Excitations in HeII

1
Excitations in He-II
In region A, at low energies, we have a linear
dispersion
This initial part of the curve is analogous to an
acoustic phonon in a crystalline solid and is
associated with the collective vibrations of the
He atoms
C200m/s
Lecture 16
2
Excitations in He-II
The dispersion relation close to the roton
minimum can be expressed as
with D/kB 8.65K, ko19.1nm-1 and mr0.16m4
mr is an effective mass, and D signifies the
presence of an energy gap for these excitations
which are called rotons
The energy gap is crucial for the occurrence of
superfluidity
Lecture 16
3
Rotons
To a certain extent the roton excitation can be
considered as a He atom moving independently and
rapidly through the superfluid with the other
atoms moving out of the way
The wavelength of the roton excitation is
comparable to the interatomic spacing
Rotons provide the only major contribution to the
thermodynamic parameters of He-II besides the
acoustic longitudinal phonon (first sound)
processes
Single particle excitations may also be present,
but these are heavily damped and are therefore
very broad and are without any appreciable
amplitude
Also
It is important to note that the entire entropy
of He-II is carried by the normal component of
the two fluids
It is therefore the normal component of the
superfluid that is associated with the thermal
excitations in He-II
Lecture 16
4
Drag in superfluid He-II
In order to see how how important the dispersion
relationship is in defining the properties of
He-II consider the concept of drag associated
with a heavy body of mass M moving through the
superfluid
To experience drag the body must create
excitations in the superfluid
This process conserves both energy and momentum,
thus
(phk, the momentum of the excitation)
Lecture 16
5
Critical (or Landau) velocity in He-II
and this is only allowed if such an excitation
exists on the dispersion curve.
This is also known as the Landau velocity
From the dispersion curve the minimum value of
w(k)/k is at the roton minimum
At this point w(k)/k1.2THz/2Å-1
Hence vcvL 60m.s-1
(cf200m.s-1 for the velocity of sound)
Lecture 16
6
The Landau criterion
The critical or Landau velocity is often smaller
than 60m.s-1 as the Landau result assumes the
liquid is streamline and therefore excludes the
possibility of forms of internal motion other
than phonons and rotons
(At low velocities it is also possible to excite
vortices)
Another way of looking at this is that if vL gt0
there can be no drag for velocities vltvL. So if
vL gt0 superfluidity is possible
This is the Landau criterion
Without a roton minimum this would occur only
along the linear portion of the dispersion curve
close to k0, ie at the velocity of sound
It could also occur for a parabolic free atom
excitation, ie at vL0. The Landau criterion says
that superfluidity cannot exist in this situation
- therefore there can be no free atom single
particle excitations
Superfluidity is therefore a consequence of the
energy gap, and of the rotons being the only
excitations
Lecture 16
7
Measurement of drag in 4He
Lecture 16
8
Quantisation of circulation
(providing the mass flow is not too great)
We can use this wave function to examine
circulation in a superfluid.
We will consider He-II occupying an annular
region, eg the space between two concentric
cylinders, assuming that T0 so the He-II is a
pure superfluid
Lecture 16
9
Quantisation of circulation
Since the superfluid wave function is single
valued, a trip around a closed loop must leave it
unchanged
DS can therefore only be zero or an integral
multiple of 2p
Lecture 16