
According to the Bohr–Wheeler fission condition [52] an isolated spherical nucleus in vacuum is stable with respect to quadrupolar deformations if
where and are the Coulomb and surface energies of the nucleus, respectively. The superscript reminds us that we are considering a nucleus in vacuum. The Bohr–Wheeler condition can be reformulated in order to be applied in the inner crust, where both Coulomb and surface energies are modified compared to the “in vacuum” values. Neglecting curvature corrections and expanding all quantities to the linear order in , where is the fraction of volume occupied by the clusters, it is found [326] that spherical clusters become unstable to quadrupolar deformation if .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].

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 rodlike and slablike 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 rodlike 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 rodlike and slablike nuclei by cooling down hot uniform nuclear matter without any assumption of the nuclear shape. They also found the appearance of intermediate spongelike structures, which might be identified with the ordered, bicontinuous, doublediamond 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 crustcore 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 Xray 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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