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Spacing of class II levels in the fission of 240Pu
Volume
30B,
number
8
PH
SPACING
OF
CLASS
II
Y S IC S
LEVELS
L E T T E R S
IN
THE
8 December
FISSION
OF
1969
240pu
D. K. SOOD and N. SARMA Bhabha Atomic Research Centre, Bombay, India
Received 5 September 1969
A modified correlation analysis of high resolution fission cross sections is suggested which would yield more reliable estimates of the spaeing, DII of the class II levels. Subsequent analysis of the reaction 239pu(n,f) gives the value of/)iI as 225 :~ 25 eV for the nucleus 240Pu.
A second m i n i m u m (1) in the d e f o r m a t i o n p o t e n t i a l of nuclei in the a c t i n i d e r e g i o n has been shown [2,3] to lead to the e x i s t e n c e of f i s s i o n i s o m e r s and to g r o s s s t r u c t u r e in f i s s i o n c r o s s s e c t i o n s n e a r the threshold. The l e v e l spacing, DII of c l a s s II l e v e l s built on the s e c o n d m i n i m u m has been d e t e r m i n e d [4-6] by applying a c o r r e l ation a n a l y s i s as s u g g e s t e d by E g e l s t a f f [7]. It has h o w e v e r been pointed out [8] that such c o r r e l a t i o n a n a l y s e s r e s u l t in a m b i g u o u s v a l u e s of DII. In this p a p e r we s u g g e s t a m e t h o d f o r the a n a l y s i s of f i s s i o n c r o s s s e c t i o n s which would yie ld m o r e r e l i a b l e e s t i m a t e s of DII" The method is then applied to the data [9] on the 239pu(n, f) reaction. Th e c o r r e l a t i o n a n a l y s i s c o m p r i s e s a study of the s e r i a l c o e f f i c i e n t s , rk(W) f o r an a v e r a g i n g i n t e r v a l W, defined as
2.0
1.0
t
aj (50)
0.0 -- ~ ~
n
j~= l aj
rk(W) =r n
(W)~+k (W) n q½
L~l(aj(w))2~l(~+k(W,)2j
(1)
w h e r e the aj(W) a r e d e r i v e d f r o m e x p e r i m e n t a l c r o s s s e c t i o n s , (rF(E) a c c o r d i n g to the f o r m u l a
jW a (IV)--
(j-fl)W
~ aF(E)4-E L~F(E)JE)-
I]
(2)
One would expect the rk(W) to v a n i s h as a r e sult of the r a n d o m n a t u r e of the compound nuc l e u s ( c l a s s I) r e s o n a n c e s which d o m i n a t e (~F(E). Highly s i g n i fi can t v a l u e s of rk(W) will a r i s e f r o m i n t e r m e d i a t e ( c l a s s II) s t a t e s in the f i s s i o n i n g nucleus. H o w e v e r the e f f e c t s due to t h e finite r a n g e of data and the B r e i t - W i g n e r f o r m of the
-I.0 ~
I0
20
30
40
50
j-.. Fig. 1. A typical plot of the aj(W) sequence for W : 50 eV. c l a s s I r e s o n a n c e s m ay a l s o give r i s e to s p u r i ous high v a l u e s of rk(W) , even in the a b s e n c e of c l a s s II l e v e l s . P a t r i c k and J a m e s [5,6] i n c o r p o r a t e high l e v e l s of s i g n i f i c a n c e to avoid such s p u r i o u s effects. A f t e r a d e t a i l e d study of c o m p u t e r - g e n e r a t e d c r o s s s e c t i o n s , P e r e z et al. [8] conclude that the u s e of c o r r e l o g r a m s [4-6] f o r d e t e r m i n i n g the widths and s p a c i n g s of c l a s s II l e v e l s y i e l d s am b i g u o u s r e s u l t s . The s t a t i s t i c a l 523
Volume 3OB, number 8
PHYSICS
LETTERS
8 December 1969
0.4 4.50
0-3
(c)
20
0-2
0"1
1
0
i
450
(b)
40
6'0
k-,
I. z
50
2o
n
r k (20) 0.4
1% Significance level
0Z I
0.3
I
450 (0)
825 0,2
I0%
50
0'1 40
I
o.o
40
I
k ~
50
-0.I
I0% Si@nificonce level -0.2
Io
i
502 kW
-0.3 L
J
L
L
200
400
600 kW
"
800
( eV)----
2. C o r r e J o g r a m s shov¢ing rk(W ) v e r s u s kW for (a) W = 20 eV and (b) W = 10 eV. k s c a l e in the two figures is adjusted to c o r r e s p o n d to kW scale. Levels of significance of 10% and 1% are shown by full lines. Fig.
t e s t s p r o p o s e d by t h e m a r e n e v e r t h e l e s s e q u a l l y i n s e n s i t i v e a n d do not l e a d b e y o n d t h e d e t e c t i o n of i n t e r m e d i a t e s t r u c t u r e . F r o m o u r s t u d y of c o r r e l o g r a m s (fig. 2), w e
524
[
1002
1502
( e v ) --~
Fig. 3. kW frequency d i s t r i bution h i s t o g r a m s for levels of significance of (a) 10%, (b) 1% and (c) 0.1%. The step width is 50 eV. Peak p o s i tions in eV are indicated by arrows.
f i n d t h a t t h e m a g n i t u d e of r k ( W ) i s v e r y s e n s i t i v e t o t h e c h o i c e of W. T h i s i m p l i e s t h a t in s p i t e of t h e i n t r o d u c t i o n of a h i g h l e v e l of s i g n i f i c a n c e , o n e m a y e r r in t h e v a l u e of D i i . It i s t h e r e f o r e
Volume 30B, number 8
PHYSICS
essent~',~l to s c a n t h e e n t i r e s u r f a c e of rk(W), k , W. W k: s u g g e s t that the m o s t a p p r o p r i a t e m e t h o d to e s t i m a t e DII is to c o n s t r u c t the f r e q u e n c y d i s tribution histogram for the product kW for all the p a i r s of k, W f o r w h i c h r k ( W ) l i e s a b o v e a p r e s e t l e v e l of s i g n i f i c a n c e , d u r i n g a r e g u l a r s c a n of t h e s u r f a c e . In s u c h h i s t o g r a m s one o b s e r v e s s e v e r a l p e a k s a p p e a r i n g a l m o s t p e r i o d i c a l l y and s u p e r i m p o s e d on a b a c k g r o u n d c o n t r i b u t e d by s p u r i o u s rk(W). T o i n v e s t i g a t e t h e r e l a t i o n s h i p of t h e s e p e a k s w i t h the p r o p e r t i e s of t h e c l a s s II l e v e l s , we s t u d i e d t h e k W f r e q u e n c y d i s t r i b u t i o n h i s t o g r a m s f o r s e v e r a l s y n t h e t i c c r o s s s e c t i o n data. A s y s t e m a t i c v a r i a t i o n of h e i g h t s , w i d t h s and p o s i t i o n s of t h e b u i l t in c l a s s II l e v e l s i n d i c a t e s that a) t h e p o s i t i o n s of t h e p e a k s a r e d e t e r m i n e d by t h e s p a c i n g d i s t r i b u t i o n and t h e f i r s t p e a k o c c u r s at k W ~ DII , and b) the p e a k h e i g h t s a r e r e l a t e d to t h e d i s t r i b u t i o n s of h e i g h t s and w i d t h s of t h e b u i l t - i n c l a s s II resonances. T h e f i r s t p e a k t h e r e f o r e is not n e c e s s a r i l y the s t r o n g e s t w h i c h r e q u i r e s that k W f r e q u e n c y d i s t r i b u t i o n h i s t o g r a m s h a v e to b e e x a m i n e d at m o r e t h a n one l e v e l of s i g n i f i c a n c e to e n s u r e that the f i r s t p e a k is not m i s s e d . With t h e s e c o n s i d e r a t i o n s , we a n a l y s e d t h e d a t a of Shunk et al. [9] on the r e a c t i o n 2 3 9 p u ( n , f). T h e s t r u c t u r e in t h e aj(W) s e q u e n c e ( s e e fig. 1) s h o u l d b e only due to c l a s s II l e v e l s s i n c e t h e a v e r a g i n g i n t e r v a l of 50 eV > D I ~ 9 eV [11 ]. _ F r o m a s t u d y of s u c h a~(W) p l o t s , we o b t a i n D I I ~ ~ 2 1 0 + 30 eV and widtti ~ 125 =L 50 eV. Fig. 2 s h o w s c o r r e l o g r a m s f o r W = 10 and 20 eV, w h e r e at k W ~ 230 eV, r k ( W ) r i s e s r a p i d l y f r o m b e l o w 10% l e v e l (fig. 2b) to 1% l e v e l (fig. 2a). Fig. 3 s h o w s the k W f r e q u e n c y d i s t r i b u t i o n h i s t o g r a m s
LETTERS
8 December 1969
f o r 10%, 1% and 0.1% l e v e l s . F o u r d i s t i n c t p e a k s a p p e a r at k W = 9 + 2, 225 ± 25, 450 ~= 50 and 825 ~ 25 eV. R a p i d v a r i a t i o n s in t h e h e i g h t s of t h e s e p e a k s ( s e e fig. 3a, b, c) i l l u s t r a t e t h e n e c e s s i t y f o r e x a m i n i n g s e v e r a l l e v e l s of s i g n i f i c a n c e b e f o r e d r a w i n g any c o n c l u s i o n s . T h e p e a k at k W = 9 eV c o r r e s p o n d s t o / ) I f o r 0 + l e v e l s [11]. A v a l u e o f D i i = 225 ± 25 eV is o b t a i n e d f r o m the a n o m a l y at k W = 225 eV w h e r e a s t h e p e a k s at 450 and 825 eV a r e p r o b a b l y due to h i g h e r " h a r m o n i c s " . By u s i n g the r a t i o DII//)I = = 25 and t h e l e v e l d e n s i t y f o r m u l a of C a m e r o n [12], we find that the s e c o n d m i n i m u m is s i t u a t e d at 1.49 MeV a b o v e t h e f i r s t m i n i m u m .
References 1. w, M. Strutinsky, Nucl. Phys. A95 (1967) 420. 2. H. Weigmann, Z. Physik 214 (1968) 7. 3. J. E. Lynn, IAEA Syrup. Phys. and Chem. Fission, Vienna (1969); Contribution SM-122/204. 4. M. Cao, E. Migneco and J. P. Theobald, Phys. Lett e r s 27B (1968) 409. 5. B.H. Patrick and G. D. James, Phys. Letters 28B (1968) 258. 6. G.D. James and B. H. Patrick, IAEA Syrup. on Phys. and chem. fission, Vienna (1969); Contribution SM122/53. 7. P.A. Egelstaff, J. Nucl. Energ~y 7 (1958) 35.
8. R.B. P e r e z , G. de Saussure and M. N. Moore, IAEA Symp. on Phys. and chem. fission, Vienna (1969); Contribution SM-122/121. 9° E. R. Shunk, W.K. Brown and R. La Banve, CONF660303 (1966) 979. 10. W. C. Hamilton, Statistics in physical science (The Ronald P r e s s Company, New York, 1964) p. 183. 11. H. Derrien et al., Proc. IAEA Conf. on Nucl. data for r e a c t o r s , Paris (1966) Vol. II, p. 195. 12. A. Gilbert and A. G. W. Cameron, Can. J. Phys, 43 (1965) 1446.
525
30B,
number
8
PH
SPACING
OF
CLASS
II
Y S IC S
LEVELS
L E T T E R S
IN
THE
8 December
FISSION
OF
1969
240pu
D. K. SOOD and N. SARMA Bhabha Atomic Research Centre, Bombay, India
Received 5 September 1969
A modified correlation analysis of high resolution fission cross sections is suggested which would yield more reliable estimates of the spaeing, DII of the class II levels. Subsequent analysis of the reaction 239pu(n,f) gives the value of/)iI as 225 :~ 25 eV for the nucleus 240Pu.
A second m i n i m u m (1) in the d e f o r m a t i o n p o t e n t i a l of nuclei in the a c t i n i d e r e g i o n has been shown [2,3] to lead to the e x i s t e n c e of f i s s i o n i s o m e r s and to g r o s s s t r u c t u r e in f i s s i o n c r o s s s e c t i o n s n e a r the threshold. The l e v e l spacing, DII of c l a s s II l e v e l s built on the s e c o n d m i n i m u m has been d e t e r m i n e d [4-6] by applying a c o r r e l ation a n a l y s i s as s u g g e s t e d by E g e l s t a f f [7]. It has h o w e v e r been pointed out [8] that such c o r r e l a t i o n a n a l y s e s r e s u l t in a m b i g u o u s v a l u e s of DII. In this p a p e r we s u g g e s t a m e t h o d f o r the a n a l y s i s of f i s s i o n c r o s s s e c t i o n s which would yie ld m o r e r e l i a b l e e s t i m a t e s of DII" The method is then applied to the data [9] on the 239pu(n, f) reaction. Th e c o r r e l a t i o n a n a l y s i s c o m p r i s e s a study of the s e r i a l c o e f f i c i e n t s , rk(W) f o r an a v e r a g i n g i n t e r v a l W, defined as
2.0
1.0
t
aj (50)
0.0 -- ~ ~
n
j~= l aj
rk(W) =r n
(W)~+k (W) n q½
L~l(aj(w))2~l(~+k(W,)2j
(1)
w h e r e the aj(W) a r e d e r i v e d f r o m e x p e r i m e n t a l c r o s s s e c t i o n s , (rF(E) a c c o r d i n g to the f o r m u l a
jW a (IV)--
(j-fl)W
~ aF(E)4-E L~F(E)JE)-
I]
(2)
One would expect the rk(W) to v a n i s h as a r e sult of the r a n d o m n a t u r e of the compound nuc l e u s ( c l a s s I) r e s o n a n c e s which d o m i n a t e (~F(E). Highly s i g n i fi can t v a l u e s of rk(W) will a r i s e f r o m i n t e r m e d i a t e ( c l a s s II) s t a t e s in the f i s s i o n i n g nucleus. H o w e v e r the e f f e c t s due to t h e finite r a n g e of data and the B r e i t - W i g n e r f o r m of the
-I.0 ~
I0
20
30
40
50
j-.. Fig. 1. A typical plot of the aj(W) sequence for W : 50 eV. c l a s s I r e s o n a n c e s m ay a l s o give r i s e to s p u r i ous high v a l u e s of rk(W) , even in the a b s e n c e of c l a s s II l e v e l s . P a t r i c k and J a m e s [5,6] i n c o r p o r a t e high l e v e l s of s i g n i f i c a n c e to avoid such s p u r i o u s effects. A f t e r a d e t a i l e d study of c o m p u t e r - g e n e r a t e d c r o s s s e c t i o n s , P e r e z et al. [8] conclude that the u s e of c o r r e l o g r a m s [4-6] f o r d e t e r m i n i n g the widths and s p a c i n g s of c l a s s II l e v e l s y i e l d s am b i g u o u s r e s u l t s . The s t a t i s t i c a l 523
Volume 3OB, number 8
PHYSICS
LETTERS
8 December 1969
0.4 4.50
0-3
(c)
20
0-2
0"1
1
0
i
450
(b)
40
6'0
k-,
I. z
50
2o
n
r k (20) 0.4
1% Significance level
0Z I
0.3
I
450 (0)
825 0,2
I0%
50
0'1 40
I
o.o
40
I
k ~
50
-0.I
I0% Si@nificonce level -0.2
Io
i
502 kW
-0.3 L
J
L
L
200
400
600 kW
"
800
( eV)----
2. C o r r e J o g r a m s shov¢ing rk(W ) v e r s u s kW for (a) W = 20 eV and (b) W = 10 eV. k s c a l e in the two figures is adjusted to c o r r e s p o n d to kW scale. Levels of significance of 10% and 1% are shown by full lines. Fig.
t e s t s p r o p o s e d by t h e m a r e n e v e r t h e l e s s e q u a l l y i n s e n s i t i v e a n d do not l e a d b e y o n d t h e d e t e c t i o n of i n t e r m e d i a t e s t r u c t u r e . F r o m o u r s t u d y of c o r r e l o g r a m s (fig. 2), w e
524
[
1002
1502
( e v ) --~
Fig. 3. kW frequency d i s t r i bution h i s t o g r a m s for levels of significance of (a) 10%, (b) 1% and (c) 0.1%. The step width is 50 eV. Peak p o s i tions in eV are indicated by arrows.
f i n d t h a t t h e m a g n i t u d e of r k ( W ) i s v e r y s e n s i t i v e t o t h e c h o i c e of W. T h i s i m p l i e s t h a t in s p i t e of t h e i n t r o d u c t i o n of a h i g h l e v e l of s i g n i f i c a n c e , o n e m a y e r r in t h e v a l u e of D i i . It i s t h e r e f o r e
Volume 30B, number 8
PHYSICS
essent~',~l to s c a n t h e e n t i r e s u r f a c e of rk(W), k , W. W k: s u g g e s t that the m o s t a p p r o p r i a t e m e t h o d to e s t i m a t e DII is to c o n s t r u c t the f r e q u e n c y d i s tribution histogram for the product kW for all the p a i r s of k, W f o r w h i c h r k ( W ) l i e s a b o v e a p r e s e t l e v e l of s i g n i f i c a n c e , d u r i n g a r e g u l a r s c a n of t h e s u r f a c e . In s u c h h i s t o g r a m s one o b s e r v e s s e v e r a l p e a k s a p p e a r i n g a l m o s t p e r i o d i c a l l y and s u p e r i m p o s e d on a b a c k g r o u n d c o n t r i b u t e d by s p u r i o u s rk(W). T o i n v e s t i g a t e t h e r e l a t i o n s h i p of t h e s e p e a k s w i t h the p r o p e r t i e s of t h e c l a s s II l e v e l s , we s t u d i e d t h e k W f r e q u e n c y d i s t r i b u t i o n h i s t o g r a m s f o r s e v e r a l s y n t h e t i c c r o s s s e c t i o n data. A s y s t e m a t i c v a r i a t i o n of h e i g h t s , w i d t h s and p o s i t i o n s of t h e b u i l t in c l a s s II l e v e l s i n d i c a t e s that a) t h e p o s i t i o n s of t h e p e a k s a r e d e t e r m i n e d by t h e s p a c i n g d i s t r i b u t i o n and t h e f i r s t p e a k o c c u r s at k W ~ DII , and b) the p e a k h e i g h t s a r e r e l a t e d to t h e d i s t r i b u t i o n s of h e i g h t s and w i d t h s of t h e b u i l t - i n c l a s s II resonances. T h e f i r s t p e a k t h e r e f o r e is not n e c e s s a r i l y the s t r o n g e s t w h i c h r e q u i r e s that k W f r e q u e n c y d i s t r i b u t i o n h i s t o g r a m s h a v e to b e e x a m i n e d at m o r e t h a n one l e v e l of s i g n i f i c a n c e to e n s u r e that the f i r s t p e a k is not m i s s e d . With t h e s e c o n s i d e r a t i o n s , we a n a l y s e d t h e d a t a of Shunk et al. [9] on the r e a c t i o n 2 3 9 p u ( n , f). T h e s t r u c t u r e in t h e aj(W) s e q u e n c e ( s e e fig. 1) s h o u l d b e only due to c l a s s II l e v e l s s i n c e t h e a v e r a g i n g i n t e r v a l of 50 eV > D I ~ 9 eV [11 ]. _ F r o m a s t u d y of s u c h a~(W) p l o t s , we o b t a i n D I I ~ ~ 2 1 0 + 30 eV and widtti ~ 125 =L 50 eV. Fig. 2 s h o w s c o r r e l o g r a m s f o r W = 10 and 20 eV, w h e r e at k W ~ 230 eV, r k ( W ) r i s e s r a p i d l y f r o m b e l o w 10% l e v e l (fig. 2b) to 1% l e v e l (fig. 2a). Fig. 3 s h o w s the k W f r e q u e n c y d i s t r i b u t i o n h i s t o g r a m s
LETTERS
8 December 1969
f o r 10%, 1% and 0.1% l e v e l s . F o u r d i s t i n c t p e a k s a p p e a r at k W = 9 + 2, 225 ± 25, 450 ~= 50 and 825 ~ 25 eV. R a p i d v a r i a t i o n s in t h e h e i g h t s of t h e s e p e a k s ( s e e fig. 3a, b, c) i l l u s t r a t e t h e n e c e s s i t y f o r e x a m i n i n g s e v e r a l l e v e l s of s i g n i f i c a n c e b e f o r e d r a w i n g any c o n c l u s i o n s . T h e p e a k at k W = 9 eV c o r r e s p o n d s t o / ) I f o r 0 + l e v e l s [11]. A v a l u e o f D i i = 225 ± 25 eV is o b t a i n e d f r o m the a n o m a l y at k W = 225 eV w h e r e a s t h e p e a k s at 450 and 825 eV a r e p r o b a b l y due to h i g h e r " h a r m o n i c s " . By u s i n g the r a t i o DII//)I = = 25 and t h e l e v e l d e n s i t y f o r m u l a of C a m e r o n [12], we find that the s e c o n d m i n i m u m is s i t u a t e d at 1.49 MeV a b o v e t h e f i r s t m i n i m u m .
References 1. w, M. Strutinsky, Nucl. Phys. A95 (1967) 420. 2. H. Weigmann, Z. Physik 214 (1968) 7. 3. J. E. Lynn, IAEA Syrup. Phys. and Chem. Fission, Vienna (1969); Contribution SM-122/204. 4. M. Cao, E. Migneco and J. P. Theobald, Phys. Lett e r s 27B (1968) 409. 5. B.H. Patrick and G. D. James, Phys. Letters 28B (1968) 258. 6. G.D. James and B. H. Patrick, IAEA Syrup. on Phys. and chem. fission, Vienna (1969); Contribution SM122/53. 7. P.A. Egelstaff, J. Nucl. Energ~y 7 (1958) 35.
8. R.B. P e r e z , G. de Saussure and M. N. Moore, IAEA Symp. on Phys. and chem. fission, Vienna (1969); Contribution SM-122/121. 9° E. R. Shunk, W.K. Brown and R. La Banve, CONF660303 (1966) 979. 10. W. C. Hamilton, Statistics in physical science (The Ronald P r e s s Company, New York, 1964) p. 183. 11. H. Derrien et al., Proc. IAEA Conf. on Nucl. data for r e a c t o r s , Paris (1966) Vol. II, p. 195. 12. A. Gilbert and A. G. W. Cameron, Can. J. Phys, 43 (1965) 1446.
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