Post on 03-Jan-2016
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K. Kaneko, et al, unpublished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
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
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
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