Spectroscopic factors of the closed shell

nuclei in the source term approach

N. K. Timofeyuk

University of Surrey

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 = 1 ( 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

SF of closed shell nuclei measured from (e,ep) reactions:(taken from compillation of Kramer et al NPA 679 (2001) 267)

A A-1 lj SIPM Sexp Sexp/SIPM

4He 3H s1/2 2 0.7-0.8 16O 15N p1/2 2 1.27(13) 0.64(5) p3/2 4 2.25(22) 0.56(11)40Ca 39K d3/2 4 2.58(19) 0.65(5) s1/2 2 1.03(7) 0.52(4)48Ca 47K s1/2 2 1.07(7) 0.54(4) d3/2 4 2.26(16) 0.57(4) d5/2 6 0.683(49) 0.11(1)208Pb 207Tl s1/2 2 0.98(9) 0.48(5) d3/2 4 2.31(22) 0.58(6) h11/2 12 6.85(68) 0.57(6) d5/2 6 2.93(28) 0.49(5) g7/2 8 2.06(20) 0.26(3)

Reduction of spectroscopic strength from knockout reactions A. Gade et al, Phys. Rev. C 77, 044306 (2008)

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)

Two ways of calculation of Ilj(r) for B = A-1:

1) Traditional way, direct evaluation (direct overlap (DO))

2) To solve the inhomogeneous equation (IE)

1

21

1/ 2

( ) ( ) ( )

ˆˆ( )A A

A

Al lj lj

lj l J Jj J

Z eT I r v r

r

v r Y r

V

source term

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:

1 1

1

1 1

ˆA A

i A A ANN i A

i Ai i

e e Z e eV r r

r r r

V= =

For the two-body NN potential the M3YE potential is used fromBertsch 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

A = 2

Veff(r) SFM3YE 0.91

Realistic SF:0.94 for AV18

SIPM SSTA Sab-initio

3H 1.5 1.21 1.304He 2.0 1.29 1.50

experiment: Shell model Reduction factor (e,e’p) 1.27 ± 0.13 0ħ (non TI) 2.0 0.64 ± 0.07p knockout 1.12 ± 0.07 0ħ (TI) 2.13(p,d) 1.48 ± 0.16 4ħ (non TI) 1.65

SSTA = 1.52

A A-1 SSM SSTA

7Li 6He 0.69 0.28 7Li 6Li 0.87 0.44 8Li 7He 1.02 0.38 8Li 7Li 1.14 0.66 8B 7Be 1.14 0.78 9Li 8Li 1.04 0.609Be 8Li 1.13 0.45 9C 8B 1.04 0.7110Be 9Li 1.93 0.8110Be 9Be 2.67 1.48 12B 11B 0.99 0.9712C 11B 2.85 1.5513C 12C 0.63 0.6314C 13C 1.87 1.8214N 13N 0.72 0.6015N 14N 1.48 1.3116O 15N 2.13 1.52

Sexp experiment

0.42(4) (e,e’p)0.74(11) (d,t)0.36(7) (d,3He)

0.89(7) p knockout0.59(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

0.420.680.580.970.971.140.731.141.041.93

Prelim

inary

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-1 lj SIPM Sexp SSTA

4He 3H s1/2 2.0 1.4-1.6 1.29 16O 15N p1/2 2.0 1.27(13) 1.52 p3/2 4.0 2.25(22) 2.60 40Ca 39K d3/2 4.0 2.58(19) 3.01 s1/2 2.0 1.03(7) 1.15 48Ca 47K s1/2 2.0 1.07(7) 1.20 d3/2 4.0 2.26(16) 2.35 d5/2 6.0 0.683(49) 3.61 208Pb 207Tl s1/2 2.0 0.98(9) 0.92 d3/2 4.0 2.31(22) 1.76 d5/2 6.0 2.93(28) 2.71

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, 152501 (2009))

Sexp = 1.74 0.19 for s1/2 removal

SIPM = 2.0 (or S = 2.18 with centre-of mass removal)

SSM(SDPF-M) = 1.769; SSM(USDB) = 1.810

Source term approach with oscillator IPM wave functions for 24O and 23O gives

SSTA = 1.64

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, 024304 (2006))

SIPM = 1.0

Sexp = 0.58 0.11 for p1/2 removal

Source term approach with oscillator IPM wave functions for 57Ni and 56Ni gives

SSTA = 0.62

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

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

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

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,

1d3/2, 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

Final nucleus J Ex (MeV) SSTA/SIPM

131Sn 3/2+ g.s. 0.83 1/2+ 0.332 0.83 5/2+ 1.655 0.82

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 ifNN 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, 242501 (2009)N.K. Timofeyuk, Phys. Rev. C 81, 064306 (2010)

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,

1d3/2, 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

Final nucleus J Ex (MeV) SSTA/SIPM

131Sn 3/2+ g.s. 0.83 1/2+ 0.332 0.83 5/2+ 1.655 0.82

131In 9/2+ g.s. 1/2 0.302 3/2 1.290