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3.3 Pastas

The equilibrium structure of nuclear clusters in neutron star crusts results from the interplay between the total Coulomb energy and the surface energy of the nuclei according to the virial Equation (49View Equation). At low densities, the lattice energy Equation (41View Equation) is a small contribution to the total Coulomb energy and nuclear clusters are therefore spherical. However, at the bottom of the crust the nuclei are very close to one another. Consequently, the lattice energy represents a large reduction of the total Coulomb energy, which vanishes in the liquid core when the nuclear clusters fill all space, as can be seen from Equation (51View Equation). In the densest layers of the crust the Coulomb energy is comparable in magnitude to the net nuclear binding energy (this situation also occurs in the dense and hot collapsing core of a supernova and in heavy ion collisions leading to multifragmentation [5657]). The matter thus becomes frustrated and can arrange itself into various exotic configurations as observed in complex fluids. For instance, surfactants are organic compounds composed of a hydrophobic “tail” and a hydrophilic “head”. In solutions, surfactants aggregate into ordered structures, such as spherical or tubular micelles or lamellar sheets. The transition between the different phases is governed by the ratio between the volume of the hydrophobic and hydrophilic parts as shown in Figure 16View Image. One may expect by analogy that similar structures could occur in the inner crust of a neutron star depending on the nuclear packing.
View Image

Figure 16: Structures formed by self-assembled surfactants in aqueous solutions, depending on the volume ratio of the hydrophilic and hydrophobic parts. Adapted from [226].

According to the Bohr–Wheeler fission condition [52] an isolated spherical nucleus in vacuum is stable with respect to quadrupolar deformations if

(0) (0) EN,Coul < 2EN,surf , (66 )
where E (0) N,Coul and E (0) N,surf are the Coulomb and surface energies of the nucleus, respectively. The superscript (0) 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 w1 ∕3, where w is the fraction of volume occupied by the clusters, it is found [326Jump To The Next Citation Point] that spherical clusters become unstable to quadrupolar deformation if w > wcrit = 1∕8 .

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 [191315Jump To The Next Citation Point] show that as the nuclear volume fraction w increases, the stable nuclear shape changes from sphere to cylinder, slab, tube and bubble, as illustrated in Figure 17View Image. 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 [274Jump To The Next Citation Point124Jump To The Next Citation Point] 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 7View Image), while other groups found the predicted sequence of pasta phases [419Jump To The Next Citation Point420Jump To The Next Citation Point206Jump To The Next Citation Point207Jump To The Next Citation Point]. The nuclear curvature energy is, thus, important for predicting the equilibrium shape of the nuclei at a given density [327].

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Figure 17: Sketch of the sequence of pasta phases in the bottom layers of ground-state crusts with an increasing nuclear volume fraction, based on the study of Oyamatsu and collaborators [315].

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, [417Jump To The Next Citation Point]). 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 17View Image (see also the discussion in Section 5.4 of [326Jump To The Next Citation Point]). Some models [274Jump To The Next Citation Point97124282] predict that the spherical clusters remain energetically favored throughout the whole inner crust. Generalizing the Bohr–Wheeler condition to nonspherical nuclei, Iida et al. [206207] 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 [293Jump To The Next Citation Point]. Nevertheless, the nuclear pastas may be destroyed by thermal fluctuations [419Jump To The Next Citation Point420Jump To The Next Citation Point]. 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 14 −3 ρ ∼ 10 g cm. Nevertheless, by filling the densest layers of the crust, they may represent a sizable fraction of the crustal mass [274Jump To The Next Citation Point] 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 [201Jump To The Next Citation Point384Jump To The Next Citation Point], 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 [274Jump To The Next Citation Point179Jump To The Next Citation Point] (see Section 11) and enhancing the heat capacity [112113135Jump To The Next Citation Point]. The elastic properties of the nuclear pastas can be calculated using the theory of liquid crystals [325Jump To The Next Citation Point419Jump To The Next Citation Point420Jump To The Next Citation Point] (see Section 7.2). The pasta phase could thus affect the elastic deformations of neutron stars, oscillations, precession and crustquakes.

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