Superfluids



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SUPERFLUIDS… flow without resistance

Nick Shuttleworth

Supervisor: Michael Forshaw

PHYS3ECP – Communication Skills
Electronic version available at: www.ucl.ac.uk/~zcapxa5/sf.doc

TABLE OF CONTENTS

  • STATEMENT OF INTENDED READERSHIP




  • ABSTRACT




  • MOTIVATING STATEMENT




  • AN EXTENSIVE HISTORY




  • PHENOMENA




  • 4He AND BEC




  • 3He AND BCS




  • SUPERFLUIDITY AND SUPERCONDUCTIVITY




  • CONCLUSION

STATEMENT OF INTENDED READERSHIP

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.



ABSTRACT

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.



MOTIVATING STATEMENT

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.



PHENOMENA

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.

HELIUM 4 AND BEC

The 4He superfluid transition (i.e. the transition from He I to He II), as explained by Fritz London, corresponds to a process known as Bose-Einstein Condensation – named for Satyendra Bose, who developed the basic theory for photons, and Albert Einstein, who extended the theory to particles.

BEC is a phase transition that occurs when bosons are cooled into the ground state. Bosons obey Bose-Einstein statistics and as such will occupy the most favourable energy state in a system regardless of whether there is another particle already in it. This means that on cooling towards absolute zero, the bosons start to collect together in the ground state. The lower the temperature falls the better the quantum mechanical wavefunction is at describing the particles and their behaviour. As the particles lose energy their de Broglie wavelengths grow larger until, at a sufficiently low temperature, they begin to overlap in space. The more the waves overlap the more likely they are to link together and form a single coherent wave, describing all of the particles simultaneously at a macroscopic scale and forming a Bose-Einstein Condensate.

Although 4He atoms are bosons (having integer spin) they do not undergo a BEC exactly as stated above because they still interact strongly with each other, even at such low temperatures. The difference between a Bose-Einstein Condensate and the superfluid state of 4He can be thought of as the same difference that exists between the more normal gas and liquid phases.



HELIUM 3 AND BCS

The immense interest in 3He as a superfluid stems from the fact that it was historically thought that only bosons could enter the superfluid state. The reason for this is that the understanding of superfluidity was based almost solely on comparisons to Bose-Einstein Condensation, and 3He atoms are not bosons but fermions. In the everyday, classical world this is a negligible difference and 3He and 4He are for all intents and purposes the same. However, at the quantum scale this difference – bosons having integer spin, fermions half-integer – changes everything.

Fermi-Dirac statistics determine the statistical distribution of fermions over the energy states in a system; They describe the probability of a given energy level to be occupied by a fermion. Fermions also obey the Pauli exclusion principle (PEP), which states that no more than one particle may occupy the same quantum state at the same time. On approaching temperatures of the order of the 3He superfluid transition, the energy of the particles is very low and the number of available quantum states is relatively small. In contrast to the 4He bosons, the 3He fermions cannot simply pile up in the lowest energy state because of the PEP and must be distributed across a range of energy states. This creates a problem for understanding 3He superfluidity in terms of BEC as described above. The solution of this problem comes from a comparison of the processes underlying superfluidity with those of superconductivity.

In 1957 John Bardeen, Leon Cooper and John Schrieffer developed a complete theoretical explanation for the phenomenon of superconductivity, for which they received the Nobel Prize in 1972. BCS theory, as it became known, demonstrated that the interaction between electrons and the lattice leads to the formation of bound pairs of electrons, Cooper pairs. Pairs of electrons can behave very differently from single electrons, which are fermions and therefore must obey the PEP. The pairs of electrons act more like bosons which can condense into the same energy level. This pairing of particles is the step which motivates an explanation of 3He superfluidity. By grouping in pairs the 3He atoms can effectively become bosons and can, as such, move into the lowest energy state together forming a sort of condensate.



SUPERFLUIDITY AND SUPERCONDUCTIVITY

Superconductivity is a phenomenon very closely related to superfluidity. At its most basic it is the conduction of electricity without resistance. It was discovered in 1911 by Heike Kamerlingh Onnes, the same man who first liquefied helium. On cooling mercury to 4K he found that its resistivity suddenly dropped to zero. Since then many elemental metals and exotic metal compounds have been found to become superconducting over a range of temperatures, and phenomena such as magnetic field expulsion seen.

There are many parallels between superconductivity and superfluidity. The first, and most obvious, is in their behaviours. Put simply, superfluidity is the flow of liquid without friction, and superconductivity is the flow of electric current without resistance. Resistance and friction are very similar things, friction being a kind of internal resistance, so the two phenomena look very much like the same thing observed from different directions. In fact, this is so much so that experiments performed on one can easily be adapted to the other; For example an electric current can be set up in a toroid of wire that will never decay. Putting a pure superfluid in a hollow toroid would show the flow never dying away. This phenomenon is known as a persistent current.

Second is their demonstrative powers. Both are seen as some of the strongest evidence for quantum mechanical effects being real, rather than merely mathematical artefacts. They are both examples of obvious macroscopic effects with obvious quantum mechanical causes.

Third is the temperatures at which they take place. Both are inherently low temperature phenomena because of their reliance upon low energy states. Superfluidity in 3He occurs at 2.4mK and in 4He at around 2.2K, whereas superconductor transition temperatures span a larger range, although still at low temperatures – type I superconductors range from rhodium at 0.3mK to lead at 7.2K; type II from AuIn3 at 50μK to (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33 at 138K.

Two connected differences exist between superfluids and superconductors: the prevalence of the phenomena and the extent of practical development. There is a big difference in the numbers of materials which exhibit superfluidity and superconductivity. There are over a hundred different materials which have clearly superconducting properties but the list for superfluidity is rather short. It is well known to exist for both 3He and 4He, and has been seen recently in parahydrogen. Then there are the formation of Bose-Einstein Condensates in alkali gases and electrons in superconductors. The BECs are of a very similar makeup to the 4He superfluid but are also quite distinct, if looked at closely, in that the BEC system is much simpler than that of 4He because of the lack of strong interatomic interactions. Electrons in superconductors are mentioned here because their behaviour does bear a strong resemblance to that of superfluid 3He, as a result of which superconductivity is sometimes considered as a special case of superfluidity in which the fluid components, in this case the electrons, are charged. Finally there are neutron stars, one of the more abstract areas affected by superfluid research. It is thought by some astrophysicists that the neutrons and protons form superfluids within the cores of neutron stars. This theory is thought to be testable based upon changes in the rotational periods of the stars due to changes in the superfluid coupling.

Practical development for superfluids has been very limited, in contrast to superconductors. Despite their being discovered at roughly the same time superconductors have a wide variety of modern uses, ranging from superconducting magnets in MRI scanners, Maglev trains and particle accelerators to microchips and uses in electricity generation, whereas superfluids are still limited to the lab. The main applications of liquid helium are in laboratory cooling experiments, although even these do not take advantage of the superfluid properties. At its best liquid 3He can be used to cool below 2 microkelvin when used in a dilution refrigerator. Dilution refrigeration takes advantage of the properties of a mixture of 3He - 4He to reduce temperature. 3He - 4He mixtures undergo a phase separation when cooled below 0.87K, resulting in two phases, a dilute phase (mainly 4He) and a concentrated phase (mainly 3He), between which the 3He atoms can be moved. The specific heat of a 3He atom is larger in the dilute phase than in the concentrated phase, so energy is used if an atom passes from the concentrated to the dilute phase. Pumping on the dilute phase will remove 3He, and because the dilute phase cannot have less than 6% 3He at equilibrium, 3He atoms from the concentrated phase cross the phase boundary to replace it. This crossing of the phase boundary can be thought of as a kind of evaporation, which removes heat from the system.

As well as the practical, superfluid helium has theoretical uses and is used as a tool to study other phenomena. In particular the formation of turbulence in the superfluid has recently been used to study how order can turn into chaos. This research may lead to a better understanding of the ways in which turbulence arises – one of the last unsolved problems of classical physics. Also, phase transitions in 3He have recently been used in attempts to simulate the formation of cosmic strings in the early universe. Some cosmologists believe that cosmic strings might have acted as the seeds for the development of early galaxies. These hypothetical strings are thought to have broken the symmetry of the original unified interaction and given rise to the four fundamental forces as they exist today.



CONCLUSION

From the above it may seem to some that superfluids are little more than a scientific point of interest, an extreme which, while noteworthy, is not of any real use. It is true that there are very few practical applications of superfluidity at present and those that there are are involved in things even more fantastical than superfluids themselves. But this is no reason for a lack of interest or of exploration. There are at present several good theories which bring this condensed matter extreme back into the clutches of our understanding, but it does not come without a fight, throwing up new and interesting phenomena all the time; An understanding is not an explanation.



REFERENCES

http://coffee.phys.unm.edu/dynamx/hel_phys.html
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http://www.yutopian.com/Yuan/TFM.html
http://cua.mit.edu/ketterle_group/Introduction_to_BEC.htm
http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents.html
http://nobelprize.org/physics/laureates/2001/phyadv.pdf
http://www.bbc.co.uk/dna/h2g2/A600968
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bcs.html
http://scienceworld.wolfram.com/physics/Fermi-DiracStatistics.html
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http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Rodney_Guritz%20Folder/properties.htm
http://superconductors.org/Uses.htm
[All references as accessed on 23/01/06]

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