Symmetry Energy within theBrueckner-Hartree-Fock

approximation

“International Symposium on Nuclear Symmetry Energy”Smith College, Northampton ( Massachusetts) June 17th-20th 2011

Isaac VidañaCFC, University of Coimbra

In collaboration with:

U. Coimbra: C. Providência, C. Ducoin

U. Barcelona: A. Polls

U. Surrey: A. Rios

IPN, Orsay: J. Margueron

Isospin asymmetric nuclear matter is present in:

Nuclei, especially those far away from thestability line & in astrophysical systems

(neutron stars)

Motivation

A well-grounded understanding of the properties of isospin-richnuclear matter is necessary for both nuclear physics & astrophysics

However, some of these properties are notwell constrained. In particular the densitydependence of the symmetry energy is stillan important source of uncertainties.

Some properties of asymmetric nuclear matter can be obtained from:

the analysis of experimental data inheavy ion collisions

(e.g., ID, double n/p ratios, GDR, …)

the analysis of existing correlationsbetween different quantities in bulk

matter & finite nuclei(e.g. δR versus L)

PREX experiment @ JLAB

A major effort is being carried out to studyexperimentally the properties of asymmetric nuclearsystems. Experiments at CSR , GSI (FAIR), RIKEN,GANIL, FRIB can probe the behavior of the symmetryenergy close and above saturation density.

Astrophysical observations of compact objects window into nuclear matter at extreme isospin asymmetries

In this talk …

Study of density dependence of the symmetryenergy within the BHF approximation andcomparison with effective models (Skyrme & RMF).

Phys. Rev. C 80, 045806 (2009)

Analysis of correlations between L and Ksym.Special attention to correlations of L with neutronskin thickness and crust-core transition point inneutron stars.

based on:

Phys. Rev. C 83, 045810 (2011)

Equation of State of Asymmetric Nuclear Matter

!

E

A(",#) = E

SNM(") + S2(")#

2+ S4 (")#

4+O(6)

Charge symmetry expansion of (E/A)ANM on even powers of isospin asymmetry β=(ρn-ρp)/(ρn+ρp)

!

S2(") =

1

2

# 2E /A

#$ 2$ = 0

,

!

S4(") =

1

24

# 4E /A

#$ 4$ = 0

!

ESNM(") =

E

A(",# = 0),

!

S2(") ~

E

A(",# =1) $

E

A(",# = 0)

In good approximation:

!

ESNM(") = E0 +

K0

2

" # "03"0

$

% &

'

( )

2

+Q0

6

" # "03"0

$

% &

'

( )

3

+O(4)

!

K0

= 9"0

2#2ESNM(")

#"2"= "0

$ 240 ± 20MeV

!

Q0 = 27"03#

3ESNM(")

#"3"= "0

$ %500 ÷ 300MeV

!

E0 = ESNM(" = "0) # $16MeV

Similarly S2(ρ) can be also characterized with few bulk parameters around ρ0

!

S2(") = Esym + L" # "03"0

$

% &

'

( ) +

Ksym

2

" # "03"0

$

% &

'

( )

2

+Qsym

6

" # "03"0

$

% &

'

( )

3

+O(4)

!

L = 3"0

#S2(")

#""= "0

!

Ksym = 9"0

2#2S2(")

#"2"= "0

!

Qsym = 27"0

3#3S2(")

#"3"= "0

Less certain & predictions of different models vary largely

ESNM(ρ) commonly expanded around saturation density ρ0

Combining the expansions of ESNM(ρ) and S2(ρ) one arrives at

where

!

"0(#) = "

0$ 3"

0

L

K0

# 2 +O(4)

!

E0(") = E

0+ Esym"

2+O(4)

!

K0(") = K0 + Ksym # 6L #Q0

K0

L$

% &

'

( ) "

2+O(4)

!

Q0(") =Q

0+ Qsym # 9L

Q0

K0

$

% &

'

( ) "

2+O(4)

!

E

A(",#) = E0(#) +

K0(#)

2

" $ "0(#)

3"0(#)

%

& '

(

) *

2

+Q0(#)

6

" $ "0(#)

3"0(#)

%

& '

(

) *

3

+O(4)

Kτ

BHF approximation of ANM

Bethe-Goldstone Equation

!

G "( ) =V +VQ

" # E # E ' + i$G "( )

Energy per particle

!

E

A(",#) =

1

A

h2k

2

2m$

+1

2Re U$ (

r k )[ ]

%

& '

(

) *

k+kF$

,$

,

!

E" (k) =h

2k

2

2m"

+ Re U" (k)[ ]

!

U" (k) =r k

r k ' G # = E" (k) + E" '

(k ')( )r k

r k '

Ak '$kF" '

%" '

%

Infinite sumation of two-hole line diagrams

Partial sumation of pp ladder diagrams

Pauli blocking Nucleon dressing

Few words on the NN and NNN forces used …

Argonne V18 (Av18) NN potential

!

Vij = Vp (rij )Oij

p

p=1,18

"

!

Oij

p=1,14 = 1,r " i #

r " j( ),Sij ,

r L #

r S ,L

2,L2 r " i #

r " j( ),

r L #

r S ( )

2$ % &

' ( ) * 1,

r + i #

r + j( )[ ]

!

Oij

p=15,18 = Tij ,r " i #

r " j( )Tij ,SijTij , $ zi + $ zj( )[ ]

Urbana IX (UIX) NNN potential

!

Vijk

UIX=Vijk

2"+Vijk

R

!

Vijk

2": Attractive Fujita-Miyazawa force

π

π

!

Vijk

2" = A Xij ,X jk{ }r # i $

r # j ,

r # j $

r # k{ } +

1

4Xij ,X jk[ ]

r # i $

r # j ,

r # j $

r # k[ ]

%

& '

(

) *

cyclic

+

!

Xij =Y m" rij( )r # i $

r # j + T m" rij( )Sij

!

Y (x) =e"x

x1" e

x2

( )

!

T(x) = 1+3

x+3

x2

"

# $

%

& ' e(x

x1( ex

2

( )2

!

Vijk

R = B T2rij( )T 2 rjk( )

cyclic

"

!

Vijk

R: Repulsine & Phenomenological

!

VNN

eff r r ij( ) = V

UIX r r i,

r r j ,

r r k( )n

r r i,

r r j ,

r r k( )d3

r r k"

Reduced to an effectivedensity-dependent 2BF

A, B fit to reproduce the saturation point

!

Un ~U0+Usym"

!

Usym =Un "Up

2#

BHF nucleon mean field in ANM

Symmetry potential

Isospin splitting of mean field in ANM

!

Up ~U0"Usym#

G-matrix gives access to in-medium NN cross sections

!

"## ' =m#*m# '*

16$ 2h4

2J +1

4$G## '%## 'LL 'SJ

2

, ## '= nn, pp,npLL 'SJ

&

Phenomenological approaches

Skyrme

Relativistic mean field models

Non-linear Walecka models (NLWM) with constant coupling constants: NL3, TM1, GM1, GM3 & FSU Density dependent hadronic models (DDH) with density dependent coupling constants: TW, DD-ME1, DD-ME2, DDHδ

Quark meson coupling model QMC

Lyon group SLy SkI family

Nuclear matter: system of non-overlaping MIT bags interacting through exchange of scalar and vector mean fields

Bulk parameters of ESNM(ρ) & S2(ρ)

0.92

0.56

0.62

1.00

1.05

0.67

0.46

0.47

0.59

0.66

0.65

γ

-339.6-159.8-27.863.135.8-225.1213.6-17.300.240BHF (2BF)

-343.8-162.8-23.466.933.6-224.9185.9-14.620.176BHF (3BFb)

-446.428.0-10.093.533.7-387.5291.0-15.700.150QMC

-332.1535.2-124.755.332.7-540.1240.1-16.250.153TW

-276.6424.1-51.360.532.6-523.4230.0-16.300.148FSU

-518.7-66.433.6110.836.8-285.2281.0-16.320.145TM1

-698.4181.2100.9118.537.4203.1271.6-16.240.148NL3

-322.5358.8-43.459.929.6-335.7250.3-16.150.162SkI4

-292.7602.8-98.443.931.8-364.2229.9-15.980.160SLy230a

-320.4520.8-119.845.331.8-362.9229.8-15.970.159SLy4

-334.7-112.8-31.366.534.3-280.9195.5-15.230.187BHF (3BFa)

KtQsymKsymLEsymQ0K0E0r0Model

!

S2(") = Esym

"

"0

#

$ %

&

' (

)

* ) =L

3Esym

HIC at intermediate energiesconsistent with

(Adapted from M. B. Tsang et al,Phys. Rev. Lett. 102, 122701 (2009))

Symmetry Energy versus L

(Adapted from D. V. Shetty & S. J.Yennello, Pramana 75, 259 (2010))

Recent extracted values of L

BHF

Density dependence of S2 and L

Spin-Isospin contributions to Esym and L

66.534.3Total29.214.3F.G.13.41.4(1,1)52.228.4(1,0)-4.3-3.9(0,1)-24.0-5.9(0,0)

LEsym(S,I)

Larger contribution to Esym and Lfrom S=1 channel (tensor)

!

L(") # 3"0

$S2(")

$"

Effect of three-body forces

3BF increase the slope of S2(ρ) astrophysical consequences:

larger Yp earlier onset of direct URCA

stiffer EoS larger Mmax of neutron star

Correlation of Ksym & Kτ with L

• BHFSkyrme RMF

• BHF

S2: crossing at ρ ~ 0.11 fm-3, S2(0.11)~24±4 MeV (expected from finite nuclei constraints at ρ<ρ0)

L : tendency to cross at ρ ~ ρ0/3

Ksym: no crossing observed

S 2 (M

eV)

S 2 (M

eV)

Neutron Skin Thickness & Symmetry Energy

Neutron skin thickness

!

"R = rn2# rp

2

Typel & Brown showed that δRcalculated in mean field modelsis very sensitive to the slope ofthe symmetry energy.

Typel-Brown correlation

Den

sity

ρ (

fm-3

)

r (fm)

Fully self-consistent finite nuclei calculation based on BHF approach toodifficult δR estimated to lowest order in the diffuseness corrections (Steineret al., Phys. Rep. 411, 325 (2005))

!

"R #3

5t

!

t ="c

#0("c )(1$"c

2)

Es

4%ro2

d# # Esym /S2(#) $1( ) ESNM (#) $ E0( )$1/ 2

o

#0 (" c )

&

d# # ESNM (#) $ E0( )1/ 2

o

#0 (" c )

&

thickness of semi-infiniteasymmetric matter

Correlation of the Neutron Skin Thickness δR with L & Ksym

Linear increase of δR with L & Ksym

!

P ",#( ) ="2

3"0

L# 2 + K0+Ksym#

2( )" $ "

0

3"0

+L

%

& '

(

) *

Not surprising:

Pressure δR

In n-rich matter pressure increases with L at fixed ρ

Neutron Stars & Symmetry Energy:Crust-Core transition density

The crust of a neutron star is very importantfor a number of observable properties :

thermal evolution glitches X-ray burst ….

(Picture from Nicolas Chamel )

It is very important to understand well thecrust-core transition region

Which constraints are set by the isospin dependence of the nuclear EoS on the transition region ?

How sensitive is to the symmetry energy ?

!

Tr(C ) > 0, Det(C ) > 0

Curvature Matrix:

Crust-core transition estimated from crossing of β-equilibrium EoSand spinodal instability line

Thermodynamical spinodal upperbound of the the real ρt ( ~ 15%larger than TF calculation ofnuclear pasta )

(Figures courtesy of C. Ducoin & C.Providência)

positive definite:

!

C =

"µn/"#n "µ

n/"#p 0

"µp /"#n "µp /"#p 0

0 0 "µe /"#e

$

%

& & &

'

(

) ) )

!

+k2

Dnn Dnp 0

Dpn Dpp 0

0 0 0

"

#

$ $ $

%

&

' ' '

+4(e2

k2

0 0 0

0 1 )1

0 )1 1

"

#

$ $ $

%

&

' ' '

ρtt: we consider first this point

ρtd

Correlation of Ypt and ρt with LPr

oton

frac

tion

Ypt

Density ρt [fm-3]

Prot

on fr

actio

n Y

pt

Prot

on fr

actio

n Y

ptDensity ρt [fm-3]L [MeV]

Den

sity

ρt [

fm-3

]

40 60 20 80 100 120 140

L [MeV]

spinodal

β−stability

Clear decreasing correlations

Dispersion of Ypt due to dispersion of Esym

Higher L { Lower Ypt

Lower ρt

!

µn "µp = 4#S2($) = µe "µ% e

&Yp1/ 3

(1" 2Yp )=

4S2($)

hc 3' 2$( )1/ 3

(

)

* *

+

,

- -

L- Pt correlation: an open issue

Xu et al., PRC, 2009 Decrease of Pt

Moustakidis et al., PRC, 2010 Increase of Pt

!

P ",#( ) ="2

3"0

L# 2 + K0+Ksym#

2( )" $ "

0

3"0

+L

%

& '

(

) *

BUT …

40 60 80 100 120 L (MeV)

ρ t (f

m-3

)

Ypt

Increase of Pwith L in n-richmatter at fixeddensity

Role of other bulk parameters Dynamical vs Themodynamical spinodal

Transition point well correlated

Fits on pairs of parameters

!

Pt= aX

1+ bX

2+ c

P dt(M

eV fm

-3)

P dt(M

eV fm

-3)

0

0.1

0.2

0.3

0.4

0.5

P dt(M

eV fm

-3)

0

0.1

0.2

0.3

0.4

0.5

L (MeV) [Esym-0.127L] (MeV)

[L01-0.343Ksym01] (MeV)

Unclear L-Pt correlation

Inclusion of Esym more reliable corelation

Good correlation with L and Ksym at ρ=0.1 fm-3 Relativistic

Skyrme

Neutron Skin Thickness & The Crust-CoreTransition Density

Inverse correlation between δR and ρt(Horowiz & Piekarewicz)

an accurate measurement of neutronskin in neutron rich nuclei can provideconsiderable & valuable information onthe crust-core transition density.

(PREX exeriment @ JLAB)

Neutron Star Crust & Neutron Skin aremade out of neutron rich matter at similardensities

Both are governed by EoS at subnucleardensities in particular by S2(ρ) & its

derivativesNeutron Star Heavy nucleus

Summary & Conclusions

Study of S2(ρ) within the BHF approximation & comparison with effective models (Skyrme & RMF).

Correlation of ρt & Ypt with L:

Robust correlation of ρt with L. Correlation L-Ypt more disperse due to dispersion on Esym.

Correlation of Pt with L:

Opposite contributions difficult prediction. Improvement with combination of L & Ksym at ρ ~ 2/3ρ0.

L=66.5 MeV compatible with values deduced from different observables.

Larger contribution to Esym & L from S=1 channels (tensor).

3BF increase the slope of S2(ρ).