高エネルギー天体現象のための現実的核力に基づく核物質状態方程式
H. Togashi (Waseda University)
H. Kanzawa, M. Takano (Waseda University)
S. Yamamuro, H. Suzuki, K. Nakazato ( Tokyo University of Science)
K. Sumiyoshi (Numazu College of Technology), H. Matsufuru (KEK)

Contents
1：Introduction
2：EOS for Uniform Nuclear Matter
3：EOS for Non-uniform Nuclear Matter
4：Summary
素核宇融合による計算基礎物理学の進展 @ 合歓の郷 2011/12/03

1. Introduction
The aim of this study is
To construct a new nuclear Equation of State (EOS)for supernova (SN) simulations
based on the realistic nuclear force.
The nuclear EOS plays an important role for astrophysical studies.
1. Lattimer-Swesty EOS : The compressible liquid drop model(NPA 535 (1991) 331)
2. Shen EOS : The relativistic mean field theory (NPA 637 (1998) 435)
・ K. Nakazato EOS : (PRD 77(2008) 103006) ・ G. Shen EOS : (PRC 83 (2010) 015806)
・ C. Ishizuka EOS : (J. Phys. G 35(2008)085201)・ M. Hempel EOS : (NPA 837 (2010) 210)
・ S. Furusawa EOS : (APJ 738 (2011) 178)

These EOSs are based on phenomenological models for uniform matter.

There is no nuclear EOS based on the microscopic many-body theory.

We aim at a new EOS for SN with the variational method.

Our Plan to Construct the EOS for SN Simulations
Uniform Nuclear Matter
EOS constructed with the cluster variational method
Non-uniform Nuclear Matter
EOS constructed withWe are here.

the Thomas-Fermi (TF) calculation

Completion of a Nuclear EOS table for SN simulations
Density ρ : 105.1  ρm 
1016.0g/cm3
Temperature T : 0  T  400 MeV
Proton fraction x : 0  x 0.65

110 point
92 point
66 point

2. EOS for Uniform Nuclear Matter
The Nuclear HamiltonianTwo-body Hamiltonian

Three-body Hamiltonian

the AV18 two-body nuclear potential

the UIX three-body nuclear potential

We assume the Jastrow wave function.
ΦF: The Fermi-gas wave function
at zero temperature
Ptsµ: Spin-isospin projection operators

fij : Correlation function
CentralTensor

Spin-orbitTwo-Body Energy
E2/N is the expectation value of H2 with the Jastrow wave functionin the two-body cluster approximation.
ρ : Total nucleon number density
ρp : Proton number density
x = ρp/ρ : Proton fraction

E2/N is minimized with respect to fCtsµ(r), fTtµ(r) and fSOtµ(r)with the following two constraints.
1. Extended Mayer’s condition

Ftsµ(r): Radial distribution functions
FFtsµ(r): Ftsµ(r) for the degenerate Fermi gas

2. Healing distance condition
Mean distance
Healing distance

between nucleonsah : adjustable parameter
ah is determined so that E2/N reproduces the resultsby APR(Akmal, Pandharipande and Ravenhall)
APR : PRC58(1998)1804rhTotal Energy per Nucleon at Zero TemperatureThree Body Energy
Constructed with the expectation valueof H3 with the Fermi gas wave function
Parameters of E3/N are
determined so that

TF calculation for atomic nuclei
reproduces the gross feature ofthe experimental data.

ρ0[fm-3]

E0[MeV]

0.16

−16.1

K[MeV] Esym[MeV]
240

30.0

Application to Neutron Star

●Causality violation
occurs

Effect of causality violation is small.

Free Energy at Finite Temperatures
We follow the prescription proposed by Schmidt and Pandharipande.
(Phys. Lett. 87B(1979) 11)

Free Energy(A. Mukherjee et al., PRC 75(2007) 035802): Approximate Internal Energy: Approximate Entropy

Approximate Internal Energy

S0/N is expressed with the averaged
occupation probabilities ni(k)chosen to be the same as at 0 MeV
E2/N : Expectation value of H2
with the Jastrow-type wave function at finite temperature.

ΦF: The Fermi-gas wave function
expressed with ni(k)Free Energy per Nucleon at Finite Temperatures

Free energy per nucleon at T=20MeV

Free energy per nucleon at T=30MeV

AM: A. Mukherjee, PRC 79(2009) 045811.

Uniform Nuclear Matter at Low Density

E/N → 0MeV is not reproduced in the limit of ρ → 0 fm−3
Because of the deuteron clusteringImprovement of Cluster Variational Method
Healing distance condition
Mean distance
between nucleonsah : adjustable parameter

New healing distance condition

rh → ahr0 (high density)(low density)
rh → ba, b, c : adjustable parameters
a = 1.76 b = 6.38 fm c = 10b, c is determined so that TF calculation for atomic nuclei keeps the gross feature.

E/N with New Cluster Variational Method

E/N → 0MeV in the limit of ρ → 0 fm−3

F/N with New Cluster Variational Method

3. Non-uniform Nuclear Matter
We follow the TF method by Shen et. al. (NPA637(1998)435)

Energy in the Wigner-Seitz (WS) cell
Bulk energy

Gradient energy
F0 = 68.00 MeV fm5Coulomb energy

ε : Energy density of uniform nuclear matter
Parameterlog10(T) [MeV]Minimum−1.24

Maximum
1.40

Mesh
0.12

Number23 + 1ζ = (1−2Yp)2

0.0

1.0

0.1

11+2nB [fm−3]

0.00010.16000.0001

1600

Nucleon density distribution(0MeV)(ζ = 0.85,0.95)

24×13×1600⋍ 500000 pointa : Lattice constant(i = p, n)
E/Vcell is minimized with respect to niout, niin Ri, ti, aat given density and proton fraction.

TF Calculation for Atomic Nuclei⊿M = MTF−Mexp

MTF : Mass by the Thomas-Fermi calculation
Mexp : Experimental data

RMS deviation (for 2226 nuclei) 2.99 MeV

TF Calculation for Atomic Nuclei

Our results are in good agreement withthe experimental data and the sophisticated atomic mass formula.

TF Calculation for Non-uniform Nuclear Matter

Phase diagram of nuclear matter at T = 0MeV

ρB = 1014.23 g/cm3

5. Summary• The EOS for uniform nuclear matter is constructedwith the cluster variational method. (zero and finite temperatures)• The EOS for non-uniform nuclear matter at zero temperature iscalculated with the Thomas-Fermi callculation.
Uniform nuclear matter
Deuteron clustering appears at low density.
E/N at low density is refined for TF calculation.
Non-uniform nuclear matter
Phase diagram at zero temperature is reasonable.

Future Plans• Construction of the EOS table for non-uniform matter• Addition of α particle contribution

Construction of the EOS for supernova simulations