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Relativistic equation of state at subnuclear densities in the Thomas- Fermi approximation Zhaowen Zhang Supervisor: H. Shen Nankai University 20th-22th Oct KIAA at Peking University, Beijing, China Z. W. Zhang and H. Shen, Astrophys. J. 788, 185 (2014).

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Motivation Methods Results Conclusion Background

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Supernova explosions Neutron star formations Equation of state(EOS) of nuclear matter is very important in understanding many astrophysical phenomena: Lots of the EOS investigations focused on the case of zero temperature or high density for uniform matter.

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Background G. Shen C. J. Horowitz S. Teige. PhysRevC, 82, (2010) The EOS for the core-collapse supernova simulations covers wide ranges of temperature, proton fraction, and baryon density. T=1 MeV T=3.16 MeV T=6.31 MeV T=10 MeV

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Background … Lattimer–Swesty Compressible liquid-drop model Lattimer, J. M., & Swesty, F. D. Nucl. Phys. A, 535, 331 (1991) Some famous nuclear EOSs H. Shen etc. Parameterized Thomas– Fermi approximation Shen, H., Toki, H., Oyamatsu, K., & Sumiyoshi, K. Prog. Theor. Phys., 100, 1013 (1998) G. Shen & Horowitz etc. Relativistic mean field theory G. Shen C. J. Horowitz S. Teige. PhysRevC, 83, (2011)

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Background Parameterized Thomas–Fermi approximation Nucleon distribution function Gradient energy F 0 = 70 MeV fm 5 is determined by reproducing the binding energies and charge radii of finite nuclei.

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Motivation Self-consistent Thomas–Fermi approximation Nucleon distribution and gradient energy are calculated self-consistently. Both droplet and bubble configurations are considered. bubbledroplet uniform matter In present work, we compare and examine the difference between PTF and STF.

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Methods Lagrangian density Equations of motion Mean field approach

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Methods Distribution function Fermi–Dirac distribution Chemical potential Wigner–Seitz cell BCC

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Methods Thermodynamic quantities Entropy density Free energy Energy density

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Methods Calculation T Y p ρ B R WS μ i σ 0 (r) ω 0 (r) ρ 0 (r) Nucleon distribution n i (r) σ(r) ω(r) ρ(r) A(r) n i (r) converge E cell S cell F cell Minimizing F cell by changing R WS Thermodynamically favored state YES NO Mmσmσ mωmω mρmρ gσgσ gωgω gρgρ g 2 (fm -1 )g3g3 c3c TM1 Parameter set Y. Sugahara and H. Toki, Nucl. Phys. A, 579, 557 (1994) different initial fields lead to different configuration

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Results Strong Yp dependence T=1 T=10 Bubble appearance Delay the transition to uniform matter Free energy & Entropy Small difference

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Results T=1 T=10 ρBρB The densities at the center are lower in the STF. The cell radius R c of STF is larger. More free nucleons exist outside the nuclei at T = 10 MeV. Nucleon distribution

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Results Numbers & Fractions T=1 T=10 Nuclei fraction Neutron gas fraction Proton gas fraction T=1 T=10 Cause by difference of nucleon distribution More nucleons can drip out of the nuclei AdAd ZdZd AdAd ZdZd XAXA XAXA XnXn XnXn XpXp Dominant

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Results T=1T=10 Y p =0.3 Y p =0.5 Neutron chemical potential The results of droplet are almost identical for STF and PTF. The sudden jumps caused by the different Coulomb potential of bubble and droplet.

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Results T=1T=10 Y p =0.3 Y p =0.5 Proton chemical potential The difference of STF and PTF may be caused by the Coulomb and surface energies. Proton is directly effected by Coulomb interaction.

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Conclusion Outlook 1.More pasta phases could be considered in STF. 2.Alpha particles will be included in the future.

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Thank you!

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