S16_FourthSound

Measurement of the Temperature Dependence of the 4^th Sound Velocity of

Superfluid Helium-4 in an Annular Resonator

Luke Molacek and Richard Spieker

Introduction

The temperature dependence of the 4th sound velocity in superfluid helium-4 in an annular resonator was measured. Previous MXP projects have used cylindrical resonators to measure 4th sound velocity. Annular resonators have additional applications such as for measuring persistent current.

Theory

At temperatures above the lambda transition (2.17 K) liquid helium 4 is a classical fluid and is denoted as helium I. As helium I is cooled through the lambda transition, it becomes helium II. Helium II exhibits unusual properties including appearing to have zero viscosity when subject to capillary flow, and non-zero viscosity when and object moves through it. The two-fluid model is used to describe helium II and explains its unusual behavior. In the two-fluid model, helium II is treated as being composed of two inseparable fluids: the normal fluid and the superfluid. The total density of helium II is thus given by the sum of the component densities [1]

.

where

is the total density, is the normal fluid density, and is the supefluid density. The normal fluid consists of thermally excited helium atoms and carries the entropy of helium II. It is this fluid that comprises the entirety of helium I. The superfluid, analogous to a Bose-Einstein condensate, consists of a macroscopic number of helium atoms in the ground state, as well as atoms which form the depletion and has no entropy. The atoms of the depletion would contribute to the population of the ground state, but are forced to slightly higher energy levels due to inter-atomic interactions. Shown below is a plot of the normalized superfluid and normal fluid densities as functions of temperature. At temperatures above 2.17 K, the normal fluid comprises the entire fluid. Just below 2.17 K, the superfluid density obtains non-zero values and increases as temperature decreases. In the limit of 0 K, the superfluid comprises the entirety of helium II [1].

Shown is a plot of the normalized superfluid and normal fluid densities as functions of temperature. Image taken from [1].

Several varieties of sound waves can propagate in helium II, including 1st, 2nd, and 4th sound. In order to understand 4th sound, knowledge of 1st and 2nd sound is necessary since the velocity of the former depends on the velocities of the latter two. 1st sound, similar to ordinary sound waves in air, consists of a pressure and density wave. In these waves, the normal and superfluid components move in phase under constant temperature and entropy. The velocity of 1st sound is represented by [1]

.

where is the velocity of 1st sound, is the pressure, and is the entropy per unit mass. 2nd sound, unique to helium II, is an entropy, or temperature wave. In this form of sound, the normal fluid and superfluid travel out of phase, maintaining constant density while the normal fluid. The velocity of 2nd sound is given by [1]

is the velocity of 2nd sound, is the absolute temperature, and is the specific heat at constant pressure. Both 1st and 2nd sound generally coexist together in helium II. 4th sound is a pressure and density wave that propagates in a superleak. A superleak is a channel through which the superfluid can flow, but the normal fluid is immobilized. The velocity of 4th sound is given by [1]

.

where is the velocity of 4th sound. A plot of 1st, 2nd, and 4th sound velocities as functions of temperature is shown below. 2nd sound and 4th sound are phenomena of helium II, and thus have non-zero velocities for temperatures below 2.17 K. The velocity of 1st sound has larger values in comparison to 2nd and 4th sound. The velocity of 2nd sound is relatively small at approximately 20 m/s, having roughly constant value for temperatures between 1.0 K and 2.0 K. The velocity of 4th sound increases rapidly below the lambda point, approaching its maximum value of approximately 240 m/s around 1.0 K [1].

Shown is a plot of the velocities of 1st, 2nd, and 4th sound for temperatures between 1.1 K and the lambda transition. Image taken from [1].

In order to measure the velocity of 4h sound, the resonance of 4th sound standing waves can be observed. When waves originate from a point along an annular resonance chamber, waves propagate clockwise and counterclockwise around the cavity, interfering constructively where they meet on the opposite side of the chamber. Under such conditions, for standing waves to exist, an integer number of half wavelengths must fit within half the circumference of the cavity (the distance from their origin to where they meet and interfere constructively). Thus, the speed of the wave is related to the frequency through the relation

is the velocity of the wave, is the length of the standing wave, is a positive integer specifying which mode is being observed, and is the frequency of the nth normal mode.

Experimental Setup

The annular resonator used is shown in a figure below. The bottom plate contained the resonance chamber which had an average diameter of 2 in. The resonance chamber was filled with 0.05 micron diameter aluminum oxide powder to create a superleak. The top plate contained four button transducers which consisted of cylindrical sections of copper held in place and electrically isolated from the rest of the resonator by epoxy and 1 mil thick circular pieces of foil placed underneath so that they were held in place between the top and bottom plates. The detecting and driving transducers were supplied with a 200 VDC bias to reduce the effects of noise on the small signal observed. The driving transducer was supplied with a 5 VRMS sinusoidal signal, causing the foil underneath to oscillate and thereby creating 4th sound waves. These waves caused the detecting transducer's foil to oscillate, causing the measured voltage to oscillate about the bias.

Shown is a cross section of the top and bottom plates of the resonator. Original figure.

The resonator was mounted on the end of a cryostat. The cryostat was placed in a double-dewar. The double dewar consisted of two concentric silvered glass dewars. The inner dewar held liquid helium. The outer dewar held liquid nitrogen. The liquid nitrogen in the outer dewar served to insulate the helium bath within the inner dewar. The inner dewar was connected to a vacuum pump. By adjusting the pumping rate, the vapor pressure above the helium bath could be reduced, thereby lowering its temperature.

When the temperature of the helium bath was stabilized through adjustment of the pumping rate and power supplied to a heater resistor, the frequency of the signal supplied to the driving transducer was swept in 10 Hz increments to locate resonances.

Results

We were able to resolve a number of normal modes for each temperature selected for data collection. We observed the first eight normal modes for one temperature as shown in a figure below. Since the width and amplitude of the higher modes remained relatively constant, there was no indication we would be unable to resolve even higher modes should a longer frequency sweep have been performed.

The first eight modes were observed at 2.092 K and are indicated by their number in the figure. Original figure.

It was observed that the higher modes were not integer multiples of the fundamental mode as they should have been. It was hypothesized that this could have resulted from inconsistencies in the packing of the aluminum oxide powder such as possible porosity gradients, and the inability to pack the powder flush to the surface of the resonance chamber without imperfections. The velocity of 4th sound was calculated using the relation between resonant frequency and speed described near the end of the theory. This was completed for every discernible peak at each temperature. The grain of powder in the superleak functioned as essentially immovable spherical scattering centers, causing the observed velocity to be reduced. This required the use of the velocity correction factor [2]

is the correction factor, is the porosity. Using this equation, the observed velocity of 4th sound is

.

where is the velocity which was measured directly. Using this, the actual measured velocity of 4th sound was determined. As was expected, difference values of the velocity were obtained when using different modes since they were not integer multiples of each other. Shown below is a plot of the 4th sound velocity measured using the 1st mode, 2nd mode, and a fit across all discernible modes at each temperature. The latter was determined by plotting resonant frequency as a function of the integer specifying the mode. From the slope, the velocity was determined.

Plot of the 4th sound velocity as a function of temperature as determined from the first mode, second mode, and a least squares fit over the observable modes at each temperature. Original figure

Conclusion

We were able resolve a number of modes, up to the 9th mode at one temperature. There was no indication we would not have been able to resolve even higher modes if a longer frequency sweep had been performed. However, the 2nd and higher modes failed to be integer multiples of the fundamental, causing the velocity as determined to be dependent on the mode used. We were not able to obtain velocity values that agreed with known values within uncertainty. This was likely the result of density gradients (causing porosity to be non-uniform) in the aluminum oxide powder, causing the observed velocity to vary within the resonator. In addition, there may have been regions with porosity low enough to allow for motion of the normal fluid, further altering the observed velocity.

References

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

[2] Shapiro, Kenneth A., and Rudnick, Isadore, "Experimental Determination of the Fourth Sound Velocity in Helium II," Physical Review Mar. 1965: Vol. 137, No. 5A, 1383-1391.

[3] Kojima, H., Veith, W., Guyon, E., and Rudnick, I. "Superfluid Density in the Presence of Persistent Current in Superfluid Helium-4." Journal of Low Temperature Physics Feb. 1976: Vol. 25, Nos, 1/2, 195-217.

[4] Brickwedde, F. G. "The 1958 Helium-4 Scale of Temperatures" Physica Sept. 1958, Vol. 24, S128-S131.