Structured Mixed Phase of Nuclear Matter

Toshiki Maruyama (JAEA)In collaboration with

S. Chiba, T. Tatsumi , D.N. Voskresensky, T. Tanigawa, T. Endo, H.-J. Schulze, and N. Yasutake

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

2

Phase transitions in nuclear matter

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, I=II.

• Many components noncongruent (e.g. water+ethanol)

Gibbs cond. TI=TII, PiI=Pi

II, iI=i

II.No Maxwell construction !

• Many charged components (nuclear matter)Gibbs cond. TI=TII, i

I=iII.

No Maxwell construction !No constant pressure !

r

rU

dr

dP iii

;

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In the mixed phase with charged particles, non-uniform “Pasta” structures are expected. [Ravenhall et al, PRL 50(1983)2066, Hashimoto et al, PTP 71(1984)320]

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

Quite differentdifferent from a bulk picturebulk picture of mixed phase.Need to take into account the effects of pasta structures when we calculate the EOS.

Surface tension & Coulomb

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(I) Low density nuclear matter (I) Low density nuclear matter

• Collapsing stage of supernova explosion (non equil) Liquid-gas phase transition T=0 (not realistic) electron gas + nuclear liquid T>0 (1~several MeV) nuclear gas + nuclear liquid

• Neutron star crust (equil) Neutron drip T=0 neutron liquid + nuclear liquid

Field equations to be solvedField equations to be solvedRelativistic 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]

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* 3

2 2 2 2 2

*

1

2

1 1 1 1 1 1( ) ( )

2 2 4 2 4 21

(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

3 41( ) ( )

4N N Ng c g

6

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

Symmetric matter: Energy minimum at B=0.16 fm and E/A= MeV

7

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

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 exp ( ) ( ) / ,

( ; , ) 1 exp ( ( )) /

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

For Fermions, we employ Thomas-Fermi approx. at finite

) (2 )

i n p i N i

e e e C

e e e i p n i

f p m T

f p V T

d p d pf f

T

r p r r

r p r

r r p r r

0 0 0 0

; , )

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

i

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

p

r r r r r r r

Chemical equilibrium fully consistent with all the density distributions and fields.

EOM for fields

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Numerical calculation of mixed-phaseNumerical 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=0.5T=0

Yp=0.1T=0

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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=0.5T=0

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Clustering mechanism of symmetric nuclear matter

(Note that the total pressure is positive due to electron pressure.)

For symmetric matter at T=0, Maxwell constr leads:negative pressure region (B0) is unstable. mixed phase (clustering)

But at some density just below 0, uniform matter is favored due to finite size effects (surface and Coulomb).

Yp=0.5

Maxwell &bulk Gibbs

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Yp=0.1T=0

In the case of asymmetric (Yp<0.5) nuclear matter,p(r) and n(r) are different.e(r) is approximately independent.

Maxwell constr does not satisfy Gibbs cond. Non congruent

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EOS and the structure size

T=0 T=0Pasta structures (droplet, rod,…) appear with the change of density.

Pasta energy gain

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mechanical instability(dP/d<0) formation of “pasta” (?)

phase coexistence(bulk Gibbs)

Instability of uniform matter at finite T

Due to the surface tension and the Coulomb interaction, the region of inhomogeneous matter should be limited ( “pasta” < coexistence ).Its relation with Mechanical instability is under investigation.

For Yp<0.5, chemical instability should be taken into account. future work

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EOS with pasta structures in nuclear matter at T>0

Pasta structure 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.

spinodal region (dP/d <0) is unstable. clusterize

Spinodal region is included in the pasta region (?).

Symmetric matter Yp=0.5

Asymmetric matter Yp=0.3

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Strong surface tension and weak Coulomb large R

Extreme case no minimum. (pasta unstable) [Voskresensky et al,PLB541(2002)93;NPA723(2003)291]

Uniform matter appear at some part of coexistence region.

size of structure R

Structure sizeStructure size

(Charge screening)

Dependence of E/A on R.

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Coulomb screening effectsCompare different treatments of Coulomb int.

Smaller structure size for “no Coulomb screening” calc.Coulomb screening enlarges the structure size.Narrower pasta region for “no Coulomb screening” calc. Coulomb screening makes pasta unstable.

(uniform electron)

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Surface tension effects Compare different nuclear surface tension

Weak surface Normal surface tension

Smaller structure size for weak surface tension.Stronger surface tension enlarges structure size.Wider pasta region for weak surface tension.Weaker surface tension makes EOS close to bulk Gibbs.

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Beta equil.

Beta-equilibrium case

Only “droplet” structure appears.

The change of EOS due to the non-uniform structure is small.

The proton fraction Yp is drastically influenced by the non-uniform structure.

T=0

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Yp-PB phase diagram of matter

• Symmetric matter has constant value of Yp=0.5. It corresponds to the Maxwell construction.• In general cases, liquid and gas phases 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.

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Mixed phase at higher densitiesMixed phase at higher densities

• Kaon condensation [PRC 73(2006)035802, RecentResDevPhys7(2006)1]• Hadron-quark transition [PRD 76(2007)123015, PLB 659(2008)192]

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From a Lagrangian with chiral symmetry

K single particle energy (model-independent form)

Threshold condition of condensation

(II) Kaon condensation(II) Kaon condensation

22 * 2 2 2

* 2 2 2

( ) 2 4 2 4

2 4

( 0)

K n p n p

K K KN p n

K n p e

p m f f

m m f

p

p

Kaon condensation in uniform matter

• With increase of density, fraction of proton and electron increases so as to decrease neutron energy.• At around 30 there appears K and neutron fraction further decreases. • At higher density, pnsince K attracts p more than n.

Due to the strong attraction between K and p, pressure decreases. (density gradient becomes negative) First-order phase transition Phase separation Non uniform matter

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Par

ticle

frac

tion

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EOM for fields (RMF model) Kaon field K(r) added.

2 2 2

2 2 20 0

2

2

2

0 0

2 2 20 0 0 0

2 * 20 0

2 2

4

2

2

4 ,

2

s sN p n K K K

N n p K K K K C K K

N n p K K K K C K K

K K C K K

C p e

K C

K

K

dUm g g m f

d

m g g m f V g g R

R m R g g m f V g g

K

R

m V g g R

V

K

e

V g

K

K K

ch ch

0 0

1/3

2 * 20 0

2

2

* 20 0

3

K K

e e C

n Fn N N N

p Fp N N N C

g R

V

k m g g R

k m g g R V

K

Kaonic pasta structure

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At 2-3 hyperons are expected to appear. Softening of EOS Maximum mass of neutron star becomes less than 1.4 solar mass. Contradicts the obs >1.5 Msol

Possibility to resolve this problemby introducing quark phase in neutron stars.

(III) Hadron-quark phase transition(III) Hadron-quark phase transition

1.4

(only pn) } hyperons

[Schulze et al, PRC73 (2006) 058801]

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Coupled equationsCoupled equations to get density profile, energy, pressure, etc of the system

2 2

4/3

3

, 2 ,

( )( , , , , , , , )

( )

( ) ( ) ( ) ( ) / 8

( ) (( )

( )

( ) 3 ( )

1/ ( )

u e d s n u d p e n e

ii

B e C

HB

Q

e e

B

i u d s p n e

V e

E A d rV

r

r

r r r r

rr

r

r r

r

hadron phase: BHF)

(quark phase: MIT bag)

3

3

2 1 1( ) ( ) ( ) ( ) ( ) ( ) 0 ( )

3 3 3

1 1 1( ) ( ) ( ) ( ) ( ) (

3 3

B

V

p u d s eV

p n u d

S S

V

d r

d rV

r r r r r r

r r r r r

average baryon density

Q-H bondary area

cell volume

total charge

1) ( ) ( )

3 s BV

r r given

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Hadron-quark dropletHadron-quark droplet

Rearrangement of chargeQuark phase is negatively charged. u quarks are attracted and ds quarks repelled. Same happens to p in the hadron phase. Localization of e into hadron phase.

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EOS of matterEOS of matter

Full calculation is close to the Maxwell construction (local charge neutral). Far from the bulk Gibbs calculation (neglects the surface and Coulomb).

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0,

4

211

41

2

0

13

2

RRRmM

dsssrmm

r

PP

r

GmP

m

Pr

r

Gm

dr

dP

r

Density at r

Mass inside rTotal mass and Radius

Pressure ( input of TOV eq.)

Solve TOV eq.

Structure of compact starsStructure of compact stars

Bulk Gbbs Full calc surf=40 MeV/fm2 Maxwell const.

Density profile of a compact star (M=1.4 solar mass)

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Mass-Radius relation of compact stars

Full calc yields the neutron star mass very close to that of the Maxwell constr.

The maximum mass are not very different for three cases. surf=40

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• Maxwell construction :Assumes local charge neutrality (violates the Gibbs cond.)and neglects surface tension.

• Bulk Gibbs calculation :Respects the balances of i between phases but neglects surface tension and the Coulomb interaction.• Full calculation: includes everything.Strong surface Large R, charge-screening effectively

local charge neutral. close to the Maxwell construction.Weak surface Small R Coulomb ineffective close to the bulk Gibbs calc.

Equation of state of mixed phase ?Equation of state of mixed phase ?

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SummarySummary

• We have studied ``Pasta’’ structures of nuclear matter at the first-order phase transitions.• Pasta structures appear in liquid-gas transition at low-density, meson condensation at high-density, and hadron-quark transition. • Coulomb screening and stronger surface tension enlarges the structure size. • If surface tension is strong, Maxwell construction is effectively valid.