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 A T D P B T+ D T

Fine capillary (100nm) through which only superfluid He-II can flow Superconductivity and Superfluidity Lecture 15

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

Superconductivity and Superfluidity Lecture 15

The fountain effect

Lecture 15

Movies courtesy of Jack Allen

Superconductivity and Superfluidity

Heat and mass transfer

Such manifestations of the thermomechanical effect show that the transfer of heat and mass in He-II are inseparable

Heater superfluid

T+ D T

normal

T

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

Superconductivity and Superfluidity Lecture 15

Temperature Waves

Heater

T+

superfluid

D T

normal

T 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

value of the ratio r s / r density which remains uniform, but the local 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 Superconductivity and Superfluidity

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 “first sound” is the normal longitudinal pressure waves which involve fluctuations in the total density at constant temperature. First sound in He-II involves the superfluid and normal components moving in phase “second sound” in He-II involves the superfluid and normal components moving out of phase. The total density remains constant, and a temperature (or entropy) wave is created. The speed of propagation depends upon r s / r n v n v s v v n s

Lecture 15 Superconductivity and Superfluidity

The Ground State of He-II

The ground state of He-II is a pure superfluid The 4 He atom has a resultant spin of zero and is therefore a boson, and an assembly of 4 He 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

High T Fermions T=0 High T Bosons T=0

Superconductivity and Superfluidity

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 4 He 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 E=ħ  At first sight it might be expected that these excitations should be single particle excitations: E  p 2 2 m perhaps with some modifications to take account of interactions k =p/ħ 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 Superconductivity and Superfluidity

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 k =0 represents the speed of sound in the solid Such dispersion curves are determined by neutron scattering : An incident neutron of wavevector |q i |=2 p / l i is incident on the sample It creates an excitation of wavevector k and energy scattered wavevector of |q f |=2 p / l f angle q E=ħ  ( k) emerging with a to the original direction at q q f q i k

Superconductivity and Superfluidity Lecture 15

k

Measuring excitatons with neutrons

k From the conservation of energy   ( k )   2 2 m ( q i 2  q 2 f ) and from conservation of momentum k 2  q 2 i  q 2 f  2 q i q f cos q So by determining q f function of q as a we obtain sets of (k) and k for the excitations Such neutron experiments are carried out using a triple axis spectrometer and q which allows q i , q f to be independently varied q q f q i

Superconductivity and Superfluidity Lecture 15

Particles and quasiparticles

A typical dispersion relation for the excitations in a solid is shown on the right 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 ħ/ t Measurements therefore have to be carried out at low temperatures to “sharpen up” the excitations and better define the dispersion relation

Lecture 15 Superconductivity and Superfluidity

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 2 Free atom excitations 1 The most important features of the He-II dispersion curves were first suggested by Landau in 1941 and confirmed later by neutron scattering 0 0 1 2 Wave number k (?

3 -1 ) 4

Superconductivity and Superfluidity Lecture 15