Volume 27B. number 4 P H Y S I C S L E T T E R S 8 July 1968

A P P L I C A B I L I T Y O F T H E S T A T I S T I C A L M O D E L T O T H E D E S C R I P T I O N O F T H E P O P U L A T I O N O F G R O U N D - S T A T E

R O T A T I O N A L L E V E L S

S. J. M I L L S and W. L. R A U T E N B A C H National Physical Research Laborator>'. CSIR. Pretoria. South Afr ica

Received 30 May 1968

Compar ison of the re la t ive in tensi t ies of ground-s ta te rotational t rans i t ions occurr ing in 178W. 180W and 182W. following (a , 2ny) reac t ions , shows that the s ta t i s t ica l model of nuclear reac t ions is not applicable to the descr ip t ion of these in tensi t ies .

T h e s t a t i s t i c a l m o d e l of c o m p o u n d n u c l e a r r e - a c t i o n s [1,2] h a s r e c e n t l y b e e n e m p l o y e d to e x - p l a i n t he r e l a t i v e p o p u l a t i o n s of l e v e l s in t he g r o u n d - s t a t e r o t a t i o n a l b a n d s of n u c l e i p r o d u c e d in n e u t r o n - e v a p o r a t i o n r e a c t i o n s [3-6] .

A p p l i c a t i o n of t h e s t a t i s t i c a l m o d e l , h o w e v e r , i m p l i e s t h a t a l l t h e p r o c e s s e s t a k i n g p l a c e d e p e n d m a i n l y on g e n e r a l ( o r s t a t i s t i c a l ) p r o p e r t i e s of t h e n u c l e u s and v e r y l i t t l e on t he f i n e r d e t a i l s of n u c l e a r s t r u c t u r e . Such g e n e r a l p r o p e r t i e s i n - c l u d e p a r t i c l e t r a n s m i s s i o n c o e f f i c i e n t s , l e v e l d e n s i t i e s , e f f e c t i v e m o m e n t s of i n e r t i a and t h e n a t u r e of t h e y - r a y c a s c a d e to t he g r o u n d s t a t e of t h e p r o d u c t n u c l e u s . N u c l e a r s y s t e m s wi th n e a r - ly i d e n t i c a l g e n e r a l p r o p e r t i e s s h o u l d t h u s y i e l d n e a r l y i d e n t i c a l e x p e r i m e n t a l r e s u l t s if t he s t a t i s - t i c a l d e s c r i p t i o n of n u c l e a r p r o c e s s e s a p p l i e s . C o n v e r s e l y , a n e c e s s a r y c o n d i t i o n f o r o b t a i n i n g m e a n i n g f u l r e s u l t s f r o m " b e s t " f i t s b e t w e e n s t a t i s t i c a l m o d e l c a l c u l a t i o n s and e x p e r i m e n t a l d a t a i s t h a t n e a r - i d e n t i c a l s y s t e m s m u s t y i e l d a p p r o x i m a t e l y t h e s a m e e x p e r i m e n t a l r e s u l t s .

T h e a p p l i c a b i l i t y of t h e s t a t i s t i c a l m o d e l to t he d e s c r i p t i o n of t h e r e l a t i v e p o p u l a t i o n s of l e v e l s in t h e g r o u n d - s t a t e r o t a t i o n a l b a n d s of d e f o r m e d n u c l e i h a s t h e r e f o r e b e e n t e s t e d by m e a s u r i n g t h e c r o s s s e c t i o n s of t he 8 + ~ 6 + and 6 + ~ 4 + t r a n s i - t i o n s r e l a t i v e to t h a t of t h e 4 + ~ 2 + t r a n s i t i o n f o r e a c h of t he n u c l e i 178W, 180W and 182W, p r o d - u c e d in (a, 2n) r e a c t i o n s . T h e s e n u c l e i a r e a d - j a c e n t e v e n - e v e n i s o t o p e s in a r e g i o n b e t w e e n c l o s e d s h e l l s and h a v e n e a r l y i d e n t i c a l g r o u n d - s t a t e r o t a t i o n a l b a n d s [7]. On t he b a s i s of t h e s t a t i s t i c a l m o d e l n e a r l y i d e n t i c a l r e s u l t s s h o u l d t h e r e f o r e b e o b t a i n e d f o r t he s a m e v a l u e of t h e a - p a r t i c l e e n e r g y E a in e x c e s s of t h e (a, 2n) t h r e s h o l d e n e r g y E t h r .

T h e e x p e r i m e n t h a s b e e n p e r f o r m e d by b o m - b a r d i n g i s o t o p i c a l l y e n r i c h e d ox ide t a r g e t s of a b o u t 5 m g / ' c m 2 w i th 17 to 32 MeV a - p a r t i c l e s . T h e y - r a y s p e c t r a w e r e m e a s u r e d w i th a 5 m m × × 4 c m 2 Ge(Li ) d e t e c t o r w i t h a t y p i c a l r e s o l u t i o n of 5 keV FWHM, p l a c e d a t 90 ° to the b e a m d i r e c - t ion . Al l t he r o t a t i o n a l t r a n s i t i o n s h a v e b e e n a s s u m e d to h a v e i d e n t i c a l a n g u l a r d i s t r i b u t i o n s w i t h i n e x p e r i m e n t a l e r r o r s [5,8]. T h e 2 + - 0 + t r a n s i t i o n s h a v e not b e e n c o n s i d e r e d due to l a r g e e x p e r i m e n t a l e r r o r s and u n c e r t a i n t i e s in t h e l a r g e i n t e r n a l c o n v e r s i o n c o r r e c t i o n .

T h e e x p e r i m e n t a l r e s u l t s a r e s h o w n in fig. 1. T h e i n d i c a t e d e r r o r s a r e e s t i m a t e d e f f e c t i v e s t a n d a r d d e v i a t i o n s , a n d do not i n c l u d e any p o s - s i b l e s y s t e m a t i c e r r o r s e x c e p t t h o s e m a d e in t he d e t e r m i n a t i o n of t he p e a k a r e a s in t he y - r a y s p e c t r a .

A l s o s h o w n in fig. 1 a r e t he t h e o r e t i c a l r e s u l t s o b t a i n e d f o r d i f f e r e n t v a l u e s of 3 / ~ R , the r a t i o of t h e e f f e c t i v e m o m e n t of i n e r t i a to t he r i g i d - b o d y m o m e n t of i n e r t i a . In t h e s e c a l c u l a t i o n s [9] we h a v e e s s e n t i a l l y a d o p t e d t he p r o c e d u r e of B i s h o p et al . [10]. H o w e v e r , a s h a s b e e n p r o p o s e d b y S a r a n t i t e s and P a t e [11], t he l eve l d e n s i t i e s h a v e b e e n e v a l u a t e d on ly fo r t h a t p a r t of t h e t o t a l e x - c i t a t i o n e n e r g y w h i c h i s a s s o c i a t e d w i th t h e r m a l m o t i o n of t he n u c l e o n s . T h e e x p r e s s i o n f o r t he l e v e l d e n s i t y g i v e n by L a n g and Le C o u t e u r [12] h a s b e e n u s e d , and t he l e v e l d e n s i t y p a r a m e t e r h a s b e e n t a k e n as a = A / 1 0 . 7 , w h e r e A i s t he m a s s n u m b e r of t h e n u c l e u s [13]. T h e e f f e c t s of p a i r i n g of p r o t o n s and of n e u t r o n s in t he n u c l e u s h a v e a l s o b e e n t a k e n in to a c c o u n t in t he way d e s - c r i b e d by V a n d e n b o s c h et al . [14]. T h e p a i r i n g e n e r g y h a s b e e n t a k e n a s 5 = 1.0 MeV [15].

A s e x p e c t e d , t he t h e o r e t i c a l p r e d i c t i o n s of the

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Volume 27B. number 4 P H Y S I C S L E T T E R S 8 July 1968

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s t a t i s t i c a l m o d e l d i f f e r v e r y l i t t l e in t he t h r e e c a s e s fo r t he s a m e v a l u e of 9 / ~ R . H o w e v e r , t he a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l r e s u l t s i s v e r y p o o r , e s p e c i a l l y f o r t h e r a t i o of t he c r o s s s e c t i o n of the 8 + - 6 + t r a n s i t i o n to t h a t of t he 4 + -- 2 + t r a n s i t i o n . F u r t h e r m o r e , " b e s t " f i t s o c c u r a t qu i t e d i f f e r e n t v a l u e s of 9 / 9 R , a nd t h e r e i s a l a r g e v a r i a t i o n in t he q u a l i t y of t h e " b e s t " f i t s ob t a ined .

T h e c o n c l u s i o n c a n t h e r e f o r e b e d r a w n t h a t t h e s t a t i s t i c a l m o d e l a l o n e i s not a l w a y s c a p a b l e of d e s c r i b i n g t he r e l a t i v e p o p u l a t i o n s of g r o u n d - s t a t e r o t a t i o n a l l e v e l s in e v a p o r a t i o n r e a c t i o n s s a t i s f a c t o r i l y . T h e r e a s o n fo r t h i s f a i l u r e p r o b - a b l y l i e s in t h e f ac t t h a t p u r e l y s t a t i s t i c a l c o n - s i d e r a t i o n s s h o u l d no t b e a p p l i e d to t he ~ - c a s c a d e t h r o u g h t he l o w e r e x c i t e d s t a t e s of t he r e s i d u a l n u c l e u s , s i n c e t he m o r e d e t a i l e d e f f e c t s of n u c - l e a r s t r u c t u r e m a y be of c o n s i d e r a b l e i m p o r t a n c e .

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physics (John Wiley and Sons, New York, London, 1952).

2. V .F .Wei s skopf , P h y s . R e v . 5 2 (1937) 294. 3. S.J$/gare, Nucl. Phys .A95 (1967)481. 4. S.J~/gare, Nuc l ,Phys . A95 (1967) 491. 5. B . J . Sheperd, C. F. Wil l iamson and I. Halpern. Phys .

Rev. Le t t e r s ] 7 (1966) 806. 6. G.B. Hansen, B. Elbek. K.A. Hagemann and W. F.

Hornyak, Nucl. Phys . 47 (1963) 529. 7. N.L. Lark and H. Morinaga, Nucl. Phys . 63 (1965)

466. 8. J .O . Newton, F .S . Stephens, R .M. Diamond, K.

Kotaj ima and E. Matthias, Nucl. Phys . A95 (1967) 357. S. J. Mills, D. Sc. thes i s , Po tchefs t room Univers i ty for Chr is t ian Higher Education, 1968 (unpublished). C. T. Bishop, J . R . Huizenga and J. P. Hummel , P h y s . R e v . 135 (1964) B401. D. G. Sarant i tes and B. D. Pate , Nucl. Phys . A93 (1967) 545. J. M. Lang and K. J . Le Couteur, Proc . Phys . Soc. 67A (1954) 586. D . B . B e a r d and A.McLel lan , Phys . Rev. 131 (1963) 2664. R. Vandenbosch, L. Haskin and J. C. Norman, Phys . Rev.137 (1965) Bl134. P. E. Nemirovsky and Y.V.Adamchuk , Nucl. Phys . 39 (1962) 551.

9 .

10.

11.

12.

13.

14.

15.

Fig. 1. Compar i son between exper imenta l r e su l t s (circles) and s ta t i s t ica l model calculat ions (lines). The open c i r c l e s and solid l ines r e fe r to the 6 +--* 4 + t r an s i - tion, and the closed c i r c l e s and dashed l ines to the 8 + ~ 6 + t rans i t ion. All the c r o s s sect ions a re given

rela t ive to that of the 4 + ~ 2 + t rans i t ion .

208