Acoustic Emission by Quartz Tuning Forks and Other Oscillating Structures in Cryogenic 4He Fluids

We report on experimental investigations of acoustic emission by quartz tuning forks resonating at frequencies 32 kHz, 38 kHz, 77 kHz and 100 kHz immersed in cold gaseous ⁴He and its normal and superfluid liquid phases. Frequency dependence of the observed low-drive-linewidth at 350 mK together with the temperature and pressure dependences (1.3 K < T < 4.2 K, 0 < p < 25 bar) of the observed damping of the high frequency (77 and 100 kHz) resonators measured in normal liquid ⁴He and its superfluid phase provide strong and direct evidence of the importance of sound emission by these tuning forks. Three analytical models of acoustic emission by vibrating tuning forks are developed and compared with the experimental results. We also discuss the importance of sound emission for experiments with the commonly used 32 kHz tuning forks as well as other oscillating structures??spheres, wires, grids and various micromachined sensors. We compare the relative importance of dissipative losses due to laminar viscous/ballistic drag and acoustic emission in liquid and 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.     

High Quality Tuning Forks in Superfluid 3He-B Below?200?��K

We have investigated the influence of the damping force acting on high quality tuning forks (Q??10⁶) of different sizes and geometries in superfluid ³He-B at temperatures below 200 ??K and a pressure of 0.1 bar. The measurements show that at low velocities, the damping of the largest fork expressed in terms of its resonance characteristic width ??f ₂ rises up as its velocity increases. This is in contradiction to the damping of the fork due to Andreev reflection and it may be caused by the interaction of this fork with excitations trapped in the Andreev bond states. We present our preliminary experimental results.     

The Influence of Acoustic Emission on the Onset of Turbulent Flow in He II

An experimental study is carried out on the influence of acoustic emission on the development of turbulent flow in He II at temperature 370 mK and at various pressures, from the saturated vapor pressure to that of ⁴He crystallization. The experimental technique of oscillating quartz tuning fork was applied for simultaneously exciting both superfluid flow, near the oscillating tuning fork prongs, and the acoustic wave emission. The flow rate of He II and the amplitude of the radiated acoustic waves were driven by the fork prongs, excited by voltage oscillations. Furthermore, the power of an acoustic wave depends on the density and sound velocity in He II as ρ/c ⁵, which in turn depends on the pressure. This allowed to measure the influence of the amplitude of acoustic emission on the flow in He II. It has been shown that the occurrence of acoustic radiation by tuning fork leads to an increased flow velocity for the beginning of nonlinear flow regime.     

The Fulde–Ferrell–Larkin–Ovchinnikov State in Pnictides

We present experimental results on crosstalk of non-electrical origin between high frequency quartz tuning forks immersed in the same volume of helium gas, liquid or superfluid. We compare these results with various observations of other groups and propose an explanation of this puzzling phenomenon. To the best of our knowledge, notable crosstalk has only been observed in superfluid helium both in the two-fluid regime and at very low temperatures, but was rarely seen to behave in a systematic way. We demonstrate some of its most significant properties—amplitude dependence within a short time span, long-term temporal instability, effects of the geometry of the setup and of obstacles placed between the tuning forks. Although the results are not fully understood, as the most likely explanation, we ascribe the observations to the coupling of tuning forks to standing acoustic modes inside the experimental volume, emphasizing the importance of second sound for understanding the observations at temperatures within the two-fluid regime (1 K<T<2.17 K). Finally, we suggest simple precautions leading to suppression of excessive acoustic crosstalk between oscillating objects in He II.     

Acoustic Resonances in Helium Fluids Excited by?Quartz Tuning Forks

Ordinary quartz tuning fork resonators, operated at about 30 or 200 kHz frequency, couple to acoustic first and second sound resonances in helium fluids under certain conditions. We have studied acoustic resonances in supercritical ⁴He, normal and superfluid ⁴He, and in isotopic mixtures of helium. Suggestive temperature, pressure, and concentration dependences are given. Furthermore, we propose a thermometric reference point device based on second sound resonances in helium mixtures, and indicate possible differences in the nature of second sound resonances in superfluid ⁴He and helium mixtures.     

Response of a Mechanical Oscillator in Solid 4He

We present the first measurements of the response of a mechanical oscillator in solid ⁴He. We use a lithium niobate tuning fork operating in its fundamental resonance mode at a frequency of around 30 kHz. Measurements in solid ⁴He were performed close to the melting pressure. The tuning fork resonance shows substantial frequency shifts on cooling from around 1.5 K to below 10 mK. The response shows an abrupt change at the bcc-hcp transition. At low temperatures, below around 100 mK, the resonance splits into several overlapping resonances.     

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.     

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.     

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.     

Analytical and finite element method design of quartz tuning fork resonators and experimental test of samples manufactured using photolithography 1—significant design parameters affecting static capacitance C0

Resonance frequency of quartz tuning fork crystal for use in chips of code division multiple access, personal communication system, and a global system for mobile communication was analyzed by an analytical method, Sezawa’s theory and the finite element method (FEM). From the FEM analysis results, actual tuning fork crystals were fabricated using photolithography and oblique evaporation by a stencil mask. A resonance frequency close to 31.964 kHz was aimed at following FEM analysis results and a general scheme of commercially available 32.768 kHz tuning fork resonators was followed in designing tuning fork geometry, tine electrode pattern and thickness. Comparison was made among the modeled and experimentally measured resonance frequencies and the discrepancy explained and discussed. The average resonance frequency of the fabricated tuning fork samples at a vacuum level of 3×10⁻² Torr was 31.228-31.462 kHz. The difference between modeling and experimentally measured resonance frequency is attributed to the error in exactly manufacturing tuning fork tine width by photolithography. The dependence of sensitivities for other quartz tuning fork crystal parameter C₀ on various design parameters was also comprehensively analyzed using FEM and Taguchi’s design of experiment method. However, the tuning fork design using FEM modeling must be modified comprehensively to optimize various design parameters affecting both the resonance frequency and other crystal parameters, most importantly crystal impedance.     

The Damping and Drag Coefficient of Quartz Tuning Fork in Superfluid $$^{3}\hbox {He}$$–$$^{4}\hbox {He}$$4He Solutions in the Laminar Flow

The \(^{3}\)He impurity influence on the oscillations of a quartz resonator and thus its drag coefficient in a laminar flow of a superfluid \(^{3}\)He–\(^{4}\)He mixture has been investigated. The temperature dependences of the resonance curves were measured on quartz tuning forks with a resonance frequency 32 kHz in vacuum in superfluid mixtures with \(^{3}\)He concentrations of \(x_{3}=0.05\) and 0.15 in a wide range of driving forces at temperatures from 0.5–2.5 K. The results obtained were used to plot the temperature dependence of the drag coefficient. With the help of the normalization on the effective area of the oscillating body, the concentration dependence of the drag coefficient of the quartz tuning fork and the vibrating sphere in superfluid solutions has been constructed and analyzed.     

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.     

Direct Measurement of the Critical Velocity Above Which a Tuning Fork Generates Turbulence in Superfluid Helium

The dynamics of an electrically-driven 8 kHz quartz tuning fork has been studied experimentally in liquid helium-4 in the temperature range 1.3<T<4.2 K under the saturated vapour pressure. The fork has relatively large dimensions compared to standard 32 kHz fork used in recent investigations. The velocity of the tip of the fork prong is measured by the indirect electromechanical equivalent method and is compared with the velocity of another 8 kHz fork (from the same batch) determined by direct optical measurement of the oscillation amplitude through Michelson interferometry. A comparison of these results has provided absolute values for the critical velocity of the transition to the turbulent state.     

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.     

Mechanical Response of <Superscript>4</Superscript>He Films Adsorbed on Graphite with a Quartz Tuning Fork

We have carried out quartz-crystal microbalance (QCM) experiments of 32 kHz quartz tuning fork for ⁴He films adsorbed on Grafoil, and have measured the temperature dependence of the resonance frequency and Q value for various areal densities. It was found that the frequency does not decrease exactly in proportion to the areal density. This means that the film still undergoes decoupling partly, although it is strongly suppressed from that of 5 MHz QCM measurements. Above the three-atom thick film, the decoupling due to the superfluidity of the overlayer is observed. In addition, we found that the competition between the superfluidity and the slippage takes place for a large oscillation amplitude. From the comparison with 5 MHz QCM measurements, it is concluded that the acceleration of substrate plays an important role in the slippage.     

Slippage of 4He Films and Precursor Phenomenon of Superfluidity

We have carried out quartz crystal microbalance (QCM) experiments for ⁴He films on an exfoliated single-crystalline graphite using a 32 kHz tuning fork, and have measured the temperature dependence of the resonance frequency and the Q value for various areal densities and oscillation amplitudes. Comparing with the previous experiments for Grafoil, the decoupling of the films due to the slippage or the superfluidity was larger than that of Grafoil, and the competition between the slippage and the superfluidity was observed in three-atom thick films. Furthermore, it was found that the slippage is suppressed gradually at higher temperature than the superfluid onset T _c , and that the relaxation time decreases at low temperatures while it obeys the Arrhenius law at high temperatures. These results suggest a precursor to the superfluidity of ⁴He films.     

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.     

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.     

Analytical and finite element method design of quartz tuning fork resonators and experimental test of samples manufactured using photolithography 2: comprehensive analysis of resonance frequencies using Sezawa's approximations

Resonance frequency of a quartz tuning fork crystal for use in chips of code division multiple access, personal communication system, and a global system for mobile communication was comprehensively analyzed by an analytical method, Sezawa's approximations and the finite element method. A comparison was also made in a more detailed and comprehensive manner among resonance frequencies calculated by the Sezawa's approximations. From the finite element method analysis results, actual tuning fork crystals were fabricated using mass-production capable positive (subtractive) photolithography, selective etching and subsequent positive (subtractive) photoresist spray coating method. A target resonance frequency of was aimed at and a general scheme of commercially available 32.768 kHz tuning fork resonators was also followed in designing tuning fork geometry, tine electrode pattern and thickness. Comprehensive comparison was made among the modeled and experimentally measured resonance frequencies and the discrepancy explained and discussed. Finite element method analysis results quite closely agreed with the experimentally measured resonance frequencies (32.676-32.933 kHz) of the fabricated tuning fork samples measured at a vacuum level of 10⁻⁵ Torr. The difference between modeling and experimentally measured resonance frequency is attributed to the error in exactly manufacturing tuning fork tine width by photolithography. However, the tuning fork design using finite element method modeling must be modified comprehensively to optimize various design parameters affecting both the resonance frequency and other crystal parameters, most importantly crystal impedance (series resistance).