Study of nuclear excited states through quantised angular

momentum

Makito Oi

Senshu University, Tokyo

Study of “rotation”

Apparent retrograde motionof Saturn in 2011 (Apr-July)… observation by MO

Eudoxus

Hipparcus

3D

1D

Nuclear rotation

• Static rotation• Uniform rotation• One-dimensional rotation

“Yrast” states (1D)

Rcoll

Collectivity: many-body Physics

• Microscopic and Macroscopic viewpoints

• Degrees of freedom: collectivity

Quadrupole deformation

prolate triaxial oblate

sphere ND SD

0.00.2 0.6

Wobbling motion as a 3D rotation

• Triaxial rotor• High spin (J >> 1)• Quasi-1D rotation (Jx ~ J)

Bohr and Mottelson (1975)

Non-abelian algebra: QM flucutation

Classical: J2 Quantum: J2 + J

Definition of K quantum number

K : quantisaion axis

| J K >

QM flucutation (II)

Example: state |J K=J >

Nuclear chiral rotation

• Valence neutron (hole)• Valence proton (particle)• Triaxial rotor (collective)

Three Spin Vectors

Rcoll

Jparticle

Jhole

Jtotal

PRL84 (2000), Dimitrov, Frauendorf, Donau

Angular momentum projection

Norm overlap kernels

NPA 80 (1966), Onishi and Yoshida

The Onishi formula

Square root of determinant

NPA 80 (1966), Onishi and Yoshida

Nodal lines

A collection of singular points PLB 606 (2005), M.Oi and Tajima

A Pfaffian formula of Robledo

Pfaffian: polynomials

Robledo, PRC 79 (2009)

Another Pfaffian formulafor odd-A systems

Chirality needs a particle and a hole

M. Oi and Mizusaki, PLB 707 (2012)

A Pfaffian formulaof Oi and Mizusaki

Several unfamiliar bipartite matrices…

M. Oi and Mizusaki, PLB 707 (2012)

Meaning of X-1

Antisymmetrised density matrix (with a bipartite form)

Mizusaki and M.OI, arXiv:1204.6531(2012)

Extended Pfaffian formulaof Mizusaki and Oi

Mizusaki and M.OI, arXiv:1204.6531(2012)

“Lewis Carroll” formulafor Pfaffian

Avez and Bender, PRC 85 (2012)

Mizusaki and M.OI, arXiv:1204.6531(2012)

Summary

• Nuclear excited states through angular momentum

• Structural transitions to exotic shape and 3D rotation

• Quantization of classical / semi-classical 3D models

• New formulae were discovered

• Ready for numerical analysis!

University of Tokyo

LB ２０４５：NaI scintillator

“Global warming effect”

“Next big ones to come…”

“Centre of the world”

“Demonic” least common multiple

FUKUSHIMA NO.4 REACTOR

Apr, 2012