We propose to measure the velocities of first, second, and fourth sound in superfluid liquid helium from temperatures of about 1.6 K to just above the lambda point. The techniques that will be used are oscillating a capacitive membrane inside one end of a resonator cell with an AC voltage and receiving a signal out from the other side with the same type of transducer, and then using resistors in place of the capacitors. By creating standing waves within the cell, we will be able to measure the frequency of the resonance, and therefore determine the velocities of the sounds. We will also be able to determine the Q value for our resonator cell at each temperature by analyzing the input and output waves.
Theory, Apparatus, and Procedure:
Since the discovery of superfluid liquid helium in 1932, physicists have been able to predict, discover, and investigate certain properties of the superfluid that make it extremely unique in comparison to otherwise "normal" substances and materials. If liquid helium is cooled through the lambda point, 2.17 K, it can be thought of as consisting of two interpenetrating components: the superfluid component mentioned above and a normal component. This is known as the two-fluid model and the liquid is often referred to as helium II. The ratio of the amount of normal fluid to the superfluid is a function of temperature (see figure 1), i.e. the more the temperature is lowered below the lambda point, the less normal component exists. The superfluid component displays properties that are somewhat counter-intuitive and is responsible for the unique behavior of He II. Some of these behaviors include zero translational viscosity and zero entropy (only present in the superfluid component), efficient heat transfer, individualized vortices within a rotating sample, and a thermo-mechanical effect. In this experiment we intended to investigate and measure three properties - first, second, and fourth sound - of liquid He4 as a function of temperature from about 1.3 K to the lambda point. However, we were only successful in measuring second sound.
First sound in liquid helium, and in all other media, is simply a pressure wave that is completely analogous to sound that we hear traveling though air. In this case, the superfluid component will cause the speed of first sound to continually decrease as the temperature is increased from about 1 K, with a discontinuity at the lambda point, as shown in figure 2. This property can be predicted by the equation for the speed of first sound,
. In this equation, which pertains to first sound in all media, P is pressure, rho is the density, and S is the entropy. The equation above can also be expressed as:
. Here, gamma is the ratio of specific heats at constant pressure to constant volume, rho is the density and beta is the compressibility coefficient. The cusp in u1 at the lambda point can be attributed to the behavior of the compressibility of He4. It has been experimentally verified that the compressibility of He4 experiences an increase and then a discontinuity as the temperature is increased through the lambda point. Since this factor is in the denominator of equation for u1, it should be expected that the first sound graph experiences the negative slope up to the lambda point as shown in figure 2.
Second sound, which is the propagation of energy or heat, is a very unique property supported by liquid helium II, unlike first sound. If a heat pulse is produced within liquid helium below the lambda point, the pulse will be able to continue to travel in a well-defined pulse instead of dispersing as it would in a normal material. This is because the superfluid component of liquid helium II has zero entropy. Thus, if heat is introduced into a region containing superfluid and normal components of liquid He4, the superfluid component will tend to flow towards the hotter area and the normal component will flow out. This process gives rise to the velocity of the two components opposing one another. Quantitatively, the speed of second sound is given by the equation
. Here, rhos and rhon are the densities of the superfluid component and normal component respectively, T is the absolute temperature, S is the entropy of the liquid (contained in the normal component), and C is the heat capacity. Recognizing that rhos and rhon add together to give the total fluid density, rhos/rhon can be written as:
rho/rhon - 1
where rho is the total fluid density. Then, the equation for second sound can be expressed as:
The graph of typical second sound values is shown in figure 2. For the region in which we will be investigating (1.3 K and above), second sound rises from about 19 m/s to just above 20 m/s and then decreases zero as the temperature approaches the lambda point. The shape of the plot in figure 2 can be predicted and calculated using the equation above and values for rho/rhon, S, and C given by J. Maynard along the SVP curve . Figure 3 shows a plot of these values along with the calculated values for second sound velocity (using the above equation) as a function of temperature. Since above T = 2.17 K the density of the superfluid component is zero (therefore rho/rhon = 1), it can be easily seen that second sound velocity does not exist above the lambda point.
Fourth sound is defined as pressure waves within only the superfluid component of the liquid helium. Due to the fact that superfluid liquid helium has zero viscosity, but the normal component has a certain amount of viscosity, we are able to manipulate a situation in order to prevent the normal component from flowing while allowing the superfluid to move about freely. To do this, the normal fluid component must be locked within the confines of a small-diameter channel, known as a superleak. The maximum diameter size can be calculated by recognizing that the viscous wavelength for the normal fluid is given by
where etan is the viscosity of the normal component., omega is the wave frequency multiplied by 2*pi, and rhon is once again the normal fluid density . At the lambda point, lambdan = 301*f -1/2 µm and becomes larger as the temperature is lowered . Thus, for frequencies less than 10 kHz, the normal fluid viscosity wavelength will have a value of at least 3.01 µm, indicating that the diameter of the superleak cannot exceed this value in order to lock the normal fluid in place.
Knowing that the velocity of fourth sound is measured only in the superfluid component, it is possible to guess that the velocity is heavily dependent on the fractional density of the superfluid component. This is, in fact, true and the speed is given by:
. Here, rhos/rho is the fractional density of the superfluid, and u1 is the velocity of first sound.
In order to measure these sounds, an apparatus that can create individual pulses of energy or pressure in the liquid helium is needed. Then, the time for a pulse to travel from one part of the apparatus to another can be measured and divided by the distance the pulse traveled in order to calculate the velocities. Another way to measure the speed of the sounds is to create a continuous sine wave within a hollow cell filled with liquid helium II. Then, if the frequency of the input wave is adjusted to the right value according to the length of the cell, standing waves will occur within the cell. This is the technique that we employed in this experiment.
We took measurements of the sounds using two different types of resonator cells. However, the submersion mechanism was the same for both resonator cells. The cell was attached to the end of a dipstick that fits into a helium double dewar. The dipstick holder was screwed onto the top of the inner dewar to form an air-tight seal. Also on the holder were connectors for our various electronics. The wires for the electrical components contained within the helium dewar were wrapped around the dipstick and attached accordingly. The inner dewar was first cooled with a liquid nitrogen jacket, then filled with liquid helium. Both set-ups included a four-wire Germanium resistive thermometer located outside the resonator cell. Temperatures were determined via calibrating the Ge resistors resistance to the pressure given by an analog pressure gauge and comparing the pressures to the accepted values of temperature vs. vapor pressure of liquid helium obtained by the National Bureau of Standards . The resistance of the Ge resistor was read in kilo-ohms with an HP-34401A multimeter set to four-wire mode. In order to reach temperatures that allow superfluid to exist, we pumped on the liquid helium with a mechanical pump and monitored the temperature with the pressure gauge and our (calibrated) resistive thermometer. The temperature was controlled by manually adjusting the valves on the pump. When taking measurements, we started at the lowest temperature possible (about 1.3 K) and took data at temperature intervals of about 0.01 K or 0.02 K up to about 2.16 K, after which the measurements became impossible to make with our equipment. The first set-up we used was one that employed a resonator cell containing resistive transducers to produce and receive heat signals in the liquid helium. The way we produced the signals in this resonator cell was to send an AC signal to the drive resistor, a strain gauge with a resistance of 121 Ohms. The power delivered to this resistor was then given by: P = V2/R where V is the input voltage and R is the resistance. Since the input voltage was an AC signal, this equation becomes:
So, the power output of the driving resistor resonated in the cell at a frequency twice that of the input. After the signals were produced by the drive and sent through the resonator cell, they were detected with an Allen Bradley EB 6805, 68Ohm, carbon-glass resistor, which was ground in half for increased sensitivity. The power being transmitted by the drive produced heat waves within the cell, which varied the resistance of the detector. Thus, by biasing the detecting resistor with a constant DC current, we were able to obtain an oscillating voltage output from the receiver. This cell could only be used to detect second sound because it was not possible for the resistive transducers to either produce or detect density waves. The electronics scheme we used for this resonator cell is shown the apparatus section. We produced the input AC signal with the built in function generator of an SR-830 lock-in amplifier. The receiving transducer was biased with 5 VDC through a 150kOhm resistor. The large difference between this resistance and that of the detector allowed a relatively constant current to flow across the detector. The signal coming out of the detector was then put through a high-pass filter with C = .1µF in order to block off any DC voltage. The filtered signal was then sent through an SR-560 pre-amplifier with the gain set to 100. The output of the pre-amp was fed into the input of the SR-830 lock-in amplifier, which was set to lock in on 2*omega. The lock-in further amplified the signal by 2 *106 and we viewed this output along with the original AC signal provided by the function generator on a Tektronix 2245A oscilloscope. In order to calculate second sound velocities for each temperature with this cell, we first manipulated the pump so that we had a stable temperature. Then, we adjusted the frequency that we sent into the cell until we saw a resonance from the cell output that oscillated at 2*omega, as stated above. The velocity could then be found by calculating: where L is the length of the resonator cell, f is the frequency of the resonance and n is the harmonic number . We determined the harmonic number by measuring two subsequent harmonics for each temperature and calculating the frequency difference between them in order to find the fundamental frequency. Then, the harmonic number was just the resonance frequency divided by the fundamental. The second resonator cell that we used was one that we built. It was built so that it had capacitors on each end instead of resistors. One of the sides of the capacitors was the flattened end of an electrode insulated from the rest of the cell with Sty-cast. The other side was made by coating one side of a 25mm diameter, 1µm-thick Nuclepore membrane with a layer of aluminum that was about five hundred angstroms thick. Then, we placed the membrane onto the surface of the brass electrode so that the electrode and aluminum were separated by the membrane. Thus, the aluminum was in contact with the sides of the cell, which were made of brass and electrically grounded. The capacitance of the membranes on each side of the resonator cell was measured by a Fluke PM6304 RCL meter to be 19 pF. The energy stored in the drive capacitor is given by where C is the capacitance of the electrode and aluminum on the membrane, and V is the input voltage. If the capacitor is biased with a DC voltage along with the time-varying voltage, then the energy equation becomes
where Vdc is the bias. By making the DC bias to be much larger than Vo, then the V2 term from the above equation can be expanded as a Taylor series (centered at zero) to yield:
Since Vdc is much larger than Vo, we can ignore the (Vocos(wt))2 term so that we are left with:
The energy of the capacitor is dissipated in oscillating the membrane to produce pressure and energy waves, and therefore producing first, second, and fourth sounds. From the equation above, it is evident that the capacitor drive will cause the energy to oscillate within the resonator cell at a frequency equal to that of the input frequency when resonance is achieved. These signals were picked up on the other side of the resonator cell by the receiving capacitor in the form of an oscillating voltage. This system was very much analogous to a speaker and a microphone.In order to operate this cell, we employed the electronics scheme shown in the apparatus section. First, we sent in white noise provided by an HP 35670A Spectrum Analyzer. The noise signal was put through a high-pass filter with C = 50nF and R = 10M. This filter was also attached to the cell bias, which biased the capacitors with 150 VDC. After the signals had gone though the cell and been picked up by the receiving capacitor, we sent the out-going signals through a high-pass filter in order to take of any DC voltage components, similar to the resistive resonator cell. The filtered signals were fed directly into the input of the spectrum analyzer, where they were averaged about 50 times. Using the white noise output of the spectrum analyzer allowed us to do an entire frequency sweep for each temperature at once. Therefore, we were able to easily determine harmonic numbers and resonance frequencies. This technique also enabled us to determine the Q of our resonator cell. To do this, we focused in on one peak of the spectrum, measured the peak width at the point where the full amplitude divided by the square root of two occurred, then divided the resonance frequency by this difference. During the course of our measurements, we decided to focus on the 8th, 9th, and 10th harmonics of second sound because when zoomed in on, they were some of the most well defined peaks in the spectrum. Once we had determined the frequencies at which these harmonics occurred, we calculated the velocity of second sound with the equation given in the case of the resistive cell:
where L was the length of the new cell, f was the frequency of the resonance, and n was the harmonic number.