International Workshop XLV on Gross Properties of Nuclei and Nuclear Excitations
Hirschegg, Kleinwalsertal, Austria, January 15-21, 2017

Discerning the symmetry energy andneutron star properties fromnuclear collective excitations
N. Paar
Department of Physics
Faculty of Science University of Zagreb, Croatia

INTRODUCTION•Nuclear matter equation of state (EOS) plays important role in nuclear physics and astrophysics
symmetric nuclear matter
neutron matter

60

E/A[MeV]

40

•

20

SYMMETRY ENERGY0EOS
beyond thesaturation
point?

-2000.1

0.2

0.3

0.4-3

ρ[fm ]•

Symmetry energy S(ρ) describes the increase in the energy ofthe N≠Z system as protons are turned into neutrons•

Symmetry energy slope parameter determines the pressure ofneutron matter•

It is important for understanding the properties of neutron/richnuclei, neutron-rich matter and neutron star radii, evolution ofcore collapse of massive star, explosive nucleosynthesis,…

•Nuclear experiments provide important constraints on thesymmetry energy at nuclear densities

A.Le Fevre et al., NPA 945, 112 (2016)

SYMMETRY ENERGY• Nuclear matter equation of state:⇢ = ⇢n + ⇢p=⇢n⇢p⇢

40

E/A[MeV]E(⇢, ) = ESN M (⇢) + Esym (⇢) 2 + ...

symmetric nuclear matter
neutron matter

6020

SYMMETRY ENERGY0• Symmetry energy term-20

Esym (⇢) ⌘ S2 (⇢) = J✏ = (⇢00L✏ +THEORY...
FRAMEWORK

0.1

⇢00.2

0.3

ρ[fm-3]⇢)/(3⇢0 )dS2 (⇢)
L = 3⇢0| ⇢0d⇢
J – symmetry energy at saturation density
L – slope of the symmetry energy
B. Tsang, NSCL0.4

SYMMETRY ENERGY AND NEUTRON SKIN208Pb=J~L• strong linear correlation between
neutron skin thickness and symmetryenergy parameters J (a4), L (p0)R.J.Furnstahl, Nucl. Phys. A 706, 85 (2002)•Parity violating electron scattering –Lead Radius Experiment(PREx) @ JLab:
Abrahamyan et al. PRL 108, 112502 (2012)

X. Roca-Maza et al.,Phys. Rev. Lett. 106, 252501 (2011)

COLLECTIVE EXCITATIONS AND THE SYMMETRY ENERGY• There are various (isovector) modes of collective excitations that also provide
constraints on the neutron skin thickness, recent experimental data available• Isovector giant dipole resonancesØ Dipole polarizability A. Tamii et al., PRL 107, 062502 (2011)
D.M. Rossi et al., PRL 111, 242503 (2013)↵D ⇠ m 1

T. Hashimoto et al., Phys. Rev. C 92, 031305(R) (2015)• Pygmy dipole resonances A. Carbone et al., PRC 81, 041301(R) (2010)
A. Klimkiewitz et al., PRC 76, 051603(R) (2007)• Isovector giant quadrupole resonances
S.S. Henshaw, M.W. Ahmed, et al, PRL 107, 222501 (2011)• Charge-exchange excitations - Anti-analog GDR, Gamow-TellerA. Krasznahorkay et al., PLB 720, 428 (2013)••…

The goal: use the knowledge on collective excitations to constrain the symmetry energy

THEORY FRAMEWORKRELATIVISTIC NUCLEAR ENERGY DENSITY FUNCTIONAL
i) Nucleons are Dirac particles coupledby the exchange mesons and the photon field Tii) Four-fermion contact interaction(Point-coupling model)σ•Density dependent couplings•

The model parameters are constrained directly by many-bodyobservables (masses, charge radii, pseudo-data, …)

ωρ• Complicated many body dynamics encoded in the functional and its empirical constants•

DIRHB -- a relativistic self-consistent mean-field framework for atomic nuclei
Relativistic Hartree Bogoliubov model T. Niksic et al., Comp. Phys. Comm. 185, 1808 (2014).

•In the small amplitude limit, self-consistent quasiparticle random phase approximation(QRPA) is used to describe nuclear excitations, etc.

CORRELATIONS: NUCLEAR MATTER vs. PROPERTIES OF NUCLEI• Covariance analysis in the EDF framework - information on relevant correlations andstatistical uncertainties

Curvature matrix:• Pearson product-moment correlation coefficient

Mij = 12 ⇥pi ⇥pj

provides a measure of the correlation (linear dependence)
between two variables A and B.

Correlation matrix between
nuclear matter properties
and several properties for
208Pb (DDME-min1)- neutron skin thickness,
properties of giant
resonances,…
strongly correlated
uncorrelated
208Pb
X. Roca Maza, N.P., G. Colò, JPG 42, 034033 (2015)

2|p0

VARIATION OF THE SYMMETRY ENERGY IN CONSTRAINING THE EDF• Establish a set of 8 relativistic point coupling interactions that span the range ofvalues of the symmetry energy at saturation density: J=29,30,…36 MeV•Adjust the properties of 72 spherical nuclei to exp. data (binding energies, chargeradii, diffraction radii, surface thickness, pairing gaps)•

Each interaction is determined independently using the same dataset supplemented
with an additional constraint on J
J[MeV]L[MeV]

29

31.9

30

37.0

25

31

44.1

20

32

52.5

15

33

62.2

10

34

72.3

535

83.4

36

94.3

NEUTRON MATTER ENERGY p.p.
35

E/N [MeV]

30

00

RNEDF-J29-36
RNEDF10.05

J0.10.15-3ρn [fm ]

0.2

0.25

CONSTRAINING THE SYMMETRY ENERGY• Isovector dipole transition strength for 208Pb using a set of relativistic point couplinginteractions which vary the symmetry energy properties (J=30,31,…,36 MeV)
J=30
J=31,32,33,34,35
J=362 2R[e fm /MeV]

20208Pb•••

Isovector giant dipole resonancePygmy dipole strengths
Dipole polarizability ( ↵D ⇠ m 1 )•The transition strength is sensitiveon the properties of symmetryenergy - (J,L)

1510è Dipole response can be used toconstrain effective nuclear
interactions (isovector channel)

568101214
E[MeV]

16182022•

Pygmystrength

Isovector giant
dipole resonance

There are exp. data available on
the dipole response in nuclei(αD, IVGDR, pygmy strength)

DIPOLE POLARIZABILITY IN 68Ni AND SYMMETRY ENERGY•

Measurement of dipole polarizabilityof unstable neutron rich 68Ni
D. Rossi et al, PRL 111, 242503 (2013)• Implementation to constrain thesymmetry energy using relativistic
point coupling interactions

4.6

4.4

68Ni

4.44.2αD [fm3]

αD [fm3]4.268Ni

43.8

43.8
3.6

3.6

3.4

3.4
3.2

283032

J [MeV]3436204060

L [MeV]

80100

DIPOLE POLARIZABILITY IN 208Pb AND SYMMETRY ENERGY••

22Using relativistic point couplinginteractions and measurement ofdipole polarizability in 208Pb

22208Pb

αD [fm3]

αD [fm3]

208Pb

21212020191918181728

A. Tamii et al., PRL 107, 062502 (2011)

3032

J [MeV]

3436

17204060

L [MeV]

80100

GIANT QUADRUPOLE RESONANCES IN 208Pb AND SYMMETRY ENERGY• Isovector quadrupole transition strength for 208Pb using a set of relativistic point couplinginteractions which vary the symmetry energy properties (J=30,31,…,36 MeV)
23.6

J=30
J=31,32,...,35
J=36E(IVGQR) [MeV]

R[fm4/MeV]

5000

208Pb

23.4

208Pb

4000
3000

23.2
2322.8
22.6
22.4

GIANT QUARUPOLE RESONANCES IN
208Pb

2000

22.2

2228

1000

30

32

343638

J [MeV]
05

1015

20

25

30

35

23.4

40

E[MeV]•Precise exp. determination of isovector giant
quadrupole resonances
S.S. Henshaw, M.W. Ahmed, G. Feldman et al,
PRL 107, 222501 (2011)

E(IVGQR) [MeV]Isovector giant
quadrupole resonance

208Pb

23.2
2322.8
22.6
22.4
22.2
22204060

L[MeV]

80100

CONSTRAINING THE SYMMETRY ENERGY (J-L)•Based on the same set of relativistic density-dependent point coupling interactions

100
IVGQR
αD (208Pb)

L [MeV]80

αD (120Sn)
αD (68Ni)

60

68
PDR ( Ni)

40
J = 31.6 ± 0.7 MeV
L = 50.6 ± 6.4 MeV

202830323436
J [MeV]384042

CONSTRAINING THE SYMMETRY ENERGY (J-L)

120HIC
QMC & neutron star100FRDM

L [MeV]

IAS

80

208Pb(p,p)

PDR (LAND 2007)60

PDR (Carbone 2010)
Lattimer & Lim (2013)
Hebeler et al. (2013)

40

RNEDF (DD-PC)

2028••••
••

30

32

34
J [MeV]3638

Lattimer & Lim, ApJ. 771, 51 (2013) – compilation from various approaches
K. Hebeler et al., AJ 773, 11 (2013) – based on nuclear interactions derived from chiral EFT
P. Danielewicz and J. Lee, Nucl. Phys. A 818, 36 (2009) – IAS
A. Carbone, G. Colo, A. Bracco, et al., Phys. Rev. C 81, 041301 (2010) – PDRA. W. Steiner and S. Gandolfi, Phys. Rev. Lett. 108, 081102 (2012) – QMC (Av8’)+ neutron starsetc.

CONSTRAINING THE SYMMETRY ENERGY: 2nd approach•

use the experimental data on collective excitations to constrain the symmetry energy
within the fitting protocol to determine the parameters of the functional (relativisticpoint coupling interaction; N.P., M. Hempel et al. in prep.)• Adjust the properties of 72 spherical nuclei to exp. data (binding energies, charge
radii, diffraction radii, surface thickness, pairing gaps)•constrain the symmetry energy S2(ρ0)=J (2%) from exp. data on dipole polarizability(208Pb) within an iterative procedure A. Tamii et al., PRL 107, 062502 (2011) + update (2015).

•constrain the nuclear matter incompressibility Knm (2%) from exp. data on ISGMR
modes (208Pb); D. Patel et al., PLB 726, 178 (2013).

• constrain the maximal neutron star mass by solving the Tolman-Oppenheimer-Volkov(TOV) equations + observational data from J. Antoniadis, et al. Science 340, 448 (2013)“Correct symmetry energy” is the one obtained for the interaction that accurately
reproduces the exp. data on dipole polarizability

CONSTRAINING THE SYMMETRY ENERGY: 2nd approach
ISOSCALAR GIANT
MONOPOLE RESONANCE

ISOVECTOR GIANTDIPOLE RESONANCE

EXP.

20

ISGMR

EXP.

208

Pb

IVGDR

208

Pb

20

R[e fm ]Exp.
αD= (19.6 ± 0.6) fm321510234

R[10 fm ]15500Dipole polarizability:
αD= (19.68 ± 0.21) fm3

10A.Tamii et al., PRL 107, 062502(2011). + update (2015).

5510 15 20
E[MeV]

253000510 15 20
E[MeV]• ISGMR energy determines
the nuclear matter incompressibility:
Knm=232.4 MeV
E (Exp.) = (13.91 ± 0.11) MeV (TAMU)
E (Exp.) = (13.7 ± 0.1) MeV (RCNP)

2530• IVGDR – αD determine thesymmetry energy for the interaction

J = 31.89 MeV
L = 51.48 MeV

SOME OTHER PROPERTIES…
Neutron skin thickness

Nuclear binding energies (calc. – exp.)
8

RNEDF1
PC-PK1
DD-PC1
DD-ME2420-2-4-6-80

50

100

A

150

200

Neutron matter

Neutron star mass-radius relationship
3.0

Gravitational Mass MG [M ]

EB

cal.

- EBexp.

[MeV]

6J0348+0432
Steiner et al.
HS(DD2)
BHB
SFHo
RNEDF1, HS(DD2) crust

2.5

2.0

1.5

1.0

0.5

0.0

91011121314

Radius R [km]15161718

NEUTRON SKINS AND NEUTRON STAR PROPERTIES• Neutron star properties• The phase transition density (nt) and pressure (Pt) at the inneredge separating the core from crust are sensitive to poorly
constrained density dependence of the nuclear matter
symmetry energy
C.J. Horowitz and J. Piekarewicz, PRL 86, 5647 (2001)(D. Page)• Relationship between the neutron-richskin of 208Pb and liquid-to-solid transition
density for neutron rich matter.

Neutron star structure and excitations in finite nuclei• Constraints on the neutron star core-to-crust transition density nt and pressure Pt• RNEDF and thermodynamic method Ch. C. Moustakidis et al., PRC 81, 065803 (2010)• Exp. data for AGDR, IVGQR and IVSMR excitation energies (208Pb), dipole polarizability(208Pb) and PDR energy weighted strength (68Ni).
nt([fm-3]

1.2
AGDR (208Pb)
αD ( 208Pb)

1RNEDF (EXC.)

0.0955

A18+δv+UIX*(A. Akmal, 1998)

0.087

EOS Friedman
Pandharipande(C.P. Lorenz, 1993)

0.096

Chiral EFT (NN+3N)(K. Hebeler, 2013)

0.076-0.088

HIC(B.A.Li, 2005)

0.040-0.065

Pt [MeVfm-3]

IVGQR (208Pb)

0.8

PDR (68Ni)
208Pb)IVSMR (0.6

MDI (Krastev, Li 2010)

0.4

DBHF+Bonn B
RNEDF (2010)

0.2

DDME

00.09

0.095

0.1

0.105

0.11

0.115

nt [fm-3]
N. P., Ch. C. Moustakidis, T. Marketin, D. Vretenar, G. A. Lalazissis, PRC 90, 011304(R) (2014)

CONCLUDING REMARKS• Dipole excitations in nuclei (PDR, αD, IVGDR, AGDR) and other modes (IVGQR,…)
provide valuable constraints for the nuclear matter symmetry energy• Accurate measurements have important implications to reduce uncertainties in thesymmetry energy – motivation for the future experiments• Small uncertainty in the calculated symmetry energy (J,L) limits the choice of currently
available equations of state used in modeling neutron stars and supernova matter.
• Neutron star core-to-crust transition density and pressure can be assessed usinginformation on collective nuclear excitations / symmetry energy
Final conclusion:
The EDF / EOS with the correct symmetry energy is the one that reproduces
the exp. data on isovector constraints from nuclear experiments