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). A B T T+DT Assume that initially A and B are at the same pressure and temperature DP Then increase the temperature of B with respect to A…. ….a pressure difference also forms Fine capillary (100nm) through which only superfluid He-II can flow This is because He-II flows through the “superleak” to the region of higher temperature in order to minimise the temperature gradient 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 Lecture 15

The fountain effect Movies courtesy of Jack Allen Lecture 15

Heat and mass transfer Such manifestations of the thermomechanical effect show that the transfer of heat and mass in He-II are inseparable T+DT T superfluid normal Heater 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 Lecture 15

Temperature Waves T+DT T superfluid normal Heater 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 density which remains uniform, but the local value of the ratio rs/r 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

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 vn vs “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 rs/rn vn vs Lecture 15

The Ground State of He-II The ground state of He-II is a pure superfluid High T T=0 Fermions The 4He atom has a resultant spin of zero and is therefore a boson, and an assembly of 4He 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. High T T=0 Bosons This creates a “condensate” in which all particles have the same wavefunction (cf the superconducting ground state) Lecture 15

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

Excitations in a solid   k qf qi 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 |qi|=2p/li is incident on the sample k q qi qf It creates an excitation of wavevector k and energy E=ħ(k) emerging with a scattered wavevector of |qf|=2p/lf at angle q to the original direction Lecture 15

Measuring excitatons with neutrons k q qi qf From the conservation of energy and from conservation of momentum So by determining qf as a function of q we obtain sets of w(k) and k for the excitations Such neutron experiments are carried out using a triple axis spectrometer which allows qi, qf and q to be independently varied 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/ Measurements therefore have to be carried out at low temperatures to “sharpen up” the excitations and better define the dispersion relation Lecture 15

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 ,Å-1 Free atom excitations ……..perhaps not too surprising as longitudinal (but not transverse) sound waves can propagate in a liquid The most important features of the He-II dispersion curves were first suggested by Landau in 1941 and confirmed later by neutron scattering Lecture 15