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Diffraction mechanism for direct nuclear reactions
Volume 7, number 5
P H YS I C S L E T T E R S
DIFFRACTION
MECHANISM
FOR
DIRECT
15 December 1963
NUCLEAR
REACTIONS*
A. DAR
Department of Nuclear Physics, The Weizrnann Institute of Science, Rehovoth, Israel Received 15 November 1963
B u t l e r ' s t h e o r y 1) of d i r e c t nuclear r e a c t i o n s r e s t r i c t s the reaction to the nuclear surface and is based on the assumption that the nucleus is t r a n s p a rent for both the incident and final unbound p a r t i c l e s . However, optical model analysis of elastic s c a t t e r ing 2) and d i r e c t nuclear r e a c t i o n s yield absorptive potentials which imply a mean f r e e path for nucleons in the nuclear surface region, of the o r d e r of 1.5 × 10-13 cm for - 15 MeV incident energy. F o r heavier nuclear p r o j e c t i l e s the mean f r e e path is obviously shorter. One m a y t h e r e f o r e conclude that nuclear t a r g e t s behave essentially like black spheres for all nuclear p r o j e c t i l e s with medium incident energy. In the case of elastic s c a t t e r i n g if we neglect external potential c o r r e c t i o n s and reflection c o r r e c t i o n s , we are than led to a diffraction-like angq)ar distribution, typical of a black disk. Some c a s e s 3) were known to fit this diffraction picture. We have now s y s t e m a t i c a l l y investigated the c r o s s sections for elastic s c a t t e r i n g of n, p, d, t, He3, q, C12, N14 and O 16 f r o m v a r i o u s t a r g e t s in the region above 10 MeV p e r particle in the incident projectile. The data show a r e m a r k a b l e diffraction behaviour which is s m e a r e d a little, probably due to external potential scattering. The analysis is based on the black sphere diffraction f o r m u l a 4) f o r the s c a t t e r i n g amplitude: f(0) = ikR 2 (1 + cos 0) J l ( l q l R) 2[qlR
'
(1)
where k is the magnitude of the momentum of the incident p r o j e c t i l e , 0 is the s c a t t e r i n g angle, Iql is the magnitude of the momentum t r a n s f e r q, and R is the "diffraction radius". The e x p r e s s i o n (1) f o r the s c a t t e r i n g amplitude can be derived 5) f r o m appropriate boundary conditions imposed on the total wave function in the f o r m a l s c a t t e r i n g theory. The same method is applied also 5) to the case of d i r e c t nuclear reactions. D i r e c t nuclear r e a c t i o n s at medium e n e r g y a r e localized at the nuclear surface 6) and the bulk of the * The research reported in this document has been sponsored in part by the Office of Scientific Research, OAR, through th.e European Office, Aerospace Research, United States Air Force.
nucleus is essentially an absorptive medium for both the projectile and the reaction products. The " s o u r c e " of the detected reaction product which is emitted in the reaction in the f o r w a r d direction is therefore localized on an annhlar region located at the plane of the line which s e p a r a t e s the shaded p a r t of the nucleus f r o m its illuminated part. The s o u r c e is also a coherent one since we a s s u m e d i r e c t interaction, and its amplitude depends on the angular coordinate ~ of the ring through Ylm (0 =~1 , ~). In the case of deuteron stripping (d,p) this dependence c o m e s through the phase of the captured neutrons, Y lnmn(O= ~1 ,~), since the detected protons are detached f r o m the neutron at the position where the latter is captured. In the 1 case of a knock-out y lm(O= ~,~o) is the amplitude of the bound particle before it was knocked out (assuming the incident projectile is captured in an s state). The Fraunhofer amplitude is t h e r e f o r e p r o portional to (1
+~cosO)
f
R+~R
R
f
2,
e-iq "r rlm(O=½,,~)rdrd~a
O
(2)
where ~ R is the width of the annular region. F o r I ql ~R << 1 the differential c r o s s section turns out to be proportional to:
(1-- + k-~ cos O)2 (2,,RAR)2 ×
T,
(2U1~
m=-l,-/+2 \ 4 , /
q-m): (l+m)~
(l-m)!!(l+m)!! [ J t m I ( i q l R ) ] 2 ,
(3)
where &i is the momentum of the incident particle and kf is the momentum of the emitted particle. KfR sin O has been replaced by l ql in o r d e r to comply with time r e v e r s i b i l i t y and s y m m e t r y r e q u i r e ments 7). The differential c r o s s section da/dG derived f r o m (3) has the following general c h a r a c t o r i s t i c s : 1) F o r a certain angular momentum l, d~/dO is proportional to linear combinations of squares of cylindrical Bessel functions of o r d e r s l , l - 2,1-4 , . .
339
Volume 7, n u m b e r 5
PHYSICS
LETTERS
For example: d q_~
for l = 0
d~
a: Jo2[iqlR)" '
p e c t i v e l y . M o r e d a t a a r e d i s c u s s e d in a f o r t h c o m i n g p u b l i c a t i o n 5). The c u r v e s w e r e n o r m a l i z e d to f i t t h e f i r s t m a x i m u m . The m a i n d e v i a t i o n f r o m the B u t l e r theory occurs for l > 1 where the latter predicts a
"
d~_~ a: J 1 2 ( I q ] R ) d~
for l = 1
f
for 1 = 2
d_..g._~~: ¼Jo2(iqlR) + } J 2 2 ( i q l R ) da
for l = 3
dcr a: ~ J 1 2 ( I q l R ) + ~ J 3 2 ( I q l R ) e . c . d~
;
f
[
[
Zr 9' (d.p) Zr s2
h0 --
Ed=10.85MEV.
~/2~/Yx c o s {Ix - ½~(/+ ½)]}.
....
w
O.B -fD
Q-'5.53MEV
BUTLER CURVE Ln=2 R=?.5fro. FFRACTION CURVE Ln=2 R='Z5fro,
/~/~~llt! ~DI
Beyond t h e i r f i r s t m a x i m a t h e B e s s e l f u n c t i o n s a p proach rather quickly their asymptotic form: Jl(X) ~
15 D e c e m b e r 1963
0.6
c
I I
(4)
0,4--
X~co
i
J
I I
I T h e r e f o r e i f the d i f f e r e n t i a l c r o s s s e c t i o n s f o r l a r g e O.2--/ a r g u m e n t s a r e p l o t t e d v e r s u s I q l R >> 1, t h o s e f o r I even a n g u l a r m o m e n t a a r e out of p h a s e with t h o s e f o r odd o n e s . T h i s p r o p e r t y a l s o r e s u l t s f r o m B o r n t i 60 90 120 30 approximation calculations. eLab The t h e o r y a b o v e h a s b e e n a p p l i e d with r e m a r k a b l e s u c c e s s to t h e a n a l y s i s of v a r i o u s d i r e c t n u c l e a r Fig. 2. Experimental results 9), Butler curve and diffraction curve for Zr91(d,p)Zr 92, stripping rereactions such as stripping, heavy particle stripping, action, E d = 10.8 MeV, Q = 5.53 MeV. p i c k up, k n o c k out a n d i n e l a s t i c c o l l i s i o n . T y p i c a l f i t s a r e i l l u s t r a t e d in f i g s . 1-5 f o r P b ( n , n ) P b e l a s t i c s c a t t e r i n g , ( d , p ) s t r i p p i n g r e a c t i o n with /n = 2, h e a v y p a r t i c l e s t r i p p i n g (r,,a) w i t h l~=O, ( d , t ) p i c k up r e a c t i o n with In =0 and (~,p) k n o c k o u t r e a c t i o n w i t h / p = 1 r e s I I I I ~0.0 [
--
I
I
l~ I0"
i
I
C 12
Pb(n,n) Pb En = 14.5 MEV DIFFRACTION CURVE R=9Ofm
k
(a,a)
C Iz
Ea=I8MEV Q=O EXPERI MENTAL RESULTS DIFFRACTION CURVE(R.P.S,) 1.a=O
....
R=4.7
/" '# ~
~0.5 i' I
fm.
rr LU I--
BUTLER CURVE
¢n z
I0
°go o
"--
o •
o
o
-o
o
o
o
io [
I
20
I
40
I
60 8C.M.
I
80
o
o
I
I00
F i g . 1. E x p e r i m e n t a l r e s u l t s 8) and d i f f r a c t i o n c u r v e f o r the d i f f e r e n t i a l e l a s t i c c r o s s s e c t i o n of ~ 14 MeV n e u t r o n on P b . 340
/
o
"~+
120
I00 °
120 =
140 °
',,
160 °
180 °
8C.M. Fig. 3. E x p e r i m e n t a l r e s u l t s 10), B u y e r c u r v e and d i f f r a c t i o n c u r v e for C12(~,~}C 12 heavy p a r t i c l e s t r i p p i n g r e a c t i o n in b a c k w a r d a n g l e s , E ~ = 18 Me~v
Volume 7, number 5 I
P H YS I C S L E T T E R S I
I
25 li:
IJJ
(/3 2 0 Z gO
:E =-
15
b
lo
Io
I
s i n g l e s p h e r i c a l B e s s e l f u n c t i o n b e h a v i o u r of the differential c r o s s section while diffraction p r e d i c t s l i n e a r c o m b i n a t i o n of c y l i n d r i c a l B e s s e l f u n c t i o n s . The d a t a show s i g n i f i c a n t d i f f r a c t i o n f e a t u r e s . If the d i f f r a c t i o n p i c t u r e holds then the / - a s s i g n m e n t of the o r b i t of the s t r i p p e d p a r t i c l e m a y a l s o c h a n g e : t r a n s i t i o n s w h i c h looked like a m i x t u r e of l = 0 and l = 2 in the B u t l e r t h e o r y m a y now b e c o m e r a t h e r p u r e l= 2 etc. As to the o v e r a l l fit in f o r w a r d d i r e c t i o n s d i f f r a c t i o n p r e d i c t i o n s a r e about a s good a s t h o s e o b t a i n e d by the m o r e c o m p l i c a t e d o p t i c a l m o d e l c a l c u l a t i o n s . It i s f e l t that e x t e r n a l p o t e n t i a l s c a t t e r i n g c o r r e c t i o n s do not p l a y s u c h in i m p o r t a n t r o l e a s i n the c a s e of e l a s t i c s c a t t e r i n g . I m p r o v e m e n t of the fit in b a c k w a r d a n g l e s m a y p r o b a b l y be o b t a i n e d by a l l o w i n g s o m e r e f l e c t i o n and the s u p e r i m p o s i n g of d i f f e r e n t m e c h a n i s m s ; t h i s w i l l be d i s c u s s e d in a f u t u r e publication.
I
F 19 (d,t) F I°
t
Ed= 14.8 MEV, Q=-4.144 MEV BUTLER L=O,
....
CURVE R = 7 frn.
DIFFRACTION CURVE R= 7.2 fro.
I I0
20
:50
40
5 0 8C.M"
The a u t h o r would like to t h a n k P r o f e s s o r s A. de-~Chalit I. T a l m i , C . A . L e v i n s o n and Dr. A. S. R e i n e r f o r h e l p f u l s u g g e s t i o n s and f r u i t f u l d i s cussions.
Fig. 4. Experimental results 11), Butler curve and diffraction curve for F19(d, t)F 18 pick up reaction. Ed= 14.8 MeV, Q=4.18 MeV. I
8
P- 7
I
I
]
I
I
I
]
ci2 (a,p) N ts
k
Q=-4.96
I
....
Mev
E d = 5 0 . 5 Mev
DIFFRACTION
CURVE
Ip=l
Z
!
R = 4 . 9 fm.
>6 ¢r ¢Y
15 December 1963
- -
BUTLER
/
CURVE
5
¢Y
<~ 4 3 "o
2
I
1
/%, ,
20
40
60
,
,
80
I00
,_04 120
,
140
160
8C.M.
Fig. 5. Experimental results 12), Butler curve and diffraction curve for C12(a,p)N15 knock-out reaetionEa=30.5 MeV, Q =-4.96 MeV. The experimental points are due to C.E. Hunting and W.S.Wall as quoted in ref. 12). 1) S.T. Butler, Proc. Roy. Soc. (London) A208 (1951) 559; 6) E . g . , N.Austern, in: Selected topics in nuclear theory Austern, Butler and McManus, Phys. Rev. 92 (1953) 350; ( I . A . E . A . , Vienna, 1963); S. T.Butler, Phys. Rev. 106 (1957) 272. I.E.MeCarthy, Phys. Rev. 128 (1962) 1237. 2) E . g . , A.E.Glassgold, Progr. Nuclear Phys. 7 (1959) 7) R . J . G l a u b e r , in: Lectures in theoretical physics, vol. page 123; 1 (Interseience Publishers, Inc. New York, 1959). F . G . P e r r y , Phys.Rev. 131 (1963) 745. 8) S. Berko et al., Nuclear Phys. 6 (1958) 210; 3) E . g . , J . S . B l a i r , Phys. Rev. 108 (1957) 827; J.H.Coon et al., Phys. Rev. 111 (1958) 250. A. I. Yavin and G.W. Farwell, Nuclear Phys. 12 (1959) 1; 9) H . J . M a r t i n et al., Phys. Rev. 125 (1962) 942. G. Igo, W. Lorenz and U. Schmidt, Rohr Report No. 29 10) J . C . C o r e l l i et al., Phys. Rev. 116 (1959) 1184; (Max Planck Institute for Nuclear Physics, 1961). S.S.M.Wong et a l . , Phys. Rev. 125 (1962) 280. 4) Bohr, Peierls, Placzeck and Bethe, Phys. Rev. 57 I1) A.I.Hamburger, Phys. Rev. 118 (1960) 1271. (1940) 1075. 12) S.T.Butler, Phys. Rev. 106 (1957) 272. 5) A.Dar, to be published in Nuclear Phys.
341
P H YS I C S L E T T E R S
DIFFRACTION
MECHANISM
FOR
DIRECT
15 December 1963
NUCLEAR
REACTIONS*
A. DAR
Department of Nuclear Physics, The Weizrnann Institute of Science, Rehovoth, Israel Received 15 November 1963
B u t l e r ' s t h e o r y 1) of d i r e c t nuclear r e a c t i o n s r e s t r i c t s the reaction to the nuclear surface and is based on the assumption that the nucleus is t r a n s p a rent for both the incident and final unbound p a r t i c l e s . However, optical model analysis of elastic s c a t t e r ing 2) and d i r e c t nuclear r e a c t i o n s yield absorptive potentials which imply a mean f r e e path for nucleons in the nuclear surface region, of the o r d e r of 1.5 × 10-13 cm for - 15 MeV incident energy. F o r heavier nuclear p r o j e c t i l e s the mean f r e e path is obviously shorter. One m a y t h e r e f o r e conclude that nuclear t a r g e t s behave essentially like black spheres for all nuclear p r o j e c t i l e s with medium incident energy. In the case of elastic s c a t t e r i n g if we neglect external potential c o r r e c t i o n s and reflection c o r r e c t i o n s , we are than led to a diffraction-like angq)ar distribution, typical of a black disk. Some c a s e s 3) were known to fit this diffraction picture. We have now s y s t e m a t i c a l l y investigated the c r o s s sections for elastic s c a t t e r i n g of n, p, d, t, He3, q, C12, N14 and O 16 f r o m v a r i o u s t a r g e t s in the region above 10 MeV p e r particle in the incident projectile. The data show a r e m a r k a b l e diffraction behaviour which is s m e a r e d a little, probably due to external potential scattering. The analysis is based on the black sphere diffraction f o r m u l a 4) f o r the s c a t t e r i n g amplitude: f(0) = ikR 2 (1 + cos 0) J l ( l q l R) 2[qlR
'
(1)
where k is the magnitude of the momentum of the incident p r o j e c t i l e , 0 is the s c a t t e r i n g angle, Iql is the magnitude of the momentum t r a n s f e r q, and R is the "diffraction radius". The e x p r e s s i o n (1) f o r the s c a t t e r i n g amplitude can be derived 5) f r o m appropriate boundary conditions imposed on the total wave function in the f o r m a l s c a t t e r i n g theory. The same method is applied also 5) to the case of d i r e c t nuclear reactions. D i r e c t nuclear r e a c t i o n s at medium e n e r g y a r e localized at the nuclear surface 6) and the bulk of the * The research reported in this document has been sponsored in part by the Office of Scientific Research, OAR, through th.e European Office, Aerospace Research, United States Air Force.
nucleus is essentially an absorptive medium for both the projectile and the reaction products. The " s o u r c e " of the detected reaction product which is emitted in the reaction in the f o r w a r d direction is therefore localized on an annhlar region located at the plane of the line which s e p a r a t e s the shaded p a r t of the nucleus f r o m its illuminated part. The s o u r c e is also a coherent one since we a s s u m e d i r e c t interaction, and its amplitude depends on the angular coordinate ~ of the ring through Ylm (0 =~1 , ~). In the case of deuteron stripping (d,p) this dependence c o m e s through the phase of the captured neutrons, Y lnmn(O= ~1 ,~), since the detected protons are detached f r o m the neutron at the position where the latter is captured. In the 1 case of a knock-out y lm(O= ~,~o) is the amplitude of the bound particle before it was knocked out (assuming the incident projectile is captured in an s state). The Fraunhofer amplitude is t h e r e f o r e p r o portional to (1
+~cosO)
f
R+~R
R
f
2,
e-iq "r rlm(O=½,,~)rdrd~a
O
(2)
where ~ R is the width of the annular region. F o r I ql ~R << 1 the differential c r o s s section turns out to be proportional to:
(1-- + k-~ cos O)2 (2,,RAR)2 ×
T,
(2U1~
m=-l,-/+2 \ 4 , /
q-m): (l+m)~
(l-m)!!(l+m)!! [ J t m I ( i q l R ) ] 2 ,
(3)
where &i is the momentum of the incident particle and kf is the momentum of the emitted particle. KfR sin O has been replaced by l ql in o r d e r to comply with time r e v e r s i b i l i t y and s y m m e t r y r e q u i r e ments 7). The differential c r o s s section da/dG derived f r o m (3) has the following general c h a r a c t o r i s t i c s : 1) F o r a certain angular momentum l, d~/dO is proportional to linear combinations of squares of cylindrical Bessel functions of o r d e r s l , l - 2,1-4 , . .
339
Volume 7, n u m b e r 5
PHYSICS
LETTERS
For example: d q_~
for l = 0
d~
a: Jo2[iqlR)" '
p e c t i v e l y . M o r e d a t a a r e d i s c u s s e d in a f o r t h c o m i n g p u b l i c a t i o n 5). The c u r v e s w e r e n o r m a l i z e d to f i t t h e f i r s t m a x i m u m . The m a i n d e v i a t i o n f r o m the B u t l e r theory occurs for l > 1 where the latter predicts a
"
d~_~ a: J 1 2 ( I q ] R ) d~
for l = 1
f
for 1 = 2
d_..g._~~: ¼Jo2(iqlR) + } J 2 2 ( i q l R ) da
for l = 3
dcr a: ~ J 1 2 ( I q l R ) + ~ J 3 2 ( I q l R ) e . c . d~
;
f
[
[
Zr 9' (d.p) Zr s2
h0 --
Ed=10.85MEV.
~/2~/Yx c o s {Ix - ½~(/+ ½)]}.
....
w
O.B -fD
Q-'5.53MEV
BUTLER CURVE Ln=2 R=?.5fro. FFRACTION CURVE Ln=2 R='Z5fro,
/~/~~llt! ~DI
Beyond t h e i r f i r s t m a x i m a t h e B e s s e l f u n c t i o n s a p proach rather quickly their asymptotic form: Jl(X) ~
15 D e c e m b e r 1963
0.6
c
I I
(4)
0,4--
X~co
i
J
I I
I T h e r e f o r e i f the d i f f e r e n t i a l c r o s s s e c t i o n s f o r l a r g e O.2--/ a r g u m e n t s a r e p l o t t e d v e r s u s I q l R >> 1, t h o s e f o r I even a n g u l a r m o m e n t a a r e out of p h a s e with t h o s e f o r odd o n e s . T h i s p r o p e r t y a l s o r e s u l t s f r o m B o r n t i 60 90 120 30 approximation calculations. eLab The t h e o r y a b o v e h a s b e e n a p p l i e d with r e m a r k a b l e s u c c e s s to t h e a n a l y s i s of v a r i o u s d i r e c t n u c l e a r Fig. 2. Experimental results 9), Butler curve and diffraction curve for Zr91(d,p)Zr 92, stripping rereactions such as stripping, heavy particle stripping, action, E d = 10.8 MeV, Q = 5.53 MeV. p i c k up, k n o c k out a n d i n e l a s t i c c o l l i s i o n . T y p i c a l f i t s a r e i l l u s t r a t e d in f i g s . 1-5 f o r P b ( n , n ) P b e l a s t i c s c a t t e r i n g , ( d , p ) s t r i p p i n g r e a c t i o n with /n = 2, h e a v y p a r t i c l e s t r i p p i n g (r,,a) w i t h l~=O, ( d , t ) p i c k up r e a c t i o n with In =0 and (~,p) k n o c k o u t r e a c t i o n w i t h / p = 1 r e s I I I I ~0.0 [
--
I
I
l~ I0"
i
I
C 12
Pb(n,n) Pb En = 14.5 MEV DIFFRACTION CURVE R=9Ofm
k
(a,a)
C Iz
Ea=I8MEV Q=O EXPERI MENTAL RESULTS DIFFRACTION CURVE(R.P.S,) 1.a=O
....
R=4.7
/" '# ~
~0.5 i' I
fm.
rr LU I--
BUTLER CURVE
¢n z
I0
°go o
"--
o •
o
o
-o
o
o
o
io [
I
20
I
40
I
60 8C.M.
I
80
o
o
I
I00
F i g . 1. E x p e r i m e n t a l r e s u l t s 8) and d i f f r a c t i o n c u r v e f o r the d i f f e r e n t i a l e l a s t i c c r o s s s e c t i o n of ~ 14 MeV n e u t r o n on P b . 340
/
o
"~+
120
I00 °
120 =
140 °
',,
160 °
180 °
8C.M. Fig. 3. E x p e r i m e n t a l r e s u l t s 10), B u y e r c u r v e and d i f f r a c t i o n c u r v e for C12(~,~}C 12 heavy p a r t i c l e s t r i p p i n g r e a c t i o n in b a c k w a r d a n g l e s , E ~ = 18 Me~v
Volume 7, number 5 I
P H YS I C S L E T T E R S I
I
25 li:
IJJ
(/3 2 0 Z gO
:E =-
15
b
lo
Io
I
s i n g l e s p h e r i c a l B e s s e l f u n c t i o n b e h a v i o u r of the differential c r o s s section while diffraction p r e d i c t s l i n e a r c o m b i n a t i o n of c y l i n d r i c a l B e s s e l f u n c t i o n s . The d a t a show s i g n i f i c a n t d i f f r a c t i o n f e a t u r e s . If the d i f f r a c t i o n p i c t u r e holds then the / - a s s i g n m e n t of the o r b i t of the s t r i p p e d p a r t i c l e m a y a l s o c h a n g e : t r a n s i t i o n s w h i c h looked like a m i x t u r e of l = 0 and l = 2 in the B u t l e r t h e o r y m a y now b e c o m e r a t h e r p u r e l= 2 etc. As to the o v e r a l l fit in f o r w a r d d i r e c t i o n s d i f f r a c t i o n p r e d i c t i o n s a r e about a s good a s t h o s e o b t a i n e d by the m o r e c o m p l i c a t e d o p t i c a l m o d e l c a l c u l a t i o n s . It i s f e l t that e x t e r n a l p o t e n t i a l s c a t t e r i n g c o r r e c t i o n s do not p l a y s u c h in i m p o r t a n t r o l e a s i n the c a s e of e l a s t i c s c a t t e r i n g . I m p r o v e m e n t of the fit in b a c k w a r d a n g l e s m a y p r o b a b l y be o b t a i n e d by a l l o w i n g s o m e r e f l e c t i o n and the s u p e r i m p o s i n g of d i f f e r e n t m e c h a n i s m s ; t h i s w i l l be d i s c u s s e d in a f u t u r e publication.
I
F 19 (d,t) F I°
t
Ed= 14.8 MEV, Q=-4.144 MEV BUTLER L=O,
....
CURVE R = 7 frn.
DIFFRACTION CURVE R= 7.2 fro.
I I0
20
:50
40
5 0 8C.M"
The a u t h o r would like to t h a n k P r o f e s s o r s A. de-~Chalit I. T a l m i , C . A . L e v i n s o n and Dr. A. S. R e i n e r f o r h e l p f u l s u g g e s t i o n s and f r u i t f u l d i s cussions.
Fig. 4. Experimental results 11), Butler curve and diffraction curve for F19(d, t)F 18 pick up reaction. Ed= 14.8 MeV, Q=4.18 MeV. I
8
P- 7
I
I
]
I
I
I
]
ci2 (a,p) N ts
k
Q=-4.96
I
....
Mev
E d = 5 0 . 5 Mev
DIFFRACTION
CURVE
Ip=l
Z
!
R = 4 . 9 fm.
>6 ¢r ¢Y
15 December 1963
- -
BUTLER
/
CURVE
5
¢Y
<~ 4 3 "o
2
I
1
/%, ,
20
40
60
,
,
80
I00
,_04 120
,
140
160
8C.M.
Fig. 5. Experimental results 12), Butler curve and diffraction curve for C12(a,p)N15 knock-out reaetionEa=30.5 MeV, Q =-4.96 MeV. The experimental points are due to C.E. Hunting and W.S.Wall as quoted in ref. 12). 1) S.T. Butler, Proc. Roy. Soc. (London) A208 (1951) 559; 6) E . g . , N.Austern, in: Selected topics in nuclear theory Austern, Butler and McManus, Phys. Rev. 92 (1953) 350; ( I . A . E . A . , Vienna, 1963); S. T.Butler, Phys. Rev. 106 (1957) 272. I.E.MeCarthy, Phys. Rev. 128 (1962) 1237. 2) E . g . , A.E.Glassgold, Progr. Nuclear Phys. 7 (1959) 7) R . J . G l a u b e r , in: Lectures in theoretical physics, vol. page 123; 1 (Interseience Publishers, Inc. New York, 1959). F . G . P e r r y , Phys.Rev. 131 (1963) 745. 8) S. Berko et al., Nuclear Phys. 6 (1958) 210; 3) E . g . , J . S . B l a i r , Phys. Rev. 108 (1957) 827; J.H.Coon et al., Phys. Rev. 111 (1958) 250. A. I. Yavin and G.W. Farwell, Nuclear Phys. 12 (1959) 1; 9) H . J . M a r t i n et al., Phys. Rev. 125 (1962) 942. G. Igo, W. Lorenz and U. Schmidt, Rohr Report No. 29 10) J . C . C o r e l l i et al., Phys. Rev. 116 (1959) 1184; (Max Planck Institute for Nuclear Physics, 1961). S.S.M.Wong et a l . , Phys. Rev. 125 (1962) 280. 4) Bohr, Peierls, Placzeck and Bethe, Phys. Rev. 57 I1) A.I.Hamburger, Phys. Rev. 118 (1960) 1271. (1940) 1075. 12) S.T.Butler, Phys. Rev. 106 (1957) 272. 5) A.Dar, to be published in Nuclear Phys.
341