Dipole Polarizability and ParityViolating Asymmetry in 208Pb

Xavier Roca-MazaINFN, Sezione di Milano, Via Celoria 16, I-20133, Milano (Italy)

The Nuclear Dipole Polarizability and its Impact on Nuclear

Structure and Astrophysics, 18-22 June 2012, Trento, Italy.

Table of contents:

Motivation

Isovector static dipole polarizability αD : Definition,Droplet model approach and Hartree-Fock + Random PhaseApproximation results for the case of 208Pb.

Parity violating elastic electron scattering: Single anglemeasurement of Apv in 208Pb within the Distorted Wave BornApproximation based on mean-field nucleon distributions.

Constraints set by Apv measured at JLab and αD measuredat RCNP on mean-field (MF) calculations.

Summary of different constraints on the density slope ofthe symmetry energy around saturation.

Conclusions

Motivation:

The importance of determining isovector properties in nuclei

In the past (and also in the present), neutron properties instable medium and heavy nuclei have been mainly measuredby using strongly interacting probes.

⇓Limited knowledge of isovector properties

At present, the use of rare ion beams has opened the possibility of

measuring properties of exotic nuclei. parity violating elastic electron scattering (PVES), a

model independent technique, has allowed to estimate theneutron radius of a stable heavy nucleus like 208Pb(PREx@JLab).

⇓Promising perspectives for the near future.

Motivation:

It is possible to connect observables with general isovectorproperties of the nuclear effective interaction?

Example:Mean-Fieldpredictions show a clearcorrelation between∆rnp of a medium andheavy nucleus and thedensity slope of thesymmetry energy(L = 3ρ0∂ρS(ρ)|ρ0 =3ρ0p0).

R.J. Furnstahl, NPA, 706, 85 (2002)

Motivation:

More generally within MF, it has been found a semi-empiricallaw: asym(A) ≈ S(ρA) with ρA = ρ0 − ρ0/(1 + cA1/3) ⇒direct and clearconnection of any groundstate isospin sensitiveobservable with theparameters of the EoS.Following the sameexample: ∆r totalnp (A, I ) =

∆rbulknp (A, I ) +

∆r surfacenp (A, I )

MSk7

D1SSk-T6

SkM*

DD-ME2FSUGold Ska Sk-Rs

Sk-T4G2 NLC

NL3*

NL-ZNL1

0 50 100 150L (MeV)

0

0.1

0.2

0.3

∆rnp

of

208 P

b (

fm)

HFB-8

SGIIHFB-17

SLy4

SkSM*SkMP

NL-SH

D1N

TM1

NL3

NL-RA1

Total fit: r=0.992, slope=1.6 fm/GeVBulk fit: r=0.993, slope=1.4 fm/GeVSurface fit: r=0.602, slope=0.2 fm/GeV

∆rbulknp (A, I ) ≈ 2r03J

L(

1− ǫAKsym

2L

)

ǫAA1/3

(

I − IC)

M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys. Rev. Lett. 102, 122502 (2009); Phys. Rev. C 80

024316 (2009); Phys. Rev. C 81 054309 (2010) and Phys. Rev. C 82, 054314 (2010)

Motivation:Observables, processes and observations known to becorrelated with the isovector properties of the nucleareffective interaction

Binding energies

Neutron distributions (proton elastic scattering, antiprotonicatoms, parity violating asymmetry,...)

Giant Resonances: Giant Dipole, Gamow-Teller, IsobaricAnalog, Spin Dipole and Anti-analog of the Giant DipoleResonances (inelastic hadron-nucleus, nucleus-nucleus andγ-nucleus scattering).

Heavy Ion Collisions (EoS — transport models)

Neutron Star properties: mass-radius relation, transitiondensity crust-core, composition,... (observational data).

Low-energy dipole response (?)

Isovector Giant Quadrupole Resonance (?)

Isoscalar Giant Resonances along isotopic chains (?)

...

Isovector static dipole polarizability

Definition: αD

The linear response or dynamic polarizability of a nuclearsystem excited from its g.s., |0〉, to an excited state, |ν〉, dueto the action of an external oscillating dipolar field of the form(Fe iwt + F †e−iwt):

FD =Z

A

N∑

i

rnY1M(rn)−N

A

Z∑

i

rpY1M(rp)

is proportional to the static dipole polarizability, αD , forsmall oscillations

αD =8π

9e2m−1 =

8π

9e2

∑

ν

|〈ν|FD |0〉|2E

where m−1 is the inverse energy weighted moment of thestrength function,

SD(E ) =∑

ν

|〈ν|FD |0〉|2δ(E − Eν)

Macroscopic approach: αD

Being E the energy of a nucleus within the Liquid DropModel, Droplet Model,... and Fext an external field (dipoleoperator) → constrained calculation keeping N and Z fixed:

δ

E (ρ, δ) + Fext(Λ, ρ, δ)− λn

∫ ∞

0ρn(r)dr − λp

∫ ∞

0ρp(r)dr

= 0

LDM1:

m−1 ≈A〈r2〉1/248J

DM2:

m−1 ≈A〈r2〉1/248J

(

1 +15

4

J

QA−1/3

)

1 A.B . Migdal, J. Phys. USSR 8 (1944) 331; J.S . Levinger, Nuclear photodisintegration (Oxford Univ . Press,London, 1960) sect. 3-1.

2 J. Meyer, P. Quentin, and B. K. Jennings, Nucl. Phys. A 386 (1982) 269.

Droplet model approach: symmetry energy and neutron skin

The symmetry energy with surface effects is written in theDM:

asym(A) =J

1 + 94JQA−1/3

,

while the neutron skin thickness (also including surfaceeffects),

∆rnp =

√

3

5

[

t− e2Z

70J

]

+∆rsurfacenp

t ≡ 3r02

J/Q

1 + 94JQA−1/3

(I − IC ),

I ≡ (N − Z)/A

IC ≡ (e2Z)/(20JR)

R = r0A1/3

∆rsurfacenp =√

(3/5)[5(b2n − b2p)/(2R)] is a

correction caused by the difference in thesurface widths bn and bp of the neutron andproton density profiles

Droplet model approach: connection between αD and theneutron skin

Combining these formulas,

αD ≈ A〈r2〉12J

1 +5

2

∆rnp +√

35e2Z70J −∆r surfacenp

〈r2〉1/2(I − IC )

For a heavy nucleus and assuming small variations† of 〈r2〉,e2Z/70J and ∆r surfacenp as compared to that of J and ∆rnp,

αDJ ≈ p1 + p2∆rnp

†Some numbers: 208Pb and assuming J = 32 ± 2 MeV, ρ0 = 0.160 ± 0.05 fm−3 and

∆rsurfacenp ≈ 0.09 ± 0.01 (MF models) ⇒ (e2Z)/(70J) − ∆rsurfacenp ≈ −0.04 ± 0.01 fm

〈r2〉1/2 ≈ 5.23 ± 0.55 fm, Ic ≈ 0.027 ± 0.003

Mean-Field + RPA results for 208Pb

J. Piekarewicz, B. K. Agrawal, G. Colo, W.

Nazarewicz, N. Paar, P.-G. Reinhard, X.

Roca-Maza and D. Vretenar, Phys. Rev. C 85

041302 (2012) (R).

0.1 0.15 0.2 0.25∆r

np (fm)

5

6

7

8

9

10

10−2

α DJ

(M

eV fm

3 ) αD

J = 3.1(2) + 18.4(9) ∆rnp

r = 0.95EDF

20.1(6) x 32(2)

MF-Region

∆rnp ≈ 0.18± 0.05 fmassuming J = 32 ± 2 MeV and αD from A. Tamii et al. Phys. Rev.

Lett. 107, 062502 (2011)

Parity violating elastic electron scattering in208Pb

From previous talks, we have seen that,

Electrons interact by exchanging a γ or a Z0 boson.

While protons couple basically to γ, neutrons do it to Z0.

Ultra-relativistic electrons, depending on their helicity,interact with the nucleons V± = VCoulomb ± VWeak.

Ultra-relativistic electrons moving under the effect of V±

where Coulomb distortions are important ⇒ solution of theDirac equation via the Distorted Wave Born Approximation(DWBA).

Input for the calculation: ρn and ρp

Refs: C. J. Horowitz, Phys. Rev. C 57 3430 (1998); C. J. Horowitz, S. J. Pollock, P. A. Souder, and R. Michaels,

Phys. Rev. C 63, 025501 (2001); M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys. Rev. C 82, 054314

(2010); X. Roca-Maza, M. Centelles, X. Vinas, and M. Warda, Phys. Rev. Lett. 106 252501 (2011) and (for the

electric proton and neutron form factors) J. Friedrich and Th. Walcher, Eur. Phys. J. A 17, 607623 (2003)

PREx data analysis:

PREx measures, model-independently, the parity violatingasymmetry at 1.06 GeV and for a single angle (∼ 5 deg.) in208Pb,

Apv =(dσ+dΩ

− dσ−dΩ

)

/

(dσ+dΩ

+dσ−dΩ

)

ρn of 208Pb is the quantity to be determined, so...

Which is the optimal neutron distribution in reproducingApv (Ebeam = 1.06 GeV,θ = 5 deg) from PREx?

Analysis I: Fix ρp with the experimental data on elastic electron

scattering.

Assume the electric proton and neutron form factors fromother experiments.

Assume a parametrized density for the neutron distributionsuch as a two parameter Fermi function, 2pF:

ρn(r) =ρ0n

1 + e(r−Cn)/an

∫ ∞

0ρn(r)dr = N

Fit the parameters (e.g. Cn and an), keeping the number ofneutrons fixed, to the data on Apv

Problem: One can only fix one of the parameters since Apv isknown at only one scattering angle.Solution: Fix a range for the other parameter based on theoreticalcalculations.Problem: Model dependence is introduced.

Analysis II: 2pF densities fitted to MF ρn andρp

D1SSGIID1N

SLy4

SkaSkMP

SkM*

HFB-8

Sk-T6

MSk7

SkSM*

Sk-T4G2

NLC

DD-ME2

NL3NL3*

NL-Z

FSUGold

NL-SH

NL1

0.08 0.09 0.1 0.11 0.12a

n− a

p (fm)

6.2

6.4

6.6

6.8

7

ALR

( 1

07 )Sk-RS

HFB-17

NL-RA1

TM1

D1SSk-

T6

SLy4

SkM*

FSUGol

d

SkSM

*SkM

P

SkaSk-Rs

Sk-T4

NL-SH

G2

NL-RA1

NL3NL3

*

NL-Z

NL1

0 0.06 0.12 0.18 0.24C

n− C

p (fm)

6.2

6.4

6.6

6.8

7

ALR

( 1

07 )

HFB-8

HFB-17SGII

D1N

DD-ME2

TM1NLC

MSk7

DWBAA

LR = 10

−7 [α + β (C

n− C

p) + γ (a

n− a

p)]

M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys. Rev. C 82, 054314 (2010)

A qualitative guide to understand the previouscorrelation:

Apv within the Plane Wave Born Approximation,

Apv =GFq

2

4πα√2

[

4 sin2 θW +Fn(q)− Fp(q)

Fp(q)

]

... which depends on Fn(q)− Fp(q) that for a 2pF andq → 0, it is approximately,

−q2

6

[

3

5(Cn − Cp)(Cn + Cp) +

7π

5(an − ap)(an + ap)

]

variation of Apv dominated by (Cn − Cp) and (an − ap).

In this case: Cp and ap fixed from the experiment; Cn from thevalue of Apv and the rather good linear dependence shown in thelower panel of the last figure; an using the correlation shown(calibrated with MF).Problem: Model dependence.

The solution to this problem: measurements of Apv at differentangles

⇓model-independent determination of ∆rnp

In case in which it is not possible, we propose the followinganalysis...

Analysis III: directcorrelations withinMF

X. Roca-Maza, M. Centelles, X. Vinas, and

M. Warda, Phys. Rev. Lett. 106 252501

(2011)

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

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

[6]

Klos [8]

Hoffmann [4

]

PC-F1

BCP

PC-PK1

RHF-PKO3

DD-PC1

G1

MF correlations allowsto determine ∆rnp and Lwithout directassumptions on ρDifferent experimentson proton elasticscattering andantirpotonic atomsagrees with thecorrelation

Constraints set by Apv measured at JLab andαD measured at RCNP on MF calculations.

5 6 7 8 9

102

x αD

J (MeV fm3)

0.68

0.69

0.7

0.71

0.72

0.73

0.74

0.75A

pv (

ppm

)Mean-Field

PREx

RCNP

MF Region

Summary of different constraints on L

20 40 60 80 100L (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

Shetty et al. PRC 76 (2007) 024606

Li et al. Phys. Rep. 464 (2008) 113

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

Masses

Masses

n−p Emission Ratios

Isoscaling

Isospin Diffusion

GDR

PDR

PDR

Optical Potentials

Antiprotonic Atoms

Nuclear Model Fit

Heavy IonCollisions

GiantResonancies

N−A scatteringCharge Ex. Reac.Energy Levels

L~61 11 MeV+−

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

X. Roca-Maza et. al. PRL 106 252501 (2011)(centered at 61 MeV)PVES 1% accuracy

PREX(Central value at L=154 MeV)

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

Conclusions:

Families of MF models predict a high linear correlationbetween αD and ∆rnp in 208Pb.

The Droplet Model suggest a linear correlation betweenαDJ and ∆rnp. (This result should be analyzed moreprecisely on the basis of microscopic calculations)

Conclusions:

A model-independent determination of ∆rnp in 208Pb viaPVES experiments would need a measurement of Apv atdifferent scattering angles.

We demonstrate a close linear correlation between Apv and∆rnp within the same framework in which the ∆rnp iscorrelated with L. This allows to determine unambiguouslythe value of ∆rnp and L within the MF framework.

Other experiments fairly agree with the correlation betweenApv and ∆rnp.

Apv measured by the PREx collaboration at JLab and αD

measured at RCNP are complementary observables that mayset tight constraints on the density dependence of thesymmetry energy.

Most of the empirical estimations of L agrees in a range forits central value that lies within 50 and 70 MeV.

Collaborators:B. K. Agrawal1

G. Colo2,3

W. Nazarewicz4,5,6

N. Paar7

J. Piekarewicz8

P.-G. Reinhard9

D. Vretenar7

Michal Warda10

Mario Centelles11

Xavier Vinas11

1 Saha Institute of Nuclear Physics, Kolkata 700064, India2 Dipartimento di Fisica, Universita degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy3 INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy4 Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA5 Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA6 Institute of Theoretical Physics, University of Warsaw, ulitsa Hoa 69, PL-00-681 Warsaw, Poland7 Physics Department, Faculty of Science, University of Zagreb, Zagreb, Croatia8 Department of Physics, Florida State University, Tallahassee, Florida 32306, USA9 Institut fr Theoretische Physik II, Universitt Erlangen-Nrnberg, Staudtstrasse 7, D-91058 Erlangen, Germany10 Katedra Fizyki Teoretycznej, Uniwersytet Marii CurieSklodowskiej, ul. Radziszewskiego 10, PL-20-031 Lublin,Poland11 Departament dEstructura i Constituents de la Materia and Institut de Ciencies del Cosmos, Facultat de Fısica,Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain

Extra Material

Correlations: J, αD and ∆rnp

0.1 0.15 0.2 0.25∆r

np (fm)

5

6

7

8

9

10

10−2

α DJ

(M

eV fm

3 ) αD

J = 3.1(2) + 18.4(9) ∆rnp

r = 0.95EDF

20.1(6) x 32(2)

MF-Region

18 19 20 21 22 23 24α

D (fm

3)

30

35

40

J (M

eV)

r = 0.65EDF

0.1 0.15 0.2 0.25∆r

np (fm)

30

35

40

J (M

eV)

r = 0.95EDF

Correlations: J, αD and ∆rnp

DM versus other calculations and experiment

J.

Meyer, P. Quentin, and B. K. Jennings, Nucl. Phys. A 386 (1982) 269

Kτ versus L from isospin difusion, GMR and neutronskin data

20 60 100 140L (MeV)

-700

-600

-500

-400

-300

Kτ

(M

eV)

isospin diffusion

GMR of Sn

isospin diffusion

Ssw= 0

Sswmin

Sswmax

Note: Kτ ≈ Ksym − 6L has been estimated as in isospin difussion and GMR data only for comparison purposesM. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys. Rev. Lett. 102, 122502 (2009)L. W. Chen, C. M. Ko, and B. A. Li, Phys. Rev. Lett. 94, 032701 (2005); Phys. Rev. C72, 064309 (2005)B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. 464, 113 (2008)

T. Li et al, Phys. Rev. Lett. 99, 162503 (2007)

Covariance analysis: χ2 test

Observables O are used to calibrate the parameters p of a givenmodel. The optimum parametrization p0 is determined by aleast-squares fit with the global quality measure,

χ2(p) =m∑

ı=1

(Otheo.ı −Oref.

ı

∆Oref.ı

)2

Assuming that the χ2 is a well behaved (analytical) function in thevicinity of the minimum and that can be approximated by anhyper-parabola,

χ2(p)− χ2(p0) ≈ 1

2

n∑

ı,

(pı − p0ı)∂pı∂pχ2(p − p0)

≡n

∑

ı,

(pı − p0ı)Mı(p − p0)

where M is the curvature matrix.

Covariance analysis: χ2 test

M provides us access to estimate the errors between predictedobservables (A(p)),

∆A =

√

√

√

√

n∑

ı

∂pıAEıı∂pıA (1)

E = M−1 and the correlations between predicted observables,

cAB ≡ CAB√CAACBB

(2)

where,

CAB = (A(p)− A)(B(p)− B) ≈n

∑

ı

∂pıAEı∂pB

Covariance analysis: SLy5-min as an example

Covariance analysis: SLy5-min as an example

ρ0

e(ρ0)

m* / m IS-GQR K0

IS-GMR

IV-M-1 S2(ρ0

)

Lκ

rn

∆rnp

IV-GDR

IV-PDR

0 0.2 0.4 0.6 0.8 1

SLy5-min: correlation with GDR

ρ0

e(ρ0)

m* / mISGQRK 0

ISGMR m−1

(GDR)

S2(ρ0)

Lκ

rn

∆rnp

IVGDRIVPDR

0 0.2 0.4 0.6 0.8 1

SLy5-min: correlation with PDR

ρ0

e(ρ0)

m* / m

ISGQR

K0 ISGMR

m−1(GDR)

S2(ρ0)

Lκ

rn

∆rnp

IVGDR

IVPDR

0 0.2 0.4 0.6 0.8 1

SLy5-min: correlation with m−1

Figure: Pearson product-moment correlation coefficient for the IVGDR(left panel), IVPDR (middle panel) and m

−1(IVGDR) (right panel) withall other studied properties as predicted by the covariance analysis ofSLy5.