1. Isovector and isoscalar pairing in low-lying states of N = Z nuclei
GuanJian Fu
Tongji University
Jun. 5, “Advances in theoretical nuclear physics”, Stockholm

2. Nucleon pair: isospin singlet / triplet
A proton and a neutron are the same kind of nucleons, with different z-components of isospin
T = 1/2, Tz = +1/2 for
= -1/2 for
Nuclear force is charge independent approximately The Coulomb interaction doesn’t play an important role in low-lying states
Isospin
conservation
singlet states and triplet states in isospin space, we have
T = 1, Tz = 1
T = 1, Tz = -1
T = 1, Tz = 0
T = 0, Tz = 0
isospin triplet states
isovector
isospin singlet state
isoscalar

3. Interactions with T = 1 attractive and short ranged: 𝛿 interaction
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Interactions with T = 1
attractive and short ranged: 𝛿 interaction
𝑗𝑗𝐽𝑇 𝑉 12 (𝛿) 𝑗𝑗𝐽𝑇
R. F. Casten, Nuclear Structure from a Simple Perspective.

4. multi-j shell space + effective interaction
Isovector J = 0 pairing
multi-j shell space + effective interaction
(generalized) seniority scheme:
collective (correlated) J = 0 pair
ground states of even-even semimagic nuclei are well described by S pair wave function!
overlap > 0.92

5. Interactions with T = 0 4 + 2 + 5 + 3 + 7 + 1 + 0 +
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Interactions with T = 0
attractive and short ranged: 𝛿 interaction
42Sc ( j = 7/2 )
0 +
1 +
7 +
3 +
2 +
4 +
5 +
𝑗𝑗𝐽𝑇 𝑉 12 (𝛿) 𝑗𝑗𝐽𝑇
effective int.
The question is whether or not a pairing correlated state or a pair condensation can be formed.
R. F. Casten, Nuclear Structure from a Simple Perspective.
S. Frauendorf and A. O. Macchiavelli, Prog. Part. Nucl. Phys 78, 24

6. Isoscalar spin-aligned pair of the g9/2 orbit

7. Isoscalar spin-aligned pair of the g9/2 orbit
Single g9/2 shell calculation for 96Cd, the spin-aligned pair: pair condensation:
|𝜑 = ( 𝐴 9,0 † × 𝐴 9,0 † ) (𝐼,0) |0
C. Qi, J. Blomqvist, T. Back, B. Cederwall, A. Johnson, R. J. Liotta, and R. Wyss, Phys. Scr. T 150,
P. van Isacker, Int. J. Mod. Phys. E 22,

8. Isoscalar spin-aligned pair of the g9/2 orbit
Single g9/2 shell calculation for 92Pd, the spin-aligned pair: pair condensation basis:
( ( ( 𝐴 9,0 † ×𝐴 9,0 † ) ( 𝐽 2 ,0) ×𝐴 9,0 † ) ( 𝐽 3 ,0) ×𝐴 9,0 † ) (𝐼,0) |0

9. NPA: a pair truncation model for multi-j shells
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NPA: a pair truncation model for multi-j shells
Shell model, the “exact” many-body solution
Nucleon-pair approximation (NPA) with isospin
the pair basis:
关于配对波函数或者基于配对基矢的研究非常少
Calculate NPA matrix elements very complicated
Pair truncation: the building block is (J,T) pairs
Validity: overlap between pair (NPA) wave functions and SM wave functions in many-j shell calculation
G. J. Fu, Y. Lei, Y. M. Zhao, S. Pittel, A. Arima, Phys. Rev. C 87, (2013).
Y. M. Zhao and A. Arima, Phys. Rep. 545, 1 (2014)

10. 96Cd: dual description S pair: J=0,T=1 D pair : J=2,T=1
K pair : J=8,T=1
A(9) pair : J=9,T=0
+ JUN45
Dual description:
Non-orthogonality feature of nucleon-pair basis!
G. J. Fu, J. J. Shen, Y. M. Zhao, and A. Arima, Phys. Rev. C 87, (2013).

11. g.s. of N=Z even-even nuclei
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g.s. of N=Z even-even nuclei
Study N = Z even-even nuclei: many-j shells + effective interaction
20Ne, 24Mg, 32S, 36Ar : USDB
44Ti, 48Cr : GXPF1
60Zn, 64Ge, 92Pd, 96Cd : JUN45
For N = Z nuclei:
G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, (2015)

12. g.s. of N=Z even-even nuclei
04:12
g.s. of N=Z even-even nuclei
Study N = Z even-even nuclei: many-j shells + effective interaction
20Ne, 24Mg, 32S, 36Ar : USDB
44Ti, 48Cr : GXPF1
60Zn, 64Ge, 92Pd, 96Cd : JUN45
For N = Z nuclei:
G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, (2015)

13. g.s. of N=Z even-even nuclei
04:12
g.s. of N=Z even-even nuclei
Study N = Z even-even nuclei: many-j shells + effective interaction
20Ne, 24Mg, 32S, 36Ar : USDB
44Ti, 48Cr : GXPF1
60Zn, 64Ge, 92Pd, 96Cd : JUN45
For N = Z nuclei:
G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, (2015)

14. g.s. of N=Z even-even nuclei
04:12
g.s. of N=Z even-even nuclei
Study N = Z even-even nuclei: many-j shells + effective interaction
20Ne, 24Mg, 32S, 36Ar : USDB
44Ti, 48Cr : GXPF1
60Zn, 64Ge, 92Pd, 96Cd : JUN45
For N = Z nuclei:
G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, (2015)

15. g.s. of N=Z even-even nuclei
04:12
g.s. of N=Z even-even nuclei
Study N = Z even-even nuclei: many-j shells + effective interaction
20Ne, 24Mg, 32S, 36Ar : USDB
44Ti, 48Cr : GXPF1
60Zn, 64Ge, 92Pd, 96Cd : JUN45
For N = Z nuclei:
For semimagic nuclei:
(> 0.92)
G. J. Fu, Y. M. Zhao, and A. Arima, Phys. Rev. C 91, (2015)

16. T = 0 states of N=Z odd-odd nuclei
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T = 0 states of N=Z odd-odd nuclei
Study low-lying T = 0 states of 3 valence protons and 3 valence neutrons: many-j shells + effective interaction
3 isovector pairs cannot form T = 0 states, have to break 1 pair;
isoscalar pair condensation may be possible
22Na, 34Cl: USDB
46V: GXPF1
62Ga, 94Ag: JUN45

17. T = 0 states of N=Z odd-odd nuclei
04:12
T = 0 states of N=Z odd-odd nuclei
Study low-lying T = 0 states of 3 valence protons and 3 valence neutrons: many-j shells + effective interaction
3 isovector pairs cannot form T = 0 states, have to break 1 pair;
isoscalar pair condensation may be possible
22Na, 34Cl: USDB
46V: GXPF1
62Ga, 94Ag: JUN45
Preliminary results:
J = 1, T = 0 pairing
for the 1+ states, overlap is
for the 3+ states, overlap is
J = 3, T = 0 pairing
for the 9+ state of 34Cl and 62Ga, overlap > 0.92
J = 5, T = 0 pairing
for the 15+ state of 46V, overlap > 0.99

18. Thanks for your attentions！
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Thanks for your attentions！

19. Charge independence
W. Benenson and E. Kashy, Rev. Mod. Phys. 51, 527 (1979).

20. Charge independence
D. D. Warner, M. A. Bentley, and P. van Isacker, Nat. Phys. 2, 311 (2006).

21. multi-j shell space + effective interaction
Isovector J = 0 pairing
(generalized) seniority scheme:
collective (correlated) J = 0 pair
ground states of even-even semimagic nuclei are well described by S pair wave function!
multi-j shell space + effective interaction
overlap > 0.92

22. Isovector J = 0 pairing overlap > 0.92
(generalized) seniority scheme:
collective (correlated) J = 0 pair
ground states of even-even semimagic nuclei are well described by S pair wave function!
ground states of odd-mass semimagic nuclei:
overlap > 0.92
V. Zelevinsky and A. Volya, Physics of Atomic Nuclei.