Thermomechanical effect

1
Thermomechanical effect
The thermal conductivity of He-II is very high
(tending to infinite for small heat currents) and
therefore it is not possible to sustain a
temperature gradient (except in situations as
shown on the left).
Assume that initially A and B are at the same
pressure and temperature
Then increase the temperature of B with respect
to A.
.a pressure difference also forms
This is because He-II flows through the
superleak to the region of higher temperature
in order to minimise the temperature gradient
Lecture 15
2
The fountain effect
An extreme example of the thermomechanical effect
is the Fountain Effect, discovered by Jack Allen
at St Andrews University in 1938
The superleak in this case is a wide tube
containing fine compressed powder.
One end is open to the He-II bath and the other
is joined to a vertical capillary
When the powder is heated, superfluid flows into
the superleak with such speed that He-II is
forced out of the capillary as a jet.
A very small amount of heat will produce a jet
30-40cm high
Lecture 15
3
The fountain effect
Movies courtesy of Jack Allen
Lecture 15
4
Heat and mass transfer
Such manifestations of the thermomechanical
effect show that the transfer of heat and mass
in He-II are inseparable
Normal fluid flows from the source to the sink of
heat, but superfluid flows from sink to source
and the total density remains constant everywhere
Only the normal fluid fraction can transport heat
- superfluid flow by itself cannot transfer heat
Lecture 15
5
Temperature Waves
If the heat supply is varied periodically (by ac
current through the heater) the two fluids
oscillate in amplitude
This has no effect on the total density which
remains uniform, but the local value of the ratio
rs/r and consequently the local temperature
undergoes oscillations
In this way He-II is able to propagate
temperature waves through the liquid, not
according to the usual fourier equation, but as
true wave motion with a wave velocity that is
independent of frequency
These temperature waves are entirely analogous to
ordinary sound waves, except that the
thermodynamic variable is temperature not
pressure
Lecture 15
6
Second Sound
Provided that the rate of heat supply is not too
large, and the frequency is not too high, the
temperature waves are propagated with virtually
no attenuation
It is also possible to transmit sharp pulses of
temperature through the He-II liquid.
In a resonance tube standing temperature waves
can also be established
The phenomenon of propagating, pulsed or standing
temperature waves is called Second Sound
Lecture 15
7
The Ground State of He-II
The ground state of He-II is a pure superfluid
The 4He atom has a resultant spin of zero and is
therefore a boson, and an assembly of 4He atoms
behaves according to Bose-Einstein statistics.
An ideal boson gas of particles with non-zero
rest mass exhibits the phenomenon known as
Bose-Einstein condensation - at low temperatures
all the particles crowd into the the same quantum
state corresponding to the lowest single-particle
energy level of the system.
This creates a condensate in which all
particles have the same wavefunction (cf the
superconducting ground state)
Lecture 15
8
Elementary excitations in He-II
The basic concept for understanding He-II and the
associated two fluid model is that of elementary
excitations
Above the l point 4He behaves like a dense
classical gas
Below the l point it behaves differently as the
de Broglie wavelengths of the He atoms are
comparable to interatomic spacings
Landau pointed out that it was necessary to
describe the atomic motion in terms of elementary
excitations
Instead the excitation spectrum looks more like
that of a crystal lattice where phonons dominate
ie it is collective motion of the He atoms that
is important
Lecture 15
9
Excitations in a solid
Remember that for a solid the collective
excitations associated with lattice vibrations
have a dispersion relation of the form
where the linear regime close to k0 represents
the speed of sound in the solid
Such dispersion curves are determined by neutron
scattering
An incident neutron of wavevector qi2p/li is
incident on the sample
It creates an excitation of wavevector k and
energy Eh?(k) emerging with a scattered
wavevector of qf2p/lf at angle q to the
original direction
Lecture 15
10
Measuring excitatons with neutrons
So by determining qf as a function of q we obtain
sets of w(k) and k for the excitations
Such neutron experiments are carried out using a
triple axis spectrometer which allows qi, qf and
q to be independently varied
Lecture 15
11
Particles and quasiparticles
The phonon model of a crystal is an example of a
general method of dealing with excitations in an
interacting system
The original particles (ie atoms) and their
interactions (ie bonding) are replaced by a set
of non-interacting or weakly interacting
quasiparticles (in this case phonons)
At low enough temperatures the density of
quasiparticles is sufficiently small to neglect
the interactions
However because thermal excitations interact with
one another they have a finite lifetime t (ie
they are damped) and have an energy uncertainty
of h/?
Measurements therefore have to be carried out at
low temperatures to sharpen up the excitations
and better define the dispersion relation
Lecture 15
12
Excitations in fluids and superfluids
For most liquids, including He above the l-point,
neutron scattering measures excitations that are
broad and ill-defined
However, below the l-point they sharpen
considerably and look similar to those of a
crystalline solid
..perhaps not too surprising as longitudinal
(but not transverse) sound waves can propagate in
a liquid
The most important features of the He-II
dispersion curves were first suggested by Landau
in 1941 and confirmed later by neutron scattering
Lecture 15