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According to the Bohr–Wheeler fission condition [52] an isolated spherical nucleus in vacuum is stable with respect to quadrupolar deformations if
where Reasoning by analogy with percolating networks, Ogasawara & Sato [308] suggest that as the nuclei fill
more and more space, they will eventually deform, touch and merge to form new structures. A long
time ago, Baym, Bethe and Pethick [39] predicted that as the volume fraction exceeds 1/2,
the crust will be formed of neutron bubbles in nuclear matter. In the general framework of
the compressible liquid drop model considering the simplest geometries, Hashimoto and his
collaborators [191, 315] show that as the nuclear volume fraction
increases, the stable nuclear
shape changes from sphere to cylinder, slab, tube and bubble, as illustrated in Figure 17
. This
sequence of nuclear shapes referred to as “pastas” (the cylinder and slab shaped nuclei resembling
“spaghetti” and “lasagna” respectively) was found independently by Ravenhall et al. [345] with a
specific liquid drop model. The volume fractions at which the various phases occur are in good
agreement with those predicted by Hashimoto and collaborators on purely geometrical grounds. This
criterion, however, relies on a liquid drop model, for which curvature corrections to the surface
energy are neglected. This explains why some authors [274
, 124
] find within the liquid drop
model that spherical nuclei remain stable down to the transition to uniform nuclear matter,
despite volume fractions exceeding the critical threshold (see in particular Figure 7
), while
other groups found the predicted sequence of pasta phases [419
, 420
, 206
, 207
]. The nuclear
curvature energy is, thus, important for predicting the equilibrium shape of the nuclei at a given
density [327].
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These pasta phases have been studied by various nuclear models, from liquid drop models to
semiclassical models, quantum molecular dynamic simulations and Hartree–Fock calculations (for the
current status of this issue, see, for instance, [417]). These models differ in numerical values of the densities
at which the various phases occur, but they all predict the same sequence of configurations shown in
Figure 17
(see also the discussion in Section 5.4 of [326
]). Some models [274
, 97, 124, 282] predict that
the spherical clusters remain energetically favored throughout the whole inner crust. Generalizing the
Bohr–Wheeler condition to nonspherical nuclei, Iida et al. [206, 207] showed that the rod-like and
slab-like clusters are stable against fission and proton clustering, suggesting that the crust layers
containing pasta phases may be larger than that predicted by the equilibrium conditions. It
has also been suggested that the pinning of neutron superfluid vortices in neutron star crusts
might trigger the formation of rod-like clusters [293
]. Nevertheless, the nuclear pastas may be
destroyed by thermal fluctuations [419
, 420
]. Quite remarkably, Watanabe and collaborators [417]
performed quantum molecular dynamic simulations and observed the formation of rod-like and
slab-like nuclei by cooling down hot uniform nuclear matter without any assumption of the
nuclear shape. They also found the appearance of intermediate sponge-like structures, which
might be identified with the ordered, bicontinuous, double-diamond geometry observed in block
copolymers [284]. Those various phase transitions leading to the pasta structures in neutron star crusts
are also relevant at higher densities in neutron star cores, where kaonic or quark pastas could
exist [281].
The pasta phases cover a small range of densities near the crust-core interface with .
Nevertheless, by filling the densest layers of the crust, they may represent a sizable fraction of the
crustal mass [274
] and thus may have important astrophysical consequences. For instance, the
existence of nuclear pastas in hot dense matter below saturation density affects the neutrino
opacity [201
, 384
], which is an important ingredient for understanding the gravitational core collapse
of massive stars in supernova events and the formation of neutron stars (see Section 12.1).
The dynamics of neutron superfluid vortices, which is thought to underlie pulsar glitches (see
Section 12.4), is likely to be affected by the pasta phase. Besides, the presence of nonspherical
clusters in the bottom layers of the crust influences the subsequent cooling of the star, hence
the thermal X-ray emission by allowing direct Urca processes [274
, 179
] (see Section 11) and
enhancing the heat capacity [112, 113, 135
]. The elastic properties of the nuclear pastas can
be calculated using the theory of liquid crystals [325
, 419
, 420
] (see Section 7.2). The pasta
phase could thus affect the elastic deformations of neutron stars, oscillations, precession and
crustquakes.
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