Study of medium-mass nuclei bylarge-scale shell model
calculations
Yusuke Tsunoda
Department of Physics, University of Tokyo
Collaborators: T. Otsuka (Univ. Tokyo)
N. Shimizu (Univ. Tokyo)
M. Honma (Univ. Aizu)
Y. Utsuno (JAEA)

Motivation
Shell structure
in stable nuclei (Z～N)
1d5/2

50

Different structure
in exotic nuclei (Z≪N, Z≫N)

Some magic numbers disappear
New magic numbers appear

0g9/2
1p3/2, 0f5/2, 1p1/2

28
0f7/2

Magic number nucleus: closed shell
Ground state has spherical shape
Excited states are higher

20
0d5/2, 1s1/2, 0d3/2

80p3/2, 0p1/2

20s1/2

open shell nucleus:
Nucleus can have deformed shape
Excited states are lower

Motivation
Phenomena in exotic nuclei (Z≪N, Z≫N)
Evolution of shell structure

Shape coexistenceStates of different shape coexist

Size of shell gaps changes
Magicity becomes strong/weak

Necessity of large-scale calculations
in large model space

Shell model calculation…

unoccupied
pfg9d5 model spacepartially occupied

50
40

28

1d5/2
0g9/2
1p1/2, 0f5/2
1p3/2
0f7/220…

core: occupied

astrophysics

This model space is wide enoughto discuss how magic numbers 28, 50
and semi-magic number 40 are retained/broken
Our effective interaction consists of
microscopic and empirical interactions

Monte Carlo shell model (MCSM)
More than 1015 (in our model space)

Direct diagonalization is
impossible in large model spaceabout 100In MCSM, we diagonalize
small Hamiltonian matrix
constructed from MCSM bases

T. Otsuka et al. PPNP47, 319 (2001)

Generated to minimize
eigenenergies
using quantum MC+ variational method

eigenstate

angular-momentum,parity projection

Slater determinantEnergy-variance extrapolation〈H〉(MeV)• Second-order
extrapolation using
energy variance〈ΔH2〉= 〈H2〉- 〈H〉2
• Points arecalculated witheach number of bases

Extrapolation of 68Ni 0+0+3

0+20+1

N. Shimizu, Y. Utsuno, T. Mizusaki, T. Otsuka,T. Abe, and M. Honma, Phys. Rev. C 82, 061305(R) (2010)
Based on T. Mizusaki and M. Imada,
Phys. Rev. C 65, 064319 (2002); 67, 041301(R) (2003)〈ΔH2〉(MeV2)

Parallel calculation• Calculational bottleneck is integration over 3D rotations
to restore the angular momentum⇒ # of total mesh points ～ 50,000• Calculated parallelly for each mesh point• Calculated on K computer (AICS),FX10 (U. Tokyo),and T2K Open Supercomputer(U. Tokyo, U. Tsukuba)
K computer• 14 hours, 7680 CPU cores for 68Ni 8+ (K computer)Results of Cr and Ni
24Cr

reduced transition
probability

Ex(2+)(MeV)

Ex(2+)(MeV)

NN

B(E2; 0+→2+)(e2fm4)

B(E2; 0+→2+):

B(E2; 0+→2+)(e2fm4)

2+ ex. energy

28NiNNB. Pritychenko, et al., arXiv:1102.3365v2 (2011)de Angelis, private communication

Neutron excitation across N=40 gap
Excitation across N=40

# ofneutrons
excitedacrossN=40(0+1)

Excitation across N=4024Cr

Change of
N=40 magicity

NN# of particles
above N=40

N
d5/2
g9/2
p1/2
f5/2
p3/2
f7/2

E (MeV)

N=40 gap

E (MeV)

Neutroneffective
singleparticle
energy28Ni

N=40 gap

N# of holes
below N=4068Ni

PES (energy contour plot of various shapes)
0+1 state of 68Ni (Z=28, N=40)• Potential energy surface(PES) is calculatedby Constrained HF• Location of circle:
quadrupole
deformation ofunprojected
MCSM basis• Area of circle:
overlap probability
between eachprojected basis andsphericalwave function

oblatetriaxial

prolate

68Ni 0+ states ⇔ different shapes

0+2 : oblate β ～ -0.2

We diagonalize
the quadrupole momentto get principal axes
β: deformation parameterβ>0: prolate
β=0: sphericalβ<0: oblate

Shapecoexistence

symmetry axis

0+1 : spherical β ～ 00+3 : prolate β ～ +0.4

Level scheme of 68Ni
Broda et al., submitted

Colors are determinedfrom the calculation

Summary• MCSM calculations for Cr, Ni nuclei in pfg9d5-shell• Analysis of nuclear shape by using
overlap and deformation of MCSM bases• N=40 magicity changes between Cr and Ni• Three 0+ states of 68Ni ⇔ three different shapes
within 3 MeV (shape coexistence)• Calculated excitation energies of 68Ni
agree with experiments
Unified description by using the same Hamiltonian
due to large-scale calculation