1. Superconductivity and Superfluidity Temperature scales Lecture 14

2. Superconductivity and Superfluidity Superfluidity in helium The two isotopes of helium, 3 He and 4 He, have the lowest normal boiling points of all known substances 3.19K and 4.21K respectively Under their saturated vapour pressures they remain liquid down to the very lowest temperatures To produce a solid phase requires application of rather high pressures (approximately 25 atmospheres) This is because of (a) the low mass of He (b) weak interatomic forces between He atoms (a) gives rise to large zero point energy: assume a He atom is confined within a sphere of radius R bounded by other atoms - from the Heisenberg Uncertainty Principle we have and for the energy and this is the zero point correction to the total energy Lecture 14

3. Superconductivity and Superfluidity Zero point effects distance energy Hard core repulsion R o (van der Waals) R o (total) Zero point contribution The already weak Van der Waals (6-12) bonding is further weakened by the zero point contribution - note the shallow minimum well removed from the origin Lecture 14

4. Superconductivity and Superfluidity Basic properties of helium Immediately below their respective boiling points both 3 He and 4 He behave like ordinary liquids with a small viscosity Their thermal and kinetic properties in this regime can be described with moderate success by a simple kinetic theory of gases At low temperatures both show marked deviations from quasi-classical behaviour In liquid 4 He the change in behaviour is marked by a characteristic anomaly in the heat capacity at 2.17K - the so called  transition because of its shape Below this temperature the 4He liquid becomes superfluid -eg it can flow through capillaries with no viscosity Lecture 14

5. Superconductivity and Superfluidity 3 He From its boiling point down to approximately 1K 3 He behaves like 4 He close to its boiling point Between 1K and 0.003K 3 He behaves differently both to 4 He and quasi- classical expectations At 2.6mK 3 He, like 4 He, also undergoes a transition to a superfluid state, signalled by an abrupt increase in the heat capacity It should be noted that while 4 He is governed by Bose statistics, 3 He is governed by Fermi statistics The superfluid transition in 3 He is associated with pairing of 3 He atoms, the resulting pairs then behaving as bosons Lecture 14

6. Superconductivity and Superfluidity Phase diagram of 3 He 0 1 2 3 4 Temperature (K) 50 100 150 0 Pressure (atmospheres) Solid BCC Solid HCP melting (after Grilly and Mills, 1959) Region of negative expansion coefficient Fluid Lecture 14

7. Superconductivity and Superfluidity Phase diagram of 4 He Solid He Upper point (1.76K,29.8atm) -line point (2.17K) evaporation Critical point (5.2K, 2.26atm) Liquid He-II Liquid He-I He gas 0 1 2 3 4 5 6 Temperature (K) Pressure (atmospheres) 10 20 30 0 40 (after London, 1954) Lecture 14

8. Superconductivity and Superfluidity Liquid He-II The -transition marks an abrupt boundary between He-I and He-II Above T bubbles of vapour form within the bulk of the liquid and the whole liquid is agitated by the bubbles rising to the surface Below T  the liquid becomes calm and still - this is because it is almost impossible to set up an appreciable temperature gradient because heat transport takes place so readily, hence all evaporation takes place only at the free surface. The -transition is sharp to within a mK. Lecture 14

9. Superconductivity and Superfluidity Viscosity and superfluidity For not too great a pressure the velocity of flow appears independent of pressure head, and is greater through tubes of smaller diameter For classical liquids this would give the same as for (b) - but He-II gives different results Measurement of the flow of He-II through fine capillaries indicate a vanishingly small viscosity (10 11 that of He-I) (a) T(K) 2 04 V (b) Viscosity can also be measured with rotation viscometers. The results give quite normal coefficients of viscosity, similar to that of He-I, but with a different temperature dependence T(K) 2 04 V (c) Viscosity can be measured by observing the damping on a disc oscillating in the liquid. T(K) 2 04 V Lecture 14

10. Superconductivity and Superfluidity Two fluid model It appears that He-II is capable of being both viscous and non-viscous at the same time This is the essence of the two-fluid model first suggested by Tisza in 1938 Note: The two fluids cannot be physically separated We cannot even regard some atoms as belonging to the normal fluid ad some atoms as belonging to the superfluid All 4 He atoms are the same! Instead the total mass density is And the total mass current density is Lecture 14

11. Superconductivity and Superfluidity Andronikashvili’s experiment (1946) Andronikashvili demonstrated the two fluid character of He-II using a torsion pendulum composed of a stack of discs close enough to drag He around with them Below T the period of oscillation decreased sharply indicating that not all the fluid in the spaces between the discs was being dragged around by the discs Torsion wire 0 12 T(K) 0.5 1.0 -point s/s/ n/n/  Lecture 14

12. Superconductivity and Superfluidity A “magic liquid” Adsorption on a surface in contact with any liquid or its saturated vapour is a common phenomenon He-II films are exceptionally thick (typical thicknesses under saturated vapour pressure is 30nm or 100 atomic layers) - sufficiently thick to permit superfluid to flow through the film - at rates approaching 20cm.s -1 ! (a) lower an empty beaker into a He-II bath The beaker begins to fill even though the rim is above the liquid (b) raise the beaker The beaker begins to empty itself (c) Raise the filled beaker above the liquid level and drops will drip Lecture 14

13. Superconductivity and Superfluidity A “magic liquid” Lecture 14