Calculation of symmetry energy using Argonne family potentials with three nucleon interaction

1. Calculation of symmetry energy using Argonne family potentials with three nucleon interaction Zahra Asadi Mohsen Bigdeli University of Zanjan, Zanjan, Iran The Modern Physics of Compact stars and Relativistic Gravity, Yerevan 2017

2. Outline • The Equation of State (EOS) of Nuclear Matter • The method of calculation of EOS • Nucleon-Nucleon Interaction • Argonne family potentials • Three nucleon interaction • Symmetry Energy • Results

3. Outline • The Equation of State (EOS) of Nuclear Matter • The method of calculation of EOS • Nucleon-Nucleon Interaction • Argonne family potentials • Three nucleon interaction • Symmetry Energy • Results

4. The Equation of State (EOS) of Nuclear Matter • The equation of state (EOS) of asymmetric nuclear matter have long been realized as acrucial parameter to understand the structural properties of neutron-rich nuclei and neutron stars. • At saturation density, the EOS of pure neutron matter is useful to extract information on the symmetry energy and its slope. • The behavior of the EOS of nuclear matter extremely depends on both nucleon-nucleon (NN) interactions and calculation methods.

5. Outline • The Equation of State (EOS) of Nuclear Matter • The method of calculation of EOS • Nucleon-Nucleon Interaction • Argonne family potentials • Three nucleon interaction • Symmetry Energy • Results

6. The method of calculation of EOS • Bethe-Brueckner-Goldstone (BBG) • The Coupled Cluster (CC) expansion • The self-consistent Green’s function (SCGF) • Brueckner-Hartree-Fock (BHF) • Different methods based on Monte-Carlo (MC) • The renormalization group (RG) • Lowest order constrained variational (LOCV)

7. Lowest order constrained variational (LOCV) • The LOCV method has been used to investigate the EOS of asymmetric nuclear matter. • This equation of state is derived from an accurate many-body calculation and is based on the cluster expansion of the energy functional. J.C. Owen, R.F. Bishop, J.M. Irvine, Nucl. Phys. A 277, 45 (1977)

8. LOCV …

9. Outline • The Equation of State (EOS) of Nuclear Matter • The method of calculation of EOS • Nucleon-Nucleon Interaction • Argonne family potentials • Three nucleon interaction • Symmetry Energy • Results

10. Nucleon-Nucleon Interaction • The Nucleon-Nucleon interactions, that are employed to access accurate computationof the EOS, have been proposed by the fitness of protons and neutrons interaction’s datawith the experimental information of the nature of nuclear force via two and three bodypotentials. • AV14 • AV18 • AV8′ • AV6′

11. Outline • The Equation of State (EOS) of Nuclear Matter • The method of calculation of EOS • Nucleon-Nucleon Interaction • Argonne family potentials • Three nucleon interaction • Symmetry Energy • Results

12. Argonne family potentials • AV18 R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. ev. C 51, 38 (1995).

13. Argonne family potentials … • AV8′ • The AV8′potentialis a re-projection of AV18 on to eight operators: B. S. Pudlineret al., Phys. Rev. C56, 1720 (1997)

14. Argonne family potentials … • AV6′ • After eliminating spin-orbit terms from AV8′ and reformulated it to maintain the deuteron binding, AV6′ is acquired. • Tensor terms of AV6′ is remained the same as AV8′, and the subtracting of the AV8′ spin-orbit potential’s radial function from the central potential in the ST = 10 channel caused to built the central potential AV6′ as: R. B. Wiringa and S. C. Pieper, Phys. Rev. Lett. 89, 182501 (2002)

15. Outline • The Equation of State (EOS) of Nuclear Matter • The method of calculation of EOS • Nucleon-Nucleon Interaction • Argonne family potentials • Three nucleon interaction (TNI) • Symmetry Energy • Results

16. Three nucleon interaction (TNI) • Adding TNI to the AV18, AV8′ and AV6′ models, • TNA: Three nucleon attractive part of TNI • TNR : Three nucleon repulsive part of TNI is given by multiplying factor of exp(−γ1ρ) I.E. Lagaris, V.R. Pandharipande, Nucl. Phys. A 359, 349(1981)

17. Outline • The Equation of State (EOS) of Nuclear Matter • The method of calculation of EOS • Nucleon-Nucleon Interaction • Argonne family potentials • Three nucleon interaction • Symmetry Energy • Results

18. Symmetry energy • The symmetry energy, Esym(ρ) is given by difference between the energy per baryon of pure neutron matter and the energy per baryon of infinite symmetric nuclear matter with equal number of proton and neutron. • Symmetry energy is very important quantity for studying the nuclear and astrophysical related problems. • Applications: • Mechanism of stability and neutron skin thickness of very neutron rich nuclei. • Reaction pathways and abundances of heavy neutron- rich nuclei produced during r-process nucleosynthesis supernova evolution. • The core-crust transition in neutron stars. • Threshold density of the kaon condensation. • The structure of neutron stars (the maximum masses and radii)

19. Symmetry energy … • The energy per baryon of isospin asymmetric nuclear matter can be expanded in even powers of x:

20. Symmetry energy …

21. Symmetry energy … • In 1998, Bordbar and Modaressinvestigated the symmetry energy versus density for different potentials and methods in left panel and versus the proton to neutron ratio (R) in right panel. G. H. Bordbar and M. Modarres, Phys. Rev. C 57 (1998) 714.

22. Symmetry energy … • In 2013, Bigdeli used the LOCV method to study the behavior of nuclear symmetry energy versus density and spin states of neutrons and protons with AV18 potential. • The symmetry energy per nucleon of the polarized nuclear matter as a function of the total number density (ρ) for different values of neutron and proton polarization. M. Bigdeli. Int. J. Mod. Phys. E 22, No. 8, 1350054 (2013).

23. Symmetry energy … • Gandolfi and et al. studied the correlation between the neutron star radius and the symmetry energy. They also combined QMC and theoretical models of the three-nucleon interactions, and recent neutron star observations to constrain the value of the symmetry energy and its density dependence. • The value of L as a function of Esym obtained from various EOS. S. Gandol , J. Carlson, S. Reddy, A. W. Steiner, R. B. Wiringa, Eur. Phys. J. A50, 10 (2014).

24. Symmetry energy… • The symmetry energy above saturation density calculated with the different EOS is pointed out by Baldo in 2017. Marcello Baldo 2017 J. Phys.: Conf. Ser. 861 012001

25. Outline • The Equation of State (EOS) of Nuclear Matter • The method of calculation of EOS • Nucleon-Nucleon Interaction • Argonne family potentials • Three nucleon interaction • Symmetry Energy • Results

26. Results • Energy per particle for symmetric nuclear matter and pure neutron matter in terms of density for different NN potentials.

27. Results • The energy per particle versus quadratic asymmetry parameter densityβ^2at ρ = 0.5 fm^-3 for different NN potentials.

28. Results • Comparison between our results for the energy per particle of symmetric nuclear matter and those of BHF and FHNC calculations with the AV8′ and AV6′ potentials.

29. Results … • The nuclear symmetry energy as a function of density fordifferent NN potentials.

30. Results … • Saturation properties of symmetric nuclear matter energy and nuclear symmetry energy. (Here ρs is given in fm^−3 and energy parameters are in MeV )

31. Results … • The nuclear symmetry energy as a function of density for different NN potentials.

32. Results … • The nuclear symmetry energy as a function of density for different EOS.

33. Thanks for your attention.