Elastic electron scattering as a revitalizedexperimental tool in modern nuclear physics:

a theoretical point of view

Xavier Roca-MazaUniversita degli Studi di Milano

May 8th 2013

Theoretical study of elastic electron scattering off stable and exotic nuclei X. Roca-Maza, M. Centelles, F.

Salvat, and X. Vinas Phys. Rev. C 78, 044332 (2008).

Electron scattering in isotonic chains as a probe of the proton shell structure of unstable nuclei X. Roca-Maza, M.

Centelles, F. Salvat, and X. Vinas Phys. Rev. C 87, 014304 (2013).

Neutron Skin of 208Pb, Nuclear Symmetry Energy, and the Parity Radius Experiment X. Roca-Maza, M.

Centelles, X. Vias, and M. Warda Phys. Rev. Lett. 106, 252501.

Motivation

In-medium nuclear (effective) interaction is not wellunderstood for extreme values of isospin asymmetry,that is, far from the stability valley

Experimental studies of elastic electron scattering byunstable nuclei:

will be feasible in rare ion beam facilities such as RIKEN(Japan) and GSI (Germany)

determine e-m charge distribution model independently better understanding of nuclei under more extreme conditions data on large isospin asymmetries

Theoretical studies of elastic electron scattering byunstable nuclei:

Physical process well understood since many years ago. Exact calculations available once the exact electromagnetic

charge distribution is known Theoretical guidance for future experiments

Motivation

In addition ...

Experimental studies of ineslastic electron scattering byunstable nuclei at forward angles that prominentlymeasure the E1 response:

will be also feasible in facilities such as SCRIT (Japan) determine the GDR in unstable nuclei (some mixing with other

resonances will reduce the accuracy) better understanding of the E1 response of unstable nuclei

Theoretical studies on the GDR in unstable nuclei: Physical process well understood Calculations available Theoretical guidance for experiments

Motivation

In-medium nuclear (effective) interaction for moderatevalues of isospin asymmetry, that is, close/within thestability valley is not precisely determined (neither)

Experimental studies of parity violating elastic electronscattering by stable medium and heavy nuclei whereisospin asymmetries are larger:

are feasible in facilities such as JLab (USA) and MAMI(Germany)

determine the weak charge distribution model independently better understanding of neutron distribution in nuclei

Theoretical studies of parity violating elastic electronscattering by stable nuclei:

Physical process well understood. Exact calculations available once the exact electromagnetic

and weak charge distributions are known Theoretical guidance for experiments

Table of contents

Elastic Scattering of Electrons by NucleiTheoryStable Nuclei: Overview

Conclusions

Exotic Nuclei: New experimental landscapeFuture experiments

How can theory help in the experimental analysis?

Results

Conclusions

Parity violating electron scatteringTheoryPast and Future ExperimentsResults on 48Ca and 208PbConclusions

Elastic Scattering of Electronsby Nuclei

Exact solution: Dirac partial-wave (also known as DWBA) calculation

of elastic scattering of electrons by nuclei. X. Roca-Maza, M. Centelles,

F. Salvat, and X. Vinas & Phys. Rev. C 78, 044332 (2008). F. Salvat et

al. Comp. Phys. Comm. 165 157-190 (2005).

Theory: study of the nuclear charge distribution

Ebeam ∼ 2π ~cλNucl.size

where λnucl.size ∼ 2〈r2〉1/2 ∼ 2 − 10 fm⇒ 100 − 600 MeV.

Relativistic treatment is needed mec2/Ebeam . 0.005.

At these energies, effect of screening by the orbitingatomic electrons is limited to scattering angles smaller than1 degree (we will not calculate them here).

The interaction potential is Vnucl.elec. calculated from ρch(parametrized, model, ... )

Vnucl.elec. = 4πZ0e2

1

r

∫ r

0ρch(u)u

2du +

∫

∞

r

ρch(u)udu

spherical symmetry assumed

Theory: direct and spin-flip amplitudes

The scattering of relativistic electrons by a central fieldV (r) is completely described by the direct scatteringamplitude, f (θ), and the spin-flip scattering amplitude, g(θ).

f (θ) and g(θ) are complex functions solutions of the Diracequation for V (r) that behave asymptotically as a plane waveplus an outgoing spherical wave.

f (θ) and g(θ) admit the so called partial-wave expansion,

f (θ) =1

2ik

∞∑

l=0

(l + 1)[

e2iδκ=−l−1 − 1]

+ l[

e2iδκ=l − 1]

Pl (cos(θ))

and,

g(θ) =1

2ik

∞∑

l=0

[

e2iδκ=l − e2iδκ=−l−1]

P1l (cos(θ))

where k is the projectile wave number (~k = p), Pl and P1l are

Legendre polynomials and δκ are the phase shifts induced by the

central potential

Theory: phase shifts δκ

The phase shifts δκ represent the large-r behavior of theDirac spherical waves, solution of the Dirac equation,

ψEκm(r) =1r

(

PEκ(r)Ωκ,m(r)iQEκ(r)Ω−κ,m(r)

)

,

where Ωκ,m(r) are the spherical spinors and PEκ(r) andQEκ(r) satisfy,

dPEκ(r)

dr= −κ

rPEκ +

E − V + 2mec2

c~QEκ

dQEκ(r)

dr= −κ

rQEκ −

E − V

c~PEκ

where κ = (l − j)(2j + 1) is the relativistic quantum number.

PEκ(r → ∞) ≈ sin(kr − lπ/2 + δκ) for finite range fields.

Attractive (repulsive) potentials give positive (negative)phase shifts.

Theory: Basic quantities

Elastic DCS per unit solid angle for spin unpolarized electrons

dσ

dΩ= |f (θ)|2 + |g(θ)|2

Spin polarization function of the electrons from an initiallyunpolarized beam (Sherman function)

S(θ) ≡ if (θ)g∗(θ)− f ∗(θ)g(θ)

|f (θ)|2 + |g(θ)|2

Theory: the Form Factor

|FDWBA(q)|2 ≡dσ/dΩ

dσpoint/dΩ

where dσpoint/dΩ is the DWBA solution for a point nucleus andc~q = 2E sin(θ/2).

This definition, as comparedto dσ/dΩ

dσMott/dΩ, disentagles

better the finite size effectsof the nucleus.

Nevertheless, it is found thatthe choice is not critical forthe low momentum transferregime.

0 0.4 0.8 1.2 1.6q (fm

−1)

10−5

10−4

10−3

10−2

10−1

1

101

102

|F(q

)|2

122Zr "point"

140Ce "point"

154Hf "point"

0 0.4 0.8 1.2 1.6q (fm

−1)

10−5

10−4

10−3

10−2

10−1

1

101

102

|F(q

)|2

122Zr "Mott"

140Ce "Mott"

154Hf "Mott"

x10

x102

Mott DCS: dσMott

dΩ =(

Ze2

2E

)2cos2 θsin4 θ

; for small angles diverges as θ−4

Theory: Energy Dependence in the e-m Form Factor

0 0.2 0.4 0.6 0.8 1 1.2

q (fm−1

)

10−4

10−3

10−2

10−1

1

|FD

WB

A(q

)|2

0.3 GeV0.5 GeV0.7 GeV0.9 GeV

208Pb

118Sn

x10−1

Test: The form factor in DWBA is almost energy-independent inthe low q-regime

FDWBA(q) is a good quantity for the study of theelectromagnetic structure of the nucleus

Stable Nuclei: Overview

Experiment versus Theory in stable nuclei

Nuclear Model (NM) provides:

0

0.02

0.04

0.06

0.08G2NL3FSUGoldDD-ME2SLy4SkM*Exp(fit)

0

0.02

0.04

0.06

ρ ch

(fm

−3)

0 2 4 6 8

r (fm)

0

0.02

0.04

0.06

16O

90Zr

208Pb

NM+DWBA provides:

G2FSUGoldDD-ME2SLy4Exp(fit)

10−8

10−6

10−4

10−2

1

|FD

WB

A(q

)|2

0 1 2

q (fm-1

)

16O

90Zr

208Pb

374.5 MeV

302.0 MeV

502.0 MeV

102

x10−2

x102

... and a more demanding test:D(A− B) ≡ (A− B)/(A+ B)

D(40Ca − 42Ca) D(40Ca − 44Ca) D(116Sn − 118Sn)

30 40 50 60 70 80 90

θ (deg)

-5

0

5

10

Exp.Exp(fit)G2NL3

30 40 50 60 70 80 90

θ (deg)

-10

0

10

20

FSUGoldDD-ME2SLy4SkM*

30 40 50 60 70 80 90

θ (deg)

-20

-10

0

10

20

30 40 50 60 70 80 90

θ (deg)

-20

0

20

30 40 50 60 70 80 90 100

θ (deg)

-10

-5

0

5

10

30 40 50 60 70 80 90 100

θ (deg)

-10

-5

0

5

10

D(40Ca − 48Ca) D(48Ca − 48Ti) D(118Sn − 124Sn)

Ability test for the models

Nucleus Ee d2w ”Best”

MeV Exp. fit DD-ME2 G2 NL3 FSUGold SLy4 SKM* model16O 374.5 11.1b 88.7 13.1 38.6 206. 191. 194. G240Ca 250.0 7.18b 3.15 16.2 13.9 0.84 24.4 24.3 FSUGold

500.0 3.48b 1.49 42.9 19.7 5.79 40.0 39.0 DD-ME248Ca 250.0 6.66b 4.85 9.74 7.14 4.08 14.9 13.6 FSUGold

500.0 3.19b 1.11 17.0 3.53 2.57 21.84 18.5 DD-ME290Zr 209.6 0.78b 0.87 2.21 1.36 0.65 6.53 5.36 FSUGold

302.0 0.86b 0.91 9.92 3.27 0.67 9.35 7.19 FSUGold118Sn 225.0 5.43a 18.4 34.8 25.5 31.8 2.75 4.20 SLy4208Pb 248.2 30.6b 44.4 154. 74.8 89.5 89.2 61.0 DD-ME2

502.0 21.2b 14.1 186. 50.5 61.1 95.9 76.5 DD-ME2

D(40Ca− 42Ca) 250.0 0.56c 9.1 28.3 16.0 11.1 9.1 12.9 DD-ME2/SLy4D(40Ca− 44Ca) 250.0 1.14c 4.5 29.6 12.2 3.88 7.08 9.13 FSUGoldD(40Ca− 48Ca) 250.0 1.06c 16.4 4.89 7.74 38.5 94.1 49.3 G2D(48Ca− 48Ti) 250.0 2.49c 18.0 19.6 31.0 37.8 71.8 64.9 DD-ME2D(116Sn − 118Sn) 225.0 2.05a 8.05 7.80 9.00 10.1 13.2 18.5 G2D(118Sn − 124Sn) 225.0 4.03a 5.35 6.98 7.50 9.22 7.05 7.18 DD-ME2

aA.S. Litvinenko et al., Nucl.Phys. A182, 265 (1972). bB. Dreher, J. Friedrich, K. Merle, H. Rothhaas and G.

Luhrs, Nucl.Phys.A235, 219 (1974). c R.F. Frosch et al. Phys. Rev. 174 (1968) 1380.

Conclusions

Theory is well understood and calculations are feasible: exactsolution of the scattering process once the nuclear e-mcharge distribution has been provided.

disagreement with the experiment due exclusively to thenuclear model

The defined Form Factor include all finite size effects is nearly energy-independent at low momentum transfer

Exist a quasi-model-independent q-regime: Up to 1 − 1.5fm−1 for the studied models and scattering processes.

New experimental landscape:e - Rare Isotope Beams

ELISe@FAIR and SCRIT@RIKEN projects

Self-Confinig Rare Isotope Target (e-RI scattering) -SCRIT Operative

ELectron-Ion Scattering in a Storage Ring (eA collider) -ELISe Still under developement

At the beginnig of next year, SCRIT collacoration will startmeasuring the e-m distribution of unstable Sn isotopes, from

N=82 to N=62

How can theory help in the experimentalanalysis? Could we find simple and general

trends for F (q) in exotic nuclei?

Helm Model: 2 parameters fitted to theoretical predictionsto mimic future experimental analysis

Helm Charge Form Factor: R0 & σ

FH(q) =

∫

e i~q~rρH(~r)d~r =3

R0qj1(qR0)e

−σ2q2/2

where σ measures the surface fall-off of the densitydistribution and R0 measures its bulk extension.

How we determine the parameters: R0: one requires that the first zero of FH occurs at the same q

of FPWBA (fourier transform of the self-consistent density).Therefore, it coincides with the sharp radius.

σ: is chosen to reproduce the height of the second maximumof |FPWBA|

Results: Z=50 and Z=20 isotopic chains

Charge Form Factor

FDWBA increases and shifts towards smaller q as the neutronnumber increases

10−7

10−6

10−5

10−4

10−3

10−2

10−1

1

40Ca

70Ca

G2Helm

0 0.5 1 1.5 2

q (fm−1

)

10−6

10−5

10−4

10−3

10−2

10−1

1

0 0.5 1 1.5 2 2.5

q (fm−1

)

56Ca

|FD

WB

A(q

)|2

100Sn

132Sn

176Sn

Methodology accurate for low-momentum transfer

Correlations: the smaller the bulk part of the nuclearcharge distribution and the compact the surface, thesmaller the form factor

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

σ2q

2

min

0.6

0.7

0.8

0.9

1

1.1

103

|FD

WB

A(

q min )

|2

Calcium Isotopes500 MeV

r = 0.982

0.4 0.45 0.5 0.55 0.6

σ2q

2

IP

6

6.2

6.4

6.6

6.8

7

7.2

7.4

103

|FD

WB

A(

q IP )

|2

r = 0.992

Tin Isotopes500 MeV

Therefore, if two or more isotopes have been measured ...

linear correlations would provide, for an unknown nucleus ofthe chain, a hint on the value expected for the square of theexperimental electric charge form factor at its firstminimum

if the value of the squared modulus of the form factor isdetermined experimentally at its first minimum, thecharge density in the Helm model can be sketched fromsimilar correlations

use of more elaborated versions of the Helm model thattake into account the central depression of the charge densityshould allow one to extend the domain of validity of ourmethod up to larger values of the momentum transfer.

Results & Correlations: N=82, N=50 andN=14 isotonic chains

Differential cross sections and form factors

0 10 20 30 40θ (deg)

10−33

10−30

10−27

10−24

10−21

dσel

astic /

dΩ (

mba

rn /

sr ) 122

Zr G2 Helm140

Ce G2Helm

154Hf G2

Helm

x10

x102

(a)

0 0.4 0.8 1.2 1.6q (fm

−1)

10−5

10−4

10−3

10−2

10−1

1

101

102

|FD

WB

A(q

)|2

x10

x102

Methodology accurate for low-momentum transfer

Charge form factors FDWBA increases and shifts towardssmaller q as the neutron number increases

0.75 0.8 0.85 0.9 0.95 1q (fm

−1)

4

6

8

10

103 |F

DW

BA(q

)|2

First inflection point

N=82 122Zr

124Mo

128Pd

132Sn

136Xe

140Ce

154Hf

The increasing rate of the form factor basically depends onthe proton level which is being filled!!

0.3 0.4 0.5 0.6 0.7σ2

q2

IP

4

6

8

10

103

|FD

WB

A(

q IP )

|2

154Hf

N=82 Isotones500 MeV

152Yb

148Dy

144Sm140

Ce

136Xe

132Sn

128Pd

120Sr

122Zr

142Nd

146Gd

150Er 1h11/2

2d5/2

1g7/2

1g9/2

2p1/2

2d3/2

and 3s1/2

also contribute

0 0.5 1 1.5q (fm

−1)

-0.2

0

0.2

0.4

0.6

0.8

1

f nlj (

q)

1g 9/21g 7/22d 5/21h 11/22d 3/23s 1/2

0.84 0.88

-0.2

-0.1

0

0.1

0.2

∆qIP

Also in lighter isotonic chains...

1.2 1.3 1.4 1.5 1.6

σ2q

2

min

0

1

2

3

4

5

6

104

|FD

WB

A(

q min )

|2

24Ne

N=14 Isotones500 MeV

26Mg

28Si

30S

32Ar

34Ca

22O1d5/2

2s1/2

1d3/2 1f

7/2 and 1f

5/2 also contribute

0 0.5 1 1.5 2 2.5 3q (fm

−1)

-0.2

0

0.2

0.4

0.6

0.8

1

f nlj (

q)

1d 5/22s 1/21d 3/21f 7/21f 5/2

∆qmin

The larger the number of protons, the larger the formfactor

...this was clear, less clear was that it is almost linear alongisotonic chains

28 30 32 34 36 38 40R

0

2 (fm

2)

4

6

8

10

12

103

|FD

WB

A(

q IP )

|2

154Hf

N=82 Isotones500 MeV

152Yb148

Dy144

Sm140Ce136

Xe

132Sn

128Pd

120Sr

122Zr

142Nd146

Gd

150Er

8 9 10 11 12 13 14R

0

2 (fm)

0

1

2

3

4

5

6

104

|FD

WB

A(

q min )

|2

24Ne

N=14 Isotones500 MeV

26Mg

38Si

30S

32Ar

34Ca

22O

Conclusions: isotopic chains

The described analysis is potentially useful for futureelectron-nucleus elastic scattering experiments,

the linear correlations shown would provide, for an unknownnucleus of a chain, a hint on the value expected for|Fexp(qmin)|2.

The exact analysis of the Coulomb phase shifts applied toa exotic nuclei and compared with future measurments could,potentially, elucidate some aspects related with the isospinasymmetry of the nuclear force.

The use of more elaborated versions of the Helm modelshould allow one to extend the domain of validity of ourmethod up to larger values of q

Conclusions: isotonic chains

Rate of change of the electric charge form factor isextremely sensitive on the proton level which is beingfilled

levels with large n and small l contribute with opposite signwith respect to levels without radial nodes and large angularmomenta.

plotting |F (q)|2 against σ2q2min magnifies such effects

Therefore, electron scattering in isotonic chains can be auseful tool to probe the proton single-particle shellstructure of exotic nuclei: filling order and occupancy of thedifferent valence proton orbitals.

Conclusions: warning...

Extensive experimental investigations more difficult becauseof the limitations arising from small production rates, short

half-lives, and small cross sections when one deals with unstablenuclei

Parity violating electronscattering

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)

Theory:

The electron interacts with a nucleus by exchanging eithera γ or a Z0 boson.

γ couples basically to protons, Qem = Z , and Z0 couplesbasically to neutrons,

QW = −N + (1− 4 sin2(θW ))Z ≈ −N + 0.1Z .

Ultra-relativistic electons interact with the Coulomb + or− the Weak potential depending on the helicity of the electrons,

Vtot = VC ± VW where VW = GFρW (r)/22/3,This produces a parity-violating amplitude in the scatteringprocess.

The effect of the parity-violating part of the weak interactionmay be isolated by measuring the parity-violating asymmetry,

APV =dσ+/dΩ− dσ−/dΩ

dσ+/dΩ+ dσ−/dΩ

where +/− indicates positive or negative helicity of e.

Parity violating elastic scattering determine the nuclearweak charge distribution in a similar way as theelectromagnetic charge distribution is determined inparity conserving elastic electron scattering

The determination of APV is model-independent.

Experiments at different anlges are not planned for thenear future ⇒ on cannot map the whole weak density innuclei in a model independent way.

Theory:

Qualitatively,

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). For q → 0, it isapproximately,

−q2

6

(

〈r2n 〉 − 〈r2p 〉)

= −q2

6

[

∆rnp(〈r2n 〉1/2 + 〈r2p 〉1/2)]

= −q2

6

(

2〈r2p 〉1/2∆rnp +∆r2np

)

variation of Apv at a fixed q dominated by the variation of∆rnp. Fp(q) well fixed by experiment

Past and Future Experiments

Past

PREx measured APV in 208Pb @ 5deg and 1.063 GeVmodel-independently → first electro-weak probe of theexcistence of a weak charge radius larger than theelectromagnetic charge radius in a heavy nucleus.

Future

PREx II: improve accuracy of PREx (JLab)

CREx: measure APV in 48Ca @ 4deg and 2.2 GeV (JLab)

Super PREx or PV-RAPTOR: APV in 208Pb with betteraccuracy than PREx and PREx II (MAMI)∗

∗ Open also to measure other nuclei if well motivated from the theory.

208Pb: directcorrelationsDWBA; no radiativecorrections or strangequark effects included

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 allows to

determine ∆rnp and L

without direct assumptions

on ρ, PREx-II and

PV-RAPTOR expected

accuracy → constrain on L

Different experiments on

proton elastic scattering

and antirpotonic atoms

agrees with the correlation

48Ca: direct correlations within MF includingradiative corrections and strange quark effects

DD

-ME

1

DD

-ME

2

DD

-ME

δ

FS

UG

old

G2

LNS

NL1

SG

IIS

III

SK

255

SK

I2

SLy

4

TM

1

0.15 0.2 0.25∆ r

np (fm)

2

2.1

2.2

2.3

2.4

Apv

(pp

m)

EDFs Gs

E(Q) = 0

r=0.98

EDFs Gs

E(Q) = −0.006±0.016 Q

2/(0.1 GeV

2)

48Ca @2.2 GeV, 4 deg.

SkM

*S

Ly5

SkI

3

PC

-F1

weightedaverage∆r

np(Exp.)

NL3

Apv decreases by around 0.005 ppm with an error of about 0.01 -0.02 ppm when G s

E (Q2) is included.

Used G sE (Q

2) from PRC 76, 025202 (2007) by Liu, McKeown, and Ramsey-Musolf Average ∆rnp from

hadronic probes: PRC12, 778 1978; PRL87, 08250113, 343 (2004); Phys. Rev. 174, 1380 (1968); Physics

Letters 57B 47 (1975); PRC 67, 054605 (2003) and PRC33 1624 (1986).

48Ca: estimation of spin-orbit effects

FS

UG

old

NL3

DD

-ME

1

DD

-ME

2

DD

-ME

δ

FS

UG

old

G2

LNS

NL1

SG

IIS

III

SK

255

SK

I2

SLy

4

TM

1

0.15 0.2 0.25∆ r

np (fm)

2

2.1

2.2

2.3

2.4

Apv

(pp

m)

Spin-Orbit included

48Ca @2.2 GeV, 4 deg.

SkM

*S

Ly5

SkI

3

PC

-F1

weightedaverage∆r

np(Exp.)

NL3

In the two tested models, spin-orbit effects shifts to lower valuesthe Apv consistently by about 0.07 ppm. This predicts a reductionof ∆rnp of about 0.05 fm.Charge density distributions including spin orbit effects provided by

J. Piekarewicz (FSU).

48Ca: Estimation of three-neutron forceseffects in comparison with other corrections

Considering errors in Gs

E and no spin-orbit contributions //

No quark strange contributions and spin-orbit effects considered (filled in orange)No quark strange or spin-orbit contributions

Considering Gs

E and no spin-orbit contributions

FSUGold / FSUGold + Shell Model

a

b

cab

c

Self-consistent nucleon distributions in 48

Ca

Self-consistent 40

Ca core + Shell Model NN for 8 neutrons in excess

Self-consistent 40

Ca core + Shell Model NN+3N for 8 neutrons in excess

errors

spin-orbit mainly from ν1f7/2

0.05

ppm

±0.0

15 p

pm

0.07

ppm

rn ~ 0.01 fm

Shell Model calculations based on χEFT with NN to N3LO (fixedto scattering data) and 3N to N2LO (fixed to B tritium and R ofalpha particle) provided by J. Menendez (TU Darmstadt).Three-neutron forces used here shifts downwards the Apv by about0.05 ppm (very similar to spin-orbit effect)

Conclusions

A precise and model-independent determination of ∆rnp in48Ca and 208Pb via PVES experiments would probe at thesame time the density dependence of the nuclear symmetryenergy and the relevance of three neutron-forces in 48Ca.Eventually, it can also provide indirect indications on theimpact of 3N in 208Pb.

We demonstrate a close linear correlation between Apv and∆rnp within the same framework in which the ∆rnp iscorrelated with L.

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

Collaborators:

Xavier Vinas

Mario Centelles

Francesc Salvat

Michal Warda

Extra material

Isotopes

Helm and self-consistent charge densities and charge radii

0 1 2 3 4 5 6 7 8

r (fm)

0

0.02

0.04

0.06

0.08

0.1

ρ ch

(fm

−3)

G2Helm

3.4 3.6 3.8 4 4.2

A1/3

3.3

3.4

3.5

3.6

3.7

Rch

(f

m)

G2Helm

Calcium Isotopes

40Ca

56Ca

70Ca

0 2 4 6 8 10

r (fm)

0

0.02

0.04

0.06

0.08

ρ ch (

fm−3

)

G2Helm

4.6 4.8 5 5.2 5.4 5.6

A1/3

4.4

4.6

4.8

5

5.2

Rch

(f

m)

G2Helm

100Sn

132Sn

176Sn

Tin Isotopes

Correlations: evolution of first minimum or inflection point

0.25 0.26 0.27 0.28 0.29 0.3

A−1/3

1.05

1.1

1.15

1.2

1.25

q min (

fm−1

)

1.05

1.1

1.15

1.2

q eff,m

in (

fm−1

)

1.02 1.05 1.08 1.11 1.14 1.17 1.2

qmin

(fm−1

)

3.4

3.5

3.6

3.7

Rch

(f

m)

G2Helm

r = 0.9997

Calcium Isotopes

0.18 0.19 0.2 0.21

A−1/3

0.8

0.85

0.9

q IP (

fm−1

)

0.72

0.76

0.8

0.84

q eff,m

in (

fm−1

)

0.78 0.8 0.82 0.84 0.86 0.88 0.9

q IP

(fm−1

)

4.4

4.6

4.8

5

5.2

Rch

(f

m)

G2Helm

r = 0.997

Tin Isotopes

Isotones

Charge densities and proton single particle levels

0 2 4 6 8r (fm)

0

0.02

0.04

0.06

0.08

ρ ch

(fm

−3)

G2Helm

146Gd

122Zr

140Ce

154Hf

2p1/2

1g9/2

1g7/2

2d5/2

2d3/2

1f5/2

2p3/2

2p1/2

1g9/2

1g7/2

2d5/2

1f5/2

2p3/2

2p1/2

1g9/2

1g7/2

2d5/2

1f7/2

1f5/2

2p3/2

2p1/2

1g9/2

1g7/2

2d5/2

-20

-15

-10

-5

0

ε nlj

(MeV

)

3s1/2

2d3/2

122Zr

1h11/2

1h11/2

2d3/2

2d3/2

1h11/2

140Ce

146Gd

154Hf

Z=50

Z=50

Z=50

Z=50

3s1/2

3s1/2

3s1/2

0 2 4 6r (fm)

0

0.02

0.04

0.06

0.08

0.1

ρ ch

(fm

−3)

G2Helm

22O

28Si

34Ca

30S

1s1/2

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

1s1/2

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

1s1/2

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

1s1/2

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

1s1/2

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

1s1/2

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

1s1/2

1p3/2

1p1/2

1d5/2

2s1/2

1d3/2

1f7/2

-40

-30

-20

-10

0

10

ε nlj

(MeV

)

Z=20

22O

28Si 30

S34

Ca

Z=20 Z=20

Z=2

Z=20 Z=20 Z=20Z=20

Z=8 Z=8 Z=8Z=8 Z=8 Z=8 Z=8

Z=2 Z=2 Z=2 Z=2 Z=2Z=2

24Ne

26Mg

32Ar

Protons (b)