S17_FourthSound

Using an Annular Resonator to Measure the Speed of Fourth Sound in Liquid Helium Below 2.17K

Elena Hafner and Amanda Epstein

University of Minnesota - Methods of Experimental Physics II

Introduction

When liquid helium is cooled to below 2.17 K, it can be modeled as a mixture of a normal fluid component and a superfluid component which can flow with zero viscosity and carries no entropy [1]. This phase of helium is generally referred to as helium II. There are several forms of wave propagation which can be observed in helium II. First sound is essentially the same as a normal sound wave. It is a pressure wave in which both components of the fluid move in phase with each other [1]. Second sound also propagates in both components of the fluid; however, in this case, the two components move 180^o out of phase. This means that there are no net density oscillations associated with second sound. Instead, the fact that only the normal fluid carries entropy means that second sound can be detected as entropy oscillations [2]. Fourth sound, in contrast to first and second sound, is a pressure wave which propagates only in the superfluid component while the normal fluid is locked in place [1]. In this experiment, an annular resonator was used to measure the velocity of fourth sound at a range of temperatures between 2.17 K and 1.5 K.

Theory

The speed of fourth sound is given by [1]

where ρ is the total density, ρ_n is the normal fluid density, ρ_s is the superfluid density, and C₁ and C₂ are the speeds of first and second sound, respectively. If a fourth sound wave is detected in the resonator, there must be an antinode at the point where the oscillations are driven and at the detector on the opposing side of the resonator; therefore, there will be an integer number of wavelengths within the circumference of the resonator. We, thus, have

where d is the average diameter of the resonator, n is an integer, and f_n is the frequency of the nth harmonic.

Apparatus

Figure 1 below shows a diagram of the resonator which was used in this experiment. The bottom plate consists of an annular cavity with an average diameter of 2 in. In order to ensure that only the superfluid component was able to flow in the resonator, the cavity was packed with 0.05 μm Al₂O₃ powder using a hydraulic press. A diaphragm made from 0.5 mil thick aluminized mylar was placed between the filled resonance chamber and the upper plate of the resonator. Two of the capacitive transducers on the top plate were supplied with a 200 V bias voltage. The driving transducer was supplied with a signal that oscillated about this bias with an amplitude of 5 V. This caused the foil to oscillate underneath the transducer and push on the fluid in the cavity, inducing fourth sound waves. The transducer on the opposite side of the resonator was used to detect the waves. As the fluid in the cavity oscillated, it pushed on the foil underneath the detector causing the voltage to oscillate around the 200 V bias.

Figure 1: Diagram of annular resonator used in the experiment. The bottom plate (left) has a resonance chamber which is filled with finely packed Al₂O₃ powder

creating a superleak so that only the superfluid can flow through it. The button transducers on the top plate (right) of the resonator are used to induce and measure

waves in the resonator. This is an original figure.

The experiment was conducted inside a double dewar, as shown in figure 2. The inner dewar was filled with liquid helium while the outer dewar was filled with liquid nitrogen to insulate the experiment from the conditions of the room. In order to lower the temperature of the helium to below 2.17 K, a vacuum pump was used to pump out the helium vapor in the dewar, reducing the pressure. The tables in reference [3] were used to determine the temperature of the helium bath from the vapor pressure. Once the helium was set to a stable temperature, a lock-in amplifier was used to supply the driving transducer with sinusoidal signals at various frequencies and to measure the response of the detecting transducer. Specifically, a frequency sweep was run at each temperature from 200 Hz to 10 kHz with 10 Hz spacing between the data points.

Figure 2: Diagram of the apparatus used in this experiment. The experiment takes place inside a double dewar with the inner dewar containing liquid helium below

2.17 K and the outer dewar containing liquid nitrogen. Lock-in amplifier 1 is used to measure the voltage of the detecting transducer. Lock-in amplifier 2 is used for

measuring the resistor and controlling the heater. This is an original figure.

Results

The plot in figure 3 below shows an example of the detecting transducer signal that was observed at 2.101 K. Each of the peaks corresponds to the n^th order resonance frequency of the cavity.

Figure 3: Plot of the measured response of the detecting transducer as a function of frequency at 2.101 K. We were able to observe the first 11 harmonics. This is an original figure.

For each temperature, a plot was made of the frequency of each resonance peak vs. the mode number, and a least-square fit was used to determine the slope. The plot corresponding to the sample data from above is shown in figure 4. The slope of the frequency vs. mode number plot was then used to calculate the velocity of fourth sound at that temperature.

Figure 4: Plot of frequency vs. mode number for each of the numbered resonance peaks in figure 4. This is an original figure.

Our experimental values of the fourth sound velocity are summarized in figure 5. It is clear that these velocities show the expected trend with respect to temperature; however, they are systematically lower than the theoretical values which calculated using accepted experimental values for the velocities of first and second sound and the relative densities of the normal and superfluid components taken from reference [4]. Shapiro and Rudnick [5] suggest that there should be a temperature independent reduction in the observed phase velocity due to the effect of the individual powder grains on the superfluid flow. This correction factor is given by

where P is the porosity of the powder filled cavity. For a porosity of 79.0±0.4% as in our experiment, this equation gives a rescaling factor of 0.9512±0.0008 for the expected values of the fourth sound velocity, corresponding to the orange curve in the figure below. This correction is, however, not sufficient to bring the measurements into agreement with the expectation within the uncertainty. Instead, applying an empirically determined rescaling factor of 0.83±0.01 to the expected values gave the closest agreement with our measurements. It would be useful for future MXP groups to look in to the dependence of the velocity of fourth sound on the porosity of the filled cavity as well as identifying other possible sources for this systematic reduction in the observed velocities.

Figure 5: Plot of the fourth sound velocity as a function of temperature. Our experimental values were determined from the slope of the frequency vs. mode

number plots. The theoretical values were calculated using values of the speeds of first and second sound and the relative densities of the normal and superfluid

components taken from reference [4]. Also shown are the expected values corrected for the porosity of the filled resonator cavity and the expected values scaled

by the factor which most closely brings them into agreement with our empirical results. This is an original figure.

Conclusion

The general trend of the temperature dependence agrees with the expectation; however, the measured velocities were systematically lower than the predicted values. Future experiments should explore the effect of the porosity of the powder filled cavity on the reduction of the measured phase velocity from the theoretical fourth sound velocity. This experiment could also be extended to observe a persistent current flow of superfluid in the resonator.

References

[1] Tilley, David R, and Tilley, John, Superfluidity and Superconductivity, 3rd edition. Accord: IOP Publishing Ltd, 1990.

[2] Wilks, J., and Betts, D. S., An Introduction to Liquid Helium, 2nd edition. Oxford: Oxford University Press, 1987.

[3] Dijk, H. Van, M. Durieux, J. R. Clement, and J. K. Logan. "The 1958 He4 scale of temperatures. Part 2. Tables for the 1958 temperature scale." Journal of Research of the National Bureau of Standards Section A: Physics and Chemistry 64A.1 (1960): 4.

[4] Donnelly, Russell J., and Carlo F. Barenghi. "The Observed Properties of Liquid Helium at the Saturated Vapor Pressure." Journal of Physical and Chemical Reference Data 27.6 (1998): 1217-274.

[5] Shapiro, Kenneth A., and Isadore Rudnick. "Experimental Determination of the Fourth Sound Velocity in Helium II." Physical Review 137.5A (1965).