Holes in a superfluid

1
Holes in a superfluid
The quantisation of circulation holds for any
contour which can be continuously defined without
passing outside the boundary of the superfluid.
The annulus considered here is an example of a
multiply connected region since it contains a
hole in the superfluid
Any regions surrounded by, but not penetrated by,
superfluid are holes and are a sufficient
conditin for quantisation to occur
(a) Holes can be provided by solid boundaries (eg
superfluid contained in an annular container
(b) In rotating He-II a hole in the form of a
cylinder can appear spontaneously in the
superfluid - stable currents are then set up in
circles around the cylinder
In this case we have a vortex line
Vortices can be measured in a similar way to
those in superconductors, by cooling rotating
He-I below the l-point
Lecture 16
2
A single vortex
The pattern of flow associated with a vortex is a
series of concentric stream lines of radius r
and the magnitude of the circulation is
independent of r
The angular momentum is quantised in units of h
Lecture 17
3
A single vortex
At large radius r the flow pattern is limited in
extent by the boundaries of the liquid or the
presence of other vortices
As r approaches zero, the equation for vs
predicts a divergence - indicating that the core
region is different to the surrounding superfluid
The divergence can be avoided by assuming that
rs?0 as r ?0, and that rs falls from its value in
the bulk to zero over a typical distance ao
ao is defined as the core radius
Experimentally ao is found to be 1 Å
Lecture 17
4
Kinetic energy of a single vortex
The energy associated with a single vortex line
is principally the kinetic energy of the
circulating superfluid
where b is the vortex separation
Lecture 17
5
One quantum per vortex
It is therefore energetically favourable for each
vortex to contain only one quantum of circulation
(cf the case of vortex lines in a superconductor)
Lecture 17
6
The Landau state
To see the implication of this condition consider
He-II contained in a cylindrical bucket, and
calculate the circulation round a typical contour
such as L2
As any contour can be reduced continuously to a
point, for this equation to hold vs must be zero
everywhere in the superfluid, and rotation is not
possible.
This state is known as the Landau State
Lecture 17
7
Rotating He-II
Andronikashvilis experiment showed that an
oscillating stack of discs entrained the normal
fluid causing it to rotate but left the
superfluid at rest
This seemed to reinforce the concept that He-II
was irrotational.
In contrast, Osborne (1950) rotated a cylindrical
bucket containing He-II and found that the
meniscus of the fluid had the same shape as that
adopted by a normal liquid undergoing rigid body
rotation
This indicated that the superfluid moved at the
same angular velocity as the normal fluid
A simple explanation of this result might have
been that superfluidity is destroyed when He-II
is given an angular velocity
.except that Andronikashvili was later able to
show that the fountain effect was still observed
in rotating He-II
Lecture 17
8
Rotation of He-II
The rotation of the superfluid can be explained
by assuming that it is threaded by a series of
parallel straight vortex lines
Remember that quantised circulation is possible
around a region from which He-II is excluded, in
which case
This implies that curlvs takes a non zero value
within the area enclosed by the vortex - in this
case we define the core of the vortex as the
region in which curlvs is finite
At first sight Osbornes superfluid appears to
occupy a singly connected region, but in fact it
is made multiply-connected by the presence of
vortex lines
Lecture 17
9
Vortex lattice
Just as in the case of superconducting vortex
lines, the vortices in He-II form an hexagonal
lattice
This lattice is also susceptible to pinning,
disorder, strain etc
Lecture 17