relativistic equation of state at subnuclear densities in the thomas- fermi approximation zhaowen...
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Relativistic equation of state at subnuclear densities in the Thomas-Fermi approximation
Zhaowen ZhangSupervisor: H. Shen
Nankai University
20th-22th Oct. 2014
KIAA at Peking University, Beijing, ChinaZ. W. Zhang and H. Shen, Astrophys. J. 788, 185 (2014).
Motivation
Methods
Results
Conclusion
Background
Background
Supernova explosions Neutron star formations
• Equation of state(EOS) of nuclear matter is very important in understanding many astrophysical phenomena:
Lots of the EOS investigations focused on the case of zero temperature or high density for uniform matter.
Background
G. Shen C. J. Horowitz S. Teige. PhysRevC, 82, 015806 (2010)
• The EOS for the core-collapse supernova simulations covers wide ranges of temperature, proton fraction, and baryon density.
T=1 MeV
T=3.16 MeV
T=6.31 MeV T=10 MeV
Background
…
Lattimer–Swesty Compressible liquid-drop model
Lattimer, J. M., & Swesty, F. D. Nucl. Phys. A, 535, 331 (1991)
• Some famous nuclear EOSs
H. Shen etc. Parameterized Thomas–Fermi approximation
Shen, H., Toki, H., Oyamatsu, K., & Sumiyoshi, K. Prog. Theor. Phys., 100, 1013 (1998)
G. Shen & Horowitz etc. Relativistic mean field theory
G. Shen C. J. Horowitz S. Teige. PhysRevC, 83, 035802 (2011)
Background
Parameterized Thomas–Fermi approximation
• Nucleon distribution function
• Gradient energy
F0 = 70 MeV fm5 is determined by reproducing the binding energies and charge radii of finite nuclei.
in ou t
3
t ou
out
,
,
1 0i
ii
i C
t
i i ii
i
rr R
n r
R r R
n n nR
n
2
3
cellce 0ll ng
pnE r n d rF r
Motivation
• Self-consistent Thomas–Fermi approximation
Nucleon distribution and gradient energy are calculated self-consistently.
Both droplet and bubble configurations are considered.
bubbledroplet uniform matter
• In present work, we compare and examine the difference between PTF and STF.
Methods• Lagrangian density
Equations of motion
3
,
2 2 3 42 3
223
2
RMF
1
2
1 1 1 1
2 2 3 41 1 1
4 2 41 1 1
4 2 4
ai a i
i p n
e e e
a a a a
i M g g g e A
i m e A
m g g
W W m c
R R m F F
L
0
30 0A A
Mean field approach
2 2 2 32 3
2 2 33
2
2
23
s
v
c
m g g g
A e
n
m c g n
m g
n
n
Methods
• Distribution functionFermi–Dirac distribution
• Chemical potential
Wigner–Seitz cell
• Wigner–Seitz cell
BCC
22 0
( ) (1
) )( k kii ifn r d r rkk f
2 *2
2 *2
1
1 exp /
1
1 exp /
ki
i
ki
i
fk M T
fk M T
*M M g
p p
n n
g g eA
g g
BCC WSV V
Methods
• Thermodynamic quantities
Entropy density
Free energy
Energy density
2 2 *22
, 0
2 2 2 3 42 3
2 2 2 43
2 2 2
2
1
1 1 1 1( )
2 2 3 41 1 1
( )2 2 41 1
( )2 21
( )2
k ki i
i p n
p n
p n
p e
dkk k M f f
m g g
m c g n n
m g n n
A eA n n
ò
22
, 0
1ln 1 ln 1
ln 1 ln 1
k k k ki i i i
i p n
k k k ki i i i
s dkk f f f f
f f f f
cell cell cellF E TS
Methods• Calculation
T Yp ρB RWS
μi σ0(r) ω0(r) ρ0(r)
Nucleon distribution ni(r)
σ(r) ω(r) ρ(r) A(r)
ni(r) converge
Ecell Scell Fcell
Minimizing Fcell by changing RWS
Thermodynamically favored state
YES
NO
M mσ mω mρ gσ
938.0 511.19777 783.0 770.0 10.02892
gω gρ g2 (fm-1) g3 c3
12.61394 4.63219 -7.23247 0.61833 71.30747
TM1 Parameter set
Y. Sugahara and H. Toki, Nucl. Phys. A, 579, 557 (1994)
different initial fields lead to different configuration
Results
Strong Yp dependence
T=1
T=10
Bubble appearance
Delay the transition to uniform matter
• Free energy & Entropy
Small difference
Results
T=1 T=10
ρB
• The densities at the center are lower in the STF.• The cell radius Rc of STF is larger.• More free nucleons exist outside the nuclei at T = 10 MeV.
• Nucleon distribution
Results• Numbers & Fractions
T=1
T=10
Nuclei fractionNeutron gas fractionProton gas fraction
cell
cell
/( ) /( ) /
A d B
n n C B
p p C B
X A NX V n R NX V n R N
T=1
T=10
Cause by difference of nucleon distribution
More nucleons can drip out of the nuclei
Ad
Zd
Ad
Zd
XA
XA
Xn
Xn
Xp
Dominant
Results
T=1 T=10
Yp =0.3
Yp =0.5
• Neutron chemical potential
• The results of droplet are almost identical for STF and PTF.• The sudden jumps caused by the different Coulomb potential of bubble and droplet.
Results
T=1 T=10
Yp =0.3
Yp =0.5
• Proton chemical potential
• The difference of STF and PTF may be caused by the Coulomb and surface energies.• Proton is directly effected by Coulomb interaction.
Conclusion
Outlook
1. More pasta phases could be considered in STF.2. Alpha particles will be included in the future.
Thank you!