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Volume 24, Number 13
PHYSICAL REVIEW LETTERS
20 November 2003
Superfluid Helium
Gray O’Byrne
Department of Physics, University of Ottawa, Ottawa Ontario, K1S 5M2 Canada
(Received 27 April 2003)
When one lowers the temperature of most substances the attractive forces between atoms eventually
prevail over the repulsive forces and a solid is formed. In helium however, these attractive forces are too
weak for this to occur. At 4.2K helium will become a liquid, but at atmospheric pressure it will never
solidify. This allows us to cool helium in the liquid phase to temperatures much lower than other
substances. At these low temperatures liquid helium develops some interesting and unique properties such
as the lack of viscosity and near perfect thermal conductivity. Kapitza, one of the leading researchers in the
field coined the word "superfluidity" to describe this behavior. The uniqueness of helium to demonstrate
these properties, as well as the quantum effects allowing them will be introduced. Also, differences and
similarities between the isotopes of helium as well as some history of superfluidity will be
discussed.[S0031-9O07(03)02379-X]
PACS numbers: 82.25.+z, 05.40.+j.83.10.Nn, 87.15.He
In 1908 Heike Kamerlingh Onnes reduced the
temperature of helium below 4.2K and so, for the first
time, was able to liquefy it.[1] (For this work, he received
the Nobel Prize in 1913) Onnes continued to lower the
temperature of helium hoping that it would eventually
solidify. His efforts however, were doomed to failure. The
fact is that helium will never solidify at atmospheric
pressure. The cause of this is Quantum Mechanics and the
Uncertainty Principle.[2] As one lowers the temperature of
a substance its atoms slow down and in a classical model
they will reach a full stop at absolute zero. In other words,
the substance will reach a state of zero kinetic energy at
absolute zero. This however, is not the full picture. The
uncertainty principle tells us that we cannot have stopped
particles neatly placed in a substance. Even at absolute
zero all particles must have a small, minimal uncertainty
on their momentum. This momentum is equivalent to these
particles moving around a little. This means they have a
small amount of kinetic energy, even at absolute zero. We
call this energy the zero-point energy. Helium is the only
substance in which the zero-point energy has a
considerable effect. The combination of the small mass of
helium and the week Van der Waals forces between atoms
allow helium to remain in the liquid phase even at its zeropoint energy. Helium actually can be solidified, as seen in
Figure.1, but it is only at high pressures (about 25
atmospheres) that its atoms are finally pushed close
enough together to do so.
I would like to note that there are actually two stable
isotopes of helium. It was 4He that was used by Onnes and
by others in early experiments with liquid helium. 3He, as
we will see later, has different properties.
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FIG. 1. Phase Diagram of 4He. One can note the
absence of a triple point which leads to the liquid phase
continuing all the way down to absolute zero.
Despite Kamerlingh Onnes reducing the temperature of
liquid helium to below 1k, it was not until 1937 that a
second phase transition was observed. This unusual
occurrence was first recorded by Peter Kapitza who noticed,
among other things, that helium below 2.17K could easily
flow through the smallest pores, even ones too small to
permit regular helium. Kapitza coined the word
“superfluid”[3] to describe this new state. He received one
half of the Nobel Prize in 1978, in part due to this
discovery. Many other discoveries were made concerning
liquid helium in following years. One of these was the
behaviour of the specific heat of the liquid (Figure.2.) We
can clearly see that the specific heat has an asymptote at
2.17K. This temperature is called the lambda point due to
the resemblance of the graph to the Greek letter lambda (.)
© 2003 The American Physical Society
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Volume 24, Number 13
PHYSICAL REVIEW LETTERS
The lambda point marks the ‘second order’ phase transition
in liquid helium. The properties of the fluids on either side
of the lambda point are so different that the liquid above
2.17K is referred to as He I, and the fluid below 2.17K is
referred to as He II.
FIG. 2. Lambda Point in 4He. This marks the phase
transition to He II.
He II has many unique properties. In their early
experiments J.F.Allen and A.D.Misener [4] measured an
upper limit of the viscosity of He II at 1.07K to be  =
4*10-9. Compared to hydrogen gas (previously thought to
be the least viscous substance) this is around 104 times less
viscous. J.F.Allen accidentally made another important
discovery by shinning a flashlight on his apparatus. When
illuminated by his flashlight the helium shot up and out of
his apparatus. This is now known as the fountain effect.[5]
Perhaps the most easily viewed property of He II is its near
perfect thermal conductivity. This is because helium
simply stops boiling when brought below the lambda point.
Boiling normally occurs because localised parts of a liquid
get heated enough to vaporise. These pockets do not form
in He II because any added heat is instantly and evenly
distributed throughout the liquid. Instead we have smooth
uniform evaporation from the surface. I would at last like
to mention two, somewhat more bizarre properties of
superfluid helium. The first is that it cannot be kept in an
open vessel because it will simply creep up the sides as a
film and climb out. Left alone, an open vessel of He II will
actually empty itself. The second is that when we rotate a
container of He II the liquid does not spin with it. Instead a
pattern of whirlpools called vortices are formed.[6] These
vortices are very small. Their diameters measure only 1000
Å to 100,000 Å.
Table.1 The Two-Fluid Model
Normal fluid
Superfluid
Density
N
s (< N)
Viscosity
N
N = 0
Entropy
SN
SN = 0
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20 November 2003
To better understand He II, I will introduce the two
fluids model. In this model He II is a homogeneous mixture
of a superfluid and a normal fluid. The properties of each
are summed up in Table 1. At the lambda point He II is
100% normal fluid, but at 1K it is already almost entirely
superfluid as shown in Figure 3. We cannot, however, think
of this as a mixture in the normal sense. For example we
cannot separate the two fluids. We cannot even really say
which part is superfluid and which normal. He II simply has
properties of both. Heating He II has the expected (and
boring) effect of converting superfluid to liquid helium, but
this leads to some interesting phenomena. As mentioned
above He II is homogeneous, and so, when locally heated,
superfluid helium from the non heated sections will
immediately flow to the heat source, effectively causing
cooling. This is why He II has such a good thermal
conductivity (several hundred times that of pure copper!)
[7] This also explains the fountain effect, when Jack Allen
pointed his flashlight at his experiment he effectively heated
a section of it and caused liquid to rush to the lighted spot.
This section soon filled and the only place left for the
helium was up and out. Another demonstration of this flow
of superfluid is the production of heat waves. By placing a
heater and a thermometer at opposite ends of a vessel filled
with helium II and applying an AC current to the heater, we
can measure heat waves in our vessel as the superfluid and
normal fluid rush to either side of the vessel.
Fig.3 Densities of the ‘super’ and normal fluids in
He II as determined by viscosity measurements.
So far we have taken the superfluidity as an
experimental fact; but why does 4He at low temperatures
possess these bizarre characteristics? The answers to this are
rooted in quantum mechanics. The Third Law of
Thermodynamics in one form states that the entropy of any
substance goes to zero as its temperature goes to zero. The
following statistical interpretation of entropy was
formulated by Boltzmann
S  k B ln( )
© 2003 The American Physical Society
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Volume 24, Number 13
PHYSICAL REVIEW LETTERS
where S is the entropy and Ω is the number of quantum
states that the system can achieve. This implies that any
zero entropy system has Ω=1. In other words any system
must collapse to a unique ground state as temperature goes
to absolute zero. But what is this state? For simplicity we
will start by considering only the case of 4He. 4He is what
is known as a boson. In brief bosons are all particles with
integer spin. Bosons do not obey the Pauli Exclusion
Principle, and so, they can share a common quantum state.
This is exactly what happens when we cool 4He, the atoms
all collapse to the same ground state. This phenomenon is
know as Bose-Einstein condensation and the collection of
atoms in the ground state is called the condensate.[8] In
this ‘condensate’ the wavefunctions of the individual
particles are all superimposed. We essentially have one
great big wavefunction that governs the condensate as a
whole. This emergence of a macroscopic wavefunction as
the atoms all collapse to the same quantum state is the
essence of superfluidity. This means that by disturbing one
side of the superfluid you are actually instantly affecting
the other side as well, even if they are a macroscopic
distance away from one another (a few inches in most
experiments!) The condensate acts somewhat like a single
giant particle. But unfortunately the macroscopic wave
function is not a complete picture of superfluidity either.
Interatomic interactions are still significant in He II.
However, the unusual properties we have described are
manifestations of this quantum effect.
20 November 2003
Superfluidity in 3He was not discovered until the early
1970s when Lee, Osheroff and Richardson were able to
cool the liquid to about 2.5mK. These three shared the
Nobel Prize in 1996 for this work.[9] The unusual phase
diagram of 3He is depicted in Figure 4. One notices not only
the absence of a triple point, but also that the solid-liquid
line has a minimum around 0.3K. This interesting property
is due to the liquid actually being more ordered than the
solid below 0.3K. Since the liquid is more ordered, it
actually takes energy to solidify it. This means that we can
cool 3He by applying pressure (which causes it to solidify.)
This method is called “Pomeranchuk cooling” and was the
method used by Lee, Osheroff and Richardson in their
experiments. But how is superfluidity in 3He even possible?
3
He atoms are fermions and thus have one half spin. This
means that they obey the Pauli Exclusion Principle and so,
should not be able to undergo Bose-Einstein condensation.
The answer as it turns out is rather simple: one half plus one
half is one. At very low temperatures the helium particles
join together in pairs known as “Cooper Pairs”. The pairing
is a result of the polarization of surrounding fluid by a
passing atom. A second atom can then come along and
sense this disturbance. Weak as this interaction may initially
sound, it allows for these pairs to effectively act as bosons
which, of course, means they can collapse to a common
quantum state.
In conclusion, research in the domain of superfluidity
continues to be important. The 2003 Nobel Prize in Physics
was awarded "for pioneering contributions to the theory of
superconductors and superfluids". Superfluids are still
helping us develop our picture of the ‘quantum world’.
*To whom correspondence should be addressed E-mail
[email protected]
Figure 4. Phase diagrams of 3He.
A-Temperature range from 0 to 3.2K. Notice the
minimum in the solid-liquid line around 0.3K.
B-Temperature range from 0. to 3mK. The
superfluid phase only begins around 2mK. This is a
temperature one thousand times lower than required for
4
He.
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[1] Nobel e-Museum
(www.noble.se/physics/laurates/1913/onnes-bio.html)
[2] Tony Guénault, Basic Superfluids, Taylor & Francis
Inc., New York (2003)
[3] Pyotr Kapitza, Nature 141, 74 (1938)
[4] J.F.Allen & A.D.Misener, Nature 141, 75 (1938)
[5] J.F.Allen & H.Jones, Nature 141, 243-244 (1938)
[6] Nobel Lectures, 1996 Prize in Physics
[7] Kazuyashi Yanaka, Masters Thesis, University of
Tsukuba (2003)
[8] D.r.Tilley & J.Tilley, Superfluidity and
Superconductivity 3rd Ed., IOP Publishing Ltd, London
(1990)
[9] Gloria B. Lubkin, Physics Today (December 1996)
© 2003 The American Physical Society
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