1. Magnetic dipole excitation and its sum rule for valence nucleon pair
Tomohiro Oishi
Dept. of Phys., Faculty of Science, Univ. of Zagreb, Croatia.
International Nuclear Physics Conference (INPC) Glasgow, Scotland UK, July 29th –August 2nd, 2019.
I would like to thank all the organizers and participants to have this workshop.
It is my great honor to have this chance to talk about our works.

2. Introduction (i) Background Our aim in this work
Magnetic-dipole (M1) excitation takes place in open-shell nuclei. In open-shell nuclei, the nuclear pairing correlation is expected to play a role.
From the form of operator, this mode may provide the information on the coupled spin in medium.
Our aim in this work
To investigate the role of pairing correlation on M1.
To introduce one case of “M1 sum rule” (its general version has not been established yet…), and its applicability to evaluate the pairing effect.

3. Introduction (ii) Matrix elements of M1 (up to the 1-body operator):
Thus, M1 transition happens mainly between the LS-split levels, e.g. f7/2  f5/2 for Ca isotopes but except 40Ca.

4. Three-body model (i) 𝑥 1 𝑟 1 𝑟 2 𝑥 2 𝑅 𝐶𝑀 𝑥 𝐶 𝑂 𝑂
We limit our targets to “shell-closed even-even core + 2N” systems in this work: e.g. 42Ca ≒ 40Ca + n + n.
The system consists of (i) a spherical, shell-closure, 0+ core and (ii) a pair of neutrons or protons.
Core is NOT active for excitation: we only discuss the M1 excitation of two valence nucleons.
 M1 from 0+ ground state (GS) to 1+ excited states is discussed.
General coordinates: V + center-of-mass coordinates: 𝑂
𝑥 2
𝑥 1
𝑥 𝐶 𝑂
𝑟 2
𝑟 1
𝑅 𝐶𝑀

5. Three-body model (ii) 𝑟 1 𝑟 2 Hamiltonian:
𝑟 1
Hamiltonian:
VC (Woods-Saxon)  energy levels of the core-nucleon subsystem.
vNN  two-nucleon separation energy.
schematic-density-dependent contact interaction,
(2) Minnesota (finite-range) interaction,
𝑟 2

6. Three-body ver. of M1 sum rule
Only 2 nucleons can be active:
When 2n are coupled to JP=0+ at the initial state, possible combinations of (L12, S12) = (1,1) or (0,0) only. Thus,
→ Sum-rule value (SRV) of BM1(Ef) is related with the spin-triplet ratio in the initial state.
+ Spin-triplet ratio depends on the pairing strength.
→ M1 SRV can be used to evaluate the pairing in medium.
Note: we discuss M1 up to the one-body-operator level in this work.
Spin-triplet ratio

7. Results for 42Ca BM1(Eγ; 0+ → 1+), Eγ= Ef − EGS
Note: EGS= -S2n= MeV in experimental data.
Numerical SRV:
Sum-rule value of BM1(Eγ) is consistent to the S12=1 ratio in the GS.
BM1(Eγ) is sensitive to the pairing-interaction models, even when those are equivalently fitted to the GS energy.
 Low-lying M1 transition can provide a good reference to validate and optimize the pairing models.

8. Results for sd-shell nuclei
16O+2n or 2p
18O, EGS(Exp.)= MeV Ne, EGS(Exp.)= MeV

9. Relativistic Meanfield Cal.
Refs:
[1] D. Vretner et al., Phys. Rep. 409, (2005);
[2] N. Paar and P. Ring, Phys. Rev. C 67, (2003).
Point-coupling Lagrangian:
QRPA:
M1 occurs between LS-split levels.
Relativistic Meanfield (RMF) concludes the LS splitting naturally. 　↓
QRPA based on the RMF (Hartree+Bogoliubov) is the suitable method for the systematic M1 calculation.

10. Result with RMF Setting: System = 42Ca EDF = DDPC1 (+ Pair.)
20 HO Shells
Spherical
Result:
M1 sumrule value (SRV)
= gs2 without pairing;
0.634 gs2 with pairing.
Note that SRV = gs2 (analytic) for no-pairing case.
BM1(E) value is consistent to the three-body model’s one.
Excitation energy depends on models.
Preliminary
VS Three-body model result:

11. Conclusions Future works
M1 transitions in C+2N systems are evaluated. The corresponding, case-limited version of sum rule is introduced.
Three-body model calculation shows the sensitivity of M1 to the pairing model.  M1 can be a good (i) probe for the pairing correlation in medium, and (ii) reference to optimize its model.
Relativistic HB + QRPA result is consistent to the three-body model’s one, except the excitation energy.
More systematic calculation with the RHB+QRPA, for other nuclei, where the experimental data are available.
Careful study on the pairing models.
Optimization of (relativistic) EDF parameters.
Meson-exchange-current effect, for which we need to take multi-body-terms of operator/interaction into account.
Future works

12. Funding
This work was supported by the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Program (Grant KK ).
For more information please visit:
The sole responsibility for the content of this presentation lies with the Faculty of Science, University of Zagreb. It does not necessarily reflect the opinion of the European Union.
I would like to thank all the organizers and participants to have this workshop.
It is my great honor to have this chance to talk about our works.
Co-financed by the European Union through the Eurpoean Regional Development Fund

13. Backup

15. 𝑟 2
𝑟 1

16. IS-IV decomposition: