### 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
where and are dynamic effective masses (for a generalization to relativistic fluids, see
Chamel [93]) so that in the crust (resp. neutron superfluid) rest frame we have
(resp. ). 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
where and are the mass densities of the neutron superfluid and confined nucleons, respectively.
This entails that the total momentum density is simply . These dynamic effective masses can
be expressed directly from the internal Lagrangian density of the system by (X=f,c)
where and is the relative velocity between the neutron superfluid and the
crust.
Alternatively, we could introduce different kinds of effective masses and , so that the
momentum and the velocity of one fluid are aligned in the momentum rest frame of the other fluid,

This shows, in particular, that these effective masses are either all greater or all smaller than the nucleon
mass . From a stability analysis, Carter, Chamel and Haensel [79] proved that these effective masses
obey the following inequalities ()
where is the total mass density. Microscopic calculations carried out by Chamel [90, 91],
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 . This is in sharp contrast to the situation
encountered in the liquid core, where effective masses are slightly smaller than (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].