This report is intended for those with a knowledge of Physics equivalent to a 1st year undergraduate level. It does not deal with the complex quantum mechanical explanations of the topic specifically for this reason.
This report deals with the phenomenon of superfluidity. It discusses the history and development of superfluids extensively, as well as their theoretical basis both in general and on an individual basis. Comparisons are drawn with superconductivity and the uses of both given.
Superfluidity is one of the strangest discoveries of modern Physics. It is amazingly counterintuitive: the most fluid things ever to exist do so as close to absolute zero as we’ve ever been able to get. In fact, only at absolute zero does a 100% superfluid truly exist… as excellent an example as you’ll ever find of the unimaginable leading to the unobtainable.
AN EXTENSIVE HISTORY
The story of superfluidity really begins with liquid helium at the very beginning of the twentieth century. In 1908 a Dutch physicist, Kamerlingh Onnes working in Leiden, first liquefied helium. Two years later he discovered that when helium was cooled below a temperature of 2.2K it would abruptly stop boiling. Onnes and Dana measured its specific and latent heat in 1923 and observed a strange discontinuity at around 2.2K and in 1927 Keesom and Wolfke too found that something very profound was happening with helium. They identified a transition between two phases at 2.1768K and named He I above it and He II below (the transition was known as the ‘lambda line’, because of the shape of the line). Work by Keesom and Clausius in 1932 also showed a strange anomaly in the heat capacity at 2.17K. All these findings were pointing towards a previously unknown physical effect at around 2.2K but next came something truly baffling.
In 1938 three groups were working on measurements of the viscosity of liquid helium. Both Allen and Meisner, and Kapitza performed experiments studying viscous flow, whereby helium was passed through narrow channels, and independently observed an unimpeded flow of helium out of the container, with a factor of 106 difference between the viscosities of He I and He II; To describe this behaviour Kapitza coined the term ‘superfluidity’, by analogy with superconductivity. By contrast, Keesom and MacWood performed measurements looking at viscous drag, using an oscillating disk immersed in the helium. They found a change in viscosity of only a factor of 10 when passing through the lambda transition.
This apparent contradiction was solved later the same year by work from Fritz London and Lev Landau. London began the process of unravelling the mysteries of the helium phenomena by realising that the transition from He I to He II corresponds to a Bose-Einstein Condensation (BEC – explained later).
London’s realisation stimulated Landau to put forward an explanation for the seemingly contradictory viscosity findings. Landau proposed modelling He II as two separate, non-interacting fluids: a normal component and a superfluid component. The normal component would be a normal Newtonian liquid with a finite viscosity, whereas the superfluid component would have zero viscosity and carry zero entropy. Above Tλ (the temperature of the superfluid phase transition) helium would consist entirely of normal component. Then, on cooling through 2.1768K, atoms would begin to convert from normal to superfluid, with the liquid being entirely superfluid at 0K. Looking again at the findings of Kapitza, Allen and Meisner, and Keesom and MacWood there was now no contradiction. In the flow experiments only the superfluid component passed through the narrow channels, so the viscosity found was that of the superfluid component alone. Similarly, only the normal, viscous component of the He II had been interacting with the oscillating disk, leading to the observation of a much larger viscosity value.
Landau went on to perform extensive research which led to a complete theory of quantum liquids at very low temperatures and published numerous papers devoted to the ‘Bose-type’ between 1941 and 1947. A little ahead of his time, he also worked on ‘Fermi-type’ liquids, from 1956 to 1958, of which 3He is one. His research yielded, amongst other things, two interconnected parts of the basis of understanding of superfluids: an explanation of their elementary excitations and the ‘Critical Velocity’.
In 1941 Kapitza saw that the heat capacity of liquid helium went, at low temperature, as the cube of the temperature – a characteristic of phonon excitations in solids – but varied exponentially above 1K on a factor Δ. This behaviour is characteristic of a dispersion curve (energy-momentum spectrum) with an energy gap (Δ). Landau suggested this was due to the dispersion curve having two branches – one for phonons and one for ‘rotons’, a new kind of excitation. With this new interpretation, the view of superfluids changed from one in which the normal and superfluid atoms were treated individually to one in which the superfluid effectively forms a background and the normal fluid is simply a collection of phonons and rotons, not corresponding to individual atoms. From this sprang the new concept of the critical velocity: the maximum speed a superfluid can flow and still remain superfluid. Initially it seems counterintuitive that superfluids, defined by their ability to have unimpeded flow, should have something intrinsic to them which limits their flow rate. This ‘speed limit’ arises because the production of rotons with energy equal to Δ destroys the superfluid and the higher the speed, the more rotons are created. For He II this critical velocity is calculated to be 60ms-1.
In the late 1940s further study into liquid helium by Lars Onsager of Yale, amongst others, revealed the existence of quantized vortices in the superfluid. A vortex in superfluid helium is much like a vortex, or eddy, in normal fluids: a circular flow around a central point. The difference here is that the flow is quantized, meaning that for a given distance away from the centre of each vortex only certain velocities are allowed i.e. there is a minimum velocity, then two times that velocity, then three times and so on; No intermediate values occur. This is explained in more detail later.
In 1954 Landau and Ginzburg Mean Field Theory described the thermal transport properties of the superfluid phase for the first time. It was a major step forward, despite predicting finite thermal conductivity at the superfluid transition temperature, something shown not to be the case in 1967.
So far 4He was the only known superfluid but that changed in 1972 with the work of Lee, Osheroff and Richardson. While looking for magnetic phase transitions in 3He Osheroff noted small anomalies in data taken from a melting sample of the solid (figure 1). As they had been looking for magnetic effects these little jumps in the data were interpreted as such but later, with further development of their technique, they found that these transitions corresponded to liquid phases, at 2.7mK and 1.8mK.
Once the data had been republished as evidence of superfluidity in 3He a group in Helsinki set about measuring viscosity. They found that the damping of a string oscillating in the fluid was reduced by a factor of 1000 when cooling from 2mK to 1mK. This confirmed the reported phase transition and showed superfluidity, although of a different kind to that found in 4He, as 3He is not a boson but a fermion. This difference is described in detail in sections below.
Later in the 1970s Anthony Leggett, working at the University of Sussex, formulated the first theory for superfluidity in 3He. This helped experimentalists to interpret their results and provided a framework for a systematic explanation. He received the 2003 Nobel Prize for his work.
Figure 1 – Note the change in slope of the curves at the points A and B. The curve is taken from a paper published by D.D. Osheroff, R.C. Richardson, and D.M. Lee in Physical Review Letters 28, 885 (1972), which gives the first description of the new phase transition in 3He.
By the 1990s all of the general theoretical concepts and rules for superfluidity had been created and many observed in the experiments performed with 4He and 3He. However, some of the ideas used as explanations for superfluid behaviour, such as Bose-Einstein Condensation, were yet to be observed. In 1995 Eric Cornell and Carl Wieman, and separately Wolfgang Ketterle, created BECs of rubidium and sodium, respectively. By a variety of laser cooling and trapping techniques they created physical systems within nanokelvins of absolute zero, and demonstrated the single quantum state idea that underpins the theory of superfluidity with macroscopic quantum interference effects.
Finally, in 2000 a paper was published in Science by Slava Grebenev et al announcing the discovery of a parahydrogen superfluid. These were excellent scientific feats with which to conclude just under a century of superfluid investigation.
There are several behaviours that can be said to be the ‘hallmarks of superfluidity’ simply because they do not occur anywhere else in nature. These are: frictionless film flow, superleaks, the fountain effect, thermal counterflow, persistent currents, quantized vortices, fast heat flow and second sound, and those shown by the Andronikashvili experiment. First we’ll take film flow and then each of the others in turn; A test tube lowered into a bath of He II will gradually fill by means of the frictionless flow of superfluid through a thin film of liquid helium coating the tube’s walls. Similarly, a full tube will empty via the film until the levels inside and out are equal. This thin film is similar to the meniscus of water in a glass – the edges are seen to be above the level of the rest of the liquid – except the severely reduced viscosity means that the film can extend much further upwards. These films are only a few atoms thick and are held together by van der Waals forces. In the case of the test tube the film reaches high enough that it covers all the vertical surface, acting as a medium for superfluid to flow in or out.
Second, superleaks. A superleak is a channel or medium through which superfluid can flow but not normal fluid. For example, an earthenware pot would hold He I but on cooling through Tλ the superfluid component of He II would pass straight through. This ability to flow unimpeded through materials that no other fluid can is unique.
Third, the fountain effect, a thermomechanical effect. Imagine two containers (A and B) of He II, in thermodynamic equilibrium, connected by a narrow superleak channel. If container A is heated then the temperature of the He II rises and some superfluid must change into normal fluid to take up the entropy, since the entropy of the superfluid is zero. The normal fluid cannot flow from A into B to redress the balance, because of the superleak, so superfluid must flow the other way instead. This results in liquid accumulating in container A which will then overflow. If the top of the container is suitably designed an incredibly thin helium fountain can form, as shown in figure 2.
Figure 2 – A helium fountain. The liquid helium in the bottle is heated by
means of infra-red radiation on small black balls, causing the temperature to rise.
Fourth, thermal counterflow. When He II is confined in a channel closed at one end with a heater, superfluid enters from the other open end and flows toward the heater. On reaching the heater, its temperature rises and normal fluid is created, which then flows back toward the open end. This setup is very similar to that of a jet engine and the normal fluid leaving the heater can be formed into a jet to turn a paddle wheel.
Fifth, persistent currents. As the superfluid component can flow without resistance, a flow of He II will persist for ever once established. Such experiments are usually made in rotating buckets or ring shaped containers. This effect is very similar to the persistent electrical currents observed in superconductivity. More parallels with superconductivity will be drawn later.
Sixth, quantized vortices. The formation of quantized vortices in superfluids seems counter intuitive; If a bucket of superfluid is rotated it would be expected that the fluid would remain stationary, due to the fluid’s lack of friction with the bucket. What actually happens is that both the superfluid and normal fluids will rotate, even though the superfluid is frictionless. This occurs due to the formation of quantized vortex lines; Vortex lines are atom-sized cores of normal fluid around which the superfluid flows. Below a critical velocity the superfluid will not rotate. At the critical velocity one vortex line appears on the axis of the bucket and an array gradually forms with increasing rotation rate.
Seventh, fast heat flow and second sound. The thermal conductivity of superfluids is exceptionally high, with thermal transfer many, many times faster than in normal fluids. This is the reason for the disappearance of boiling when helium is cooled through 2.2K; The heat transfer is fast enough to deliver the heat to the atoms located at the surface of the liquid which then evaporate, taking the heat with them, rather than bubbles forming. The heat transfer is even fast enough to allow for a phenomenon called ‘second sound’. Second sound is a temperature wave borne on variations in the normal and superfluid densities, as opposed to acoustic sound which is a fluctuation in total density. For second sound heaters behave like loudspeakers, and thermometers can act like microphones.
Eighth and finally, the Andronikashvili experiment. This was the definitive experiment measuring both the density and viscosity of the normal component and verifying the two-fluid model. It consisted of an oscillating stack of aluminium disks immersed in He II. The most important aspect of the experiment was the separation of the disks. By making the separation less than the viscous penetration depth – the distance over which a nearby moving surface causes motion in the fluid – the normal component was trapped and the superfluid component decoupled completely. Effectively this left the oscillating piece with an increased moment of inertia moving in superfluid only. The possibility of getting this result is integral to proving a superfluid.