Helmholtz Resonator in Superfluid Helium

Shane Allen and Ian Young

University of Minnesota

Methods of Experimental Physics Spring 2013

Abstract

In a double-chambered resonator, measurements of Helmholtz resonances were taken by submerging in liquid helium cooled below the lambda point, 2.18 K, at which superfluid components appear. Fractional abundance of superfluid helium component to normal fluid component were calculated. A previous attempt at this measurement in the spring 2012 MXP II course was not able to produce a sensible result. Frequency responses were found at varying temperatures but the expected shift in frequency for Helmholtz resonance was not detected.

Introduction

The effect known as Helmholtz resonance for air takes place when a force pushes on a bottleneck of a cavity. As pressure in the cavity increases, air pushes up on the bottleneck, causing a portion of air to oscillate back and forth between the two sides of the cavity opening, as shown in Figure 1.

Figure 1: Helmholtz resonance - a packet of air oscillates in a neck, driven on the outside by a force and inside by reaction force [Original Figure]

Helium-4 undergoes a phase change when cooled below 2.17 K, known as the lambda point, which introduces an element with zero viscosity known as the superfluid component (1). The density ratio of this component rises as temperature drops, as seen in Figure 2.

Figure 2: The lambda point of liquid helium is the temperature at which the superfluid density, ρ_s, of liquid helium begins to appear (2).

In this experiment, an effect analogous to the Helmholtz resonance in air takes place in a chamber housing superfluid helium. The superfluid oscillates between two chambers separated by a thin porous membrane, which inhibits the transfer of normal fluid due to its viscosity while the non-viscous superfluid is free to oscillate (1). This experiment is a continuation of efforts from a project for the 2012 methods of experimental physics course at the University of Minnesota. The resonance frequency is dependent on the dimensions of the two chambers, the pores and the density ratio of the superfluid component (3). Measurement of the superfluid density ratio was planned by using the resulting Helmholtz resonance frequencies vs. temperature.

Theory

The Helmholtz resonance frequency for the dual-chambered resonator is shown in Figure 3.

Figure 3: Helmholtz resonance frequency in dual-chambered resonator.

Here, ρ_s/ρ is the density ratio of superfluid and normal components, Ω is total volume of both chambers, A is the effective cross sectional area of the pores, â„“ is the effective length of the pores, and dP/dρ is the speed of first sound in liquid helium squared.

Experimental Setup

A diagram of the experimental cell is shown in Figure 4. The experimental cell housed two chambers with a porous membrane epoxied over a hole through the dividing wall. The porous membrane allowed the superfluid component of the liquid to oscillate between the chambers without allowing the normal component to transfer due to its viscosity. This oscillation was driven by an electrode above the top chamber, which exerted a push/pull coulomb force on a metalized kapton diaphragm. Another diaphragm in the lower chamber was pushed on by the resulting pressure waves and a second electrode picked up the voltage response for measurement.

Figure 4: The dual-chamber resonator apparatus for oscillating the superfluid component. Not pictured are four 4-40 screws, which held the components in place (not to scale) [Original Figure]

Each kapton diaphragm was epoxied to its respective kapton diaphragm to prevent leaks from the top and bottom of the chambers. An indium gasket was coated in vacuum grease and press sealed between the copper rings of the cell to prevent radial leaks from the sides. A hole in the side of the upper chamber was drilled by the previous year’s MXP group to ensure fluid fill of the cell.

Calibrations for the thermometry equipment was done by taking several bath resistor readings at varying pressures above and below the lambda point. The relation between pressures and temperatures was found in a vapor pressure table for He-4 (5). Fitting the resistance values to the temperature gives a thermometry fit as shown in Figure 5.

Figure 5: The temperature calibration curve for He-4. Temperature is graphed versus the resistance value. Error bars not visible due to y scale.

Results

Of the peaks found, shown on Figure 6; peaks 4 and 5 were the only to come within less than 10 sigma away from any frequency prediction at that temperature. This occurred at (2.100±.001) K with the peak frequencies of (960±10) Hz and (990±10) Hz, that are 3.6 and 6.6 sigma off, respectively, from the predicted 924 Hz resonance. This can be seen in Figure 7.

Figure 6: Resonant frequencies with temperature compared to the expected Helmholtz resonance. Only one reading from Peak 4 and one from Peak 5 came within 10 sigma from theoretical frequencies, these are circled in red. Error bars obscured by scale.

Figure 7: Resonances discovered in frequency sweeps. Distinct peaks are labeled in red. Upper: Sweep at (2.100 ±.001) K; Lower: Sweep at (1.796±.001) K.

Conclusion and Notes for Future Projects

Due to the large quantity of resonance peaks and lack of proper temperature shift, the Helmholtz resonance could not clearly be identified. While two peaks were near the expected values for the Helmholtz resonance, their temperature shifts did not match.

A likely cause of these numerous resonance peaks would be leakage in the cell. It cannot clearly be stated as the only cause, because if these were all leakages, they would be Helmholtz leakages and would therefore shift with temperature. It may be the case that these temperature shifts are a more complicated calculation due to the large number of resonances and factors unknown. While efforts were made to seal the two chambers from leakage, it could be possible that due to the large change in temperature, the system leaked.

It may be that the membrane was incorrectly epoxied onto the copper plate. The plate was viewed under a microscope to determine whether the epoxy had covered the membrane, and was deemed acceptable. Photos of the membrane from both sides can be found in Appendix A. If the membrane were not properly epoxied, it would destroy the potential for Helmholtz resonance in the system. The quality of the epoxy on the membrane could be verified in the future by using a flow test on the membrane itself to see if fluid can flow through it.

Another possibility for these detected measurements could be small oscillations in components of the cell or in the Dewar itself, which were vibrating at a detectable resonance frequency. This is less likely due to the tightness of the cell components, which were bolted down using spring washers to avoid loosening upon cooling of the system.

If further research was to be done with this theory, an in depth flow analysis of the porous membrane would be necessary, as well as leak testing the cell. Other suggestions for future changes could be to manipulate the size of the capillary hole in the side of the upper copper plate. If reductions in frequency resonance peaks were seen with reduction of capillary radius, it would show some Helmholtz effect.

Acknowledgments

We would like to acknowledge our external advisor, Professor William Zimmermann for his invaluable assistance and expertise in liquid helium research. We would also like to acknowledge our internal advisor Gregory Pawloski and our MXP II group for their support and critique on assignments during the semester. On top of this, we would also like to thank all faculty involved, particularly Kurt Wick and Clem Pryke for their innumerable contributions to the course.