1. Coupling of (deformed) core and weakly bound neutron M. Kimura (Hokkaido Univ.)

2. Introduction We are now able to access to 1. Weakly bound neutron-rich with A ～ 40 2. Heavier unstable nuclei with N ～ 28, 50,… What will we find there? Theoretical predictions by Antisymmetrized Molecular Dynamics

3. Description of deformed core AMD method

4. AMD Framework Variational wave function Variational calculation after parity projection A-body Hamiltonian Gogny D1S effective interaction, Exact removal of spurious c.o.m. motion Single particle wave function is represented by a deformed Gaussian wave packet

5. AMD Framework Initial wave function (randomly generated) Variation (deformed) shells clustered AMD model wave function is flexible to describe various kinds of structure (shells & clusters) without assumption

6. AMD Framework 2. Angular momentum projection 1. Energy variation with the constraint on the Quadrupole deformation  Solve Hill-Wheeler eq. to obtain eigenvalue and eigenfunction 3. GCM Configuration mixing between the states with different deformation and configurations

7. AMD Framework 1. Energy variation with the constraint on the Quadrupole deformation  Single particle energy and wave function Construct single particle Hamiltonian from variational results and diagonalize it. 2. Angular momentum projection3. GCM G. Neyens, PRC84, 064301 (2011) Coexistence of many particle-hole states at very small excitation energy has been predicted by AMD Recent experiments such as p and n-knockout, n-transfer and  -decays revealed corresponding states Coexistence of many particle-hole states with different deformations (shape coexisting phenomena) is now establishing M. Kimura, Phys.Rev. C 75, 041302 (2007)

8. Description of weakly bound neutron AMD+RGM method for Core + n and 2n systems

9. AMD + RGM (core + 1n, 2n system) Solve core + 1n, 2n system (Coupled Channnel Core + n RGM) : Wave function of the core described AMD+GCM method (In the case of the 30 Ne+n system, the core is 30 Ne. is a linear combination of J  projected Slater determinants) : Valence neutron (In the case of the Core+2n system, there are two ) : Coefficient of each channels, and relative wave function between the core and valence neutrons (They are the unknown variables (functions) to be calculated by this method)

10. AMD + RGM (core + 1n, 2n system) In the practical calculation, the RGC wave function is transformed to the GCM wave functions. (straightforward but CPU demanding ) The core is a linear combination of different shapes (AMD+GCM w.f) ++ …= The basis wave functions of AMD+RCM And, we diagonalize total Hamiltonian for Core + n (2n) system

11. AMD + RGM (core + 1n, 2n system): O isotopes AMD Results (Blue Symbols) Correct description of neutron drip-line (Gogny D1S) Underestimation of even-odd staggering (Pairing correlation is not enough?) Underestimation of Sn for 23 O and 24 O (1s orbit) AMD+RGM Results (Green Symbols) Better staggering ( (1s 1/2 ) 2 and (0d 3/2 ) 2 pairs ) Improvement of the last neutron(s) orbital in 23 O and 24 O (1s orbit).

12. AMD Results (Blue Symbols) Overestimation for light isotopes Monotonic increase of radii in the calculation, while 23 O and 24 O show drastic increase in the observation AMD+RGM Results (Green Symbols) Almost no effect for light isotopes (d 5/2 ) dominance Slight increase in 23 O and 24 O (1s 1/2 ). But not enough to explain the observation. AMD + RGM (core + 1n, 2n system): O isotopes

13. Beyond island of inversion Toward neutron-dripline

14. 1n Halo of 31 Ne(N=21) Coulomb breakup, and enhanced B(E1) Observed large cross section can be explained with l= 1, 2 Large Interaction cross section M. Takechi, et. al., Nucl. Phys. A 834, (2010), 412 T. Nakamura, et. al., PRL103, 262501 (2009)

15. Wave function of 30 Ne is AMD w.f., relative motion between 30Ne and n is solved All states below 10MeV of 30 Ne are included as the core wave function of 31 Ne ► AMD result shows particle ( p3/2) + rotor ( 30 Ne(g.s.)) nature ► AMD + RGM tends to weak coupling between 30 Ne and neutron AMD + RGM for 31 Ne AMD + RGM config. 0+ × p 3/256% 2+ × p 3/224% 2+ × f 7/29% 1- × s 1/25% AMD config. 0+ × p 3/237% 2+ × p 3/241% 2+ × f 7/212% 1- × s 1/25% Sn=250 keV → 450keV Talk by Minomo K. Mimono, et al., PRC84, 034602 (2011) K. Mimono, et al., in preparation.

16. “Parity Inversion” and “Neutron-halo” near drip-line – 1n separation energy is around or less than 1MeV – 37 Mg is the heaviest odd mass Magnesium QUESTIONS – Island of inversion is extended in this region ? – Neutron Halos? 35 Mg and 37 Mg

17. 35 Mg (N=23): (fp) 3 config. vs. (fp) 4 (sd) -1 config. 1. neutron single particle level density is very large around 0 energy 2. 0p 3/2 orbit also intrudes due to the high single particle density and increase of fermi energy (larger neutron #) 3. (fp) 3, (fp) 4 (sd) -1 and (fp) 5 (sd) -2 configuration compete ⇒ possible parity inversion

18. 35 Mg (N=23): (fp) 3 config. vs. (fp) 4 (sd) -1 config. (fp) 4 (sd) -1 becomes the ground state and the parity is inverted. Stronger n-n correlation in fp shell than sd Experimental information is not enough A. Gade et al., PRC83, 044305 (2011)

19. 37 Mg (N=25): (fp) 5 vs. (fp) 6 (sd) -1 vs. (sdg) 1 (fp) 6 (sd) -2 1. Further increase of single particle level density. 2. 0g 9/2 orbit also intrudes across N=28 shell gap ! due to larger neutron # and weak binding 3. (fp) 5, (fp) 6 (sd) -1 and (g) 1 (fp) 6 (sd) -2 configurations compete 4. 1/2 + state with (g) 1 (fp) 6 (sd) -2 comes down

20. 37 Mg (N=25): (fp) 5 vs. (fp) 6 (sd) -1 vs. (sdg) 1 (fp) 6 (sd) -2 1. The ground state is normal configuration (end of island of inversion?) 2. Positive parity state with 0g 9/2 appears at small excitation energy 3. The ground state density does not reproduce the observed cross section ⇒ Need to improve the tail part of wave function.

21. 37 Mg (N=25): AMD+RGM : AMD+GCM w.f. of 36 Mg 1/2 + gains extra biding energy by RGM and degenerate with 5/2- shows better agreement with the observed Reaction cross section Strong deformed core and weak binding lowers intruding orbit from g 9/2 Need to extract core-n interaction from RGM Need to solve resonaces and scattering states l = 0 l = 2 + + …

22. Summary and Outlook Summary Microscopic description of deformed core by AMD Description of weakly bound neutron by RGM Better description of Sn and R rms of Oxygen isotopes There are still discrepancy between experiments and calculation. (new data for 24 O is in need) Possible parity-inversion in 35 Mg (Interaction dependence) 2s 1/2 neutron configuration with a halo with deformed core of 36 Mg Strong deformation of the core assists the lowering of 2s 1/2 configuration Outlook Application of R-matrix method to AMD+RGM Phase shifts, equivalent Core-n local potential, Development in more efficient calculation method Application to deformed core + 2n system