Investigation of the Temperature Dependence of Relative Superfluid Density in Helium-4

Alex Jarecki & Alan Traczyk

University of Minnesota

School of Physics and Astronomy

Minneapolis, MN 55455 Pressure oscillations were induced in liquid helium at various temperatures below 2.17 Kelvin across a super leak membrane in a dual-chambered resonation cell. The frequency at which Helmholtz resonance occurred was then converted into a relative superfluid density and compared to empirical data.

Introduction

Super-fluidity is a phase of matter experienced by Helium-4 at temperatures below the lambda transition temperature, 2.17 Kelvin at saturated vapor pressure. [1] This phase exhibits the unique characteristic of zero-viscosity capillary flow; meaning pores that are small enough to inhibit normal fluid flow due to viscous interactions do not prevent the flow of superfluid Helium. Stranger yet, the superfluid does show evidence of viscosity when tested in other fashions, such as fluid damping of a torsional oscillator. [1] To remedy the contradiction, the two-fluid model is adopted: Liquid Helium-4 below the lambda temperature is considered to be a quantum mixture of a superfluid component and a normal fluid component. The total density,

, is the sum of the superfluid component density,, and the normal fluid component density,.

(1)

The ratio of super fluid density to total density, or relative super fluid density, is thought to be inversely proportional to the temperature, as seen in Figure 1.

(2)

Figure 1: This plot illustrates the increase in the relative super fluid density ratio and the corresponding decrease in the relative normal fluid density ratio as temperature is decreased [2].

The unique property of zero viscosity in capillary flow allows the two fluid model’s predictions to be tested by inducing fluid oscillation through the pores of a super leak membrane – a membrane with pores small enough to only allow for only the passage of the superfluid component – in a process analogous to Helmholtz resonance. In the familiar example of Helmholtz resonance, blowing over the top of a bottle, breath serves to apply a pressure on the pocket of air in the neck of the bottle, forcing it towards main cavity. This, in turn, increases the pressure in the main cavity, causing the air pocket to oscillate in the neck of the bottle and produce the familiar hum at a resonation frequency based on the geometry of the bottle. In this experiment, a piezoelectric driver was used to induce pressure waves across the pores of the super leak membrane separating the halves of a dual-chambered resonation cell. The pores of the super leak membrane served as the “neck of the bottle” from the familiar example. A second piezoelectric was used on the opposite side of the cell to detect the oscillation and search for the resonation frequency. The use of a super-leak membrane ensured the resonation was only due to the superfluid component. The resonation frequency shifts as temperature decreases and more of the liquid becomes superfluid. In this way, the frequency of resonation and geometry of the resonation cell determine the relative superfluid density.

Theory

In the dual-chambered resonating cell, the piezoelectric driver applies pressure to the top of the upper chamber, causing pressure waves to propagate through the fluid and carry mass back and forth across the super leak separating the chambers. Since the total mass is conserved, we know that the mass current can be written as a function of the difference in pressure and change in temperature in the lower chamber. [2] Only the normal fluid component contributes to the entropy, since the superfluid component is in its lowest possible energy state, and since the normal fluid cannot flow across the superleak membrane, the entropy in each chamber is conserved, allowing the mass current also to be related to entropy. Combining the results from both chambers results in an expression for the total mass flow in terms of entropy. [1] This mass flow through the pores is analagous to electric current flowing through inductors [5]. Treating it as such allows for an expression for the time derivative of the mass flow. Solving the equations for mass current found through mass conservation and "entropy conservation" simultaneously and substituting the results for temperature and pressure into the time derivative found by the analogy to inductors, a second order differential relation is revealed. Assuming a solution of the form

yields a solution for the Helmholtz resonating frequency:

is the effective volume of the resonation chamber,is the speed of first sound, andis the ratio of the cross-sectional pore area to pore length. This expression is a first approximation, considering only the inductance of the pores themselves. To improve, additional inductance terms for the hole over which the membrane is epoxied, as well as the ends of the pores, where the chemical potential difference has an effective inductance. Combining these inductance terms in reciprocal results in the final expression for the Helmholtz resonating frequency in terms of the geometry of the pores, the relative superfluid density, and the speed of first sound [5].

(4)

The notation yet undefined are the length and cross-sectional area of the hole over which the membrane is epoxied, l_h and A_h, and the total unblocked area of the membrane, A_o.

Apparatus and Proceedure

The piezzoelectric drivers differ from the driving sources used by previous students in this investigation. A metal disc is partially coated with piezzoelectric ceramic material. A potential difference incurred between the metal and the ceramic causes the disc to expand, thereby displacing fluid in the vicinity and creating a pressure wave. Figure 2 depicts one of the drivers used in the current experiment.

Figure 2.

The resonation cell used by previous groups did not have endcaps to support these new drivers, so new endcaps were designed and machined. A schematic of the chamber and the end caps is shown in Figure 3

Figure 3. Left: Cross section of resonation chamber. Right: Modified endcaps for new drivers.:

The resonation chamber was fastened to a cryostat, which was submerged in the liquid Helium-4 bath, contained in a double Dewar apparatus. The outermost Dewar jacket was permanently vacuumed for insulation. The outer Dewar chamber was filled with liquid nitrogen to cool the inner Dewar chamber as well as providing insulation. The inner Dewar jacket was filled with a torr of gaseous nitrogen the night before the experiment to ensure the inner Dewar chamber was able to cool by transfering heat to the liquid nitrogen. When the liquid Helium-4 was added to the inner Dewar chamber, it is at roughly 4 Kelvin, which freezes out the gaseous nitrogen jacket, effectively providing another layer of vacuum insulation. To induce further cooling of the liquid Helium, a variable vacuum pump was used to cool by evaporation. A rough temperature calibration was performed by relating the measured vapor pressure to the pressure-temperature table provided by the University of Oregon [4]. A temperature-sensitive Germanium resistor in the bath was configured in an AC Wheatstone bridge so that its resistance could be determined at various temperatures determined by the vapor pressure. This served as a third parameter by which the temperature uncertainty was reduced, as well as providing the differential output as a signal for a heater regulation system utilizing a second resistor in the bath. At various temperatures below the lambda transition temperature, frequency sweeps implemented by a lock-in amplifier using GPIB interfacing signaled the piezzoelectric driver and reciever. The recieved signal was then plotted in both the frequency and phase domains. A schematic depiciting the interfacing is shown below.

Figure 4

Most measurements of the cell's geometry were done by past experiments. However, the superleak membrane was replaced, so the unblocked membrane area and pore area needed to be determined. This was done by performing a gas flow test across the membrane, using the manufacturer's data of unblocked area against gas flow rate. The open pore area was quoted by the manufacturer as 15.7% of the unblocked area. The values determined this way were then used in the final calculation of the relative density ratio.

Results

Figure 5 is a graph that depicts the relative superfluid density against temperature as expected from the University of Oregon’s tabulated data along with the relative superfluid densities calculated from measurements with driving amplitudes of 0.2, 0.5, and 1.0 volts.

Our results showed a systematic deviation from the expected curve. Further investigation revealed the derivation neglected to accommodate any driving amplitude dependence of the resonation frequency. Taking data at a few driving amplitudes for each temperature, we were able to discern a shift in the observed resonation frequency with driving amplitude. This shift itself displayed temperature dependence, being more pronounced at lower temperatures.

Conclusion

By using a piezoelectric driver and receiver with a lock-in amplifier controlled via GPIB interfacing, we were able to induce and detect Helmholtz resonance in liquid Helium-4 cooled below the lambda transition temperature of 2.17K. The liquid was cooled using a vacuum pump and the measured vapor pressure used to calibrate the temperature with the resistance determined by balancing an AC Wheatstone bridge of a temperature-sensitive Germanium resistor in the Helium bath. Known measurements of the Helmholtz resonation dual-chambered cell were used along with data from gas flow tests on the super leak membrane and the resonation frequency found by analyzing the collected data in order to determine the relative superfluid density at various temperatures. Our results for the relative superfluid density roughly display the inverse temperature dependence predicted by the two-fluid model and empirically tabulated by the University of Oregon, although a systematic deviation from the empirical data is present.

The systematic deviation is likely an artifact of the shift in measured resonation frequency with driving amplitude. The amount of shifting with a given change in driving amplitude was observed to increase as temperature decreased, qualitatively improving our results.

A suggestion for future groups working on a similar project is to take measurements at each temperature with five to ten driving amplitudes as a means of better characterizing the shift in measured Helmholtz resonation frequency. More detailed analysis and preliminary tests to verify the voltage range in which the piezoelectric response is linear could help understand the discrepancy as well.

References

References

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^[2] Atkins, K. (1959). Liquid Helium. Cambridge: Cambridge University Press.

^[3] Brooks, J.S., B.B. Sabo, P.C. Schubert, & W. Zimmerman. Helmholtz-Resonator

Measurements of the Superfluid Density of Liquid ⁴He in Sub-micrometer-Diameter Channels. Physical Review B. Vol. 19, Num. 9. May 1979.

^[4] Atkins, K. Third and Fourth Sound in Liquid Helium II. Physical Review. Vol. 113, Num. 4. February 1959.

^[5] Allen, S. & Young, I. (2013) Helmholtz Resonator in Superfluid Helium. Minneapolis: University of Minnesota.

^[6] Tyler, A., Reppy, J.D., &Cho, H.A. Superflow and Dissipation in Porous Glasses. Journal of Low Temperature Physics. Vol. 89, 1992.

^[7]Moore, I. Sawhney, M. Speed of Fourth sound in Liquid Helium II. Minneapolis: University of Minnesota.