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10.4 Neutron superfluidity

As already discussed in Section 8.3.2, a superfluid is locally characterized by condition (155View Equation), which, in the 4D covariant framework [7475], reads
ϖf = 0 . (251 ) μν
However, at length scales that are large compared to the intervortex spacing, the neutron vorticity 2-form will not vanish. Since the vorticity of superfluid is carried by quantized vortex lines (as illustrated in Figure 60View Image), mathematically this means that the Lie derivative of the vorticity 2-form along the 4-velocity vector μ uυ of the vortex lines, vanishes:
⃗uυ£ ϖfμν = 0 . (252 )
This condition is equivalent to
μ f uυϖμν = 0 . (253 )
View Image

Figure 60: Sketch of the 2-surface swept out by a quantized vortex line moving with the 4-velocity u υ ; Ω is the superfluid angular velocity.

Let us remark that the definition of neutrons that have to be counted as “free” is not unique and there is some arbitrariness in the above model. Nonetheless, it can be shown that the 4-momentum co-vector πfμ of the neutron superfluid is invariant under such “chemical” readjustments and the above superfluidity conditions are well defined [79]. Note also that these conditions are valid for both relativistic and nonrelativistic superfluids.

As discussed by Chamel & Carter [94Jump To The Next Citation Point], there are two cases, which are consistent with the nondissipative models considered here. The first possibility is that the neutron vortices are free and are co-moving with the superfluid, i.e. u μ= uμ υ f. On a sufficiently short time scale, it may be further assumed that the free neutron current is conserved, which implies from Equations (221View Equation) and (253View Equation) that the force f fμ vanishes. Since no external force is supposed to be exerted on the system, the force acting on the confined nucleons also vanishes. However, on longer time scales, as discussed in Section 10.2, it would be more appropriate to replace Equation (240View Equation) by the equilibrium condition Equation (241View Equation). In this case, there will still be a force acting on the superfluid (hence, also a force, acting on the confined nucleons) owing to the conversion of free neutrons into confined protons and vice versa. The other possibility is that the vortices are pinned to the crust, so that uμυ = uμc. As shown by Chamel & Carter [94Jump To The Next Citation Point], the pinning condition uμcϖfμν = 0 is equivalent to imposing that the individual vortices be subject to the corresponding Magnus force.

The dynamics of the neutron superfluid vortices in the crust play a major role in the understanding of pulsar glitches and post-glitch relaxation. Due to entrainment effects, the distribution of vortices is not simply given by Equation (163View Equation), but also depends on the angular velocity of the crust according to the following formula derived by Chamel & Carter [94Jump To The Next Citation Point]:

( ⋆ ) n = 2m-Ωf-+ 2m-- m-f − 1 (Ω − Ω ), (254 ) υ π ℏ πℏ m f c
assuming that the neutron superfluid and the crust are uniformly rotating around the same axis with angular velocities Ωf and Ωc, respectively. In addition, spatial variations of the effective masses are neglected. Since the superfluid rotates faster than the crust and since the dynamic effective neutron mass is larger than the bare mass, the entrainment effects increase the surface density of the vortices (whereas the entrainment effects decrease the surface density of neutron vortices in the liquid core).

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