Structure of neutron-rich A~60 nuclei: A theoretical perspective

Yang SunShanghai Jiao Tong University, China

KAVLI-Beijing, June 26, 2012

Disappearance of traditional magic numbers? Appearance of new magic numbers?

New magic number 14 appears in neutron-rich region Amplitude of the Z=14 gap changes with neutron-number

Fridmann et al., Nature 435 (2005)

922

‘Magic’ nucleus 42Si （ Z=14， N=28

）

New shell effects in neutron-rich nuclei

Interests in studying neutron-rich A~60 nuclei

Study changes in shell structure when more neutrons added Neutron shell evolution coupled with proton shell filling Properties of subshell gap at N~40 Physics of the neutron intruder orbit g9/2

Answer questions of how the structure changes will influence the r-process nucleosynthesis Need shell-model wavefunctions to calculate decay and

capture rates Incorporate important structure information into network

calculations

General features of neutron-rich nuclei with protons filling mid-shell

large deformation at N ~ 40 For Cr and Fe: Small 2+ excitation, strong B(E2) Ni, Zn, Ge isotopes do not show similar trend

Softness near ground state No well-defined shape in ground state Possible prolate-oblate shape competition Prolate shape stabilized when nuclei rotate (I>6)

Where can we see the neutron g9/2 physics? Back-bending in MoI along yrast line Negative-parity 2-qp bands in even-even and odd-odd isotopes 9/2+ band in odd-neutron isotopes

Shell structure changes around N=40

Experimental data for Ni (Z=28), Zn (Z=30), Ge (Z=32), Cr (Z=24)

Upper: B(E2, I=20) Lower: first 2+ energy

Figures from

K. Kaneko, et al. Phys. Rev. C 78 (2008) 064312

The role of neutron g9/2 intruder

The best place to study the role of neutron g9/2 orbit is from I = 9/2+ state (band) in odd-neutron nuclei

R. Ferrer et al. PRC 81 (2010) 044318

Possible coupling of shapes near ground state

K. Kaneko et al., Phys. Rev. C78 (2008) 064312 A. Gade et al., Phys. Rev. C81 (2010) 051304

Neutron-rich Fe isotopes: Spherical shell model calculations

K. Kaneko, et al, unpublished

Model space: (f7/2, p3/2, f5/2, p1/2) protons

(f7/2, p3/2, f5/2, p1/2, g9/2) neutrons

Wavefunctions with and without g9/2 orbit

Shell models in smaller bases may reproduce energy levels, but the wave functions (B(E2)’s) can be wrong.

Y. Sun, Y.-C. Yang et al., Phys. Rev. C 80, 054306 (2009).

J. Ljungvall et al., Phys. Rev. C 81, 061301(R) (2010).

B(E2,2-->0)=214(26) e2 fm4

for N=36B(E2,2-->0)=470 (210) e2 fm4

for N=38

Essential factors for a model

Single particle states (mean field part) Reflect shell structure (spherical, deformed) Adjust to experiment

Two- (and high order) body interactions (residual part) Mix configurations (do not have in mean field models) Transition probabilities are sensitive test

Model space (configurations) Large enough to cover important parts of physics If not possible, introduce effective parameters

Three factors are closely related, it is difficult to decouple them.

Nuclear structure models

Shell-model diagonalization method Most fundamental, quantum mechanical Growing computer power helps extending applications A single configuration contains no physics Huge basis dimension required, severe limit in applications

Mean-field method Applicable to any size of systems Fruitful physics around minima of energy surfaces No configuration mixing States with broken symmetry, cannot be used to calculate

transition, decay, and capture rates

Bridge between shell-model and mean-field method

Projected shell model Use more physical states (e.g. solutions of a deformed mean-

field) and angular momentum projection technique to build shell model basis.

Perform configuration mixing (a standard shell-model concept)

• K. Hara, Y. Sun, Int. J. Mod. Phys. E 4 (1995) 637

The method works in between conventional shell model and mean field method, hopefully can take the advantages of both.

The projected shell model

Shell model based on deformed basis

Take a set of deformed (quasi)particle states (e.g. solutions of HFB, HF + BCS, or Nilsson + BCS)

Select configurations (deformed qp vacuum + multi-qp states near the Fermi level)

Project them onto good angular momentum (if necessary, also particle number, parity) to form a basis in lab frame

Diagonalize a two-body Hamiltonian in projected basis

What the projected shell model may tell?

For a model calculation, if the three factors are coupled together, it is difficult to figure out what is the most important one.

The projected shell model uses a large model space, to ensure that all the important orbits are all included.

The residual interactions (quadrupole-quadrupole, plus higher-orders, and pairing) are well-understood, so that they are under better control.

Property of deformed single-particle states in exotic mass regions are a less- known factor.

Traditional deformed picture valid?

Nilsson parameters are taken from T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436 (1985) 14

Softness near g.s.: Projected energy surface calculations

Sun et al., Phys. Rev. C80 (2009) 054306

Possible shape coexistence No well-defined shape at I=0

Neutron-rich Fe isotopes: Projected shell model calculations

Sun et al., Phys. Rev. C80 (2009) 054306

Variation of moments of inertia in neutron-rich Fe isotopes

Comparison of calculated moments of inertia with data Irregularity at I ~ 8: alignment of g9/2 neutrons

at I ~ 16: alignment of f7/2 protons

Sun et al., Phys. Rev. C80 (2009) 054306

Transition properties of neutron-rich Fe isotopes

B(E2) reflects band-crossings at I = 8 and 16 g-factor shows a sudden drop at I = 8: neutron alignment

g-factor data: East, Stuchbery et al., PRC 79 (2009) 024304

Negative-parity states in neutron-rich Fe isotopes

Experimental bandheads start at I = 5 (60,62Fe) or 7 (58,64Fe) Predicted 2-qp states are low K-states (f5/2

- coupled to g9/2+)

Sun et al., Phys. Rev. C80 (2009) 054306

Recent experiment confirms the prediction

D. Steppenbeck et al,

PRC 85 (2012) 044316

Explore the nature of 9/2+ band in odd-neutron isotopes

Does the observed 9/2+ isomer in odd-mass Fe and Cr nuclei have a K = 9/2 component of neutron g9/2 state with prolate deformation? Not possible

Or a K = 9/2 component of neutron g9/2 state with oblate deformation? (59Cr) Deacon et al., Phys. Lett. B 622

(2005) 151

Possible shape of 9/2+ band

Y.-C. Yang, H. Jin, Y. Sun, K. Kaneko, Phys. Lett. B700 (2011) 44

Nature of 9/2+ band in odd-mass Cr and Fe isotopes

PSM calculation shows it has a prolate deformation, mainly of K=1/2 component of neutron g9/2 state

An intruder large j orbit with small K component (K=1/2) is a strongly decoupled state which shows decoupling effect

Bandhead has a larger I Low-spin members lie higher Only a favored branch is

observed

Comparison of 9/2+ band in 59Cr with prolate and oblate deformation

Y.-C. Yang, H. Jin, Y. Sun, K. Kaneko, Phys. Lett. B700 (2011) 44

Comparison of 9/2+ band in 59Cr with prolate and oblate deformation

Odd-odd Mn isotopes

D. Steppenbeck et al.

PRC 81 (2010) 014305

Low-spin “shell-model” states of positive-parity,formed by one fp proton and one fp neutron

High-spin rotational band, possibly formed by one fp proton and one g9/2 neutron

Rotational feature of high-spin bands in odd-odd Mn isotopes

The observed high-spin rotational bands have negative parity, formed by one fp proton and one g9/2 neutron

Spherical shell-model calculations without neutron g9/2 orbit

D. Steppenbeck et al.

PRC 81 (2010) 014305

fp shell model space

“Shell model states” classified by intrinsic structures

Calculations by projected shell model Y. Sun et al, Phys. Rev. C85 (2012) 054307

Calculated 2-qp bands by the projected shell model

K = 6 isomer states have a structure of 2-qp states of f7/2

protons

Y. Sun et al, Phys. Rev. C85 (2012) 054307

Collaboration

(China) Y.-C. Yang （杨迎春） H. Jin （金华） (Japan) K. Kaneko （金子和也） M. Hasegawa （长谷川宗武） S. Tazaki （田崎茂） T. Mizusaki （水崎高浩）

D. Steppenbeck, S. J. Freeman, A. N. Deacon The Argonne experimental group

Summary

A correct description of neutron-rich nuclei near N=40 needs to include the neutron g9/2 orbit.

Properties of this orbit, including the splitting into deformed K-components and interplay with other nearby orbitals have been studied by the Projected Shell Model.

Possible experimental tests are: Back-bending in MoI along yrast line Negative-parity 2-qp bands in even-even and odd-odd

isotopes 9/2+ band in odd-neutron isotopes g factor measurements

Hamiltonian and single particle space

The Hamiltonian Interaction strengths

is related to deformation by

GM is fitted by reproducing moments of inertia

GQ is assumed to be proportional to GM with a ratio ~ 0. 20

Single particle space Three major shells in the model space

N = 2, 3, 4 for both protons and neutrons

PPGPPGQQHH QM

20

ppnn QQ 00

3/2

Building blocks: a.-m.-projected multi-quasi-particle states

Even-even nuclei:

Odd-odd nuclei:

Odd-neutron nuclei:

Odd-proton nuclei:

,0ˆ,0ˆ,0ˆ,0ˆ I

MKIMK

IMK

IMK PPPP

,0ˆ,0ˆ,0ˆ,0ˆ I

MKIMK

IMK

IMK PPPP

,0ˆ,0ˆ,0ˆ I

MKIMK

IMK PPP

,0ˆ,0ˆ,0ˆ I

MKIMK

IMK PPP

Model space constructed by angular-momentum projected states

Wavefunction:

with a.-m.-projector:

Eigenvalue equation:

with matrix elements:

Hamiltonian is diagonalized in the projected basis

IM K

IM Pf

DDdI

P IM K

IM K

28

12

0''

''

fENH II

'''''' I

KK

II

KK

I PNPHH

IM KP