Non-congruence of liquid-gas phase transition of asymmetric nuclear matter

T. Maruyama (JAEA) & T. Tatsumi (Kyoto U)

1

• Mixed phase at first-order phase transitions. • Its non uniform structures. • Its equation of state.

T. Maruyama, N. Yasutake and T. Tatsumi, Prog. Theor. Phys. Suppl. 186, 69 (2010)

T. Tatsumi, N. Yasutake, T. Maruyama, arXiv:1107.0804

M. Okamoto, T. Maruyama, K. Yabana, T. Tatsumi, arXiv:1106.3407

2

Phase transitions in nuclear matter

In stellar objects, there are many phase transitions considered:

liquid-gas, neutron drip, meson condensation, hyperon mixture,

quark deconfinement, color super-conductivity, etc.

Some of them are the first-order mixed phase

EOS of mixed phase

• Single component congruent

(e.g. water)

Maxwell construction satisfies the

Gibbs cond. TI=TII, PI=PII, mI=mII.

• Many components non-congruent

(e.g. water+ethanol)

Gibbs cond. TI=TII, PiI=Pi

II, miI=mi

II.

No Maxwell construction !

• Many charged components (nuclear matter)

Gibbs cond. TI=TII, miI=mi

II.

No Maxwell construction !

No constant pressure !

r

rU

dr

dP iii

;

3

129Xe + Sn at 50 MeV/u AMD A.Ono et al. PRC 66, 014603(2002)

Collision Hot matter Gas Cool down Free nucleons

+ Liquid clusters

First-order phase transition mixed phase. Clustering of low-density nuclear matter may be the result of mixed phase.

• Multi fragmentation in heavy-ion collisions

Liquid-gas phase transition at T>0

dilute phase + dense phase (fragments)

Formation of

fragments

Finite system

Mechanical instability is important !

4

Depending on the density, geometrical structure of mixed phase changes from droplet, rod, slab, tube and to bubble

configuration. [Ravenhall et al, PRL 50 (1983) 2066]

•Matter in the collapsing stage of supernova

Liquid-gas phase transition at T>0

dilute phase + dense phase

• Neutron star crust

Neutron drip at T~0

neutron phase + nuclear phase

Formation

of “Pasta”

structures

Macroscopic system (matter in the stellar objects)

Chemical instability

Mechanical & chemical instability

5

Stability condition

3 3

3

2

2

2 2

2

3

2 2

2

3 3

,

0

0

p n p n

f P

f

f f

f f

m m

Mechanical stability + chemical stability (congruence)

,

, ,

0

0, or 0

pT Y

p n

p pT P T P

P

Y Y

m m

Theoretical model for nuclear matter

Relativistic Mean Field (RMF) model： Lorentz-covariant Lagrangian L with baryon densities, meson fields , , , electron density and the Coulomb potential, is determined.

Local density approx for Fermions： Thomas-Fermi model for baryons and electron

Consistent treatment for potentials and densities： Coulomb screening by charged particles

[T.M. et al,PRC72(2005)015802; Rec.Res.Dev.Phys,7(2006)1]

6

* 3

2 2 2 2 2

*

1

2

1 1 1 1 1 1( ) ( )

2 2 4 2 4 2

1(

4

1, ( )

3

,

,

, )

N M e

N N N N

M

e e e e

N N N

L L L L

L i m g g b e V

L m U m R R m R R

L V V i m e V F F F

m m g U bm

m m m mm m m m

m m m mm m m m m

m m mm m m m m m

3 41( ) ( )

4N N Ng c g

7

Choice of parameters -- Properties of uniform matter and nuclei--

Symmetric matter:

Energy minimum at B=0.16 fm3

and E/A=16 MeV

8

Bulk calculation of phase coexistence in nuclear matter

Gibbs conditions:

TI=TII

PiI=Pi

II

miI=mi

II. should be fulfilled

among two nuclear

matter with different

densities.

coexistence curve

bary

on p

ressure

proton fraction

Results ---

9

Bulk calculation of phase coexistence in nuclear matter

When the system enters the forbidden region, it

splits into 2 phases with different components.

Gibbs conditions:

TI=TII

PiI=Pi

II

miI=mi

II. should be fulfilled

among two nuclear

matter with different

densities.

coexistence curve

forbidden

proton fraction

bary

on p

ressure

Results ---

10

Yp-PB phase diagram of matter

congruent

non congruent

non congruent

Yp Dependence of the Congruence

• Symmetric nuclear matter is congruent and

has constant value of Yp=0.5.

Maxwell construction.

• In general cases, liquid and gas phases are

not congruent and have different values of Yp.

• In the case of small Yp, the retrograde

transition (gas-mix-gas) may occur.

But by surface tension and the Coulomb

interaction, it might be suppressed.

how 2 phases are

chemically same.

11

Symmetry-potential dependence of phase-coexistence curve

Narrower region with

weak symmetry-potential.

Strong attraction

between p and n

non-congruence of

nuclear matter

Only symmetric nuclear

matter can be congruent.

(normal x 0.1)

12

Temperature dependence of the phase-coexistence curve

At higher temperatures, the region of

mixed phase becomes narrow, and

the congruence is enhanced.

13

2 2 ( ) ( )

2 20 0

2 20 0

2 2ch ch

( ) ( ) ( ) ( ( ) ( )),

( ) ( ) ( ( ) ( )),

( ) ( ) ( ( ) (

From 0,   ( , , , , ),

)),

( ) 4 ( ), ( (

( )

) ( )

s sN n p

N p n

N p n

C p

dUm g

d

m g

R m R g

V e

R V

m m m

mm

r r r r r

r r r r

r r r r

r r r r

L L

( ))e r

12 * 2

,

1

3 3

,3 30 0

( ; , ) 1 e

For fermions, we em

xp ( ) ( ) ,

( ; , ) 1 exp ( ( ))

( ) 2 ( ; , ), ( ) 2 ( ;(

ploy Thomas-Fermi

2 ) (

approx. at finite

2 )

i n p i N i

e e e C

e e e i p n i

T

f p m T

f p V T

d p d pf f

m

m m

m

r p r r

r p r

r r p r r p

0 0 0 0

, )

( ) ( ) ( ), ( ) ( ) ( ) ( ),

i

n n N N p p N N Cg g R g g R V

m

m m r r r r r r r

Chemical equilibrium fully

consistent with all the density

distributions and fields.

EOM to be solved

Mixed phase with finite-size effects (“pasta” matter)

14

Numerical calculation of mixed-phase

• Assume regularity in structure: divide whole space into equivalent and neutral cells with a geometrical symmetry (3D: sphere, 2D : cylinder, 1D: plate). Wigner-Seitz cell approx. • Give a geometry (Unif/Dropl/Rod/...) and a baryon density B. • Solve the field equations numerically. Optimize the cell size (choose the energy-minimum). • Choose an energy-minimum geometry among 7 cases (Unif (I), droplet, rod, slab, tube, bubble, Unif (II)).

WS-cell

Density profiles in WS cell

Pasta structures in matter (case of fixed Yp)

Yp=0.5 T=0

Yp=0.1 T=0

15

16

In the case of symmetric (Yp=0.5)

nuclear matter,

p(r) and n(r) are almost equal.

e(r) is approximately independent.

p & n are congruent.

Maxwell construction may be

possible for pn matter with uniform e.

But baryon and electron are not

congruent.

Maxwell constr. of pne matter is

impossible.

Yp=0.5 T=0

Case of symmetric nuclear matter

17

Yp=0.1 T=0

In the case of asymmetric (Yp<0.5)

nuclear matter,

p(r) and n(r) are different.

e(r) is approximately independent.

p & n are non-congruent

Maxwell constr does not satisfy

Gibbs cond.

Case of asymmetric nuclear matter

18

EOS with pasta structures in nuclear matter at T 0

Symmetric matter Yp=0.5

Asymmetric matter Yp=0.3

Pasta structure appear at T 10 MeV

coexistence region (Maxwell for Yp=0.5 and bulk Gibbs for Yp<0.5) is meta-

stable. Uniform matter is allowed in some coexistence region due to finite-size

effects.

19

EOS with pasta structures in nuclear matter at T 0

Symmetric matter Yp=0.5

Asymmetric matter Yp=0.3

Pasta structure appear at T 10 MeV

coexistence region (Maxwell for Yp=0.5 and bulk Gibbs for Yp<0.5) is meta-

stable. Uniform matter is allowed in some coexistence region due to finite-size

effects.

20

Summary

We have studied liquid-gas phase transition of nuclear matter. [Bulk calculation] •The non-congruence of nuclear matter comes from strong symmetry potential. • Symmetric nuclear matter behaves like single-component matter, while asymmetric matter does not. [Considering matter structures] • Pasta structures appear in liquid-gas transition. • The region of pasta is narrower than the phase-coexistence region due to the finite-size effects (surface and Coulomb).

Future

• More understanding of pasta matter in stellar objects. • Its implications to stellar phenomena.

21

22

Pasta structures at T=10 MeV vs T=0

Mixed phase of liquid

and gas.

The surface is more

vague than T=0 case.

weaker surface

tension

smaller size

Electron distribution is

more uniform.

less screening

smaller size

T=10 MeV T=0