1. N. K. Timofeyuk University of Surrey
Spectroscopic factors of the closed shell nuclei in the source term approach
N. K. Timofeyuk
University of Surrey

2. I(r) = A|A-1 ( = A-1/2 nlj(r) in the IPM )
Closed shell nuclei: all single-particle orbits are fully occupied.
Spectroscopic factors of closed shell nuclei
in the independent particle model (IPM):
For A|A-1: Slj = 2j+1 ( times (A/(A-1))2n+l, if centre of mass motion is excluded)
For A|A+1: Slj = ( times ((A+1)/A)2n+l, if centre of mass motion is excluded)
Reminder:
I(r) = A|A-1 ( = A-1/2 nlj(r) in the IPM )
I(r) =  (lm 1/2| j mj) (JA-1 MA-1 j mj | JA MA) Ilj(r) Ylm(ȓ) 1/2
Slj is equal to the number of nucleons in the shell nlj A
m mj MA-1

3. SF of closed shell nuclei measured from (e,ep) reactions:
(taken from compillation of Kramer et al NPA 679 (2001) 267)
A A lj SIPM Sexp Sexp/SIPM
4He H s1/
16O N p1/ (13) (5)
p3/ (22) (11)
40Ca 39K d3/ (19) (5)
s1/ (7) (4)
48Ca 47K s1/ (7) (4)
d3/ (16) (4)
d5/ (49) (1)
208Pb 207Tl s1/ (9) (5)
d3/ (22) (6)
h11/ (68) (6)
d5/ (28) (5)
g7/ (20) (3)

4. Reduction of spectroscopic strength from knockout reactions
A. Gade et al, Phys. Rev. C 77, (2008)

5. Source Term Approach (STA)
Contradiction:
Knockout experiments seem to indicate that single-particle orbits in closed shell nuclei are only half filled!!!  They are not closed shell nuclei!!!
Systematics of binding energies and other observables indicates that closed-shell nuclei exist.
A possible way to resolve this contradictions is
Source Term Approach (STA)

6. Two ways of calculation of Ilj(r) for B = A-1:
Traditional way, direct evaluation (direct overlap (DO))
2) To solve the inhomogeneous equation (IE)
source term

7. Source term approach:
Model wave functions A and B are taken from the 0ħ oscillator shell model,
(which for closed shell model are the same as in the IPM)
Interaction V:
For the two-body NN potential the M3YE potential is used from
Bertsch et al, Nucl. Phys. A 284 (1977) 399
VST= V1,ST exp(–a1,STr)/r +V2,ST exp(–a2,STr)/r+V3,STexp(–a3,STr)/r + spin-orbit+tensor…
Coefficients Vi,ST and ai,ST have been found by fitting the matrix elements derived in Brighton from NN elastic scattering data

9. A = 2
Veff(r) SF
M3YE
Realistic SF:
0.94 for AV18

10. SIPM SSTA Sab-initio 3H
4He

11. SSTA = 1.52 experiment: Shell model Reduction factor
(e,e’p) ± ħ (non TI) ± 0.07
p knockout ± ħ (TI)
(p,d) ± ħ (non TI)

12. A A SSM SSTA
7Li He
7Li Li
8Li He
8Li Li
8B Be
9Li Li
9Be 8Li
9C B
10Be 9Li
10Be 9Be
12B B
12C B
13C C
14C C
14N N
15N N
16O N
Sexp experiment
0.42(4) (e,e’p)
0.74(11) (d,t)
0.36(7) (d,3He)
0.89(7) p knockout
0.59(15) (d,t)
0.77(6) p knockout
0.40(6) (d,p)
1.72(11) (e,e’p)
0.54(8) (d,p)
1.07(22) (d,p)
0.48(8) (p,d)
0.93(15) (d,p)
1.27(13) (e,e’p)
SVMC
0.42
0.68
0.58
0.97
1.14
0.73
1.04
1.93

13. SF of double-closed shell nuclei obtained from STA calculations:
Oscillator IPM wave functions are used with ħ = 41A-1/3 - 25A-2/3
and the M3YE NN potential
A A lj SIPM Sexp SSTA
4He H s1/
16O N p1/ (13)
p3/ (22)
40Ca 39K d3/ (19)
s1/ (7)
48Ca 47K s1/ (7)
d3/ (16)
d5/ (49)
208Pb 207Tl s1/ (9)
d3/ (22)
d5/ (28)
Preliminary

14. Preliminary Shell closure away from beta-stability
New magic nucleus: 24O (C.R.Hoffman et al, Phys.Lett. B 672, 17 (2009))
Neutrons occupy shells: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2
Protons occupy shells: 0s1/2, 0p3/2, 0p1/2
One-Neutron Removal Measurement 12C(24O, 23O), E=920 MeV/A
(R.Kanungo et al, Phys.Rev.Lett. 102, (2009))
Sexp = 1.74  for s1/2 removal
SIPM = (or S = 2.18 with centre-of mass removal)
SSM(SDPF-M) = 1.769; SSM(USDB) =
Source term approach with oscillator IPM wave functions for 24O and 23O gives
SSTA = 1.64
Preliminary

15. Preliminary Double magic N=Z nucleus: 56Ni
Fully occupied shells: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d 3/2, 0f7/2
57Ni has one valence neutron above double closed shell core 56Ni
One-Neutron Removal Measurement 9Be(57Ni,56Ni+γ )X
(K. L. Yurkewicz et al, Phys.Rev. C 74, (2006))
SIPM = 1.0
Sexp = 0.58  for p1/2 removal
Source term approach with oscillator IPM wave functions for 57Ni and 56Ni gives
SSTA = 0.62
Preliminary

16. Preliminary Double magic 60Ca? Fully occupied shells:
Protons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d 3/2
Neutrons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2
Preliminary

17. Preliminary Double magic 78Ni? Fully occupied shells:
Protons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d 3/2, 0f7/2
Neutrons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g 9/2
Preliminary

18. Preliminary Double magic 100Sn
Fully occupied shells: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g 9/2
Proton knockout Neutron knockout
Preliminary

19. Preliminary Double magic 132Sn Fully occupied shells:
Neutrons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g9/2, 0g7/2, 1d5/2, d3/2, 2s1/2 , 0h11/2
Protons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g 9/2
Preliminary
Final nucleus J Ex (MeV) SSTA/SIPM
131Sn / g.s 1/ 5/

20. Implications for the meaning of spectroscopic factors:
Conclusions:
STA can reconcile reduction of spectroscopic strength in double closed shell nuclei with double magic nature of these nuclei.
STA employs IPM wave function but gets reduced spectroscopic factors if
NN interaction is chosen correctly.
Implications for the meaning of spectroscopic factors:
SFs are the measure of strength of the interaction of the removed nucleon rather than the measure of the shell occupancies.
Publications:
N.K. Timofeyuk, Phys. Rev. Lett. 103, (2009)
N.K. Timofeyuk, Phys. Rev. C 81, (2010)

21. Preliminary Double magic 132Sn Fully occupied shells:
Neutrons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g9/2, 0g7/2, 1d5/2, d3/2, 2s1/2 , 0h11/2
Protons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g 9/2
Preliminary
Final nucleus J Ex (MeV) SSTA/SIPM
131Sn / g.s 1/ 5/
131In / g.s.
1/2
3/2