Third Sound in Superfluid 3He
Results: New Information - c3and rs
Difficulties in the Third Sound Spectra
Third Sound Speed c3
Superfluid Density rs
For more information
Difficulties in the third sound spectra
Third Sound Spectra
Difficulties interpreting the third sound spectra:
- Q: For thinner films, and for higher temperatures for all the films, the Q's of the observed resonances are low, making it difficult to distinguish their resonant frequencies. - Based on the work of other groups, we hope to enhance the superfluidity in the film, and the resonances, by pre-coating the surface with an extremely thin layer of 4He. See Future Experiments page.
- Mode Structure - mixing: At low temperatures, for the thickest films, the mode structure agrees plausibly with our simple model. However, at higher temperatures, the identified modes shift relative to each other, merge with each other, or even disappear. We do not see a simple series of modes in the ratios of the zeros of the Jm Bessel functions as expected. -This may indicate some new physics in the film, or unanticipated properties of the system. These questions will be addressed in future research.
Despite these difficulties, we are able to use a simple model to interpret reasonably well the spectra we observe. The model incorporates the expected ratios of the Bessel zeros for the frequencies, is motivated by the temperature dependence expected from a basic Ginzburg-Landau theory, and produces values for the third sound speed and superfluid density which semi-quantitatively fit our expectations.
Third sound speed c3
What are the mode numbers for these resonances?
To determine the speed of third sound from these spectra, we need to identify the modes which correspond to the peaks we see in the data. The wave equation gives us the dispersion relation w = c3 k, or c3 = w / k. The boundary condition on the surface motion at the perimeter determines the wave-number, kmn = amn / R. So to determine the speed c3, we need to associate a peak which occurs at
some frequency f with the correct mode numbers (m,n) corresponding to it. Then, we'd have the speed:
Below is the graph of the spectra again, and a graph showing the frequencies of the peaks in the spectra as a function of temperature.
The black dots on the right-hand graph are the actual data points; the red and blue lines are the fit to the data of a simple model. As we look at the spectra, we observe that the peaks at lower frequencies appear to mix, or shift relative to each other, in complicated ways. In contrast, the highest frequency peak in this data-set shows no such complications; this is the case for all the film thicknesses which we studied. The simple model we give now allows us to fit as mooth curve to this peak, and then generate smooth curves for the other expected modes.
Ginzburg-Landau temperature dependence: There is a temperature Tc film, less than Tc bulk, at which the film first becomes superfluid. According to a simple Ginzburg-Landau theory, the superfluid density should rise linearly from this point, so that:
Since the temperature dependence in the speed of third sound waves, and therefore in the frequencies, comes only from the superfluid density, we find that the frequencies should vary as the square root of 1 - T / Tcfilm:
- Tcfilm is the measured temperature at which the film's motion in response to an applied force disappears. The value of the constant A03 is determined by fitting the data for the highest frequency peak to this function. The data fit this function well for each set of spectra taken, over the entire range of film thicknesses studied.
- Having obtained A03 from this fit to the (0,3) peak, we then can predict the frequencies for each of the lower peaks, simply by multiplying by the ratio of the Bessel zero amn to the zero a03. These predicted frequencies are graphed as lines on the graph above, red for (0,n) azimuthally symmetric modes, blue for (1,n) azimuthally asymmetric modes. The fit between the data and these predicted lines, particularly as the temperature falls, gives us confidence in our identification of the highest peak as the (0,3) mode.
- Finally, to determine the speed c3, we use the (0,3) mode in the formula above: c3 = f03* 2pR/a03. The results of this calculation are shown for all studied film thicknesses below.
Speed of third sound c3 vs temperature T for film thicknesses from 92 nm to 281 nm.
- Magnitude: The speed of third sound varies from 0 to 5 cm/sec, which is the range we expected from prior work on 3He films and third sound.
- Monotonic in T: For each film thickness, the speed of third sound increases monotonically as T decreases. This is what we expected, since the superfluid density increases monotonically as the film cools, and this causes the wave speed to increase.
- Complex dependence on d: The dependence of c3 on d is more complicated. (Thickness increases from black to red to blue points.) At T /Tcbulk = 0.6, the speed increases rapidly with d for thin films, then stalls at about 170 nm. At lower temperatures, the speed increases until about 17 nm, then turns around and begins to fall. The graphic below helps to explain this behavior.
Third sound speed c3 and the film thickness
As the film thickness increases, there are two competing influences which govern the wave speed.
- Van der Waals force: The greater distance from the film's surface to the substrate caused the van der Waals restoring force to fall, tending to reduce c3.
- Suppression of superfluidity near the substrate: The superfluid density is suppressed for the part of the film within a coherence length x of the substrate. A d becomes larger than x, less of the film is subjected to this effect, so the average superfluid density rises, tending to incread c3.
x= 65 nm / (1-T/Tc)1/2
Superfluid density rs
We can obtain the average superfluid density of the film by inverting the formula for the speed of third sound, to obtain:
where we know
- c3 from the data above
- d from measuring the film thickness using the parallel plate capacitors
- avdW, the van der Waals acceleration, from our measurement of the van der Waals constant a. (Recall that the film thickness is determined by the equilibrium between gravity and the van der Waals force, requiring that gh = ga3/d3.
We obtain the following data for the average superfluid density.
Average superfluid density of 3He film:
- T-dependence: As T falls, the average superfluid density rises for all film thicknesses.
- d-dependence: As d rises, the average superfluid density rises for all temperatures.
- Critical temperature: As the films get thicker, the temperature at which the film becomes superfluid rises.
- Numerical agreement with theory: For the thicker films, there is a reasonably good agreement between the data and the theoretical expectations. The discrepancy grows as the films thin, however, which is a subject of our current efforts.
For more information
D. R. Tilley and J. Tilley, Superfluidity and Superconductivity, Adam Hilger Ltd., Bristol, 1986.
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