S16_SuperfluidTransition

Introduction

Helium-4 condenses to a liquid at 4.22 K and is called helium I until it reaches 2.17 K. At this temperature, the lambda point or Tλ, the fluid is called helium II and becomes a superposition of two fluid states, a normal component and a superfluid component. Superfluid helium, a macroscopic quantum system, exhibits significantly different characteristics than classical fluids. These unique characteristics include the ability for the fluid to flow through very fine capillaries with no apparent viscosity, and a singularity in the heat capacity of the fluid at the transition temperature. We treat superfluid helium as a pseudo-mixture of the two fluid states under a theory known as “the two-fluid model” [1]. In this model the densities of the normal fluid and super fluid components always add up to the total density of the fluid; see Figure 1. The superfluid component is treated as having no viscosity or entropy. As seen in Figure 1, the ratio of normal fluid to superfluid decreases rapidly as temperature drops below Tλ, tending to zero as temperature approaches 0 K.

Figure 1: Normalized graph of the relative densities of the superfluid

and normal components of liquid helium II as predicted by the two-fluid model.

Points plotted are at the temperatures at which data was collected.

Original figure, with data from [2].

The superfluid component is treated as having no viscosity or entropy. In the two fluid model, the normal component of the fluid continues to increase in viscosity as temperature decreases, however the increasing presence of the superfluid component dominates this increasing viscosity leading to a diminishing overall viscosity. At the same time, the bulk mass density rho_T of the fluid rises sharply at the transition temperature, dropping only slightly below the transition temperature [2]. These behaviors are illustrated in Figure 2.

Figure 2: Plots of the total mass density ρT and viscosity η of He4 as a function of temperature. Note

the dramatic changes in physical properties occurring at the transition temperature. Errors in density and

viscosity data are given to be on the order of 1%, too small to be seen at this scale. Original figure, data

taken from [2].

We conducted an experiment which used the fundamental oscillatory mechanics of a driven cantilever beam combined with elementary fluid mechanics to determine the superfluid transition temperature, Tλ. By exploiting the unusual viscous properties of superfluids, we observed the change in damping forces due to a transition from classical fluid to a superfluid. By driving the reed with a periodic electrostatic force and measuring the rate of oscillation of the reed dependent on temperature, we can accurately determine Tλ. Additionally, the change in resonance frequency of the reed, as well as the change in quality of the resonance of the reed provide information on the damping forces on the reed at each temperature.

Apparatus

A simplified setup is depicted in figure 3. The experiment took place at the end of a cryostat which was submerged in liquid helium. A small stainless steel reed was elecrostatically driven by a handmade driving electrode supplied with a sinusoidal signal from a lock-in amplifier at controllable frequencies. The driving electrode was biased at 200V DC to increase the driving force seen by the reed, which is proportional to the square of the applied voltage. The vibration of the reed was picked up by capacitive coupling with a detection electrode, also biased at 200V DC in order to minimize detection of capacitance from undesired sources, particularly cross-talk between the driving and detecting electrodes. Proper shielding throughout the apparatus was observed to be very beneficial for this purpose. The resonance signal picked up by the detection electrode was then sent through a pre-amplifier to further filter and boost the signal before being sent back to the lock-in amplifier. This detection signal was sent to a LabVIEW program and repeated over a specified range of frequencies, producing a frequency sweep with a resonance curve centered at the current resonance frequency of the reed.

The cryostat was provided by Dr. William Zimmerman, but the apparatus parts were self-machined. For a view of the assembled machined parts see Figure 4. All parts made of brass were machined using tools in the Mechanical Engineering Student Shop. We elected to build them instead of requesting to have them built in order to maximize the amount of time spent on the project, gain valuable experience machining small parts, and to ensure the parts were completed on the necessary schedule.

Figure 4: Apparatus mounted on the end of the cryostat. From this angle it appears that the

electrode is touching the reed; though the electrodes were ideally placed as near to the reed

as possible, great care was made to ensure no contact was made during the experiment.

Results and Conclusion

By submerging the apparatus in liquid helium and driving the reed were we able to determine the temperature at which the liquid helium underwent a phase transition. At a constant temperature the amplitude of the reed as a function of frequency was collected, the signal belonging solely to the reed was then extracted. By fitting the square of the signal to a Lorentzian distribution the resonant frequency at each temperature was determined. A plot of this data, resonant frequency as a function of temperature can be seen in Figure 5.

Figure 5: Plot of the main measurement of this experiment, the resonance frequency

of the reed as a function of temperature. A clear cusp can be seen at 2.171 K,

where we determine the phase transition to occur. Original figure.

By approximating the reed as having an oscillatory spherical solution to Stoke's formula and assuming steady state operation, use of the damped-driven oscillator equation allows us to arrive at an equation which describes the relevant physics in our experiment; a full derivation can be found in [3]:

where rho_T, is the total density of liquid helium, omega_v is the vacuum frequency of the reed, rho n is the normal component density in liquid helium II, eta is the kinematic viscosity of the fluid, and omega is the resonant frequency at any given temperature. The only measured quantity in this equation is omega. All other quantities are tabulated by Donnelly and Barenghi [2]. If this equation is plotted against itself, i.e. the left hand side on the y-axis and the right hand side on the x-axis we should see a linear behavior. There will be a slope corresponding to C and an intercept corresponding to D, however these values result from the spherical approximation and represent geometric constants which are irrelevant to this experiment. We expect a linear behavior both above and below the transition temperature since the only approximations made are dependent upon the reed, not on the properties of the fluid. A plot of the results of our resonance data used in equation can be seen in Figure 6.

Figure 6: Plot of the linearization of our main equation.

Clearly the linear behavior is exhibited below the lambda transition; using a least squares regression the line gives a reduced chi squared value of 2.94 However, above the lambda transition the behavior deviates strongly from linearity. We expected perhaps some extra noise in this data due to mechanical interference from boiling above the transition point, or perhaps even a discontinuity at the transition followed by a change in slope. Neither of these scenarios seem to be the case; the data simply is not linear in any sense of the word. This implies that the approximations made in order to reach this final equation nay be invalid at temperatures above 2.17 K, yet are somehow valid below the lambda transition. Aside from the various assumptions made to arrive at our main equation, nothing from the derivation suggests an immediately obvious reason why the mechanics above the lambda point would not hold. At the same time, the quality of the fit below the transition point, as well as the quality of the resonance curves in general give no reason to suggest that something is wrong with the data. Therefore, there must be something fundamentally missing from the theory that does not affect the fluid below the lambda point.

A good candidate for the missing piece of theory may be rooted in heat transfer, as one of the characteristics of the lambda transition is a significant rise in specific heat capacity. Heat transfer would affect the overall energy of the system more above the lambda point than below it -- our equation as it stands is rooted purely in mechanics and is not able to take this into account.

Although the use of our main equation exhibits unexpected behavior, other properties of our data behave exactly as expected. Figure 7 compares the Lorentzian resonance peaks associated with three different temperatures.