Dipole Polarizability and ParityViolating Asymmetry in 208Pb

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

Nuclear Structure and Dynamics II. July 9th to 13th 2012.

Opatija, Croatia.

Table of contents:

Motivation

Isovector static dipole polarizability αD : Definition,Hartree-Fock + Random Phase Approximation results for thecase 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 in 208Pb on mean-field (MF) calculations.

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 measured by

using strongly interacting probes.⇓

Limited knowledge of isovector properties

Isovector channel of currenteffective theories need to befixed by other means.Example: fixed to ab-initiocalculations of the N-Ninmedium interaction.X. Roca-Maza, X. Vinas, M. Centelles, P. Ring and P.

Schuck, Phys. Rev. C 84 054309 (2011).

0 0.05 0.1 0.15 0.2 0.25ρ ( fm

−3 )

-15

0

15

30

E /

A (

MeV

)

DD-MEδBHF

0 0.1 0.2ρ (fm

−3)

50

100

150m

p* −

mn*

(MeV

)

DBHF

Kτ = −255.15 MeV

ρ = 0.1520 fm−3

E / A = −16.12 MeVK = 219.1 MeVJ = 32.35 MeVL = 52.85 MeV

Motivation:

At present,The use of rare ion beams has opened the possibility ofmeasuring properties of exotic nuclei.Parity violating elastic electron scattering (PVES), a modelindependent technique, has allowed to estimate the neutronradius of a stable heavy nucleus like 208Pb (PREx@JLab).

⇓Promising perspectives for the near future.

http://www.lsw.uni-heidelberg.de/nic2010/talks/Kruecken.pdf

http://hallaweb.jlab.org/experiment/HAPPEX/

Motivation:

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

Example:Mean-Field predictionsshow a clear correlationbetween∆rnp = 〈rn〉1/2 − 〈rp〉1/2of a medium and heavynucleus and the densityslope L of the symmetryenergy,S(ρ) ≈ eneut(ρ)− esym(ρ):L = 3ρ0∂ρS(ρ)|ρ0 = 3ρ0p0

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

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)

Some considerations on the αD in nuclei

The restoring force in the isovector dipole response isproportional to the symmetry energy, S.

Larger symmetry energies at saturation shift the excitationenergies to lower values (MF).

The strength increases as a consequence of previous pointand the conserved m1 sum rule.

Since m−1 is not a conserved quantity, and weights morethe low energy region, the effect of the the low-lying stateson the αD is not negligible.

“Edipole ∼(

E 2unpert + c × Spot.

)1/2” (Bohr & Mottelson) →

Edipole expected to be correlated with the m∗ (Eunpert

depends on the level density and, therefore, on the effectivemass) and the symmetry energy parameters J and/or L.

Qualitatively: assuming small variations of m∗ and J in MFcalculations when compared to the theoretical spread in L →suggest a correlation between αD and ∆rnp.

Mean-Field + RPA results for 208Pb

∆rnp(208Pb) ≈ 0.168 ± 0.022 fm

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3

3

33

0.12 0.16 0.20 0.24 0.28 0.32[Rn-Rp]Pb

(fm)

0.12

0.16

0.2

0.24

0.28

[Rn-R

p] C

a (fm

) PREx

NL3/FSUDD-MESkyrme

Skyrme (SV)

CAB

models=0.852

∆rnp(48Ca) ≈ 0.176 ± 0.018 fm and

αD (48Ca) ≈ 2.3 ± 0.1 fm3

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).

Parity violating elastic electron scattering in208Pb

Theoretical bases of PVES:

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.

Coulomb distortions should be taken into account: DWBAcalculations give ∼30% correction with respect to PWBA.

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

)

Input for the calculation: ρn and ρp

ρp of 208Pb is well known from other experiments

ρn of 208Pb is the quantity to be determined

Problem: In the analysis, one can only fix one paramter of theadopted neutron distribution to the data on Apv .Solution: Fix a range for the other parameter/s based ontheoretical calculations.Problem: Model dependence is introduced.Solution: measurements of Apv at different angles (measuringmore nuclei would also help).

In case in which a measurement of Apv at different angles isnot possible/available,

we propose the following analysis:

Direct correlationswithin MF

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 ρ

Different experimentson proton elasticscattering andantirpotonic atomsagrees with thecorrelation

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

LNSSGII

SIII

SK255

SkI3

SKM*

SkP

Sly4

SkO’

18 19 20 21 22 23 24

αD

(fm3)

0.68

0.7

0.72

0.74

0.76

0.78A

pv (

ppm

)NL3/FSUDD-MESkyrmeSkyrme (SV)

PREx

RCNP

Conclusions: Families of MF models predict a high linear correlation

between αD and ∆rnp in 208Pb (m∗ and other propertiesexcept J and L have been fixed).

Further exeprimental and theoretical studies on αD areneeded for a better physical understanding on theproperties of the nuclear effective interaction (m∗, J, L,...)that are determing this observable.

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

We demonstrate a linear correlation between Apv and ∆rnp. Other experiments fairly agree with the correlation between

Apv 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.

Collaborators:

B. K. Agrawal1

M. Centelles2

G. Colo3,4

W. Nazarewicz5,6,7

N. Paar8

J. Piekarewicz9

P.-G. Reinhard10

P. Ring11

P. Schuck12,13,14

X. Vinas2

D. Vretenar8

M. Warda15

1 Saha Institute of Nuclear Physics, Kolkata 700064, India, 2 Departament dEstructura i Constituents de laMateria and Institut de Ciencies del Cosmos, Facultat de Fısica, Universitat de Barcelona, Diagonal 647, E-08028

Barcelona, Spain, 3 Dipartimento di Fisica, Universita degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy, 4

INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy, 5 Department of Physics and Astronomy, University

of Tennessee, Knoxville, Tennessee 37996, USA, 6 Physics Division, Oak Ridge National Laboratory, Oak Ridge,

Tennessee 37831, USA, 7 Institute of Theoretical Physics, University of Warsaw, ulitsa Hoa 69, PL-00-681 Warsaw,

Poland, 8 Physics Department, Faculty of Science, University of Zagreb, Zagreb, Croatia, 9 Department of

Physics, Florida State University, Tallahassee, Florida 32306, USA, 10 Institut fr Theoretische Physik II, Universitt

Erlangen-Nrnberg, Staudtstrasse 7, D-91058 Erlangen, Germany, 11 Physikdepartment, Technische Universitat

Munchen, D-85748 Garching, Germany, 12 Institut de Physique Nucleaire, CNRS, UMR8608, Orsay, F-91406,

France, 13 Universite Paris-Sud, Orsay, F-91505, France, 14 Laboratoire de Physique et Modelisation des Milieux

Condenses, Grenoble, F-38042, France. 15 Katedra Fizyki Teoretycznej, Uniwersytet Marii CurieSklodowskiej, ul.Radziszewskiego 10, PL-20-031 Lublin, Poland.

Extra Material

Why 208Pb?

208Pb is a stable, spin zero nucleus with large neutronasymmetry

208Pb has the advantage that it has the largest knownsplitting to the first excited of any heavy nucleus.

Well known structure since it has been extensively studied(spherical, no pairing, no deformation...)

EDF are expected to be accurate in the description ofaverage porperties (Kohn-Sham)

Charge radii (average property) in the region of Pb are welldescribed by EDF.

The correlation between the neutron skin thickness and theslope of the nuclear symmetry energy have beendemosntrated to exist within the EDF framework.

Most of the existent EDF have been fitted to spherical andalso quite frequently semi- or double-magic nuclei.

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 f

m3 ) α

DJ = 3.1(2) + 18.4(9) ∆r

npr = 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

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* / mISGQRK0

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 PDR

ρ0

e(ρ0)

m* / m

ISGQR K0

ISGMRm−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.