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).
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Motivation
Methods
Results
Conclusion
Background
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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.
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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
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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)
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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
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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.
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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
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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
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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
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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
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Results
Strong Yp dependence
T=1
T=10
Bubble appearance
Delay the transition to uniform matter
• Free energy & Entropy
Small difference
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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
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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
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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.
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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.
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Conclusion
Outlook
1. More pasta phases could be considered in STF.2. Alpha particles will be included in the future.
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Thank you!