# Presentation: 193747

## Application of the Adiabatic Self-Consistent Collective Coordinate (ASCC) Method to 

**Application of the Adiabatic Self-Consistent Collective Coordinate (ASCC) Method to **
**Shape Coexistence/ Mixing Phenomena**

- Nobuo Hinohara (Kyoyo)
- Takashi Nakatsukasa (RIKEN)
- Masayuki Matsuo (Niigata)
- Kenichi Matsuyanagi (Kyoto)

## 日野原 伸生

**Microscopic Description of ****Nuclear Large-Amplitude Collective Motion ****by Means of ****the Adiabatic Self-Consistent****Collective Coordinate Method**

- Hinohara Nobuo

- Sunny Field

- Life lively grows

- doctoral dissertation defense

- The major part of this
- thesis will appear in
- Prog. Theor. Phys.
- Jan. 2008
- within a few days

- 学位論文公聴会

- 2008.1.17

## Shape coexistence in N~Z~40 region

**50**

**28**

**20**

**34**

**34**

**38**

**42**

**36**

**40**

- neutron single particle energy

-  oblate-prolate shape coexistence
-  oblate ground state
-  shape coexistence/mixing

- Z,N = 34,36 (oblate magic numbers)
- Z,N = 38 (prolate magic number)

- Bouchez *et al. *Phys.Rev.Lett.**90**(2003) 082502.

- Fischer *et al*. Phys.Rev.**C67** (2003) 064318.

- Skyrme-HFB: Yamagami *et al. *Nucl.Phys.**A693** (2001) 579.

- 68Se

- 72Kr

## Many-Body Tunneling between Different Vacua 

**Prolate**

**Oblate **

**Spherical **

**Many-Body Tunneling between Different Vacua**

- Why localization is possible for such a low barrier ?!

**Basic question**

## I am going to report the first application of 

- I am going to report** **the** ****first**** ****application **of
**the microscopic theory of large amplitude collective motion****,**
**based on the time-dependent mean-field (TDHFB)**** ****theory****,**** ****to real nuclear structure phenomena **
**in nuclei with superfluidity**.

**Coexistence/mixing of **
**oblate and prolate shapes**** **
**is ****a typical phenomenon of **
**large amplitude collective motion****.**

**Main points**

## Main points

**and momentum are microscopically derived; **
**i.e.,**** ****self-consistently extracted **
**from huge-dimensional TDHFB phase space.**

**In     Se and      Kr,**** the collective paths,**
**connecting the oblate and prolate minima****, **
**run in the triaxially deformed region.**

**72**

**68**

**Main points**

## Collective paths obtained by means of the ASCC method

**Collective paths obtained by means of the ASCC method**

**Comparison with the**
**axially symmetric path**

- Collective coordinate

## Main points

** ****is quantized and**** ****excitation spectra, E2 transitions **
** ****and quadrupole moments are evaluated**** ****for the first time.**

** ****The result indicates that the oblate and prolate shapes**
** ****are**** ****strongly mixed at I=0, but the mixing rapidly **
** ****decreases with increasing angular momentum****. **

**Main points**

- Calculated spectra

## ASCC

**a way for wide ****applications of large-amplitude theory **
**is now open.**

**    ****SCC  and **
**quasiparticle SCC**

**   ****ATDHF and**
**     ****ATDHFB**

**ASCC**

- Villars, Kerman-Koonin, Brink,
- Rowe-Bassermann, Baranger-Veneroni,
- Goeke-Reinhard, Bulgac-Klein-Walet,
- Giannoni-Quentin, Dobaczewski-Skalski
- and many colleagues,  reviewed in
- G. Do Dang, A. Klein and N.R. Walet,
- Phys. Rep. **335** (2000), 93.

- Marumori-Maskawa-Sakata-
- Kuriyama, Yamamura,
- Matsuo, Shimizu-Takada,
- and many colleagues,
- reviewed in
- Prog. Theor. Phys. Supplement
- 141 (2001).

## Time dependent mean-field

**time-dependent **
**variational principle**

**collective momentum p**

**collecive coordinate   q**

**Adiabatic expansion**
**      ****(ATDHFB)**

- Find an optimum direction at every point of q

## Slide 11

## ASCC Basic Equations

**ASCC Basic Equations**

- moving-frame Hamiltonian

- _Local harmonic equations_ (moving-frame QRPA equations)

- (from 1st-order in p)

- (from 2nd-order in p)

- (from 0-th order in p)

- Terms not included in QRPA

- Not included in HFB

- _Collective Hamiltonian_

## Basic Scheme of the ASCC method (1)

**Basic Scheme of the ASCC method (1)**

- 1st Step:  Solve ASCC equations  and find collective path.

- collective potential

- collective mass

**Moving-frame HFB eq.**

**Moving-frame QRPA**** ****eq.**

**Double iteration for each q**

**Thouless-Valatin Moment of Inertia**** for**
**moving-frame Hamiltonian**

## Gauge invariant ASCC method.

- The ASCC  method was proposed in
-       M. Matsuo, T. Nakatsukasa and K. Matsuyanagi ,
-        Prog. Theor. Phys. 103 (2000) 959.

**Quite recently, it was found that its basic equations are**
**invariant against gauge transformations**** associated with**
**pairing correlations.**

**Gauge invariant ASCC method**.

**Choosing an appropriate ****gauge fixing condition****, **
**numerical instabilities encountered previously **
**are now completely removed****.**

- N. Hinohara et al.,  Prog. Theor. Phys. 117 (2007) 451

## Collective path in 68Se

- Collective path in 68Se

- Collective potential

- Collective mass

- Moving-frame QRPA frequency

- G2=0: Kobayasi *et al*., PTP**113**(2005), 129.

- γ

- β

- Moment of Inertia

-  Triaxial deformation connects two local minima
-  **Enhancement of the collective mass and MoI by the quadrupole pairing**

- collective path

**due to the time-odd pair field**

- Prog.Theor.Phys.115(2006)567.

- P0+P2+QQ interaction

## Collective path in 72Kr

- Collective potential

- Collective mass

- Moving-frame QRPA frequency

- G2=0: Kobayasi *et al*., PTP**113**(2005), 129.

- oblate

- prolate

- γ

- β

- β

- β

- β

- γ

- γ

- γ

- Collective path in 72Kr

- Moment of Inertia

-  **Dynamical symmetry breaking of the path**
-  Triaxial degrees of freedom: important
-  Enhancement of the collective mass and MoI
-     by the quadrupole pairing

- q=0

- P0+P2+QQ interaction

## Basic Scheme of the ASCC Method (2)

**Basic Scheme of the ASCC Method (2)**

- 2nd Step: Requantize the collective Hamiltonian.

**vibrational wave functions**

**Collective Schrodinger eq.**

**rotational wave functions**

## Excitation spectra of 68Se

** ****two rotational bands**
** ****0****2****+**** state **
** ****quadrupole pairing lowers ex.energy**

**(  ) ...B(E2) e****2**** fm****4**
**effective charge:  e****pol**** = 0.904**

- EXP:Fischer et al., Phys.Rev.**C67** (2003) 064318.

## Collective wave functions in 68Se

- Collective wave functions in 68Se

-  I = 0: oblate and prolate shapes are strongly mixed via triaxial degree of freedom
-   ground band: mixing of different K components、excited band: K=0 dominant
-  oblate-prolate mixing: strong in 0+ states, decreases as angular momentum increases

**G****2 ****= 0**

**G****2**** = G****2****self**

**localization**

## Excitation Spectra of 72Kr

- Excitation Spectra of 72Kr

**(  ) ...B(E2) e****2**** fm****4**

**effective charge is adjusted to this value**

- EXP:Fischer et al., Phys.Rev.**C67** (2003) 064318,  Bouchez, et al., Phys.Rev.Lett.90 (2003) 082502.
-              Gade, et al., Phys.Rev.Lett.**95** (2005) 022502,  **96** (2006) 189901

-  two rotational bands
-  small inter-band B(E2): shape mixing rather weak

## Collective wave functions in 72Kr

- Collective wave functions in 72Kr

**G****2 ****= 0**

**G****2**** = G****2****self**

- oblate

- prolate

-  01+ state: well localized around oblate shape
-  02+ state: weak oblate-prolate shape mixing
-  other states: well defined shape character

- oblate

- prolate

## Spectroscopic quadrupole moments

**positive**

**positive**

**negative**

**negative**

**ground  band**

**excited  band**

**ground band **

**excited  band**

- angular momentum

- angular momentum

## Probabilities of the Oblate and Prolate Components

- ground band

- excited band

- prolate

- prolate

- prolate

- prolate

- oblate

- oblate

- oblate

- oblate

**as a function of**
**angular momentum**

**as a function of**
**angular momentum**

- (different definitions)

## Summary

** ****quantizing the collective Hamiltonian derived by the ASCC method**

** ****The result indicates interesting properties of **
** ****the oblate-prolate shape mixing dynamics, like**
** ****decline of mixing with increasing angular momentum****. **

**Summary**

- Calculated spectra

**Wide applications can be envisaged in the coming years**