Systematics of the First 2 Excitation in Spherical

Systematics of the First 2+ Excitation in Spherical Nuclei with Skyrme-QRPA J. Terasaki Univ. North Carolina at Chapel Hill 1. Introduction 2. Procedure 3. Softness parameter 4. Energy 5. Transition strength 6. Comparison with other methods 7. Summary Cf. J. T. , J. Engel and G. F. Bertsch Phys. Rev. C, 78 044311 (2008)

Introduction Progress of computer resources Application of nuclear density functional theory (DFT) over the entire nuclear chart (statistical properties). We want to study dynamical properties based on DFT. The method: QRPA We choose first 2+ states Our Aim: 1) to assess strengths and weaknesses of the method by calculating as many nuclei as we can (even-even spherical) 2) to compare results with those of two other systematic studies that used different methods

Procedure 1. Choose a Skyrme parameter set. 2. Make a list of spherical nuclei initial candidates : even-even Ne - Th i) potential-energy-curve calculation (ev 8) ii) a few unconstraint calculations around Q=0 3. HFB calculation of spherical nuclei for QRPA (hfbmario) 4. Calculation of interaction-matrix elements 5. Diagonalization of QRPA Hamiltonian matrix. 6. Check of solutions

Creation operator of an excited state of QRPA: : linear combination of and of single-particle It happens that is a main component of Physically, it corresponds to a final state of particle transfer. We checked if the lowest solutions were really of the nuclei considered.

Approximate difference in particle number ΔN ≈ 2 : 40, 48 Ca, 68 Ni, 80 Zr and 132 Sn 40 Ca does not have ph-main solution up to tail of GR. 2 nd lowest solution of the other 3 nuclei : acceptable

Softness parameter Assume that matrix elements of a transition operator = 1 We define softness parameter: We wanted C=1 if Y are zero.

Potential-energy curves of Sn (arbitrarily shifted vertically, ev 8 used)

SLy 4 including those >4

Energy Histogram of ln 1. 1 = 0. 095 ln 2. 0 = 0. 693 Distribution of Exp: S. Raman et al. , At. Data Nucl. Data Tables 78, 1 (2001).

Data set All spherical Low |ΔN|=0. 5 High|ΔN| SLy 4 Low softness C=2 High softness Common All spherical Low softness Sk. M* High softness Common Num. nuclei 155 0. 33 77 0. 29 78 0. 38 106 0. 47 49 0. 04 129 0. 26 178 0. 11 115 0. 27 63 – 0. 17 129 0. 14 0. 51 0. 47 0. 54 0. 48 0. 44 0. 40 0. 44 0. 35 0. 45 0. 38

Well reproduced: cal exp

Transition strength Histogram of Distribution of

SLy 4 -0. 32 0. 42 Sk. M* -0. 29 0. 53

Comparison with other methods B. Sabbey, M. Bender, G. F. Bertsch, and P. -H. Heenen, Phys. Rev. C 75, 044305 (2007). G. F. Bertsch, M. Girod, S. Hilaire, J. -P. Delaroche, H. Goutte, and S. P´eru, Phys. Rev. Lett. 99, 032502 (2007). GCM-Hill-Wheeler(HW) SLy 4+density-dep. pair both spherical and deformed GCM-5 -dimensional collective Hamiltonian (5 DCH) Gogny both spherical and deformed

Theory QRPA(SLy 4) 0. 33 0. 51 -0. 32 0. 42 GCM-HW(SLy 4) 0. 67 0. 33 0. 16 0. 41 QRPA(Sk. M*) 0. 10 0. 45 -0. 29 0. 51 GCM-5 DCH(Gogny) 0. 19 0. 43 0. 22 0. 27 Theory Comparison was done for common spherical nuclei.

Exp. ○ QRPA(Sk. M*) □ GCM-5 DCH(Gogny) ●

Summary • Systematic QRPA calculations have been done of first 2+ states of even-even spherical Ne–Th using two Skyrme interactions plus volume-type pairing interaction, and energies and transition strengths were investigated. • Skyrme QRPA is very good for energies of doubly-magic and near-doubly-magic nuclei. • Shortcomings of this method are i) there is no first 2+ state at 40 Ca to compare with experiment, ii) energy is overestimated, and transition strength is underestimated on average,

iii) energies of transitional and “well-spherical” regions are not reproduced simultaneously, iv) breaking of particle-number conservation affects energy on average, v) dispersion of discrepancy from data is not very small. • In comparison with other methods, it turned out that i) QRPA is better than the other methods for doublymagic and near-doubly-magic nuclei, ii) QRPA and GCM-5 DCH are better than GCM-HW in terms of energy, and iii) GCM methods overestimate both energy and transition strength on average. • List of spherical nuclei depends on interaction.