Temperature Dependence of Second Sound in He-II

Dan Codoluto & Devin Dykhoff

Method of Experimental Physics – University of Minnesota

I. Goals

In this experiment, the temperature dependence of the speed of second sound in He-II was observed by mechanically producing temperature harmonics in a brass cylindrical cavity near temperatures of ~ 1.56 – 2.17 degrees Kelvin. It was found that longitudinal modes of resonance were indeed observed at twelve temperatures and a quantitative temperature dependence was obtained for the speed of second sound.

II. Concepts

One peculiar phenomenon that is observed in He-II is the propagation of heat in the form of temperature variations with some speed defined at a given temperature that is below the lambda point. This phenomenon is known as second sound. Second sound is analogous to pressure variations in a medium, which is rightly deemed first sound, in that its motion is described by a wave equation.

The velocity of second sound in this experiment was determined by analyzing the phase of second sound standing waves observed in a cylindrical cavity. The resonant frequency of the waves in the cavity is related to its geometry along with the velocity of the wave. By observing resonance at a known driving frequency, the velocity of second sound was calculated.

One of the methods for detecting second sound consists of placing a receiver at the opposing end of the cylindrical cavity where a detector is present, which directly measures the amplitude of the second sound wave. By measuring the resonant frequencies of standing second sound waves, it was possible to obtain values of second sound velocities near the lambda point.

III. Theory

Superfluid helium is modeled by a mixture of a superfluid and a normal fluid, whose respective densities are temperature dependent (right). The total density is represented by the sum of the supposed densities of the normal and superfluid:

The following relationship represents the model that predicts the existence of harmonic temperature variations with velocity c2 (below):

where T is the temperature of He-II, S is the total entropy of the normal component, and C is the total heat capacity of He-II. The temperature dependence of second sound is illustrated in the plot below [3]. Notice that as temperature approaches zero, the speed approaches about 135 m/s. The dashed line represents the region of temperature at which it is very difficult to measure second sound.

If we define a cavity with height h, we may use the equation for resonance in a cavity which specifies the velocity of a wave and its frequency,

where n = 1, 2, 3,… is the vibrational mode of the wave. Hence, if harmonics are in fact being observed during this experiment, it is expected that the mode of resonance and the location of the frequency peak should exhibit a linear relationship.

IV. Apparatus

A brass cylindrical cavity with inner radius r, outer radius R and height h (left) has at the opposing ends of the cavity an aluminized membrane with 1 micron pores (right) wedged between two end caps. The membranes act as a piezoelectric when an AC voltage is applied.

A schematic illustration of the complete resonating cell is shown. There are two pores that allow fluid to enter the cavity once the cell is submerged in the dewar. Two screws at either end of the cell secure a voltage supply. The cell is attached to a cryostat module whose backing plate contains BNC leads that supply this voltage.

This device is submerged in a dewar (below) containing liquid Helium cooled below ~ 2.17 K. One membrane is driven with an oscillating voltage with an amplitude of one V while the other receives thermal vibration via second sound. These signals are controlled by a lock-in-amplifier and are observed in a LabVIEW program. A high pass and a low pass filter separate an AC and a DC voltage as it travels to the resonating cell. The DC voltage is necessary so that the membranes remain firm when oscillating, while the AC voltage supplies the driving signal to the resonator.

Temperatures below the lambda point are reached via cooling by process of evaporation. This process consists of several steps of evacuating the dewar and bleeding in gaseous nitrogen to dilute any water vapor that might exist in either chamber. Eventually, gaseous helium is allowed in the inner chamber for the same purpose. After these steps are complete, liquid nitrogen is pumped into the outer chamber of the dewar, while liquid helium is transferred into the inner chamber.

A thermal resistor that is already submerged in the helium bath is used for regulating the temperature of the contents of the dewar. It has a small driving current sent through it so that it can detect slight variations in temperature. The signal passes through a standard AC Wheatstone bridge so that the temperature can be stabilized when slight variations are detected.

V. Results

Once the desired pressure in the dewar was obtained, the thermometry was balanced using a variable resistor. The thermometry bridge was balanced at multiple pressures to obtain a temperature resistance relationship (below). A temperature measurement was obtained by using a standard vapor pressure table for He-II. Thermometry calibration was necessary so that a precise measurement of temperature could be extracted for the second sound measurements.

Frequency sweeps where resonance was known to occur for the first 5 - 7 modes were perfmored at twelve different temperatures. Second sound resonance can be seen in the fitted peaks for 1.56 K shown at the left below. Since our cell is designed to detect longitudinal modes or resonance, we fit only these modes to obtain the temperature dependence. Once the location of the peaks was obtained at each temperature, it was determined whether harmonics were indeed being observed. The location of the resonances were plotted against the known mode number (right). Clearly, the relationship appears linear, indicating harmonics were observed.

Using the location of the peak, the known mode number, and the known height of the cavity, it was possible to produce a detailed temperature dependence of second sound over a temperature range of ~ 1.56 – 2.17 K with high precision (below).

VI. Conclusion

By producing resonance inside a cylindrical cavity containing He-II, it was possible to obtain a quantitative model of the temperature dependence of the speed of second sound in He-II over a temperature range of about 1.56 – 2.17 K. Harmonics were indeed observed since the relationship between the location of the resonance peaks and the known mode numbers was linear. The method of the resonating cavity and oscillating membrane proved to be an appropriate method for observing second sound resonance in He-II at temperatures below the critical temperature of ~ 2.17 K. Since the method of this experiment has been well established and has produced reliable results, future experimentalists may find interest in investigating possible variations of second sound detection. One such possible method involving single transducer detection was investigated. A specially designed bridge was used to balance impedances due to the bridge and cell for frequencies that fell outside resonance. Single transducer detection failed using this method, but it may be possible in the future to produce variations in the design. This detection method may allow for the application of high voltages. Specifically, applying a high voltage to a single oscillating membrane can be useful in applying a large electric field to superfluid helium so that theorized microscopic polarization of helium atoms can be observed and characterized

VII. Acknowledgements

A special thanks to the assistance of Professor William Zimmermann, Professor Gregory Pawloski, and Professor Kurt Wick.

VIII. References

[1] Tilley, David R., and Tilley, John. Superfluidity and Superconductivity, 2nd ed. Bristol, Boston: Published in association with University of Sussex Press, 1986.

[2] Atkins, K.R, Liquid Heluim. Cambridge: Cambridge University Press, 1959.

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

[4] H. van Dijk, M. Durieux, J. R. Clement, and J.K Logan. Tables for the 1958 Temperature Scale. Washington, D.C.

[5] Greywall, Dennis S., Ahlers, Guenter. Second Sound and Superfluid Density in 4He under Pressure near Tλ. Physical Review A. V7N6. June, 1973.

[6] Maynard, J., Determination of the thermodynamics of He II from sound-velocity data. Physical Review B. V14N9. 1976.