1. 1 Proton-neutron pairing by G-matrix in the deformed BCS Soongsil University, Korea Eun Ja Ha Myung-Ki Cheoun

2. Motivation Formalism - Deformed Woods-Saxon (MF) - Deformed Bardeen Cooper Schrieffer (DBCS) Results - Ge isotopes Summary 2 Contents

3. 3 MotivationsResultsSummaryFormalism  The proton-neutron (pn) pairing correlations are important in nuclear structure and decay for proton-rich nuclei with N ≈ Z : proton and neutrons occupy identical orbitals and have maximal spatial overlap.  In contrast to the proton-proton(pp) and the neutron-neutron(nn) pairing, the pn-pairing may exist in isoscalar(T=0) and isovector(T=1) pairing. Chen and Goswami, Nucl. Phys. 11, 263(1979)  Both single- and double-beta decay transitions are affected by the pn-pairing. M-K. Cheoun, Nucl. Phys. A561,74(1993), A564,329(1993), Phys.Rev.C53, 695(1996)  We examine isovector (T=1 ) and isocalar (T=0 ) pn-correlations for the ground state of even-even Ge isotopes with mass number A=64~76 within the deformed BCS approach.  Are the pn-pairing correlations restricted only to the vicinity of the N = Z ? Usually, pn-pairing correlations in medium and heavy nuclei were neglected on the ground that two Fermi levels are apart. Motivations

4. 4 MotivationsResultsSummaryFormalism   Does the pn-pairing strength depend on the deformation parameter β 2 ?  We have been calculated the strength distribution of Gamow-Teller of even-even nuclei within the deformed QRPA without pn-pairing correlations. NPA 934(2015)73- 100  The aim of this work is to consider pn-pairing correlations in deformed BCS process.

5. In many-body system, the Hamiltonian can be written as : deformed axially symmetric WS potential where, = single particle state = sign of the angular momentum projection of state = projection of the total angular momentum j on the nuclear symmetry axis  Spherical symmetry is broken.  J is a not good quantum number any more in deformed basis. ` 5 MotivationsResultsSummaryFormalism

6. 6  In experimental side, β 2 can be extracted from E2 transition probability B(E2).  We use the β 2 value calculated from RMF in this work.  How to include the deformation? MotivationsResultsSummaryFormalism Deformed Woods-Saxon potential (cylindrical WS, Damgaard et al 1969) distance function surface function  In our calculation, β 2 value is the input parameter.

7. 7 MotivationsResultsSummaryFormalism  To exploit G-matrix elements, which is calculated on the spherical basis, deformed bases are expanded in terms of the spherical bases.  Deformed single particle state (SPS)

8. 8  Shell evolution in the neutron-rich nuclei. 0f7/2 : 3301/2 0d3/2 : 2001/2 The breaking of magic number comes from the burrowing of f7/2 state below d3/2 state by the deformation. MotivationsFormalismResultsSummary E. Ha and MKC, Phys. Rev. C88(2013)

9.  Deformed BCS 9 MotivationsResultsSummaryFormalism J=0 T=1 j m j -mΩ -Ω J=0,1, 2, 3 ∙∙∙ T=0 J=1, 3, 5, ∙∙∙ T=1 J=0, 2, 4, ∙∙∙ BCS deformed BCS Ω= ½ j ≥ Ω  Since the deformed SPS are expanded in terms of the spherical SP bases the different total angular momenta of the SP basis states would be mixed. K=0 K J Laboratory frame Intrinsic frame

10. 10 Motivations Results SummaryFormalism We obtain the following Deformed HFB (DHFB) equation :  The pairing potentials ∆ are calculated in the deformed basis by using the G-matrix calculated from the realistic Bonn CD potential for the N-N interaction.  In order to renormalize the G-matrix, strength parameters, g is multiplied to the G-matrix by adjusting the pairing potentials to the empirical pairing potentials. J= even J= odd In spherical formalism

11. 11 Motivations Results SummaryFormalism No coexistence of T=0 and T=1 pn-pairing modes were found. There is very little known about the T=0 and T=1 strength of the pn-pairing. The T=0, 3 S pairing force is expected to be strong in comparison with T=1, 1 S pairing force. A strong evidence of this is that the deuteron and many other double even Z=N nuclei prefer this type of coupling due to the strong tensor force contribution. In our calculation, the pn-pairing strength g repesent T=0 mode.

12. 12 MotivationsResultsSummaryFormalism with pn-pairing without pn-pairing  Empirical pairing gaps The empirical pairing potentials of proton and neutron are evaluated by the following symmetric five term formula for neighboring nuclei. p-n pairing interaction energy δ Z ≈ N An attractive short-range residual interaction between one unpaired proton and neutron is considered to be the origin of the proton-neutron interaction energy within macroscopic pairing models.

13. 13 MotivationsSummary  The values of p-n interaction energies δ pn emp are not negligible even for large neutron excess isotopes.  The pn-pairing interaction is expecting to play a significant role in construction of the quasiparticle mean field for these nuclei.  It is supposed that the origin of this phenomenon is associated with the deformation effect, which is changing the distribution of proton and neutron SP levels. ResultsFormalism  Empirical pairing gaps for 64 Ge ~ 76 Ge

14. 14  Pairing gap for 64 Ge (Z=N) MotivationsSummaryFormalismResults  β 2 =0.217 (RMF),  In (a), below some critical value (~0.97), there are only pp & nn-pairing modes, which seems to be a result of a simple monopole pair(K=0) Hamiltonian.  Above this value, the system prefers to form only pn-pair.  pp, nn, and pn-pairs coexist in the narrow region(blue square).  In (b), the coexistence region is more wide and the phase transition becomes less sharp. Preliminary

15. 15  Pairing gap for 70 Ge (Z ǂ N) MotivationsSummaryFormalismResults  β 2 = - 0.261 (RMF),  There is a less sharp phase transition to the pn-pairing in comparison with 64 Ge.  pn-pairing mode does exist only in coexistence with pp & nn-pairing mode. Preliminary

16. 16  Pairing strengths for 64 Ge ~ 76 Ge MotivationsSummaryFormalismResults  The T=0 p-n force (g pn ) is larger in comparison with T=1 pp and nn (g pp and g nn ) forces for Ge isotope in both (a) and (b).  The largest differences among g pp (T=1), g nn (T=1), and g pn (T=0) forces are visible for maximal value of N−Z=12 ( 76 Ge), which undergoes double β-decay.  Double-beta decay transitions might be affected by the pn-pairing. Preliminary

17. 17 MotivationsSummary Formalism Results  Does the g pn depend on the deformation parameter β 2 ?  g pn (T=0 ) is sensitive to the change of the deformation parameter β 2.  In (b), g pn is symmetric on β 2 =0 for three nuclei. Preliminary

18. 18 MotivationsResultsFormalismSummary  We examined isovector(T=1) and isoscalar(T=0) pn-pairing correlations for the ground state of even-even Ge isotopes, A=64–76, within the deformed BCS approach.  For N=Z 64 Ge pure T=0 pairing mode is found and a sharp phase transition from the pp(nn)-pairing mode to the pn-pairing mode is observed.  For other nuclear systems N>Z a coexistence of T=0 and T=1 pairs in the BCS wave function is observed.  The T=0 pn-pairing correlations should be considered also for medium- heavy nuclei with large neutron excess since the pn-pairing effect is not negligible.  The single- or double-beta decay observables might be influenced by the T=0 pn-pairing.  We are in progress to study the effects of pn-pairing and deformation on beta-decay within a deformed BCS + QRPA approach. Summary

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20. 20 They used the constant and as a pairing strength. In principle, these strength have to be evaluated from the scattering of two particles in deformed mean field. Therefore, we will exploit the Brueckner reaction matrix G, which is obtained by solving the Bethe-Goldstone equation a,b,c,d : single nucleon basis states(oscillator wave functions with s.p.e from a Woods-Saxon potential) w : starting energy V ab,cd : one boson exchange potential of the Bonn group Q p : Pauli operator H 0 : harmonic oscillator Hamiltonian G-matrix

21. 21 MotivationsResultsSummaryFormalism The HFB transformation for each α state has the following simple form : Particle creation (annihilation) operator Quasiparticle creation (annihilation) operator u and v are the occupation amplitudes Here, u 1p,v 1p,u 2n,v 2n are real and u 1n,v 1n,u 2p,v 2p are complex. If there is no p-n pairing (u 1n,v 1n,u 2p,v 2p =0), this transformation reduces to two conventional BCS two-dimensional transformation :