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10.3 Entrainment and effective masses

The momenta of the free superfluid neutrons and of the confined nucleons can be expressed in terms of the velocities as
pfppff = m ⋆fvvvfff + (m − m ⋆f)vvvccc, (243 )
ccc ⋆ ⋆ ppp = m cvvccvc + (m − m c)vvffvf, (244 )
where m ⋆ f and m ⋆ c are dynamic effective masses (for a generalization to relativistic fluids, see Chamel [93Jump To The Next Citation Point]) so that in the crust (resp. neutron superfluid) rest frame we have pppfff = m ⋆vvv f fff (resp. ccc ⋆ ppp = m cvvvccc). These effective masses arise because of the momentum transfer between the free neutrons and the nuclear lattice (Bragg scattering). Due to Galilean invariance, these effective masses are not independent but are related by
⋆ ρc ⋆ m f − m = ρ (m c − m ), (245 ) f
where ρf and ρc are the mass densities of the neutron superfluid and confined nucleons, respectively. This entails that the total momentum density is simply ρfvvvfff + ρcvvvccc. These dynamic effective masses can be expressed directly from the internal Lagrangian density Λint of the system by (X=f,c)
mX⋆-= 1 + 2--∂Λint, (246 ) m ρX ∂ω2
where ρX = nXm and ωωω = vvvfff − vvvccc is the relative velocity between the neutron superfluid and the crust.

Alternatively, we could introduce different kinds of effective masses m ♯f and m ♯c, so that the momentum and the velocity of one fluid are aligned in the momentum rest frame of the other fluid,

m ♯− m = -m- (m ⋆ − m ), (247 ) f m ⋆c f
♯ m-- ⋆ m c − m = m ⋆(m c − m ) . (248 ) f
This shows, in particular, that these effective masses are either all greater or all smaller than the nucleon mass m. From a stability analysis, Carter, Chamel and Haensel [79Jump To The Next Citation Point] proved that these effective masses obey the following inequalities (X = f,c)
m ⋆X ρX ----> ---, (249 ) m ρ
m-♯X- -ρ- m < ρ , (250 ) X
where ρ = ρf + ρc is the total mass density. Microscopic calculations carried out by Chamel [90Jump To The Next Citation Point91Jump To The Next Citation Point], using for the first time the band theory of solids (see Section 3.2.4), show that these effective masses can be much larger than the bare nucleon mass m. This is in sharp contrast to the situation encountered in the liquid core, where effective masses are slightly smaller than m (see, for instance, Chamel & Haensel [95] and references therein). Note, however, that the relativistic effective neutron mass can be slightly larger than the bare mass in the liquid core, as recently shown by Chamel [93].
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