Density dependence of the nuclear symmetryenergy estimated from neutron skin thickness

in finite nuclei

X. Roca-Mazaa,b

X. Vinasb

M. Centellesb

M. Wardab,c

a INFN sezione di Milano. Via Celoria 16, 20133 Milano, ItalybDepartament d’Estructura i Constituents de la Materia and Institut de Ciencies del Cosmos,

Universitat de Barcelona, Barcelona, SpaincKatedra Fizyki Teoretycznej, Uniwersytet Marii Curie-Sklodowskiej, Poland

M. Centelles, X. Roca-Maza, X. Vinas and M. Warda,Phys. Rev. Lett. 102 122502 (2009)

M. Warda, X. Vinas, X. Roca-Maza and M. Centelles,Phys. Rev. C80 024316 (2009)

Introducing myself...

• Degree in Physics: University of Barcelona from 2000 to 2005

• Phd thesis: University of Barcelona from 2005 to 2010• Title: Isospin asymmetry in stable and exotic nuclei.• Advisors: X. Vinas and M. Centelles• Defense: 6 May 2010

• Post-doc: University of Barcelona from May to October 2010.

• Post-doc: INFN Milano from November 2010

Works in which I have participated

The symmetry energy and the outer crust in colaborationwith J. Piekarewicz

• Impact of the symmetry energy on the outer crust ofnon-accreting neutron stars.Phys. Rev. C 78 (2008) 025807.

The symmetry energy and the neutron skin thickness ofnuclei in colaboration with X. Vinas, M. Cenetelles and M.

Warda

• Single particle shell effects in the neutron skin thickness of nucleiwithin mean-field modelsIn preparation, writing...

• Origin of the neutron skin thickness of 208Pb in nuclear mean-fieldmodelsAccepted in Phys. Rev. C.

• Analysis of bulk and surface contributions in the neutron skin ofnuclei.Phys. Rev. C81 (2010) 054309.

• Neutron skin thickness in droplet model with surface widthdependence: indications of softness of the nuclear symmetry energy.Phys. Rev. C80 (2009) 024316.

• Nuclear symmetry energy probed by neutron skin thickness of nuclei.Phys. Rev. Lett. 102 (2009) 122502.

Electron scattering in colaboration with X. Vinas, M.Cenetelles and F. Salvat

• Parity violating electron scattering at the kinematics of thePREX experiment and the neutron skin thickness of 208Pb.In preparation, writing...

• Theoretical study of elastic electron scattering along N = 16,N = 50 and N = 82 isotonic chains.In preparation, writing...

• Theoretical study of elastic electron scattering off stable andexotic nucleiPhys. Rev. C 78 (2008) 044332.

The symmetry energy and the GMR in colaboration with X.Vinas, M. Cenetelles, S.K. Patra, B.K. Sharma, P.D.

Stevenson

• Influence of the symmetry energy on the giant monopoleresonance of neutron-rich nuclei.J. Phys. G. 37 (2010) 075107.

DDMEδ, new mean field effective interaction in colaborationwith X. Vinas, M. Cenetelles, P. Ring and P. Schuck

• Relativistic mean field interaction with density dependentmeson-nucleon vertices based on microscopical calculations.In preparation, writing...

Density dependence of the nuclear symmetryenergy estimated from neutron skin thickness

in finite nuclei

X. Roca-Mazaa,b

X. Vinasb

M. Centellesb

M. Wardab,c

a INFN sezione di Milano. Via Celoria 16, 20133 Milano, ItalybDepartament d’Estructura i Constituents de la Materia and Institut de Ciencies del Cosmos,

Universitat de Barcelona, Barcelona, SpaincKatedra Fizyki Teoretycznej, Uniwersytet Marii Curie-Sklodowskiej, Poland

c INFN sezione di Milano. Via Celoria 16, 20133 Milano, Italy

M. Centelles, X. Roca-Maza, X. Vinas and M. Warda,Phys. Rev. Lett. 102 122502 (2009)

M. Warda, X. Vinas, X. Roca-Maza and M. Centelles,Phys. Rev. C80 024316 (2009)

Why is important the nuclear symmetry energy ?

The nuclear symmetry energy is a fundamental quantity in NuclearPhysics and Astrophysics because it governs, at the same time, importantproperties of very small entities like the atomic nucleus ( R ∼ 10−15 m )and very large objects as neutron stars ( R ∼ 104 m )

• Nuclear Physics: Neutron skin thickness in finite nuclei, stablenuclei, Heavy-Ion collisions, Giant Resonances...

• Astrophysics: Supernova explosion, Neutron emission and cooling ofprotoneutron stars, Mass-Radius relations in neutron stars,Composition of the crust of neutron stars...

Equation of State in asymmetric matter

e(ρ, δ) = e(ρ, 0) + csym(ρ)δ2 +O(δ4)

(δ =

ρn − ρp

ρ

)Around the saturation density we can write

e(ρ, 0) ' av +1

2Kv ε

2 and csym(ρ) ' J−Lε+1

2Ksymε2

(ε =

ρ0 − ρ

3ρ0

)

ρ0 ≈ 0.16fm−3, av ≈ −16MeV , Kv ≈ 230MeV , J ≈ 32MeV

However, the values of

L = 3ρ∂csym(ρ)/∂ρ|ρ0 and Ksym = 9ρ2∂2csym(ρ)/∂ρ2|ρ0

which govern the density dependence of csym near ρ0 are less certain and

predictions vary largely among nuclear theories.

Experimental constraints

• Recent reseach in heavy-ion collisions at intermediate energy isconsistent with csym(ρ) = csym(ρ0).(ρ/ρ0)

γ at ρ < ρ0.

• Isospin difussion γ = 0.7–1.05 (L = 88± 25 MeV).

• Isoscaling γ = 0.69 (L ∼ 65 MeV)

• Inferred from nucleon emision ratios γ = 0.5(L ∼ 55 MeV).

• The GDR of 208Pb analyzed with Skyrme forces suggests aconstraint csym(0.1 fm−3)=23.3–24.9 MeV (γ ∼ 0.5–0.65).

• The study of the PDR in 68Ni and 132Sn predicts L=49-80 MeV.

• The Thomas-Fermi model of Myers and Swiatecki fitted veryprecisely to binding energies of 1654 nuclei predicts an EOS thatyields γ = 0.51

• NEUTRON SKIN THICKNESS ?

Neutron skin thickness

What is experimentally know about neutron skin thickness innuclei ?

• The neutron skin thickness is defined as S = 〈r2n 〉

1/2 − 〈r2p 〉

1/2,

where 〈r2n 〉

1/2and 〈r2

p 〉1/2

are the rms radii of the neutron andproton distributions respectively.

• 〈r2p 〉

1/2is known very accurately from elastic electron scattering

measurements (e.g. rch(208Pb) = 5.5010± 0.0009 fm [Angeli

(2004)]).

• 〈r2n 〉

1/2has been obtained with hadronic probes such as:

(a) Proton-nucleus elastic scattering (5.522fm < rn(208Pb) < 5.550

fm [Clark (2003)]).(b) Inelastic scattering excitation of the giant dipole and spin-dipoleresonances (rn(

208Pb) = 5.67± 0.07 fm [Krasznahorkay (1990)]).(c) Antiprotonic atoms: Data from antiprotonic X rays andradiochemical analysis of the yields after the antiproton annihilation(rn(

208Pb) = 5.66± 0.02 fm) [Trzcinska (2001)].

0 0.05 0.1 0.15 0.2I

-0.1

0

0.1

0.2

S (f

m)

S = (0.9± 0.15)I + (−0.03± 0.02) fmA. Trzcinska et al, Phys. Rev. Lett. 87, 082501 (2001)

CAN S OF 26 STABLE NUCLEI, FROM 40Ca TO 238U, ESTIMATEDUSING ANTIPROTONIC ATOMS DATA HELP IN CONSTRAINING

THE SLOPE AND CURVATURE OF csym ?

Symmetry energy and neutron skin thicknessin the Liquid Drop Model

• Symmetry Energy

asym(A) =J

1 + xA, xA =

9J

4QA−1/3

Esym(A) = asym(A)(I + xAIC )2A

where

I = (N − Z )/A, IC = e2Z/(20JR), R = r0A1/3

.• Neutron skin thickness

S =√

3/5

[t − e2Z/(70J) +

5

2R(b2

n − b2P)

]where

t =3r02

J/Q

1 + xA(I − IC)

Neutron skin thickness

t =2r03J

[J − asym(A)] A1/3 (I − IC)

Table: Value of asym(A) and density ρ that fulfils csym(ρ) = asym(A) forA = 208, 116 and 40 in MF models. J and asym are in MeV and ρ is infm−3.

A = 208 A = 116 A = 40Model J asym ρ asym ρ asym ρNL3 37.4 25.8 0.103 24.2 0.096 21.1 0.083NL-SH 36.1 25.8 0.105 24.6 0.099 21.3 0.086FSUGold 32.6 25.4 0.098 24.2 0.090 21.9 0.075TF-MS 32.6 24.2 0.093 22.9 0.085 20.3 0.068SLy4 32.0 25.3 0.100 24.2 0.091 22.0 0.075SkX 31.1 25.7 0.102 24.8 0.096 22.8 0.082SkM* 30.0 23.2 0.101 22.0 0.093 19.9 0.078SIII 28.2 24.1 0.093 23.4 0.088 21.8 0.077SGII 26.8 21.6 0.104 20.7 0.096 18.9 0.082

The csym(ρ)-asym(A) correlation

• There is a genuine relation between the symmetry energycoefficients of the EOS and of nuclei: csym(ρ) equals asym(A)of heavy nuclei like 208Pb at a density ρ = 0.1± 0.01 fm−3.

• A similar situation occurs down to medium mass numbers, atlower densities.

• We find that this density can be very well simulated by

ρ ≈ ρA = ρ0 − ρ0/(1 + cA1/3) ,

where c is fixed by the condition ρ 208 = 0.1 fm−3.

• Using the equality csym(ρ) = asym(A) and the LDM , theneutron skin thickness can be finally written as:

t =

√3

5

2r03

L

J

(1− ε

Ksym

2L

)εA1/3

(I − IC

)

Neutron skin thickness

t =

√3

5

2r03

L

J

(1− ε

Ksym

2L

)εA1/3

(I − IC

)

Fitting procedure and results

• We optimize

t =

√3

5

2r03

L

J

(1− ε

Ksym

2L

)εA1/3

(I − IC

)using

csym = 31.6(ρ

ρ0)γMeV , ε =

1

3(1 + cA1/3), ρ0 = 0.16fm−3

and taking as experimental baseline the neutron skins measured in26 antiprotonic atoms.

• We predict (bn ≈ bp): L = 75± 25 MeV

0 0.1 0.2I = (N−Ζ) / Α

-0.1

0

0.1

0.2

0.3

S (

fm)

40

20Ca 58

28Ni

54

26Fe

60

28Ni

56

26Fe 59

27Co

57

26Fe

106

48Cd

112

50Sn

90

40Zr

64

28Ni

116

50Sn

122

52Te

124

52Te

48

20Ca

96

40Zr

120

50Sn

116

48Cd

126

52Te

128

52Te

124

50Sn130

52Te

209

83Bi208

82Pb

232

90Th

238

92U

experimentlinear averageof experimentOur fit

S = (0.9± 0.15)I + (−0.03± 0.02) fmA. Trzcinska et al, Phys. Rev. Lett. 87, 082501 (2001)

Influence of the surface width (bn 6= bp)

S =√

3/5

[t − e2Z/(70J) +

5

2R(b2

n − b2P)

]bn and bp are obtained at the ETF level.

Surface contribution to the neutron skin thickness

√3

5

5

2R(b2

n − b2p) = σsw I = (0.3

J

Q+ c)I

Fit and results

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25

∆Rn

p (

fm)

I

c = 0.07c = -0.05

EXP0.9 I - 0.03

With ρ0 = 0.16 fm−3 and 28 . J . 35 MeV, and −0.05 . c . 0.07 fmJQ = 0.6− 0.9

Neutron skin thickness

NL3Λν2

NL3Λν1

TM1

SVI

T6 SGII

SkM*

NL3Λν3

FSUGold

NL3

SLy4

SIIID1S

NL-SH

NL1

SkP

SkX

0.4 0.8 1.2 1.6J / Q

0.1

0.2

0.3

∆Rnp

(fm

) in

208 P

b

SVI

T6 SGII

SLy4

NL3Λν3

FSUGold

NL3Λν2

NL3Λν1

SkP SkX

TM1

SkM*

NL3

SIII D1S

NL-SH

NL1

0 40 80 120L (MeV)

SVI

T6 SGII SkM*

FSUGoldNL3Λν2

NL3Λν3

NL3

SLy4

NL3Λν1

SkP

TM1

SkX

SIII

D1S

NL-SH

NL1

0.4 0.8 1.2 1.6J / Q

0

50

100

150

L (

MeV

)

L = 30− 80 MeV

Constraints on the slope of the symmetry energy

From the very small to the very big: the symmetry energyand the outer crust of a neutron star

Introduction

R (Km) ρ (gr/cm3) vscape/c g/gEarth (surface) P (dyn/cm2)

10 1014 − 1015 0.5 1011 0− 1035

∗Orientative properties of a typical neutron star of mass M = MSun.

Formalism

Total energy per nucleon

e(A,Z , ρ = ρn + ρp) = eN(A,Z ) + elat(A,Z , ρ) + eel(ρ)

The different contributionseN(A,Z ) = M(A,Z)

A

elat(A,Z , ρ) = −ClatZ2

A4/3pF

where Clat = 0.00341 andpF = (3π2ρ)1/3 = pFel

(A/Z )1/3 (Nel = Z )

eel(ρ) =m4

el8π2ρ

(xF yF (x2

F + y2F )− ln(xF + yF )

)where xF ≡ pFel

and yF ≡εFelmel

=√

1 + x2F

Composition of the outer crust

20

30

40

50

60

70

80

90

Com

posi

tion Protons

Neutrons

10-4

10-3

10-2

10-1

100

101

ρ(1011

g/cm3)

20

30

40

50

60

70

80

Com

posi

tion

FSUGold (a)

N=50

Mo

NL3 (b) N=82

N=32Fe

Sr

Kr Se

Ge Zn Ni

ZrSrKr

N=82

N=50

Ni

Sr

Kr Se

Sn Cd

Pd

Kr

Ru

Mo

Zr

Sr

The stiffer the symmetry energy the moreexotic the composition of the outer crust andthe larger the neutron skin of medium and

heavy elements

∆RNL3np (208Pb) = 0.28 fm and ∆RFSUGold

np (208Pb) = 0.20 fm

Summary and Conclusions

• We have described a generic relation between the symmetry energyin finite nuclei and in nuclear matter at subsaturation.

• We take advantage of this relation to explore constraints on csym(ρ)from neutron skins measured in antiprotonic atoms. Theseconstraints points towards a soft symmetry energy.

• We discuss the L values constrained by neutron skins in comparisonwith most recent observations from reactions and giant resonances.

• We learn that in spite of present error bars in the data ofantiprotonic atoms, the size of the final uncertainties in L iscomparable to the other analyses.

• The generic relation between the symmetry energy in finite nucleiand in nuclear matter at subsaturation plausibly encompasses otherprime correlations of nuclear observables with the density content ofthe symmetry energy as e.g. the constrains of csym(0.1) from theGDR of 208Pb (L. Trippa et al. Phys. Rev. C77, 061304(R) (2008)).

Thank you for your attention

Extra material

Some technical details• The surface stiffness coeficient Q and the surface widths bn and bp

are obtained from self-consistent calculations of the neutron andproton density profiles in asymmetric semi-infinite nuclear matter.

• To this end one has to minimize the total energy per unit area withthe constraint of conservation of the number of protons andneutrons with respect to arbitrary variations of the densities.

Econst

S=

∫ ∞

−∞

[ε(z)− µnρn(z)− µpρp(z)

]dz ,

where ε(z) is the nuclear energy density functional.• In the non-relativistic framework the densities ρn and ρp obey the

coupled local Euler-Lagrange equations:

δε(z)

δρn− µn = 0,

δε(z)

δρp− µp = 0.

The relative neutron excess δ = (ρn − ρp)/(ρn + ρp) is a function ofthe z-coordinate. When z → −∞ , the densities ρn and ρp

approach the values of asymmetric uniform nuclear matter inequilibrium with a bulk neutron excess δ0.

• From the calculated density profiles one computes:

zoq =

∫∞−∞ zρ′q(z)dz∫∞−∞ ρ′q(z)dz

,

b2q =

∫∞−∞(z − z0q)

2ρ′q(z)dz∫∞−∞ ρ′q(z)dz

.

• From the relation

t = z0n − z0p =3r02

J

Qδ0,

one can evaluate Q from the slope of t at δ0 = 0.• The distance t and the surface widths bn and bp in finite nuclei with

neutron excess I = (N − Z )/A are obtained using δ0 given by:

δ0 =I +

3

8

c1

Q

Z 2

A5/3

1 +9

4

J

QA−1/3

.