Superfluids

Type of physical science: Condensed matter physics

Field of study: Liquids

Superfluidity is a state of matter in which a fluid at extremely low temperatures, such as a principal isotope of helium or hydrogen, has zero viscosity and exhibits other properties of macroscopic quantum order not predicted from classical atomic and hydrodynamic theory. Although superfluidity is primarily an area of theoretical and experimental study in cryogenic condensed-matter physics, the electronic gas in a superconducting metal is also a superfluid, and theories of superconductivity and superfluidity are closely linked.

Overview

Helium 4 was first liquefied by the Dutch physicist Heike Kamerlingh Onnes in 1908. In 1924, Kamerlingh Onnes and Josef Dana continued the cryogenic studies of helium that first revealed behaviors not consistent with then-current theories of solid and liquid matter. Just below 2.2 kelvins, they found values of specific heat so abnormally high that they suspected major experimental error. However, in 1930, Willem Keesom and Klaus Clusius of the Leiden Low Temperature Laboratories confirmed Kamerlingh Onnes and Dana's results. In addition, when they plotted their measurements of density and specific heat versus temperature for helium 4, they discovered a tripartite transition whose specific heat peak occurs exactly at the same temperature as the density maximum. Because of its visual similarity to the Greek letter λ (lambda), this transition is known as the lambda point. Above the lambda point, helium 4 behaves as a dense classical gas; below it, its behavior is radically different and unexpected. Keesom distinguished two forms of helium 4: helium I, which is helium 4 above the lambda point, and helium II, below it.

Between 1930 and 1936, extensive research discovered additional unusual low-temperature thermal fluid behaviors. As Kurt Mendelssohn's book The Quest for Absolute Zero (1977) recounts, almost all key experimental and theoretical results were published in volume 141 of the British journal Nature in 1939. University of Cambridge physicist Rudolf Peierls and others not only reconfirmed the Leiden results but also discovered that net heat flow through helium II is not proportional to temperature difference; in other words, the lower the temperature difference, the greater the heat conductivity, which suggested that when supplying heat to helium II, liquid flows toward, not away from, the source of the heat. It was also noted that below the lambda point, helium II's fluid viscosity decreases significantly. In the same Nature issue, John Allen and Austrin Misener of the University of Cambridge and Soviet physicist Pyotr Kapitsa describe viscosity experiments that measured liquid helium II's friction by flow through fine capillary tubes and narrow plates. The experiments showed that the narrower the orifice through which the liquid had to pass, the less resistance it exhibited. Allen subsequently found that net flow in helium II is nonlinear in relation to temperature, with pressure gradients existing proportional to heat input into the fluid; these effects are called fountain pressure or superleaks. Kapitsa suggested the name "superfluidity" for the collective phenomena, as an analogy to previously discovered superconductivity.

The theoretical physicist Fritz London proposed the first theory of superfluidity. In the 1920s, Albert Einstein had shown that an ideal gas, obeying the alternative quantum-mechanical statistics of the Indian physicist Satyendra Nath Bose, undergoes unusual change when cooled to sufficiently low temperatures. Because fluid particles are subject to quantum as well as classical mechanical forces, kinetic energy does not vanish at absolute zero even in theory. This residual zero-point energy is a result of the fact that both positions and momenta of fluid particles are subject to Werner Heisenberg's uncertainty relation: Δx Δp = h/2π.

In quantum mechanics, there are two statistical species of particles. Bosons have symmetric wave functions, such that any number can occupy the same quantum state simultaneously. By contrast, fermions have antisymmetric wave functions with a rigid exclusion principle, developed by and named for Wolfgang Pauli, that precludes a given state from being occupied by more than one fermion. Examples of bosons include photons, phonons, and helium-4 atoms. Examples of fermions include electrons, neutrons, and helium-3 atoms. London revived the Bose-Einstein theory to assert that a temperature is eventually reached at which some gas particles condense, not in the usual position space to form a solid, but in the quantum-mechanical space of velocities, thereby forming a superfluid.

Shortly thereafter, Hungarian physicist Laszlo Tisza examined experimental results of disks oscillating in superfluid helium II (which, unlike capillary-flow tests, showed definite finite viscosity) and suggested that helium 4 be considered a mixture of two fluids, normal and superfluid, analogous to the two-conductor superconductivity model. Tisza also predicted, prior to later experimental confirmation, that heat impulses injected into superfluid helium II would travel through the liquid as phonons, rather than as ordinary acoustic pulses. London then demonstrated that superfluids do not carry entropy. In several modified forms—notably the BCS theory, proposed in 1957 by John Bardeen, Leon N Cooper, and John Robert Schrieffer—this two-fluid model of superfluidity became a basic framework for many experiments in low-temperature physics for decades afterward.

In the early 1940s, Soviet physicist Lev Landau developed a different approach to theoretical superfluidity. Landau described normal fluids via quantized phonons and rotons, representing local thermal equilibrium states. In developing specific hydrodynamic equations to describe superfluid motions, Landau concluded that, strictly speaking, superfluids are incapable of rotation. He further predicted what became known as the Landau critical velocity, which is a characteristic speed limit for objects moving through a superfluid. For speeds below the Landau critical velocity, the liquid retains its superfluid properties; above this velocity, it behaves in many respects like a normal fluid. Landau's theory predicted other subsequently observed phenomena, such as the slower wave known as "second sound," a temperature-entropy wave of constant pressure and density, in contrast to ordinary sound waves. Normal and superfluid components of a liquid oscillate in opposite antiphased relations, such that the fluid density remains basically constant.

In 1941, Anton Bijl, Jan de Boer, and Antonious Michels demonstrated that the maximum velocity possible for superfluid flow in a thin film of helium apparently obeys the quantum critical velocity condition of v_c ≈ (h/2π) / md, where h is the Planck constant, m is helium's atomic mass, and d is film thickness. London later proposed that the angular velocity of helium II is subject to similar quantum conditions, such that Ω < (h/2π) / mr², where Ω is the angular velocity and r is the helium container's radius. Further confirmation of Landau's theory was developed by Kapitsa in 1941, when he observed large temperature discontinuities between a copper cryogenic heater and its superfluid helium bath. This became known as the Kapitsa resistance, a thermal boundary resistance arising from the acoustic impedance mismatch between the contacting materials, and was associated with the anomalously high phonon transmission from solid metal into liquid helium.

In contrast to the descriptive phenomenological two-fluid model, the excitation model developed by Landau for superfluid helium considers the liquid below the lambda point to be analogous to a crystal at low temperatures. The helium is treated as an inert background medium in which a gas of phonons and rotons moves. In terms of the dispersion relation, which plots helium excitation temperature versus its momentum, phonons and rotons correspond to absolute and local dispersion-curve minima associated with the Landau critical velocity. Theories of superfluids usually seek to reproduce this dispersion curve as a form of verification.

The fact that critical velocities measured for helium II are usually much smaller than sound velocities in normal liquid has been used as evidence for the existence of quantized microscopic vortices. Quantized vortices also play a major role in explaining the apparent rotation of theoretically irrotational (nonrotating) helium II in a rotating container. For containers with sufficiently large angular velocities, both normal and superfluid components of helium 4 appear to rotate. What actually occurs, however, is that many quantized vortices, parallel to the container's rotation axis, cause the superfluid component to simulate solid-body rotation. Henry Hall and William Vinen first detected these vortex lines in 1956. In 1964, George William Rayfield and Frederick Reif examined vibrations of a fine submerged wire parallel to the rotation axis, thereby demonstrating circulatory vortex quantization. In 1979, Edward Yarmchuk and others developed an optical imaging technique for photographing individual vortices.

A major prediction of Landau's superfluid theory was the proposed existence of another superfluid transition in helium at an even lower temperature. Between the late 1940s and the end of the 1960s, many experiments sought to confirm or disprove Landau's prediction of a superfluid transition temperature of helium 3, variously thought to be 1.5 kelvins, 0.1 kelvin, and 0.005 kelvin. The transition temperature of 0.0026 kelvin for superfluidity in helium 3 was accidentally discovered by Douglas Osheroff, Robert Richardson, and David Lee of Cornell University in 1972.

Liquid helium 3's specific heat undergoes an unusual finite discontinuity at its superfluid transition that differs from the lambda-point transition of liquid helium 4 and is similar to the normal-to-superconducting transition for a metal outside magnetic fields. Theoretically, helium-3 superfluidity arises through the pairing of electrons into groups known as Cooper pairs. The nonzero internal angular momentum predicted by Landau's quantum fluid theory of Cooper pairs has been confirmed as the cause of many of helium 3's unusual properties. Another, related theoretical concept that applies to helium-3 superfluidity is that of the order parameter. The macroscopic order parameter is the large-scale analogue of a single-particle quantum-mechanical wave function. In helium 3, the superfluid order parameter comprises nine individual quantum states, in contrast to only one in superfluid helium 4. This order parameter results in two helium-3 superfluid phases, designated helium 3A and helium 3B.

In contrast to superfluid helium 4, superflow circulation in the helium-3A phase is not quantized. As a result, rotating samples of helium 3A exhibit rotary motions intermediate between those of ordinary classical viscous liquids and superfluid helium 4. The high-pressure superfluid helium-3A phase has been shown to be similar to ordinary liquid crystals. The lower-pressure helium-3B phase lacks the coupling and liquid-crystal properties of the 3A phase.

Experimental studies using nuclear magnetic resonance (NMR) have shown that helium-3B vortices are hundreds of times larger than those in superfluid helium 4. NMR studies have also shown that at temperatures equal to 0.6 of the superfluid transition temperature for helium 3, both temperature and gyromagnetic discontinuities occur as a result of quantum properties of helium 3B's half spin. Both helium 3A and helium 3B have characteristic frequencies that are associated with oscillations of the macroscale order parameter around its equilibrium position. These resonant ringings, measured by John Wheatley of the University of California and Richardson's Cornell University group, are called the Leggett effect, named after the British theoretical physicist Anthony Leggett, who first predicted them. Several other unexpected vortex properties observed in both the A and B phases of superfluid helium 3 differ from those of helium 4. A-phase vortices are multiply quantized, possessing continuously distributed values of vorticity. Vortices in the B phase are inherently magnetized, spontaneously possessing a dipole moment and unusual vortex properties.

The superfluid phase transition and behavior of thin films differ notably from those in bulk superfluid helium 3. Because the long-range order parameter cannot exist in two-dimensional samples, a new kind of quantum-mechanical order arises from binding quantized vortices of opposite charge. Thin films of superfluid helium 3 have offered physicists the chance to examine "gapless" anisotropic superfluids. In investigations of superfluid helium-3 thin films, local disorders have often been found to result from crystalline defects and roughness in the helium container walls. These problems can destroy Cooper pairings and effectively lower the superfluid transition temperature.

In superfluid helium 3 and superconducting metals, Cooper-pair formation produces a gap in this excitation spectrum, so that a specific energy is required to break up a Cooper pair. This gap, which is isotropic in superconductors, is directionally dependent (anisotropic) in superfluid helium 3A. The gap is evident in the A phase's specific-heat behavior, which does not decrease exponentially with temperature, as it does with superconductors. A rough surface, however, notably alters the number and energy distribution of helium-3 Cooper pairs. For roughness that is sufficiently large with respect to Cooper-pair dimensions, Cooper pairs can be broken using almost no energy. This gapless superfluid phase is characterized by helium 3's linear specific-heat decrease, as in ordinary helium 3.

Applications

Unlike cryogenic and higher-temperature superconductors, superfluids have been found to possess very few direct or immediate industrial application. In order to achieve, maintain, and measure the milli- and microkelvin temperatures necessary to create superfluids, however, numerous advances have occurred in cryogenic refrigeration, thermometry, and materials.

Soviet physicists Isaak Pomeranchuk (in 1950) and Dmitri Anufriyev (in 1965) developed a very-low-temperature cooling method for helium 3, using the adiabatic compression of a liquid-solid mix along its melting curve. With suitable precooling, such as that provided by dilution refrigerators, and the application of pressure, liquid helium 3 can be brought to the solid phase, resulting in an energy release known as Pomeranchuk cooling. The cooling power of this method is the total amount of heat absorbed when a given quantity of liquid helium 3 is converted into solid at constant temperature. Pomeranchuk cooling successfully achieves temperatures of about 0.001 kelvin (1 millikelvin).

Further cooling via adiabatic demagnetization employs spin states of salts and metals. Applying a magnetic field to weakly interacting magnetic dipoles yields reduced entropy, since magnetized states have greater order. This technique was originally performed using the magnetic dipoles of the electrons of paramagnetic salts. The magnetic field is applied under constant temperature, releasing heat that is then removed by placing the spin system in thermal contact with a helium-4 bath. When steady-state conditions are achieved, the magnetic state is isolated and the field is reduced under adiabatic conditions. Temperatures of less than 0.1 millikelvin have been achieved with this method. A more recent variation is nuclear adiabatic demagnetization, which uses the magnetic dipoles of nuclei rather than electrons and can achieve temperatures of less than 1 microkelvin (10^-6 kelvin). However, this requires extremely large magnetic fields, often three teslas and possibly as great as ten, which are not always readily achieved.

The vapor pressure of liquid helium 3 itself can be used to measure its temperature down to 0.3 kelvin. For measurement of lower temperatures the small heat capacities of most materials preclude techniques that themselves introduce significant heat to disturb the original system state. Semiconductor resistance thermometers can be used accurately to about 0.01 kelvin. Paramagnetic and nuclear-orientation methods can be employed at even lower temperatures. In NMR thermometry, which is accurate between 2 kelvins and 300 nanokelvins (or 3 x 10^-11 kelvins), continuous-wave or pulse magnetic fields are applied to platinum or copper nuclei. Although spin-lattice thermal relaxation times can be very long in the milli- and microkelvin range, NMR relaxation measurements accurately record temperatures existing immediately prior to measurement. Another milli- or microkelvin-applicable thermometer is gamma-ray anisotropy, which measures directional differences of cobalt and manganese radioactivity counts.

Perhaps the most direct application of superfluidity is in high-resolution ultrasonic microscopy (cryo-ultra-microtomy). Because sound-wave attenuation increases notably for short wavelengths of higher frequencies in conventional fluids, superfluid helium 4 at temperatures below 0.2 kelvin permits wavelength resolutions as small as 30 nanometers, which is over twenty times shorter than red light (620 to 740 nanometers). Superfluid helium 4 has negligible sound attenuation as a result of decreasing residual scattering of phonons by other thermal vibrations.

For finer resolution, superfluid helium 3 is used for temperatures to 0.3 kelvin. Still finer resolution is possible using dilution refrigeration for cooler, less attenuating superfluids. In the millikelvin temperature and gigahertz frequency ranges, sound scattering from helium zero-point motion is the last remaining source of attenuation, reducible by maintaining temperature and increasing superfluid pressure.

Superfluidity also plays a role in superfluid vacuum theory (SVT), a quantum-mechanical approach that treats the physical vacuum of the universe as a superfluid.

Context

The discovery, theory, and subsequent refinements of scientists' understanding of superfluidity have often involved complex and not always congruent interrelations of statistical and quantum mechanics, fluid dynamics, and solid-state physics, as well as mathematical theories such as the renormalization group. The fact that superfluidity underscores the reality of macroscopic quantum ordering is an important verification of quantum mechanics. Superfluidity provides yet another example of how semiformal or phenomenological theories often not only precede but also supplement more formal rigorous mathematical explanations.

Much research in superfluidity has focused on replicating superfluid behavior in matter other than helium. A 2010 study by Patricio Leboeuf and Simon Moulieras demonstrated that light, specifically photons, can be made to exhibit superfluid behavior as well.

Hydrogen has also been studied as a potential superfluid. The triple point of hydrogen is 13.8 kelvins, meaning that under normal circumstances, it, like the other elements apart from helium, solidifies before it can reach a temperature low enough to form a superfluid. Scientists were able to form a Bose-Einstein condensate from spin-polarized atomic hydrogen and hoped that a superfluid state might be achieved as well, but success remained elusive. One possible solution is parahydrogen, which is molecular hydrogen in which the two protons spin in opposite directions. In 2000, researchers at the Max Planck Institute for Fluid Dynamics in Göttingen reported that they had successfully cooled miniscule droplets of parahydrogen—fourteen to sixteen molecules each—to 0.15 kelvin, at which point they began exhibiting superfluid properties. More recently, in 2010, Hui Li and colleagues managed to achieve 85 percent superfluidity in clusters of molecular parahydrogen.

Principal terms

BOSE-EINSTEIN STATISTICS: a form of quantum mechanical statistics used to describe an assembly of bosons, which are particles whose spin is an even multiple of Planck's constant, h, in a noninteracting configuration at low temperatures

DISPERSION CURVE: a plot of the energies and frequencies of permitted excitations of an acousto-thermo-mechanical system, usually shown as energy versus inverse frequency (wave number)

FERMI-DIRAC STATISTICS: a form of quantum mechanical statistics used to describe an assembly of fermions, which are particles whose spin is a half-integral multiple of Planck's constant

LIQUID CRYSTALS: liquids that exhibit many physical properties characteristic of solid crystals, such as nonisotropic (directional) molecular orientation and optical birefringence

NUCLEAR MAGNETIC RESONANCE: a phenomenon of atomic nuclei whereby nuclei in a static magnetic field absorb radio-frequency energy at characteristic (resonant) frequencies, the precessional behavior of which can be used as an accurate measure of temperature and system order

PHONON: the quantum of acoustic-vibrational energy in a crystal or in low-temperature fluids; a quasiparticle that travels at the local velocity of sound

ROTON: a quantum of acoustic-vibrational energy, similar to but distinct from phonons, that occurs near the minimum of the dispersion curve for low-temperature helium 2

SPECIFIC HEAT: the quantity of heat required to raise a unit mass of uniform material one degree in temperature

SUPERCONDUCTIVITY: the state of zero electrical resistance in a material, frequently but not always at low temperatures

SUPERLEAK: a phenomenon in superfluids such as helium whereby a superfluid can easily pass or leak through a medium with fine pores or flow in a fountain-like fashion through capillaries that are resistive or impermeable to any normal fluid with viscosity

Bibliography

Benneman, K. H., and J. B. Ketterson, eds. Novel Superfluids. 2 vols. Oxford: Oxford UP, 2013. Print.

Bennemann, K. H., and J. B. Ketterson, eds. The Physics of Liquid and Solid Helium. 2 vols. New York: Wiley, 1976–78. Print. The most general introduction to pre- and post-superfluid helium-3 scenarios in low-temperature physics. Includes a review paper by D. Lee and R. Richardson that is particularly recommended for its conceptual and experimental background, helping readers who are confused by the variety of phenomenological and mathematical models for superfluidity.

Betts, David S. An Introduction to Millikelvin Technology. New York: Cambridge UP, 1989. Print. Provides an excellent introduction to refrigeration and thermometry in experimental superfluidity and superconductivity. Features many sections that consider the historical development of low-temperature scientific and engineering instrumentation. Extremely well referenced to applications as well as theoretical and experimental cryogenic literatures.

Galasiewicz, Zygmunt M., ed. Helium 4. New York: Pergamon, 1971. Print. Collected papers that are notable largely for their historical value. Covers the many rapid successions of discoveries about superfluid helium 4, London and Landau's theories, and the protracted search for the superfluid helium-3 transition. Includes a valuable editor's introduction that provides a good overview of the papers' contents and their historical import.

Grebenev, Slava, et al. "Evidence for Superfluidity in Para-hydrogen Clusters inside Helium-4 Droplets at 0.15 Kelvin." Science 289.5484 (2000): 1532–35. Print.

Kagan, M. Yu. Modern Trends in Superconductivity and Superfluidity. Dordrecht: Springer, 2013. Print.

Keesom, Willem H. Helium. Amsterdam: Elsevier, 1942. Print. Originally conceived as an intermediate-level (undergraduate science) monograph documenting the then-present state of the art; can be read profitably as a very detailed account of the history of cryogenics from the late nineteenth century through 1940. A good balance between recounting experimental and theoretical results. Supplemented by Lifshitz and Andronikashvili's work A Supplement to Helium (1959).

Khalatnikov, Isaak M. An Introduction to the Theory of Superfluidity. Trans. Pierre C. Hohenberg. New York: Benjamin, 1985. Print. A very comprehensive if rather advanced (undergraduate) account of all phenomenological and formal theories of superfluidity. Contains many useful illustrations, figures from original Soviet papers, and an extensive bibliography. Requires caution as a result of nonstandard nomenclatures and a biased technical emphasis. Should be read after McClintock, Meredith, and Wigmore's work and Richardson and Smith's work.

Kidnay, A. J., and M. J. Hiza, eds. Cryogenic Properties: Processes and Applications. New York: Amer. Inst. of Chemical Engineers, 1986. Print. A good general-audience treatment of most normal and some selected superfluid applications to physical chemistry and allied research. Offers a sketch of a number of industrial and applied research areas that may benefit from spin-offs of present and future superfluidity research results.

Leboeuf, Patricio, and Simon Moulieras. "Superfluid Motion of Light." Physical Review Letters 105.16 (2010): 1–4. Web. 10 Jan. 2014.

Li, Hu, et al. "Molecular Superfluid: Nonclassical Rotations in Doped Para-hydrogen Clusters." Physical Review Letters 105.13 (2010): 1–4. Web. 10 Jan. 2014.

Lifshitz, E. M., and Elevter L. Andronikashvili. A Supplement to Helium. New York: Consultants, 1959. Print. Updates Willem H. Keesom's account of physical knowledge about superfluidity and superconductivity for the post–World War II period, concentrating on Soviet theoretical and experimental progress. Largely of historical interest.

McClintock, P. V. E., D. J. Meredith, and J. K. Wigmore. Matter at Low Temperatures. New York: Wiley, 1984. Print. Gives a then-current and -complete English-language account of solid and fluid normal and superstate properties. Provides many illustrations of experimental results and initial efforts at finding industrial applications for these results.

Mendelssohn, Kurt. The Quest for Absolute Zero: The Meaning of Low-Temperature Physics. 2nd ed. New York: Halsted, 1977. Print. A classic historical account of most theoretical and experimental cryogenic work up to the late 1960s (prior to superfluid helium's discovery), written for the general reader. Includes many illustrations and special sections that help make technical aspects easy to grasp without prior background.

Richardson, Robert C., and Eric N. Smith. Experimental Techniques in Condensed Matter Physics at Low Temperatures. Redwood City: Addison, 1988. Print. An intermediate-to-advanced-level (undergraduate) reference that recapitulates the conceptual and technical development of low-temperature refrigeration, thermometry, and adjunct materials science beyond that given in previous standard accounts by Edward White and Oli Lounasmaa. Includes accounts of the accidental discovery of the superfluid helium-3 states by Richardson, Osheroff, and Lee and their subsequent work.

The Interpretation of Quantum Mechanics