Variational Study of a Nuclear Equation of State
for Core-Collapse Supernovae
H. Togashi (Waseda University)
H. Kanzawa, M. Takano (A02, A03: Waseda University)
S. Yamamuro, H. Suzuki, K. Nakazato ( A03: Tokyo Univ. of Science)
K. Sumiyoshi (A03:Numazu College of Technology)
H. Matsufuru (A04: KEK)

Contents
1：Introduction
2：EOS for Uniform Nuclear Matter

3：EOS for Non-uniform Nuclear Matter
4：Summary
QUCS 2012 @ Nara, 14 Dec. 2012

1. Introduction
The aim of this study isTo 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
2. Shen EOS : The relativistic mean field theory
・ K. Nakazato EOS : (PRD 77(2008) 103006)(NPA 535 (1991) 331)
(NPA 637 (1998) 435)

・ 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 with the Thomas-Fermi (TF) calculation
1. EOS for non-uniform matter at zero temperature★We are here.★

2. EOS for non-uniform matter at finite temperatureCompletion of a Nuclear EOS table for SN simulations

Density r : 105.1 ≤ rm ≤ 1016.0g/cm3

110 point

Temperature T : 0 ≤ T ≤ 400 MeV

92 point

Proton fraction x : 0 ≤ x ≤ 0.65

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.
FF: The Fermi-gas wave function
at zero temperature
Pts: 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.
r : Total nucleon number density
rp : Proton number density
x = rp/r : Proton fraction

E2/N is minimized with respect to fCts(r), fTt(r) and fSOt(r)with the following two constraints.

1. Extended Mayer’s conditionFts(r): Radial distribution functions
FFts(r): Fts(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 results byAPR(Akmal, Pandharipande and Ravenhall)
APR : PRC58(1998)1804

rhTwo Body Energy

ah = 1.76

APR : PRC58(1998)1804

Our results are in good agreement with the results byAPR (FHNC method).

Three-Body Energy

UIX potential:2-pion exchange part:Repulsive partThree Body Energy

Correction term
Expectation value with the Fermi-gas wave function
a,b,g,d : adjustable parameters
Parameters of E3/N are determined so as to reproduce the empirical data.
TF calculation for atomic nuclei reproduces the gross feature ofthe experimental data.Total Energy per Nucleon at Zero Temperature
Total energy per nucleon

r0[fm-3]

E0[MeV]K[MeV]

Esym[MeV]

0.16-16.1

240

30.0

Application to Neutron Star
2.5
J1614-2230

M/M

2.0

1.97±0.0
4

1.5
1.0
0.5
0.0
68

10121416

R [km]The NS mass-radius relation is consistent withobservational data .

Free Energy at Finite Temperatures I
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
E2T/N :Two-body internal energy at finite temperatures

Correlation function fij is chosen to be the same at 0MeV.

Frozen-Correlation Approximation

Free Energy at Finite Temperatures IIThe averaged occupation probability(i = p, n)
i is determined with the normalization condition.
ei(k): Single particle energy
mi*: Effective mass of nucleons

Approximate Entropy

Free energies are minimized with respect to mp* and mn*

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

Validity of the Frozen-Correlation Approximation

Full minimization
F/N is minimized with respect tofCts(r), fTt(r), fSOt(r) and mi*with
1.Extended Mayer’s condition
and2. Healing distance condition.

The free energies with the frozen-correlation approximation
are in good agreement with those with the full minimization.

Internal Energy and Entropy

Internal energy at T=20MeVEntropy at T=20MeV

Entropies are in good agreement with the approximate entropies.

This variational calculation is Self Consistent.

Pressure and Critical Temperature

Pressure
Critical Temperature TC is
defined byCritical temperature3. EOS for Non-uniform Nuclear Matter
We follow the TF method by Shen et. al. (NPA637(1998)435)

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

Gradient energy
F0 = 68.00 MeV fm5
Coulomb energy

Nucleon density distributiona : Lattice constant(i = p, n)

f : Free energy density of uniform nuclear matter
ParameterMinimumlog10(T) [MeV]-1.24

Maximum
1.40

Number23 + 1x0.0

0.5

213

r [fm-3]0.000001

0.18

1980

24×213×1980= 10121760 points

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 = 0MeVrB = 1014.23 g/cm3

TF Calculation for Non-uniform Nuclear Matter

Phase diagrams of nuclear matter at finite temperatures
F/Vcell is minimized with respect to niout, niin Ri, ti, aat given density and proton fraction.

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 is constructedin the Thomas-Fermi approximation. (zero and finite temperatures)

Uniform nuclear matter
The obtained thermodynamic quantities are reasonable.
This variational calculation is self consistent.
The validity of the frozen-correlation approximation is confirmed.
Non-uniform nuclear matter
RMS deviation for atomic nuclei is 2.99 MeV.
Phase diagrams are reasonable at zero and finite temperatures.

Ongoing Calculations• Construction of the EOS table for non-uniform matter• Contribution of the a-particle mixing

Construction of the EOS for supernova simulations