Nuclear “pasta.” The darker regions show the liquid phase, in which protons and neutrons coexist (i.e., the nuclear matter region); the lighter ones the gas phase, which is almost free of protons. Sequence (a)-(e) shows that of nuclear shape changes with increasing density. This figure is taken from Ref. [45]. 

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... 3 . An explicit calculation in Ref. [ 49] shows that values of the elastic constants are B ∼ w C+L , C ≃ w C+L and K 3 ≃ 0 . 05 w C+L l 2 . Using the Lindemann criterion and the above typical values for l and w C+L , one can estimate the melting temperature T m 10 MeV for supernova matter and T m 1 MeV for neutron star matter. In the above estimate, however, decrease of the surface tension due to the evaporation of nucleons and thermal broadening of the nuclear density profile is not taken into account. Thus the melting temperature for supernova matter would be smaller than the above value in the real situation. More elaborated calculations on the Landau-Peierls instability and of the melting temperature can be seen in Refs. [43, 64, 65]. In closing the present section, let us briefly mention several phases which are not shown in Fig. 3. In block copolymer melts, various phases with complicated structures, e.g., gyroid phase, perforated lamellar phase and double diamond phase, etc., have been observed. As will be seen in Section 4, our simulations suggest that phases with multiply connected structures also appear in the nuclear systems (see also Ref. [50]). This point should be examined in the future study 3 . Since the seminal works by Ravenhall et al. [53] and Hashimoto et al. [20], properties of the pasta phases in equilibrium states have been investigated using various nuclear models. They include studies on phase diagrams at zero temperature [34, 45, 47, 59, 64, 65, 70] and at finite temperatures [32]. These earlier works have confirmed that, for various nuclear models, the nuclear shape changes in the way of Eq. (14) as predicted by Refs. [20, 53]. In these earlier works, however, a liquid drop model or the Thomas-Fermi approximation is used with an assumption on the nuclear shape (except for Ref. [70]). Thus the phase diagram at subnuclear densities and the existence of the pasta phases should be examined without assuming the nuclear shape. It is also noted that at temperatures of several MeV, which are relevant to the collapsing cores, effects of thermal fluctuations on the nucleon distribution are significant. However, these thermal fluctuations cannot be described prop- erly by mean-field theories such as the Thomas-Fermi approximation used in the previous work [32]. In contrast to the equilibrium properties, dynamical or non-equilibrium aspects of the pasta phases had not been studied until recently except for some limited cases [24, 50]. Thus it had been unclear even whether or not the pasta phases can be formed dynamically within the time scale of the cooling of neutron stars nor whether or not the formation ...

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... phases and that of the nucleon is not large. Due to this small gap, we can study pasta phases using some microscopic approach: the quantum molecular dynamics (QMD) is one possibility. We have explained the theoretical framework of molecular dynamics calculations for nucleons including QMD and then given an overview of our previous work. Our QMD simulations have shown that the pasta phases can be formed dynamically by cooling down hot uniform nuclear matter in a finite time scale ∼ O (10 3 − 10 4 ) fm /c , which is much shorter than the cooling time of a neutron star. We have also shown that transitions between the pasta phases can occur by compression during the collapse of a star. Our latest result strongly suggests that the pasta phase with rod-like nuclei can be formed by compressing a bcc lattice of spherical nuclei; this result will be examined in detail. As we have shown, nuclear pasta is an interesting system for a wide spectrum of researchers not only to nuclear astrophysicists. Physics of the pasta phases draws on nuclear physics, condensed matter physics, and astrophysics. The pasta phases are important for understanding stellar collapse and neutron star formation, which are long-standing mystery in the Universe. The authors are grateful to Kazuhiro Oyamatsu for giving us permission to use Fig. 3. They also thank Chris Pethick for helpful comments. The research reported in this article grew out of collaborations with Kei Iida, Toshiki Maruyama, Katsuhiko Sato, Kenji Yasuoka and Toshikazu Ebisuzaki. Further research currently in progress is performed using the RIKEN Super Combined Cluster System with MDGRAPE-2. This work was supported in part by the Nishina Memorial Foundation, by a JSPS Postdoctoral Fellowship for Research Abroad, by a JSPS Research Fellowship for Young Scientists, by the Ministry of Education, Culture, Sports, Science and Technology through Research Grant No. 14-7939, and by RIKEN through Research Grant No. J130026. The integral mean curvature and the Euler characteristic (see, e.g., Ref. [39] and references therein for details) are powerful tools for extracting the morphological characteristics of the structure of nuclear matter. Suppose there is a set of regions R , where the density is higher than a given threshold density ρ th . The integral mean curvature and the Euler characteristic for the surface of this region ∂R are defined as surface integrals of the mean curvature H = ( κ 1 + κ 2 ) / 2 and the Gaussian curvature G = κ 1 κ 2 , respectively; i.e., ∂R HdA and χ ≡ ∂R GdA/ 2 π , where κ 1 and κ 2 are the principal curvatures and dA is the area element of the surface of R . The Euler characteristic χ depends only on the topology of R and is expressed ...

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... nuclear matter, which favors a spherical nucleus, being greater than those due to the electrical repulsion between protons, which tends to make the nucleus deform. When the density of matter approaches that of atomic nuclei, i.e., the normal nuclear density ρ 0 , nuclei are closely packed and the effect of the electrostatic energy becomes comparable to that of the surface energy. Consequently, at subnuclear densities around ρ ρ 0 / 2 , the energetically favorable configuration is expected to have remarkable structures as shown in Fig. 3; the nuclear matter region (i.e., the liquid phase of mixture of protons and neutrons) is divided into periodically arranged parts of roughly spherical (a), rod-like (b) or slab-like (c) shape, embedded in the gas phase and in a roughly uniform electron gas. Besides, there can be phases in which nuclei are turned inside out, with cylindrical (d) or spherical (e) bubbles of the gas phase in the liquid phase. As mentioned in the previous section, these transformations are expected to occur in the deepest region of neutron star inner crusts and in the inner cores of collapsing stars just before the star rebounds. Since slabs and rods look like “lasagna” and “spaghetti”, the phases with non-spherical nuclei are often referred to as “pasta” phases and such non-spherical nuclei as nuclear “pasta.” Likewise, spherical nuclei and spherical bubbles are called “meatballs” and “Swiss cheese”, respectively. More than twenty years ago, Ravenhall et al. [53] and Hashimoto et al. [20] inde- pendently pointed out that nuclei with such exotic shapes 2 can be the most energetically stable due to the subtle competition between the nuclear surface and Coulomb energies as mentioned above. Let us here show this statement by a simple calculation using an incompressible liquid-drop model (see also Ref. [46]). We consider five phases depicted in Fig. 3, which consist of spherical nuclei, ...

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... nuclear matter, which favors a spherical nucleus, being greater than those due to the electrical repulsion between protons, which tends to make the nucleus deform. When the density of matter approaches that of atomic nuclei, i.e., the normal nuclear density ρ 0 , nuclei are closely packed and the effect of the electrostatic energy becomes comparable to that of the surface energy. Consequently, at subnuclear densities around ρ ρ 0 / 2 , the energetically favorable configuration is expected to have remarkable structures as shown in Fig. 3; the nuclear matter region (i.e., the liquid phase of mixture of protons and neutrons) is divided into periodically arranged parts of roughly spherical (a), rod-like (b) or slab-like (c) shape, embedded in the gas phase and in a roughly uniform electron gas. Besides, there can be phases in which nuclei are turned inside out, with cylindrical (d) or spherical (e) bubbles of the gas phase in the liquid phase. As mentioned in the previous section, these transformations are expected to occur in the deepest region of neutron star inner crusts and in the inner cores of collapsing stars just before the star rebounds. Since slabs and rods look like “lasagna” and “spaghetti”, the phases with non-spherical nuclei are often referred to as “pasta” phases and such non-spherical nuclei as nuclear “pasta.” Likewise, spherical nuclei and spherical bubbles are called “meatballs” and “Swiss cheese”, respectively. More than twenty years ago, Ravenhall et al. [53] and Hashimoto et al. [20] inde- pendently pointed out that nuclei with such exotic shapes 2 can be the most energetically stable due to the subtle competition between the nuclear surface and Coulomb energies as mentioned above. Let us here show this statement by a simple calculation using an incompressible liquid-drop model (see also Ref. [46]). We consider five phases depicted in Fig. 3, which consist of spherical nuclei, ...

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... the modified URCA process, a bystander neutron absorbs the momentum and relaxes kinematical constraints. Pasta nuclei suppress the bremsstrahlung rate [ 27, 51]. If we assume that neutrino bremsstrahlung is dominated by the lowest reciprocal lattice vectors for which form factors do not vanish, the number of these vectors for each dimensional lattice leads to the neutrino emission rate of spherical, cylindrical and planar nuclei in the ratio of 6:3:1. If the nucleons in the core undergo a transition to a superfluid and/or superconducting state, the modified URCA process is suppressed by a factor of ∼ e − ∆ /k B T , where ∆ is the superfluid energy gap. In this case bremsstrahlung is dominant and the neutrino emission is suppressed efficiently as a consequence of the presence of pasta phases. Nuclear systems can be regarded as complex fluids of nucleons. Complex fluids are the systems in which scales of the fluid and their constituents cannot be separated clearly; both scales are strongly connected with each other. Such characteristic can be seen in various soft condensed matter systems. For example, polymer systems are one of the typical complex fluids (description of polymer systems in the present section is mainly based on an instructive review [28]). Actually, similar spatial structures to nuclear pasta as shown in Fig. 3 can be seen in some polymer systems such as domain structures in solutions and melts of block copolymers. Thus it would be interesting to compare the hierarchical structure of nuclear systems with that of polymer systems. Polymers are made up of monomers connected by covalent bondings and their degrees of polymerization are quite large ( > 100 ). Consequently, in polymer systems, the scale of constituents is as large as that of macroscopic phenomena such as phase separation of the domain structures. Thus polymer systems behave as complex fluids. In nuclear systems, the nature of a complex fluid is caused by strong and short range nuclear force. In Fig. 6, we show the hierarchical structures of nuclear systems and of polymer systems. Nuclear pasta phases and domain structures of block copolymers can be classified in the semimacroscopic scale. Thus theoretical methods based on the mesoscopic or semimacroscopic scales are practical for studying these systems. In the study of polymer systems, the density functional theory and the Ginzburg-Landau theory correspond to ...