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4He(p, t)2p,4He(p,3He)pn and4He(p,3He)d reactions at 156 MeV and the final - state interaction
Volume 25B, number 4
4He(p, t)2p, AT 156
PHYSICS
4He(p, 3He)pn MeV AND THE
LETTERS
AND 4He(p, FINAL-STATE
4 September 1967
3He)d REACTIONS INTERACTION
M. BERNAS, J. K. LEE *, D. BACHELIER, I. BRISSAUD, C. DETRAZ, P. RADVANYI and M. ROY Institut
de Physiqzde
NuclLaire. Laboratoire Orsay, France
Joliot-Cum’e.
Received 21 July 1967
The p-n and p-p final-state interaction in the (p, 3He) and (p, t) reactions on 4He at 156 MeV has been measured for several t and 3He angles and analysed in terms of the Watson-Migdal theory. Agreement is rather good for the shapes of the energy spectra, but not for their relative magnitudes. The differential cross section of 4He+p dd +3He has been measured from 8d = 4O to 1770 c. m.
Final-state interactions have been studied in recent years in the lo-50 MeV range on several nuclear reactions, mainly the single-nucleon transfer and the charge exchange reactions on mass 2 and 3 nuclei [l]. The analysis of these experiments is generally made using the WatsonMigdal treatment [2], but agreement with theory varies significantly according to the reactions and nuclei involved. With two-nucleon transfer reactions on 4He at 156 MeV, our purpose was to extend these studies to higher energies, where the Watson-Migdal assumptions should be more valid. In these experiments with three bodies in the final state, measurements can be made on the upper part of the energy spectrum of one of these particles. In this condition, the two remaining particles have low relative momentum q and can interact strongly. Watson and Migdal [2] have then shown that, with several simplifying assumptions, the differential cross section corresponding to the energetic particle emitted at angle 0 with energy E can be factorised as follows:
,pTnscG(@)S(6,, 4) ,
(1)
isospins where p(E) contains the phase space spectrum; G(0) corresponds to the primary reaction (here the double-nucleon transfer reaction) and is assumed to be independent of E for the limited part of the energy spectrum under study, which should be more valid at higher incident energies; the * Present address: University of Toronto, Department of Physics. 260
‘He + p _ EP
-
d + ‘He
155.4MN
0.1
0.01
L
0
30
60
90
120
150
180 ed CM
Fig. 1. Angular distribution of the 4He(p, d)3He reaction at 156 MeV. The backward-angle values were obtained by the study of the 4He(p, 3He)d reaction at forward angles. last term corresponds to the low energy S-wave scattering between the two remaining particles with phase shifts 6, (singlet and/or triplet); it is the last term which gives rise to the characteristic peak at the upper part of the energy spectrum; c represents the statistical and coupling coefficients arising from the averaging over initial states and summing over final states. The Wat son-Migdal formalism neglects the final-state interaction between the detected particle and one of the remaining particles; this should be actually valid in our experimental situation at 156 MeV where the only important final:state interaction
Volume 25B, number 4
PHYSICS
LETTERS
,z , ? ,
IO0 I
4He ( p , t )
~,
,
,~
,
4
, e , ,
4 S e p t e m b e r 1967
6
, ?
,
z
,
2p
2~J~ cr
80~
?(5-)
,2 , o,(,o.) NeV E ~ ( ~ )
Ep =f,??.:,?eV
•
A i
*T
x~ 6 0
~ 4o
i~ 2o
io2
i04
~06
108
No
112
114
ii6
i18
120
Ei MeV
Fig. 2. a) Energy spectra of 4He(p, t)2p at 2 °, 5 °, 10 ° (gaseous target) and 15 ° (liquid target). The fits by the Watson-Migdal analysis are indicated for 2 ° and 10 ° (normalization for these fits is made at the maximum of the experimental spectrum with app=-7.78 fm and ro=2.71 fm; for 2 ° is also shown the fit calculated with the G~0) extracted from the 4He(p, 3He)d c r o s s section; the experimental energy resolution has been folded in the calculated curves); p-p relative energy scales are given for the three first angles, b) Energy spectra of 4He(p, 3He)pn at 2o (gaseous target). The peak on the right corresponds to the final p-n triplet bound state (4He(p, 3He)d reaction), the broad peak on the left to the final p-n scattering state. The dashed lines indicate the Watson-Migdal fits, normalized at the maximum of the experimental spectrum with aos =-23.68 fro, ros =2.48 fro, aot=5.39 fm and rot = 1.70 fm and calculated with the G(8) extracted from the bound-state c r o s s section. The upper scale gives the p-n relative energy. s h o u l d take p l a c e b e t w e e n the two u n o b s e r v e d p a r t i c l e s w i t h low r e l a t i v e e n e r g y . We u s e d a p r o t o n b e a m of about 10 -2 # A , m o n i t o r e d by a s e c o n d a r y e m i s s i o n c h a m b e r o r a He i o n i s a t i o n c h a m b e r , i m p i n g i n g on a He t a r g e t of 1.5 c m t h i c k n e s s and 4 × 8 cm-Z a r e a , s i t u a t e d at the b o t t o m of a l i q u i d - h e l i u m c r y o s t a t . T h i s t a r get could be f i l l e d with l i q u i d o r l o w - t e m p e r a t u r e g a s e o u s h e l i u m , a c c o r d i n g to the r e s o l u t i o n needed. Background contributions were subtracted using a dummy target. The overall f. w. h. m. i n s t r u m e n t a l e n e r g y r e s o l u t i o n at s m a l l a n g l e was about 0.8 M e V f o r ( p , p ' ) , 1.3 M e V f o r (p,d), 1.5 M e V f o r (p, 3He) and 0.9 M e V f o r ( p , t ) . T h e t o t a l a n g u l a r r e s o l u t i o n was 0.9 ° . The e m i t t e d p a r t i c l e s w e r e i d e n t i f i e d and a n a l y s e d by a m a g n e t i c s p e c t r o m e t e r f o l l o w e d by a p l a s t i c s c i n t i l l a t o r t e l e s c o p e e m p l o y i n g both r a n g e and AE/Ax d i s c r i m i n a t i o n t e c h n i q u e s . We m e a s u r e d at s e v e r a l a n g l e s the e n e r g y s p e c t r a of the e m i t t e d p r o t o n s , d e u t e r o n s , t r i t o n s and 3He. The p r o t o n s p e c t r a s h o w e d no o t h e r i n e l a s t i c p e a k than the b r o a d one a r o u n d 23 M e V e x c i t a t i o n e n e r g y o b s e r v e d p r e v i o u s l y at U p p s a l a [3]. T h e d e u t e r o n s p e c t r a and the h i g h e s t p a r t of
the 3He s p e c t r a (fig. 2b) g a v e us the t o t a l a n g u l a r d i s t r i b u t i o n ( f r o m 3.7 ° to 177Oc.m.) of the 4 H e ( p , d ) 3 H e r e a c t i o n s h o w n on fig. 1. T h i s m a y be c o m p a r e d with the f o r w a r d a n g l e m e a s u r e m e n t s at 93 M e V [4]. The observed triton spectra exhibit, near maxi m u m e n e r g y (fig. 2a), an a s y m m e t r i c p e a k m o s t i m p o r t a n t at s m a l l a n g l e s . Its s h a p e has b e e n c a l c u l a t e d u s i n g f o r m u l a (1) with ~ c S(So, q) = = 2 s i n 2 5o/9C207)q2; 5o can be e x p r e s s e d in t e r m s of the s i n g l e t p - ~ s c a t t e r i n g l e n g t h app and e f f e c t i v e r a n g e t o ; CZ07) is the C o u l o m b p e n e t r a t i o n f a c t o r [5]. On the 3He s p e c t r a (fig. 2b), at the h i g h e r e n e r g y end the p e a k a p p e a r s c o r r e s p o n d i n g to the bound t r i p l e t d e u t e r o n ; the b r o a d p e a k to the left c o r r e s p o n d s to the unbound n - p s i n g l e t and t r i p l e t s c a t t e r i n g s t a t e s . To fit the s h a p e of t h i s p e a k , f o r m u l a (1) w a s u s e d with: ~ c S ( 5 o , q) = = s i n 2 5 o s / 9 q 2 + s i n 2 6 o t / 3 q 2, w h e r e 5os and 5ot c a n now be e x p r e s s e d in t e r m s of the s i n g l e t and t r i p l e t p a r a m e t e r s a s , a t and r o s , rot [5]. In e a c h c a s e , we h a v e f o l d e d in the c a l c u l a t e d s p e c t r a the e x p e r i m e n t a l f i n i t e e n e r g y r e s o l u t i o n and c o n v e r t e d to the l a b o r a t o r y s y s t e m , in o r d e r to be a b l e to fit the e x p e r i m e n t a l data. 261
Volume25B. number 4
PHYSICS
T h e g e n e r a l a g r e e m e n t o b s e r v e d in t h e s e f i t s between the experimental shapes and the shapes calculated with the Watson-Migdal treatment seems reasonably good, at least at small angles. T h i s m i g h t b e r e l a t e d to t h e f a c t t h a t t h e e n e r g y b e t w e e n t h e t, o r t h e 3He, a n d e a c h of the two remaining nucleons is large, whereas the relat i v e e n e r g y b e t w e e n t h e s e two n u c l e o n s i s s m a l l ; a n d a l s o , f o l l o w i n g P h i l l i p s a r g u m e n t s [6], t h a t we h a v e a d o u b l e p i c k - u p p r i m a r y on a n u c l e u s a s t i g h t l y b o u n d a s 4He w i t h a s m a l l r . m . s . r a dius, which means that the primary interaction has effectively a short range. The deviation observed at larger relative energy could come from a p-wave contribution or from a slight dep e n d e n c e of G(O) o n t h e o b s e r v e d p a r t i c l e e n e r g y E. T h e a b o v e c a l c u l a t i o n s do not g i v e a b s o l u t e v a l u e s of t h e c r o s s s e c t i o n s , b u t it i s q u i t e i n t e r e s t i n g to c o m p a r e t h e t h r e e e x p e r i m e n t a l p e a k s . T o do t h i s we e x t r a c t e d , i n t h e way f o l l o w e d by W a t s o n [2] f o r the p + p ~ 7 r + + p + n r e a c t i o n (final t r i p l e t p n s c a t t e r i n g s t a t e ) , t h e v a l u e of G(O) f r o m t h e e x p e r i m e n t a l c r o s s s e c t i o n of ~he 4He3(P, 3 H e ) d re4action , a n d i n s e r t e d it in t h e H e ( p , H e ) p n a n d H e ( p , t)2p c a l c u l a t e d c r o s s sections, making the simplifying assumption that G(O) i s t h e s a m e i n t h e t h r e e r e a c t i o n s (the c o u p l i n g c o e f f i c i e n t s a p p e a r i n g in t h e f a c t o r c). F o r the p-n final-state interaction spectrum, the diff e r e n t i a l c r o s s s e c t i o n c a l c u l a t e d in t h i s way i s b e l o w t h e e x p e r i m e n t a l o n e b y a b o u t 40%. T a k i n g into account the experimental errors and the fact t h a t t h e t r i p l e t c o n t r i b u t i o n to t h e s c a t t e r i n g i s rather large, one may conclude that the above a r g u m e n t f o r G(O) i s v a l i d f o r the f i n a l p - n t r i p l e t s t a t e . But t h e c o m p a r i s o n w i t h t h e p - p f i n a l - s t a t e interaction shows that the calculated cross sect i o n i s m u c h to low (by a f a c t o r of a b o u t 4). T h i s w o u l d i n d i c a t e t h a t t h e c o r r e s p o n d i n g G(O) a r e n o t t h e s a m e f o r t h e (p, t) a n d (p, 3He) r e a c t i o n s , o r f o r f o r m a t i o n of t r i p l e t a n d s i n g l e t s t a t e s . O n e s h o u l d k e e p in m i n d t h a t t h i s c o m p a r i s o n h a s b e e n m a d e u n d e r t h e a s s u m p t i o n t h a t G(O) c a n b e fac~orised in the cross sections. A n o t h e r i n t e r e s t i n g p r o b l e m i s t h e v a r i a t i o n of
262
LETTERS
4 S e p t e m b e r 1967
the a b o v e c r o s s s e c t i o n s w i t h a n g l e 0. A c a l c u l a t i o n of G(0) a t o u r e n e r g y f o r t h e t w o - n u c l e o n p i c k - u p w o u l d g i v e a p r e d i c t i o n of t h i s v a r i a t i o n a n d c o u l d a l s o l e a d to i m p r o v e m e n t s in the f i t s . It is a p p a r e n t f r o m f o r m u l a (1) t h a t t h e s h a p e of t h e f i n a l - s t a t e i n t e r a c t i o n s h o u l d not v a r y v e r y m u c h a t 156 M e V f o r 0 ~. 30 ° . In f a c t , t h e e x p e r i m e n t a l s h a p e s t a y s a b o u t the s a m e f r o m 2 ° to 10 °. It i s y e t d i f f i c u l t to s a y if the g r a d u a l d i s a p p e a r a n c e of t h e p e a k f r o m 15 ° to l a r g e r a n g l e s ( w h e r e we c o u l d o n l y u s e t h e l i q u i d t a r g e t ) i s due to a w o r s e e n e r g y r e s o l u t i o n , o r to s o m e c o n t r i b u t i o n of a c o n t i n u o u s s p e c t r u m a r i s i n g f r o m t h e r e a c t i o n m e c h a n i s m s (for i n s t a n c e t h e k n o c k - o u t r e a c t i o n ) , w h i c h s h o u l d b e m o r e d i f f i c u l t to d i s t i n guish at small angles. It i s a p l e a s u r e to t h a n k M. S. B u h l e r a n d h i s g r o u p f o r t h e c o n s t r u c t i o n a n d p r e p a r a t i o n of t h e h e l i u m c r y o s t a t a n d t a r g e t , a n d M. J. L. B o y a r d f o r h i s h e l p in t h e m e a s u r e m e n t s .
References 1. H.E. Conzett, E. Shield, R . J . Slobodrian and S. Y a m a be, Phys. Rev. L e t t e r s 13 (1964) 625; M . C e r i n e o , K. Ilakovac, I.glaus. P . T o m a ~ , V . V a l kovid. Phys. Rev. 133 (1964) B948; M. Jakobson, J . H . Manley and R. H. Stokes, Nucl. Phys. 70 (1965) 97; E. M. Henley, F. C. R i c h a r d s and D. U. L. Yu, Phys. L e t t e r s 15 (1965) 331; T.A. Tombrello and A. D. Bacher, Phys. L e t t e r s 17 (1965) 37; A. Langsford, P.H. Bowen, G.C. Cox, P . E . Dolley, R. A. J. Riddle and M. J. M. S a l t m a r s h , Nucl. Phys. A99 (1967) 246; B. J. Morton, E . E . G r o s s , J . J . Malanify and A. Zucker, Phys. Rev. L e t t e r s 18 (1967) 1007; H. Brtickmann, W. Kluge and L.Schlinzler, Phys. L e t t e r s 24B (1967) 649. 2. K . M . W a t s o n , Phys. Rev. 88 (1952) 1163; A . B . M i g d a l , Zh. Exp. i T e o r . Fiz. 28 (1955) 3; Soviet Phys. J E T P 1 (1955) 2. 3. H . T y r e n , G . T i b e l l and T h . A . M a r i s , Nucl. Phys. 4 (1957) 277. 4. W. Selove and J . M . T e e m , Phys. Rev. 112 (1958) 1658. 5. J.D. Jackson and J. M. Blatt, Revs. Mod. Phys. 22 (1950) 77. 6. R . J . N . Phillips, Nucl. Phys. 53 (1964) 650.
4He(p, t)2p, AT 156
PHYSICS
4He(p, 3He)pn MeV AND THE
LETTERS
AND 4He(p, FINAL-STATE
4 September 1967
3He)d REACTIONS INTERACTION
M. BERNAS, J. K. LEE *, D. BACHELIER, I. BRISSAUD, C. DETRAZ, P. RADVANYI and M. ROY Institut
de Physiqzde
NuclLaire. Laboratoire Orsay, France
Joliot-Cum’e.
Received 21 July 1967
The p-n and p-p final-state interaction in the (p, 3He) and (p, t) reactions on 4He at 156 MeV has been measured for several t and 3He angles and analysed in terms of the Watson-Migdal theory. Agreement is rather good for the shapes of the energy spectra, but not for their relative magnitudes. The differential cross section of 4He+p dd +3He has been measured from 8d = 4O to 1770 c. m.
Final-state interactions have been studied in recent years in the lo-50 MeV range on several nuclear reactions, mainly the single-nucleon transfer and the charge exchange reactions on mass 2 and 3 nuclei [l]. The analysis of these experiments is generally made using the WatsonMigdal treatment [2], but agreement with theory varies significantly according to the reactions and nuclei involved. With two-nucleon transfer reactions on 4He at 156 MeV, our purpose was to extend these studies to higher energies, where the Watson-Migdal assumptions should be more valid. In these experiments with three bodies in the final state, measurements can be made on the upper part of the energy spectrum of one of these particles. In this condition, the two remaining particles have low relative momentum q and can interact strongly. Watson and Migdal [2] have then shown that, with several simplifying assumptions, the differential cross section corresponding to the energetic particle emitted at angle 0 with energy E can be factorised as follows:
,pTnscG(@)S(6,, 4) ,
(1)
isospins where p(E) contains the phase space spectrum; G(0) corresponds to the primary reaction (here the double-nucleon transfer reaction) and is assumed to be independent of E for the limited part of the energy spectrum under study, which should be more valid at higher incident energies; the * Present address: University of Toronto, Department of Physics. 260
‘He + p _ EP
-
d + ‘He
155.4MN
0.1
0.01
L
0
30
60
90
120
150
180 ed CM
Fig. 1. Angular distribution of the 4He(p, d)3He reaction at 156 MeV. The backward-angle values were obtained by the study of the 4He(p, 3He)d reaction at forward angles. last term corresponds to the low energy S-wave scattering between the two remaining particles with phase shifts 6, (singlet and/or triplet); it is the last term which gives rise to the characteristic peak at the upper part of the energy spectrum; c represents the statistical and coupling coefficients arising from the averaging over initial states and summing over final states. The Wat son-Migdal formalism neglects the final-state interaction between the detected particle and one of the remaining particles; this should be actually valid in our experimental situation at 156 MeV where the only important final:state interaction
Volume 25B, number 4
PHYSICS
LETTERS
,z , ? ,
IO0 I
4He ( p , t )
~,
,
,~
,
4
, e , ,
4 S e p t e m b e r 1967
6
, ?
,
z
,
2p
2~J~ cr
80~
?(5-)
,2 , o,(,o.) NeV E ~ ( ~ )
Ep =f,??.:,?eV
•
A i
*T
x~ 6 0
~ 4o
i~ 2o
io2
i04
~06
108
No
112
114
ii6
i18
120
Ei MeV
Fig. 2. a) Energy spectra of 4He(p, t)2p at 2 °, 5 °, 10 ° (gaseous target) and 15 ° (liquid target). The fits by the Watson-Migdal analysis are indicated for 2 ° and 10 ° (normalization for these fits is made at the maximum of the experimental spectrum with app=-7.78 fm and ro=2.71 fm; for 2 ° is also shown the fit calculated with the G~0) extracted from the 4He(p, 3He)d c r o s s section; the experimental energy resolution has been folded in the calculated curves); p-p relative energy scales are given for the three first angles, b) Energy spectra of 4He(p, 3He)pn at 2o (gaseous target). The peak on the right corresponds to the final p-n triplet bound state (4He(p, 3He)d reaction), the broad peak on the left to the final p-n scattering state. The dashed lines indicate the Watson-Migdal fits, normalized at the maximum of the experimental spectrum with aos =-23.68 fro, ros =2.48 fro, aot=5.39 fm and rot = 1.70 fm and calculated with the G(8) extracted from the bound-state c r o s s section. The upper scale gives the p-n relative energy. s h o u l d take p l a c e b e t w e e n the two u n o b s e r v e d p a r t i c l e s w i t h low r e l a t i v e e n e r g y . We u s e d a p r o t o n b e a m of about 10 -2 # A , m o n i t o r e d by a s e c o n d a r y e m i s s i o n c h a m b e r o r a He i o n i s a t i o n c h a m b e r , i m p i n g i n g on a He t a r g e t of 1.5 c m t h i c k n e s s and 4 × 8 cm-Z a r e a , s i t u a t e d at the b o t t o m of a l i q u i d - h e l i u m c r y o s t a t . T h i s t a r get could be f i l l e d with l i q u i d o r l o w - t e m p e r a t u r e g a s e o u s h e l i u m , a c c o r d i n g to the r e s o l u t i o n needed. Background contributions were subtracted using a dummy target. The overall f. w. h. m. i n s t r u m e n t a l e n e r g y r e s o l u t i o n at s m a l l a n g l e was about 0.8 M e V f o r ( p , p ' ) , 1.3 M e V f o r (p,d), 1.5 M e V f o r (p, 3He) and 0.9 M e V f o r ( p , t ) . T h e t o t a l a n g u l a r r e s o l u t i o n was 0.9 ° . The e m i t t e d p a r t i c l e s w e r e i d e n t i f i e d and a n a l y s e d by a m a g n e t i c s p e c t r o m e t e r f o l l o w e d by a p l a s t i c s c i n t i l l a t o r t e l e s c o p e e m p l o y i n g both r a n g e and AE/Ax d i s c r i m i n a t i o n t e c h n i q u e s . We m e a s u r e d at s e v e r a l a n g l e s the e n e r g y s p e c t r a of the e m i t t e d p r o t o n s , d e u t e r o n s , t r i t o n s and 3He. The p r o t o n s p e c t r a s h o w e d no o t h e r i n e l a s t i c p e a k than the b r o a d one a r o u n d 23 M e V e x c i t a t i o n e n e r g y o b s e r v e d p r e v i o u s l y at U p p s a l a [3]. T h e d e u t e r o n s p e c t r a and the h i g h e s t p a r t of
the 3He s p e c t r a (fig. 2b) g a v e us the t o t a l a n g u l a r d i s t r i b u t i o n ( f r o m 3.7 ° to 177Oc.m.) of the 4 H e ( p , d ) 3 H e r e a c t i o n s h o w n on fig. 1. T h i s m a y be c o m p a r e d with the f o r w a r d a n g l e m e a s u r e m e n t s at 93 M e V [4]. The observed triton spectra exhibit, near maxi m u m e n e r g y (fig. 2a), an a s y m m e t r i c p e a k m o s t i m p o r t a n t at s m a l l a n g l e s . Its s h a p e has b e e n c a l c u l a t e d u s i n g f o r m u l a (1) with ~ c S(So, q) = = 2 s i n 2 5o/9C207)q2; 5o can be e x p r e s s e d in t e r m s of the s i n g l e t p - ~ s c a t t e r i n g l e n g t h app and e f f e c t i v e r a n g e t o ; CZ07) is the C o u l o m b p e n e t r a t i o n f a c t o r [5]. On the 3He s p e c t r a (fig. 2b), at the h i g h e r e n e r g y end the p e a k a p p e a r s c o r r e s p o n d i n g to the bound t r i p l e t d e u t e r o n ; the b r o a d p e a k to the left c o r r e s p o n d s to the unbound n - p s i n g l e t and t r i p l e t s c a t t e r i n g s t a t e s . To fit the s h a p e of t h i s p e a k , f o r m u l a (1) w a s u s e d with: ~ c S ( 5 o , q) = = s i n 2 5 o s / 9 q 2 + s i n 2 6 o t / 3 q 2, w h e r e 5os and 5ot c a n now be e x p r e s s e d in t e r m s of the s i n g l e t and t r i p l e t p a r a m e t e r s a s , a t and r o s , rot [5]. In e a c h c a s e , we h a v e f o l d e d in the c a l c u l a t e d s p e c t r a the e x p e r i m e n t a l f i n i t e e n e r g y r e s o l u t i o n and c o n v e r t e d to the l a b o r a t o r y s y s t e m , in o r d e r to be a b l e to fit the e x p e r i m e n t a l data. 261
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T h e g e n e r a l a g r e e m e n t o b s e r v e d in t h e s e f i t s between the experimental shapes and the shapes calculated with the Watson-Migdal treatment seems reasonably good, at least at small angles. T h i s m i g h t b e r e l a t e d to t h e f a c t t h a t t h e e n e r g y b e t w e e n t h e t, o r t h e 3He, a n d e a c h of the two remaining nucleons is large, whereas the relat i v e e n e r g y b e t w e e n t h e s e two n u c l e o n s i s s m a l l ; a n d a l s o , f o l l o w i n g P h i l l i p s a r g u m e n t s [6], t h a t we h a v e a d o u b l e p i c k - u p p r i m a r y on a n u c l e u s a s t i g h t l y b o u n d a s 4He w i t h a s m a l l r . m . s . r a dius, which means that the primary interaction has effectively a short range. The deviation observed at larger relative energy could come from a p-wave contribution or from a slight dep e n d e n c e of G(O) o n t h e o b s e r v e d p a r t i c l e e n e r g y E. T h e a b o v e c a l c u l a t i o n s do not g i v e a b s o l u t e v a l u e s of t h e c r o s s s e c t i o n s , b u t it i s q u i t e i n t e r e s t i n g to c o m p a r e t h e t h r e e e x p e r i m e n t a l p e a k s . T o do t h i s we e x t r a c t e d , i n t h e way f o l l o w e d by W a t s o n [2] f o r the p + p ~ 7 r + + p + n r e a c t i o n (final t r i p l e t p n s c a t t e r i n g s t a t e ) , t h e v a l u e of G(O) f r o m t h e e x p e r i m e n t a l c r o s s s e c t i o n of ~he 4He3(P, 3 H e ) d re4action , a n d i n s e r t e d it in t h e H e ( p , H e ) p n a n d H e ( p , t)2p c a l c u l a t e d c r o s s sections, making the simplifying assumption that G(O) i s t h e s a m e i n t h e t h r e e r e a c t i o n s (the c o u p l i n g c o e f f i c i e n t s a p p e a r i n g in t h e f a c t o r c). F o r the p-n final-state interaction spectrum, the diff e r e n t i a l c r o s s s e c t i o n c a l c u l a t e d in t h i s way i s b e l o w t h e e x p e r i m e n t a l o n e b y a b o u t 40%. T a k i n g into account the experimental errors and the fact t h a t t h e t r i p l e t c o n t r i b u t i o n to t h e s c a t t e r i n g i s rather large, one may conclude that the above a r g u m e n t f o r G(O) i s v a l i d f o r the f i n a l p - n t r i p l e t s t a t e . But t h e c o m p a r i s o n w i t h t h e p - p f i n a l - s t a t e interaction shows that the calculated cross sect i o n i s m u c h to low (by a f a c t o r of a b o u t 4). T h i s w o u l d i n d i c a t e t h a t t h e c o r r e s p o n d i n g G(O) a r e n o t t h e s a m e f o r t h e (p, t) a n d (p, 3He) r e a c t i o n s , o r f o r f o r m a t i o n of t r i p l e t a n d s i n g l e t s t a t e s . O n e s h o u l d k e e p in m i n d t h a t t h i s c o m p a r i s o n h a s b e e n m a d e u n d e r t h e a s s u m p t i o n t h a t G(O) c a n b e fac~orised in the cross sections. A n o t h e r i n t e r e s t i n g p r o b l e m i s t h e v a r i a t i o n of
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the a b o v e c r o s s s e c t i o n s w i t h a n g l e 0. A c a l c u l a t i o n of G(0) a t o u r e n e r g y f o r t h e t w o - n u c l e o n p i c k - u p w o u l d g i v e a p r e d i c t i o n of t h i s v a r i a t i o n a n d c o u l d a l s o l e a d to i m p r o v e m e n t s in the f i t s . It is a p p a r e n t f r o m f o r m u l a (1) t h a t t h e s h a p e of t h e f i n a l - s t a t e i n t e r a c t i o n s h o u l d not v a r y v e r y m u c h a t 156 M e V f o r 0 ~. 30 ° . In f a c t , t h e e x p e r i m e n t a l s h a p e s t a y s a b o u t the s a m e f r o m 2 ° to 10 °. It i s y e t d i f f i c u l t to s a y if the g r a d u a l d i s a p p e a r a n c e of t h e p e a k f r o m 15 ° to l a r g e r a n g l e s ( w h e r e we c o u l d o n l y u s e t h e l i q u i d t a r g e t ) i s due to a w o r s e e n e r g y r e s o l u t i o n , o r to s o m e c o n t r i b u t i o n of a c o n t i n u o u s s p e c t r u m a r i s i n g f r o m t h e r e a c t i o n m e c h a n i s m s (for i n s t a n c e t h e k n o c k - o u t r e a c t i o n ) , w h i c h s h o u l d b e m o r e d i f f i c u l t to d i s t i n guish at small angles. It i s a p l e a s u r e to t h a n k M. S. B u h l e r a n d h i s g r o u p f o r t h e c o n s t r u c t i o n a n d p r e p a r a t i o n of t h e h e l i u m c r y o s t a t a n d t a r g e t , a n d M. J. L. B o y a r d f o r h i s h e l p in t h e m e a s u r e m e n t s .
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