Properties of nuclear pastas

Abstract

In this review we study the nuclear pastas as they are expected to be formed in neutron star crusts. We start with a study of the pastas formed in nuclear matter (composed of protons and neutrons), we follow with the role of the electron gas on the formation of pastas, and we then investigate the pastas in neutron star matter (nuclear matter embedded in an electron gas).

Nuclear matter (NM) at intermediate temperatures (1 MeV ≲ T ≲ 15 MeV), at saturation and sub-saturation densities, and with proton content ranging from 30% to 50% was found to have liquid, gaseous and liquid–gas mixed phases. The isospin-dependent phase diagram was obtained along with the critical points, and the symmetry energy was calculated and compared to experimental data and other theories. At low temperatures (T ≲ 1 MeV) NM produces crystal-like structures around saturation densities, and pasta-like structures at sub-saturation densities. Properties of the pasta structures were studied with cluster-recognition algorithms, caloric curve, the radial distribution function, the Lindemann coefficient, Kolmogorov statistics, Minkowski functionals; the symmetry energy of the pasta showed a connection with its morphology.

Neutron star matter (NSM) is nuclear matter embedded in an electron gas. The electron gas is included in the calculation by the inclusion of an screened Coulomb potential. To connect the NM pastas with those in neutron star matter (NSM), the role the strength and screening length of the Coulomb interaction have on the formation of the pastas in NM was investigated. Pasta was found to exist even without the presence of the electron gas, but the effect of the Coulomb interaction is to form more defined pasta structures, among other effects. Likewise, it was determined that there is a minimal screening length for the developed structures to be independent of the cell size.

Neutron star matter was found to have similar phases as NM, phase transitions, symmetry energy, structure function and thermal conductivity. Like in NM, pasta forms at around T ≈ 1.5 MeV, and liquid-to-solid phase changes were detected at T ≈ 0.5 MeV. The structure function and the symmetry energy were also found to depend on the pasta structures.

Keywords

nuclear pasta / neutron star matter / nuclear symmetry energy / molecular dynamics / nuclear phase transitions

References

[1] D. G. Ravenhall, C. J. Pethick, and J. R. Wilson, Structure of matter below nuclear saturation density, Phys. Rev. Lett. 50(26), 2066 (1983)

[2] K. Oyamatsu, Nuclear shapes in the inner crust of a neutron star, Nucl. Phys. A 561(3), 431 (1993)

[3] T. Maruyama, K. Niita, K. Oyamatsu, T. Maruyama, S. Chiba, and A. Iwamoto, Quantum molecular dynamics approach to the nuclear matter below the saturation density, Phys. Rev. C 57(2), 655 (1998)

[4] C. P. Lorenz, D. G. Ravenhall, and C. J. Pethick, Neutron star crusts, Phys. Rev. Lett. 70(4), 379 (1993)

[5] K. S. Cheng, C. C. Yao, and Z. G. Dai, Properties of nuclei in the inner crusts of neutron stars in the relativistic meanfield theory, Phys. Rev. C 55(4), 2092 (1997)

[6] T. Kido, T. Maruyama, K. Niita, and S. Chiba, MD simulation study for nuclear matter, Nucl. Phys. A663–664, 877c (2000)

[7] G. Watanabe, K. Iida, and K. Sato, Thermodynamic properties of nuclear “pasta” in neutron star crusts, Nucl. Phys. A 676(1–4), 455 (2000)

[8] R. D. Williams and S. E. Koonin, Sub-saturation phases of nuclear matter, Nucl. Phys. A 435(3–4), 844 (1985)

[9] M. Hashimoto, H. Seki, and M. Yamada, Shape of Nuclei in the Crust of Neutron Star, Prog. Theor. Phys. 71(2), 320 (1984)

[10] P. N. Alcain and C. O. Dorso, The neutrino opacity of neutron rich matter, Nucl. Phys. A 961, 183 (2017)

[11] D. Page, J. M. Lattimer, M. Prakash and A. W. Steiner, Minimal Cooling of Neutron Stars: A New Paradigm, Astrophys. J. Suppl. 155, 623 (2004)

[12] B. Schuetrumpf, G. Martínez-Pinedo, M. Afibuzzaman, and H. M. Aktulga, Survey of nuclear pasta in the intermediate-density regime: Shapes and energies, Phys. Rev. C 100(4), 045806 (2019)

[13] B. Schuetrumpf, G. Martínez-Pinedo, and P. G. Reinhard, Survey of nuclear pasta in the intermediate-density regime: Structure functions for neutrino scattering, Phys. Rev. C 101(5), 055804 (2020)

[14] G. Watanabe, K. Sato, K. Yasuoka, and T. Ebisuzaki, Microscopic study of slablike and rodlike nuclei: Quantum molecular dynamics approach, Phys. Rev. C 66(1), 012801 (2002)

[15] C. J. Horowitz, M. A. Perez-García, and J. Piekarewicz, Neutrino-“pasta” scattering: The opacity of nonuniform neutron-rich matter, Phys. Rev. C 69(4), 045804 (2004)

[16] B. Schuetrumpf and W. Nazarewicz, Twist-averaged boundary conditions for nuclear pasta Hartree–Fock calculations, Phys. Rev. C 92(4), 045806 (2015)

[17] F. J. Fattoyev, C. J. Horowitz, and B. Schuetrumpf, Quantum nuclear pasta and nuclear symmetry energy, Phys. Rev. C 95(5), 055804 (2017)

[18] C. O. Dorso, P. A. Giménez Molinelli, and J. A. López, in: “Neutron Star Crust”, Eds. C. A. Bertulani and J. Piekarewicz, Nova Science Publishers, ISBN 978-1620819029 (2012)

[19] P. N. Alcain, P. A. Giménez Molinelli, and C. O. Dorso, Beyond nuclear “pasta”: Phase transitions and neutrino opacity of new “pasta” phases, Phys. Rev. C 90(6), 065803 (2014)

[20] C. J. Horowitz, M. A. Pérez-García, J. Carriere, D. K. Berry, and J. Piekarewicz, Nonuniform neutron-rich matter and coherent neutrino scattering, Phys. Rev. C 70(6), 065806 (2004)

[21] C. O. Dorso, P. A. Giménez Molinelli, and J. A. López, Topological characterization of neutron star crusts, Phys. Rev. C 86(5), 055805 (2012)

[22] I. Tanihata, Preprint RIKEN-AF-NP-229, 1996; P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, New parametrization for the nuclear covariant energy density functional with a point-coupling interaction, Phys. Rev. C 82(5), 054319 (2010); M. Dutra, O. Lourenço, J. S. Sá Martins, A. Delfino, J. R. Stone, and P. D. Stevenson, Skyrme interaction and nuclear matter constraints, Phys. Rev. C 85(3), 035201 (2012)

[23] S. Kumar and Y. G. Ma, Investigation of compressibilities using neutron-rich projectiles fragmentation at intermediate energy, Nucl. Phys. A 898, 59 (2013)

[24] P. Danielewicz, R. Lacey, and W. G. Lynch, Determination of the equation of state of dense matter, Science 298(5598), 1592 (2002)

[25] W. D. Myers and W. J. Swiatecki, The nuclear Thomas–Fermi model, Acta Phys. Pol. B 26, 111 (1995)

[26] A. Barrañón, J. Escamilla Roa, and J. A. López, Entropy in the nuclear caloric curve, Phys. Rev. C 69(1), 014601 (2004)

[27] P. J. Siemens, Liquid–gas phase transition in nuclear matter, Nature 305(5933), 410 (1983); P. J. Siemens, Macroscopic behaviour of nuclear matter, Nature 336(6195), 110 (1988)

[28] J. A. López and C. O. Dorso, Lecture Notes on Phase Transitions in Nuclear Matter, World Scientific, 2000

[29] H. Müller and B. Serot, Phase transitions in warm, asymmetric nuclear matter, Phys. Rev. C 52(4), 2072 (1995)

[30] J. A. López, A. Gaytán Terrazas, and S. Terrazas Porras, Isospin-dependent phase diagram of nuclear matter, Nucl. Phys. A 994, 121664 (2020)

[31] See, e.g., https://www.ks.uiuc.edu/Research/vmd/current/ug/node73.html, retrieved Sep. 2, 2019

[32] J. A. López, E. Ramírez-Homs, R. González, and R. Ravelo, Isospin-asymmetric nuclear matter, Phys. Rev. C 89(2), 024611 (2014)

[33] J. A. López and S. Terrazas Porras, Symmetry energy in the liquid–gas mixture, Nucl. Phys. A 957, 312 (2017)

[34] K. Hagel, J. B. Natowitz, and G. Röpke, The equation of state and symmetry energy of low-density nuclear matter, Eur. Phys. J. A 50(2), 39 (2014)

[35] S. Kowalski, J. B. Natowitz, S. Shlomo, R. Wada, K. Hagel, J. Wang, T. Materna, Z. Chen, Y. G. Ma, L. Qin, A. S. Botvina, D. Fabris, M. Lunardon, S. Moretto, G. Nebbia, S. Pesente, V. Rizzi, G. Viesti, M. Cinausero, G. Prete, T. Keutgen, Y. E. Masri, Z. Majka, and A. Ono, Experimental determination of the symmetry energy of a low density nuclear gas, Phys. Rev. C 75(1), 014601 (2007)

[36] R. Wada, K. Hagel, L. Qin, J. B. Natowitz, Y. G. Ma, G. Röpke, S. Shlomo, A. Bonasera, S. Typel, Z. Chen, M. Huang, J. Wang, H. Zheng, S. Kowalski, C. Bottosso, M. Barbui, M. R. D. Rodrigues, K. Schmidt, D. Fabris, M. Lunardon, S. Moretto, G. Nebbia, S. Pesente, V. Rizzi, G. Viesti, M. Cinausero, G. Prete, T. Keutgen, Y. El Masri, and Z. Majka, Nuclear matter symmetry energy at 0.03≤ρ/ρ0≤0.2, Phys. Rev. C 85(6), 064618 (2012)

[37] L. W. Chen, C. M. Ko, and B. A. Li, Isospin-dependent properties of asymmetric nuclear matter in relativistic mean field models, Phys. Rev. C 76(5), 054316 (2007)

[38] E. L. Medeiros and J. Randrup, Thermostatic properties of Seyler–Blanchard nuclei, Phys. Rev. C 45(1), 372 (1992)

[39] C. J. Horowitz and A. Schwenk, Cluster formation and the virial equation of state of low-density nuclear matter, Nucl. Phys. A 776(1–2), 55 (2006)

[40] J. Xu, L. W. Chen, B. A. Li, and H. R. Ma, Temperature effects on the nuclear symmetry energy and symmetry free energy with an isospin and momentum dependent interaction, Phys. Rev. C 75(1), 014607 (2007)

[41] P. A. Giménez Molinelli, J. I. Nichols, J. A. López, and C. O. Dorso, Simulations of cold nuclear matter at subsaturation densities, Nucl. Phys. A 923, 31 (2014)

[42] A. Vicentini, G. Jacucci, and V. R. Pandharipande, Fragmentation of hot classical drops, Phys. Rev. C 31(5), 1783 (1985); R. J. Lenk and V. R. Pandharipande, Disassembly of hot classical charged drops, Phys. Rev. C 34(1), 177 (1986); R. J. Lenk, T. J. Schlagel, and V. R. Pandharipande, Accuracy of the Vlasov–Nordheim approximation in the classical limit, Phys. Rev. C 42(1), 372 (1990)

[43] G. Raciti, R. Bassini, M. Begemann-Blaich, S. Fritz, S. J. Gaff, N. Giudice, C. Gross, G. Immé, I. Iori, U. Kleinevoss, G. J. Kunde, W. D. Kunze, U. Lynen, M. Mahi, T. Möhlenkamp, W. F. J. Müller, B. Ocker, T. Odeh, J. Pochodzalla, G. Riccobene, F. P. Romano, A. Sajia, M. Schnittker, A. Schüttauf, C. Schwarz, W. Seidel, V. Serfling, C. Sfienti, W. Trautmann, A. Trzcinski, G. Verde, A. Wörner, H. Xi, and B. Zwieglinski, A systematic study of the nuclear caloric curve, Nuovo Cim. 111(8-9), 987 (1998)

[44] H. Sonoda, G. Watanabe, K. Sato, K. Yasuoka, and T. Ebisuzaki, Phase diagram of nuclear “pasta” and its uncertainties in supernova cores, Phys. Rev. C 77(3), 035806 (2008)

[45] C. O. Dorso, G. Frank, and J. A. López, Phase transitions and symmetry energy in nuclear pasta, Nucl. Phys. A 978, 35 (2018)

[46] C. J. Horowitz, Links between heavy ion and astrophysics, Eur. Phys. J. A 30(1), 303 (2006)

[47] G. Watanabe and K. Iida, Electron screening in the liquid– gas mixed phases of nuclear matter, Phys. Rev. C 68(4), 045801 (2003)

[48] T. Maruyama, T. Tatsumi, D. N. Voskresensky, T. Tanigawa, and S. Chiba, Nuclear “pasta” structures and the charge screening effect, Phys. Rev. C 72(1), 015802 (2005)

[49] C. J. Horowitz, M. A. Perez-Garcia, D. K. Berry, and J. Piekarewicz, Dynamical response of the nuclear “pasta” in neutron star crusts, Phys. Rev. C 72(3), 035801 (2005)

[50] J. Piekarewicz and G. T. Sánchez, Proton fraction in the inner neutron-star crust, Phys. Rev. C 85(1), 015807 (2012)

[51] J.A. López and E. Ramírez-Homs, Effect of an electron gas on a neutron-rich nuclear pasta, Nuc. Sci. Tech. 26, S20502 (2015)

[52] A. S. Schneider, C. J. Horowitz, J. Hughto, and D. K. Berry, Nuclear “pasta” formation, Phys. Rev. C 88(6), 065807 (2013)

[53] K. Binder, B. J. Block, P. Virnau, and A. Tröster, Beyond the van der Waals loop: What can be learned from simulating Lennard–Jones fluids inside the region of phase coexistence, Am. J. Phys. 80(12), 1099 (2012)

[54] C. J. Horowitz, D. K. Berry, C. M. Briggs, M. E. Caplan, A. Cumming, and A. S. Schneider, Disordered nuclear pasta, magnetic field decay, and crust cooling in neutron stars, Phys. Rev. Lett. 114(3), 031102 (2015)

[55] C. Dorso, G. Frank, and J. A. López, Symmetry energy in neutron star matter, Nucl. Phys. A 984, 77 (2019)

[56] J. A. López, J. A. Muñoz, C. O. Dorso, and G. Frank, Machine learning Minkoswki functionals of neutron star crusts, J. Phys. Conf. Ser. (2019); J. A. López and J. A. Muñoz, Analytical expression and neural network study of the symmetry energy, CERN Proc. 1, 29 (2019)

[57] P. N. Alcain, Dependencia en el isospín de la ecuación de estado de la materia nuclear, Ph.D. Thesis, Universidad de Buenos Aires, 2019

[58] D. Frenkel yB. Smit, Understanding Molecular Simulations, 2nd Ed., Academic Press, 2002

[59] A. Deibel, A. Cumming, E. F. Brown, and S. Reddy, Latetime cooling of neutron star transients and the physics of the inner crust, Astrophys. J. 839(2), 95 (2017)

[60] E. F. Brown, A. Cumming, F. J. Fattoyev, C. J. Horowitz, D. Page, and S. Reddy, Rapid neutrino cooling in the neutron star MXB 1659-29, Phys. Rev. Lett. 120(18), 182701 (2018)

[61] A. S. Schneider, D. K. Berry, M. E. Caplan, C. J. Horowitz, and Z. Lin, Effect of topological defects on “nuclear pasta” observables,Phys. Rev. C 93(6), 065806 (2016)

[62] R. Nandi and S. Schramm, Transport properties of the nuclear pasta phase with quantum molecular dynamics, Astrophys. J. 852(2), 135 (2018)

[63] C. J. Horowitz, and D. K. Berry, Shear viscosity and thermal conductivity of nuclear “pasta”, Phys. Rev. C 78(3), 035806 (2008)

[64] J. M. Dunn, Nanoscale phonon thermal conductivity via molecular dynamics, Ph.D. Thesis, Purdue University, 2016

[65] F. Müller–Plathe, A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity, J. Chem. Phys. 106(14), 6082 (1997)

[66] A. Barrañón, C. O. Dorso, J. A. López, and J. Morales, LATINO: A semi-classical model to study nuclear fragmentation, Rev. Mex. Fis. 45(suppl. 2), 110 (1999)

[67] A. Chernomoretz, L. Gingras, Y. Larochelle, L. Beaulieu, R. Roy, C. St-Pierre, and C. O. Dorso, Quasiclassical model of intermediate velocity particle production in asymmetric heavy ion reactions, Phys. Rev. C 65(5), 054613 (2002)

[68] A. Barrañón, C. O. Dorso, and J. A. López, Searching for criticality in nuclear fragmentation, Rev. Mex. Fís. 47(sup. 2), 93 (2001)

[69] A. Barrañón, C. O. Dorso, and J. A. López, Time dependence of isotopic temperatures, Nucl. Phys. A 791(1–2), 222 (2007)

[70] A. Barrañón, R. Cárdenas, C. O. Dorso, and J.A. López, The critical exponent of nuclear fragmentation, Acta Physica Hungarica A: Heavy Ion Phys. 17(1), 59 (2003)

[71] C. O. Dorso and J. A. López, Selection of critical events in nuclear fragmentation, Phys. Rev. C 64(2), 027602 (2001)

[72] A. Barrañón, J. Escamilla Roa, and J. A. López, The transition temperature of the nuclear caloric curve, Braz. J. Phys. 34(3A), 904 (2004)

[73] C. O. Dorso, C. R. Escudero, M. Ison, and J. A. López, Dynamical aspects of isoscaling, Phys. Rev. C 73(4), 044601 (2006)

[74] C. A. Dorso, P. A. G. Molinelli, and J. A. López, Isoscaling and the nuclear EoS, J. Phys. G 38(11), 115101 (2011); C. O. Dorso, P. A. G. Molinelli, and J. A. López, Searching for the origin of isoscaling: Confinement and expansion, Rev. Mex. Phys. S57 (1), 14 (2011)

[75] T. M. Nymand and P. Linse, Ewald summation and reaction field methods for potentials with atomic charges, dipoles, and polarizabilities, J. Chem. Phys. 112, 6152 (2000)

[76] P. N. Alcain, P. A. Giménez Molinelli, J. I. Nichols, and C. O. Dorso, Effect of Coulomb screening length on nuclear “pasta” simulations, Phys. Rev. C 89(5), 055801 (2014)

[77] B. L. Holian, A. F. Voter, and R. Ravelo, Thermostatted molecular dynamics: How to avoid the Toda demon hidden in Nosé–Hoover dynamics, Phys. Rev. E 52(3), 2338 (1995)

[78] H. C. Andersen, Molecular dynamics simulations at constant pressure and/or temperature, J. Chem. Phys. 72(4), 2384 (1980)

[79] S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys. 81(1), 511 (1984)

[80] J. A. López, S. Terrazas Porras, and A. Rodríguez Gutiérrez, Thermodynamics of neutron-rich nuclear matter, AIP Conf. Proc. 1753, 050001 (2016)

[81] B. A. Li, L. W. Chen, and C. M. Ko, Recent progress and new challenges in isospin physics with heavy-ion reactions, Phys. Rep. 464(4–6), 113 (2008)

[82] B. A. Li, A. Ramos, G. Verde, and I. Vidana, Topical issue on Nuclear Symmetry Energy, Eur. Phys. J. A 50(2), 9 (2014)

[83] J. B. Natowitz, G. Röpke, S. Typel, D. Blaschke, A. Bonasera, K. Hagel, T. Klähn, S. Kowalski, L. Qin, S. Shlomo, R. Wada, and H. H. Wolter, Symmetry energy of dilute warm nuclear matter, Phys. Rev. Lett. 104(20), 202501 (2010)

[84] S. Typel, H. H. Wolter, G. Röpke, and D. Blaschke, Effects of the liquid–gas phase transition and cluster formation on the symmetry energy, Eur. Phys. J. A 50(2), 17 (2014)

[85] M. Dutra, O. Lourenço, J. S. Sá Martins, A. Delfino, J. R. Stone, and P. D. Stevenson, Skyrme interaction and nuclear matter constraints, Phys. Rev. C 85(3), 035201 (2012)

[86] M. Dutra, O. Lourenço, S. S. Avancini, B. V. Carlson, A. Delfino, D. P. Menezes, C. Providência, S. Typel, and J. R. Stone, Relativistic mean-field hadronic models under nuclear matter constraints, Phys. Rev. C 90(5), 055203 (2014)

[87] M. Colonna, V. Baran, M. D. Toro, and H. H. Wolter, Isospin distillation with radial flow: A test of the nuclear symmetry energy, Phys. Rev. C 78(6), 064618 (2008)

[88] Y. Zhou, B. Anglin, and A. Strachan, Phonon thermal conductivity in nanolaminated composite metals via molecular dynamics, J. Chem. Phys. 127(18), 184702 (2007)

[89] J. Dunn, E. Antillon, J. Maassen, M. Lundstrom, and A. Strachan, Role of energy distribution in contacts on thermal transport in Si: A molecular dynamics study, J. Appl. Phys. 120(22), 225112 (2016)

[90] K. H. Lin and A. Strachan, Thermal transport in SiGe superlattice thin films and nanowires: Effects of specimen and periodic lengths, Phys. Rev. B 87(11), 115302 (2013)

[91] F. A. Lindemann, The calculation of molecular vibration frequencies, Phys. Z. 11, 609 (1910)

[92] Z. W. Birnbaum, Numerical tabulation of the distribution of Kolmogorov’s statistic for finite sample size, J. Am. Stat. Assoc. 47(259), 425 (1952)

[93] E. Gosset, A three-dimensional extended Kolmogorov– Smirnov test as a useful tool in astronomy, Astron. Astrophys. 188, 258 (1987)

[94] G. Fasano and A. Franceschini, A multidimensional version of the Kolmogorov–Smirnov test, Mon. Not. R. Astron. Soc. 225(1), 155 (1987)

[95] G. J. Babu and E. D. Feigelson, Astronomical Data Anal ysis Software and Systems XV, Eds. C. Gabriel, et al., ASP Conference Series, 351, 127 (2006)

[96] K. Michielsen and H. De Raedt, Integral-geometry morphological image analysis, Phys. Rep. 347(6), 461 (2001)

[97] B. Schuetrumpf, M. A. Klatt, K. Iida, J. A. Maruhn, K. Mecke, and P. G. Reinhard, Time-dependent Hartree–Fock approach to nuclear “pasta” at finite temperature, Phys. Rev. C 87(5), 055805 (2013)

[98] A. Strachan and C. O. Dorso, Time scales in fragmentation, Phys. Rev. C 55(2), 775 (1997); A. Strachan and C. O. Dorso, Fragment recognition in molecular dynamics, Phys. Rev. C 56(2), 995 (1997)

[99] C. O. Dorso and J. Randrup, Early recognition of clusters in molecular dynamics, Phys. Lett. B 301(4), 328 (1993)

[100] P. N. Alcain and C. O. Dorso, Dynamics of fragment formation in neutron-rich matter, Phys. Rev. C 97(1), 015803 (2018)

[101] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117(1), 1 (1995)

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