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As a consequence of its quantum nature, su- perfluid helium supports six distinct sound modes (ordinal numbers zero through five); Izzy’s research group discovered two of them, and he exploited all six to elucidate the behavior of quantum fluids.
Helium is the only element that remains liq- uid at temperatures down to absolute zero. This is because helium is so light that it has a large quantum mechanical zero-point mo- tion and because the helium atom is spheri- cally symmetrical so that it has a very small interatomic attractive potential. This also means that liquid helium is very pure be- cause any contaminant would become a fro- zen solid and fall out of solution.
At atmospheric pressure, liquid helium
has a temperature of 4.2 K. If the pressure
is lowered, the liquid helium is cooled by
evaporation. At the “lambda temperature”
(Tλ) of 2.172 K (−271°C), the fluid under-
goes a second-order phase transition and a
macroscopic fraction of the fluid condenses
into a single macroscopic quantum state (i.e., a Bose-Ein- stein condensate); the quantum mechanical wave function of that ground state occupies the entire container. As the temperature is further decreased, a larger fraction of the fluid enters this superfluid ground state.
The Two-Fluid Model of Superfluidity
The phenomenological theory that describes the dynamics of superfluid helium was proposed by Lev Landau, who re- ceived the Nobel Prize in Physics in 1962 for his “two-fluid theory of superfluidity.” This theory treats the superfluid as two independent interpenetrating fluids: a normal-fluid component and a superfluid component. Each component has its own temperature-dependent mass density (ρn and ρs, respectively), with the total mass density being simply their sum (ρ = ρs+ρn). Above the superfluid transition tempera- ture (Tλ), the normal-fluid fraction is one (ρn/ρ = 1). As the temperature is decreased below Tλ, the superfluid fraction increases monotonically.
Both components carry mass and both move in response to pressure gradients according to ordinary fluid mechanics. The normal-fluid component and the superfluid component each move independently according to their own velocity
Figure 2. Schematic representation of four of the six sound modes that can be excited in superfluid helium. Red lines and arrows, velocity of the normal-fluid component (vn); blue lines and arrows, velocity of the superfluid component (vs). a: First sound is restored by a pressure difference (∆p). b: Second sound is restored by a temperature difference (∆T). c: Third sound is a surface wave in atomically thin helium layers of thickness (h) adsorbed on a solid substrate. The surface wave’s restoring force is pro- vided by the van der Waals attraction (ΔU) between the helium and the substrate. Only the superfluid can move (i.e., vn = 0) because the normal fluid is immobilized by its nonzero viscosity. d: Fourth sound propagates in porous media where the small pore size immobilizes the normal-fluid component, whereas the superfluid component can oscillate because it has no viscosity. Pressure differences (∆p) provide the restoring force for fourth sound. Fifth sound is not shown, but it also is present in porous media if the pressure is relieved by a free surface or in thick helium films. Temperature gradients provide the restoring force for fifth sound.
fields ( and , respectively). ThThe normal-flfluid component acts like an ordinary fluid; it has viscosity and can transport entropy. The superfluid component has no viscosity and be- ing described by a single macroscopic quantum state, it has no disorder and hence zero entropy.
Both components carry mass and both move in response to forces and, typically in acoustics, in differences in pres- sure (p) according to ordinary fluid mechanics. However, having the superfluid component in a macroscopic quan- tum ground state, with the normal-fluid atoms in thermally excited states, statistical mechanics can be involved in fluid motion; in particular, the superfluid component can move in response to differences in temperature (T) as well as in pressure. For ordinary fluids, temperature gradients lead to thermal energy diffusion, but for superfluids, temperature gradients can produce energy advection (i.e., mass flows). As demonstrated in the Second Sound section, this means that there can be “heat waves” (Holland et al., 1963) in su- perfluids as well as in sound waves.
Far more detail than can be included in this article is provid- ed in the excellent review that Izzy wrote for an Enrico Fer- mi Summer School held in Varenna, Italy (Rudnick, 1976).
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