s19 Velocity of 1st Sound Near the Lambda Point & Solid Helium

*Solid Helium Information can be found at the bottom

Measuring the Velocity of 1st Sound Near the Lambda Point in ⁴He II

Sam Dietterich and Brock Grafstrom

University of Minnesota - Methods of Experimental Physics II

Introduction

The behavior of liquid helium has been studied extensively over the past century due to its unique physical properties at extremely low temperatures, with superfluidity (due to Bose-Einstein Condensation) being the most significant property. Helium's bizarre physical characteristics were first noticed a short time after the gas was initially liquefied by Heike Onnes, who cooled helium gas below 4.2K.[1] The plethora of research that resulted eventually lead to the discovery of an anomaly near the lambda point at saturated vapor pressure where there is a sudden drop in the speed of 1st sound. This is thought to be related to the known process of Bose-Einstein Condensation, but currently no theoretical models exist to describe the behavior of acoustic sound waves in superfluid helium.

For this experiment, we had the capacity to measure the speed of 1st sound at saturated vapor pressure in liquid helium at specific temperatures between 1.8K and 2.2K. We did this with a cylindrical first sound resonator with a longitudinal driver and detector transducer configuration, which could be used to detect the resonant frequencies in liquid helium. We aimed to use this capacity to verify this 1st sound anomaly at the lambda point (2.17 K). We also attempted to improve on the accuracy of existing data. By measuring the corresponding resonant frequencies across a range of temperatures, a graph could be created to illustrate the overall decrease in the speed of 1st sound as the temperature approached the lambda point.

Theory

As the temperature drops below 2.17K at saturated vapor pressure (known as the lambda point) 4He nuclei start to transition to their quantum ground state and form what is called a Bose-Einstein condensate. This Bose-Einstein condensed component of the fluid is called the superfluid component, which possess the interesting property that it has zero entropy due to the fact that all of the helium nuclei are in the same quantum state (recall that the quantum ground state has multiplicity one here, and the entropy is proportional to the log of the multiplicity).[3] Due to their very low energy and light mass, the de Broglie wave lengths of the helium nuclei are very large and overlap, one another until the system behaves as one coherent wave. This is what causes the unique properties of the superfluid phase, including essentially zero viscosity, and extremely high thermal conductivity.[4] [5]

Figure 1 - Accepted Empirical Data

This plot illustrates the mysterious "dip" in the speed of 1st sound around the lambda point at 2.17 K. Note that this plot corresponds to the empirical data obtained from Donnelly and Barenghi, and that of a theoretical model.^[3]

The speed of sound measurements were carried out with a cylindrical cavity resonator. The basic idea is that the frequency of acoustic resonances depends not only on the shape and size of the resonator, but also on the speed of sound in the material carrying the acoustic vibrations. The relation between the speed of first sound and the resonant frequencies can be found by solving the wave equation in cylindrical coordinates with cylindrical Dirichlet boundary conditions. The wave equation in cylindrical coordinates is,

where 𝜈1st is the velocity of first sound. The cylindrical boundary conditions for the acoustic cavity are,

where the longitudinal frequency 𝜔pz and radial frequency 𝜔mnr are given by,

By rearranging the equation above, the velocity of first sound can be described by,

While this relationship accurately describes the velocity of first-sound for longitudinal and radial resonances, the radial resonance term can be omitted. This is because the driver and detector transducers were placed at the ends of the resonator facing one another. This means that only the longitudinal resonances were strongly activated. Therefore, the radial part can be ignored and the velocity formula simplifies to,

Experimental Methods

A standard helium filled, double Dewar with a vacuum pump was used to cool the cryostat and resonator down to temperatures of about 1.8K. The inner Dewar was filled with liquid helium at 4.2K before vacuum induced evaporative cooling, while the outer Dewar was filled with liquid nitrogen at 77K. The liquid nitrogen served two purposes; first it helped shield the inner Dewar from external heat, and second, it helped to precool the inner Dewar before any liquid helium was added. The entire inner wall of both Dewar’s was lined with a silver coating to prevent radiative heat from entering (Fig. 2).

Figure 2 - Experimental Apparatus

A LabVIEW program was used to control the output frequency of the lock-in amplifier. This incoming signal was then sent through coaxial cables into the cryostat and connected to the driving transducer. The detecting resonator’s signal was also sent out of the cryostat via coaxial cables into a preamplifier. The resulting output signal was then passed into a Stanford Research SR-183A Lock-In Amplifier, and the resulting data was recorded in the same LabVIEW program.

Two resistors were also placed near the bottom of the inner Dewar, one (a carbon resistor) to act as a heater, and the other (a germanium doped semiconductor resistor) for temperature measurements with an attached Wheatstone bridge. This allowed the helium bath to maintain a constant temperature for extended periods of time. Inside the Dewar was a cryostat, which supported a cavity resonator and all necessary electrical connections. The cavity resonator was used for speed of sound measurements. To generate and detect 1^st sound during the experiment, two circular transducers were mounted parallel to each other at each end of a cylindrical resonator. The transducers were comprised of a brass plate and a piece of aluminum coated Mylar. By applying a 200V DC bias voltage across both transducers, a charge build-up could be generated on both Mylar films, much like a capacitor. A 5.00V (amplitude) ac voltage could then be applied to one transducer (called the drive transducer) to cause the aluminum coated Mylar film to oscillate and produce sound. The film in the other transducer would then be vibrated by the sound coming through the helium, which would produce a measurable ac voltage.

A day before any measurements were recorded, the cryostat was sealed in the dewar, which was then back filled with nitrogen twice. The inner dewar jacket was then vacuumed out and back filled with 1 Torr of nitrogen gas 3 times. The outer dewar was filled with liquid nitrogen to pre-cool the inner dewar. On the following day, helium gas was used to backfill the inner dewar, after which liquid helium was transferred into the inner dewar. A vacuum pump was then used to evaporatively cool the liquid helium down to 1.8K. As the helium was cooling, the temperature bridge circuit was constantly balanced, and the decreasing pressure was recorded, in order to construct a temperature calibration curve. We then lowered the temperature until it had stably reached the lowest value we could achieve. Then the drive transducer was driven across a range of frequencies to determine the speed of sound. Here is a YouTube video of our first liquid helium run, where we performed this procedure. The video starts at the helium transfer step. To see the nitrogen transfer, and some of the other steps that occur before the helium transfer, watch the solid helium video located at the bottom this page.

From the spectrum it is clear that the sharp peaks in the detector voltage correspond to the harmonics for the resonant frequency of liquid helium at 2.011K. The temperature was determined by comparing the pressure gauge reading to a known helium vapor pressure table.

Results

After calculating the mean velocity and standard error of the mean for each of the 24 data points, MatLab was used to plot the experimental values as a function of temperature, along with the accepted literature values (Fig. 4). At 2.167 K, the calculated velocity of 1st sound was determined to be 218.50 ± 0.46 𝑚/s, which was found to be well within agreement (0.02𝜎) with the accepted value of 218.48 𝑚/𝑠.

Figure 4 - Final Data

After examining our final data (Fig. 4) it became clear that some systematic error was affecting our velocity measurements, as the data points from 1.8 – 2.1K fall below the accepted velocity data by a value greater than 1σ but less than 2σ (except for the lowest temperature measurement). We are not certain of the cause for our discrepancies, but we speculate that the location of the pressure gauge may have been a critical factor for measuring lower temperatures.The location of the pressure gauge and vacuum pump are directly adjacent to one another, and because the vacuum pump was required to pump at a higher rate in order to achieve lower temperatures, a slight Bernoulli Effect could have occurred and caused the pressure gauge to read vapor pressures that were approximately 0.2 Torr lower than the actual pressure of the helium bath. Note that all of the previous liquid helium experiments have experienced this trend. Therefore, it is in the best interest of future experimenters to determine if the location of the pressure gauge is a critical factor for temperature measurements below 2.1K !

Conclusion

By using an acoustic cavity that was comprised to copper transducers, the longitudinal resonances of superfluid helium were excited across a temperature range of 1.8 – 2.17K. The required temperatures were achieved by submerging the resonator cavity into a liquid helium bath inside of a double-walled Dewar, and by using a combination of evaporative cooling and resistor heating elements to stabilize the bath to a specific temperature. While some systematic errors effected the measurements at low temperatures (below 2.10 K) the calculated velocities for the speed of 1st sound were determined to be within one-sigma across the temperature range of 2.10 – 2.17 K. The strong agreement (0.02𝜎) between the calculated velocity (218.50 m/s) the accepted value (218.48 m/s) at 2.168 K also indicates that our resonance measurements have the capability of being extremely accurate for temperatures near the lambda point.^[2] However, future experiments should consider moving the location of the pressure gauge to observe whether its location relative to the pump has any noticeable effect on temperature measurements below 2.10 K. Currently this setup can only measure the velocity of 1st sound for temperatures between 1.5 and 2.17K, so other experiments should attempt to design an apparatus that can accurately measure the velocity of 1st sound for temperatures above 2.17K. By measuring the speed of 1st sound above the lambda point, future experimenters could achieve a more comprehensive study of the region around the lambda point, while likely increasing the precision for the measured velocities of 1st sound in the superfluid region.

A Method For Solidifying Helium

Sam Dietterich and Brock Grafstrom

University of Minnesota - Methods of Experimental Physics II

Preface:

Dear Reader,

This report should hopefully serve you as a guide for what the experimental set-up, procedure, and data analysis consist of. Unfortunately, we did not have the time to trouble-shoot all aspects of our apparatus, and therefore we did not take any data for a high-pressure solid helium run. Fortunately, much of the techniques used for the 1st sound velocity experiment can be applied here, so we would highly recommend that you read our report located at the top of this wiki, before proceeding. Sam has also created a number of videos that explain all of the steps that we performed during our semester, if you want to watch exactly how we performed our experiments. With that being said; Good Luck!

Introduction

Out of the 118 known elements that comprise the physical world, Helium is the only element that remains as a liquid at absolute zero. While the behavior of condensed liquid helium has been studied extensively over the course of the past century due to its unique physical properties (e.g. superfluidity), solid helium research by comparison has been scarcely touched. Despite being first solidified in 1926 by Willlem Keesom, the difficulties with studying solid helium have lead to uncertainties about the substance's properties. Currently, the most hotly-contested topic about solid helium is whether solid helium possesses a super-solid phase akin to the superfluid phase for its liquid counterpart. As of the writing of this article, the general consensus is that supersolidity does not exist in the solid phase of helium, but more research is needed before this can become a foregone conclusion. The primary technique for detecting the solidification of helium relies on torsional oscillator experiments like the ones performed by [cite] [cite]. While these experiments have been shown to reliably produce good results, they lack a means for accurately detecting the exact moment of solidification. This is because torsional oscillators require a significant mass of solid helium to form, before their period can be affected by the increased moment of inertia. Fortunately, torsional oscillators are not the only means to detect solidification, as acoustic resonances can also be used to measure the precise moment when helium crystallizes into the solid phase. This method also has the added benefit of being fairly simple to construct and easy to operate, which opens up the possibility for more research to be done on solid helium than what is currently being performed. By using resonance measurements to determine the speed of first-sound travelling through helium, this article hopes to convey the accuracy, and validity of using such a method for detection the exact phase transition of solid helium for a range of temperatures and pressures.

Theory

Understanding exactly why solidification fails to occur is a little more complicated than understanding superfluid formation, but it is an important part of understanding the physical setting in which the anomaly of interest shows up. The basic reason for why solidification fails to occur, is that the quantum zero-point energy of the helium nucleus is too high to allow for solid formation in such an inert, low atomic mass material. We do not mean to suggest that the quantum zero-point energy of the helium nucleus is unprecedented compared to other elements. It is merely too high for solidification to occur in the context of helium’s chemical inertness, and low atomic mass. The low atomic mass makes for larger quantum fluctuations for a given amount of energy, which holds the material apart, and the truly unprecedented chemical inertness of helium means very little is holding the material together (extremely weak Vander Waals forces). As one cools helium below the lambda point, helium atoms start moving into this solid resistant quantum ground state resulting in the formation of an ever-purer superfluid (i.e. a larger fraction of the liquid helium density is comprised of the superfluid component) instead of crystallizing. To summarize, helium remains a liquid at absolute zero due to its...

• Low atomic Mass Which Results in

• Weak Interatomic Forces (Vander Waals) ===> - A failure to solidify below 25 atm

• A High Quantum Zero-Point Energy

The result of helium nuclei having large de Broglie wavelengths and large zero-point energies is a Bose Einstein Condensate, which fundamentally remains in the liquid phase at 0K. That is until a pressure of 25atm, or greater is applied, at which point liquid helium begins to crystallize.

Figure 1 - Helium-4 Phase Diagram

The Liquid He I region corresponds to the normal component of helium, while the liquid He II region corresponds to helium in the superfluid state. Helium nuclei can solidify into the 𝛂 - phase (hcp) or the 𝛄 - phase (bcc) to form different crystal structures depending on what the given temperature and pressure conditions are at any given point. (Figure published by Vignos and Fairbank)^[7]

Unsurprisingly, solidification is accompanied with sudden changes in both density and rigidity. This has a large effect on the speed of first sound in helium. Vignos and et. Al. determined 320m/s for the speed of first sound in the superfluid in our experimental pressure/temperature range, and the speed of sound in the solid, just above the transition has been determined to be around 540m/s.[6] Our experiment will work within the pressure range of 25 – 31 atm, and the temperature range of 1.6 – 1.8K, which corresponds to the transition between the superfluid (He-II) region of the phase diagram into the -phase solid. Therefore, the large increase in the velocity of first sound allows for an effective way to accurately detect the transition within this temperature range. It is important to note that within part of the temperature and pressure ranges we achieved in this project, a solid to solid phase transition occurs which also represents a change in density and a change in rigidity. Immediately after solidification, the helium is in the gamma phase (bcc, less dense solid), and with further increased pressure recrystallizes to the alpha phase solid (hcp, denser solid). A discontinuous jump in the speed of sound also accompanies this transition, however, the difference in density and especially rigidity is radically smaller for this transition, and there is therefore no risk of it being confused with the liquid to solid transition. ^[2] Figure 2 illustrates this concept.

Figure 2 - Theoretical Prediction for the Speed of 1st Sound

^{Note that the liquid phase velocities are approximately 380 m/s, while the speed of first sound in solid helium has been measured around 540m/s.[7]}

Note that the relationships between the speed of first sound and the resonant frequencies are still given by,

which is partly derived in the liquid helium report listed above. Therefore, by using the equation listed above, it should be clear to see that one will see a dramatic increase in the speed of first-sound when solidification begins to occur. By measuring the resonant frequencies as the pressure is continually increased (at a constant temperature) it should be possible to graph the velocity of sound as a function of pressure. By doing this, the discrete data points can be "fit" to some non-linear polynomial (e.g. quadratic or cubic) to obtain the inflection point. Much like an acid-base titration curve in chemistry, the inflection point for this graph serves as a way to systematically measure helium's transition from liquid to solid. Or more specifically, the inflection point should correspond to the pressure (at a given temperature) for when the helium is 50% liquid and 50% solid. By repeating this process across a range of temperatures, it should be possible to plot the pressure and temperature for each inflection point. By using non-linear fitting, it should be possible to construct the phase-line that corresponds to the liquid He II phase transition into the 𝛄-solid (as seen in Fig. 1).^[7]

Experimental Methods

The experimental setup for achieving solid helium makes use of the same equipment that is described in our liquid helium report. The only addition that you will have to make is the addition of the high pressure system. To achieve the necessary pressures of 25atm and above, a fresh helium gas cylinder was connected to a high-pressure regulator that was rated for a maximum pressure of 1000psi. The high-pressure regulator was then connected to a cold trap via a copper tube. The cold trap used liquid nitrogen to freeze out some impurities in the helium. Additionally, such low temperatures also helped other impurities stick an activated-charcoal filter contained in the trap. The filtered helium was then sent through a t-fitting that lead to a 20L ballast tank. The ballast tank only served to increase the overall volume of the system, which helped to minimize pressure fluctuations due to small gas leaks. Also attached to the t-fitting is a release valve which leads to a small vacuum pump that was used to vacuum out all of the air in the pressure system before use.

A second t-fitting near the ballast tank was also used to attach a high-pressure gauge, which denoted pressure differences of 5psi for the overall system. Another 5ft length of copper tubing, was used to connect the high-pressure system to the capillary cryostat. The cryostat utilized brass capillaries to transfer the helium gas into an experimental cell that was machined out of 303 grade stainless steel (Fig. 3) which housed the same resonator that was used for the 1st-Sound liquid helium experiments.

Figure 3 - High-Pressure System

Copper tubing (drawn as a line) is ¼ inch in diameter, as are all of the t-fittings that are used to connect separate components. The cell on the far right was housed in a capillary cryostat located in the helium dewar (cyrostat and dewar omitted for clarity).

Experimental Procedure

A day before any measurements were recorded, the capillary cryostat should be sealed in the dewar. The inner dewar jacked should then be vacuumed out and back filled with 1 Torr of nitrogen gas. The process of vacuuming and backfilling with nitrogen should be performed 3 times. After this is done, the outer dewar can be filled with liquid nitrogen to pre-cool the inner dewar. Make sure to record the final level of liquid nitrogen before you leave the lab!

On the following day, helium gas should be used to backfill the inner dewar, after which liquid helium can be transferred into the inner dewar. In our experience, the large vacuum pump could be used to evaporatively cool the liquid helium down to 1.6K. As the helium is cooling, constantly balance the temperature bridge circuit, while recording the corresponding pressure. By recording the pressure and resistor values, it is possible to construct the temperature calibration curve.

*Note: This last paragraph describes what the procedure “ought” to be if everything works according to plan. Therefore, you will likely need to adapt this procedure if/when complications arise.

While the experimental cell is cooling down to around 1.6K for the first solidification trial, the high-pressure regulator should be used to add a small volume of helium gas into the experimental cell, and ultimately the resonator cavity. The drive transducer can then driven across a range of frequencies to determine the speed of sound. This is solely for the purpose of indicating when the cavity is full of liquid helium. Once full, the high-pressure regulator should opened further to output a pressure that is approximately 10psi larger than the fill test resonance measurements. This process of gradually increasing the pressure can repeated until pressure measurements at the liquid-solid equilibrium (and beyond) have been made. The point of solidification (should) manifest itself as a sharp shift in the location of the resonances, which corresponds to a sudden increase in the speed of sound. To decrease the pressure of the system, the release valve can be opened to allow for a controlled decrease in pressure. The pressure of the system should then be decreased until it returns to the value that corresponds below the melting pressure. The temperature of the system can then be increased to a new constant temperature via the heating resistor that is located at the bottom of the helium bath. The pressure of the system can then increased in the same manner, in order to measure the point of solidification at a new temperature. Repeat this process until you have detected solidification across a range of temperatures and pressures.

Watch the video for a better idea of what is outlined above...

As the video conveys, we did not manage to solidify helium due to some poor electrical connections and lack of time. Nevertheless, we hope that your project doesn't meet the same fate. Best of luck in your endeavors with solid helium!

References

[1] Onnes, H. K, The Liquefaction of Helium, KNAW, Amsterdam 11, 168-185 (1908)

[2] Donnelly, R.J., Barenghi, C., Crogenic Helium Turbulence Laboratory, University of Oregon (1998)

[3] Pekeris, C. L., The Zero-Point Energy of Helium, Phys. Rev. 79, 884 (1950).

[4] London, F., Phys. Rev. 54, 947 (1938)

[5] Tisza, L. The Theory of Liquid Helium ,72, 9 (1947)

[6] Belanger, M., Edens, M., Speed of Transverse 2^nd Sound Waves in Liquid Helium II, MXP Wiki (2013)

[7] Vignos, J.H. and Fairbank, H.A., Phys. Rev. Lett. 6, 265 (1961)

[8] Sasaki, S., Caupin, F. & Bailbar, S. J Low Temp Phys (2008) 153: 43