superfluidity


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superfluidity,

tendency of liquid helium below a temperature of 2.19°K; to flow freely, even upward, with little apparent friction. Helium becomes a liquid when it is cooled to 4.2°K;. Special methods are needed to cool a substance below this temperature, which is very near absolute zero (see Kelvin temperature scaleKelvin temperature scale,
a temperature scale having an absolute zero below which temperatures do not exist. Absolute zero, or 0°K;, is the temperature at which molecular energy is a minimum, and it corresponds to a temperature of −273.
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; low-temperature physicslow-temperature physics,
science concerned with the production and maintenance of temperatures much below normal, down to almost absolute zero, and with various phenomena that occur only at such temperatures.
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). When the temperature reaches 2.19°K;, the properties of liquid helium change abruptly, so much so that ordinary helium is known as helium I and helium below this temperature is known as helium II. The transition temperature between helium I and helium II is known as the lambda point because a graph of certain properties of helium takes a sharp turn at this temperature and resembles the Greek letter lambda (&Lgr;). Liquid helium II flows easily through capillary tubes that resist the flow of ordinary fluids (see capillaritycapillarity
or capillary action,
phenomenon in which the surface of a liquid is observed to be elevated or depressed where it comes into contact with a solid. For example, the surface of water in a clean drinking glass is seen to be slightly higher at the edges, where
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) and a Dewar flaskDewar flask
[for Sir James Dewar], container after which the common thermos bottle is patterned. It consists of two flasks, one placed inside the other, with a vacuum between. The vacuum prevents the conduction of heat from one flask to the other.
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 filled with helium II from a larger container will empty itself back into the original container because the liquid helium flows spontaneously in an invisible film over the surface of the flask. The behavior of helium II can be partially understood in terms of certain quantum effects (see quantum theoryquantum theory,
modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles; the quantum theory and the theory of relativity together form the theoretical basis of modern physics.
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). Helium stays a liquid down to absolute zero because its zero-point energy is such that it cannot become a solid without giving up an amount of energy that is less than that allowed by the quantum theory. Similarly, quantum restrictions keep helium II from behaving like a normal fluid because the energy interactions associated with friction and viscosity in normal fluid flow involve amounts not possible for helium II.

Superfluidity

The frictionless flow of liquid helium at low temperature; also, the flow of electric current without resistance in certain solids at low temperature (superconductivity).

Both helium isotopes have a superfluid transition, but the detailed properties of their superfluid states differ considerably because they obey different statistics. 4He, with an intrinsic spin of 0, is subject to Bose-Einstein statistics, and 3He, with a spin of ½, to Fermi-Dirac statistics. There are two distinct superfluid states in 3He called A and B.

The term “superfluidity” usually implies He II or the A and B phases of 3He, but the basic similarity between these and the “fluid” consisting of pairs of electrons in superconductors is sufficiently strong to designate the latter as a charged superfluid. Besides flow without resistance, superfluid helium and superconducting electrons display quantized circulating flow patterns in the form of microscopic vortices. See Bose-Einstein statistics, Liquid helium, Quantized vortices, Second sound, Superconductivity

Superfluidity

 

the special state of a quantum fluid wherein the fluid flows through narrow slits and capillary tubes without friction. The part of the fluid that flows in this way possesses zero entropy.

For many years the only representative of the family of superfluids was assumed to be liquid helium 4He. It becomes a superfluid at the temperature Tλ = 2.17°K, when the saturated vapor pressure is ps = 37.8 mm Hg. Superfluid 4He, which is known as He II (seeHELIUM), was discovered by P. L. Kapitsa in 1938. Between 1972 and 1974, liquid 3He was discovered to exhibit superfluidity at temperatures below Tc = 2.6 X 10–3°K on its melting curve. The transition from the normal 4He and 3He fluids into the superfluid state is a second-order phase transition.

A superfluid cannot be described as a nonviscous liquid because experiments on the torsional oscillations of a disk immersed in He II have shown that the damping of the oscillations at temperatures close to Tx, which is called the lambda point, differ little from the damping of similar oscillations in He I, which does not exhibit superfluidity.

Theory of He II superfluidity. L. D. Landau provided a theoretical explanation of the superfluidity of He II in 1941. The Landau theory, which is sometimes called two-fluid hydrodynamics, is based on the notion that at low temperatures the properties of He II as a weakly excited quantum system are due to the presence of elementary excitations, or quasiparticles. According to this theory, He II can be regarded as consisting of two interpenetrating components—a normal component and a superfluid component. At temperatures not too close to Tλ, the normal component is a set of two kinds of quasiparticles —phonons, or sound quanta, and rotons, which are quanta of short-wavelength excitations with a higher energy than that possessed by phonons. When T = 0°K, the density of the normal component ρn = 0, since at this temperature a quantized system is in the ground state and excitations (quasiparticles) are absent. At temperatures between absolute zero and 1.7°-1.8°K, the set of elementary excitations in 4He can be treated as an ideal quasiparticle gas. At temperatures still closer to Tλ, the ideal-gas model becomes inapplicable because quasiparticle interactions are appreciably more intense. The interaction of the quasiparticles with one another and with the walls of the container is responsible for the viscosity of the normal component.

Figure 1. Diagram illustrating the two-fluid model of He II: (T) absolute temperature, (ρn/ρ) ratio of the density of the normal component to the density of He II

The other component of He II, the superfluid component, has no viscosity and therefore flows freely through narrow slits and capillary tubes. Its density ρs = ρ - ρn, where ρ is the density of the liquid. When T = 0°K, ρs = ρ. As the temperature increases, the concentration of quasiparticles increases, and, consequently, ρs decreases. When T = Tλ, ρs vanishes. As Figure 1 shows, superfluidity disappears at the lambda point. According to the Landau theory, the liquid also ceases to be a superfluid when its flow velocity exceeds a certain critical value, at which rotons begin forming spontaneously (seeQUANTUM FLUID). When the rotons are formed, the superfluid component loses momentum equal to the momentum of the emitted rotons and consequently slows down. The experimental value of the critical velocity, however, is substantially smaller than the value required by the Landau theory for the destruction of superfluidity.

From the microscopic point of view, the appearance of superfluidity in a liquid consisting of atoms with integral spin (bosons), such as 4He atoms, is associated with the transition, when T < Tλ, of an appreciable number of atoms into a state of zero momentum. This phenomenon is known as Bose-Einstein condensation, and the set of atoms that have gone into the new state can thus be called a Bose-Einstein condensate. As N. N. Bogoliubov showed in 1947 and 1963, the existence of atoms in He II that possess different characteristics of motion (the condensate atoms and the atoms not in the condensate) leads to the Landau two-fluid hydrodynamics. The state of all particles in the Bose-Einstein condensate is described by the same quantum-mechanical wave function ψ = n01/2e, where n0 is the density of the condensate and Φ is the phase of the wave function. In the case of weakly interacting atoms n0 coincides with ρs. Because of strong atomic interactions, n0 in He II is only a few percent of ρs when T = 0°K. The relation between the velocity vs of the superfluid component and Φ is given by the equation vs= (ħ/m)∇Φ, where is the gradient of the function Φ, m is the mass of the 4He atom, and ħ = h/2 π (h is Planck’s constant). Thus, the superfluid component moves irrotationally (seePOTENTIAL FLOW) and, consequently, does not experience resistance from the objects past which it flows or from the walls of the channel or container.

According to theoretical results obtained by L. Onsager in 1948 and R. Feynman in 1955, the irrotational nature of the flow of the superfluid component is violated at the axes of quantized vortices. These vortices differ from vortices in ordinary fluids (seeVORTICAL MOTION) in that the circulation about the vortex axis is quantized. The quantum of circulation is equal to h/m. The interaction between the superfluid and normal components of a superfluid occurs through the quantized vortices. This interaction results in a weak but finite slowing of the flow of the superfluid in a closed channel. At some velocity of the superfluid component relative to the velocity of the normal component or to the walls of the container, quantized vortices begin forming intensely enough to cause the superfluidity to disappear. According to this theory, superfluidity disappears at velocities that are considered lower than those predicted by the Landau theory and that are much closer to the actual value of the critical velocity. Quantized vortices have been observed experimentally in rotating containers containing He’ll. Moreover, ring-shaped quantized vortices have been detected in experimental studies of ions injected into He II.

Superfluidity of 3He. Under certain conditions, superfluidity can also be displayed by a system of atoms with half-integral spin. Such atoms are fermions, and the system is a Fermi liquid. Superfluidity appears when the attractive forces between the fermions lead to the formation of Cooper pairs, which are bound pairs of fermions (seeCOOPER EFFECT). Cooper pairs have integral spins and therefore can form a Bose-Einstein condensate. This type of superfluidity is exhibited by electrons in some metals; in this case, the superfluidity is known as superconductivity. A similar situation occurs in the case of liquid 3He, whose atoms have spin 1/2 and form a typical quantized Fermi liquid. The properties of the Fermi liquid can be described as the properties of a gas of fermion quasiparticles whose effective masses are approximately three times as large as the mass of a 3He atom. The attractive forces between the quasiparticles in 3He are very small, and only at temperatures of the order of a few millidegrees Kelvin are conditions in 3He suitable for the formation of quasiparticle Cooper pairs and the appearance of superfluidity.

Figure 2. Phase diagram for 3He at low temperatures: (7) absolute temperature in millidegrees Kelvin, (p) pressure in atmospheres

The discovery of superfluidity in 3He was fostered by the development of efficient methods for attaining low temperatures—specifically, the Pomeranchuk effect and magnetic cooling. These techniques were used to clarify the characteristic features of the 3He phase diagram at ultralow temperatures (Figure 2). Unlike the 4He phase diagram (see Figure 1 in HELIUM), the 3He phase diagram exhibits two superfluid phases—A and B. The transition of the normal Fermi liquid into the A phase is a second-order phase transition. In other words, the heat of transition is equal to zero. The Cooper pairs formed in the A phase have spin 1 and nonzero angular momentum. The A phase may exhibit regions in which the spins and moments of all pairs have the same relative orientation. For this reason, the A phase is an anisotropic liquid. In a magnetic field, the A phase splits into the two phases A1 and A2, each of which is also anisotropic. The transition from the superfluid A phase into the superfluid B phase is a first-order phase transition. The heat of transition is ~ 1.5 × 10-6 joule/mole, or ~15 ergs/mole. The magnetic susceptibility of 3He drops suddenly at the A → B transition and continues to decrease as the temperature is lowered. The B phase appears to be isotropic.

Effects accompanying superfluidity. Ordinary, or first, sound, which consists in periodic density variations, can be propagated through a superfluid as in any gas or liquid. Unique to a superfluid, however, is the propagation of second sound, which is sound in a gas of quasiparticles. Second sound involves fluctuations in the density of the quasiparticles and, consequently, in temperature.

A superfluid has an anomalously high thermal conductivity. This thermal conductivity is accounted for by convection—the heat is transferred by the macroscopic motion of the quasiparti-cle gas.

When He II is heated in one of two containers connected by capillary tubes, a pressure difference arises between the containers. This phenomenon is known as the thermomechanical effect and has been attributed to an increase in the concentration of quasiparticles in the container with the higher temperature. Because the narrow capillary tube does not pass the viscous normal component, the quasiparticle gas acquires an additional pressure similar to the osmotic pressure of a solution.

The reverse phenomenon, known as the mechanocaloric effect, also exists. When He II flows rapidly out of a container through a capillary tube, the temperature within the container increases (since the concentration of quasiparticles increases), and the effluent helium becomes cooler.

The superfluid helium film that forms over the solid walls of the helium’s container also has interesting properties. For example, the film can equalize the He II level in containers sharing a common wall.

REFERENCES

Kaptisa, P. L. Eksperiment, teoriia, praktika. Moscow, 1974.
Khalatnikov, I. M., and 1. A. Fomin. “Sverkhtekuchest’ i fazovye perekhody v zhidkom gelii-3.” Priroda, 1974, no. 6.
Khalatnikov, I. M. Teoriia sverkhlekuchesti. Moscow, 1971.
Kvantovye zhidkosti: Teoriia: Eksperiment. Moscow, 1969.
Mendelssohn, K. Na puti k absoliuinomu nuliu. Moscow, 1971. (Translated from English.)
Keller, W. E. Helium-3 and Helium-4. New York, 1969.

G. E. VOLOVIK

superfluidity

[¦sü·pər·flü′id·əd·ē]
(cryogenics)
The frictionless flow of liquid helium at temperatures very close to absolute zero through holes as small as 10-7 centimeter in diameter, and for particle velocities below a few centimeters per second.
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