1. ２０００ Neutron Distribution observed with Electron Scattering
Mar , 2019, ELPH
Neutron Distribution observed with Electron Scattering
ー　through electromagnetic interaction
Haruki Kurasawa , Chiba Univ.,
and Toshio Suzuki , ELPH , Tohoku Univ.
２０００

2. Purpose
( In addition to Change of the Proton Distribution due to Excess Neutrons )
The possibility to observe neutron distribution in electron scattering
and its implication for the detailed study of the nuclear surface structure
Key Points
１． Charge distribution of neutrons
is not observable,
at q ＝　0 2
and 　　　in < r > c , mean square charge radius of nuclei,
but is observable,
at q　＝　0 4
and 　　　in < r > c　, ・・・・・・・・
２． Relativistic corrections due to the magnetic density of neutrons
Anomalous magnetic moment μ = n
Charged neutron spin-density
Spin-orbit density

3. e’ γ e Relativistic Description of Charge Density F （ｑ ） F （ｑ ）
2000 Kurasawa and Suzuki
for Charge Radius and
Phase Shift Analyses e’ N’
virtual 2
F 　（ｑ　） １ μ
Dirac form factor γ 2
F 　（ｑ　） 2 μ
Pauli form factor
Bjorken-Drell e N μ ２
/４M τ
Sachs form factor μ τ
R. G. Sachs, Phys. Rev. 126, 2256 (1962)
Point nucleon density
Spin-orbit density
(relativistic corrections)

4. { Point nucleon density Spin-Orbit Density (relativistic corrections)
Effective mass　M* = M – V (r) ～０．６ M s { Z N
＝ n.
= 0
(Non-relativistic ?)

5. r-x r x Charge Density Nucleon form factor from experiment (<2000)
　＝　 0
for q 　＝　 0
T. Eden et al., Phys. Rev. C 50, 1749(1994); M. Meyerhoff et al., Phys. Lett. B 327, 201 (1994); S. Platchkov et al., Nucl. Phys. A510, 740 (1990).

6. 1. Relativistic expression of the charge radius2
independent of <r > n
charge radius 2
depends on <r > n k
k-2
k-4
<r > c ～
<r >n ,
<r >n ,
・・・・・・

7. + 3/4M Comment on D-F correction ? Double counting ? consistency ? 2
<r > c
J. L. Friar etal., P. R. A 56, (1997)
R. Sanchez et al. P.R.L. 96, (2006)
A.Ong et al., P. R. C 82, (2010)
(S. Abrahamyan et al. P.R.L. 108, (2012))
D-F correction ?
Double counting ?
consistency ?
relativistic 2
+ 3/4M
+0.033 =
-0.090
ours
NL3
0.026
-0.157 =

8. Self-consistent Non-rel. four-current
Nishizaki, Kurasawa and Suzuki, Phys. Lett. B209(1988) 6
should satisfy the continuity equation 　[ H , ρ ] = q ・ J　.
κ　anomalous magnetic moment
Four-component
Two-component ： Foldy -Wouthuysen-Tani transformation in terms of 2
Up to 1/(M*)
Darwin-Foldy term ＝Sachs　
Spin-Orbit term
Example
Not so good approx.
See also Nishizaki, Kurasawa and Suzuki, Phys. Lett. B171(1) (1986) 1

9. ? ? 2 Bjorken-Drell Relativistic expression ( PWBA (m / E ) ＜ 1) e
W. BERTOZZI, J. FRIAR, J. HEISENBERG and J.W. NEGELE、P. L.41B, 408(1972)
G. DoDang and N. V. Giai, P.R.C30, 731(1984) 2
To the order we are working ( l/m ),
Candra et al, P.R.C13, 245 (1976) ? 2
Bjorken-Drell Relativistic　expression ( PWBA (m / E ) ＜ 1) e ＜
J. L. Friar et al., P. R. A 56, (1997)
Darwin-Foldy term ? ?
Darwin-Foldy correction for Zitterbewegung of the proton
A.Ong et al., P. R. C 82, (2010)

10. Ca48 is the best example as neutron-rich nuclei.%
6.87
13.0
We note k k
< r > ＝ < r >
But each contribution is model-dependent. c p 4
Loosely bound neutrons contribute to charge < r > - moment appreciably.
Ca70　　　< r > ≒< r > 　　R2p ≒ － (R2n +R4wn) 4 4 c p
～8.2％+4.8％=13％

11. Sly4 is better than NL3 ??? （ ) Exp 2.16% 3.80 % (6.39%) 1.75% (2.81%)
（ )
2.16%
3.80 % (6.39%)
1.75% (2.81%)
Li11　 = % (Rp =2.213, Rn=2.936)
(Exp. Rp =2.467
P.R.L. 96,033002(2006)
O = %
O = %
Sn = %
2.10%
3.43%
1.66%
Exp
Ca40
Frosch（Fermi)
Ca48
FBA 　by　Suda
Pb208
Sly4 is better than NL3 ???

12. <r > ・・・・・・・・・・ are observed.
PWBA
full
<r > ・・・・・・・・・・ are observed.　 6 c

13. １９６８ 2. Phase Shift Analyses with Non-rel. and Rel. Models Experiment
Date: Fri, 12 May :19:
2. Phase Shift Analyses with Non-rel. and Rel. Models
１９６８
Experiment
50 years ago
E = 249:5MeV
40Ca Δc = 0:00 Δw = 0:00 Δz = −0:00, ⟨r2⟩ = 3:481
48Ca Δc = 0:00 Δw = 0:00 Δz = −0:00, ⟨r2⟩ = 3:476
Ca40
Ca48　 2 c ｚ ｗ
√<　r 　>
Ca40
3.6758
0.5851
3.4869
Ca48
3.7444
0.5255
-0.03
3.4762
Δ rms ( Ca40 – Ca48) =
（ｆｍ）

14. Ca40 － Ca4８ Difference Frosch et al. Δ rms ( Ca40 –Ca48 ) = + 0.0107=
（ｆｍ） 2
r ×Δ（　ρ（Ca40 ）　– ρ（ Ca４８）　）
D(θ)＝
Fermi

15. １９７２ ρ only ! ｐ full 2 I/M Exp. up to 1/M Inconsistent?
See also Candra et al,
P.R.C13, 245 (1976) 2
I/M Exp. up to 1/M ρ
only !
Inconsistent? ｐ
full
J.W. Negele, Phys. Rev. Cl (1970) 1260
Density –dependent HF

16. + 30 years ２００３ １９７２ 10 parameters Density Functional Theory ？
SK1
6 parameters
+ 30 years
２００３
D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626
10 parameters
Density　Functional Theory ？
　Phenomenological Model　!？

17. １９９８
SLｙ４
10 parameters ?

18. ? Frosch etal. Charge radius
H. Kurasawa and T. Suzuki, P.R.C62, (2000)
Phase Shift Analysis for Relativistic and Non-relativistic Charge Density
with four-component description of the electron ( m ≠ 0 )
D. R. Yennie, D. G. Ravenhall, and R. N. Wilson, Phys. Rev. 95, 500 (1954)

19. １９９８ １９７２ SLｙ４ 10 parameters SK1 rms p 3.49 40Ca , 3.51 48Ca
Exp (Fermi)
１９７２
SK1
6 parameters
D. Vautherin and D.M. Brink, Phys. Rev. C 3 (1972) 626
Exp. rms c Ca , Ca
rms p Ca , Ca

20. １９６８ Bohr and Mottelson Vol.1 Woods-Saxon <r > fm (√<r > )
Exp (Fermi)
<r > fm (√<r > ) 2 2 2
Δ 　 　 (+0.042)

21. １９７６ Relativistic Nuclear Model F only ! full1p
full
Relativistic Nuclear Model
the theoretical justifications for the inclusion of these effects are impeccable !

22. １９９７ Relativistic DFT ？ ？ Incompressibility ? NL3
Nuclear Physics A368 (1981)
C.J. HOROWITZ" and BRIAN D. SEROT
１９９７
parameters 4+3
Relativistic DFT ？
NL3
PHYS. REV. C 55
540
New parametrization for relativistic mean field theory
Phenomenological Model　!？
G. A. Lalazissis, J. Konig, and P. Ring
parameters 6+3
Charge radius？
proton radius ? ？
Incompressibility ?

23. ２０００ NL3 proton radius ? <r > fm (√<r > ) ρ ～ W E=249.5MeV
2000 Kurasawa and Suzuki
NL3
２０００
E=249.5MeV
Exp. (Fermi)
proton radius ?
<r > fm (√<r > ) 2 2 2
Ring
3.469
3.470
Δ　 　 　 　
ρ　　～　W ｃｎ ｃｎ

24. <r > fm (√<r > )
NL-SH
Date: Mon, 29 May :59:
E=249.5MeV
Exp. (Fermi)
Ruta
<r > fm (√<r > ) 2 2 2
Δ 　 　 　
Δp (<0) changes the sign Δtotal (>0).

25. E=499.5MeV
NL3
NL-SH

26. <r > fm (√<r > )
K=211MeV
Exp.(Fermi)
E=249.5MeV
<r > fm (√<r > ) 2 2 2
Δ　 　 　 　

27. <r > fm (√<r > )2 2 2
Δ　 　 　 　
NL-SH
<r > fm (√<r > ) 2 2 2
Δ 　 　 　
What is the difference?
Δp (<0) changes the sign Δtotal (>0).
D(θ) is sensitive to the difference.

28. P.R.L .95, (2005)
B. G. Todd-Rutel and J. Piekarewicz
‘‘FSUGold’’
K 230 MeV
No-sea RPA！？
Phys. Rev. C 68, 　!
D. Vretenar,
T. Niksˇic´ and
P. Ring
K=250 MeV represents the lower limit

29. Serious Problems remain in Relativistic Models.
(No-Free term Approx.)

30. ２００1 No-sea RPA Incompressibility K ～ 270 MeV ? Example 1
Z.Ma, N. Van Giai, A. Wandelt, D. Vretenar and P. Ring
Nucl. Phys. A686 (2001) 173 , ・・・・・・・・・・・・・・・・・・・・
Giant Monopole States 0+
Fully Self-consistent ? Relativistic RPA
Fermi Sea
Dirac Sea
No-sea RPA

31. ２０15 ∞
－　B
－　B

32. ２０13 The energy‐weighted sum value is 0 ?? Example 2
Fermi Sea
Dirac Sea
Goto-Inamura Paradox
Schwinger Term by hand
中西　場の量子論　（培風館）ｐ１３６,
　Itzykson –Zuber　Quantum Field Theory （McGrow－Hill)　p530
No Sea RPA yields S = 0, but is not relevant to this famaus paradox !
２０13
The paradox was solved analytically by

33. ｎ ｐ 遷移 ・ ・ ・ A + ( ∞ － B ) － ( ∞ － C ) = 3 ( N － Z ) N > Z Ｐ N
Other Examples ・
Current　Conservation　 ○
Kurasawa and Suzuki, N.P. A445, 685 (1985) y
141 ・
C.M. Motion 　　　　　　　 ×
Dawson and Furnstahl, P. R. C42, 2009(1990)
115
ｎ　　　　ｐ　遷移 ・
N > Z
　 A ( ∞ － B ) － ( ∞ － C )　 = 　3 ( N － Z ) Ｐ N
Ｄｉｒａｃ　Sea N Ｐ
A = 3(N-Z) (1-2v /3) 2
5 ～12％ F

34. The non-rel. Coulomb sum value is Quenched by 20 ～ 30%.
World Data
Meziani et al., Phys. Rev.Lett. 52 (1984)2130
H. Kurasawa and T. Suzuki, N. P. A490 (1988) 571 ; P. T. P. 86 (1991) 773.
Morgenstern and Meziani, Phys. Lett. B515 (2001)269
Renormalized
Hartree＋RPA
Mean Field+RPA
Hartree
Mean Field

35. ω core core γ ＝ Ｍ * < M Ｐｒｏｔｏｎ ΔＥΔｔ > ｈ/２＋ γ 2
F 　（ｑ　）
Ｍ * < M １
Ｐｒｏｔｏｎ ＝ ＋ ＋ ＋
・・・・・・・・・ ω
core
core
ΔＥΔｔ　>　ｈ/２
Kurasawa and Suzuki,　Prog. Theor. Phys. 86(1991)773
Swelling of the Proton size 2
0.209 fm
0.811fm fm
No calculation with renormalization in finite nuclei yet

36. ２０１５ Chiral Effective Field Theory ? Ab initio？
Theoretical uncertainties,
Parameters,
Limited space,
Conversion …….

37. Ab initio model？
State-of-Art EDF Theory ?
Exp.
3.37
How many versions ?
< 5% ?
(D-F correction : double counting !)
experiment
Ab initio

38. p only ? √１２．０４４ ＝ ３．４７０ density Ms =520MeV (surface thickness)
Neutron charge-
Spin-orbit density
density
p only ?
Ms =520MeV
(surface thickness)
525 MeV decreases r from to fm
and increases the binding energy per nucleon
by 0.13 MeV(15.75MeV).
√１２．０４４　＝　３．４７０

39. Horowitz-Serot

40. Summary
Electron scattering yields rich and fundamental information on nuclear structure.
１. Not only proton distribution, but also structure of excess neutrons
in neutron-rich nuclei would be investigated through electron scattering.
1.1 Together with the density profile, its kth-component provides us with k k
k-2
k-4
< r > ＝ < r > +
<r >n ,
<r >n ,
・・・・・・ c p 2 2 2
< r > :
< r > < r > c n
s-o
～　2.5　%
p-density dominates 4 2 4
< r > :
< r > < r > c n
s-o
～　10 %
n-density is not negligible .
1.2 Together with cross sections, their difference between isotopes gives details of
Δρ　　　,　
Δρ　　　, W　　　　
c p
c n
c p, n
2. New data on neutron-rich nuclei would be useful
for detailed investigation with phenomenological nuclear models .
3. Not only proton form factor, but also neutron form factors are necessary
for studying excess neutrons, and also nuclear response.
precise e – d or e-n scattering is required.
Thank you