Variational Calculation for the Equation of State of Nuclear Matter H. Kanzawa (Waseda Univ.) K. Oyamatsu (Aichi Shukutoku Univ.) K. Sumiyoshi (Numazu C.T.) M. Takano (Waseda Univ.) OMEG07 5th Dec.’07

The Aim of This Study To construct the Equation of State (EOS) of Nuclear Matter for supernova (SN) simulations starting from the realistic nuclear force with the variational method. Introduction: In the SN simulations, the nuclear EOS is essential. There are only two EOS’s available for SN simulations. Latimer-Swesty EOS: compressible liquid drop model Shen EOS: relativistic mean field theory No nuclear EOS based on a microscopic many-body theory. We aim at a nuclear EOS for SN using variational method .

Our Plan Nuclear EOS table for SN simulation This talk Construction of the uniform EOS We construct the EOS for uniform nuclear matter at zero and finite temperatures . Ref.) H. Kanzawa et. al., Nucl. Phys. A791(2007),232. Atomic nuclei We tune the parameters in the EOS for uniform nuclear matter by calculating atomic nuclei using Thomas-Fermi(TF) approximation. Nonuniform nuclear matter We extend to nonuniform nuclear matter at zero and finite temperatures (TF calculation). Nuclear EOS table for SN simulation →Application to the SN simulation

１．Uniform Nuclear matter at Zero Temperature Ref.) H. Kanzawa, K. Oyamatsu, K. Sumiyoshi, M. Takano, Nucl. Phys. A791(2007),232.

Nuclear Hamiltonian The Hamiltonian UⅨ pot. AV18 pot. (isoscalar part) Two-body Hamiltonian Three-body Hamiltonian Three-body 　potential: UⅨ pot. Repulsive part 2-pion exchange part Two-body potential: AV18 pot. (isoscalar part)

Two-body Energy (step1) For uniform EOS, we take 2steps. First, we consider only H2. Second, we consider H3. The correlation function The Jastrow-type wave function is assumed :The Fermi gas wave function step1 Minimization of Expectation value of H2 with the Jastrow-type wave function In the two-body cluster approximation →Euler-Lagrange eqs. for fC, fT, fLS are solved.

Two constraints rh 1) Extended Mayer’s condition Normalization condition 2) Healing Distance condition Healing Distance rh adjustable parameter

(Fermi Hypernetted Chain Calculation) Two-body energies A.Akmal, V.R.Pandharipande and D.G.Ravenhal, 　Phys.Rev.C58(1998 ),1804 (Fermi Hypernetted Chain Calculation) adjustable parameter

Three-Body Energy (step2) (for neutron matter) (for symmetric matter) correction term　 repulsive part 　 　 2-pion exchange part α，β，γ，δ： adjustable parameters The expectation values with the Fermi gas wave function ↓ determined so that E/N reproduces the empirical values Total Energy:

Total energies The empirical values: α = 0.41 β = -0.2185 Esym(MeV) ρ0(fm-3) E0(MeV) K(MeV) 0.16 -15.8 250 29.9 　 α = 0.41 　 β = -0.2185 γ = -1604 MeV･fm6 　　δ = 13.93 fm3

Application to Neutron Stars 2.16M ☉ 2.22M ☉ Causality violation occurs

２．Uniform Nuclear matter at Finite Temperatures Ref.) H. Kanzawa, K. Oyamatsu, K. Sumiyoshi, M. Takano, Nucl. Phys. A791(2007),232.

At Finite Temperatures Minimizing free energy per nucleon: following the prescription of K.E.Schmidt and V.R.Pandharipande, Phys. Lett., 87B (1979), 11 T : Temperature Approximate internal energy per nucleon E3 is the same as at zero temperature The expectation value of H2 at finite temperatures Approximate entropy per particle n(k) :occupation probability m*:effective mass is minimized with respect to m* in n(k)

Free Energies per Nucleon Ref.) B.Friedman and V. R. Pandharipande, Nucl. Phys. A361 (1981), 502.

Pressure 　 ＊critical temperature Tc～18MeV

３．Thomas-Fermi Calculations for Atomic Nuclei

Thomas-Fermi approximation Toward SN-EOS: TF calculation for the nonuniform SN matter Before treating the nonuniform matter Isolated atomic nuclei (Low density limit) Tuning of the parameters(a,b,g,d) in the EOS for uniform nuclear matter so as to reproduce the empirical data of atomic nuclei. ・masses and radii of the β-stable nuclei ・β-stability line (symmetric matter) Three-body energy in the EOS has uncertainty. We make use of this uncertainty to construct more realistic EOS.

Thomas-Fermi approximation Binding energy of the nucleus (N,Z) gross term gradient term Coulomb term Energy density: using the uniform EOS. Normalization condition The distribution of protons and neutrons ( i =p, n ) Minimizing －B(N, Z) with respect to tn ,Rn ,tp ,Rp for each nuclei Ref) K. Oyamatsu, NPA561(1993)431

Deviation from the empirical values ΔM=M(TF)-Mexp ΔrRMS=rRMS (TF)-rRMSexp ΔZβ=Zβ (TF)-Zβ(emp) Masses b-stability line RMS radii ・masses and RMS radii are improved significantly ・β-stability line is fairly improved

＊saturation value:ρ0, E0, K and Esym Esym: symmetric energy Parameter-tuned EOS Esym(MeV) ρ0(fm-3) E0(MeV) K(MeV) Before tuning 0.160 -15.8 250 29.9 After tuning 0.170 -16.2 247 32.4 APR 0.161 -16.0 280 34.1 ＊saturation value:ρ0, E0, K and Esym also change with α～δ ρ0: saturation density E0:saturation energy K: incompressibility Esym: symmetric energy

Summary Future Plans We constructed the uniform nuclear EOS toward supernova simulations. Summary Uniform matter at zero temperature ・Our simpler variational method is successful. ・The neutron star maximum mass is 2.2M ⊙ Uniform matter at finite temperatures ・Our EOS is stiffer than Friedman and Pandharipande. TF Calculation for atomic nuclei ・We perform the fine tuning of the parameters in the EOS for uniform nuclear matter . Future Plans ・Extension to nonuniform SN matter toward EOS table for SN simulation

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