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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

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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

Gabriela Popa

outline
Outline

Introduction

What we know about the nucleus

Characteristic energy spectraTheoretical Model

Configuration space

System HamiltonianResults Conclusion and future work

Gabriela Popa

chart of the nuclei

Chart of the nuclei

Z (protons)

N (number of neutrons)

Gabriela Popa

nuclear vibrations
3

3

Nuclear vibrations

2

2

1

1

-vibration

-vibration

Gabriela Popa

schematic level schemes of spherical and deformed nuclei
Schematic level schemes of spherical and deformed nuclei

8

6

6

5

6

0, 2, 4

4

4

3

2

8

0

2

2

6

4

2

0

0

spherical

deformed

E = n 

E = I(I+1)/(2)

single particle energy levels
Single particle energy levels

N = 5

1h

h11/2

s1/2

3s

d3/2

d5/2

2d

N = 4

g7/2

1g

f9/2

p1/2

2p

f5/2

N = 3

p5/2

1f

f7/2

2s

d3/2

N = 2

s1/2

1d

d5/2

p1/2

1p

N = 1

p3/2

1s

N = 0

s1/2

l·s

l·l

Gabriela Popa

particle distribution
Particle distribution

Valence space: U(Wp) U(Wn)

totalnormal unique

Wp =32 Wnp =20 Wup =12

Wn =44 Wnn =30 Wun=14

  • Particle distributions: (l,m)
  • protons: neutrons: p n total
  • 152Nd 10 6 4 10 6 4(12,0)(18,0) (30, 0)
  • 156Sm 12 6 612 6 6(12,0)(18,0) (30, 0)
  • 160Gd 14 8 614 8 6(10,4)(18,4) (28, 4)
  • 164Dy 16 10 616 10 6(10,4)(20,4) (30, 4)
  • 168Er 18 10 818 10 8(10,4)(20,4) (30, 4)
  • 172Yb 20 12 820 12 8(36,0)(12,0) (24, 0)
  • 176Hf 22 14 822 14 8(8,30)(0,12) (8, 18)

Gabriela Popa

wave function
Wave Function

| =  Ci | i 

|i = |{; } (,)KL S; JM

 ( = , ) = n [f ] ( ,  ), S 

( ,  ) ( ,  ) =  (, ) 

Gabriela Popa

direct product coupling
Direct Product Coupling

Coupling proton and neutron

irreps to total (coupled) SU(3):

(lp, mp)  (ln, mn)

 (lp+ln , mp+mn)

+ (lp+ln- 2,mp +mn+ 1)

+ (lp+ln+1,mp+mn- 2)

+ (lp+ln- 1,mp+mn- 1)2

+ ...

 S m,l (lp+ln-2m+l,mp+mn+m-2l)k

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

Gabriela Popa

21 st su 3 irreps corresponding to the highest c 2 values were used in 160 gd
21st SU(3) irreps corresponding to the highestC2 values were used in 160Gd

(10,4)

(18,4)

(28,8)

(29,6)

(30,4)

(31,2)

(32,0)

(26,9)

(27,7)

(10,4)

(20,0)

(30,4)

(10,4)

(16,5)

(26,9)

(27,7)

(10,4)

(17,3)

(27,7)

(12,0)

(18,4)

(30,4)

(12,0)

(20,0)

(32,0)

(8,5)

(18,4)

(26,9)

(27,7)

(9,3)

(18,4)

(27,7)

(7,7)

(18,4)

(25,11)

(26,9)

(27,7)

(7,7)

(20,0)

(27,7)

(4,10)

(18,4)

(22,14)

(,)(,)

(,)

Gabriela Popa

tricks
Tricks

Invariants Invariants

Rot(3) SU(3)

Tr(Q2)  C2

Tr(Q3)  C3

b2~ l2+ lm + m + 3(l + m+1)

g =tan-1 [3 m / (2 l + m+3)]

Gabriela Popa

system hamiltonian
System Hamiltonian

SU(3) conserving Hamiltonian:

H = c Q•Q + aL2 + b KJ2 + asym C2 + a3C3

+ one-body and two-body terms

+ Hs.p.p + Hs.p.n+Gp HPp +Gn HPn

Rewriting

this Hamiltonian becomes ...

H = Hspp +Hspn+Gp HPp +Gn HPn+ cQ•Q

+ aL2 + b KJ2 + asymC2 + a3 C3

Gabriela Popa

.

parameters of the pseudo su 3 hamiltonian
Parameters of the Pseudo-SU(3) Hamiltonian

From systematics:

 = 35/A 5/3 MeV

Gp = 21/A MeV

Gn = 19/A MeV

h = 41/A 1/3 MeV

p= 0.0637 p = 0.60

n= 0.0637 n= 0.60

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

Gabriela Popa

energy spectrum for 160 gd
Energy spectrum for 160Gd

Exp. Th.

Exp. Th.

Exp. Th.

2.5

+

8

2

+

160

8

+

Gd

6

+

+

7

4

1.5

+

6

+

2

+

5

+

0

+

4

p

+

+

=

K

0

1

3

+

2

+

8

2

p

+

= 2

K

+

0.5

6

+

4

+

2

+

0

0

p

+

=

K

0

Gabriela Popa

direct product coupling1
Direct Product Coupling

Coupling proton and neutron

irreps to total (coupled) SU(3):

(lp, mp)  (ln, mn)

 (lp+ln , mp+mn)

+ (lp+ln- 2, mp +mn+ 1)

+ (lp+ln+1, mp+mn- 2)

+ (lp+ln- 1, mp+mn- 1)2

+ ...

m,l  (lp+ln-2m+l,mp+mn+m-2l)k

Scissors

Twist

Scissors + Twist

… Orientation of the p-n system is quantized

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

Gabriela Popa

m1 transition strengths m 2 n in the pure symmetry limit of the pseudo su 3 model
M1 Transition Strengths [m2N]in the Pure Symmetry Limit of the Pseudo SU(3) Model

Nucleus

B(M1)

mode

160 ,162

Dy

(10,4)

(18,4)

(28,8)

(

29,

6)

0.56

t

(

26,

9)

1.77

s

1

(27

;

7)

1.82

s+t

2

(27

;

7)

0.083

t+s

164

Dy

(10,4)

(20,4)

(30,8)

(31,6)

0.56

t

(28,9)

1.83

s

(29,7)

1.88

s+t

(29,7)

0.09

t+s

()() ()

()1+

Gabriela Popa

energy levels of 164 dy
Energy levels of 164Dy

2.5

2

1.5

1

0.5

0

-0.5

Exp. Th. Exp. Th. Exp. Th. Exp. Th.

+

4

+

2

+

6

+

+

+

6

4

0

+

+

4

+

4

2

+

+

+

2

2

0

+

0

+

164

0

Dy

Energy [MeV]

K = 0

+

K = 0

+

6

6

+

+

5

5

+

+

4

4

+

3

+

3

+

2

+

+

2

6

+

6

K = 2

+

+

4

e)

c)

4

+

2

+

2

+

+

0

0

g.s.

… and M1 Strengths

Gabriela Popa

energy levels of 168 er
Energy Levels of 168Er

3

2.5

2

1.5

1

0.5

0

Exp. Th.

Exp. Th.

Exp. Th.

Exp. Th.

+

6

168

Er

+

4

+

2

+

0

+

6

+

8

Energy [MeV]

+

8

+

4

+

8

+

6

+

6

+

7

+

+

2

7

+

+

4

4

+

+

+

6

0

6

+

2

+

2

+

+

5

5

+

0

+

0

K = 0

+

+

4

4

+

+

8

8

+

+

3

3

+

+

2

2

+

+

6

6

K = 0

a)

+

4

+

4

K = 2

+

2

+

2

+

+

0

0

g.s.

… and M1 Strengths

Gabriela Popa

total b m1 strength n 2
Total B(M1) strength (N2)

Nucleus

Calculated

Experiment

Pure

S

U

(3)

Theory

160

Dy

2.48

4.24

2.32

162

Dy

3.29

4.24

2.29

164

Dy

5.63

4.36

3.05

B(M1)[N2]

Gabriela Popa

su 3 content in 172 yb
SU(3) content in 172Yb

0

2

0

gs

2

100

100

80

80

60

60

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

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

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

40

40

20

20

0

0

0

2

0

gs

2

Gabriela Popa

conclusions
Conclusions

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
  • B(E2) transitions within the g.s. band well
  • reproduced
  • 1+ energies fall in the correct energy range
  • fragmentation in the B(M1) transition probabilities correctly predicted

Gabriela Popa

conclusions1
Conclusions
  • 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.
  • A proper description of collective properties of the first excited K = 0+andK=2+states must take into account the mixing of different SU(3)-irreps, which is driven by the Hamiltonian.

Gabriela Popa

future work
Future work
  • In some nuclei total strength of the M1 distribution
  • is larger than the experimental value
  • Consider the states with J = 1 in the configuration space
  • Consider the abnormal parity levels
  • There are new experiments that determine the inter-band B(E2) transitions
  • Improve the model to calculate these transitions
  • Investigate the M1 transitions in light nuclei
  • Investigate the energy spectra in super-heavy nuclei

Gabriela Popa

slide31
Thank you!

Gabriela Popa

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