S15FourthSound

Determination of Temperature Dependence of the Fourth Sound Velocity in He-II

Methods of Experimental Physics Spring 2015

Brendon Jones and Yijia Liu

Abstract

We intend to find the temperature dependence of fourth sound in He-II with a cylindrical resonator packed with fine powder near temperatures of 1.6K to 2.1K. The resonance frequency of fourth sound is detected by the transducer mounted on the resonator. The measured resonance frequency was used to calculate the velocity of the fourth sound at different temperatures. The values obtained in the experiment were compared to the accepted values, with the results agreeing to within 2.03% difference.

Theory

Fig.1(left):Predicted velocity of sound waves in He II vs. the absolute temperature from 1.1K to 2.2K.Fig.2(right): The plot of density ratio vs. temperature below the lambda point.Figure from K.R. Atkins.

Liquid Helium 4 has two distinct states separated by the lambda point (2.172K) at a moderate pressure, known as He-I and He-II. He-I behaves like a normal fluid but He-II does not. He-II is described by the “two fluid model”. It behaves as if it consists of two components: a normal component and a superfluid component.The superfluid component exhibits unique properties like zero viscosity and zero entropy. In He II there are various modes of sounds; first sound is a density wave that propagates in both He I and He II ; second sound is an entropy wave which propagates in He II only; fourth sound is a pressure wave only in He II when the normal component is inhibited. Experimentally, fourth sound is achieved in a cylindrical resonator packed with a fine powder.When the oscillation of He-II occurs in the packed powder, first sound and second sound suffer attenuation in velocity due to the inhibition to the normal component.

.When the powder density approaches the limit of completely stop the normal component in place, both sound waves can only propagate in the superfluid component.This new non-attenuated wave propagation is known as fourth sound, which is predicted by Pellam's formula shown below, where C₁,C₂ and C₄ refer to the velocity of mode of sound respectively and ρ_s/ρ and ρ_n/ρ refer to the density ratio of superfluid component and normal component.

Eq.(1):

Apparatus

0.05 μm aluminium oxide powder is tightly packed in the cavity to clamp the normal component of the liquid helium. Two transducers consisting of electrodes and diaphragms are used in the experiment. One transducer is used as a density wave generator and the other is used as a receiver to detect the frequency and amplitude of the sound wave. An important component of the transducer is a metallic diaphragm which is used to convert electrical oscillations into mechanical oscillations or vice versa.

Fig.3:The cross section of the brass cylindrical resonator with the transducers plugged in. The inner diameter of the sound chamber is (1.6±0.1) cm and the height of the resonator is (6.3 ±0.1)cm. The two transducers are mounted on the two ends of the sound chamber.

Therefore, the velocity of fourth sound is given by:

To maintain a cold temperature a system of dewars will be used. A larger, outer dewar will hold a liquid nitrogen bath that the inner, smaller dewar will be placed in. The resonator attached to the end of a cryostat module will be placed into the inner dewar where the liquid helium is held. The nitrogen bath will lower the temperature to about 77K. To lower the temperature below the lambda point , a pump is used to pump away the helium vapor so that the liquid helium can make the transition into its superfluid state. The temperature of the liquid helium is determined by the vapor pressure of the inner dewar which is controlled by the pumping rate. A vapor-pressure manometer is used to calibrate a thermal resistor attached to the cryostat which is also used to stabilize the temperature of the cryostat by a Wheatstone bridge circuit.

The SR 830(Lock-in amplifier) is set to drive and detect fourth sound. The frequency of the input voltage is swept from 0Hz to 8kHz, with about 5Hz incremented each time. The amplitude vs. frequency is plotted for each temperature. The temperature ranges from approximately 1.5K to the lambda point. These measurements are automated by a !LabVIEW program.

Results

Fig.4: The plot of amplitude vs. frequency for the detected signals at 1.624 K and 1.803K. At 1.624K, the first harmonic showed up at 1610.0± 5.0Hz and the second harmonic showed up at 3010.0±5.0 Hz. At 1.803K, the first harmonic and the second harmonic occured at !1465.0±5.0 Hz and !2725.0±5.0Hz

Fig.5:The data points with error represent the experimentally found velocity at a certain temperature. The model line is from the accepted results tabulated by Brooks and Donnelly [4].

The table compares the expected fourth sound velocities at varying temperatures to our experimentally determined result obtained with P=87.8%. The expected velocity is obtained from Brooks and Donnelly [4].

Due to the inconsistency or absence of the upper harmonics the fourth sound velocity was calculated using with the frequency of the first resonant peak,using Equation (1). However, due to the powder in the resonator the fourth sound waves experience a path length difference that results in a decrease of the velocity. A correction factor was multiplied to to Equation(1) to account for this distortion, where P is the porosity of the packed power, defined as the fraction of volume occupied by He-II.

Conclusion

The velocity of fourth sound agreed well with the expected results despite that the higher harmonics cannot be identified clearly. When looking for the resonant frequencies in the helium filled cavity there were some aspects of the data that were not expected. The two most notable were the large amplitude signals at low frequencies and the lack of reliable harmonics after the first resonant peak. It is possible that both of these effects could arise from the packing density of the powder. If the powder is not packed densely enough the normal component and the powder itself could move, introducing complicated combinations of the different types of sounds in the helium.

References

[1] K.R. Atkins, Liquid Helium. Cambridge University Press (1959).

[2] K.A. Shapiro and I. Rudnick, Experimental Determination of the Fourth Sound Velocity in Helium II, A1381, A1391(1964).

[3] I Rudnick and K.A. Shapiro, Phys. Rev. Letters 9,5,191(1962).

[4] J. Brooks and R. Donnelly, The Calculated Thermodynamic Properties of Superfluid Helium-4, Journal of Physical and Chemical Reference Data, Vol 6 (1997).

[5] H. van Dijk, M. Durieux, J.R. Clement, and J.K. Logan, The 1958 He-4 Scale of Temperatures, Journal of Research of the National Bureau of Standards-Physics and Chemistry, Vol 64A (1960).

Acknowledgement

We would like to thank to the assistance of Professor William Zimmermann and Professor Elias Puncher.

Main.liux1267 - 11 May 2015