Third Sound in Superfluid 3He
Third Sound in Superfluid 3He
Results: Third Sound Spectra
In order to understand the spectra we observe more clearly, let me first describe the spectra we obtained from a simple model of the 3He film in our cell.
Wave equation with driving term and damping:
- The left-hand side of this is just the wave equation, as we derived on a previous page.
- To this, we add a term, F(r, j, t) which represents the driving force applied. In this case, the driving force is provided by the voltage applied between the drive plate and the substrate, and we can easily model the forcing term which results.
- We also add a term to represent the dissipation. There are several damping mechanisms which might a priori be responsible for the dissipation observed in the spectra for 3He third sound (this is itself an interesting question we are currently studying). For this model, we use a generic linear damping term, where b can be adjusted to represent a particular mechanism.
Solve for the response recorded by our detector:1. Express the known driving force in terms of the eigenfunctions of the problem, those drum-head modes which are solutions of the wave equation in cylindrical coordinates. This determines the strength of the driving force on each particular resonant mode.
2.Put in a particular damping form, again expressed in terms of individual modes.
3.Solve for the film motion for each mode separately, then find the total motion of the film by adding all these motions together.
4.Calculate the response of the detector due to the total motion of the film.
Results of the model:
A typical result of the model is shown below. On the figure, I've noted which resonant drum-head mode (m,n) is responsible for producing each peak. (Here I've chosen c3 = 5 cm/sec, b = 1 Hz, and a tilt of 1 mrad to provide sensitivity to the azimuthally asymmetric modes.)
Things to note:
- The fundamental (lowest frequency) mode has the largest amplitude.
- The amplitudes of the peaks falls off as you go to higher frequency, with the radial (0,n) modes falling in sequence and the azimuthal (1,n) modes likewise falling in sequence.
- The shape of the spectra is distinctive, with a high-Q fundamental mode, a sharp rise to the (1,1) mode, then another gradual rise to the second radial mode (0,2).
Experimental data:
Here I show a spectra taken for a 233 nm thick 3He film, at 320 mK.
Compare with the model spectra:
- There is a qualitative similarity between the experimental spectra and the model above, which helps us to identify the modes responsible for each peak
- The data shows a strong fundamental mode, corresponding to the (0,1) mode in the model.
- The shape of the spectra above the (0,1) mode, with the sharp rise into the second peak and the gentle curve from there to the third peak, suggests that they are respectively the (1,1) and (0,2) modes in the model.
- The shapes of the higher frequency modes does not match as well, but the amplitudes behave as expected, falling from (0,1) to (0,2) to (0,3), and from (1,1) to (1,2).
Below I show a set of spectra at different temperatures, taken for the same 233 nm thick 3He film. The spectra are offset from each other for clarity, with the temperature ranging from 790 mK at the bottom, to 320 mK at the top. The temperature spacing is roughly even.
Conclusions from this data:
- There are clearly well-defined surface-wave modes.
- The average surface displacement is less than 3 nm, as expected.
- The modes fall in the expected frequency range, about 1-4 Hz.
- As the temperature rises (going down the figure), the frequencies of the modes fall, asexpected since the speed is proportional to the square root of the superfluid density, which, falls as T rises.
- As the temperature reaches some value near 790 mK, the motion dies out completely. (The suppression of the critical temperature, below the critical temperature Tc = 930 mK in a 3He bath, is well-known for films or slabs of 3He.)
We've taken similar spectra for a range of film thicknesses from 90 nm up to 280 nm, all showing comparable features.This data is the first evidence for the existence of third sound in films of superfluid 3He.
D. R. Tilley and J. Tilley, Superfluidity and Superconductivity, Adam Hilger Ltd., Bristol, 1986.
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