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!