Third Sound in Superfluid 3He
Third Sound in Superfluid 3He
Experimental Design
Deriving the wave - equation for third sound waves
Experimental
Below we have a schematic of the experimental cell we built.
- The 3He film forms on a polished copper substrate, which is sitting in a bath of helium whose free surface is a height h below the film.
- The equilibrium film thickness, d, varies as we change the height, h, approximately as d= a / h1/3. (a is a constant measuring the strength of the van der Waals attraction; this relation occurs because the equilibrium is a balance between the gravitation potential energy, U = mgh, and the van der Waals potential energy.) Typical values are d ~ 100 nm and h ~ 1 mm.
- The height h is measured with a vertical annular capacitor, whose capacitance changes linearly as the dielectric helium liquid fills it.
- Above the film, there are two circular plates, an annular "drive plate" and a disk-shaped "detector plate." Each of these forms a parallel plate capacitor with the substrate.
Drive and detection
Once we have put the film on the plate, we then excite and detect resonant standing waves on the surface of the film.
- We excite standing waves by applying a voltage V = Vdc +Vac sinwt between the annular drive plate and the substrate. This produces an electric field between the plates, and since the film is dielectric, it feels a force proportional to V2. Since Vdc >> Vac, the ac component of the force is 2 Vdc Vacsinwt, with a frequency w.
- We then monitor the response of the film's surface measuring the changing capacitance between the inner plate and the substrate. We lock-in on this signal at the drive frequency w, and then sweep this frequency. The result is a spectrum showing the film's response as a function of frequency, which should be dominated by the resonant frequencies of the standing waves on the film's surface.
Deriving the wave-equation for third sound waves
In the simplest case, there are two controlling equations which govern a third sound wave:
Conservation of mass: a divergence of the mass current must be accounted for by a change in the thickness of the film.
Equation of motion: a superfluid acceleration parallel to the substrate is caused by a gradient in the chemical potential m, which is caused by a gradient in Z, the displacement of the surface.
Definitions:
Z(r, t) = surface displacement at a position r and time t
vs = superfluid velocity
d = thickness of film
r = density of liquid
= average superfluid density, averaged over the thickness of the film
= chemical potential due to van der Waals force
= acceleration at surface due to van der Waals force
Taking the time derivative of the first equation, and the spatial derivative of the second, we obtain the equation governing third sound
waves:
This is just the usual wave equation, describing waves whose speed is given by c3, where
We can compare this to the speed of waves crossing shallow water, as you might see at a beach or on a pond:
Both speeds depend on the square root of the thickness (or depth) of the liquid; both also depend on the square root of the acceleration due to the restoring force, in one case gravity, in the other the van der Waals force. The speed of third sound also includes an additional factor of the superfluid density fraction, due to the fact that only the superfluid component of the liquid is moving.
Standing wave resonances: drum-head modes
We can solve the wave equation to find the shape of the standing waves. Our system has cylindrical symmetry, so a generic solution of the wave equation takes the form:
The complete solution will be a sum of terms like this one, with Bessel functions of the radial coordinate r, sines or cosines of the azimuthal coordinate j, and a time-dependence given for each mode m,n by the resonant frequency wmn. This is identical to the motion of the membrane on a drum-head.
Shapes of modes given by mode numbers (m,n):
(0,1)
(0,3)
(1,1)
Resonant Frequences
Boundary Conditions
The resonant frequencies are determined by the boundary condition on the wave. In 4He films, we have shown that the film obeys the boundary condition that the surface displacement is zero at the perimeter of the disk. This same boundary conditions best fits our data on 3He films, as well.
Imposing the condition that Z(R,j,t) = 0, where R is the radius of the substrate, gives the following dispersion relation, or expression for the resonant frequencies:
where Jm(amn) = 0. amn is called the nth zero of the Bessel function Jm, and a0n = 2.405, 5.520, 8.654, etc., a1n = 3.832, 7.016, etc.
Ratios of Bessel zeros
At any given temperature and film thickness, c3 should be constant, and R = 19 mm is a constant. So, the ratios of frequencies for different resonant modes should be the same as the ratio of zeros of the Bessel functions, amn. For instance, the frequencies of the peaks corresponding to the (0,1) and (0,2) modes should be in the ratio of 2.405 to 5.520. This feature allows us to use the ratios of the frequencies we observe in the spectrum to identify the modes.
Azimuthal symmetry
Our cell appears to have circular symmetry, including the substrate, the detector plate, and the drive plate. If this were true, we would not be able to drive or detect modes where m was not equal to zero. In fact, we observe modes corresponding to m=1, in both the 4He and 3He films we have studied. We believe that this is caused by a slight tilt of the upper electrodes with respect to the substrate. A 1 mrad tilt would put the plates 20 microns closer on one side than the other, producing an asymmetry sufficient to explain our observation of non-circularly symmetric modes.
The phenomena we are looking for doesn't exist unless the film is cooled to less than one milli-degree above absolute zero. We are able to cool the cell to a few hundred micro-degrees above absolute zero by using a combination of cooling techniques.
- The experimental cell is cooled down to 10 mK using a 3He-4He dilution refrigerator. The cell sits on top of a copper bar, called a nuclear stage,which has been magnetized in the 7 Tesla field of a superconducting magnet.
- The cell and the nuclear stage are then thermally isolated from the dilution refrigerator, so that the copper bar can be used as a nuclear demagnetization refrigerator. When isolated, the temperature of the copper nuclei is proportional to the strength of the applied magnetic field. Turning down the applied field cools the nuclei in the copper bar; they can easily be cooled below 100 mK.
- The copper nuclear stage then acts as an ice cube to cool the liquid helium inside the cell. Thermal contact between the two is enhanced by a silver-sintered heat exchanger built into the nuclear stage, which is good enough to allow the 3He to reach about 310 mK. The substrate has another heat exchanger built into it to help the 3He cool off the internal components in the cell.
- The temperature of the 3He is measured by pulsed NMR thermometry on platinum powder, which is immersed in the helium.
Below is a schematic of the cell with refrigeration components indicated:
Below is a schematic of the electrical circuit we used to measure the third sound spectra.
- Capacitance bridge: In the center of the diagram, you can see the internal components of the cell, the substrate, detector plate, and drive plate. The detector plate and substrate form a parallel plate capacitor, which is part of a capacitance bridge. When the average film thickness changes, it produces a capacitance change which shows up at the output of the pre-amplifier as a voltage change. In 1 sec, we can measure an average film thickness change of about 20 pm.
- Third sound drive: We apply a driving voltage V = Vdc + Vac sinwt between the substrate and the drive plate, where Vdc ~ 9 V and Vac ~ 0.35 V, sweeping the frequency from 0.2 to 5 Hz.
- Synchronous detection: We use a lock-in amplifier to measure the response of the film, at the drive frequency. We record the magnitude and phase of the response as a function of frequency; these are the "spectra" which reveal resonances of third sound being excited on the film's surface.
For more information
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1975.
D. R. Tilley and J. Tilley, Superfluidity and Superconductivity, Adam Hilger Ltd., Bristol, 1986.
P. M. Morse, Vibration and Sound, McGraw-Hill Book Company, Inc., New York, 1948.
F. Pobell, Matter and Methods at Low Temperatures, Springer-Verlag, Berlin, 1992.
O. V. Lounasmaa, Experimental Principles and Methods Below 1K, Academic Press, London, 1974.
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