Response of a Mechanical Oscillator in Solid 4He
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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.
KeywordsSolid 4He Tuning fork Viscoelastic properties
The mechanical properties of solid 4He have received considerable attention in recent years following claims of a possible ‘supersolid’ response in torsional oscillator experiments . There is now growing evidence that supersolidity was not observed in these measurements, and that the effects observed are more likely to be associated with changes in the mechanical properties of the solid, possibly related to the dynamics of defects [2, 3, 4]. Here we present preliminary measurements of the response of a lithium niobate miniature tuning fork immersed in solid 4He. The device provides a simple technique to study the mechanical properties of solid helium over a very wide temperature range.
Miniature tuning forks have been used for many research applications, such as scanning probe microscopy  and high performance characterization of liquids . In recent years they have been used in quantum fluids research to study many different properties such as viscosity [7, 8], turbulence [9, 10], cavitation , Andreev scattering [12, 13], and acoustic modes [14, 15].
For many applications it is important to minimise intrinsic losses, so miniature tuning forks are often made from quartz. To study solid 4He it is more important to optimise the piezoelectric properties to allow large driving forces and large signal currents for small velocities. For this purpose we have made tuning forks with lithium niobate which has a much larger piezoelectric modulus .
2 Experimental Details
The lithium niobate tuning fork is shown in Fig. 1b. It has prongs of length L=6 mm, width W=1 mm, thickness T=1.1 mm and separation, D=0.6 mm. The tuning fork was produced in-house by mechanically cutting a LiNbO3 Y-cut wafer and using a parallel arrangement of electrodes  as shown in Fig. 1b (right). The fork was coated with Cr/Au using an RF sputtering system to increase the electromechanical coupling, shown in Fig. 1b (left).
Measurements in helium were made using isotopically pure 4He produced in Lancaster . The solid samples were grown at various constant temperatures ranging from below 10 mK to around 1.6 K. The solid samples were grown by increasing the pressure until the fill-line located at the very top of the cell became blocked. It is inevitable that, after filling, small pockets of liquid will remain trapped, most likely in the top corners of the cell. The measurements described below were therefore taken at, or close to, the melting pressure. This was confirmed by measurements of the two pressure sensors.
3 Fork Operation and Characterisation
The tuning fork was driven by a function generator whose output was attenuated to give a sufficiently small drive voltage. The signal current was measured with a custom-made current-to-voltage converter  and a lock-in amplifier. Measurements in vacuum at 4 K found the fundamental resonance to have a frequency of 24.581 kHz and a width of Δf 2=0.18 Hz. The fork constant, using Eq. (3) is found to be a=4.2×10−4 NV−1.
4 Measurements Around the bcc-hcp Phase Transition
Measurements were made by sweeping the frequency at a constant drive amplitude whilst measuring the signal current. We measure the components of the signal which are in-phase and out-of-phase with the driving voltage. After subtracting the background signals due to the measurement circuit, we obtain the signal due to the fork motion which gives the prong tip velocity using Eq. (2).
5 Measurements in the hcp Phase Down to Very Low Temperatures
The resonance frequency in the solid is significantly higher than the vacuum frequency. The large frequency shift corresponds to the extra elastic restoring force provided by the solid. This will depend on both the shear modulus and bulk modulus of the solid helium surrounding the prongs. The elastic properties of solid helium are quite anisotropic particularly at low temperatures  so we expect the frequency to depend on the precise structure and orientation of each crystal which will vary for different samples. Indeed we find a significant variation in the resonant frequency in both phases for different samples. The resonance frequencies can vary by about 10 kHz under nominally identical conditions. We always observe a discontinuity in the response at the bcc-hcp phase transition although the precise behaviour varies for different samples.
The tuning fork motion in the solid is quite heavily damped as evident from the large frequency widths of the response. This indicates viscoelastic behavior. Damping may result from the motion of defects and/or vacancies in the solid. We have made further studies of the viscoelastic properties of solid 4He using the ‘floppy’ wire shown in Fig. 1. These measurements will be discussed in an accompanying paper.
The temperature at which the tuning fork resonance splits into multiple narrower resonances in the hcp phase is similar to the temperatures where changes in the mechanical properties have been observed by many groups [1, 2, 3, 4]. It has been suggested that the elasticity of the solid may change at low temperatures due to pinning of defects by 3He impurities . This may be responsible for the low temperature behaviour of the tuning fork. We believe that the multiple resonances correspond to resonances of the entire crystal, defined by the geometry of the cell. We note that the samples studied here were grown from isotopically pure 4He, but the gas handling system was previously used for 3He samples so there may be a small residue of 3He.
To conclude, we have presented the first measurements of the response of a lithium niobate tuning fork vibrating in solid 4He over a wide temperature range. The response of the tuning fork is sensitive to the viscoelastic properties of the solid and is dependent on the crystal structure and orientation. The device offers a very simple and versatile tool to complement other studies of solid helium.
We thank S.M. Holt, A. Stokes and M.G. Ward for excellent technical support. We also thank J. Bennett and L. Matsiev for discussions on the LiNbO3 tuning fork design. This research is supported by the Leverhulme Trust, the UK EPSRC and by the European FP7 Programme MICROKELVIN Project, no. 228464.
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