Systematics of the First 2 + Excitation in Spherical Nuclei with Skyrme-QRPA

Systematics of the First 2+ Excitationin Spherical Nuclei with Skyrme-QRPA

J. TerasakiUniv. North Carolina at Chapel Hill

1. Introduction2. Procedure3. Softness parameter4. Energy5. Transition strength6. Comparison with other methods7. Summary

Cf. J. T., J. Engel and G.F. Bertsch Phys. Rev. C, 78 044311 (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: QRPAWe choose first 2+ states

Procedure1.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 elements5. Diagonalization of QRPA Hamiltonian matrix.6. Check of solutions

ijijijjiij aaYaaXO

Creation operator of an excited state of QRPA:

ia : linear combination of

ic icand

of single-particle

It happens that ii cc is a main component of O

Physically, it corresponds to a final state of particletransfer.

We checked if the lowest solutions were really of the nuclei considered.

Approximate difference in particle number

ij

jiijij vvYXN )()( 2222 12

ΔN ≈ 2 :40,48Ca, 68Ni, 80Zr and 132Sn

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

ij

ijij YXC 2

We define softness parameter:

Assume that matrix elements of a transition operator = 1

)()(

strength Transition

hpphhphpphph

phphph

phphph

YXYXYX

YX

2

2

Softness parameter

We wanted C=1 if Y are zero.

2 ijij YXC

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

SLy4

including those> 4

Histogram of

exp

callnEE

RE

Distribution of

Energy

ln1.1 = 0.095ln2.0 = 0.693

Exp:S. Raman et al., At. Data Nucl. Data Tables 78, 1 (2001).

) ,( calexp EE

Data set Num. nuclei

SLy4

All spherical 155 0.33 0.51Low |ΔN| 77 0.29 0.47High|ΔN| 78 0.38 0.54

Low softness 106 0.47 0.48High softness 49 0.04 0.44

Common 129 0.26 0.40

SkM*

All spherical 178 0.11 0.44Low softness 115 0.27 0.35High softness 63 –0.17 0.45

Common 129 0.14 0.38

ER E

|ΔN|=0.5

C=2

Well reproduced:

exp

cal

0.350.27Low softness

0.240.21These nuclei

0.350.27Low softness

0.240.21These nuclei

EER

Histogram of

exp

cal

)()(

ln

22

EBEB

RQ

Distribution of))( )(( calexp 22 EBEB ,

Transition strength

SLy4 -0.32 0.42

SkM* -0.29 0.53

QR Q

Comparison with other methods

B. Sabbey, M. Bender, G. F. Bertsch, and P.-H. Heenen, Phys. Rev. C 75, 044305 (2007).

GCM-Hill-Wheeler(HW)SLy4+density-dep.pairboth spherical and deformed

G. F. Bertsch, M. Girod, S. Hilaire, J.-P. Delaroche,H. Goutte, and S. P´eru, Phys. Rev. Lett. 99, 032502 (2007).

GCM-5-dimensionalcollective Hamiltonian(5DCH)Gognyboth spherical anddeformed

Theory

QRPA(SLy4) 0.33 0.51 -0.32 0.42

GCM-HW(SLy4) 0.67 0.33 0.16 0.41

ER E QR Q

Theory

QRPA(SkM*) 0.10 0.45 -0.29 0.51

GCM-5DCH(Gogny) 0.19 0.43 0.22 0.27

ER E QR Q

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 40Ca 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.