- Email: [email protected]
Alpha decay and the structure of β-vibrational states
Volume 24B, n u m b e r 5
PIIYSICS
LETTERS
References 1. H.Moringa, Phys. Rcv. 101 (1956) 259; T. Engeland, Nuclear P h y s i c s 72 0965) 68; G. E. Brown, Congr. Intern. de Physique nucl~aire, Vol. I ( P a r i s , 1964) 129. 2. W . H . B a s s i e h i s and G.Ripka, Phys. L e t t e r s 15 (1965} 320. 3. G . E . Brown and A . M . G r e e n , Nuclear Phys. 75 (1966) 401. 4. L . S . C e l e n z a , R . M . D r e i z l e r , A.Klcin and G . J . D r e i s s , Phys. L e t t e r s 23 0966) 241.
ALPHA
DECAY
AND
THE
STRUCTURE
6 March 1967
5. E . B o e k e r , Phys. Letters 21 (1966) 69. 6. G . F . B e r t s c h , Phys. L e t t e r s 21 (1966) 70. 7. F . A z j e n b e r g - S e l o v c and T. Lavintsen, Nuclear Phys. 11 (1959) 1. 8. P . F e d e r m a n a n d I.Kelson, Phys. Rev. I.etters 17 (1966) 1055. 9. J. tlayward, Nuclear Phys. 81 (1966) 193. 10. J . B a r T o u v a n d C . A . L e v i n s o n , Phys. Rev.. to be published. 11. M.K. B a n e r j e e and G . J . S t e p h e n s o n ,Jr., to be published.
OF
fl-VIBRATIONAL
STATES
A. S , ~ N D U L E S C U * a n d O. D U M I T R E S C U * *
Research Institute for Theoretical Physics, University of IIelsinki, Helsinki, Findland Received 15 J a n u a r y 1967
The h i n d r a n c e f a c t o r s and the reduced wave amplitudes for alpha t r a n s i t i o n s to f l - s t a t e s a r e computed. It is shown that the alpha i n t e n s i t i e s a r e strongly enhanced by the quadrupole f o r c e s and that only the superfluid model with pairing and quadrupole-quadrupole r e s i d u a l interactions can explain the experimental data, i.e. fls t a t e s a r e not pure pairing vibrational ones.
In t h e l a s t y e a r s a s a t i s f a c t o r y d e s c r i p t i o n of the alpha decay process has been achieved conn e c t i n g t h e d e c a y r a t e s w i t h t h e l e v e l s of t h e b o u n d n u c l e o n s . On t h e o t h e r h a n d t h e v i b r a t i o n a l s t a t e s in n u c l e i h a v e b e e n e x p l a i n e d a s s u m i n g f i e l d s w h i c h c r e a t e o r a n i h i l a t e two p a r t i c l e s . R e c e n t l y S o l o v i e v [1] o b t a i n e d d e t a i l e d m i c r o s copic vibrational wave functions for deformed even-even nuclei, using the superfluid model with 42 n e u t r o n a n d 42 p r o t o n N i l s s o n l e v e l s a n d r e s i d u a l i n t e r a c t i o n s of p a i r i n g a n d q u a d r u p o l e - q u a drupole types. In t h e p r e s e n t p a p e r we h a v e c o m p u t e d t h e h i n d r a n c e f a c t o r s (HF)fi a n d r e d u c e d w a v e a m p l i t u d e s (bL}/3 f o r a l p h a t r a n s i t i o n s t o / 3 - v i b r a t i o n a l s t a t e s in t h e t r a n s u r a n i u m r e g i o n , u s i n g t h e s e w a v e f u n c t i o n s , by m e a n s of t h e r e l a t i o n s (HF)fl = [~ i k o l ( S ) ~10(g.s.)] 2 [~l kOl(B')~lO(fls.)] -2
(1)
(bLM = [ ~ I k L I ( B ' ) ~ ' I O ~ S ) ] [~1&0itB')v20t~s~l-X(2) * Institute for Atomic P h y s i c s , B u c h a r e s t , Romania. ** C h e m i s t r y Faculty, B u c h a r e s t U n i v e r s i t y , Buchar e s t , Romania.
212
w h e r e TI0 a r e the a m p l i t u d e s of t h e r e d u c e d w i d t h s f o r t h e g r o u n d - , r e s p e c t i v e l y f l - s t a t e . In t h e h y p o t h e s i s t h a t t h e s u p e r f l u i d p r o p e r t i e s of the initial and final nuclei are the same the red u c e d w i d t h s in (2N) (2~ (ff2/MRo)]½ u n i t s a r e g i v e n b y t h e f o l l o w i n g r e l a t i o n s [2] I0 Y l 0 ( g . s . ) = ~ A+_,+_ (vv/ww) ~v ~w l~¢d
TiO(fls.)=-2 -½
(3)
~ A 10 (vv'/ooW)fvv'~w + VpTO0 +-,+A 10 (vvlwco') ~vfcow' +-~+-
(4)
f s s ' : % s ' vs Vs' - ~ss' us us'
(~)
+ ~
l~O)O~t where
~s : Us vs;
a n d k i i , (B) t h e F r 6 m a n m a t r i x . N o t e t h a t f o r t h e d e c a y to t h e g r o u n d s t a t e , eq. (3), t h e r e i s a c o m p l e t e c o h e r e n c e w i l l a l l t e r m s of t h e s a m e s i g n , w h e r e a s t h e s u m m a t i o n f o r t h e alpha amplitude to the fl-vibrational state may inv o l v e t e r m s of b o t h s i g n s .
Volume 24B, number 5
PFtYSICS
LETTERS
6 March 1967
Table 1 The neutron (proton) levels in the physical zone. t l e r c O~U) is the number of the individual level for neutron (proton) s y s t e m , "~'[06'qu) the corresponding number of nucleons in the Nilsson model with no residual £nteraction, [ N n z A E ] t h e u s u a l Nilssonquantum numbers, Ew(Eu) the single particle e n e r g i e s in ~'i~o units and qcow (quv) the quadrupole moments of Nilsson levels.
w
Nw
INn z AE]
ew
qww
u
18
138
631+
0.71
1.139
17
90
b,~
INn z AE]
~u
quu
651+
0.68
1.830 1.628
19
140
752*
0.72
1.673
18
92
530+
0.75
20
142
633-
0.78
0.!}86
19
94
642+
0.83
1.28!}
21
144
743+
0.85
1.058
20
96
523-
0.85
0.418
22
146
631-
0.90
1.389
21
98
521+
0.98
0.59S
2:3
148
622+
0.97
0.189
22
100
633+
0.99
0.619
24
150
624-
1.03
0.089
25
152
734.+
1.10
0.342
26
154
620+
1.17
0.469
-C~ 0~0
030
/~_/2-~= A
lAY
/"
/I /LA\
020
MO
+I+~-
Fig. 1. The quadrupole moments qcow and the "superfluid weights" ~¢0¢0 for the alpha transition to the ground state and those fw6o to the/3-vibrational state of 236pu as a function of the 42 single-neutron e n e r g i e s ECO for some values of the quadrupole strength p a r a m e t e r I< = p A-~ //Wo. F o r t h e c a l c u l a t i o n of t h e c o e f f i c i e n t s the reduced alpha amplitude a s s o c i a t e d with p r o t o n t r a n s f e r f r o m o r b i t a l s VT, V'T' a n d n e u t r o n t r a n s f e r f r o m wcr, w'cr', we u s e d t h e s i m p l i f i e d p r o c e d u r e [3] fl = a , ct b e i n g the n u c l e a r w e l l o s c i l l a t o r c o n s t a n t a n d [3 b e i n g t h e G a u s s i a n s i z e p a r a m e t e r of t h e a l p h a p a r t i c l e . In a s e p a r a t e p a p e r w e h a v e s h o w n , by d e f i n i n g a mean square deviation coefficient, that this m e t h o d c a n b e a p p l i e d w i t h c o n f i d e n c e to t h e a c -
AITOT' ga' ( u u ' / W W ' ) ,
tual wave functions. F o r the f a v o u r e d alpha t r a n s i t i o n s of e v e n - e v e n d e f o r m e d n u c l e i we h a v e o b t a i n e d g o o d a g r e e m e n t with the p r e v i o u s c a l c u l a t i o n s a n d e x p e r i m e n t a l d a t a [4]. T h e a l p h a p a r t i c l e can be f o r m e d f r o m two n u c l e o n s s i t u a t e d in d i f f e r e n t l e v e l s (s' ¢ s ) o r s i t u a t e d in t h e s a m e l e v e l (s' = s). T h e a s y m p t o t i c b e h a v i o u r (Es+ Es, >> w a n d [ E s - ~tI, >> &) i s
213
Volume 24B, n u m b e r 5
PHYSICS
LETTERS
6 March 1967
Table 2 The reduced wave amplitudes b 2 f o r / 3 - s t a t e s as a function of the strength p a r a m e t e r p and deformation p a r a m e t e r 77. F o r a given isotope, the f i r s t value is the amplitude b 2 for the Fr~iman p a r a m e t e r B' = 0 and the second one for B' = 0.9. The FrSman p a r a m e t e r B' = 0.9 is the s a m e as for the alpha t r a n s i t i o n s to the ground state.
p=7.3
232U 2341J 236U 238pu 240Pu
p=8.4
r/=4
~/=6
r/=4
?7=6
0.417
0.452
0.677
0.790
0.765
1.029
0.452
0.471
0.814
0.780
0.284 0.642
P=8.8 7/4
r/=6
77=4
r/:6
0.697
0.615
0.635
0.678
0.689
1.001
0.973
0.940
1.022
0.986
0.473
0.487
0.533
0.541
0.630
0.633
0.835
0.794
0,892
0.846
0.980
0.933
0.437
0.284
0.437
0.497
0.539
0.570
0.586
0.777
0.642
0.777
0.868
0.861
0.934
0.898
0.057
0.212
0.432
0.453
0.462
0.466
0.508
0.503
0.300
0.485
0.777
0.752
0.800
0.752
0.846
0.788
0.035
0.086
0.372
0.370
0.449
0.437
0.514
0.498
0.239
0.288
0.689
0.642
0.773
0.715
0.841
0.777
qSS' fss'
~ -const
i~ s _ X I + IEs, _X i ×
s i g n (E s, - ~)
s i g n (E s - X)] f o r s ' ¢ s (6)
1
fss ---const
-
IE s _ % [ +
s i g n (E s - k) c o n s t '
qss [E s - k'~
2
f o r s'= s (7)
w h e r e ~, i s t h e c h e m i c a l p o t e n t i a l , A t h e gap, E s t h e s i n g l e p a r t i c l e e n e r g i e s , e s the q u a s i p a r t i c l e energies and q s s ' the quadrupole matrix elements. In t h e f i r s t c a s e ( s ' ¢ s ) t h e r a t i o f s s ' / q s s ' i s p o s i t i v e f o r a l l c o m b i n a t i o n s of l e v e l s w i t h both situated below the Fermi surface, or with s b e l o w a n d s ' a b o v e f o r !E s ' - ~ l < ! E s - x l ors' b e l o w a n d s a b o v e f o r [E s , - ~ ! > i E s -X]" T h e r a t i o i s n e g a t i v e f o r t h e c o m b i n a t i o n s of l e v e l s such that both are situated above the Fermi surface, or such that s is below and s' above for IEs' - ~1 > I E s - ~ ! and s i s a b o v e a n d s ' b e l o w f o r I E s , _ ~ l ~ --ol - X i . F o r t h i s c a s e we h a v e a cancellation effect. In t h e s e c o n d c a s e ( s ' = s) t h e l a s t t e r m , p r o p o r t i o n a l to t h e q u a d r u p o l e m o m e n t , i s a d d e d f o r the levels below the Fermi surface and subtracted for those above. These rules are visualised in fig. 1, w h e r e t h e qww a n d f w o j - v a l u e s f o r 214
p=9.2
exp.
< 0.27 [8-10] < 0.55 [8-101
0.54 [9] 0.77 [8,10]
2 3 6 p u a r e d r a w n a s a f u n c t i o n of t h e 42 s i n g l e n e u t r o n e n e r g i e s . A l s o f r o m t h i s p i c t u r e we c a n s e e t h a t t h e f s s - v a l u e s a r e s t r o n g l y e n h a n c e d by t h e q u a d r u p o l e f o r c e s a n d t h a t the a l p h a i n t e n s i t y to t h e 2 - s t a t e i s c o m p a r a b l e w i t h the i n t e n s i t y to t h e g r o u n d s t a t e ( s e e t h e 2-½ ~ w - v a l u e s f r o m fig. 1), if t h e s t r e n g t h of t h e q u a d r u p o l e f o r c e K = pA-t I~oo i s c h o s e n to fit t h e e x p e r i m e n t a l 2 energies. In t h e p h y s i c a l z o n e , t h e n e u t r o n ( p r o t o n ) l e v e l s 1 8 - 2 6 ( 1 7 - 2 2 ) a r e d i v i d e d in two z o n e s w i t h r e s p e c t to t h e v a l u e of t h e q u a d r u p o l e m o m e n t ( s e e t a b l e 1 a n d fig. 1). T h e f i r s t one i s 1 8 - 2 2 ( 1 7 - 1 9 ) a n d h a s q s s ~ 1.2 (1.5), t h e s e c o n d one i s 2 3 - 2 6 ( 2 0 - 2 2 ) a n d h a s q s s ~ 0.2 (0.5) w h i c h i s 6(3) t i m e s s m a l l e r t h a n f o r t h e f i r s t z o n e . P a s s i n g f r o m t h e f i r s t z o n e to t h e s e c o n d one, t h e f s s - v a l u e s c h a n g e a p p r o x i m a t e l y in t h e s a m e r a t i o . T h i s e x p l a i n s t h e l a c k of t h e e x p e r i m e n t a l d a t a f o r N ~ 148 a n d Z >~ 96 s i n c e t h e h i n d r a n c e factors for these cases must be very large. W e h a v e c o m p u t e d f o r 27 n u c l e i in t h e t r a n s u r a n i u m r e g i o n , t h e h i n d r a n c e f a c t o r s (HF)(~ a n d t h e r e d u c e d w a v e a m p l i t u d e s (bL)2, f o r two d e f o r m a t i o n v a l u e s 07 = 4 a n d 6) of t h e A +I0- , + - (vv'/~o0)') c o e f f i c i e n t s . T h e r e s u l t s f o r U a n d P u i s o t o p e s a r e g i v e n in f i g s . 2 a n d 3 f o r V = 4 together with the experimental data [7-10]. Practically the same results are obtained for 71= 6 ( s e e t h e m o r e d e t a i l e d r e f . 4). T h e e x p e r i mental hindrance factors can be explained assuming that p = constant for a given element. F r o m fig. 2 we c a n s e e t h a t a good a g r e e m e n t : s o b t a i n e d w i t h p = 7.3 f o r U i s o t o p e s a n d p = 8.4
Volume 24B, n u m b e r 5
PtIYSICS
I E (MeVJ •[
~. . . . . .
I[
........ .i, I . . . . .I : ~
.
.~
LETI"EI~S
6 March 1967
P'Z3
P-Z3
I
I
t
Ip.Ta
...........
'--
~
.I .
I
P.9.2 ~.......
¢
I --.
ii\ P.73
~1, \v P-84
"
p.8~\ ~'~,o=e,g
L
5u L, "P 8/,
,a; 5i b
[. i
X i
lii ~]/j/
k , , I
el " P 73 /:' : ~,''/i
L
,•
I i
P= 9
"
F [
05 I L I
.f
',
L
\\
\,I'=88~
P:88
\
i
Is~O
",
i
< II ~
14g
U
'II
144
l~b
"- /,;Z
1,~h,
k
U
& ),
I
L
142
I,.4.
[ ~III
<.5
l
~,~
ll
J I I I L
.;::
~Y
Fig. 3. The hindrance factors for the ~ - v i b r a t i o n a l states with the phonon of the final nucleus (N, Z) and with the phonon of the nucleus ( N , Z + 2 ) together with the e x p e r i mental data [7-10]. The values of a given clement are connected by lines, solid for the final phonon (N, Z} and dashed for the phonon (N, Z +2).
! .; !
/
.[
i
,
P:86/~,/"
/
/ /
t p. 84
i:
,
7C
/I/
•
i
/~#
[4#
Fig. 2. The e n e r g i e s and the hindrance factors for the /3-vibrational s t a t e s for some values of the p a r a m e t e r p. Wc denoted by lines (-) the experimental values [7-10], by })lack (e) and ()pen (o) c i r c l e s t h e t h c o r c t i c a l values for )7=4 with the F r b m a n p a r a m e t e r B' =0 r e s pectively B' : 0..9. In the upper part of the figures a r e givcn the corresponding experimental (+) and t h e o r e t i cal (I) values of t h e / 3 - e n e r g i e s . The values of -1 giwm element a r e conneetcd by solid lines. o r 8.8 f o r P u i s o t o p e s . T h e l a r g e v a r i a t i o n of t h e h i n d r a n c e f a c t o r s f o r 2 3 8 p u a n d 2 4 0 p u i s due to the fact that these nuclei are situated at the front i e r of two n e u t r o n z o n e s w i t h d i f f e r e n t q w w v a l u e s ( s e e fig. 1). F o r s m a l l v a l u e s of t h e parameter p the hindrance factors become very l a r g e i m p l y i n g t h a t / ~ - v i b r a t i o n a l s t a t e s a r e not p u r e p a i r i n g v i b r a t i o n a l o n e s [6]. T h e i n t r o d u c t i o n of a F r 0 m a n m a t r i x f o r B - s t a t e s w i l l d e crease the hindrance factors. The results for the s a m e F r 0 m a n p a r a m e t e r [3] B = 0.9 a s f o r t h e g r o u n d s t a t e a r e s h o w n by a r r o w s a n d o p e n c i r c l e s in f i g . 2. No e s s e n t i a l d i f f e r e n c e i s o b t a i n e d .
A d e t a i l e d a n a l y s i s of the i n f l u e n c e of t h e c o u p l i n g b e t w e e n g r o u n d a n d ~ % s t a t e s on a l p h a i n t e n sities must be effectuated. W h e n t h e n u m b e r of p r o t o n s Z i n c r e a s e s t h e hindr,'mce factors for p = constant increase very m u c h . T h i s l e a d s to s o m e d o u b t s c o n c e r n i n g t h e v a l i d i t y of the a s s u m p t i o n t h a t t h e p h o n o n v a c u u m is identical for the initial and final nucleus. Indeed computing the hindrance factors with the p h o n o n s t a t e w a v e f u n c t i o n s f o r the n u c l e i (N, Z + + 2), t h e o b t a i n e d v a l u e s a r e m u c h d i f f e r e n t f r o m t h o s e c o r r e s p o n d i n g to t h e n u c l e i (N, Z), w h e r e N a n d Z a r e t h e n u m b e r of t h e n e u t r o n s r e s p e c t i v e l y of t h e p r o t o n s of t h e f i n a l n u c l e i ( s e e fig. 3). T h i s i m p l i e s t h a t a m o r e s o p h i s t i c a t e d t h e o r y of alpha decay which takes into account the differenc e s in t h e b a c k g o i n g d i a g r a m s of t h e i n i t i a l a n d final nuclei is needed. T h e r e d u c e d w a v e a m p l i t u d e s b2 f o r f l = v i b r a t i o n a l s t a t e s a r e g i v e n in t h e t a b l e 2 f o r B ' = 0 a n d B ' = 0.9 t o g e t h e r w i t h t h e e x p e r i m e n t a l d a t a [ 7 - 1 0 ] . T h e r e a r e no e s s e n t i a l d i f f e r e n c e s f o r d i f f e r e n t v a l u e s of t h e d e f o r m a t i o n p a r a m e t e r 77 = 4 o r 7/= 6. A s t r o n g d e p e n d e n c e of t h e b 2 = v a l u e s w i t h t h e •215
Volume 24B, number 5
PHYSICS
F r O m a n p a r a m e t e r B ' is o b t a i n e d . T h e t h e o r e t i cal v a l u e s , f o r the s a m e v a l u e s of the p a r a m e t e r P a s f o r the e x p l a n a t i o n of the h i n d r a n c e f a c t o r s , a r e in a g r e e m e n t with the e x p e r i m e n t a l data if we a s s u m e that the F r O m a n p a r a m e t e r B ' is i n c r e a s i n g with the m a s s n u m b e r s t a r t i n g with z e r o f o r 232U and b e c o m i n g a p p r o x i m a t e l y 0.9 for 240pu, the s a m e v a l u e a s for the g r o u n d s t a t e a l p h a t r a n s i t i o n s . No m i x i n g of the g r o u n d a n d / 3 - b a n d s was t a k e n into a c c o u n t . We would like to thank P r o f e s s o r s A. B o h r , B. M o t t e l s o n , D. B~s, J . O . R a s m u s s e n and V. G. S o l o v i e v f o r t h e i r i n t e r e s t in this p a p e r . We a l s o thank P r o f e s s o r K. V. L a u r i k a i n e n f o r the kind h o s p i t a l i t y at the R e s e a r c h I n s t i t u t e f o r T h e o r e t i cal P h y s i c s . MmW t h a n k s a r e due a l s o to M r . E. R i i h i m a k i f o r p e r f o r m i n g the n u m e r i c a l c a l c u l a tions.
RELATION
BETWEEN
I, E T T E R S
6 March 1967
References
1. K.M.Zheleznova, A.A.Korneichuk, V.G.Soloviev, P. Vogel and G. Yungklausen, Dubna, Preprint JINR, D-2157 (1965). 2. A.Shndulescu and O.Dumitrescu, Physics I,etters 19 (1965) 404. 3. A.S~induleseu and F.Staneu, Acta Phys. Polon. 27 {1965) 655. 4. O. Dumitrescu, E.Riihim~ikiandA.SSndulescu, to be published. 5. E. Riihim~iki and A.Shnduleseu, to be published. 6. D.R.B~s and R . A . B r o g l i a . Pairing Vibrations, P r e print Nordita 1965. 7. S.Bj0rnholm, M.I.ederer, F . A s a r o a n d I . P e r l m a n , Phys. Rev. 130 (1963) 2000. 8. C . M . L e d e r e r , UCRL-11028, Thesis, University of California (1963) unpublished. 9. J . K . Poggenburg, UCRI,-16187. Thesis, University of California (1965) unpublished. 10. S.Bj¢rnholm, Nuclear excitations in even isotopes of the heaviest elements (Munksgaard, University of Copenhagen, Copenhagen, 1965).
PROTON-NUCLEUS INTERACTION AT
20
AND GeV
PROTON-NUCLEON
W. E. F R A H N * International Centre f o r Theoretical Physics, Trieste, Italy
Rcceivcd 15 January 1967
It is shown that the parameters for proton-nucleus scattering at 19.3 GeV/c are closely related to those for proton-nucleon scattering at the same energy.
R e c e n t p a r t i a l - w a v e a n a l y s e s [1,2] of the d i f f e r e n t i a l and t o t a l c r o s s s e c t i o n s of 19.3 GeV p r o tons s c a t t e r e d by c o m p l e x n u c l e i [3] h a v e r e v e a l ed that the s i g n of the r e a l p a r t of the c o h e r e n t n u c l e a r s c a t t e r i n g a m p l i t u d e at t h i s e n e r g y is n e g a t i v e , c o r r e s p o n d i n g to a p r e d o m i n a n t l y r e p u l s i v e n u c l e o n - n u c l e u s i n t e r a c t i o n . One e x p e c t s that t h i s p r o p e r t y is c l o s e l y c o n n e c t e d with the f a c t that the e l a s t i c p r o t o n - n u c l e o n a m p l i t u d e h a s an a p p r e c i a b l e n e g a t i v e r e a l p a r t in the m u l t i GeV r e g i o n [4]. In the p r e s e n t n o t e we show, by m e a n s of a s i m p l e i m p u l s e a p p r o x i m a t i o n , that the p a r a m e t e r s of the c o h e r e n t p r o t o n - n u c l e u s s c a t t e r i n g a r e in f a c t q u a n t i t a t i v e l y r e l a t e d to t h o s e to p r o t o n - n u c l e o n e l a s t i c s c a t t e r i n g . T h e p r o t o n - n u c l e u s s c a t t e r i n g d a t a at 19.3 G e V / c h a v e b e e n d e s c r i b e d in t e r m s of a p a r a m e t e r i z e d s c a t t e r i n g f u n c t i o n ~l = S 0,) = 216
= exp[i25(~)] of the f o r m [5] S(X) = g(~) + E[1 - g ( ~ ) ] + i # d g / d k ,
(1)
w h e r e ~,~(~) = [1 + e x p ( A - X ) , / h ] -1 and ~ = l+½. T h e p a r a m e t e r s A and A a r e s e m i c l a s s i c a l l y r e l a t e d to the r a d i u s R and the d i f f u s e n e s s d of the i n t e r a c t i o n r e g i o n by the r e l a t i o n s A = k R , A = kd, w h e r e k is the c . m . m o m e n t u m , ¢ is the t r a n s p a r e n c y , and # m e a s u r e s the s t r e n g t h of the r e a l p a r t of the p r o t o n - n u c l e u s i n t e r a c t i o n . F o r the f o l l o w i n g c a l c u l a t i o n s we m a y n e g l e c t C o u l o m b and s p i n - d e p e n d e n t t e r m s . In t h i s c a s e the c o h e r e n t p r o t o n - n u c l e u s s c a t t e r i n g a m p l i t u d e r e s u l t i n g f r o m eq. (1) i s g i v e n by * Permanent address: Physics Department, University of Cape Town.
PIIYSICS
LETTERS
References 1. H.Moringa, Phys. Rcv. 101 (1956) 259; T. Engeland, Nuclear P h y s i c s 72 0965) 68; G. E. Brown, Congr. Intern. de Physique nucl~aire, Vol. I ( P a r i s , 1964) 129. 2. W . H . B a s s i e h i s and G.Ripka, Phys. L e t t e r s 15 (1965} 320. 3. G . E . Brown and A . M . G r e e n , Nuclear Phys. 75 (1966) 401. 4. L . S . C e l e n z a , R . M . D r e i z l e r , A.Klcin and G . J . D r e i s s , Phys. L e t t e r s 23 0966) 241.
ALPHA
DECAY
AND
THE
STRUCTURE
6 March 1967
5. E . B o e k e r , Phys. Letters 21 (1966) 69. 6. G . F . B e r t s c h , Phys. L e t t e r s 21 (1966) 70. 7. F . A z j e n b e r g - S e l o v c and T. Lavintsen, Nuclear Phys. 11 (1959) 1. 8. P . F e d e r m a n a n d I.Kelson, Phys. Rev. I.etters 17 (1966) 1055. 9. J. tlayward, Nuclear Phys. 81 (1966) 193. 10. J . B a r T o u v a n d C . A . L e v i n s o n , Phys. Rev.. to be published. 11. M.K. B a n e r j e e and G . J . S t e p h e n s o n ,Jr., to be published.
OF
fl-VIBRATIONAL
STATES
A. S , ~ N D U L E S C U * a n d O. D U M I T R E S C U * *
Research Institute for Theoretical Physics, University of IIelsinki, Helsinki, Findland Received 15 J a n u a r y 1967
The h i n d r a n c e f a c t o r s and the reduced wave amplitudes for alpha t r a n s i t i o n s to f l - s t a t e s a r e computed. It is shown that the alpha i n t e n s i t i e s a r e strongly enhanced by the quadrupole f o r c e s and that only the superfluid model with pairing and quadrupole-quadrupole r e s i d u a l interactions can explain the experimental data, i.e. fls t a t e s a r e not pure pairing vibrational ones.
In t h e l a s t y e a r s a s a t i s f a c t o r y d e s c r i p t i o n of the alpha decay process has been achieved conn e c t i n g t h e d e c a y r a t e s w i t h t h e l e v e l s of t h e b o u n d n u c l e o n s . On t h e o t h e r h a n d t h e v i b r a t i o n a l s t a t e s in n u c l e i h a v e b e e n e x p l a i n e d a s s u m i n g f i e l d s w h i c h c r e a t e o r a n i h i l a t e two p a r t i c l e s . R e c e n t l y S o l o v i e v [1] o b t a i n e d d e t a i l e d m i c r o s copic vibrational wave functions for deformed even-even nuclei, using the superfluid model with 42 n e u t r o n a n d 42 p r o t o n N i l s s o n l e v e l s a n d r e s i d u a l i n t e r a c t i o n s of p a i r i n g a n d q u a d r u p o l e - q u a drupole types. In t h e p r e s e n t p a p e r we h a v e c o m p u t e d t h e h i n d r a n c e f a c t o r s (HF)fi a n d r e d u c e d w a v e a m p l i t u d e s (bL}/3 f o r a l p h a t r a n s i t i o n s t o / 3 - v i b r a t i o n a l s t a t e s in t h e t r a n s u r a n i u m r e g i o n , u s i n g t h e s e w a v e f u n c t i o n s , by m e a n s of t h e r e l a t i o n s (HF)fl = [~ i k o l ( S ) ~10(g.s.)] 2 [~l kOl(B')~lO(fls.)] -2
(1)
(bLM = [ ~ I k L I ( B ' ) ~ ' I O ~ S ) ] [~1&0itB')v20t~s~l-X(2) * Institute for Atomic P h y s i c s , B u c h a r e s t , Romania. ** C h e m i s t r y Faculty, B u c h a r e s t U n i v e r s i t y , Buchar e s t , Romania.
212
w h e r e TI0 a r e the a m p l i t u d e s of t h e r e d u c e d w i d t h s f o r t h e g r o u n d - , r e s p e c t i v e l y f l - s t a t e . In t h e h y p o t h e s i s t h a t t h e s u p e r f l u i d p r o p e r t i e s of the initial and final nuclei are the same the red u c e d w i d t h s in (2N) (2~ (ff2/MRo)]½ u n i t s a r e g i v e n b y t h e f o l l o w i n g r e l a t i o n s [2] I0 Y l 0 ( g . s . ) = ~ A+_,+_ (vv/ww) ~v ~w l~¢d
TiO(fls.)=-2 -½
(3)
~ A 10 (vv'/ooW)fvv'~w + VpTO0 +-,+A 10 (vvlwco') ~vfcow' +-~+-
(4)
f s s ' : % s ' vs Vs' - ~ss' us us'
(~)
+ ~
l~O)O~t where
~s : Us vs;
a n d k i i , (B) t h e F r 6 m a n m a t r i x . N o t e t h a t f o r t h e d e c a y to t h e g r o u n d s t a t e , eq. (3), t h e r e i s a c o m p l e t e c o h e r e n c e w i l l a l l t e r m s of t h e s a m e s i g n , w h e r e a s t h e s u m m a t i o n f o r t h e alpha amplitude to the fl-vibrational state may inv o l v e t e r m s of b o t h s i g n s .
Volume 24B, number 5
PFtYSICS
LETTERS
6 March 1967
Table 1 The neutron (proton) levels in the physical zone. t l e r c O~U) is the number of the individual level for neutron (proton) s y s t e m , "~'[06'qu) the corresponding number of nucleons in the Nilsson model with no residual £nteraction, [ N n z A E ] t h e u s u a l Nilssonquantum numbers, Ew(Eu) the single particle e n e r g i e s in ~'i~o units and qcow (quv) the quadrupole moments of Nilsson levels.
w
Nw
INn z AE]
ew
qww
u
18
138
631+
0.71
1.139
17
90
b,~
INn z AE]
~u
quu
651+
0.68
1.830 1.628
19
140
752*
0.72
1.673
18
92
530+
0.75
20
142
633-
0.78
0.!}86
19
94
642+
0.83
1.28!}
21
144
743+
0.85
1.058
20
96
523-
0.85
0.418
22
146
631-
0.90
1.389
21
98
521+
0.98
0.59S
2:3
148
622+
0.97
0.189
22
100
633+
0.99
0.619
24
150
624-
1.03
0.089
25
152
734.+
1.10
0.342
26
154
620+
1.17
0.469
-C~ 0~0
030
/~_/2-~= A
lAY
/"
/I /LA\
020
MO
+I+~-
Fig. 1. The quadrupole moments qcow and the "superfluid weights" ~¢0¢0 for the alpha transition to the ground state and those fw6o to the/3-vibrational state of 236pu as a function of the 42 single-neutron e n e r g i e s ECO for some values of the quadrupole strength p a r a m e t e r I< = p A-~ //Wo. F o r t h e c a l c u l a t i o n of t h e c o e f f i c i e n t s the reduced alpha amplitude a s s o c i a t e d with p r o t o n t r a n s f e r f r o m o r b i t a l s VT, V'T' a n d n e u t r o n t r a n s f e r f r o m wcr, w'cr', we u s e d t h e s i m p l i f i e d p r o c e d u r e [3] fl = a , ct b e i n g the n u c l e a r w e l l o s c i l l a t o r c o n s t a n t a n d [3 b e i n g t h e G a u s s i a n s i z e p a r a m e t e r of t h e a l p h a p a r t i c l e . In a s e p a r a t e p a p e r w e h a v e s h o w n , by d e f i n i n g a mean square deviation coefficient, that this m e t h o d c a n b e a p p l i e d w i t h c o n f i d e n c e to t h e a c -
AITOT' ga' ( u u ' / W W ' ) ,
tual wave functions. F o r the f a v o u r e d alpha t r a n s i t i o n s of e v e n - e v e n d e f o r m e d n u c l e i we h a v e o b t a i n e d g o o d a g r e e m e n t with the p r e v i o u s c a l c u l a t i o n s a n d e x p e r i m e n t a l d a t a [4]. T h e a l p h a p a r t i c l e can be f o r m e d f r o m two n u c l e o n s s i t u a t e d in d i f f e r e n t l e v e l s (s' ¢ s ) o r s i t u a t e d in t h e s a m e l e v e l (s' = s). T h e a s y m p t o t i c b e h a v i o u r (Es+ Es, >> w a n d [ E s - ~tI, >> &) i s
213
Volume 24B, n u m b e r 5
PHYSICS
LETTERS
6 March 1967
Table 2 The reduced wave amplitudes b 2 f o r / 3 - s t a t e s as a function of the strength p a r a m e t e r p and deformation p a r a m e t e r 77. F o r a given isotope, the f i r s t value is the amplitude b 2 for the Fr~iman p a r a m e t e r B' = 0 and the second one for B' = 0.9. The FrSman p a r a m e t e r B' = 0.9 is the s a m e as for the alpha t r a n s i t i o n s to the ground state.
p=7.3
232U 2341J 236U 238pu 240Pu
p=8.4
r/=4
~/=6
r/=4
?7=6
0.417
0.452
0.677
0.790
0.765
1.029
0.452
0.471
0.814
0.780
0.284 0.642
P=8.8 7/4
r/=6
77=4
r/:6
0.697
0.615
0.635
0.678
0.689
1.001
0.973
0.940
1.022
0.986
0.473
0.487
0.533
0.541
0.630
0.633
0.835
0.794
0,892
0.846
0.980
0.933
0.437
0.284
0.437
0.497
0.539
0.570
0.586
0.777
0.642
0.777
0.868
0.861
0.934
0.898
0.057
0.212
0.432
0.453
0.462
0.466
0.508
0.503
0.300
0.485
0.777
0.752
0.800
0.752
0.846
0.788
0.035
0.086
0.372
0.370
0.449
0.437
0.514
0.498
0.239
0.288
0.689
0.642
0.773
0.715
0.841
0.777
qSS' fss'
~ -const
i~ s _ X I + IEs, _X i ×
s i g n (E s, - ~)
s i g n (E s - X)] f o r s ' ¢ s (6)
1
fss ---const
-
IE s _ % [ +
s i g n (E s - k) c o n s t '
qss [E s - k'~
2
f o r s'= s (7)
w h e r e ~, i s t h e c h e m i c a l p o t e n t i a l , A t h e gap, E s t h e s i n g l e p a r t i c l e e n e r g i e s , e s the q u a s i p a r t i c l e energies and q s s ' the quadrupole matrix elements. In t h e f i r s t c a s e ( s ' ¢ s ) t h e r a t i o f s s ' / q s s ' i s p o s i t i v e f o r a l l c o m b i n a t i o n s of l e v e l s w i t h both situated below the Fermi surface, or with s b e l o w a n d s ' a b o v e f o r !E s ' - ~ l < ! E s - x l ors' b e l o w a n d s a b o v e f o r [E s , - ~ ! > i E s -X]" T h e r a t i o i s n e g a t i v e f o r t h e c o m b i n a t i o n s of l e v e l s such that both are situated above the Fermi surface, or such that s is below and s' above for IEs' - ~1 > I E s - ~ ! and s i s a b o v e a n d s ' b e l o w f o r I E s , _ ~ l ~ --ol - X i . F o r t h i s c a s e we h a v e a cancellation effect. In t h e s e c o n d c a s e ( s ' = s) t h e l a s t t e r m , p r o p o r t i o n a l to t h e q u a d r u p o l e m o m e n t , i s a d d e d f o r the levels below the Fermi surface and subtracted for those above. These rules are visualised in fig. 1, w h e r e t h e qww a n d f w o j - v a l u e s f o r 214
p=9.2
exp.
< 0.27 [8-10] < 0.55 [8-101
0.54 [9] 0.77 [8,10]
2 3 6 p u a r e d r a w n a s a f u n c t i o n of t h e 42 s i n g l e n e u t r o n e n e r g i e s . A l s o f r o m t h i s p i c t u r e we c a n s e e t h a t t h e f s s - v a l u e s a r e s t r o n g l y e n h a n c e d by t h e q u a d r u p o l e f o r c e s a n d t h a t the a l p h a i n t e n s i t y to t h e 2 - s t a t e i s c o m p a r a b l e w i t h the i n t e n s i t y to t h e g r o u n d s t a t e ( s e e t h e 2-½ ~ w - v a l u e s f r o m fig. 1), if t h e s t r e n g t h of t h e q u a d r u p o l e f o r c e K = pA-t I~oo i s c h o s e n to fit t h e e x p e r i m e n t a l 2 energies. In t h e p h y s i c a l z o n e , t h e n e u t r o n ( p r o t o n ) l e v e l s 1 8 - 2 6 ( 1 7 - 2 2 ) a r e d i v i d e d in two z o n e s w i t h r e s p e c t to t h e v a l u e of t h e q u a d r u p o l e m o m e n t ( s e e t a b l e 1 a n d fig. 1). T h e f i r s t one i s 1 8 - 2 2 ( 1 7 - 1 9 ) a n d h a s q s s ~ 1.2 (1.5), t h e s e c o n d one i s 2 3 - 2 6 ( 2 0 - 2 2 ) a n d h a s q s s ~ 0.2 (0.5) w h i c h i s 6(3) t i m e s s m a l l e r t h a n f o r t h e f i r s t z o n e . P a s s i n g f r o m t h e f i r s t z o n e to t h e s e c o n d one, t h e f s s - v a l u e s c h a n g e a p p r o x i m a t e l y in t h e s a m e r a t i o . T h i s e x p l a i n s t h e l a c k of t h e e x p e r i m e n t a l d a t a f o r N ~ 148 a n d Z >~ 96 s i n c e t h e h i n d r a n c e factors for these cases must be very large. W e h a v e c o m p u t e d f o r 27 n u c l e i in t h e t r a n s u r a n i u m r e g i o n , t h e h i n d r a n c e f a c t o r s (HF)(~ a n d t h e r e d u c e d w a v e a m p l i t u d e s (bL)2, f o r two d e f o r m a t i o n v a l u e s 07 = 4 a n d 6) of t h e A +I0- , + - (vv'/~o0)') c o e f f i c i e n t s . T h e r e s u l t s f o r U a n d P u i s o t o p e s a r e g i v e n in f i g s . 2 a n d 3 f o r V = 4 together with the experimental data [7-10]. Practically the same results are obtained for 71= 6 ( s e e t h e m o r e d e t a i l e d r e f . 4). T h e e x p e r i mental hindrance factors can be explained assuming that p = constant for a given element. F r o m fig. 2 we c a n s e e t h a t a good a g r e e m e n t : s o b t a i n e d w i t h p = 7.3 f o r U i s o t o p e s a n d p = 8.4
Volume 24B, n u m b e r 5
PtIYSICS
I E (MeVJ •[
~. . . . . .
I[
........ .i, I . . . . .I : ~
.
.~
LETI"EI~S
6 March 1967
P'Z3
P-Z3
I
I
t
Ip.Ta
...........
'--
~
.I .
I
P.9.2 ~.......
¢
I --.
ii\ P.73
~1, \v P-84
"
p.8~\ ~'~,o=e,g
L
5u L, "P 8/,
,a; 5i b
[. i
X i
lii ~]/j/
k , , I
el " P 73 /:' : ~,''/i
L
,•
I i
P= 9
"
F [
05 I L I
.f
',
L
\\
\,I'=88~
P:88
\
i
Is~O
",
i
< II ~
14g
U
'II
144
l~b
"- /,;Z
1,~h,
k
U
& ),
I
L
142
I,.4.
[ ~III
<.5
l
~,~
ll
J I I I L
.;::
~Y
Fig. 3. The hindrance factors for the ~ - v i b r a t i o n a l states with the phonon of the final nucleus (N, Z) and with the phonon of the nucleus ( N , Z + 2 ) together with the e x p e r i mental data [7-10]. The values of a given clement are connected by lines, solid for the final phonon (N, Z} and dashed for the phonon (N, Z +2).
! .; !
/
.[
i
,
P:86/~,/"
/
/ /
t p. 84
i:
,
7C
/I/
•
i
/~#
[4#
Fig. 2. The e n e r g i e s and the hindrance factors for the /3-vibrational s t a t e s for some values of the p a r a m e t e r p. Wc denoted by lines (-) the experimental values [7-10], by })lack (e) and ()pen (o) c i r c l e s t h e t h c o r c t i c a l values for )7=4 with the F r b m a n p a r a m e t e r B' =0 r e s pectively B' : 0..9. In the upper part of the figures a r e givcn the corresponding experimental (+) and t h e o r e t i cal (I) values of t h e / 3 - e n e r g i e s . The values of -1 giwm element a r e conneetcd by solid lines. o r 8.8 f o r P u i s o t o p e s . T h e l a r g e v a r i a t i o n of t h e h i n d r a n c e f a c t o r s f o r 2 3 8 p u a n d 2 4 0 p u i s due to the fact that these nuclei are situated at the front i e r of two n e u t r o n z o n e s w i t h d i f f e r e n t q w w v a l u e s ( s e e fig. 1). F o r s m a l l v a l u e s of t h e parameter p the hindrance factors become very l a r g e i m p l y i n g t h a t / ~ - v i b r a t i o n a l s t a t e s a r e not p u r e p a i r i n g v i b r a t i o n a l o n e s [6]. T h e i n t r o d u c t i o n of a F r 0 m a n m a t r i x f o r B - s t a t e s w i l l d e crease the hindrance factors. The results for the s a m e F r 0 m a n p a r a m e t e r [3] B = 0.9 a s f o r t h e g r o u n d s t a t e a r e s h o w n by a r r o w s a n d o p e n c i r c l e s in f i g . 2. No e s s e n t i a l d i f f e r e n c e i s o b t a i n e d .
A d e t a i l e d a n a l y s i s of the i n f l u e n c e of t h e c o u p l i n g b e t w e e n g r o u n d a n d ~ % s t a t e s on a l p h a i n t e n sities must be effectuated. W h e n t h e n u m b e r of p r o t o n s Z i n c r e a s e s t h e hindr,'mce factors for p = constant increase very m u c h . T h i s l e a d s to s o m e d o u b t s c o n c e r n i n g t h e v a l i d i t y of the a s s u m p t i o n t h a t t h e p h o n o n v a c u u m is identical for the initial and final nucleus. Indeed computing the hindrance factors with the p h o n o n s t a t e w a v e f u n c t i o n s f o r the n u c l e i (N, Z + + 2), t h e o b t a i n e d v a l u e s a r e m u c h d i f f e r e n t f r o m t h o s e c o r r e s p o n d i n g to t h e n u c l e i (N, Z), w h e r e N a n d Z a r e t h e n u m b e r of t h e n e u t r o n s r e s p e c t i v e l y of t h e p r o t o n s of t h e f i n a l n u c l e i ( s e e fig. 3). T h i s i m p l i e s t h a t a m o r e s o p h i s t i c a t e d t h e o r y of alpha decay which takes into account the differenc e s in t h e b a c k g o i n g d i a g r a m s of t h e i n i t i a l a n d final nuclei is needed. T h e r e d u c e d w a v e a m p l i t u d e s b2 f o r f l = v i b r a t i o n a l s t a t e s a r e g i v e n in t h e t a b l e 2 f o r B ' = 0 a n d B ' = 0.9 t o g e t h e r w i t h t h e e x p e r i m e n t a l d a t a [ 7 - 1 0 ] . T h e r e a r e no e s s e n t i a l d i f f e r e n c e s f o r d i f f e r e n t v a l u e s of t h e d e f o r m a t i o n p a r a m e t e r 77 = 4 o r 7/= 6. A s t r o n g d e p e n d e n c e of t h e b 2 = v a l u e s w i t h t h e •215
Volume 24B, number 5
PHYSICS
F r O m a n p a r a m e t e r B ' is o b t a i n e d . T h e t h e o r e t i cal v a l u e s , f o r the s a m e v a l u e s of the p a r a m e t e r P a s f o r the e x p l a n a t i o n of the h i n d r a n c e f a c t o r s , a r e in a g r e e m e n t with the e x p e r i m e n t a l data if we a s s u m e that the F r O m a n p a r a m e t e r B ' is i n c r e a s i n g with the m a s s n u m b e r s t a r t i n g with z e r o f o r 232U and b e c o m i n g a p p r o x i m a t e l y 0.9 for 240pu, the s a m e v a l u e a s for the g r o u n d s t a t e a l p h a t r a n s i t i o n s . No m i x i n g of the g r o u n d a n d / 3 - b a n d s was t a k e n into a c c o u n t . We would like to thank P r o f e s s o r s A. B o h r , B. M o t t e l s o n , D. B~s, J . O . R a s m u s s e n and V. G. S o l o v i e v f o r t h e i r i n t e r e s t in this p a p e r . We a l s o thank P r o f e s s o r K. V. L a u r i k a i n e n f o r the kind h o s p i t a l i t y at the R e s e a r c h I n s t i t u t e f o r T h e o r e t i cal P h y s i c s . MmW t h a n k s a r e due a l s o to M r . E. R i i h i m a k i f o r p e r f o r m i n g the n u m e r i c a l c a l c u l a tions.
RELATION
BETWEEN
I, E T T E R S
6 March 1967
References
1. K.M.Zheleznova, A.A.Korneichuk, V.G.Soloviev, P. Vogel and G. Yungklausen, Dubna, Preprint JINR, D-2157 (1965). 2. A.Shndulescu and O.Dumitrescu, Physics I,etters 19 (1965) 404. 3. A.S~induleseu and F.Staneu, Acta Phys. Polon. 27 {1965) 655. 4. O. Dumitrescu, E.Riihim~ikiandA.SSndulescu, to be published. 5. E. Riihim~iki and A.Shnduleseu, to be published. 6. D.R.B~s and R . A . B r o g l i a . Pairing Vibrations, P r e print Nordita 1965. 7. S.Bj0rnholm, M.I.ederer, F . A s a r o a n d I . P e r l m a n , Phys. Rev. 130 (1963) 2000. 8. C . M . L e d e r e r , UCRL-11028, Thesis, University of California (1963) unpublished. 9. J . K . Poggenburg, UCRI,-16187. Thesis, University of California (1965) unpublished. 10. S.Bj¢rnholm, Nuclear excitations in even isotopes of the heaviest elements (Munksgaard, University of Copenhagen, Copenhagen, 1965).
PROTON-NUCLEUS INTERACTION AT
20
AND GeV
PROTON-NUCLEON
W. E. F R A H N * International Centre f o r Theoretical Physics, Trieste, Italy
Rcceivcd 15 January 1967
It is shown that the parameters for proton-nucleus scattering at 19.3 GeV/c are closely related to those for proton-nucleon scattering at the same energy.
R e c e n t p a r t i a l - w a v e a n a l y s e s [1,2] of the d i f f e r e n t i a l and t o t a l c r o s s s e c t i o n s of 19.3 GeV p r o tons s c a t t e r e d by c o m p l e x n u c l e i [3] h a v e r e v e a l ed that the s i g n of the r e a l p a r t of the c o h e r e n t n u c l e a r s c a t t e r i n g a m p l i t u d e at t h i s e n e r g y is n e g a t i v e , c o r r e s p o n d i n g to a p r e d o m i n a n t l y r e p u l s i v e n u c l e o n - n u c l e u s i n t e r a c t i o n . One e x p e c t s that t h i s p r o p e r t y is c l o s e l y c o n n e c t e d with the f a c t that the e l a s t i c p r o t o n - n u c l e o n a m p l i t u d e h a s an a p p r e c i a b l e n e g a t i v e r e a l p a r t in the m u l t i GeV r e g i o n [4]. In the p r e s e n t n o t e we show, by m e a n s of a s i m p l e i m p u l s e a p p r o x i m a t i o n , that the p a r a m e t e r s of the c o h e r e n t p r o t o n - n u c l e u s s c a t t e r i n g a r e in f a c t q u a n t i t a t i v e l y r e l a t e d to t h o s e to p r o t o n - n u c l e o n e l a s t i c s c a t t e r i n g . T h e p r o t o n - n u c l e u s s c a t t e r i n g d a t a at 19.3 G e V / c h a v e b e e n d e s c r i b e d in t e r m s of a p a r a m e t e r i z e d s c a t t e r i n g f u n c t i o n ~l = S 0,) = 216
= exp[i25(~)] of the f o r m [5] S(X) = g(~) + E[1 - g ( ~ ) ] + i # d g / d k ,
(1)
w h e r e ~,~(~) = [1 + e x p ( A - X ) , / h ] -1 and ~ = l+½. T h e p a r a m e t e r s A and A a r e s e m i c l a s s i c a l l y r e l a t e d to the r a d i u s R and the d i f f u s e n e s s d of the i n t e r a c t i o n r e g i o n by the r e l a t i o n s A = k R , A = kd, w h e r e k is the c . m . m o m e n t u m , ¢ is the t r a n s p a r e n c y , and # m e a s u r e s the s t r e n g t h of the r e a l p a r t of the p r o t o n - n u c l e u s i n t e r a c t i o n . F o r the f o l l o w i n g c a l c u l a t i o n s we m a y n e g l e c t C o u l o m b and s p i n - d e p e n d e n t t e r m s . In t h i s c a s e the c o h e r e n t p r o t o n - n u c l e u s s c a t t e r i n g a m p l i t u d e r e s u l t i n g f r o m eq. (1) i s g i v e n by * Permanent address: Physics Department, University of Cape Town.