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

2. Motivation Methods Results Conclusion Background

3. 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.

4. Background G. Shen C. J. Horowitz S. Teige. PhysRevC, 82, 015806 (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

5. 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, 035802 (2011)

6. 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.

7. 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.

8. Methods Lagrangian density Equations of motion Mean field approach

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

10. Methods Thermodynamic quantities Entropy density Free energy Energy density

11. 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 938.0511.19777783.0770.010.02892 gωgω gρgρ g 2 (fm -1 )g3g3 c3c3 12.613944.63219-7.232470.6183371.30747 TM1 Parameter set Y. Sugahara and H. Toki, Nucl. Phys. A, 579, 557 (1994) different initial fields lead to different configuration

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

13. 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

14. 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

15. 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.

16. 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.

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

18. Thank you!