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, (2008)

Introduction 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 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

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 (ev8) 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 thatis 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 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. 2nd lowest solution of the other 3 nuclei : acceptable 40 Ca does not have ph-main solution up to tail of GR. 2nd lowest solution of the other 3 nuclei : acceptable

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

Potential-energy curves of Sn (arbitrarily shifted vertically, ev8 used)

SLy4 including those > 4

Histogram of Distribution of Energy ln1.1 = ln2.0 = ln1.1 = ln2.0 = Exp: S. Raman et al., At. Data Nucl. Data Tables 78, 1 (2001).

Data set Num. nuclei SLy4 All spherical Low |ΔN| High|ΔN| Low softness High softness Common SkM* All spherical Low softness High softness63– Common |ΔN|=0.5 C=2

Well reproduced: exp cal

Histogram of Distribution of Transition strength

SLy SkM*

Comparison with other methods B. Sabbey, M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys. Rev. C 75, (2007). GCM-Hill-Wheeler(HW) SLy4+density-dep.pair both spherical and deformed G. F. Bertsch, M. Girod, S. Hilaire, J.-P. Delaroche, H. Goutte, and S. P´eru, Phys. Rev. Lett. 99, (2007). GCM-5-dimensional collective Hamiltonian (5DCH) Gogny both spherical and deformed

Theory QRPA(SLy4) GCM-HW(SLy4) Theory QRPA(SkM*) GCM-5DCH(Gogny) Comparison was done for common spherical nuclei.

● Exp. ○ QRPA(SkM*) □ GCM-5DCH(Gogny)

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, Summary

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 doubly- magic and near-doubly-magic nuclei, ii) QRPA and GCM-5DCH 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.