Quantum Statistics
229
at 2.17 K. Above 2.17 K, it behaves like a normal liquid and is known as helium I.
Below this temperature, it acquires some unusual properties, e.g. it flows through
capillaries without any apparent viscosity. This form is known as helium II and
many of its properties can be described by regarding it as a mixture of two
fluids, one a normal fluid and the other a superfluid which has no viscosity.
This mixture is similar to a Bose-Einstein gas with some condensation, the
superfluid corresponding to the particles in the ground state. This would explain
the zero viscosity. The identification of the two phenomena is further
strengthened by the observation that the specific heat of
4
He also shows a singular
behaviour at 2.17 K. The observed specific heat has the shape of λ [see Fig. 7.4
(b)] and hence the transition is called a λ-transition while the transition
temperature is called the λ-point. It should be noted however that careful
experiments indicate that the specific heat has a logarithmic infinity at the
λ-point T λ . This however may be due to the fact that the particles considered in
Bose-Einstein condensation were noninteracting which is certainly no the case
for the atoms of liquid helium. Finally, using V = 27.6 cm
3
/mole for liquid
helium in Eq. (7.78), one obtains T c = 3.13 K compared with T λ = 2.17 K. These
observations strongly suggest that the λ-transition is a form of Bose-Einstein
condensation.
Liquid
3
He: Helium has an isotope
3
He which is a fermion (it has 2 protons,
1 neutron and 2 electrons) and which liquifies at 3.2 K. It is found that
3
He,
though a fermion, undergoes a transition to the superfluid state at 2.6 × 10
–3
K.
This arises from the fact that two
3
He atoms interact with each other and produce
a weakly-bound system at low temperatures. This bound system is a boson
which can undergo a transition to the superfluid state.
Hydrogen: The atoms of about half of the elements are bosons, i.e. they
obey Bose statistics. Even then, Bose-Einstein condensation is not a common
phenomenon. The reason for this is that the condensation takes its simplest
form only for an ideal gas in which the atoms do not interact with each other. In
real atoms, the electromagnetic interaction tends to bind them and most
substances go into the solid state long before the critical temperature for BoseEinstein condensation is reached. Therefore, condensation is expected in only
those systems where the interaction between the atoms is weak compared to the
zero-point energy of the atoms, e.g. in helium. An interesting possibility that is
being currently considered is the Bose-Einstein condensation of atomic hydrogen
[see Silvera and Walraven, Sc. Am. 246, 1, 56 (1982)]. It is true that under
ordinary conditions, the interaction between the hydrogen atoms is quite strong
and binds them into molecules in which the spins of the two electrons are
antiparallel. However, if the atoms with parallel electron spins are isolated, for
example, by using strong inhomogeneous magnetic fields. Pauli’s exclusion
229
at 2.17 K. Above 2.17 K, it behaves like a normal liquid and is known as helium I.
Below this temperature, it acquires some unusual properties, e.g. it flows through
capillaries without any apparent viscosity. This form is known as helium II and
many of its properties can be described by regarding it as a mixture of two
fluids, one a normal fluid and the other a superfluid which has no viscosity.
This mixture is similar to a Bose-Einstein gas with some condensation, the
superfluid corresponding to the particles in the ground state. This would explain
the zero viscosity. The identification of the two phenomena is further
strengthened by the observation that the specific heat of
4
He also shows a singular
behaviour at 2.17 K. The observed specific heat has the shape of λ [see Fig. 7.4
(b)] and hence the transition is called a λ-transition while the transition
temperature is called the λ-point. It should be noted however that careful
experiments indicate that the specific heat has a logarithmic infinity at the
λ-point T λ . This however may be due to the fact that the particles considered in
Bose-Einstein condensation were noninteracting which is certainly no the case
for the atoms of liquid helium. Finally, using V = 27.6 cm
3
/mole for liquid
helium in Eq. (7.78), one obtains T c = 3.13 K compared with T λ = 2.17 K. These
observations strongly suggest that the λ-transition is a form of Bose-Einstein
condensation.
Liquid
3
He: Helium has an isotope
3
He which is a fermion (it has 2 protons,
1 neutron and 2 electrons) and which liquifies at 3.2 K. It is found that
3
He,
though a fermion, undergoes a transition to the superfluid state at 2.6 × 10
–3
K.
This arises from the fact that two
3
He atoms interact with each other and produce
a weakly-bound system at low temperatures. This bound system is a boson
which can undergo a transition to the superfluid state.
Hydrogen: The atoms of about half of the elements are bosons, i.e. they
obey Bose statistics. Even then, Bose-Einstein condensation is not a common
phenomenon. The reason for this is that the condensation takes its simplest
form only for an ideal gas in which the atoms do not interact with each other. In
real atoms, the electromagnetic interaction tends to bind them and most
substances go into the solid state long before the critical temperature for BoseEinstein condensation is reached. Therefore, condensation is expected in only
those systems where the interaction between the atoms is weak compared to the
zero-point energy of the atoms, e.g. in helium. An interesting possibility that is
being currently considered is the Bose-Einstein condensation of atomic hydrogen
[see Silvera and Walraven, Sc. Am. 246, 1, 56 (1982)]. It is true that under
ordinary conditions, the interaction between the hydrogen atoms is quite strong
and binds them into molecules in which the spins of the two electrons are
antiparallel. However, if the atoms with parallel electron spins are isolated, for
example, by using strong inhomogeneous magnetic fields. Pauli’s exclusion