# Presentation: 913059

## K. Nomura1, N. Shimizu1, and T. Otsuka1,2,3,4

**LACM-EFES-JUSTIPEN Workshop @ ORNL **

- based on Phys. Rev. Lett. **101**, 142501 (2008)

**1****Univ. Tokyo/****2****CNS/****3****RIKEN/****4****MSU**

**New microscopic foundation of the IBM and the structure of exotic nuclei**

**Notes:**
Good morning.
My name is Kosuke Nomura, from Japan.
Today I am talking about the new theoretical method to study the structures of neutron-rich exotic nuclei, which was already refereed to by Iachello on Monday.
This work has done in collaboration with Dr. Shimizu and Prof. Otsuka at The University of Tokyo.

## Outline

- Motivation: IBM and Mean Field
- Applications to exotic nuclei
- Applications of the Wavelet analysis
- Summary

## Interacting Boson Model (IBM)

**Interacting Boson Model (IBM)**

- Arima and Iachello (1974)

-  - dynamical symmetries
-  - parameters are adjusted to experiments

-  Otsuka, Arima, Iachello and Talmi (1978)
-  Otsuka, Arima, and Iachello (1978)

-  - applied to spherical shapes
-  - open question for general cases

-  a model of the low-lying collectivity with sd bosons as “building blocks”

-  shell-model foundation (OAI method)

**Notes:**
The quadrupole collectivity has been extensively studied in terms of the Interacting Boson Model.
In this model, nuclear many body system is described by using s-bosons and d-bosons as basic building blocks.
There are analytic solutions for limited cases, based on the group theory.
But for most of the nuclei, the parameters of the IBM have been determined so as to reproduce the experimental data.

On the other hand, IBM has its own microscopic foundation starting from the fermion system.
Usually this has been studied using the nuclear shell model.
OAI mapping method is an well-known example.
This method has been applied only to spherical nuclei.
Thus, for general cases, the microscopic foundation of the IBM remains to be an open question.
In this study, we propose a novel way of determining the Hamiltonian of the IBM, based on the mean-filed theory.

## So far, both have nothing to do with each other

-  - describes intrinsic properties
-  - it is not easy to obtain excitation spectra

-  - calculates levels precisely
-  - needs microscopic basis for general cases

**Mean Field (Skyrme) model**

**IBM**

**So far, both have nothing to do with each other**

**But, **

***It may be interesting to construct an IBM Hamiltonian based on the Skyrme model***

**Notes:**
This is the basic idea of the present study.
The mean-field theory, including the Skyrme model, calculates systematically the ground state properties of the 
nuclei in the wider area on the nuclear chart.

It may be interesting to construct the IBM Hamiltonian from the Skyrme model.

## Mean-Field Derivation of IBM

**Mapping**

**fermionic PES**

**bosonic PES**

**Energy expectation values in the ****intrinsic states**** (PES’s) are compared to simulate the effects of ****nuclear force**** & ****Pauli principle****.**

**Mean-Field Derivation of IBM**

## constrained Skyrme Hartree-Fock calculation

**constrained Skyrme Hartree-Fock calculation**

-  - zero-range pairing in the BCS approximation
-  - mass quadrupole constraint
- 	with triaxial degrees of freedom

- in mesh representation (ev8 code: Bonche et al.)

**fermionic PES**

** ****coherent state formalizm**

**bosonic PES**

- Dieperink and Scholten, etc. (1980)

-  - IBM-2 Hamiltonian:

-  - Assumptions:

**Notes:**
\begin{align*}
  \hat{H}=\epsilon(\hat{n}_{d\pi}+\hat{n}_{d\nu})
  +\kappa\cal{\hat{Q}}^{\chi_{\pi}}_{\pi}\cdot\cal{\hat{Q}}^{\chi_{\nu}}_{\nu}
\end{align*}

\begin{align*}
 |\Phi\rangle\propto\prod_{\rho=\pi,\nu}
 \Big[s^{\dagger}_{\rho}+\sum_{\mu=0,\pm 1,\pm 2}
 \alpha_{\rho\mu}d^{\dagger}_{\rho\mu}\Big]^{n_{\rho}}|0\rangle
\end{align*}

\begin{eqnarray*}
\left\{
\begin{array}{l}
\alpha_{\rho\pm 2}=\frac{1}{\sqrt{2}}\beta_{\rho}\sin{\gamma_\rho}}\\
\alpha_{\rho\pm 1}=0\\
\alpha_{\rho 0}=\beta_{\rho}\cos{\gamma_{\rho}}\\
\end{array}
\right.
\end{eqnarray*} 

\begin{eqnarray*}
\left\{
\begin{array}{l}
\beta_{\pi}=\beta_{\nu}\equiv\beta_{\rm b}\\
\gamma_{\pi}=\gamma_{\nu}\equiv\gamma_{\rm b}\\
\end{array}
\right.
\end{eqnarray*} 

\begin{eqnarray*}
 E(\beta_{\rm b},\gamma_{\rm b})
 &=&\frac{\epsilon(n_{\pi}+n_{\nu})}{(1+\beta_{\rm b}^2)}+
 \frac{2\kappa n_{\pi}n_{\nu}\beta_{\rm b}^2}{(1+\beta_{\rm b}^2)^2}\nonumber \\
 &\times&\Big[2-2\sqrt{\frac{2}{7}}
(\chi_{\pi}+\chi_{\nu})\beta_{\rm b}\cos{3\gamma_{\rm b}}
+\frac{1}{7}\chi_{\pi}\chi_{\nu}\beta_{\rm b}^2\Big]
\end{eqnarray*} 

\begin{eqnarray*}
 Q_I
 &=&q\langle\cal{\hat{Q}}^{\chi_{\pi}}_{\pi}+\cal{\hat{Q}}^{\chi_{\nu}}_{\nu}\rangle \nonumber \\
 &=&\frac{q\Big[2(n_{\pi}+n_{\nu})\beta_{\rm b}-\sqrt{\frac{2}{7}}
(n_{\pi}\chi_{\pi}+n_{\nu}\chi_{\nu})\beta^2_{\rm b}\Big]}{1+\beta_{\rm b}^2}
\end{eqnarray*} 

\begin{eqnarray*}
 V_{\rm Skyrme}&=&t_0(1+x_0{\hat P}_{\sigma})\delta({\bf
  r_1}-{\bf r_2})
+\frac{1}{2}(1+x_1{\hat P}_{\sigma})[\hat{{\bf k}^{\prime}}^2\delta({\bf
 r_1}-{\bf r_2})+\delta({\bf r_1}-{\bf r_2})\hat{\bf k}^2] \nonumber \\
&+&t_2(1+x_2{\hat P}_{\sigma})\hat{\bf k}^{\prime}\delta({\bf
  r_1}-{\bf r_2})\hat{\bf k}
+iW_0(\hat{\bf \sigma_1}+\hat{\bf \sigma_2})\cdot\hat{\bf k}^{\prime}\times\delta({\bf
  r_1}-{\bf r_2})\hat{\bf k} \nonumber \\
&+&\frac{1}{6}(1+x_3{\hat P}_{\sigma})\delta({\bf r_1}-{\bf
  r_2})\rho^{\alpha}\Big(\frac{{\bf r_1}+{\bf r_2}}{2}\Big)
\end{eqnarray*}

\begin{align*}
 |\Phi\rangle\propto\prod_{\rho=\pi,\nu}
 \Big[s^{\dagger}_{\rho}+\beta_{\rho}\Big\{d^{\dagger}_{\rho 0}\cos{\gamma_{\rho}}+\frac{1}{\sqrt{2}}(d^{\dagger}_{\rho +2}+d^{\dagger}_{\rho -2})\sin{\gamma_{\rho}}\Big\}\Big]^{n_{\rho}}|0\rangle
\end{align*}

\begin{eqnarray*}
\left\{
\begin{array}{l}
q_1=Q_0\cos{\gamma_{\rm f}}-\frac{\sqrt{3}}{3}Q_0\sin{\gamma_{\rm f}}}\\
q_2=\frac{2\sqrt{3}}{3}Q_0\cos{\gamma_{\rm f}}}\\

\end{array}
\right.
\end{eqnarray*}

\begin{align*}
Q_0=\sqrt{\frac{16\pi}{5}}\frac{3}{4\pi}A(1.21A^{1/3})^2\beta_{\rm f}
\end{align*}

##  - Skyrme PES is simulated by IBM PES using

-  - Skyrme PES is simulated by IBM PES using

**bosonic PES**

-  - relationships of deformation variables

**Notes:**
Considering these, we can obtain the IBM-PES like this with a few parameters.
We also take into account the conditions on deformation parameters, which were already introduced briefly by Iachello last Monday.
gamma_b equals to gamma_BM, since both have the same meanings limited to 0 to 60 degrees.
beta_b is, for simplicity, assumed to be proportional to beta_BM with the coefficient C_beta.

## Comparisons of PES’s

**minimum :**

**148****Sm**

**154****Sm**

**132****Ba**

**208****W**

**134****Ba**

**U(5)**

**SU(3)**

**O(6)**

**E(5)**

**E(5)**

**Comparisons of PES’s**

-  - minimum
-  - global pattern
-      of HF PES are reproduced
-    up to 2 MeV

- IBM

- HF

**Notes:**
They are PES’s for typical dynamical symmetries, calculated by Skyrme model and IBM.
For 148Sm, which is an well-known example of U(5) symmetry in IBM, IBM-PES reproduces the position and the global pattern of Skyrme-PES.
For 154Sm, Skyrme-PES has the pronounced sharp minimum. IBM-PES describes this.
Next one is 132Ba, an example of O(6) symmetry. Skyrme-PES has the large flat area in triaxial region as gamma-instability.
IBM-PES reproduces this quite well except for the deviation of the position of the minimum.
The last one is 208W, which is considered to be close to the E(5) symmetry. It also has the gamma-instability. IBM-PES describes it quite nicely.
Then, we can conclude ....

## Evolutions of IBM parameters (Sm)

- Neutron Number

- U(5)

- X(5)

- SU(3)

**Evolutions of IBM parameters (Sm)**

**Skyrme SLy4**

- gradual variation

- drastic change around N=90 ( spherical-deformed transition )

- evolve linearly with N

- kept constant

**Notes:**
This is the evolutions of IBM parameters with neutron number in the case of Sm isotopes.
Each parameters change gradually. 
epsilon, kappa and chi change rather suddenly around N=90.
This shows nothing but a transition from spherical to deformed states.
chi pi is kept constant.
C_beta is determined so as to reproduce the position of the Skyrme-PES.

## Low-lying levels for Sm

- data: taken from NuDat 2.4

- U(5)

- X(5)

- SU(3)

- IBM-SLy4

**Low-lying levels for Sm**

**Notes:**
This is a result of level evolution in Sm isotopes.
Our calculation reproduces the experimental trends especially for yrast levels.
Around N=86, levels look like phonon states, namely the U(5) symmetry.
As N increases, 2+ and 4+ states come down, but 0 and 2 are pushed for both experiment and the calculation.
Around N=92, levels look like rotational band, namely the SU(3) limit.
We can also see the structure of the X(5) CPS around N=90.
The ratio of 4+ to 2+.

## PES’s for Ba isotopes (SkM*)

**PES’s for Ba isotopes (SkM*)**

- Spherical

- Rotational

- Rotational

- O(6)

- E(5)

- Spherical

- HF

- HF

- IBM

- IBM

- N=54-66

- N=68-80

## Low-lying levels for Ba

**SkM***

- Signature of spherical  gamma-soft  deformed transition

**SLy4**

**E(5) nucleus – ****134****Ba**

**Low-lying levels for Ba**

## Outline

- Motivation: IBM and Mean Field
- Applications to exotic nuclei
- Applications of the Wavelet analysis
- Summary

## PES’s for exotic W isotopes

**PES’s for exotic W isotopes**

- SLy4

- HF

- SkM*

- IBM

- IBM

- HF

- N=130

- N=132

- N=134

- N=136

- N=138

- N=140

- N=130

- N=132

- N=134

- N=136

- N=138

- N=140

**E(5) symmetry **

## Low-lying levels (IBM from SkM*)

**Low-lying levels (IBM from SkM*)**

**208****W**

****** ****N**

- U(5)

- E(5)/O(6)

- SU(3)?

**E(5)-O(6) pattern **
**is maintained **
****** ****never been seen in stable nuclei**

**208****W******** quite similar to ****134****Ba**

- No experimental data

**Notes:**
This is a prediction on n-rich W isotopes with A>200.
This panel shows the level evolution in W isotopes.
As N increases, levels change from U(5) structure to O(6) limit.
The important point is, the E(6)/O(6) pattern is kept from N=130 to 136. 
There is an ambiguity in shape phase transition.
This has never been seen in stable nuclei.
This panel is the level scheme for 208W. 
This has the similar structure of the E(5) symmetry like 134Ba.
More importantly, there is no data for this region.

## Outline

- Motivation: IBM and Mean Field
- Applications to exotic nuclei
- Applications of the Wavelet analysis
- Summary

## Wavelet analysis on PES fit

- Scale (frequency)

- Position

- Wavelet function

- PES (axially symmetric)

** ****- superposition of PES using appropriate wavelet**
** ****- useful for localized signals**

**PES fit is carried out in wavelet space**

**Wavelet analysis on PES fit**

**Notes:**
\begin{align*}
  C(\delta\beta,\beta_x)=\frac{1}{\sqrt{\delta\beta}}\int
  E(\beta,\gamma=0)\Phi^{\star}\Big(\frac{\beta_x-\beta}{\delta\beta}\Big)d\beta
\end{align*}

\begin{align*}
  C(\textcolor[rgb]{1,0,0}{\delta\beta},\textcolor[rgb]{0.2,0.2,1}{\beta})=\frac{1}{\sqrt{\textcolor[rgb]{1,0,0}{\delta\beta}}}\int
  E(\beta^{\prime},\gamma=0)\textcolor[rgb]{0,0.6,0}{\Phi}^{\star}\Big(\frac{\textcolor[rgb]{0.2,0.2,1}{\beta}-\beta^{\prime}}{\textcolor[rgb]{1,0,0}{\delta\beta}}\Big)d\beta^{\prime}
\end{align*}

## PES fit in wavelet space (Sm)

- HF

- HF

- IBM

- IBM

- Morlet wavelet

**PES fit in wavelet space**** ****(Sm)**

- |C|^2

## Level evolutions for Sm

- Paul

- uniquely determined

## Summary

**Connection between ****mean-field**** (Skyrme) model and ****IBM**
- 	*Skyrme model **determines IBM parameters, while **IBM **calculates levels precisely.*
- 	   -  typical cases of dynamical symmetries
-        -  signatures of critical-point symmetries
**Applications to exotic nuclei**
-    *intriguing features that has never been seen in stable nuclei  *
-        - e.g. sustained E(5)-O(6) patterns in W & Os
*	*                to be examined experimentally

**END**

**Notes:**
To summarize,
First, we propose the ....
In this study, Skyrme model plays a role to provide the parameters of IBM. IBM calculates observables such as ... which cannot be obtained by Skyrme model.
The present method describes quantitatively three dynamical symmetries and E(5) and X(5) CPS.
Next, ....
The usual shell-model calculation cannot treat such a heavy region.
For example, we can find the exotic E(5)/O(6) pattern in n-rich isotopes, which has never ....
This should be examined in the future experiments.

## Backup Slides

## Relationships of deformation variables

- Relationships of deformation variables

- negligible

- proportional to beta_f

- both are limited to 0 to 60 degrees and have the similar meanings

## Quantum shape/phase transitions

- Intermediate nuclei are numerically studied

**Casten’s symmetry triangle**

**Figs**. : taken from R.F. Casten, Nature Physics **2** 811 **(**2006)

**Quantum shape/phase transitions**

##  - boson coherent state

-  Dieperink and Scholten ; Ginocchio and Kirson ; Bohr and Mottelson, 1980

-  - boson coherent state

- Casten’s triangle

-  - IBM-2 Hamiltonian

**Notes:**
First we perform the constrained Skyrme HF calculations.
PES is obtained as functions of deformation parameters, denoted as beta_BM and gamma_BM.

IBM-PES is calculated in the framework of coherent state formalism.
The standard form of IBM Hamiltonian is used with proton and neutron degrees of freedom.

## Preliminary results (Sm isotopes)

**Preliminary results (Sm isotopes)**

- determined automatically

## PES fit in wavelet space (2)

**PES fit in wavelet space (2)**

- HF

- HF

- IBM

- IBM

- Mexican Hat

## PES fit in wavelet space (3)

**PES fit in wavelet space (3)**

- HF

- HF

- IBM

- IBM

- Paul

## PES’s for Ru isotopes (SLy4)

**PES’s for Ru isotopes (SLy4)**

- (near-)
- spherical

- N=54-66

- N=68-80

- IBM

- HF

- IBM

- HF

- gamma-unstable

- (near-)
- spherical

## Levels of Ru isotopes

**IBM from SkM***

**Levels of Ru isotopes**

**IBM from SLy4**

**containing predictions on N>70**