## PACKARD GROUP

Third Sound in Superfluid He-3

WHAT IS THIRD SOUND?

Superfluids

Water flowing down a pipe experiences viscous drag, which causes it to lose energy and slow down. Liquid helium (both 3He and 4He) and the electrons in a superconductor, have the amazing property that they can flow without this energy loss. This superfluid behavior is extremely interesting, as much for its numerous practical applications as for the beauty of the theories which have been developed to explain it.

Two-fluid hydrodynamics

To understand third sound, we need a simple model of a superfluid. The two-fluid model says that we can think of the superfluid as being composed of two interpenetrating liquids, a superfluid component and a normal-fluid component.

The superfluid component is free to flow without viscosity, while the normal-fluid component experiences viscous drag. As we get colder, or "more superfluid," the fraction of liquid in the superfluid component, rs/r, gets larger and the fraction in the normal-fluid component, r n/ r, gets smaller. (Here rs and r n are the superfluid and normal-fluid densities, respectively, and r = r s + r n is the total density).

Each component moves at its own velocity, vs and vn, respectively, and the total current is just the sum of these the two mass currents, given by j =rs vs +r n vn.

Sound Modes in Bulk Liquid

Since the two components move independently, there are several new types of sound modes which are possible.

• First sound: the superfluid and normal fluid components move in-phase with each other.
• Second sound: the superfluid and normal fluid components move out-of-phase with each other.

These two modes are very different. First sound looks like the ordinary "sound" wave, a pressure or density oscillation. In contrast, second sound maintains a constant density while creating an entropy or temperature wave. The speeds of the two types of waves are very different; in particular, the speed of second sound depends strongly on the densities rs and rn.

Both first and second sound have been observed and studied in bulk samples of 4He and  3He.

Third sound

Instead of looking at waves in a bath of liquid helium, let's look at the motion of a thin film on a substrate.

Normal fluid near a wall, such as the substrate above, tends to move with the wall. The normal liquid only moves freely when it is farther from the wall than the viscous penetration depth. For our situation, the film thickness d is much, much less than the viscous penetration depth, so the normal-fluid velocity vn is zero. The normal-fluid component is clamped to the substrate.

However, the superfluid component is free to move and it does, oscillating parallel to the substrate. This creates surface waves, which are very analogous to shallow-water waves such as you might see at a beach. This wave motion is called third sound.

Third sound was first seen in 4He in 1962,1 and has been a valuable probe of superfluid films for the last three decades.2-10 Third sound had not been detected in 3He until this work.

Speed of a Third Sound Wave

The speed of a gravity wave travelling across the surface of a shallow pool of water depends on its depth, d, and the acceleration due to gravity g = 9.8 m/s2:

The gravitational acceleration g comes into play because when a point on the water's surface rises above the equilibrium height, gravity acts as the restoring force to pull it back down. For a third sound wave, the role of restoring force is played by the van der Waals force between the helium atoms and the substrate. The van der Waals force is electrical in nature, but attracts inert, neutral atoms. It is responsible for holding the film onto the substrate, and for these films it is much stronger than gravity.

For a third sound wave, we have an analogous expression:

Comparing the two expressions, we see that:

Comparing the two expressions, we see that:

• The depth, or film thickness, d, appears exactly the same in each case.
• The gravitational acceleration is replaced by the acceleration due to the van der Waals force. (This acceleration depends on the film thickness approximately as d-4.)
• There is an additional factor for the third sound speed, the average superfluid density of the film. This reflects the fact that only the superfluid component of the film is moving.

Approximate values for these parameters are known from previous work on 3He.11,12 For films of 3He which are about 100 nm thick, at about 400 mK, we expected the speed of third sound to be about c3 ~ 4 cm/sec, and designed our experimental apparatus accordingly.

What's it good for?

Third sound has proven itself to be an excellent probe of superfluid films, partly due to the fact that its speed depends on the superfluid density, a key parameter for understanding superfluid behavior. For example, it has been used to study:

•           "critical velocity" of flowing films2
•           persistence of super-currents4
•           phase transitions, including the Kosterlitz-Thouless 2-D transition3
•           the limits of superfluidity in extremely thin films7
•           vortices in two dimensional systems9

More generally, what these experiments and others have shown is that third sound can be used to do very sensitive, high-Q measurements on the superfluid state in 4He. Third sound is a dynamical probe, which should exist inherently in any superfluid. However, it had never been detected in films of superfluid 3He until now.

D. R. Tilley and J. Tilley, Superfluidity and Superconductivity, Adam Hilger Ltd., Bristol, 1986.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1975.

References

1.C. W. F. Everitt, K. R. Atkins, and A. Denenstein, "Third Sound in Liquid Helium Films." Phys. Rev. 136, A1494-A1499 (1964).

2.K. A. Pickar and K. R. Atkins, "Critical Velocity of a Superflowing Liquid-Helium Film Using Third Sound." Phys. Rev. 178, 389-399 (1969).

3.I. Rudnick, "Critical Surface Density of the Superfluid Component in 4He Films." Phys. Rev. Lett. 40, 1454-1455 (1978).

4.D. T. Ekholm and R. B. Hallock, "Studies of the Decay of Persistent Currents in Unsaturated Films of Superfluid 4He." Phys. Rev. B 21, 3902-3912 (1980).

5.F. M. Ellis, R. B. Hallock, M. D. Miller, and R. A. Guyer, "Phase Separation in Films of 3He -4He Mixtures." Phys. Rev. Lett. 46, 1461-1464 (1981).

6.J. G. Brisson, J. C. Mester, and I. F. Silvera, "Third Sound of Helium on a Hydrogen Substrate." Phys. Rev. B 44, 12453-12462 (1991).

7.P. J. Shirron and J. M. Mochel, "Atomically Thin Superfluid Helium Films on Solid Hydrogen." Phys. Rev. Lett. 67, 1118-1121 (1991).

8.K. S. Ketola, S. Wang, and R. B. Hallock, "Anomalous Wetting of Helium on Cesium." Phys. Rev. Lett. 68, 201-204 (1992).

9.F. M. Ellis and L. Li, "Quantum Swirling of Superfluid Helium Films." Phys. Rev. Lett. 71, 1577-1580 (1993).

10.P. A. Sheldon and R. B. Hallock, "Third Sound and Energetics in 3He-4He Mixture Films." Phys. Rev. B 50, 16082-16085 (1994).

11.J. Xu and B. C. Crooker, "Very Thin Films of 3He: A New Phase?" Phys. Rev. Lett. 65, 3005-3008 (1990).

12.J. G. Daunt, R. F. Harris-Lowe, J. P. Harrison, A. Sachrajda, S. Steel, R. R. Turkington, and P. Zawadski, "Critical Temperature and Critical Current of Thin-Film Superfluid 3He." J. Low Temp. Phys. 70, 547-568 (1988).