Holes in a superfluid The quantisation of circulation

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Holes in a superfluid The quantisation of circulation holds for any contour which can

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 core (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 lpoint Lecture 16 streamlines Superconductivity and Superfluidity

A single vortex The pattern of flow associated with a vortex is a series

A single vortex The pattern of flow associated with a vortex is a series of concentric stream lines of radius r The circulation around each contour of radius r is v. S~1/r and the magnitude of the circulation is independent of r Also which is the same as for a vortex in a classical liquid but now we have the quantisation condition The angular momentum is quantised in units of ħ Lecture 17 r Superconductivity and Superfluidity

A single vortex At large radius r the flow pattern is limited in extent

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 v. S rs/ rn v. S~1/r 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 ao r Superconductivity and Superfluidity

Kinetic energy of a single vortex The energy associated with a single vortex line

Kinetic energy of a single vortex The energy associated with a single vortex line is principally the kinetic energy of the circulating superfluid The kinetic energy per unit length of core is core centre at a distance r from the At a distance r the mass per unit length of circulating superfluid is The kinetic energy at a distance r is therefore kinetic energy per unit length of core is and the total where b is the vortex separation Using vs=k/2 pr this can be evaluated as But Lecture 17 so Superconductivity and Superfluidity

One quantum per vortex Now if each vortex contained only one quantum, but there

One quantum per vortex Now if each vortex contained only one quantum, but there were n vortices in the superfluid the total energy would be 1 But if there were only one vortex containing n quanta the total energy would be 2 For n>1 it is clear that 2 > 1 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 Superconductivity and Superfluidity

The Landau state When the two fluid model of liquid He was first suggested

The Landau state When the two fluid model of liquid He was first suggested it was generally accepted that it would be difficult to set the superfluid fraction into rotation because superfluid flow was characterised by the irrotationality condition introduced by Landau (1941): curl vs = 0 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 L 2 By Stoke’s theorem the circulation can be written as an integral over the surface A enclosed by the contour: L 2 He-II so for any contour in the continuous fluid. 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 Superconductivity and Superfluidity

Rotating He-II Andronikashvili’s experiment showed that an oscillating stack of discs entrained the normal

Rotating He-II Andronikashvili’s 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 Superconductivity and Superfluidity

Rotation of He-II The rotation of the superfluid can be explained by assuming that

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 Osborne’s superfluid appears to occupy a singly connected region, but in fact it is made multiply-connected by the presence of vortex lines Lecture 17 Superconductivity and Superfluidity

Vortex lattice Just as in the case of superconducting vortex lines, the vortices in

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 Superconductivity and Superfluidity