# Presentation: 867489

## Microscopic interpretation of the excited K = 0+, 2+ bands of deformed nuclei

- Microscopic interpretation of the excited K = 0+, 2+ bands of deformed nuclei

*Gabriela Popa*
*Rochester Institute of Technology*
*Collaborators:*
- J. P. Draayer
*Louisiana State University*
- J. G. Hirsch
*UNAM, Mexico*
- Georgieva
*Bulgarian Academy of Science*

## Outline

- 	Introduction
- 	What we know about the nucleus
- 	Characteristic energy spectraTheoretical Model
- 	Configuration space
- 	System HamiltonianResults  	Conclusion and future work

## Chart of the nuclei

- N (number of neutrons)

- Z (protons)

## Nuclear vibrations

- -vibration

- 3

- 1

- 2

- 3

- -vibration

- 1

- 2

## Schematic level schemes of spherical and deformed nuclei

**0, 2, 4**

**0**

**0**

**2**

**4**

**6**

**8**

**6**

**8**

**2**

**0**

**2**

**4**

**6**

**2**

**3**

**4**

**5**

**6**

- spherical

- deformed

- E = n 

- E = I(I+1)/(2**)

## Experimental energy levels

## Experimental energy spectra of 162Dy

## Single particle energy levels

- 1s

- 1p

- 2s

- 1g

- N = 3

- N = 2

- N = 1

- N = 0

- N = 4

- 1d

- l·l

- l·s

- d5/2

- d3/2

- s1/2

- 1f

- 2p

- 3s

- 2d

- 1h

- N = 5

- s1/2

- p1/2

- p3/2

- f7/2

- f5/2

- p5/2

- f9/2

- p1/2

- d3/2

- d5/2

- h11/2

- s1/2

- g7/2

## Particle distribution

**Valence space: U(************) ******** ****U(************)**
**	****total**** ****	****	****normal ****	****unique  **
************** ****=32****	****	********n******** =20****	********u********=12**
************** ****=44****	****	********n******** =30****	********u************=14**

**Particle distributions:                  ****                  ******
**               ****protons:****	****     ****neutrons:           ********total******
**  ****152****Nd    ****10      6      4     ****   10****	****    6      4****     ****(12,0)****  ****(18,0****)  (30, 0)**
******156****Sm    ****12      6      6****        ****12     6      6****     ****(12,0)****  ****(18,0****)  (30, 0)******
** ****160****Gd    ****14      8      6****        ****14     8      6****     ****(10,4)****  ****(18,4****)  (28, 4)******
** ****164****Dy     ****16    10     6****        ****16    10      6****     ****(10,4)****  ****(20,4****)  (30, 4)******
** ****168****Er     ****18     10     8****        ****18    10     8****     ****(10,4)****  ****(20,4****)  (30, 4)******
** ****172****Yb    ****20     12      8****        ****20    12     8****     ****(36,0)****  ****(12,0****)  (24, 0)******
** ****176****Hf     ****22     14      8****        ****22    14     8****     ****(8,30)****  ****(0,12****)  (8, 18)******
****

**Particle distribution**

## Wave Function

**Wave Function**

- | =  Ci | i 
- |i = |{; } (,)KL S; JM
-  ( = , ) = n [f ] ( ,  ), S 
- ( ,  ) ( ,  ) =  (, ) 

## Direct Product Coupling

**Coupling ****proton ****and ****neutron**** **
**irreps to total (coupled) SU(3):**

****************************************************
- **************+********************************+**************
- **************+******************-************** +******************+
- **************+************************************+******************-
- **************+******************-******************+******************-
- 
******m,l ****************+**************-2m+l**************+************+**m-2l)k

**with the multiplicity denoted by k = k(m,l)**

**Direct Product Coupling**

## 21st SU(3) irreps corresponding to the highestC2 values were used in 160Gd

- (,)(,)

- (,)

**21****st**** SU(3) irreps corresponding to the highest****C****2**** values were used in ****160****Gd**

## Tricks

**Invariants   ********   ****Invariants**
**Rot(3)                SU(3)**
**Tr(Q****2****)     ********         C****2     **
**Tr(Q****3****)     ********         C****3     **

**	****************~****************+ ********+ ********+ ********+ ******
******=********tan************+ ******

## System Hamiltonian

**SU(3) conserving Hamiltonian:**
**H  = ******** Q******•******Q  + ****a**** ****L****2**** + ****b**** K****J****2**** + ****a****sym ****C****2**** + ****a****3**** ****C****3**

**this Hamiltonian becomes ...**

**+ one-body and two-body terms**

**+ H****s.p.****************H****s.p.************************G******** H****P****************G******** H****P**********

**Rewriting**

** **
**H = H****sp********+********H****sp************+********G******** H****P********+********G******** H****P************+********Q**•**Q******
**    ****+  ****a**** ****L****2**** + ****b**** K****J****2**** + ****a****sym**** ****C****2**** + ****a****3 ****C****3**
** **

- .

**System Hamiltonian**

## Parameters of the Pseudo-SU(3) Hamiltonian

- Fit to experiment(fine tuning): a, b, asym and a3

**********  ****= 0.0637   ************ **** = 0.60**
**********  ****= 0.0637   ************= 0.60**

- From systematics:
-   = 35/A 5/3 MeV
- G****= 21/A MeV
- G****= 19/A MeV
- h = 41/A 1/3 MeV

## Energy spectrum for 160Gd

- 0

- 0.5

- 1

- 1.5

- 2

- 2.5

**K**

****

****

**0**

**+**

**K**

****

****

**+**

**K**

****

****

**0**

**2**

**+**

**160**

**Gd**

- 0

- +

- 6

- +

- 0

- +

- 2

- +

- 2

- +

- 2

- +

- 4

- +

- 4

- +

- 8

- +

- 6

- +

- 3

- +

- 8

- +

- 7

- +

- 6

- +

- 4

- +

- 5

- +

- 8

- +

- Exp. Th.

- Exp. Th.

- Exp. Th.

- Energy spectrum for 160Gd

## Direct Product Coupling

**Coupling ****proton ****and ****neutron**** **
**irreps to total (coupled) SU(3):**

****************************************************
******************+********************************+******************
******************+********************-**************** +********************+******
******************+************************************+********************-******
******************+********************-********************+********************-**********
****
******m,l ****************+****************-2m+l********************+************+m-2l)****k**

**Twist**

**Scissors**

**Scissors + Twist**

**... ****Orientation of the ********-******** system is quantized **
**     ****with the multiplicity denoted by k = k(m,l)**

**Direct Product Coupling**

## M1 Transition Strengths  [ 2N]in the Pure Symmetry Limit of the Pseudo SU(3) Model

- () () ()

- ()1+

**M1 Transition Strengths  [**** ********2****N****]****in the Pure Symmetry Limit of the ****Pseudo SU(3) Model**

## Energy levels of 164Dy

- 164

- Dy

- e)

- K = 2

- K = 0

- K = 0

- 2

- +

- 4

- +

- 6

- +

- 0

- +

- 2

- +

- 4

- +

- 6

- +

- 0

- +

- 2

- +

- 4

- +

- 6

- +

- 0

- +

- 2

- +

- 4

- +

- 0

- +

- 2

- +

- 4

- +

- 0

- +

- 2

- +

- 4

- +

- 6

- +

- 0

- +

- 2

- +

- 4

- +

- 6

- +

- 3

- +

- 5

- +

- 2

- +

- 4

- +

- 6

- +

- 3

- +

- 5

- +

- g.s.

- c)

- Energy [MeV]

- Exp. Th.   Exp. Th.   Exp. Th.   Exp. Th.

**Energy levels of ****164****Dy**

**... ****and M1 Strengths**

## Energy Levels of 168Er

- 168

- Er

- K = 0

- K = 0

- K = 2

- 0

- +

- 2

- +

- 8

- +

- 6

- +

- 4

- +

- 0

- +

- 2

- +

- 8

- +

- 6

- +

- 4

- +

- 0

- +

- 2

- +

- 8

- +

- 6

- +

- 4

- +

- 0

- +

- 2

- +

- 4

- +

- 2

- +

- 8

- +

- 6

- +

- 4

- +

- 7

- +

- 3

- +

- 5

- +

- 2

- +

- 8

- +

- 6

- +

- 4

- +

- 7

- +

- 3

- +

- 5

- +

- 0

- +

- 2

- +

- 6

- +

- 6

- +

- 4

- +

- 0

- +

- 2

- +

- 6

- +

- 4

- +

- g.s.

- Energy [MeV]

- Exp. Th.

- Exp. Th.

- Exp. Th.

- Exp. Th.

- a)

**Energy Levels of ****168****Er**

**... ****and M1 Strengths**

## Total B(M1) strength (N2)

- B(M1)[******N****2**]

**Total B(M1) strength (********N****2****)**

## First excited K=0+ and K=2+ states

## Slide 23

## Slide 24

## Slide 25

## Slide 26

## SU(3) content in 172Yb

- 0

- 20

- 40

- 60

- 80

- 100

- 0

- 20

- 40

- 60

- 80

- 100

- 0

- gs

- 2

- 0

- 2

- 0

- gs

- 2

- 0

- 2

- a = (36,0)(12,0)x(24,0)

- b = (28,10)(12,0)x(16,10)

- c = (20,20)(4,10)x(16,10)

**SU(3) content in**** 172****Yb**

## Conclusions

- B(E2) transitions within the g.s. band well
- reproduced

- ground state,  , first and second excited K = 0+ bands well described by a few representations

- calculated results in good agreement with  the low-energy spectra

- 1+ energies fall in the correct energy range
- fragmentation in  the B(M1) transition probabilities correctly predicted

## Conclusions

- A proper description of collective properties of the first excited K = 0+*** ***and*** ***K=2+*** **** *states must take into account the mixing of different SU(3)-irreps, which is driven by the Hamiltonian.

- A microscopic interpretation of the relative position of the collective band, as well as that of the levels within the band, follows from an evaluation of the primary SU(3) content of the collective states. The latter are closely linked to nuclear deformation.
-  If the leading configuration is triaxial (nonzero ), the ground and  bands belong to the same SU(3) irrep;
-  if the leading SU(3) configuration is axial (=0), the K =0 and  bands come from the same SU(3) irrep.

## Future work

-  In some nuclei total strength of the M1 distribution
- is larger than the experimental value

- Investigate the energy spectra in super-heavy nuclei

- Investigate the M1 transitions in light nuclei

-  There are new experiments that determine the inter-band B(E2) transitions
-  Improve the model to calculate these transitions

-  Consider the abnormal parity levels

-  Consider the states with J = 1 in the configuration space

## Thank you!