viscometer. With the silver powder sintered heat exchanger sandwich-like design, the thermal relaxation time is as short as a few minutes at the base temperature. The low temperature (below 1.2 K) reproducibility has been tested and is satisfied within experimental errors.     

Turbulence generated by vibrating wire resonators in superfluid 4He at low temperatures

No Heading We have measured responses of vibrating wire resonators in superfluid ⁴He at millikelvin temperatures. We find evidence for turbulence generation above a critical velocity on the order of a few cm/s. At the critical velocity for the onset of turbulence, the resonator velocity abruptly decreases and shows hysteretic behavior. Surprisingly we find that the resonant frequencies of the resonators increase in the turbulent regime. We also find that the critical velocity may be reduced by the presence of turbulence generated by a neighboring vibrating wire resonator, allowing the detection of existing turbulence in the low temperature regime.PACS numbers: 67.40 Vs, 67.40 Pm     

Hysteresis,Switching and Anomalous Behaviour of a Quartz Tuning Fork in Superfluid 4He

We have been studying the behaviour of commercial quartz tuning forks immersed in superfluid ⁴He and driven at resonance. For one of the forks we have observed hysteresis and switching between linear and non-linear damping regimes at temperatures below 10 mK. We associate linear damping with pure potential flow around the prongs of the fork, and non-linear damping with the production of vortex lines in a turbulent regime. At appropriate prong velocities, we have observed metastability of both the linear and the turbulent flow states, and a region of intermittency where the flow switched back and forth between each state. For the same fork, we have also observed anomalous behaviour in the linear regime, with large excursions in both damping, resonant frequency, and the tip velocity as a function of driving force.     

Novel Oscillating Aerogel Experiments in Superfluid 3He at Ultralow Temperatures

We present novel experiments on a disk of 98% aerogel oscillating in superfluid ³ He at ultralow temperatures. The aerogel dik is attached to a goal post shaped vibrating wire resonator and immersed in liquid ³ He cooled by a Lancaster style nuclear cooling stage. At low pressures we see no evidence for superfluidity within the aerogel down to our base temperature of below <0.11T_c. At higher pressures we observe large temperature dependent frequency shifts, reminiscent of torsional oscillator experiments. We find the transition temperature at 5 bar to be around 600K. The response of the resonator is highly non linear when the helium in the aerogel is superfluid. The resonant frequency decreases strongly with increasing wire amplitude. This offers an exciting new technique for measuring the superfluid properties of ³ He in aerogel in the ultralow temperature regime.     

The Dynamic Texture of Superfluid 3He-B at Very Low Temperatures and in High Magnetic Fields

No Heading We report experiments using a pair of crossed vibrating wire resonators (VWRs) in the B phase of superfluid ³He in the zero temperature ballistic limit and in magnetic fields up to the B to A phase transition field of 340 mT. The VWRs are sensitive mechanical probes of energy gaps, textures and turbulent flow. In high magnetic fields the energy gap is no longer isotropic but significantly distorted, and the damping measured by the VWRs increases. With the crossed pair, we find that we can reduce the damping measured by one VWR when we increase the drive on the other one. We suggest that the reduction arises from the orientation of the surrounding texture by superflow and the screening of quasiparticles by the creation of superfluid turbulence.PACS numbers: 67.57.Fg, 67.57.Hi.     

Andreev Reflection of Quasiparticles by a Vortex Tangle in Superfluid 3He-B?

We have made the surprising discovery that the thermal damping of a vibrating wire resonator in superfluid ³ He-B at ultra low temperatures is considerably depressed when a second wire in the vicinity is driven supercritically. The damping of a vibrating wire resonator at low velocities in the B-phase arises from the scattering of quasiparticle excitations and has a temperature dependence proportional to the Boltzmann factor exp(–/kT) at low temperatures. At higher velocities (v>v_L/3), the wire breaks Cooper pairs and emits a quasiparticle beam. At first sight it seems paradoxical that heating the superfluid can reduce the quasiparticle flux on a neighbouring wire. We can only understand this on the basis that vorticity emitted by the supercritical wire shields, via Andreev reflection, part of the background quasiparticle flux from reaching the other wire. If this interpretation is correct, these techniques will provide a sensitive probe of vortex dynamics in the ultra low temperature regime.     

Non-linear Mechanical Response of the A-like Phase of Superfluid 3He in Aerogel

We present measurements of the response of the A-like phase of superfluid ³He in aerogel to an applied flow. The measurements are made using a cylindrical piece of 98% silica aerogel attached to a vibrating wire resonator. The resonator is immersed in superfluid ³He at low temperatures and relatively high magnetic fields such that the aerogel confined superfluid is in the A-like phase, while the surrounding fluid is in the bulk B-phase. We observe a variety of interesting non-linear and hysteretic effects when the resonator is driven to higher velocities. We present some of our preliminary findings and speculate on their implications.     

Boundary effects in fluid flow. Application to quantum liquids

The effect of boundaries on the flow of rarefied gases is considered. For an excitation gas of arbitrary statistics and energy-momentum relationship we determine the magnitude of the slip length and the flow between parallel plates mostly by variational methods. Our approximate method avoids the need to solve integral equations numerically and yields in the stationary case better than 1% agreement with known exact results for the classical Maxwell-Boltzmann gas. Our general results are primarily applied to normal and superfluid Fermi liquids. We calculate the surface impedance of an oscillating plate and determine the frequency-dependent slip length for frequencies ranging from the hydrodynamic to the collisionless limit. Our results are applied to the analysis of viscosity measurements based on a torsional oscillator or a vibrating wire. The slip effects are shown to be very important for realistic experimental parameters, especially at low temperatures in the superfluid B phase of liquid ³He.     

The Thermal Damping of an Aerogel Resonator in Superfluid 3He-B at Ultra Low Temperatures

No Heading We present measurements of the thermal damping of a cylindrical aerogel sample oscillating in superfluid. ³He-B in the low temperature regime. The measurements are made at low pressures where the ³He confined in the aerogel is normal. As in the case of conventional vibrating wire resonators, the thermal damping arises from quasiparticle collisions at the wire surface and is enhanced by many orders of magnitude by Andreev scattering from the superfluid backflow around the resonator. However, in the case of aerogel, incoming quasiparticles must be absorbed and thermalised within the aerogel before being re-emilled.PACS numbers: 67.57.Bc, 67.57.De, 67.57.Hi, 67.57.Pq.     

NMR Study of Superfluid Phases of 3He Confined in 97.5% Porous Silica Aerogel

We have studied the scattering effect from aerogel strands on superfluid phases of ³He by a cw NMR method at 920 kHz. Liquid ³He at a pressure of 13 bar was confined in 97.5% porous aerogel from the same batch as that of a recent 4th sound study. The NMR experiment was performed in a magnetic field of 28.4 mT down to 0.3 mK. As temperature decreased, the NMR resonant frequency increased below 0.76 mK. The temperature of 0.76 mK agrees with the superfluid transition temperature T ^aerogel _c observed in the 4th sound study at the same pressure. Below T ^aerogel _c the behavior of thefrequency shift as a function of temperature indicates that there is no phasetransition to the other superfluid phase down to about 0.4 T ^aerogel _c . Owing to a very large surface solid ³He magnetization, we could not determine the superfluid phase of ³He in the aerogel in the magnetization measurement.     

Temperature dependence of the superfluid density in restricted geometries determined by an oscillating U-tube technique

The temperature dependence of the superfluid density of ⁴ He in restricted geometries has been determined at temperatures ranging from 66 mK to 1.7 K by use of an oscillating U-tube technique. Compressed powders were used to produce geometries with pore radii from about 10 to 20 Å. In contrast with bulk helium, an appreciable variation in the superfluid mass is observed down to temperatures as low as 66 mK. For the geometries investigated the superfluid density displays a linear temperature dependence for temperatures less than about 0.6 K.Work supported in part by the National Science Foundation under Grant Number GH-37658, and by the National Science Foundation under Grant Number GH-33637, through the Cornell Materials Science Center, MSC Report # 2308.