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)