Speed of Transverse Second Sound Waves in Liquid Helium II

Max Belanger and Michael Edens

University of Minnesota

Methods of Experimental Physics Spring 2013

Abstract

We mechanically stimulate transverse second sound waves in liquid Helium to determine the speed of second sound by measuring the frequencies at which resonances are produced in a cylindrical cavity. We obtained the speed of second sound for ten temperatures ranging from 1.75 K to 2.171 K. The speed of second sound at 2.171 K (0.005 K away from the Lambda Point) was calculated to be 4.47±0.044 m/s. This value was 2.59 ’s away from the expected value of 4.533 m/s. We also analyzed the temperature dependent characteristics of a resonance peak in frequency space whose mode is defined as m=4, n=0.

Introduction

Liquid helium 4 exists in two forms, He I which exhibits the properties of a normal liquid, and He II which exists in temperatures below the Lambda Point of 2.17K and exhibits the properties of a super fluid [1]. In superfluid helium, heat is able to propagate as a wave of entropy known as second sound. Second sound can be defined as a wave of entropy which propagates by variation of the density of the normal and superfluid components such that the total density of the fluid remains constant[1].

Second sound waves will be generated mechanically by driving a selectively porous membrane (pore size 1 micron) at a known frequency. When the membrane moves, the superfluid component will pass through the pores because of its low viscosity while the normal component is displaced. Similarly, the amplitude of second sound waves will be determined measuring the displacement of a detector membrane. The resonant frequency is related to the speed of second sound and the shape of the cavity containing the driving membrane; therefore by identifying the resonance peaks in frequency space, the speed of second sound may be determined [1]. The speed of the second sound waves will be measured because there is an interest in understanding such waves at temperatures near the Lambda Point to assist in future research applications.

The detection of second sound resonance near the Lambda Point becomes difficult because of effects which diminish its intensity relative to non-resonant frequencies. Two such effects are an increase in attenuation and the superfluid component reducing to zero as the Lambda Point is approached [2]. Therefore, the ability to detect resonance at temperatures near the Lambda Point depends on the ability to reduce noise and detect small signals accurately.

Theory

The two-fluid model states that a superfluid, such as He II, is composed of a normal fluid component and a superfluid component such that the total density, , can be separated into a normal density component, rho_n, and a superfluid density component, rho_s,

(1).

Similarly the total current density, j, is related to the products of density and velocity and of the normal and superfluid components respectively

(2).

In second sound, the total current density is zero but the relative densities of the normal and superfluid components oscillate. The viscosity of the normal component causes a dissipation of fluid flow. This dissipation is proportional to the velocity of 2^nd sound and the fraction of the normal component. As can be seen in figure 2, when the velocity is large, the fraction of the normal component is small and the fraction of the superfluid component is large. The resulting effect of that dissipation can be ignored and entropy conservation can be assured.

A wave equation for entropy, S, can then be found with the speed of second sound, v2nd, by

(3).

The solution for in equation (3) in cylindrical coordinates take the form

(4)

where J_m is the m^th Bessel function and and are the tangential and radial angular velocities respectively. Cylindrical coordinates are used because the cavity is cylindrical. From equation (4), the allowed transverse wavelengths are characterized by

(5)

where is the cavity radius and is the solution to

(6).

The vanishing derivative in equation 6 is significant be these points corresponds to maxima of the Bessel function which are the resonance peaks. From this condition it was determined that the allowed resonant frequencies are

(7)

where

(8)

and

(9).

Here is the height of the cavity and is the cavity radius. will be ignored in equation 8 as the cavity is designed such that the wavelengths of the transverse waves will be significantly larger than the fundamental wavelength for longitudinal waves. In this way, the frequency of the longitudinal waves will be significantly higher than the frequency of any transverse mode analyzed. Thus equation 7 reduces to

(10)

which gives

(11)

when solved for . We are able to determine the velocity of second sound using equation (11) by measuring the resonant frequencies. Figure 2a depicts the modes for the first 4 alpha values. Note that the m=0 n=1 mode specifies a radial wave. Figure 2b is a table of values .

Experimental Design

The experiment utilizes a flat cylindrical cavity, shown in figure 3, designed such that the wavelengths of the transverse waves will be significantly larger than the fundamental wavelength for longitudinal waves. The inner cavity has a radius and height of 0.102 ± 0.002 cm and 1.905 ±.002 cm respectively. In this way, the frequency of the longitudinal waves will be above the frequency of any transverse mode analyzed.

The diaphragm consists of a thin piece of polycarbonate film with pores measuring approximately 5μm in diameter. Last year’s MXP students evaporated a strip of aluminum across the diaphragm with a thickness of approximately .05μm allowing the diaphragm to be driven electrostatically without affecting the ability of the superfluid component to pass through the pores [3].

The diaphragm is driven by a function generator which creates an oscillating charge difference between the diaphragm and a separate contact on the electrodes. The detector side of the diaphragm, which is mechanically isolated from the driver side by a strip of epoxy attached to the upper wall, detects the displacement by electrostatic potential. The resonator cavity is submerged in liquid helium in a glass double dewar, as shown in figure 4a. The outer dewar contains liquid nitrogen which acts as an insulator and radiation shield for the helium, located in the inner dewar. A pump is then used to create a vacuum in the inner cell which further reduces the temperature of the liquid helium to the desired range by evaporation. Temperature is maintained by a Wheatstone bridge circuit which varies the pumping speed, and therefore applied vacuum, to maintain the desired temperature.

A Stanford Research SR830 function generator is used to drive the second sound generating membrane at frequencies ranging from 100Hz to 3000Hz in 2Hz increments. The detector membrane's movement in response to second sound waves is converted to a signal that is received by a lock-in amplifier. The driver bias box adds a 200V DC offset to the driving signal which pushes the diaphragm away from the electrodes and incorporates a low-pass filter to reduce signal noise [4]. The complete circuit is depicted in figure 4b and is driven by a LabView program. Data on applied frequency and detected amplitude will be recorded and exported to a excel spreadsheet for analysis by the LabView program.

Discussion

We obtained the speed of second sound for ten temperatures ranging from 1.75 K to 2.171 K. The speed of second sound at 2.171 K (0.005 K away from the Lambda Point) was calculated to be 4.47±0.044 m/s. This value was 2.59 ’s away from the expected value of 4.533 m/s. We noticed a trend that the measured data was consistently lower than the expected data, but were not able to determine what caused this offset. Also, we analyzed the temperature characteristics of our most prominent Resonance frequency; the m=4, n=0 mode. Given more time, we would have liked to investigate a possible systematic error caused be the damping forces at the walls inside the cell. This was not pursued due to time constraints and the complexity of the math involved. For future improvements I would recommend buying new low noise BNC cables to help reduce the power harmonics during data taking. Another recommendation to improve upon our experiment would be to redesign the lower portion of the cell by reducing the thickness ( ) of the inner cavity shown in figure 3. Equation 8 shows that reducing ‘ ’ raises the longitudinal resonance frequencies causing less interference with the lower transverse frequencies. Finally, it would be interesting to further the analysis to higher frequency modes than the six modes that we analyzed. We were not successful in extrapolating higher frequencies modes, but this could be the focus of another experiment.

References

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

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

[3] Giolas, Lucas. Speed of Second Sound Transverse Waves in Liquid Helium. 2012.

[4] Wicklund, Andrew and Mecklenburg, Matthew. 2004. Retrieved March 4, 2013 from http://mxp.physics.umn.edu/s04/Projects/s04he/

[5] "Helium Vapor Pressure/Temperature Calculator." Quantum Design. N.p.. Web. 6 May 2013. <http://www.qdusa.com/techsupport/hvpCalculator.html>.