Quartz Tuning Forks as Cryogenic Vacuum Gauges

We report measurements of the mechanical \(Q\) of a 32.7 kHz quartz tuning fork as a function of pressure for helium and argon at T  \(=\)  300 K and for helium in the temperature range 7.0–0.7 K. In the low pressure ballistic regime, the damping due to the surrounding gas is inversely proportional to \(P\) , while for higher pressures, a hydrodynamic treatment accounts for most of the variation of \(Q\) with \(P\) . We have combined the ballistic and hydrodynamic models together with calculations of the thermal transpiration correction to correlate the tuning fork \(Q\) at low temperature with the pressure measured with a room temperature pressure gauge. The fork was found to be useful as an in situ pressure gauge for pressures above \(\sim \) 0.1 mTorr. A dissipation peak and frequency drop associated with the superfluid transition in the adsorbed helium film is also observed for \(T<1.4\)  K.     

Quartz Tuning Forks and Acoustic Phenomena: Application to Superfluid Helium

Immersed mechanical resonators are well suited for probing the properties of fluids, since the surrounding environment influences the resonant characteristics of such oscillators in several ways. Quartz tuning forks have gained much popularity in recent years as the resonators of choice for studies of liquid helium. They have many superior properties when compared to other oscillating bodies conventionally used for this purpose, such as vibrating wires. However, the intricate geometry of a tuning fork represents a challenge for analyzing their behavior in a fluid environment—analytical approaches do not carry very far. In this article the characteristics of immersed quartz tuning fork resonators are studied by numerical simulations. We account for the compressibility of the medium, that is acoustic phenomena, and neglect viscosity, with the aim to realistically model the oscillator response in superfluid helium. The significance of different tuning fork shapes is studied. Acoustic emission in infinite medium and acoustic resonances in confined volumes are investigated. The results can aid in choosing a quartz tuning fork with suitable properties for experiments, as well as interpreting measured data.     

Measuring the Prong Velocity of Quartz Tuning Forks Used to Probe Quantum Fluids

Recently, quartz tuning forks have been used to probe the dynamics of quantum fluids. For many of these measurements it is important to know the velocity amplitude of the tips of the vibrating fork prongs. We have used different techniques to establish, with an accuracy of a few percent, the relationship between the electrical and mechanical properties of several commercial quartz tuning forks with fundamental resonant frequency ～32 kHz. The velocity is usually inferred from an electro-mechanical calibration that models a quartz prong as a clamped, rectangular cantilever beam. We have tested the accuracy of this calibration using three methods: measurement of the amplitude at which the fork prongs touch each other; direct optical measurement of the moving fork prongs using strobe microscopy; and a Michelson interferometry technique operating with a 670 nm laser. All three methods yield consistent results. The velocity so determined is found to be 10% lower than that of the standard electro-mechanical calibration.     

Studies on Helium Liquids by Vibrating Wires and Quartz Tuning Forks

We present results of low-temperature experiments on dilute mixtures of ³He in ⁴He and on pure ³He, obtained by means of two kinds of mechanical oscillators immersed in the liquid sample: vibrating wires and quartz tuning forks. The helium sample was cooled either by adiabatic demagnetization of an immersed copper nuclear stage or by adiabatic melting of ⁴He in superfluid ³He. The measured effect of the surrounding fluid on the mechanical resonance of the oscillators is compared with existing theories. We also discuss resonances of second sound and the state of supersaturation, both observed by a tuning fork in helium mixtures.     

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.     

Frequency Characteristics of a Quartz Tuning Fork Immersed in He II

The effect of dissipation on frequency characteristics of tuning forks was measured, the dissipation being induced by acoustic radiation of different wavelengths, excited by tuning forks. The tuning forks have been immersed in the superfluid helium. The fork resonance frequencies 32, 77 and 99 kHz have been measured at T=370 mK in the pressure range between SVP and 24.9 atm. Most of the tuning forks have been studied in a commercial can. It is found that at wavelength λ>0.6 cm the frequency dependence is determined by the relationship between density and pressure. It is established that a decrease in wavelength enhances influence of the acoustic radiation on the fork oscillation frequency. In the case where the sound wavelength is equal to the can internal diameter an acoustic resonance occurs. The frequency reaches values higher than the fork frequency in vacuum. Further reduction of the sound wavelength leads to the situation when the resonant frequency is similar to the frequency at long wavelengths.     

Thermometry in Normal Liquid 3He Using a Quartz Tuning Fork Viscometer

We have developed the use of quartz tuning forks for thermometry in normal liquid ³He. We have used a standard 32 kHz tuning fork to measure the viscosity of liquid ³He over a wide temperature range, 6 mK<T<1.8 K, at SVP. For thermometry above 40 mK we used a calibrated ruthenium oxide resistor. At lower temperatures we used vibrating wire thermometry. Our data compare well with previous viscosity measurements, and we give a simple empirical formula which fits the viscosity data over the full temperature range. We discuss how tuning forks can be used as convenient thermometers in this range of temperatures with just a single parameter needed for calibration.     

Nucleation of Solid 4He by Acoustic Waves

When an acoustic wave pulse was applied to metastable superfluid ⁴He above the megting pressure, a bulk ⁴He crystal was nucleated in the superfluid at low temperatures. Overpressure in which the metastable liquid could exist was investigated by changing the acoustic wave power. A larger power pulse could nucleate the crystal in smaller overpressures, which decreased linearly in the acoustic wave power.     

Acoustic Turbulence in Superfluid 4He

Recent work on nonlinear second sound wave propagation and acoustic turbulence in superfluid ⁴He is reviewed. Observations of direct and inverse turbulent energy cascades are described. The direct cascade arises due to the huge nonlinear dependence of the second sound wave velocity on its amplitude. The flux of energy injected at the driving frequency is transformed via successively higher harmonics until it is eventually attenuated by viscous dissipation at the short wavelength edge of the spectrum. The onset of the inverse cascade occurs above a critical driving energy density, and it is accompanied by giant waves that constitute an acoustic analogue of the rogue waves that occasionally appear on the surface of the ocean. The theory of the phenomena is outlined and shown to be in good agreement with the experiments.     

Acoustic Resonances and Non-linearity in Solid 4He

Direct measurements of the low frequency shear modulus of solid ⁴He show a remarkable increase below 150?mK, a regime where torsional oscillator (TO) experiments show evidence of mass decoupling. Acoustic resonance measurements at higher frequencies confirm the unusual elastic behavior. A striking feature of both TO and shear modulus measurements is their amplitude dependence—the low temperature anomalies are reduced as the drive amplitudes increase. We have studied the amplitude dependence of acoustic resonances in both standard and isotopically pure ⁴He crystals. The resonance peaks shift to lower frequencies as the drive amplitude increases. At high amplitudes the peaks are asymmetric and exhibit both bistability and hysteresis—classic features of nonlinear oscillators. At the lowest amplitudes the peak frequency and shape are independent of drive amplitude and are non-hysteretic. The threshold for nonlinear behavior is lower for the isotopically pure crystal. We also studied the effects of annealing on the acoustic resonances. The nonlinear behavior was qualitatively unchanged but annealing often affected the shape of the resonance peaks, which often had a complicated structure at low temperatures.     

Homogeneous Nucleation of 4He Crystals by Acoustic Waves

We have recently found experimental evidence for the homogeneous nucleation of crystals in metastable liquid ⁴He at high pressure. For this we combined the focusing of a high intensity spherical acoustic wave with a simple light scattering technique. We discuss the analysis used to distinguish between nucleation of bubbles in the negative pressure swings of the wave from nucleation of crystals in the positive swings. We also discuss the interest of our results and future developments of our experiment in the general context of the study of nucleation and instability limits in phase diagrams.     

Propagation of Quantized Vortices Driven by an Oscillating Sphere in Superfluid 4He

We performed numerical simulation of quantum turbulence at 0 K generated from remnant vortices attached to an oscillating sphere. The remnant vortices are extended by the sphere motion and form a tangle with emitting vortex loops. As time passes, the length of vortices in a computational volume becomes statistically steady. We investigate in the statistical steady state the distribution of the length of vortex loops and anisotropy of their propagation direction caused by the sphere oscillation. The propagation direction of the emitted vortex loops is anisotropic along the oscillation direction of the sphere. The obtained results are consistent with results obtained in the experimental study using vibrating wires in superfluid ⁴He.     

The Frequency Dependence of the Added Mass of Quartz Tuning Fork Immersed in He II

We measured the dependences of the resonance frequency of tuning forks immersed in liquid helium at \(T = 0.365\hbox { K}\) in the pressure interval from saturated vapor pressure to 24.8 atm. The quartz tuning forks have been studied with different resonance frequencies of 6.65, 8.46, 12.1, 25.0 and 33.6 kHz in vacuum. The measurements were taken in the laminar flow regime. The experimental data allow us to determine the added mass of a quartz tuning fork in He II. It was found that the added mass per unit length of the prong fork is frequency dependent. Some possible qualitative explanations for such dependence are proposed. In addition, we observed, at \(T = 0.365\hbox { K}\), the changes in added mass with pressure according to the pressure dependence of He II density.     

Deconvolution of Acoustic Emission and Other Causal Time Series

A new technique, root projection (RP), is given for quantitative deconvolution of causal time series in the presence of moderate amounts of noise. Deconvolution is treated as a well-conditioned but underdetermined problem and a priori information is employed to obtain comparable noise reduction to that achieved by singular value decomposition (SVD) techniques while providing more accurate frequency information about the inverse. Two detailed examples arc given. The first gives noise analysis for alternate methods for deconvolution with a Gaussian kernel. The second example presents a model acoustic emission transducer calibration problem with typical noisy and incomplete output data. This example is treated by the use of a robust cross-cutting algorithm combining both the RP and SVD methods.     

Quartz Oscillator Study of Monolayer 4He and 3He - 4He Mixture Films Adsorbed on Solid H2

The superfluid transition in submonolayer and monolayer ⁴ He films and ³ He - ⁴ He mixture films on solid H ₂ has been studied using a quartz crystal microbalance technique. Kosterlitz-Thouless (KT) transitions were observed in submonolayer ⁴ He films with density greater than 0.062 ± 0.002 Å ^–2 . We determine a binding energy of ⁴ He to 0.241 Å ^–2 H ₂ of –15.7 K in the. presence of 1 monolayer of ⁴ He. At several ⁴ He coverages, a range of submonolayer ³ He coverages was studied (n ₃ 0.0567 Å ^–2 ). With each increase in the ³ He coverage, the KT transition temperature decreased. For the higher coverage mixture films studied (n ₄ 0.0726 Å ^–2 ) we observed an apparent second decoupling of the film from the quartz oscillator frequency in addition to the KT transition. We have studied the. coverage dependence of this new feature.     

Nucleation of 4He Crystal at Melting Pressure by Acoustic Waves

No Heading When an acoustic wave pulse of several msec was applied to superfluid ⁴He at melting pressure, ⁴He crystal was nucleated on the transducer and rapidly disappeared after the pulse. We measured thresholds of the acoustic wave power which induced a nucleation in the temperature range between 0.09 K and 1 K. The treshold of the nucleation exhibited no temperature dependence below 0.3 K. The nucleation was not induced above 0.6 K even if the injected acoustic pulse was strong. 0.6 K is close to the inversion temperature at which the direction of the force by acoustic wave was inverted on the solid-liquid interface of ⁴He [Phys. Rev. Lett. 90, 075301 (2003)]. Our observation indicates that the effect of acoustic radiation pressure contributes to the heterogeneous nucleation by acoustic waves.PACS numbers: 67.80.–s, 68.08.–p, 81.10.–h, 43.25.Qp.     

Thermoviscous Effects in Steady and Oscillating Flow of Superfluid 4He: Experiments

The correct interpretation of superfluid flow experiments relies on the knowledge of thermal and viscous effects that can cause deviations from ideal behavior. The previous paper presented a theoretical study of dissipative and reactive(nondissipative) thermoviscous effects in both steady and oscillating flow of an isotropic superfluid through small apertures and channels. Here, a detailed comparison is made between the theory and a wide array of experimental data. First, the calculated resistance to steady superflow is compared with measurements taken in a constant pressure-head flow cell. Second, the resonant frequency and Q of three different helmholtz oscillators are compared with predictions based on the calculated frequency response. The resonant frequency and Q are extracted numerically from the frequency response, and analytical results are given in experimentally important limits. Finally, the measured and calculated frequency response are compared at a temperature where the Helmholtz oscillator differs significantly from a simple harmonic oscillator. This difference is used to explain how the thermal properties of the oscillator affect its response. The quantitative agreement between the theory and experiment provide an excellent check of the previously derived equations. Also, the limiting expressions shown in this paper provide simple analytical expressions for calculating the effects of the various physical phenomena in a particular experimental situation.