High phonon states of quadrupole vibration in spherical nuclei

Volhme2, number 6

HIGH

PHYSICS

PHONON

LETTERS

15 October 1962

STATES OF QUADRUPOLE IN SPHERICAL NUCLEI

VIBRATION

Y a s u k a z u YOSHIZAWA Institute for Theoretical Physics, Copenhagen, Denmark Received 20 September 1962

R e c e n t l y m a n y s t r o n g low e n e r g y g a m m a r a y s (< 1 MeV) h a v e b e e n o b s e r v e d b e t w e e n e x c i t e d s t a t e s of p d l 0 6 b y S m i t h 1), C d l 0 8 b y K a t o h et a l . 2), c d l l 0 b y K a t o h et a l . 3 , 4 ) and C d l l 4 b y S m i t h e r 5). In c o m p a r i s o n with t h e b r a n c h i n g r a t i o s obtained from the Weisskopf estimate, these trans i t i o n s (< 1 MeV) a r e m u c h s t r o n g e r than high e n e r gy t r a n s i t i o n s (> 1 MeV). T h e s p h e r i c a l v i b r a t i o n m o d e l 6) s u g g e s t s t h a t one phonon t r a n s i t i o n s a r e e n h a n c e d , w h i l e two o r h i g h e r phonon t r a n s i t i o n s a r e h i n d e r e d . To e x p l a i n t h i s f a c t f o r t r a n s i t i o n s f r o m s e c o n d 2+ s t a t e s , s e v e r a l n u c l e a r m o d e l s h a v e b e e n p r o p o s e d . In t h i s p a p e r , an a t t e m p t i s m a d e to a s s i g n h i g h e r e x c i t e d s t a t e s in t e r m s of t h e s p h e r i cal vibrational model. In t h e v i b r a t i o n a l r e g i o n t h e E2 c o m p o n e n t of t h e t r a n s i t i o n b e t w e e n t h e s e c o n d and t h e f i r s t 2 + s t a t e s i s p r e d o m i n a n t , i . e . , M 1 / E 2 ~ 0.01. T h i s c o m p o n e n t is stronger than the crossover transition, i.e., B ( E 2 , 2 ' ~ O ) / B ( E 2 , 2 " * 2 ) = 0.001 - 0 . 1 , w h e r e 0, 2 and 2' d e n o t e t h e g r o u n d , t h e f i r s t 2 + and t h e s e c o n d 2+ s t a t e , r e s p e c t i v e l y . A s s u m i n g t h a t M 1 / E 2 << 1 f o r t r a n s i t i o n s f r o m higher excited states as well as for the transition f r o m t h e s e c o n d 2+ s t a t e , r e l a t i v e r e d u c e d t r a n s i tion p r o b a b i l i t i e s of g a m m a r a y s e m i t t e d f r o m t h e same excited state are evaluated from experimental i n t e n s i t i e s . F i g . 1 s h o w s the l e v e l s c h e m e s of e i g h t n u c l e i e x t e n s i v e l y s t u d i e d and t h e r e l a t i v e r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s . R e f e r e n c e s a r e m a d e to N u t l e y and G e r h a r t 7) f o r P d 104, S m i t h 1) f o r p d l 0 ~ , K a t o h et a l . 2) f o r Cd 108, K a t o h et a l . 3 , 4 ) f o r c d l l 0 , S m i t h e r 5) f o r Cd 114, R o b i n s o n et a l . 8) f o r Xe 132, J o h n s o n et a l . 9) f o r Xe 134 and J o h n s and N a b l o 10) and B a g g e r l y e t a l . 11) f o r P t 192. A c c o r d i n g to the s e l e c t i o n r u l e f o r t h e t o t a l o s c i l l a t o r n u m b e r N , i . e . , AN = 1, a s s i g n m e n t s f o r t h i s n u m b e r can be m a d e to h i g h e r e x c i t e d s t a t e s . F o r t h e low e n e r g y t r a n s i t i o n s (< 0.3 MeV), even if t h e r a t i o of t h e r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s , B(M1)/B(E2), i s the s a m e a s t h a t f o r t h e 2 ' - 2 t r a n s i t i o n , the i n t e n s i t y r a t i o M 1 / E 2 b e c o m e s * On leave of absence from Faculty of Science, Osaka University, Osaka, Japan.

l a r g e , b e c a u s e of the e n e r g y d e p e n d e n c e of t h e t r a n s i t i o n p r o b a b i l i t y . In a d d i t i o n , t h e E2 and M1 transitions are hindered for the AN = 0 transition with t h e M1 t r a n s i t i o n s b e i n g t h e s t r o n g e r a s e x p e c t e d f r o m c o n f i g u r a t i o n m i x i n g . We t h u s n e g l e c t the B(E2) v a l u e of the low e n e r g y t r a n s i t i o n in t h e a s s i g n m e n t . T h e v a l u e s shown in fig. 1 a r e n o r m a l i s e d to t h e l a r g e s t r e l a t i v e B(E2) v a l u e a m o n g t r a n s i t i o n s h i g h e r than 0.3 MeV. F o r c o m p a r i s o n we t a k e t h e e n e r g y r a t i o of t h e e x c i t e d s t a t e to the f i r s t 2+ s t a t e i n s t e a d of t h e l e v e l e n e r g y . S t e l s o n and McGowan 12) and E c c l e s h a l l et al. 13,14) h a v e r e p o r t e d l a r g e B(E2) v a l u e s of t h e t r a n s i t i o n s f r o m t h e s e c o n d 2 + and t h e f i r s t 4 + s t a t e s to t h e f i r s t 2+ s t a t e . T h e r e f o r e t h e s e s t a t e s a r e a s s i g n e d a s N = 2. To d i s c u s s t h r e e phonon s t a t e s N = 3, e x p e r i m e n t a l and t h e o r e t i c a l 15) t r a n s i t i o n p r o b a b i l i t i e s b e t w e e n t h r e e and two phonon s t a t e s a r e s u m m a r i s e d in t a b l e s 2 and 3, r e s p e c t i v e l y . Table 1 Relation between phonon numbers and energy ratios of excited states to the first 2+ state. N

Ra~o

R~io~

1.00 1.9-2.5 2.8-3.9 3.8-4.8 4.8-

1.00 0.9-1.3 0.9-1.3 0.9-1.2 0.9-

The r e m a r k a b l e p o i n t s a r e a s f o l l o w s : 1. The e n e r g i e s of t h e v i b r a t i o n a l l e v e l s f a l l into d i s t i n c t i n t e r v a l s c h a r a c t e r i z e d by t h e phonon n u m b e r N, a s shown in t a b l e i . T h e v a l u e s of the e n e r g y r a t i o s a r e in q u a l i t a t i v e a g r e e m e n t w i t h t h e m o d e l of h a r m o n i c v i b r a t i o n a b o u t s p h e r i c a l e q u i librium. 2. T h e r a t i o s of t h e r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s show t h a t B (E2, AN=2)/B(E2, AN=l) = 0.001 0.1. T h i s i s s i m i l a r to the r a t i o B ( E 2 , 2 ' ~ 0 ) / B(E2,2'~2). 3. The a g r e e m e n t i s f a i r l y good b e t w e e n t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s of t h e r e l a t i v e r e 261

Volume 2, number 6

PHYSICS

6

LETTERS

N I~ ~_.

5

15 October 1962

N

°

N I~

I~

lllll

8S ~d ~n

4

II'l|

'~q

(4) 4

ill

o .,, I

3 44

%3

3

4 5,6"

i

, I

t

6+

2~

iiiiii 3 ~4

1'1 i

!

e~

24.

2

2+

~11

1

2+

0

O*

,

1[11, 1

o. I .,I

II

i

! I I(~

2

0

00~

1'

0

I/

O*

~'

p d 106

p d 104

5

Cd 108

Cd 110

1~

N I=

N lR

(2*)T~

2 (2~)

2 2+

1

1

c~

4

%3 2

2 ~)

2,,

~ J

2*

2* -~

I

I

.

0

o~

Cd 114

If

Y v

~' V

X e 132

o o,

o o÷--J

X e 134

p t 192

Fig. 1. Level schemes and relative reduced transition probabilities. N denotes the estimated total oscillator number and In the experimental spin and parity. Numbers with arrows mean the relative B(E2) values. Thick and thin lines show the transitions having the relative r e duced probabilities l a r g e r than 0.1 and less than 0.1, respectively. d u c e d t r a n s i t i o n p r o b a b i l i t i e s f r o m t h r e e to two phonon s t a t e s e x c e p t in a f e w c a s e s . F r o m t h e s e v a l u e s an a t t e m p t i s m a d e to p r e d i c t t h e s p i n s of 262

t h e t h r e e p h o n o n s t a t e s , a s shown in t a b l e 2. T h e a s s i g n m e n t s f o r t h e phonon n u m b e r a r e m a d e to m o s t of t h e known s t a t e s . S i n c e o n l y high

Vol,u~ne 2, number 6

PHYSICS

LETTERS

15 October 1962

Table 2 Transitions from N = 3 states to N = 2 states. Initial state Nucleus

Energy (MeV)

pdl06

1.557

cdll0

Cd114

3, 4 +

Transition energy (MeV)

Relative gamma intensity

Relative B(E2) 4

1.127

2+

1.229

4+

0.430 0.328 0.803 0.703

2O 1~1 ~ 0 . 6 13 ±3 9 +2

1.0 0.4 ± 0 . 2 1.0O 0.75±0.3

8 4 6 10

1.0 0.8 +_0.2 1.0 2.7 ± 0 . 6

3,4 +

1.127 1.229

2+ 4+

2.162

3, 4 +

3,4 +

2+ 4+ 2+ 4+

0.688 0.619

2.218

1.474 1.542 1.474 1.542

1.730

2, 3, 4+

1.208 1.282 1.133 1.208 1.282

2+ 4+ 0+ 2+ 4+

0.522 0.448 0.707 0.631 (0.558)

1.62 1.74 1.62 1.74

2+ 4+ 2+ 4+

0.75 0.61 1.00 0.89

0.612 0.785 0.612 0.785

2+ 4+ 2+ 4+

0.309 0.136 0.589 0.417

I, 2+

2.34 2.64

pt192

Spin parity

Energy (MeV)

1.931

1.840

Xe134

Spin parity

Final state

0.921 1.201

3, 4+ 4+

Table 3 Relative reduced transition probabilities between three and two phonen states. The symbol Bo(E2D means BCE2, N = I ~ N= 0).

Initial state (N = 3) Spin and parity

Final state (N = 2) Spin and parity

B(E2) ~ Relative Bo (E 2) 4 B(E2)

0+

2+

3

1.00

2+

0+ 2+ 4+

7/5 4/7 36/35

2.45 1.00 1.80

3+

2+ 4+

15/7 6/7

1.00 0.40

4+

2+ 4+

11/7 10/7

1.00 0.91

e n e r g y g a m m a r a y s have been o b s e r v e d for s e v e r a l e x c i t e d s t a t e s , t h e s e s t a t e s cannot be a s s i g n e d . We e x p e c t m o s t of t h e s t a t e s without a s s i g n m e n t to be of p a r t i c l e e x c i t a t i o n . The r e l a t i v e r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s a r e l a r g e r than t h e unit f o r s o m e low e n e r g y t r a n s i t i o n s (~ 0.3 MeV). T h e s e t r a n s i t i o n s h a v e AN = 0 o r r e l a t e to the s t a t e s w i t h o u t th e phonon n u m b e r a s s i g n m e n t s . T h e r e f o r e , it i s p r o b a b l e that m o s t of t h e l a r g e v a l u e s r e s u l t f r o m p r e d o m i n a n t M1 t r a n s i tions. Exceptions a r e the t r a n s i t i o n s f r o m the third 2 + s t a t e to the f i r s t 0 + s t a t e in Cd 114 and f r o m t h e f i r s t 4 + s t a t e to t h e s e c o n d 2 + s t a t e in P t 192. It i s c o n c l u d e d that in t h e r e g i o n s i n v e s t i g a t e d

0.745 0.676

_+1 +1 +1 + 1.5

0.49 + 0.05 0.15 + O.01 1.6 + 0.2 0.16 _+0.02 (< 40) 1.3 19 +4 7 +_3 ~6

35

+_7

0.19 + 0.i0 7.1 _+1.4 1.6 _+0.8

Proposed spin of initial state

3or4

1.00 0 . ~ ±0.08 5.6 ± I . 0

(2)

1.00 1.00 42 (2. 20) 1.0 1.5 (± 1.0) 1.00 0.30 +_0.16 1.0

1.3

+ 0.7

h e r e t h e phonon n u m b e r i s a good q u a n t u m n u m b e r up to N = 4 o r 5, t h o u g h r a t i o s of M 1 / E 2 a r e u n certain. The a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e t h a n k s

to P r o f . N i e l s B o h r f o r h o s p i t a l i t y in h i s I n s t i t u t e and to P r o f . A. B o h r and P r o f . B. R. M o t t e l s o n f o r valuable discussions. 1) W.G.Smith, Phys. Rev. 122 (1961) 1600. 2) T.Katoh et al., Nuclear Physics 36 (1962) 394. T. Katoh, H. Gotoh and Y. Yoshizawa, J. Phys. Soc. Japan, to be published. 3) T. Katoh and Y. Yoshizawa, Nuclear Physics 32 (1962) 5. 4) T. Katoh, M. Nozawa and Y. Yoshizwa, Nuclear Phys. 32 (1962) 25. 5) R . K . S m i t h e r , Phys. Rev. 124 (1961) 183. 6) A. Bohr and.B.R. Mottleson, Dan. Mat. Fys. Medd. 27 (1953) no. 16. 7) H. Nutley and J . B . G e r h a r t , Phys. Rev. 120 (1960) 1815. 8) R.L.Robinson, E . E i c h l e r and N.R.Jolmson, Phys. Rev. 122 (1961) 1863. 9) N.R.Johnson et al., Phys. Rev. 122 (1961) 1546. 10) M.W. Johns and S.V. Nablo, Phys. Rev. 96 (1954) 1599. 11) L . L . Baggerly et al., Phys. Rev. 100 (1955) 1364. 12) P.H.Stelson and F.K. McGowan, Phys. Rev. 121 (1961) 209. 13) D. Eccleshall, B.M. Hinds and M . J . L . Yates, Nuclear Phys. 32 (1962) 190. 14) D. E ccleshall, B.M. Hinds, M . J . L . Yates and N. MacDonald, private communication. 15} B.R. Mottelson, private communication. 263