S18_Measurement of Fourth Sound Velocity of Liquid Helium

Experimental Determination of the Speed of Fourth Sound in Liquid Helium Below 2.17 Kelvin

Arpit Arora and Mikhail Borisov

University of Minnesota - Methods of Experimental Physics II

Introduction

Liquid helium is used intensively in industry and the academic world. The scope of its applications incorporates fundamental fields such as aviation industry, cryogenics, electronics and healthcare. It is consequently critical to comprehend properties of liquid helium in order to improve effectiveness of its usage in existing applications and to find new ways of using it [1]. This is what we have tried to achieve in our project, namely we worked on experimental determination and comprehension of an acoustic property of liquid helium: the fourth sound velocity.

Liquid helium at temperatures below the lambda point, 2.17 Kelvin, can be modeled as a mixture of a normal fluid component and a superfluid component, where the latter can flow with zero viscosity and has no entropy. This phase of liquid helium is referred as Helium II. Several modes of sound propagation exist in Helium II. The first sound mode is a pressure and density wave. In this mode both components of Helium II move in phase with each other. In contrast, the second sound mode represents a temperature and entropy wave, where components of Helium II move out of phase. Finally, the fourth sound mode is referred to a pressure and density wave that propagates only in the superfluid component of Helium II with the normal component being locked. By locking we mean containment of Helium II in a matrix of a porous medium where a drag force exerted by medium's constituents prevent normal component from moving.

Conducted experiment determines the fourth sound velocity in Helium II within the range of temperatures between 1.47 K and 2.14 Kelvin. An annular resonator packed with porous Al2O3 powder is used for measurements. In addition, the experiment reveals the hydrodynamics of Helium II via exploring effects of the powder porosity and the quality of packing on the measured fourth sound velocity compared to values of the velocity obtained via an alternative method. This method utilizes a definition of the squared fourth sound velocity as a superposition of squared velocities of the first and second modes [2].

Finally, a geometry of an annular resonator permits observation of rotation effects. These effects are manifested in splitting of resonant peaks that are used in determination of the fourth sound velocity. In order to provide enough resolution for identifying this splitting, a certain sharpness of resonant peaks is required. Thus, the sharpness of peaks from acquired experimental data is examined on potential in exhibiting rotation effects.

Theory

At temperatures below the lambda point transition, 2.17 K, liquid helium exhibits interesting properties that can be described by a two-fluid model [3]. In this model, it consists of a superfluid component and a normal fluid component, and the total density of a fluid ⍴ is given by addition of two separate component densities as:

where ⍴s is the density of the superfluid component, and ⍴n is the density of the normal fluid component [4]. The normal fluid component carries the entropy of the system and comprises of thermally excited helium atoms. On the other hand, the superfluid component consists of helium atoms in the ground state with no associated entropy. Figure 1 plots normalized densities of superfluid and normal fluid components as a function of temperature below the lambda point. At temperatures above the lambda point, only the normal component exists. Superfluid component density increases with a decrease in temperature until in the limit of 0 K the superfluid component is the only component present in liquid helium.

Figure 1: Helium II normalized superfluid and normal fluid component densities as a function of temperature [4].

Different sound waves including first, second and fourth sound waves propagate in Helium II. The first sound wave corresponds to an ordinary sound wave as pressure and density fluctuations. In this case both fluid components move in phase at a constant temperature and entropy. The second sound wave is an entropy and temperature wave where the normal component and superfluid component travel out of phase maintaining constant density. First and second sound waves coexist in Helium II, and the fourth sound velocity is interdependent on both of the initial sound wave velocities. Specifically, the fourth sound wave is a pressure and density oscillations that propagate only in the superfluid component of Helium II while the normal component is immobilized [3]. The velocity of fourth sound wave is given by:

where C4 is the fourth, C1 is the first, and C2 is the second sound velocities. Figure 2 graphs the fourth sound velocity as a function of temperature. The velocity is computed according to the above equation from accepted empirical data of the first and second sound velocities. This accepted data is also illustrated in the figure. The fourth sound velocity is expected to increase as the temperature decreases, and it peaks at 235 m/s at about 1.0 K [3].

Figure 2: The fourth sound velocity as a function of temperature. The fourth sound velocity is computed from accepted empirical data of the first and second sound velocities [3].

The fourth sound velocity can be measured from resonant frequencies of standing waves induced inside a cavity of an annular resonator. When fourth sound waves propagate along an annular cavity, they interfere constructively. Antinodes are formed at the point where oscillations are driven and at the point on the opposite side of a resonator. Under these circumstances, the wavelength 𝜆 associated with a fundamental frequency of oscillations must correspond to a doubled propagation distance. This distance is defined as an average circumference of an annular region computed from an average diameter of a resonator. According to the reasoning above, the velocity of a fourth sound wave becomes:

where C4 is the fourth sound velocity, d is the average diameter of the resonator's annular cavity, and fn is a frequency of n'th harmonic with n being a harmonic number respectively. The formula is derived from the wave equation solution for an annular symmetry. An approximation of the average diameter is embedded in the solution, and it arises from treating the difference of annular radii being much less than a characteristic annular circumference.

Apparatus

In the experiment we used a device called resonator that has an annular geometry. Figure 3 illustrates a schematics of this device, and figure 4 shows its photograph. The resonator used in the experiment has the average diameter of cavity D = 5.004 ± 0.001 cm. The cavity was packed with alumina abrasive (Al2O3) Linde B 0.05 micron powder under 1 metric ton pressure using a hydraulic press. Packing was performed in stages such that powder uniformly filled the cavity finely locking normal component of liquid helium from flowing. The porosity P of packed powder i.e. the free space volume inside the cavity normalized by the total volume of the cavity, was calculated from the volume of the resonator's cavity, powder density and mass of the packed powder. The calculation yields the porosity: 79.0 ± 0.4 %. In words, slightly less than 80 % of the cavity consisted of voids created by powder particles. These voids correspond to the space for free flow of superfluid component through the resonator.

Figure 3: Schematics of the annular resonator containing transducers marked in a darker color. The average diameter of the annular resonator is D = 5.044 ± 0.001 cm.

In addition, this distance corresponds to the spacing between driving and detecting transducers.

Figure 4: Annular resonator used for resonance frequencies detection. Note that the annular cavity of resonator is packed with powder.

A 0.25 mil double layer diaphragm composed of mylar and aluminum was placed between the resonator cavity and the lid of the resonator. Both isolated electrodes on the resonator lid and the diaphragm facing the electrodes with the mylar surface formed transducers. A transducer is a device that transforms an acoustic signal into an electric one and vice versa. Figure 3 illustrates how transducers are incorporated in the resonator. Two diametrically opposite transducers were supplied 200 V DC bias voltage. One of the transducers was driven with a 5 VRMS sine wave signal with varying frequency in the range between 10 Hz and 8,000 Hz. This signal induced vibrations of the diaphragm, hence to a fourth sound wave in Helium II contained inside the cavity of the resonator. The other transducer was used to translate the amplitude of a resulting acoustic signal on the other end of the resonator into an electric voltage amplitude. The system response was measured at different frequencies. A lock-in amplifier, interfaced with a LabView program, was used to acquire resulting amplitude versus frequency setting data. From this data, the velocity of a fourth sound wave induced in the resonator was later computed.

The measurements were conducted for a set of representative temperatures below the lambda point temperature. We used double glass double dewar system to cool an experimental cell to required temperatures. Detailed description of the cooling procedure can be found in the work of White on experimental techniques in low-temperature physics [5]. Figure 5 provides a schematic representation of the refrigeration system. The experimental cell along with a resistance-based thermometer and a heating inductor was attached to a cryostat. The cryostat was then placed inside the inner dewar intended for liquid helium containment. Temperature equilibrium required for conducting measurements was reached by means of the heating inductor and cooling, by the method of evaporation, vacuum pump. These two devices were connected in a negative feedback loop to atomize the temperature equilibrium maintenance. Finally, temperatures were measured via a calibrated thermal resistor attached to a resistance bridge interfaced with a lock-in amplifier.

Figure 5: Diagram of the apparatus used in the experiment. During the experiment resonator was contained inside the inner dewar filled with liquid helium at temperatures below 2.17 K.

Lock-in amplifier 1 was used to measure voltage amplitudes exiting the resonator as a function of driver's frequency.

Lock-in amplifier 2 was used for balancing the resistance bridge for temperature determination. This is an original figure.

Results

We begin the discussion of the results by showing calibration of thermal resistors. Resistance values for two thermal resistors were acquired for helium vapor pressures ranging between 35.0 ± 0.05 Torr and 5.0 ± 0.05 Torr. According to helium specifications, vapor pressures were mapped to corresponding temperatures [6]. It was noted that the pressure gauge used for measuring vapor pressures was offset by 0.4 ± 0.05 Torr. The offset was determined by physically observing the lambda point transition and recording the pressure at which it occurred. The reading was then translated to temperature scale and compared to the accepted value. Taking the offset into account, temperatures were plotted against resistances and fitted to a power functions. Figure 6 shows the resulting fits. Equations of fits are the following:

Figure 6: Temperature calibration curves used for thermometry in the conducted experiment.

In the process of conducting the experiment, we have acquired resonance patterns for 14 representative temperatures between 1.47 - 2.14 K. At each of the temperature settings, frequency range from 10 Hz to 8,000 Hz was considered. Figure 7 plots a sample data set obtained at 2.054 ± 0.001 K. The fundamental frequency resonance and its harmonics can be clearly observed on the graph. Peaks are labeled by harmonic numbers. For few data sets resonance patterns contained degenerate peaks. This degeneracy occurs randomly and rarely, and the most probable explanation for its appearance is the interference with harmonics of the power line.

Figure 7: Plots the raw resonance data collected for the temperature of 2.054 ± 0.001 K.

For each data set we extracted frequencies at which resonances occur. Uncertainty in frequencies was estimated as a step size: 5 Hz. These frequencies were plotted against harmonic numbers, and fitted to a straight line using the method of least squares. For example, χ distribution and the resulting linear fit for the data set acquired at 2.089 ± 0.001 K are shown in the figure 8. The slope of the linear fit was used in evaluating the fourth sound velocity while an uncertainty in the slope, resulting from the χ squared fitting, was used in calculation of uncertainty in values of the fourth sound velocity.

Figure 8: Shown are χ distribution and the linear fit evaluated for 2.089 ± 0.001 K data set.

The above mentioned analysis yielded the velocity of fourth sound at several temperature settings. Values of velocity were plotted as a function of temperature, and the resulting graph can be found in figure 9. It is clear that these values show the expected trend with respect to temperature; however, they are systematically lower than the theoretical values calculated from accepted experimental data for the velocities of first and second sound and the relative fluid component densities. 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. However, this correction is not sufficient to bring the measurements into an agreement with the expectation as the rescaled curve from the figure indicates. Instead, application of an empirically determined rescaling factor of 0.83±0.01 to the expected values gave the closest agreement with our measurements.

Figure 9: Plot of the fourth sound velocity as a function of temperature. Measured values were determined from the slope of the frequency versus mode number plots.

The theoretical values were calculated using values of the speeds of first and second sound and relative densities of the normal and superfluid components taken from reference [4].

Also, shown is the curve of measured values corrected for the effect of individual powder grains on a superfluid flow. This is an original figure.

Finally, we investigated quality of our data from a perspective of potential in exhibiting rotation effects. Sharpness of resonance peaks serves as a measure of the quality. Therefore, we computed Q - factors associated with each resonance harmonic and compared the results from different temperature data sets. Figure 10 shows data obtained for the second harmonic resonant peak. An interesting behavior of Q - factor is observed with decreasing temperature: sharpness of resonance peaks increases. Motivated by this observation, we evaluated another characteristic, namely full-width normalized by a harmonic number, at a lower temperature tail of the experimental data. According to literature, value of 5 or less corresponds to a sufficient sharpness of a resonance for rotation effects resolution [7]. However, the smallest value that was extracted from the measured resonances in our experiment accounted for 7. This observation suggests that additional steps are required in enhancing resonance sharpness.

Figure 10: Plots the calculated Q factor of the 2nd resonant peak for all temperature data sets.

Conclusion

To conclude, fourth sound velocity in liquid helium at 14 different temperature ranging between 1.47 K to 2.14 K was measured. It was observed that the velocity increased with the decrease in temperature as expected. Moreover, general trend of increase matches the expectation based on an alternative method of fourth sound velocity determination. The measured values were systematically lower than the predicted values by 8% for the low temperatures tail and 30% near the lambda point in terms of percentage difference. These reductions are partly accounted by porosity of the powder filled cavity on the superfluid flow; however, this correction does not bring the measured values in agreement with the prediction. Potential cause for a high order reduction near the lambda point is unknown. The obtained resonance peaks possess increase in sharpness with decrease in temperature, but we were not able to observe peaks that are sufficiently sharp for rotation effects observation.

This project, leads to several interesting possibilities for future MXP groups. This work could be improved by focusing on the measurement of the fourth sound velocity near the lambda point in order to identify the reason behind rather anomalous deviation of the measurement and prediction in this range of temperatures. Studying a dependence on various parts in the experiment, such as diaphragm and porosity, may result in a justification of overall observed systematic reduction. Finally, conducted experiment also leads to a possibility for future experiments on measurement of bulk rotation modes of the liquid helium and detection of a persistent currents in the annular resonator [7].

References

[1] 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.

[2] Tilley, David R, and Tilley, John, Superfluidity and Superconductivity, 2nd edition. Accord: Adam Hilger Ltd, 1986.

[3] K. A. Shapiro and I. Rudnick, Experimental Determination of the Fourth Sound Velocity in Helium II. Phys. Rev. 137, A1383 (1965).

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

[5] G. K. White, Experimental Techniques in Low Temperature Physics (Oxford University Press, Oxford, England, 1959).

[6] Properties of Helium. Brookhaven National Laboratory.

[7] Kojima, H., et al. “Superfluid Density in the Presence of Persistent Current in superfluid4He.” Journal of Low Temperature Physics, vol. 25, no. 1-2, 1976, pp. 195–217., doi:10.1007/bf00654829.