構造を仮定しない計算による
低密度原子核物質の非一様構造
Non-uniform structure of low-density nuclear matterby three-dimensional calculation
without any assumption of the structures
Minoru Okamoto1,2 , Toshiki Maruyama2
Kazuhiro Yabana1,3 , Toshitaka Tatsumi4

Graduate School of Pure and Applied Science, University of Tsukuba1
Advanced Science Research Center, Japan Atomic Energy Agency2
Center of Computational Sciences, University of Tsukuba3
Graduate School of Science, Kyoto Universyty41

・MenuⅠ. Pasta structure
Ⅱ. Relativistic mean field theory

Ⅲ. Resultsⅰ) Fixed proton fractionⅱ) β-equilibrium
Ⅳ. Conclusion2

Pasta Structure : “Mixed phase with structure” in first phase transition“meat ball” “spaghetti”
dropletrod~0.2ρ0

Suggested by D.G.Ravenhall, C.J.Pethick, J.R.Wilson in 1983
“lasagna”
slab
Blue：nuclear matter
Balance with Surface tension
cyan：gas nuclei
ρ0：normal nucleus density～0.16fm-3“cheese”
bubbleand Coulomb repulsion

2EC  ES (Equilibrium condition)

K.Oyamatsu, NPA 561 (1993) 431~0.7ρ0tube“macaroni”

Pasta structure in Neutron star・Half mass of Crust region→ Influence on EOS of Crust・GlitchC.P.Lorenz et al, PRL 70 (1993) 4

Rapid increasing of rotational speed of Neutron Star
Whirlpool ofsuper fluidity ofneutron pinned
to nuclei

NucleiSuper fluidity(Neutron)

It may occur from the collapse of whirlpool ofneutron super fluidity on lattice points?
→ It may depend on the crystalline of nuclei andmore exotic structure like “pasta”

・Neutrino cross-section in supernova and collapsing stellar cores
H.Sonoda, G.Watanabe et al PRC 75 042801(2007)3

Conventional Studies
Wigner-Seitz(WS) cell approximation

Calculation in small cellWhole space is divided into equivalent cells.
These are imposed geometrical symmetry as follow
3D → sphere
2D → cylinder
1D → plate0.00“droplet”

Merit ： All calculations are 1D’s↓
Fast calculation & low cost“rod”

“slab”

0.02

0.04“tube”

0.06 fm-3“bubble”

Proton density distribution (based on RMF)

・ Fast calculation & low cost
・ Several complex structures appear as meta
stable state・ Restriction of periodic boundary condition

But …
Only simple structure appear

・Existence of Non-typical pasta structure
M.Matsuzaki PRC 73, 028801 (2006)
K.Nakazato,K.Oyamatsu PRL 103, 132501 (2009)
G.Watanabe et.al PRC 68, 035806 (2003)

・Crystalline of Pasta structure
Conventional studies“droplet” “bubble”
body-centered cubicdumbbell(farfalle)

Gyroid double-diamond“rod” “tube”
honeycomb

K.Oyamatsu, NPA 561 (1993) 431

Full real space
3D calculation
4

Relativistic mean field theoryL   i     m  g   g      g    a  a 

Mean field approximation

111
1
33     m2 2  bm  g    c g  
2234
11 1

 m2                  
22 2

11 m2 R R   R  R  2
41 1
    A   A   A   A 2 2











   , 0 R   ＝  , 0 R  Thomas-Fermi approximation&Zero temperaturedU ( )  2 (r )  m2 2 (r )   gN  ns (r )   ps (r )d  20 (r )  m2 0 (r )  gN  p (r )   n (r ) 
  2 R0 (r )  m2 R0 (r )  g N  p (r )   n (r ) 
 2V (r )  4e 2 (  p (r )  e (r )) p   B  e n   B k F2 , p  m*N2  gN 0  g N R0  Vk F2 , n  m*N2  gN 0  g N R03
V (r )  e  e (r )  3 25

Solutions••

As an initial condition, randomly distribute fermions (n, p, e) over the grid
We solve coupled differential equations, and simultaneously relax fermions density
distributions to attain the uniformity of their chemical potential


2 



k F2 , p (r )  m*N (r )  gN 0 (r )  g N  0 (r )  V (r )

2 



k F2 , n (r )  m*N (r )  gN 0 (r )  g N  0 (r )

 p (r )  n (r ) ρB③μ① ＞ μ②⇒ρB

ρ①：ρ①‐δ ρ②：ρ②＋δ

⑤
① ②μB

④μB
Repeat the same comparison to ③, ④, ⑤•

In process…

BA

CExchanging the baryon density, we solve coupled differential equation again.
•We adjust the particle densities between distant grid points chosenrandomly so as to avoid making separate droplets with different chemical
potentials•

To save the calculation cost, we did not compare the chemical potential
with all grids point and performed the comparison of the chemical
potentials with the small regions frequently.
6

Results : Fixed proton ratio Yp=Z/A(A: total mass , Z: proton number)
Yp=N/A=0.5 (A: total mass, N: proton number)
0.00

proton

0.02

0.04

0.06

0.08 fm-3

electron“droplet”[fcc]
ρB=0.012 fm-3

•
•

“rod”
[honeycomb]
0.024 fm-3

“slab”
0.05 fm-3“tube”“bubble”[honeycomb][fcc]
0.08 fm-3
0.094 fm-3

We obtained the typical pasta structures as ground states.
Crystalline of Pasta structure

New results••

Crystalline of droplet :“fcc” is energetically more favorable than “bcc”
Size of droplet and lattice constant :Smaller than WS cell approximation

Effect of the different size of droplet

7

In the previous study of “Pasta structure” …
D.G. Ravenhall et.al, PRL. 50, 2069 (1983)
K.Oyamatsu et.al, PTP. 72, 2 (1984)

Liquid drop
model

Total Energy  (bulk)  (Surface)  (Coulomb)
Assuming the same size of droplet for “bcc” and “fcc”

All the energy difference is the Coulomb Energy of “bcc” and “fcc”

In our calculation …
ρB [fm-3]

0.010

0.0120.014

0.016

0.018

R(fcc) [fm]

6.33

6.67

6.79

7.02

7.31

R(bcc) [fm]

6.88

6.95

7.17

7.45

7.62

Radius of droplet

Energy difference between “fcc” and “bcc”：
0.2～0.8 MeV/A
The ratio of Coulomb energy difference is 20％

Need to calculate self-consistently including
the size of “Pasta structure”
8Yp=0.3
0.00

proton

0.02

0.04

0.06

0.08 fm-3

neutron“droplet”
“rod”
[fcc]
[simple]ρB=0.016 fm-3 0.030 fm-3

“slab”
0.05 fm-3“tube”[simple]0.066 fm-3

“bubble”[fcc]
0.080 fm-3

Yp=0.1
0.00

proton

0.02

0.04

0.06

0.08 fm-3

neutron“droplet”[fcc]
ρB=0.020 fm-3

“rod”
[simple]0.040fm-3“slab”
0.05 fm-3“tube”[simple]0.066 fm-3

“bubble”[fcc]
0.070 fm-3

For any proton fraction, typical pasta structures appear as ground states

9For β-equilibrium
ρB③

μ① ＞ μ②⇒ρBρ①：ρ①‐δ ρ②：ρ②＋δ

⑤
① ②μB

μB

④

Repeat the same comparison to ③, ④, ⑤β-equilibrium ：  n (r )   p (r )  eρoldμn＞μp+μe

YesNoPreliminary

ρn- δ
ρp+ δρe+ δρnewρn+ δρp- δρe- δ

For WS cell approximation and small cell calculation,
droplet only appear as ground state・“droplet” and “rod” appear as ground states
・The comparison of energy between “droplet” and “rod”
is not yet carried out.
10

・Summary
We explored the pasta structures of low-density nuclear matter based on RMF
without any assumption for the structures.
 In 3D calculation, we obtained the typical pasta structures as ground state We explored the crystalline structure of pasta
 ”fcc” is energetically more favorable than ”bcc” for droplet.
⇒ Difference of droplet’s size for “bcc” and ”fcc”
 In β-equilibrium, we obtained “droplet” & “rod” structure as ground state.

・Future plans
 Explore of the EOS of β-equilibrium nuclear matter in detail Calculation of Shear modulus in the crust region of Neutron star Extension to the higher density and finite temperature nuclear matter Comparing with QMD calculation result for supernova compression
process and local minimum states of our calculation

11