Shell evolution in N∼28 unstable nuclei studied by the shell model
Yutaka Utsuno
Advanced Science Research Center, Japan Atomic Energy Agency
List of collaboration
Takaharu Otsuka (Univ. Tokyo/RIKEN/MSU)
Alex Brown (MSU)
Michio Honma (Aizu Univ.)
Takahiro Mizusaki (Senshu Univ.)
Acknowledgment
W.D.M. Rae for letting us use his NuShellX code V4.0R2 (http://knollhouse.org/) MSHELL by T. Mizusaki was also used.
Introduction—shell structure in exotic nuclei
• Shell structure in nuclei
– One‐body potential (Woods‐Saxon) Evolution of the shell structure
by the Woods‐Saxon potential
• gives overall agreement with experiment near stable nuclei.
• Slow and monotonic change • Study of exotic nuclei
– Much information on the change of the shell structure is available by keeping the Fermi surface unchanged.
putting protons
how the gap
is changed
Taken from A. Bohr and B.R. Mottelson, Nuclear Structure, vol. 1
Case of the “island of inversion”
Neutron shell gap by SDPF‐M
Na
Na
16
20
0d5/2
Y. Utsuno et al., Phys. Rev. C 60, 054315 (1999).
Strong T=0 monopole
interaction between Extending the island of inversion
d5/2 and d3/2
Shell evolution by the tensor force
• Origin of the drastic change
– Spin dependence (T. Otsuka et al., Phys. Rev. Lett. 87, 082502 (2001).)
– Develops into the tensor force
• Unique features of the tensor shell evolution
T. Otsuka et al., Phys. Rev. Lett. 95, 232502 (2005).
– Evolution sensitive to the location of the Fermi Surface
– The opposite direction between j> and j<
turning
• The magnitude is doubled.
Steeper and non‐monotonic shell evolution is expected.
T. Otsuka et al., Phys. Rev. Lett. 97, 162501 (2006).
A new interaction for the sd‐pf shell
• Aim of the study
– To construct and test an interaction without the correction of monopole interaction (The origin of the correction has been mostly included in the shell evolution by such as tensor force.)
– 0hw and 1hw states are the scope of the interaction.
• Components of the interaction
– sd part + pf part + cross‐shell part
– USD as the sd part (with a slight modification as adopted in SDPF‐M: changing magic number from N=16 to 20)
– GXPF1B as the pf part (with a slight modification in the f7/2 pairing and q‐
pairing matrix elements; improving the 2+1 of Si isotopes around N=22)
– A newly constructed interaction for the cross‐shell interaction
• central + (two‐body) LS + tensor
Tensor force in the cross‐shell interaction
•  with cutoff at 0.7 fm is adopted.
• Justification for the use
T=0 monopole centroid of the tensor force (MeV)
Quite similar

MK
0.223
0.210
0.080
p3
0.036
0.035
0.013
f7
f5
‐0.335
‐0.315
‐0.120
f7
p1
‐0.073
‐0.070
‐0.026
p3
p3
0.092
0.150
0.064
p3
f5
‐0.048
‐0.046
‐0.017
p3
p1
‐0.229
‐0.376
‐0.160
f5
f5
0.382
0.360
0.137
f5
p1
0.097
0.093
0.034
p1
p1
0.306
0.501
0.213
i
j
– Successful in the Sb isotope chain with spherical calculation
f7
f7
f7
– The monopole centroid of : very similar to that of GXPF1
• The tensor part of GXPF1: extracted by using the spin‐
tensor transformation (M.W. Kirson, Phys. Lett. B 47, 110 (1973).).
• Comparison with another
– Millener‐Kurath (MK): only about 1/3
GXPF1
Central force in the cross‐shell interaction
• Simplicity in the monopole interaction from GXPF1
Monopole centroids of the central
force for the pf‐shell
– T=0: dependence on node
• Very similar between the same (n1, n2)
• n1=n2 case is stronger than n1≠n2
– Enables to fit with a simple force
• Present interaction
– Gaussian with density (R=(r1+r2)/2) dependence
– Better than MK (Yukawa)
No direct fitting to experimental data
Semi‐quantitative property of the monopole interaction responsible for shell evolution • Tensor force
(n,l,j)
– Spin dependence (direction of j and j’)
– The larger l and l’ are, the larger the strength is. • Central force
– Node dependence
– The interaction between the same node is more attractive.
(n’,l’,j’)
Shell evolution from N=20 to 28
• The effect of the cross‐shell interaction
– (sd) orbits are of interest.
• Neutron: f7/2
–
Vm(f
7/2, sd) • To be discussed
1. Z=16 gap: single hole states in 19K isotopes
2. Effects on collectivity: deformation in 42Si28
3. Reduction of the LS splitting: distribution of the spectroscopic factor
0f7/2
0d3/2
1s1/2
0d5/2
Evolution of d3/2‐s1/2 gap in K isotopes
• Energy levels
– Significance of the tensor force is clear.
K isotopes
– Directly reflect the gap between (d3/2) and (s1/2) at N=20 and 28
– 1/2+1 has a large mixing with (d3/2) ⊗(2+) in N=22, 24, and 26.
∼1 MeV
Monopole interaction in K levels
• 0d3/2 vs. 1s1/2 from N=20 to 28
= Vm(0f7/2, 0d3/2) vs. Vm(0f7/2, 1s1/2)
0f7/2
0d3/2
1s1/2
• Central vs. tensor – Both the central and the tensor contribute to almost the same extent.
Sharp change of the gap
p‐n monopole centroid (in MeV)
f7/2
d3/2
s1/2
difference
central
‐1.10
‐0.88
‐0.22
tensor
‐0.21
0
‐0.21
strength scaled at A=42
Collectivity of Si isotopes: N=28 magicity
• Energy levels N≤26
– 2+1 is dominated by (f7/2)2
• Pairing and q‐pairing in f7/2 are more sensitive.
• Large difference at N=28
– Disappearance of the magic number
Exp.) 40Si: C.M. Campbell et al., Phys. Rev. Lett. 97, 112501 (2006). 42Si: B. Bastin et al. Phys. Rev. Lett. 99, 022503 (2007).
Potential energy surface (PES) for 42Si
• PES: constrained (Q0) Hartree‐
Fock calculation in the shell model space
– Successful in the shape coexistence in 56Ni (T. Mizusaki et al., Phys. Rev. C 59, R1846 w/o tensor
(1999).)
• Effect of the tensor force: large
• Oblate deformed g.s. with tensor
– Consistent with calculated Q moment of the 2+1: +23 e2fm4
w/ tensor
Comparison of the effective SPE Proton shell gap as function of N
Neutron shell gap as function of Z
w/o tensor
w/ tensor
reduction
by tensor
L‐S closure
j‐j closure j‐j closure
L‐S closure
• Coherent quenching of proton and neutron shell gaps which increase toward the j‐j closure
Evolution of the f7/2‐p3/2 gap from Z=14 to 16
• The f7/2‐p3/2 gap: increases from S (Z=16) to Si (Z=14)
– Minimum at Z=16
– Is there any evidence?
Ex.(3/2‐1) by the present calc.:
1.22 MeV (35Si) and 0.56 MeV (37S)
S. Nummela et al., Phys. Rev. C 63, 044316 (2001).
3/2‐
646
7/2‐
0
37 S
16 21
Table of isotopes,
8th edition
Difference between tensor and central
tensor force
central force
f7/2
f7/2
d3/2
16
s1/2
14
d3/2
16
narrowing
s1/2
nearly
constant
14
d5/2
d5/2
• Both tensor and central affect the reduction of the Z=16 gap.
• Almost only tensor contributes to the reduction of the LS splitting.
Spectroscopic factor for 1p removal from 48Ca
• d5/2 hole state
Present interaction (w/ tensor)
– Ex.: high
– Fragments into many states
• Spectroscopic factor
– The centroid gives the single particle energy.
• Comparison between experiment and calculation
– Quenching factor 0.7 is needed.
– Very good : both position and strength
(e,e’p): G.J. Kramer et al., Nucl. Phys. A 679, 267 (2001).
What happens without the tensor force?
d3/2
s1/2
w/o tensor in the cross shell int.
Without tensor
d5/2
• d3/2
– The position of the single‐hole state shifts to the left.
• d5/2
– 5/2+ levels exist from around 3 MeV, but the strength shifts to higher excitation energy.
d3/2‐s1/2 gap
d5/2‐s1/2 gap
Summary
• Shell evolution in the N ∼ 28 region by the shell model
– Quantitative and comprehensive description from a new interaction without fitting the monopole interaction to experiment
• Tensor: responsible for spin dependence
• Central: responsible for node dependence
– From N=20 to N=28: • Evolution of the d3/2‐s1/2 gap: interplay between tensor and central
• Deformation in 42Si: coherent shell quenching in a j‐j closed nuclei
• Spectroscopic factor for 1p removal from 48Ca: reduction of LS splitting dominated by tensor