Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 47 (2014)

Competition between T =1 and T =0 pairing in pf shell nuclei withN = Z†
H. Sagawa,∗1,∗2 Y. Tanimura,∗3 and K. Hagino∗3−0.2

The role of the neutron-proton isoscalar spin-triplet(T =0, S=1) pairing interaction in ﬁnite nuclei has
been a topic of discussion for long.1–3) The isoscalar
spin-triplet pairing interaction is known to be stronger
than the isovector spin-singlet (T =1, S=0) one innuclear matter.4) Nevertheless, nuclei favor the spinsinglet T =1 pairing between identical particles. A
straightforward explanation for this contradiction isthat most stable nuclei have diﬀerent numbers of neutrons and protons; thus, protons and neutrons occupy
diﬀerent single-particle orbits near the Fermi surface,which leads to the inhibition of T =0 pairing. It wasalso suggested that the nuclear spin-orbit ﬁeld largely
suppresses the spin-triplet pairing, much more than
the spin-singlet pairing.5,6)
To clarify the role of T = 0 pairing, we diagonalize the Hamiltonian with the spin-singlet and spintriplet pairing terms in pf shell model conﬁgurations
for nuclei with the same number of protons and neutrons, N = Z. The pairing correlation energies ofthe (J π = 0+ , T =1) and (J = 1+ , T =0) states areshown in Fig. 1 as a function of the scaling factor
f for the T = 0 pairing. The lowest energy statewith J π =0+ for the l = 3 case acquires more binding
energy than the J π =1+ state for the strength factor
f < 1.5. In the case of strong T =0 pairing, that is,f ≥ 1.6, the J π =1+ state acquires more binding energy than the lowest J π =0+ state. These results arelargely attributed to the quenching of the T =0 pairing matrix element by the transformation coeﬃcient
corrsponding to a change of the scheme from the jj
coupling to LS coupling. This quenching never happens for the T =1 pairing matrix element, since themapping of the two-particle wave function between thetwo √coupling schemes is simply implemented by a factor j + 1/2. For the l = 1 case, there is a competition between the J π =0+ and the J π =1+ states as
seen in Fig. 1. Because of smaller spin-orbit splittingin this case, the couplings among the available conﬁgurations are rather strong, and the lowest J π =1+ state
acquires more binding energy than the J π =0+ state
when f ≥ 1.4. These results are consistent with thespins observed for N = Z odd-odd nuclei in the pf
shell, where all the ground states have the spin-parity
58
J π = 0+ , except for 58
29 Cu. The ground state of 29 Cuπ+has J = 1 , because the odd proton and odd neutron†∗1∗2∗3Pairing Correlation Energy (MeV)

−0.4
−0.6
−0.8
−1−1.2−1.4−1.6−1.8−20.8

l=1 with T=0
l=3 with T=0
l=1 with T=1
l=3 with T=11

1.2

T=0f=G

T=1

1.4

1.6

1.8/GFig. 1. (Color online) Pairing correlation energies for thelowest (J π = 0+ , T =1) and (J = 1+ , T =0) states withthe l = 3 and l = 1 configurations as a function of thescaling factor f of the T = 0 pairing. The strengthof the spin-singlet T =1 pairing interaction is fixed atG(T =1) =24/A MeV with mass A=56, while the strength
for the spin-triplet T =0 pairing interaction, G(T =0) , isvaried with the factor f multiplied by G(T =1) .

occupy mainly the 2p orbits, wherein the spin-orbit
splitting is expected to be much smaller than in 1forbits.
In summary, by diagonalizing the pairing Hamiltonian, we have shown that the spin-triplet pairing correlation energy in the 1f shell conﬁguration becomeslarger than the spin-singlet pairing energy when thestrength of the spin-triplet pairing is larger than thatof the spin-singlet pairing by a factor of 1.6 or more.
However, for the 2p conﬁguration, the spin-triplet pairing correlation becomes dominant even when the factor
f is approximately 1.4.
References
1) A. L. Goodman, Nucl. Phys. A186, 475(1972); Phys.
Rev. C60, 014311(1999).
2) A. O. Macchiavelli et al., Phys. Rev. C61, 041303(R)(2000);
A. O. Macchiavelli et al., Phys. Lett. B480, 1(2000).
3) A. F. Lisetskiy et al., Phys. Rev. C68, 034316 (2003).
4) M. Baldo, U. Lombardo, and P. Schuck, Phys. Rev.
C52, 975(1995); L. G. Cao, U. Lombardo, and P.
Schuck, Phys. Rev. C74,064301(2006).
5) A. Poves and G. Martinez-Pinedo, Phys. Lett. B430,203(1998).
6) G. F. Bertsch and Y. Luo, Phys. Rev. C81,064320(2010).

Condensed from the article in Phys. Rev. C 87, 034310(2013)RIKEN Nishina Center
Center for Mathematics and Physics, the University of Aizu
Department of Physics, Tohoku University- 54 -