1. 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

2. 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 .

3. 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

4. １．Uniform Nuclear matter at Zero Temperature
Ref.) H. Kanzawa, K. Oyamatsu, K. Sumiyoshi, M. Takano,
Nucl. Phys. A791(2007),232.

5. 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)

6. 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.

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

8. (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

9. Three-Body Energy (step2)
(for neutron matter)
(for symmetric matter)
correction term　
repulsive part 　 　 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:

10. 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 =
γ = MeV･fm6
　　δ = fm3

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

12. ２．Uniform Nuclear matter at Finite Temperatures
Ref.) H. Kanzawa, K. Oyamatsu, K. Sumiyoshi, M. Takano,
Nucl. Phys. A791(2007),232.

13. 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)

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

15. Pressure 　
＊critical temperature Tc～18MeV

16. ３．Thomas-Fermi Calculations for Atomic Nuclei

17. 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.

18. 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

19. 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

20. ＊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

21. 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

22. Thank you for your attention!