SUPERFLUIDS… Flow Without Resistance Nick Shuttleworth Supervisor Michael Forshaw

superfluids… flow without resistance nick shuttleworth supervisor: michael forshaw phys3ecp – communication ski

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
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STATEMENT OF INTENDED READERSHIP
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ABSTRACT
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MOTIVATING STATEMENT
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AN EXTENSIVE HISTORY
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PHENOMENA
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4He AND BEC
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3He AND BCS
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SUPERFLUIDITY AND SUPERCONDUCTIVITY
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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
http://quench-analysis.web.cern.ch/quench-analysis/phd-fs-html/node48.html
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
http://en.wikipedia.org/wiki/Fermi-Dirac_statistics
http://www-kyoryu.scphys.kyoto-u.ac.jp/iutam/program/pdf/27-02.tsubota.pdf
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]