The Nuclear Equation of Stateand the Symmetry Energy

Xavier Roca-Maza

Università degli Studi di Milano e INFN, sezione di Milano

TNPI2016 - XV Conference on Theoretical Nuclear Physics in Italy,

Pisa, April 20th-22nd 2016.

1

STRENGTH: STructure and REactions of Nuclei:towards a Global THeory.

First let me recall that my reasearchat Milano is framed in a national initiative called

STRENGTH: STructure and REactions ofNuclei: towards a Global THeory

network research of main Italian theoretical groups work-ing at the INFN in the fields of low-energy nuclear struc-

ture and reactions: Catania, LNS, Milano, Napoli, Padovaand Pisa.

2

STRENGTH: STructure and REactions of Nuclei:towards a Global THeory.

→ Our goal: develop a more quantitative and unified

understanding of nuclear structure and reactions for

stable and exotic nuclei that are studied nowadays in RIB

facilities

→ In this talk:

• The symmetry energy, which rules the isovector channelof the nuclear Equation of State, is not well contrained innuclear effective models.

• I will review part of our work in which we have tried tounderstand which nuclear observables are more sensitiveto the properties of the symmerty energy. Such aninvestigation is of crucial importance for building morereliable EDFs with larger predictive power and may helpin setting the basis for our goal.

3

The Nuclear Equation of Stateand the Symmetry Energy

Xavier Roca-Maza

Università degli Studi di Milano e INFN, sezione di Milano

TNPI2016 - XV Conference on Theoretical Nuclear Physics in Italy,

Pisa, April 20th-22nd 2016.

4

Table of contents:

→ Introduction

→ The Nuclear Many-Body Problem→ Nuclear Energy Density Functionals→ EoS: Symmetry energy

→ The impact of the symmetry energy on nuclear andastrophyiscs observables

→ Some examples: Neutron stars outer crust, neutron skinthickness, parity violating asymmetry, pygmy states (?),GDR, dipole polarizability, GQR, AGDR ...

→ Conclusions

5

INTRODUCTION

6

The Nuclear Many-Body Problem:

→ Nucleus: from few to more than 200 strongly interactingand self-bound fermions.

→ Underlying interaction is not perturbative at the(low)energies of interest for the study of masses, radii,deformation, giant resonances,...

→ Complex systems: spin, isospin, pairing, deformation, ...

→ Many-body calculations based on NN scattering data inthe vacuum are not conclusive yet:

→ different nuclear interactions in the medium are founddepending on the approach

→ EoS and (recently) few groups in the world are able toperform extensive calculations for light and medum massnuclei

→ Based on effective interactions, Nuclear Energy Density

Functionals are successful in the description of masses,

nuclear sizes, deformations, Giant Resonances,...

7

Nuclear Energy Density Functionals:Main types of successful EDFs for the description of masses,

deformations, nuclear distributions, Giant Resonances, ...

Relativistic mean-field models, based on Lagrangians where effective mesons carry the interaction:

Lint = ΨΓσ(Ψ, Ψ)ΨΦσ +ΨΓδ(Ψ, Ψ)τΨΦδ

−ΨΓω(Ψ, Ψ)γµΨA(ω)µ −ΨΓρ(Ψ, Ψ)γµτΨA

(ρ)µ

−eΨQγµΨA(γ)µ

Non-relativistic mean-field models, based on Hamiltonianswhere effective interactions are proposed and tested:

VeffNucl = V

long−rangeattractive + V

short−rangerepulsive + VSO + Vpair

→ Fitted parameters contain (important) correlations

beyond the mean-field

→ Nuclear energy functionals are phenomenological → not

directly connected to any NN (or NNN) interaction

8

The Nuclear Equation of State: Infinite System

0 0.05 0.1 0.15 0.2 0.25

ρ ( fm−3

)

-20

-10

0

10

20

30e

( M

eV )

S(ρ)~

e(ρ,δ=1)neutron matter

symmetric mattere(ρ,δ=0)

Saturation

(0.16 fm−3

, −16.0 MeV)

E

A(ρ, β) =

E

A(ρ, β = 0) + S(ρ)β2 + O(β4)

→ NuclearMatter

[

β =ρn − ρp

ρ

]

9

The Nuclear Equation of State: Infinite System

0 0.05 0.1 0.15 0.2 0.25

ρ ( fm−3

)

-20

-10

0

10

20

30e

( M

eV )

S(ρ)~

e(ρ,δ=1)neutron matter

symmetric mattere(ρ,δ=0)

Saturation

(0.16 fm−3

, −16.0 MeV)

E

A(ρ, β) =

E

A(ρ, β = 0) + S(ρ)β2 + O(β4)

→ NuclearMatter

→ SymmetricMatter

[

β =ρn − ρp

ρ

]

10

The Nuclear Equation of State: Infinite System

0 0.05 0.1 0.15 0.2 0.25

ρ ( fm−3

)

-20

-10

0

10

20

30e

( M

eV )

S(ρ)~

e(ρ,δ=1)neutron matter

symmetric mattere(ρ,δ=0)

Saturation

(0.16 fm−3

, −16.0 MeV)

E

A(ρ, β) =

E

A(ρ, β = 0) + S(ρ)β2 + O(β4)

→ NuclearMatter

→ SymmetricMatter

→ Symmetry energy

[

β =ρn − ρp

ρ

]

11

The Nuclear Equation of State: Infinite System

0 0.05 0.1 0.15 0.2 0.25

ρ ( fm−3

)

-20

-10

0

10

20

30e

( M

eV )

S(ρ)~

e(ρ,δ=1)neutron matter

symmetric mattere(ρ,δ=0)

Saturation

(0.16 fm−3

, −16.0 MeV)

E

A(ρ, β) =

E

A(ρ, β = 0) + S(ρ)β2 + O(β4)

=E

A(ρ, β = 0) + β2

(

J+ Lx+1

2Ksymx2 + O(x3)

)

[

β =ρn − ρp

ρ; x =

ρ− ρ0

3ρ0

]

12

The Nuclear Equation of State: Infinite System

0 0.05 0.1 0.15 0.2 0.25

ρ ( fm−3

)

-20

-10

0

10

20

30e

( M

eV )

S(ρ)~

e(ρ,δ=1)neutron matter

symmetric mattere(ρ,δ=0)

Saturation

(0.16 fm−3

, −16.0 MeV)

E

A(ρ, β) =

E

A(ρ, β = 0) + β2

(

J + L x+1

2Ksym x2 + O(x3)

)

→ S(ρ0) = J → d

dρS(ρ)

∣

∣

∣

∣

ρ0

=L

3ρ0=

P0

ρ20→ d2

dρ2S(ρ)

∣

∣

∣

∣

ρ0

=Ksym

9ρ20

[

β =ρn − ρp

ρ; x =

ρ− ρ0

3ρ0

]

13

The Nuclear Equation of State: Infinite System

0 0.05 0.1 0.15 0.2 0.25

ρ ( fm−3

)

-20

-10

0

10

20

30e

( M

eV )

S(ρ)~

e(ρ,δ=1)neutron matter

symmetric mattere(ρ,δ=0)

Saturation

(0.16 fm−3

, −16.0 MeV)

E

A(ρ, β) =

E

A(ρ, β = 0) + β2

(

J + L x+1

2Ksym x2 + O(x3)

)

→ S(ρ0) = J → d

dρS(ρ)

∣

∣

∣

∣

ρ0

=L

3ρ0=

P0

ρ20→ d2

dρ2S(ρ)

∣

∣

∣

∣

ρ0

=Ksym

9ρ20

The uncertainties on S(ρ) impacts on many nuclearphysics and astrophysics observables.

[

β =ρn − ρp

; x =ρ− ρ0

]

14

The impact of thesymmetry energy on

nuclear and astrophyiscsobservables

15

Relevance of the neutron star crust on the star evolution and

dynamics (brief motivation)

→ The crust separates neutron star interior from thephotosphere (X-ray radiation).

→ The thermal conductivity of the crust is relevantfor determining the relation between observedX-ray flux and the temperature of the core.

→ Electrical resistivity of the crust might beimportant for the evolution of neutron starmagnetic field.

→ Conductivity and resistivity depend on thestructure and composition of the crust

→ Neutrino emission from the crust may significantly contribute to total neutrino losses from stellar interior(in some cooling stages).

→ A crystal lattice (solid crust) is needed for modelling pulsar glitches, enables the excitation of toroidalmodes of oscillations, can suffer elastic stresses...

→ Mergers (binary systems that merge) may enrich the interstellar medium with heavy elements, created by arapid neutron-capture process.

→ In accreting neutron stars, instabilities in the fusion light elements might be responsible for the phenomenonof X-ray bursts

Source: Pawel Haensel 2001

16

The symmetry energy and the structure and composition of a

neutron star outer crust

→ span 7 orders of magnitude in denisty (from

ionization ∼ 104 g/ cm to the neutron drip

∼ 1011 g/ cm)

→ it is organized into a Coulomb lattice ofneutron-rich nuclei (ions) embedded in arelativistic uniform electron gas

→ T ∼ 106 K ∼ 0.1 keV → one can treat nuclei andelectrons at T = 0 K

→ At the lowest densities, the electroniccontribution is negligible so the Coulomb lattice

is populated by 56Fe nuclei.

→ As the density increases, the electroniccontribution becomes important, it isenergetically advantageous to lower its electronfraction by e− + (N,Z) → (N+ 1,Z− 1) +νe

and therefore Z ↓ with constant (approx) numberof N

→ As the density continues to increase, penaltyenergy from the symmetry energy due to theneutron excess changes the composition to a different N−plateauZ

A≈ Z0

A0−

pFe

8asymwhere (A0,Z0) = 56Fe26

→ The Coulomb lattice is made of more and moreneutron-rich nuclei until the critical neutron-dripdensity is reached (µdrip = mn).

[M(N,Z) +mn < M(N+ 1,Z)]

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

N=50

Mo

NL3 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

Physical Review C 78, 025807 (2008)

The faster the symmetry energyincreases with density (L ↑), themore exotic the composition ofthe outer crust.

17

The symmetry energy and the neutron skin in nuclei

v090M

Sk7H

FB-8

SkP

HFB

-17

SkM*

DD

-ME2

DD

-ME1

FSUG

oldD

D-PC

1Ska

PK1.s24 Sk-R

sN

L3.s25Sk-T4G

2

NL-SV2PK

1N

L3N

L3*

NL2

NL1

0 50 100 150 L (MeV)

0.1

0.15

0.2

0.25

0.3

∆rnp

(fm

)

Linear Fit, r = 0.979Mean Field

D1S

D1N

SG

II

Sk-T6

SkX S

Ly5

SLy4

MS

kAM

SL0

SIV

SkS

M*

SkM

P

SkI2S

V

G1

TM1

NL-S

HN

L-RA

1

PC

-F1

BC

P

RH

F-PK

O3

Sk-G

s

RH

F-PK

A1

PC

-PK

1

SkI5

Physical Review Letters 106, 252501 (2011)

The faster the symmetry energy increases with density, thelargest the size of the neutron skin in (heavy) nuclei.

∆rnp ∼1

12

IR

JL

[Exp. from strongly interacting probes: 0.18± 0.03 fm (Physical Review C 86 015803 (2012))].18

The symmetry energy and parity violating electron scattering

(Apv : relative difference between the elastic cross sections of right- and left-handed electrons)

v090HFB-8

D1S

D1N

SkXSLy5

MSkA

MSL0

DD-ME2

DD-ME1

Sk-Rs

RHF-PKA1

SVSkI2

Sk-T4

NL3.s25

G2SkI5

PK1

NL3*

NL2

0,1 0,15 0,2 0,25 0,3∆ r

np (fm)

6,8

7,0

7,2

7,4

107

Apv

Linear Fit, r = 0.995Mean Field From strong probes

MSk7

HFB-17SkP

SLy4

SkM*

SkSM*SIV

SkMP

Ska

Sk-Gs

PK1.s24

NL-SV2NL-SH

NL-RA1

TM1, NL3

NL1

SGIISk-T6

FSUGoldZenihiro

KlosHoffm

ann

PC-F1

BCP

PC-PK1

RHF-PKO3

DD-PC1

G1

Tarbert

Physical Review Letters 106, 252501 (2011)

(Calculation at a fixed q equal to PREx)

→ Electrons interact by exchanging a γ (couples top) or a Z0 boson (couples to n)

→ Ultra-relativistic electrons, depending on theirhelicity (±), will interact with the nucleus seeinga slightly different potential: Coulomb ± Weak

→ Apv ≡ dσ+/dΩ−dσ−/dΩ

dσ+/dΩ+dσ−/dΩ∼

Weak

Coulomb

→ Input for the calculation are the ρp and ρn

(main uncertainty) and nucleon form factors forthe e-m and the weak neutral current.

→ In PWBA for small momentum transfer:

Apv ≈ GFq2

4√2πα

(

1−q2〈r2p〉1/2

3Fp(q)∆rnp

)

The largest the size of the neutron distribution in nuclei, thesmaller the elastic electron parity violating asymmetry.[Exp. from ew probes: 0.302± 0.175 fm (Physical Review C 85, 032501 (2012))].

19

Isovector Giant Resonances (some considerations)

→ In isovector giant resonances neutrons and protons

“oscillate” out of phase

→ Isovector resonances will depend on oscillations of thedensity ρiv ≡ ρn − ρp ⇒ S(ρ) will drive such “oscillations”

→ The excitation energy (Ex) within a Harmonic Oscillator

approach is expected to depend on the symmetry energy:

ω =

√

1

m

d2U

dx2∝

√k → Ex ∼

√

δ2e

δβ2∝√

S(ρ)

where β = (ρn − ρp)/(ρn + ρp)

→ The dipole polarizability (α ∼

∫σγ−abs

Energy2∼ IEWSR)

measures the tendency of the nuclear charge distributionto be distorted, that is, from a macroscopic point of view

α ∼electric dipole moment

external electric field

20

The symmetry energy and the Giant Dipole Resonance

(Ex ≈ f(0.1) ∝√

S(0.1fm−3) )

11 12 13 14 15 16E-1 [MeV]

4.6

4.8

5

5.2

5.4

5.6

5.8

6

6.2

6.4

f(0.

1)=

S(0

.1)(

1+k

1/2 [M

eV1/

2 ]

Physical Review C 77, 061304 (2008)

The faster the symmetry energy increases with density around

saturation

[

S(ρA) ≈ J− Lρ0 − ρA

3ρ0

]

, the smaller the excitation

energy of the Giant Dipole Resonance (GDR).21

The symmetry energy and the Pygmy Dipole Resonance

(Pygmy: low-energy excited state appearing in the dipole response of N , Z nuclei)

Physical Review C 81, 041301 (2010)

The faster the symmetry energy increases with density, thelarger is the energy (E) times the probability (P) of exciting thePygmy state ⇒ larger the Energy Weighted Sum Rule (EWSR)∝ E× P.

WARNING: we lack of a clear understanding of the physicalreason for this correlation

22

Dipole polarizability and the symmetry energy

0.12 0.16 0.2 0.24 0.28 0.32∆r

np (fm)

5

6

7

8

9

10

10−2

α DJ

(M

eV f

m3 ) r=0.97

FSUNL3DD-MESkyrmeSVSAMiTF

Physical Review C 85 041302 (2012); 88 024316 (2013); 92, 064304 (2015)

Macroscopic model:

→ Using the dielectric theorem: m−1 moment canbe computed from the expectation value of theHamiltonian in the constrained (D dipoleoperator) ground state H

′ = H+ λD

→ Assuming the Droplet Model (heavy nucleus):

αD ≈ αbulkD

[

1+1

5

L

J

]

where

αbulkD ≡ πe2

54

A〈r2〉J

(Migdal first derived)

→ L ≈α

expD −αbulk

D

αbulkD

5J

By using the Droplet Model one can also find:

αDJ≈ πe2

54A〈r2〉

[

1+5

2

∆rnp−∆rcoulnp −∆rsurf

np

〈r2〉1/2(I−IC)

]

For a fixed value of the symmetry energy at saturation, thefaster the symmetry energy increases with density, the largerthe dipole polarizability.

23

IV-IS GQRs and the symmetry energy

Within the Quantum HarmonicOscillator approach

EIVx = 2 hω0

√

1+5

4

h2

2m

Vsym〈r2〉( hω0)

2〈r4〉and EDF calculations, one candeduceVsym ≈ 8(S(ρA) − Skin(ρ0))

Skin(ρ0) ≈ εF0/3 (Non-Rel) 0 10 20 30

E (MeV)

0

2

4

6

B(E

2;IV

) (1

03 f

m4 M

eV-1)

S(ρA) =εF0

3

A2/3

8ε2F0

[

(

EIVx

)2

− 2(

EISx

)2]

+ 1

S(ρ = 0.1 fm−3 for 208Pb) ≈ 23.3± (0.6)exp ± theory ∼ 10% MeV

The faster the symmetry energy increase with density around

saturation, the smallest the difference between the IS and IV

excitation energy differences

24

E1 transitions in CER and the symmetry energy

1+,T

0

∆S=1

∆L=1,∆S=1

1-,T

0-1

0-,1

-,2

-,T

0-1

1+,T

0-1

0+,T

0

TZ=T

0-1

1-,T

0

Target nucleus

TZ=T

0

0+,T

0

E1

Daughter nucleus

IAS

GT

IVSGDR

AGDR

strong

(p,n)

∆L=1

M1

0.12 0.16 0.20 0.24 0.288.0

8.5

9.0

9.5

10.0

EA

GD

R-E

IAS(M

eV

)

Rn-R

p(fm)

Exp. Data from Ref. [56]

Exp. Data from Ref. [53]

Phys. Rev. C 92, 034308 (2015)

AGDR (∆Jπ = 1− with ∆L = 1 and ∆S = 0) is the T0 − 1

component of the charge-exchange of the GDR.

EAGDR −EIAS ≈ 5

√

5

3

J

I

1+γ

αHZ

hc

m〈r2〉1/2

[

(

1−εF∞

3J

)

I−3

2

(

∆Rnp −∆Rsurfnp

〈r2〉1/2

)

−3

7IC

]

EAGDR −EIAS ≈ ε

∆EC(EIVGDR − ε)

mAGDR0

mIVGDR0

∆rnp ≈ 0.21± 0.01 fm

The faster the symmetry energy increases around saturation,the smaller the excitation energy of the IVGDR and the difference between the excitation energies of AGDR - IAS

25

CONCLUSIONS

26

Conclusions: EoS around saturation

→ The isovector channel of the nuclear effective interaction isnot well constrained by current experimental

information.

→ Many observables available in current laboratories aresensitive to the symmetry energy. Problems: accuracy and

model dependent analysis. Systematic experiments mayhelp.

→ Exotic nuclei more sensitive to the isovector properties(due to larger neutron excess). Problems: more difficult to

measure, accuracy and model dependent analysis.Systematic experiments may help.

→ The most promissing observables to constraint thesymmetry energy are the neutron skin thickness and thedipole polarizability in medium and heavy nuclei.

27

Thank you for yourattention!

28

Available constraints on L

20 40 60 80 100 120 140L (MeV)

Centelles et al. PRL 102 (2009) 122502

Warda et al. PRC 80 (2009) 024316

Danielewicz NPA 727 (2003) 233

Myers et al. PRC 57 (1998) 3020

Famiano et al. PRL 97 (2006) 052701

Masses

B. Tsang et al. PRL 103 (2009) 122701

Trippa et al. PRC 77 (2008) 061304(R)

Klimkiewicz et al. PRC 76 (2007) 051603(R)

Carbone et al. PRC 81 (2010) 041301(R)

Xu et al. PRC 82 (2010) 054607.

Neutron Skins

Neutron Skins

Kortelainen et al. PRC 82 (2010) 024313Masses

n−p Emission Ratios

Masses

Isospin Diffusion

GDR

PDR

PDR

Optical Potentials

Antiprotonic Atoms

Nuclear Model Fit

Heavy IonCollisions

GiantResonancies

N−A scatteringCharge Ex. Reac.Energy Levels

Vidaña et al. PRC 80 (2009) 045806BHF Bare N−N Potential

X. Roca-Maza et. al. PRL 106 252501 (2011)(arb. cent.)PVES

1% acc.

PREX

Lie-Wen Chen et al. PRC 82 (2010) 024321

PREx Collab. et. al. PRL 108 112502 (2012)

p & α scattering ch-exch excitations

Neutron Skins Sn

A. W. Steiner et al. Astrophys. J. 722 (2010) 33Neutron-Star Observations Mass-Radius

H. Kebeler et al. PRL105 (2010) 161102and s. Gandolfi et al. PRC85 (2012) 032801 (R)

Chiral Lagrangian and Quantum Montecarlo Neutron Matter

AIP Conference Proceedings 1491, 101 (2012)

29