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8.3 Superfluid hydrodynamics

8.3.1 Superflow and critical velocity

The basic property of superfluid is that it can flow without dissipation. In a normal fluid, friction and viscosity arise because particles are randomly scattered. Such scattering events are forbidden in superfluid because energy and momentum cannot be simultaneously conserved. The key argument of Landau is that in a superfluid like helium-4, the particles are strongly correlated so that the concept of single particles becomes meaningless. However, at low enough temperatures, the system is still assumed to be described in terms of noninteracting “quasiparticles”, which do not correspond to material particles but to many-body motions (excitations). The energy spectrum of these quasiparticles can be very different from that of single particles. Using this idea, Landau [247Jump To The Next Citation Point] was able to explain the origin of nondissipative superflow and the existence of a critical velocity beyond which superfluidity disappears. The argument is the following. Let us consider a macroscopic body of mass M flowing through the superfluid. At low temperatures, its velocity VVV can only be changed in scattering processes where one or more quasiparticles are created, assuming that the flow is not turbulent. For a quasiparticle of energy E (p) and momentum p to be created, energy conservation implies that

1-M V 2 > 1M V ′2 + E (p ), (140 ) 2 2
where VVV′′′ is the body’s velocity after the event. However, momentum must also be conserved,
′′′ MVVV = MVVV + ppp . (141 )

These two conditions can only be satisfied if

p2 E (p) < VVV ⋅ppp − ----. (142 ) 2M

Since the perturbing body contains a macroscopic collection of particles, the mass M is very large so that the second term can be neglected. The resulting inequality cannot be satisfied unless the velocity exceeds some critical value

{ } E (p) Vc = min ----- , (143 ) p
where min {x} is the smallest value of x in a set {x}. This means that for velocities smaller than Vc, the creation of quasiparticles is forbidden and therefore the fluid flows without dissipation. In a normal liquid, the single particle energy is given by an expression of the form
p2-- E (p ) = 2m , (144 )
where m is an effective mass suitably renormalized to include many-body effects. Consequently, the critical velocity, according to Landau’s criterion (143View Equation), is zero, Vc = 0. Since liquid helium-4 is superfluid, Landau [247Jump To The Next Citation Point246Jump To The Next Citation Point] postulated a different energy spectrum. At low momenta, the quasiparticle excitations are sound waves (phonons) as illustrated in Figure 50View Image. The dispersion relation is thus given by
E (p) = csp , (145 )
where cs is a sound speed.
View Image

Figure 50: Schematic picture illustrating collective motions of particles associated with a low momentum quasiparticle (phonon).

At very high momenta, the dispersion relation coincides with that of a normal liquid, Equation (144View Equation). In between, the dispersion relation exhibits a local minimum and is approximately given by

2 E (p) = Δ + (p −-p0)-. (146 ) r 2mr

The quasiparticles associated with this minimum were dubbed “rotons” by I.E. Tamm as reported by Landau [247Jump To The Next Citation Point]. Landau postulated that these rotons are connected with a rotational velocity flow, hence the name. These rotons arise due to the interactions between the particles. Feynman [145] argued that a roton can be associated with the motion of a single atom. As the atom moves through the fluid, it pushes neighboring atoms out of its way forming a ring of particles rotating backwards as illustrated in Figure 51View Image. The net result is a vortex ring of an atomic size.

View Image

Figure 51: Schematic picture illustrating collective motions of particles associated with a roton quasiparticle according to Feynman’s interpretation.

The roton local minimum has also been interpreted as a characteristic feature of density fluctuations marking the onset of crystallization [200334307Jump To The Next Citation Point]. According to Nozières [307], rotons are “ghosts of Bragg spots”. Landau’s theory has been very successful in explaining the observed properties of superfluid helium-4 from the postulated energy spectrum of quasiparticles.

In weakly interacting dilute Bose gases, as in ultra-cold Bose atomic gases, the energy of the quasiparticles are given by (see for instance Section 21 of Fetter & Walecka [142])

∘ --------------- ( p2)2 E (p) = ---- + p2c2s , (147 ) 2m
where cs is the speed of sound. At low momentum p, it reduces to Equation (145View Equation), while at high momentum it tends to Equation (144View Equation). Dilute Bose gases, thus, have only phonon excitations. Quite remarkably, the ideal Bose gas (which is characterized by the dispersion relation (147View Equation) with cs = 0) exhibits a Bose-Einstein condensation at low enough temperatures, but is not superfluid since its critical velocity is equal to zero.

Owing to the specific energy spectrum of quasiparticles in both atomic Bose gases and helium-4, the critical velocity does not vanish, thus explaining their superfluid properties. Landau’s critical velocity, Equation (143View Equation), of superfluid helium-4 due to the emission of rotons, is given by

V = Δr-≃ 60 m s− 1. (148 ) c p0
This value has been confirmed by ion propagation experiments [137]. However, in most experiments, much smaller critical velocities are measured due to the existence of other kinds of excitations. The critical velocity of atomic Bose gases has also been measured, using laser beams instead of a macroscopic object [235]. Again, velocities smaller than Landau’s velocity (which is equal to the velocity of sound in this case) have been found.

The previous discussion of the critical velocity of Bose liquids can be easily extended to fermionic superfluids. In the BCS theory, fermions form bound pairs, which undergo Bose condensation when the temperature falls below a critical temperature (Section 8.2). The quasiparticle energies for a uniform Fermi system are given by

∘ ------------------- E (p) = (ε(p ) − μ )2 + Δ (p)2, (149 )
where p = ℏk, ε(p) is the single particle energy, μ the chemical potential and Δ (p) the pairing gap at momentum p. According to Landau’s argument, the critical velocity (143View Equation) is equal to
Δ Vc = --F-. (150 ) ℏkF

This expression can be derived more rigorously from the microscopic BCS theory [35]. It shows that a system of fermions is superfluid (i.e. the critical velocity is not zero) whenever the interactions are attractive, so that the formation of pairs becomes possible. It is also interesting to note that the BCS spectrum can be interpreted in terms of rotons. Indeed, expanding Equation (149View Equation) around the minimum leads, to lowest order, to an expression similar to Equation (146View Equation). In this case, p0 is obtained by solving ε(p) = μ. The other parameters are given by Δr = Δ (p0) and 2 mr = Δ (p0)∕v 0 where v0 = dε∕dp is the group velocity evaluated at p0.

The presence of an “external” potential affects superfluidity. This issue has recently attracted a lot of theoretical, as well as experimental, interest in the field of optically-trapped ultra-cold atomic Bose gases [297]. It is also relevant in the context of neutron stars, where the solid crust is immersed in a neutron superfluid (and possibly a proton superconductor in the liquid crystal mantle, where nuclear “pastas” could be present; see Section 3.3). In the BCS approximation (128View Equation), considering a periodic potential (induced by the solid crust in neutron stars), the quasiparticle energies still take a form similar to Equation (149View Equation). However, the dependence on the momentum is no more isotropic. As shown by Carter, Chamel & Haensel [77Jump To The Next Citation Point], Equation (150View Equation) for the critical velocity should then be replaced by

{ } -Δ-(kkk)-- Vc = miFSn m ⋆v(kkk) , (151 )
where kkk is the Bloch wave vector (Section 3.2.4), −1 vvv(kkk) = ℏ ∇∇∇kkk ε(kkk) is the group velocity of the fermions and m ⋆ is an effective mass, which arises from the interactions between the particles and the lattice as discussed in Section 8.3.6. The subscript FS means that the minimum is to be searched on the Fermi surface7. In the neutron star crust, the effective mass of the unbound neutrons could be very large m ⋆ ≫ mn, as shown by Chamel [90Jump To The Next Citation Point91Jump To The Next Citation Point]. In the limit of homogeneous matter, the single particle energies are given by an expression of the form (144View Equation) with m = mn and ppp = ℏkkk. Consequently, m ⋆ = mn and vvv = ℏkkk∕mn, so that Equation (151View Equation) reduces to Equation (150View Equation) using the extended Brillouin zone scheme. Equation (151View Equation) shows that superfluidity disappears whenever the pairing gap vanishes at some point on the Fermi surface.

The real critical velocity is expected to be smaller than that given by Equation (151View Equation) due to finite temperature and many-body effects beyond the mean field. Likewise, the critical velocity of superfluid helium-4 obtained in Landau’s quasiparticle model is only an upper bound because in this model, the quasiparticles are assumed to be noninteracting. The experimentally-measured critical velocities are usually much smaller, in particular, due to the nucleation of vortices. Indeed, relative motions between superfluid and the vortices lead to mutual friction forces and, hence, to dissipative effects. In general, any curved vortex line does not remain at rest in the superfluid reference frame and, therefore, induces dissipation. Feynman [143Jump To The Next Citation Point] derived the critical velocity associated with the formation of a vortex ring in a channel of radius R,

ℏ ( R ) Vc ≃ ----log --- , (152 ) mR rυ
where r υ is the radius of the vortex core. This result shows that the critical velocity depends, in general, on experimental set-up. More generally, the critical velocity scales like Vc ∝ ℏ∕mL, where L is a characteristic length scale in the experiment. The theoretical determination of the breakdown of superfluidity is still an open issue, which requires a detailed understanding of superfluid dynamics and, in particular, the dynamics of vortices.

8.3.2 Rotating superfluid and vortices

Superfluidity is closely related to the phenomenon of Bose–Einstein condensation as first envisioned by Fritz London [272]. In the superfluid phase, a macroscopic collection of particles condense into the lowest quantum single particle state, which (for a uniform system) is a plane wave state with zero momentum (therefore a constant). Soon after the discovery of the superfluidity of liquid helium, Fritz London introduced the idea of a macroscopic wave function Ψ (rrr), whose squared modulus is proportional to the density n Ψ of particles in the condensate. This density n Ψ, which should not be confused with the superfluid density n 𝒮 introduced in the two-fluid model of superfluids (Section 8.3.6), can be rigorously defined from the one-particle density matrix [322]. The wave function is defined up to a global phase factor. The key distinguishing feature of a superfluid is the symmetry breaking of this gauge invariance by imposing that the phase φ(rrr) be local. The macroscopic wave function thus takes the form

√ --- Ψ (rrr ) = n Ψeiφ(rrr) . (153 )

Applying the momentum operator − iℏ∇ to this wave function shows that superfluid carries a net momentum (per superfluid particle)

ppp = ℏ∇∇∇φ . (154 )

This implies that the superflow is characterized by the condition

∇∇∇ × ppp = 0 . (155 )

In the absence of any entrainment effects (as discussed in Section 8.3.6), the momentum is given by ppp = m vvv, where m is the mass of the superfluid “particles”8 and vvv is the velocity of superfluid. Equation (155View Equation) thus implies that the flow is irrotational. This means, in particular, that a superfluid in a rotating bucket remains at rest with respect to the laboratory reference frame. However, this Landau state is destroyed whenever the rotation rate exceeds the critical threshold for the formation of vortices given approximately by Equation (152View Equation). Experiments show that the whole superfluid then rotates like an ordinary fluid. The condition (155View Equation) can therefore be locally violated as first suggested by Onsager [310] and discussed by Feynman [143]. Indeed, since the phase of the macroscopic wave function is defined modulo 2 π, the momentum circulation over any closed path is quantized

∮ ppp ⋅dddℓℓℓ = N h , (156 )
where N is any integer and h is Planck’s constant. Equation (156View Equation) is simply the Bohr–Sommerfeld quantization rule. This rule can be equivalently deduced from the following argument. Considering superfluid as a macroscopic quantum state, its momentum p is given by p = h ∕λ, where λ is the de Broglie wavelength. The quantization rule (156View Equation) thus follows from the requirement that the length of any closed path must be an integral multiple of the de Broglie wavelength.

In a rotating superfluid the flow quantization implies the formation of vortex lines, each carrying a quantum ℏ of angular momentum, the quantum number N being the number of vortices (the formation of a single vortex carrying all the angular momentum is not energetically favored). The size of the core of a vortex line is roughly on the order of the superfluid coherence length (see Section 8.2.3 for estimates of the coherence length). In some cases, however, it may be much smaller [110134Jump To The Next Citation Point], so that the coherence length is only an upper bound on the vortex core size. In the presence of vortices, Equation (155View Equation) must therefore be replaced by

∇∇∇ × ppp = m κκκ , (157 )
κκκ being the circulation of the N vortex lines. The existence of quantized vortices was demonstrated by Vinen [415] in 1961, however, they were not observed until much later, in 1974 at Berkeley [423432]. At length scales much larger than the superfluid coherence length, the finite size of the vortex core can be neglected and the circulation is, thus, given by
N κκκ = ∑ δ(2)(rrr − rrr lllppphhhaaa )κκκ , (158 ) aaa ααα α=1
where δ(2) is the two-dimensional Dirac distribution, and κκκααα is the circulation of a given vortex α
∮ 1 h κκκααα = κˆˆκˆκααα -- ppp ⋅dddℓℓℓ =--ˆκˆκˆκααα , (159 ) m m
ˆκˆκˆκααα being a unit vector directed along the vortex line. Equation (157View Equation) is formally similar to Ampere’s law in magnetostatics. The momentum ppp induced by the vortex lines is, thus, given by the Biot-Savart equation, substituting 1 ∕c with the mass m divided by 4π in Gaussian cgs units (or replacing the magnetic permeability μ0 by the mass m in SI units) and the electric current I by κ,
∫ ′′′ p r m-- ′κ r′′′ -rrr-−-rrr-- pp (rr ) = 4π dr κκ(rr) × |r − r′|3 , (160 )
where the integral is taken along the vortex lines as shown in Figure 52View Image. The analogy between hydrodynamics and magnetostatics shows, in particular, that a vortex ring should move along its symmetry axis with a velocity inversely proportional to its radius.
View Image

Figure 52: Momentum ppp(rrr) induced at a position rrr by the vortex line with circulation κκκ.

As shown by Tkachenko [405406], quantized vortices tend to arrange themselves on a regular triangular array. Such patterns of vortices have been observed in superfluid helium and more recently in atomic Bose–Einstein condensates. The intervortex spacing d υ is given by

∘ -------- d = √-h----, (161 ) υ 3m Ω
where Ω is the angular frequency. At length scales much larger than the intervortex spacing dυ, as a result of the superposition of the flow pattern of all the vortex lines, superfluid flow mimics rigid body rotation. Since at this scale a fluid element is threaded by many vortex lines, it is relevant to smoothly average the hydrodynamic equations governing the flow of the superfluid. In particular, Equation (157View Equation) now reads
∇∇∇ × ppp = mn υκκκ , (162 )
where nυ is the surface density of vortices given (in the absence of entrainment effects) by
n = m-Ω-, (163 ) υ πℏ
and the vector κκκ, whose norm is equal to h∕m, is aligned with the average angular velocity (generalization of Equation (163View Equation) to account for entrainment effects is discussed in Section 10.4).

8.3.3 Type II superconductors and magnetic flux tubes

Let us remark that the condition (155View Equation) for superfluids also applies to superconductors, like the proton superconductor in the liquid core and possibly in the “pasta” mantle of neutron stars (Section 3.3). The momentum of a superconductor is given by p v A pp ≡ m vv + qAA (in this section, we use SI units), where m, q, and v are the mass, electric charge and velocity of “superconducting” particles respectively9, and AAA the electromagnetic potential vector. Introducing the density n of superconducting particles and their electric current density (referred to simply as “supercurrent”) 𝒥𝒥𝒥 = nqvvv, Equation (155View Equation) leads to the London equation

nq2- ∇∇∇ × 𝒥𝒥𝒥 = − m BBB , (164 )
where BBB = ∇∇∇ × AAA is the magnetic field induction. According to the Bohr–Sommerfeld quantization rule (156View Equation), the London Equation (164View Equation) corresponds to N = 0. Situations with N > 0 are encountered in type II superconductors. Considering a closed contour outside a sample of such a superconductor for which 𝒥 = 0 and integrating the momentum ppp along this contour, leads to the quantization of the total magnetic flux into N flux tubes
∮ hN AAA ⋅dddℓℓℓ = ----. (165 ) q

These flux tubes tend to arrange themselves into a triangular lattice, the Abrikosov lattice, with a spacing given by

∘ ------- 2h dυ = √----- . (166 ) 3qB
Averaging at length scales much larger than d υ, the surface density of flux tubes is given by
nυ = qB-. (167 ) h

8.3.4 Superfluid vortices and magnetic flux tubes in neutron stars

For neutron superfluid in neutron stars, superfluid particles are neutron pairs, so that m = 2mn. As early as in 1964, Ginzburg & Kirzhnits [161162] suggested the existence of quantized vortex lines inside neutron stars. The critical velocity for the nucleation of vortices can be roughly estimated from V ∼ ℏ∕m R c n where R is the radius of a neutron star. For R = 10 km, 3 Vc ∼ 10 fm ∕s, which is, by several orders of magnitude, smaller than characteristic velocities of matter flows within the star. The interior of neutron stars is thus threaded by a huge number of vortices. The intervortex spacing is

∘ --2--−1- dυ ≃ 3.4 × 10−3 10--s-- cm , (168 ) Ω
which is much larger than the coherence length, so that the assumption of infinitely-thin vortex lines in Equation (158View Equation) is justified. Assuming that neutron superfluid is uniformly co-rotating with the star, the density of vortices per square kilometer is then given by 14 n υ ≃ 10 ∕P, where P is the rotation period in seconds. With a rotation period of 33 milliseconds, a pulsar like the Crab is threaded by an array of about 1018 vortex lines (assuming a radius of 10 km)! Likewise, proton superconductors (assumed to be of type II [41Jump To The Next Citation Point]) in the core of neutron stars (and possibly in the pasta layers; see Section 3.3) is threaded by quantized magnetic flux tubes each carrying a magnetic field on the order of 1015 G. The surface density of flux tubes is about 1013 times that of neutron vortices. The interactions between the neutron vortices and the flux tubes are likely to affect the dynamic evolution of the star [354Jump To The Next Citation Point355Jump To The Next Citation Point].

8.3.5 Dynamics of superfluid vortices

In this section, we will discuss the nonrelativistic dynamics of superfluid vortices. The generalization to relativistic dynamics has been discussed in detail by Carter [71]. According to the Helmholtz theorem, the vortex lines are frozen in superfluid and move with the same velocity unless some force acts on them. The dynamics of a vortex line through the crust is governed by different types of forces, which depend on the velocities vvvfff, vvvυυυ and vvvccc of the bulk neutron superfluid, the vortex and the solid crust, respectively.

All forces considered above are given per unit length of the vortex line. Let us remark that even in the fastest millisecond pulsars, the intervortex spacing (assuming a regular array) of order −3 −4 dυ ∼ 10 – 10 cm is much larger than the size of the vortex core rυ ∼ 10– 100 fermis. Consequently the vortex-vortex interactions can be neglected.

The dynamic evolution of a vortex line is governed by

m υdvvvυυυ -------= ℱℱℱddd + ℱℱℱttt + ℱℱℱmmm , (173 ) Lυ dt
where m υ is the inertial mass of the vortex line and L υ its length. Since the free neutron density inside the vortex core is typically much smaller than outside (unless the line is pinned to nuclei) [134Jump To The Next Citation Point43327Jump To The Next Citation Point], the motion of the vortex line is accompanied by a rearrangement of the free neutrons. The inertial mass of the vortex line is approximately equal to the mass density of the neutron superfluid ρf times the volume of the line πr2υL υ so that m υ∕L υ ∼ ρfπr2υ.

On a scale much larger than the intervortex spacing, the drag force ℱℱℱddd acting on every vortex line leads to a mutual friction force between the neutron superfluid and the normal constituents. Assuming that the vortex lines are rigid and form a regular array, the mutual friction force, given by fffddd = n υℱℱℱddd, can be obtained from Equation (173View Equation) after multiplying by the surface density nυ. Since 2 nυ ∼ 1∕πd υ, the inertial term on the left-hand side of Equation (173View Equation) is proportional to (rυ∕dυ)2 ≪ 1 and can be neglected. Solving the force balance equation yields the mutual friction force (per unit volume) [18Jump To The Next Citation Point]

--ℬ2--- --ℬ---- fdffdd = nυℱℱℱddd = 1 + ℬ2 ρfnυκκκ × (vvvNNN − vvvfff) + 1 + ℬ2ρfn υˆκˆκˆκ × κκκ × (vvvccc − vvvfff), (174 )
where nυ is the surface density of vortices in a plane perpendicular to the axis of rotation and ℬ is a dimensionless parameter defined by
-ℛ-- ℬ = ρ κ . (175 ) f

Different dissipative mechanisms giving rise to a mutual friction force have been invoked: scattering of electrons/lattice vibrations (phonons)/impurities/lattice defects by thermally excited neutrons in vortex cores [141189222], electron scattering off the electric field around a vortex line [48], and coupling between phonons and vortex line oscillations (Kelvin modes) [139223]. In the weak coupling limit ℬ → 0, the vortices co-rotate with the bulk superfluid (Helmholtz theorem), while in the opposite limit ℬ → +∞, they are “pinned” to the crust. In between these two limits, in a frame co-rotating with the crust, the vortices move radially outward at angle atan (ℛ) with respect to the azimuthal direction. The radial component of the vortex velocity reaches a maximum at ℛ = 1.

Vortex pinning plays a central role in theories of pulsar glitches. The strength of the interaction between a small segment of the vortex line and a nucleus remains a controversial issue [513833313412012112227]. The actual “pinning” of the vortex line (i.e., vvvυυυ = vvvccc) depends not only on the vortex-nucleus interaction, but also on the structure of the crust, on the rigidity of lines and on the vortex dynamics. For instance, assuming that the crust is a polycrystal, a rigid vortex line would not pin to the crust simply because the line cannot bend in order to pass through the nuclei, independent of the strength of the vortex-nucleus interaction! Recent observations of long-period precession in PSR 1828–11 [387Jump To The Next Citation Point], PSR B1642–03 [371Jump To The Next Citation Point] and RX J0720.4–3125 [180Jump To The Next Citation Point] suggest that, at least in those neutron stars, the neutron vortices cannot be pinned to the crust and must be very weakly dragged [372266].

Let us stress that the different forces acting on a vortex vary along the vortex line. As a consequence, the vortex lines may not be straight [198]. The extent to which the lines are distorted depends on the vortex dynamics. In particular, Greenstein [175] suggested a long time ago that vortex lines may twist and wrap about the rotation axis giving rise to a turbulent flow. This issue has been more recently addressed by several groups [323324Jump To The Next Citation Point28819]. In such a turbulent regime the mutual friction force takes the form [173]

𝒞d- 2 fffddd = κ (vvvccc − vvvfff) (vvccvc − vvvfff), (176 )
where 𝒞d is a dimensionless temperature-dependent coefficient, assuming a dense random tangle of vortex lines.

8.3.6 Superfluid hydrodynamics and entrainment

One of the striking consequences of superfluidity is the allowance for several distinct dynamic components. In 1938, Tisza [404] introduced a two-fluid model in order to explain the properties of the newly discovered superfluid phase of liquid helium-4, which behaves either like a fluid with no viscosity in some experiments or like a classical fluid in other experiments. Guided by the Fritz London’s idea that superfluidity is intimately related to Bose–Einstein condensation (which is now widely accepted), Tisza proposed that liquid helium is a mixture of two components, a superfluid component, which has no viscosity, and a normal component, which is viscous and conducts heat, thus, carrying all the entropy of the liquid. These two fluids are allowed to flow with different velocities. This model was subsequently developed by Landau [247246] and justified on a microscopic basis by several authors, especially Feynman [144Jump To The Next Citation Point]. Quite surprisingly, Landau never mentioned Bose–Einstein condensation in his work on superfluidity. According to Pitaevskii (as recently cited by Balibar [33]), Landau might have reasoned that superfluidity and superconductivity were similar phenomena (which is indeed true), incorrectly concluding that they could not depend on the Bose or Fermi statistics (see also the discussion by Feynman in Section 11.2 of his book [144]).

In Landau’s two-fluid model, the normal part with particle density n 𝒩 and velocity vvv𝒩𝒩𝒩 is identified with the collective motions of the system or “quasiparticles” (see Section 8.3.1). The viscosity of the normal fluid is accounted for in terms of the interactions between those quasiparticles (see, for instance, the book by Khalatnikov [236] published in 1989 as a reprint of an original 1965 edition). Following the traditional notations, the superfluid component, with a particle density n𝒮 and a “velocity” vvv𝒮𝒮𝒮, is locally irrotational except at singular points (see the discussion in Section 8.3.2)

∇∇∇ × vvv = 0. (177 ) 𝒮𝒮𝒮
As pointed out many times by Brandon Carter, unlike vvv𝒩𝒩𝒩 the “superfluid velocity” is not a true velocity but is defined through
ppp ≡ m vvv𝒮𝒮𝒮 , (178 )
where ppp is the true momentum per particle of the superfluid and m is the mass of a helium atom. Although deeply anchored in the Lagrangian and Hamiltonian formulation of classical mechanics, the fundamental distinction between velocities and their canonical conjugates, namely momenta, has been traditionally obscured in the context of superfluidity. Note also that the superfluid density n 𝒮 coincides neither with the density n of helium atoms (except at T = 0) nor with the density nΨ of atoms in the condensate10. The density of helium atoms is given by
n = n𝒮 + n𝒩 . (179 )

The confusion between velocity and momentum is very misleading and makes generalizations of the two-fluid model to multi-fluid systems (like the interior of neutron stars) unnecessarily difficult. Following the approach of Carter (see Section 10), the two-fluid model can be reformulated in terms of the real velocity vvv of the helium atoms instead of the superfluid “velocity” vvv 𝒮𝒮𝒮. The normal fluid with velocity vvv𝒩𝒩𝒩 is then associated with the flow of entropy and the corresponding number density is given by the entropy density. At low temperatures, heat dissipation occurs via the emission of phonons and rotons. As discussed in Section 8.3.1, these quasiparticle excitations represent collective motions of atoms with no net mass transport (see, in particular, Figures 50View Image and 51View Image). Therefore, the normal fluid does not carry any mass, i.e., its associated mass is equal to zero.

Following the general principles reviewed in Section 10, the momentum ppp of the superfluid helium atoms can be written as

ppp = m ⋆vvv + (m − m ⋆)vvv𝒩𝒩𝒩 . (180 )
Note that the momentum ppp of the helium atoms is not simply equal to m vvv due to the scattering of atoms by quasiparticles. In the rest frame of the normal fluid, in which v = 0 𝒩, the momentum and the velocity of the superfluid are aligned. However, the proportionality coefficient is not the (bare) atomic mass m of helium but an effective mass m ⋆. This effective mass m ⋆ is related to the hydrodynamics of superfluid and should not be confused with the definitions employed in microscopic many-body theories. Before going further, let us remark that in the momentum rest frame of the normal component, the relation ppp = m vvv holds! This can easily be shown from Equation (230View Equation), by using the identity (234View Equation) and remembering that the normal fluid is massless.

Comparing Equations (180View Equation) and (178View Equation) shows that the “superfluid velocity” in the original two-fluid model of Landau is not equal to the velocity of the helium atoms but is a linear combination of both velocities vvv and vvv𝒩𝒩𝒩

m ⋆ m ⋆ vvv𝒮𝒮𝒮 = m--vvv + (1 − m--)vvv𝒩𝒩𝒩 . (181 )
The current of helium atoms is given by the sum of the normal and superfluid currents
n vvv = n vvv + n vvv . (182 ) 𝒮 𝒮𝒮𝒮 𝒩 𝒩𝒩𝒩
Substituting Equation (181View Equation) into Equation (182View Equation) yields the relations
( ) n = n m--, n = n 1 − m-- , (183 ) 𝒮 m ⋆ 𝒩 m ⋆
which clearly satisfies n = n 𝒮 + n𝒩. The superfluid and normal densities can be directly measured in the experiment devised by Andronikashvili [21]. A stack of disks, immersed in superfluid, can undergo torsional oscillations about its axis. Due to viscosity, the normal component is dragged by motion of the disks, while the superfluid part remains at rest. The normal and superfluid densities can, thus, be obtained at any temperature by measuring the oscillation frequency of the disks. Since n 𝒮(T)∕n = m ∕m ⋆(T ), the dynamic effective mass m ⋆(T ) can be determined experimentally. In particular, it is equal to the bare mass at T = 0, m ⋆(T = 0 ) = m, and goes up as the temperature is raised, diverging at the critical point when superfluidity disappears. At any temperature T > 0 the dynamic effective mass of an helium atom is therefore larger than the bare mass.

Entrainment effects, whereby momentum and velocity are not aligned, exist in any fluid mixtures owing to the microscopic interactions between the particles. But they are usually not observed in ordinary fluids due to the viscosity, which tends to equalize velocities. Even in superfluids like liquid Helium II, entrainment effects may be hindered at finite temperature11 by dissipative processes. For instance, when a superfluid is put into a rotating container, the presence of quantized vortices induces a mutual friction force between the normal and superfluid components (as discussed in Section 8.3.5). As a consequence, in the stationary limit the velocities of the two fluids become equal. Substituting vvv = vvv𝒩𝒩𝒩 in Equation (180View Equation) implies that ppp = m vvv, as in the absence of entrainment.

8.3.7 Entrainment effects in neutron stars

A few years after the seminal work of Andreev & Bashkin [20Jump To The Next Citation Point] on superfluid 3He – 4He mixtures, it was realized that entrainment effects could play an important role in the dynamic evolution of neutron stars (see, for instance, [363] and references therein). For instance, these effects are very important for studying the oscillations of neutron star cores, composed of superfluid neutrons and superconducting protons [14]. Mutual entrainment not only affects the frequencies of the modes but, more surprisingly, (remembering that entrainment is a nondissipative effect) also affects their damping. Indeed, entrainment effects induce a flow of protons around each neutron superfluid vortex line. The outcome is that each vortex line carries a huge magnetic field ∼ 1014 G [8]. The electron scattering off these magnetic fields leads to a mutual friction force between the neutron superfluid and the charged particles (see [18] and references therein). This mechanism, which is believed to be the main source of dissipation in the core of a neutron star, could also be at work in the bottom layers of the crust, where some protons might be unbound and superconducting (as discussed in Section 3.3).

It has recently been pointed out that entrainment effects are also important in the inner crust of neutron stars, where free neutrons coexist with a lattice of nuclear clusters [79Jump To The Next Citation Point78Jump To The Next Citation Point]. It is well known in solid state physics that free electrons in ordinary metals move as if their mass were replaced by a dynamic effective mass me⋆ (usually referred to as an optical mass in the literature) due to Bragg scattering by the crystal lattice (see, for instance, the book by Kittel [241]). The end result is that, in the rest frame of the solid, the electron momentum is given by

pppeee = me⋆vvveee, (184 )
where vvveee is the electron velocity. This implies that in an arbitrary frame, where the solid (ion lattice) is moving with velocity vvv III, the electron momentum is not aligned with the electron velocity but is given by
peee ev e v pp = m ⋆vveee + (me − m ⋆)vvIII , (185 )
which is similar to Equation (180View Equation) for the momentum of superfluid Helium II. The concept of dynamic effective mass was introduced in the context of neutron diffraction experiments ten years ago [434] and has only recently been extended to the inner crust of neutron stars by Carter, Chamel & Haensel [79Jump To The Next Citation Point78Jump To The Next Citation Point]. While the dynamic effective electron mass in ordinary metallic elements differs moderately from the bare mass me⋆ ∼ 1– 2me (see, for instance, [204]), Chamel [90Jump To The Next Citation Point] has shown that the dynamic effective mass of free neutrons in neutron star crust could be very large, mf⋆ ∼ 10– 15mn. The dynamics of the free neutrons is deeply affected by these entrainment effects, which have to be properly taken into account (see Section 10). Such effects are important for modeling various observed neutron star phenomena, like pulsar glitches (see Section 12.4) or neutron star oscillations (see Sections 12.5 and 12.6).

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