Superfluid pdf

R-mode instability window for superfluid neutron stars. The Kepler normalized frequency is displayed for two neutron star masses 1. Figure adapted from [7,18]. Universe , 7, 59 15 of 22 7. Thermal Conductivity Another transport coefficient that we would like to discuss is the thermal conductivity due to superfluid phonon collisions.

In this case, the thermal conductivity is given by [9]:! This matrix contains different elements that are multidimensional integrals with thermally weighted phonon scattering matrix.

Note that, as in the case of bulk viscosities, the thermal conductivity requires the dispersion law at NLO, because the thermal conductivity vanishes with a linear dispersion law [40]. We find that for T. The proportionality factor depends on the EoS. This temperature independent behavior was also observed for the color-flavor locked superfluid [40]. Note that close to Tc , higher order corrections in energy and momentum might be expected in the phonon dispersion law and self-interactions.

Associated with the thermal conductivity, one can also determine the mfp for phonons, which is different from the one coming from the shear viscosity see Equations 17 and 18 to compare.

For comparison, we also show the radius of the star of 10 km with a horizontal line. One could then define a ballistic thermal conductivity and carry out a similar approach as that of the shear viscosity in Section 4. Figures adapted from [9,18]. Again, our results must be compared to those coming from electrons and muons in neutron stars.

The electron-muon contribution to the thermal conductivity was analyzed in [41]. Compared to these results, we find that phonons in the hydrodynamic regime dominate the thermal conductivity in neutron stars [9]. We also conclude that if the contri- bution of electrons-muons and phonons to the thermal conductivity become comparable, electron-phonon collisions could play an important role.

Simple estimates were performed in [42]. This topic deserves further studies. Neutrino Emissivity and the Superfluid Phonon The cooling of a neutron star is very much affected if superfluidity is achieved in its core. However, at much lower temperatures, these processes are exponentially suppressed, and the neutrino emissivity is dominated by scatterings involving the Goldstone modes of the system, as these couple with the Z0 electroweak gauge boson.

In [14,46], the neutrino emissivity involving the angulons, the Goldstone modes associated with the spontaneous breaking of the rotational symmetry that occur in a 3 P2 neutron superfluid phase, were considered, while the contribution associated with the superfluid phonons was not taken into account.

Later results on pair breaking emissivity were given in [47]. The superfluid phonon interacts with the electroweak Z0 gauge boson, which, in turn, can decay into a neutrino-antineutrino pair. Note that in [14], f 0 was the superfluid decay constant, different from the constant used here.

The difference comes in how the superfluid field is normalized with respect to the phase of the neutron-neutron condensate, which in our case is not the same as that used in [14]; see Equation 3. Energy and momentum conservation prevent the possibility that a superfluid phonon may decay into a neutrino-antineutrino pair. This process has been considered in the color-flavor locked superfluid quark matter [48]. Superfluid phonons do not thus play any relevant role in the cooling of the star by neutrino emission.

Superfluid Phonons in the Presence of Gravity and in a Moving Background In the computation of the transport coefficients in superfluid neutron stars associated with superfluid phonons, we assumed that these move in a static medium, and we ignored the effect of the gravitational field. These two effects might be easily incorporated into our EFT approach. Let us explain how. In what follows, we deal with the case of a relativistic superfluid, where all the EFT techniques used for the non-relativistic case also apply, with minor changes.

Even if we ignore the effects of a gravitational metric, as we will do in what follows, taking into account the effects of a moving superfluid medium in the superfluid phonons is much more conveniently done with the use a gravitational analogue model [50—52].

This was first suggested in [53], but we will follow a different approach here, using the superfluid phonon EFT. From the expression of the phonon Lagrangian in terms of X, it is possible to derive the EFT of the phonons moving in the background of the superfluid. The superfluid phonon is the Goldstone boson associated with the breaking of the U 1 symmetry, and it can be introduced as the phase of the quantum condensate. Then, it should be possible to decompose the phase of the condensate into two fields, the first describing the hydrodynamic variable and the second describing the quantum fluctuations associated with the phonons.

Thus, in the background of a moving superfluid, the superfluid phonon propagates as in the background of the so-called acoustic metric. The transport equation associated with the superfluid phonons should then incorpo- rate the effect of the acoustic metric.

The Christoffel symbols of the phonon transport equation are those related to the acoustic metric. This is reflected in the second term in the l. In all our developments presented in this review article, we ignored all the above effects to simplify the computations.

We might expect that if the mfp of the phonons is shorter than the variation of both the gravitational potential andof the speed of sound, we can simply ignore the effects discussed in this Section. Unfortunately, the phonon mfp tends to increase when the temperature drops.

Thus, the evaluation of transport should be reformulated along the lines discussed here. An interesting observation was formulated in [54]. While it was generally believed that sound waves do not transport mass, in that reference, it was claimed that they do carry gravitational mass if non-linear order effects are considered. That is to say, they are affected by gravity, as we have just seen, and also generate a tiny gravitational field.

In particular, this applies to the superfluid phonon field. Summary We present an overview of the computation of the shear and bulk viscosities together with the thermal conductivity due to superfluid phonons inside neutron stars, based on an effective field theory for the interaction among superfluid phonons.

The effective field theory approach is universal and valid for different superfluid systems. As the superfluid phonons couple to the Z electroweak gauge boson, they open a channel to neutrino emissivity, but we checked that it is very much suppressed, so it can hardly affect the cooling of the star.

Whether the temperature dependence is a universal feature, the evolution of the shear viscosity with density is determined by the equation of state under beta-stable equilibrium. As for the bulk viscosities in superfluid neutron matter, we see that the bulk viscosity coefficients are highly dependent on the superfluid neutron matter gap. Nevertheless, phonon- phonon collisions rule the bulk viscosity over the Urca and modified Urca processes.

We further studied the r-mode instability in neutron stars and the consequences of the shear viscosity coming from superfluid phonons. Furthermore, the thermal conductivity is dominated by phonon-phonon interactions in comparison with electron-muon collisions for densities in the core of neutron stars.

We finally discuss how the superfluid phonon effective field theory, and ultimately their interactions, is modified in the presence of a gravitational field, or by taking into account that the superfluid is not at rest. In particular, given that the mean free path of phonons tends to increase when the temperature drops, it would be interesting to evaluate the effect of gravity on the transport coefficients we reviewed.

Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References 1. Landau, L. The theory of superfuidity of helium II. USSR , 5, 71— Migdal, A. Superfluidity and the moments of inertia of nuclei. Sedrakian, A. Superfluidity in nuclear systems and neutron stars. A , 55, Page, D. Stellar Superfluids. Thermal and transport properties of the neutron star inner crust.

Manuel, C. Shear viscosity due to phonons in superfluid neutron stars. D , 84, Shear viscosity and the r-mode instability window in superfluid neutron stars.

D , 88, Bulk viscosity coefficients due to phonons in superfluid neutron stars. Such an interaction can have quite a subtle cause.

Consider the case of conduction electrons in metals. Clearly one must expect these identical charged particles to repel each other by simple electrostatics. And they do so at short range, although one must recog- nise that the metal also contains positive ions and is overall neutral.

However in some metals, but it seems not in all, another effect holds sway at low enough temperatures. This is the formation of so-called Cooper pairs of electrons, a vital ingredient of the BCS theory of super- conductivity for which John Bardeen, Leon N. Cooper and J. Robert Schrieffer were awarded the Nobel Prize for Physics. A hand- waving description of the pairing is as follows. It is based on the simple Coulomb attraction between an electron and a positive ion.

Another electron, in fact one travelling in the opposite direction, can then sense the disturbance left by the first one even though the two electrons are mainly far apart.

In the superconducting ground state, states are occupied in these correlated pairs; the pairs may be thought of essentially as bosons, and the ground state as a condensate.

It is relevant to note that this Cooper pairing is favoured in a metal with a strong electron—phonon interaction. The group 1 metals alkali metals and noble metals probably never become superconducting, however cold they become. A feature of the BCS theory and indeed of our understanding of all superfluids, whether formed from fermion pairs or from bosons is that the condensate is described by a simple coherent macroscopic wave function.

A dramatic demonstra- tion of this large-scale coherence is found in what we now call the Josephson effects. Hence one atom moving through the liquid polarises the surrounding fluid.

Thus, pairs of atoms can be formed, even though there is no permanent binding between them, and 3He undergoes a superfluid transition. However, like superconductivity, this is a subtle and small effect. The superfluid transition in 4He is at around 2 K, but in 3He TC occurs only at 1—2 mK, three orders of magnitude lower. Its discovery in by David M. Lee, Douglas D. Osheroff and Robert C.

Richardson was finally recognised by the Nobel Prize. Superfluid 3He has some similarities with superconductivity, in that it is a pairing superfluid in a Fermi system, but important differences also. It is a highly pure substance and there is no lattice to complicate matters.

The atom is electrically neutral, but has magnetic properties. And even more critically, the pairing takes place between atoms with parallel spins whereas in conventional superconductivity the pair is formed from opposite spin electrons.

As we shall see in Chapter 5, these features give a rich behaviour to the superfluid state which makes it an exciting test-bed for our understanding of much basic physics. Superconductivity is not confined to simple metals and alloys.

It has been discovered in many other materials, even in some organic materials. In the s, a whole new class of superconductors appeared. These are layered ceramic oxides rather than metals, with compositions close to a metal-insulator transition. Available transition temperatures TC to superconductivity almost instantly shot up from around 20 K for the best known alloys to over 90 K for the new ceramics.

Great excitement was generated, since superconductivity was suddenly within the range of liquid nitrogen, and new engineering possibilities appeared. The important breakthrough was made by J. Georg Bednorz and K.

These materials are not only important for applications, but are also still not theoretically understood. Pairs are certainly involved again, but the nature of the pairing mechanism remains a matter of active controversy.

Finally, it is believed that pairing fermion superfluids play their part in astrophysics. A typical neutron star has 1. The Fermi energy of these neutrons is so high, around kB times K, that the interior of the hot star, at only around a million degrees Kelvin, is thought to consist of superfluid neutrons with TC around K. Pulsars are thought to be rotating neutron stars which have strong magnetic fields, so two further Nobel prizes here should complete the list Anthony Hewish, and Russell A.

Hulse and Joseph H. Taylor Jr, ! At the start of Section 1. But what does the word superfluidity mean, and how is it observed? This implies not only a coherent ground state, which can be deformed a little to generate a current; it also requires that such a deformed ground state does not immediately dissipate itself by forming excitations of the fluid above the ground state.

Often this requirement is met by there being an energy gap between the ground state and the excited states of the substance. In other words, the nature of the excitations is important as well as the nature of the ground state. Of course there are many experiments which could be considered here, but what is important is the relative velocity between the fluid and its sur- roundings. The simplest case to consider is the interaction between a very heavy object and the fluid.

This could imply that the fluid moving at velocity v and the surrounding walls of effectively infinite mass are stationary. Or it could imply a stationary fluid with a heavy projectile being dragged through it at velocity v.

Consider a projectile of mass M which is much larger than the masses of the bosons which make up the condensate of the substance, injected at velocity v.

Whether or not the projectile feels a drag force depends on the availability of suitable excited states in the fluid. For an excitation to be created, and hence for dissipation to occur, the spectrum of excited states must allow the process to conserve energy and momentum.

The answer is that superfluidity occurs when the Landau critical velocity is non-zero. This is because if vL is zero, then the least flow will cause dissipation. The question thus centres on the spectrum, i. As indicated in equation 1.

The Landau velocity is thus found as the line of minimum slope which intersects the dispersion relation graph. First consider Bose systems. We can illustrate the situation very easily by discussing the ideal Bose gas. Here, the excited states are simply states of an independently moving gas molecule. In other words, they have the energy equal to the kinetic energy of a molecule. Any flow can generate excitations of very low energy and hence dissipate see Figure 1.

This led Landau to propose that the dispersion relation for the interacting boson liquid 4He must be of an entirely different sort. Note that without other excitations this would give a critical velocity equal to c. We shall see in Chapter 2 that this remarkable picture is robust and accurate. For Fermi systems, a similar idea operates. The idea goes as follows. The ground state of the Fermi system is no longer a state of zero energy, since the Exclusion Principle demands that the one-particle states are filled up to the Fermi energy EF.

It is often pictured as a Fermi sea, with particles occupying all states with momen- tum up to the Fermi momentum pF. This situation is illustrated in Figure 1.

A low energy disturbance of the smooth Fermi sea occurs when a particle below the surface i. Thus excitations are generated in hole-particle pairs, with both partners being close to the Fermi surface.

In this description, all energies are measured relative to EF, not to the vacuum. Also the picture takes care of the fact that all the states below EF are already filled in the ground state. The excitation picture in an ideal normal Fermi gas is illus- trated in Figure 1. With reference to the Landau criterion, equation 1.

At an arbitrarily low velocity we can generate an excitation pair at the Fermi surface, with energies of each excitation as close to zero as we like, and with momenta essentially equal to pF. At last, we come to consider what happens in a Fermi system when the pairing interaction causes the formation of a condensate. The principal result is that the ground state becomes of lower energy than the sum of the kinetic energies of the individual fermions.

The small attractive inter- action, when all the particles cooperate, gives a modest stability to the condensed state. Excitations again occur in pairs, since excitation demands breaking a Cooper pair from the condensate. But now, this excitation pair no longer has zero threshold energy.

A Free particles in the vacuum picture; B Free particles in the excita- tion picture; C The excitation picture when pairing causes an energy gap. This is illustrated in Figure 1. In brief: 1 At a low enough temperature, everything which remains in thermal equilibrium must enter an ordered state. This relates to the Third Law of Thermodynamics. An energy gap in the excitation spectrum can do the trick.

In the rest of this book, we shall explore three of these superfluids in some detail. In Chapter 2 we look at superfluidity in liquid 4He. Chapter 3 is a diversion to introduce the physics of some low temperature techniques, before investigating superfluid fermion systems. Superconductivity is discussed in Chapter 4 and finally liquid 3He in Chapter 5. Books with a low temperature material include: J. Finn, Thermal Physics, Nelson Thornes, Dittman and M.

Mandl, Statistical Physics, Wiley, Material for investigating the history and background of the subject includes: K. Mendelssohn, The Quest for Absolute Zero, Taylor and Francis, out of print but an interesting read if available at your library. Royal Holloway University of London have introductory low temperature material produced under a Partnership for Publication Awareness project.

Explore this problem with examples. Chapter 2 Liquid 4 He The history of the discovery of superfluidity in liquid 4He makes an extraordinary tale. Helium was first liquefied by Kamerlingh Onnes in Leiden in At the time this feat was a remarkable achievement in itself, but Onnes continued to cool the newly discovered liquid further, in an attempt to produce solid 4He. To do this he reduced the pressure, expecting solid to be produced as soon as the triple point was reached compare Figure 1.

As we now know, he was doomed to frustration in this endeavour, because there is no triple point Figure 1. Nevertheless, he did in fact cool the liquid to around 1 K, well into the superfluid phase, but did not recognise the phenomenon. Over the next 30 years, this remained the situation.

Various clues were found, but not understood and not always published. Principal among these clues were the specific heat maximum at 2. Even before that Onnes himself had noted a maximum in the density of the liquid. In the event it was not until that the nature of He II was recognised as a manifestation of superfluidity. This arose from measurements of viscosity.

Shortly afterwards Fritz London worked out the theory of an ideal Bose—Einstein gas, explaining the phenomenon of Bose—Einstein condensation.

It is fascinating to speculate with the hindsight of history. I suppose that the discovery by Kamerlingh Onnes in of superconductivity produced enough excitement to occupy the mind. It is also important to realise what a small community was involved in these matters, with only a handful of laboratories performing these studies notably Leiden, Toronto, Cambridge, Oxford, Moscow!

In this chapter we shall first outline briefly some basic experimental properties of liquid 4He and their description by the two-fluid model. Next we discuss the behaviour of excitations in liquid helium and the breakdown of superfluidity. Finally we describe the idea of pure superfluid helium as a quantum fluid and the quantisation of vorticity. Below the transition tempera- ture, the gas consists of 1 a substantial fraction of its particles in the ground state and 2 the remainder occupying excited states.

The ground state fraction f varies with temperature from zero at the transition to unity at the absolute zero. What Tisza and Landau appreciated was that many of the peculiar properties observed in liquid He II can start to make sense in the context of a similar two-fluid model.

We shall now see how these postulates may be used to understand the observed properties of He II. It is used sometimes to mean only the superfluid fraction, but at other times to mean the whole fluid when it is below its transition temperature. In this section we shall use it in the first sense only, and refer to the whole fluid as He II. Otherwise, the context must decide. Liquid 4 He 27 Table 2. Low viscosity means ready flow, high viscosity means reluctant flow.

Flow through tubes The classic method is to establish a pressure dif- ference between the ends of a tube and to measure the rate of flow of fluid through the tube. The viscosity is essentially a measure of the flow resistance, i. What happens in He II is quite different. Furthermore, it is not proportional to applied pressure, rather the flow becomes large at the smallest applied pressure difference and then saturates, staying effectively constant when further pressure is applied. Flow method Drag method Figure 2.

Damping of a vane The other well-known method to determine the viscosity of a fluid is to measure the drag force on a moving object. The cleanest experiment is to observe a torsional oscillator, usually a flat circular disc suspended by an axial fibre. The disc is thus free to oscillate without displacing any fluid. However, the fluid in contact with the disc must move with the disc, causing a shear strain rate in the nearby fluid.

The associated viscous shear forces resist any movement of the disc and thus cause the oscillation to decay. Hence the damping of the disc is related directly to the viscosity of the fluid.

This result is in dramatic contrast to method 1. We may note that as the temperature is lowered further, there is a small and gradual decrease in damping, but nothing sudden or dramatic. These observations are precisely those to be expected from a two-fluid model. In any mixture of two fluids, method 1 must be dominated by the smallest of the two viscosities, since the thinner fluid can find its way through the tube much more readily than the thicker one.

On the other hand, method 2 will be dominated by the largest of the two viscosities, since the large drag force exerted by the thicker fluid will prevail over the smaller force from the thinner one. In an electrical circuit analogy, method 1 represents two resistors in parallel, and the current is dominated by the channel of least resistance.

Method 2 is like having two resistors in series, since the disc must plough its way through both fluids, and the larger resistance will dominate. Of course, the two fluids in helium are not two chemically distinguishable fluids. The situation is more subtle than that, since all the helium atoms are identical and the two fluids cannot even in principle be separated. Nevertheless, the two experiments are well described using the proper- ties in Table 2. The additional ingredient is that there must be a mechanism which limits the flow in method 1.

This mechanism implies a breakdown in superfluidity. In practice in this type of measurement the breakdown occurs by the creation of a tangle of vortices in the fluid when the flow velocity becomes large enough. This uses a set-up similar to method 2 for viscosity, but instead of using a single disc as the oscillating object, a whole stack of discs was used. The experiment is illustrated schematically in Figure 2. The measurement to be made is now not the damping of the oscillator, but its frequency as a function of temperature T.

The torsion constant does not vary in the experiment, so that the frequency directly yields the moment of inertia, and hence the mass, of whatever is oscillating.

The point is that although the normal fluid is clamped to the discs, the superfluid is not. Instead it remains stationary in the surrounding vessel and thus does not contribute to the effective moment of inertia of the oscillator. Typical results are shown in Figure 2. This experiment convincingly verifies the idea of the two-fluid model.

Many liquids wet the surface of their container, and when they do so a thin surface film covers the surface, held there by surface tension forces. Helium is no exception. Gravitational syphon action can allow significant flow through the film. For instance a bucket of He II suspended above the free liquid surface gradually empties itself, even without there being any hole in the bucket.

For instance a bucket with a knife-edge rim at the top would lose much less helium than one with a rounded rim, simply because the film will be thinner over the highly curved surface of the knife-edge. As remarked earlier, the anomaly is more violent than that in an ideal Bose—Einstein condensation. In relating these observations to the two-fluid model, it is worth making two points. Heat capacity full curve and Entropy dashed curve. It certainly gives a valuable way of thinking about the superfluid component.

However, the evidence from the heat capacity and entropy is otherwise. This section then contains a gentle warning not to take the two-fluid model too literally. Nevertheless it will remain very valuable in thinking about heat transport and other effects which we discuss in the following sections. We should remark that heat transport in liquid He II is not included in this section as an adjunct to heat capacity. Heat flow in He II turns out not to bear any relation to conventional thermal conduction in a normal fluid, but a new mechanism operates which is a specifically two-fluid effect.

The answer from the two-fluid model is devastatingly simple. The heater merely converts superfluid to normal fluid. Zero entropy superfluid is converted to the entropy-carrying normal fluid at a rate sufficient to absorb the applied energy. Thus an excess of normal fluid and a deficiency of superfluid exists near the heater. As a result a counterflow of normal and superfluid is set up as illustrated in Figure 2. Superfluid is drawn towards the heater, converted into normal fluid, thus causing a flow of normal fluid away from the heater.

This straightforward idea turns out to have dramatic consequences for the properties of He II, some of which follow. Liquid 4 He 33 1. Heat transport The first property, already mentioned, is the abnormally efficient ability of He II to transfer heat. The explanation follows directly from Figure 2.

The almost frictionless counterflow of normal and superfluid in the neighbourhood of a heater represents a very efficient transport of thermal energy in the liquid. So it is that a local hot spot in the liquid simply cannot occur; the counterflow mechanism irons out any temperature inhomogeneity. Hence the very striking observation that bubbles which indicate local hot spots in a boiling liquid are never seen in boiling He II.

Instead there is a rapid but smooth evaporation from the surface. Of course, the heat transfer is never infinite. The counterflow is limited. In practice this velocity is a few millimetres per second, and represents the velocity at which self-sustaining tangles of vortex lines are generated in the experimental geometry. The fountain effect The heat transport property has demonstrated that a temperature gradient provides a driving force for superfluid to be drawn towards the high temperature end of a vessel.

This can be demon- strated and measured directly as follows. Consider the experiment shown schematically in Figure 2. This illustrates two helium reservoirs con- nected by a superleak. The superleak consists of a fine powder packed tightly in a tube, or some other restriction, which is able to allow the passage of superfluid but which clamps the normal fluid.

When heat is applied to one reservoir, then the mechanism already mentioned means that superfluid is converted to normal fluid in that reservoir. Thus there is a concentration gradient of superfluid across the superleak, and the superfluid flows through it into the heated reservoir.

In the classic foun- tain effect experiment, illustrated in Figure 2. Suppose that, in the controlled experiment of Figure 2. The flow through the superleak now takes place until a steady state is reached, in which the pressure increase in the heated vessel the so-called fountain pressure is just enough to stop the flow through the superleak. We discuss the magnitude of these effects in Section 2. The inverse fountain effect In the fountain effect, a temperature difference is seen to generate a pressure difference.

But the inverse is also true. Without heat applied, the two reservoirs in Figure 2. Now if we apply a pressure difference to one side, we know that only superfluid will pass through the superleak. A The principle; B A demonstration experiment. Hence we will deplete the superfluid on the high pressure side, i. Correspondingly we obtain an excess of superfluid on the low pressure side, so that the liquid there cools. We thus have a simple mechanically operated reversible heat pump.

A pressure difference generates a temperature difference. It is amusing to consider the consequences of this further. If we allow a vessel of He II to empty by gravity through a superleak, then the leak- ing helium must be pure superfluid. Hence, in an ideal world, it would be at absolute zero. At a first glance, this sounds like a good method of refrigeration.

Liquid 4 He 35 4. Second sound The almost frictionless counterflow of normal and superfluid results in a new type of propagating wave being possible in He II. An arrangement to demonstrate and measure second sound is sketched in Figure 2. It shows a closed cylindrical tube, with a heater at one end and a thermometer at the other.

We have already noted that heat causes a counterflow of normal and superfluid components of He II. When an AC current of frequency f is applied to the electrical heater, the conventional Joule heating has a strong component at frequency 2f. This thus generates a counterflow, which also has a frequency compon- ent at 2f. The tube can act as a resonator, just like an organ pipe does for normal sound. A simple experiment can demonstrate the existence of these resonant modes in the tube, and hence the velocity of the second sound can readily be determined.

Results are illustrated in Figure 2. Second sound is strong where there is plentiful normal and superfluid.

On the other hand at low temperatures there is little normal fluid to move, so little counterflow is possible and the mode merges with first sound. Actually the resonator tube experiment can be used to observe both first and second sound with the same equipment. In the set-up of Figure 2. This consists of a metallised piece of a very fine filter called nucleopore which acts as a capacitor plate.

The nucleopore can be driven or detected electrostatically by application of suitable voltages to the capacitor. As it moves, it allows superfluid to pass through its fine pores whilst the normal fluid cannot do so; thus a combination of counterflow motion second sound and bulk flow first sound can be excited. Other sound modes The existence of the two fluids which can move independently opens up the possibility of yet more propagating modes in addition to first and second sound.

Unsurprisingly two more are called third and fourth sound. In these the normal fluid is fully clamped. Third sound occurs in the helium film on a surface. The film is suffi- ciently thin that the normal fluid is clamped to it. However, application of a temperature gradient causes superfluid to flow towards the hot part of the film.

This can occur with virtually no pressure difference simply by the film becoming thicker where it is heated and thinner where it is cooled.

Thus a thickness—temperature wave can be sustained in the film, and this is called third sound. Fourth sound is a bulk effect in a superleak material. Nondiagonal cross-transport phenomena in a magnetic field. This brief paper supplements the review by A F Barabanov et al. Physics—Uspekhi 58 concerning the Hall and Righi—Leduc effects. Both effects are diagonal in the sense that the initial … Expand.

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Download PDF. Subjects Atomic and molecular physics Condensed-matter physics Quantum physics. Abstract In this work, a Josephson relation is generalized to a multi-component fermion superfluid. Figure 1. Full size image. Figure 2.