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EXCITATION OF NUCLEAR HOLE STATES BY THE (e,e’p) REACTION
L. Lapikás, P.K.A. de Witt Huberts
To cite this version:
L. Lapikás, P.K.A. de Witt Huberts. EXCITATION OF NUCLEAR HOLE STATES BY THE (e,e’p) REACTION. Journal de Physique Colloques, 1984, 45 (C4), pp.C4-57-C4-68.
�10.1051/jphyscol:1984406�. �jpa-00224071�
JOURNAL DE PHYSIQUE
Colloque Ct, supplément au n°3, Tome 45, mars 1984 page C4-57
EXCITATION OF NUCLEAR HOLE STATES BY THE ( e , e ' p ) REACTION
L. Lapikas and P.K.A. de Witt Huberts
National Institute for Nuclear Physics and High-Energy Physics (NIKHEF section K) , P.O. Box 4395, 1009 AJ Amsterdam,
The Netherlands
Résumé : Nous discutons les divers aspects théoriques et expérimentaux de la réaction (e,e'p). Nous décrivons les performances de l'ensemble de l'appa- reillage de NIKHEF-K, et nous les illustrons à l'aide de fonctions spectrales mesurées dans des réactions (e,e'p) sur les noyaux 1 2C , 2 7A1 et 5 1V.
A b s t r a c t : V a r i o u s t h e o r e t i c a l and e x p e r i m e n t a l a s p e c t s o f t h e ( e , e ' p ) r e a c t i o n a r e d i s c u s s e d . P e r f o r m a n c e o f t h e NIKHEF-K a p p a r a t u s i s d e s c r i b e d and i l l u s t r a t e d w i t h s p e c t r a l f u n c t i o n s f o r t h e ( e , e ' p ) r e a c t i o n on 1 2C ,2 7A 1 a n d 5 1V n u c l e i .
I . INTRODUCTION
I n t h e p a s t y e a r s a l a r g e amount o f d a t a h a s b e e n c o l l e c t e d on p r o t o n - h o l e s t a t e s i n n u c l e i from r e a c t i o n s w i t h h a d r o n i c p r o b e s , e . g . ( d , t ) ,
( t , a ) a n d ( p , 2 p ) . The S a c l a y work on t h e ( e , e ' p ) r e a c t i o n h a s a l s o c o n t r i b u t e d / l / t o o u r k n o w l e d g e o f t h e s e h o l e s t a t e s . However, t h e
( e , e ' p ) r e a c t i o n was h a m p e r e d by a l i m i t e d e n e r g y r e s o l u t i o n o f a b o u t 1 MeV a n d by low c r o s s s e c t i o n s c a u s e d by t h e w e a k n e s s of t h e e l e c t r o - m a g n e t i c i n t e r a c t i o n . C o n s e q u e n t l y a t t e n t i o n h a s b e e n f o c u s s e d m o s t l y on i n t e g r a l p r o p e r t i e s / 2 / ( s p e c t r o s c o p i c f a c t o r s , sum r u l e s ) w i t h a
few e x c e p t i o n s f o r l i g h t n u c l e i / l / . With t h e p r e s e n t l y a v a i l a b l e ( e , e ' p ) i n s t r u m e n t a t i o n / 3 / a t NIKHEF-K a m i s s i n g - e n e r g y r e s o l u t i o n o f 200 keV i s a t t a i n a b l e , c o m p a r a b l e t o t h a t o f h a d r o n i c r e a c t i o n s . I n v a r i o u s a s p e c t s t h e ( e , e ' p ) r e a c t i o n i s a d v a n t a g e o u s o v e r t h e h a d r o n i c r e a c t i o n s . F i r s t l y , t h e f i n a l - s t a t e i n t e r a c t i o n (FSI) b e t w e e n p r o t o n and r e s i d u a l n u c l e u s p r o d u c e s o n l y a s m a l l d i s t o r t i o n / 4 , 5 , 6 / w h i l e t h e a b s o r p t i o n ( t y p i c a l l y 3 0 - 7 0 % i n t h e c r o s s s e c t i o n ) i s p r a c t i c a l l y c o n s t a n t o v e r a l a r g e r e g i o n o f t h e s p e c t r a l f u n c t i o n . T h i s c o n t r a s t s w i t h t h e s i t u a t i o n i n h a d r o n i c r e a c t i o n s w h e r e t h e s e e f f e c t s a r e l a r g e a n d s e n s i t i v e t o t h e c h o i c e o f v a r i o u s p a r a m e t e r s o f t h e o p t i c a l p o t e n t i a l / 7 , 8 , 9 / . M o r e o v e r s m a l l c h a n g e s i n t h e r a d i a l s h a p e o f t h e b o u n d - s t a t e wave f u n c t i o n c a u s e l a r g e v a r i a t i o n s i n t h e d e d u c e d s p e c t r o s c o p i c f a c t o r s / I 0 , l l , 1 2 / . I n a d d i t i o n t h e w e a k - n e s s of t h e e l e c t r o m a g n e t i c i n t e r a c t i o n makes t w o - s t e p p r o c e s s e s c o n s i d e r a b l y l e s s i m p o r t a n t . Thus a more s t r a i g h t f o r w a r d a n a l y s i s of t h e h o l e s t r e n g t h i n t e r m s o f s p e c t r o s c o p i c f a c t o r s i s p o s s i b l e . T h i s w i l l p l a y an i m p o r t a n t r o l e a t h i g h e x c i t a t i o n e n e r g i e s , w h e r e t h e c o n t r i b u t i o n o f s u c h p r o c e s s e s i n h a d r o n i c r e a c t i o n s may b e d o m i n a t i n g / 1 3 / . F i n a l l y t h e o f f - s h e l l e l e c t r o n - p r o t o n c r o s s s e c t i o n , w h i c h e n t e r s i n t h e a n a l y s i s o f ( e , e ' p ) p r o c e s s e s , i s a c c u r a t e l y known from q u a n t u m e l e c t r o d y n a m i c s / 1 4 , 1 5 , 1 6 / , w h e r e a s i t s h a d r o n i c e q u i v a l - e n t i s s u b j e c t t o much more u n c e r t a i n t y / 1 7 / . T h e s e a d v a n t a g e s e n a b l e o n e t o map o u t t h e momentum d e n s i t i e s o f e a c h f i n a l s t a t e a n d t o c o m p a r e t h e m d i r e c t l y w i t h s i n g l e - p a r t i c l e wave f u n c t i o n s i n momentum s p a c e / 1 8 / .
The d i s a d v a n t a g e o f l o w e r c r o s s s e c t i o n s i n t h e ( e , e ' p ) r e a c t i o n h a s p a r t l y b e e n o v e r c o m e / 3 / by a good c o i n c i d e n c e - t i m e r e s o l u t i o n (< 1 n s ) Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984406
C4-58 J O U R N A L DE PHYSIQUE
and a l a r g e d u t y - f a c t o r beam ( a few p e r c e n t ) , Thus t h e beam c u r r e n t can be l a r g e w h i l e k e e p i n g t h e r a t i o o f a c c i d e n t a l - t o - r e a l c o i n c i d e n c e e v e n t s low.
F i n a l l y we mention t h a t t h e ( e , e t p ) r e a c t i o n h a s n o j - s e l e c t i v i t y , t h a t i s o n l y t h e 2-value o f t h e p r o t o n t r a n s f e r c a n b e d e t e r m i n e d from t h e shape o f t h e momentum d e n s i t y . T h i s i s done i n a model-
d e p e n d e n t way, a s i s t h e c a s e i n h a d r o n i c r e a c t i o n s . However, r e a c t i o n s t h a t employ p o l a r i z e d p r o j e c t i l e s ( e . g . t h e ( h , ~ ) r e a c t i o n /19/) h a v e t h e p o s s i b i l i t y t o d i s c r i m i n a t e between knock-out from s t a t e s t h a t a r e s p i n - o r b i t p a r t n e r s ( j = 2 ? P ) . These r e a c t i o n s a r e t h e r e f o r e o f a t l e a s t complementary i n t e r e s t , and should be employed in the (ee'p) data interpretation.
Below we g i v e some e l e m e n t a r y c o n s i d e r a t i o n s on t h e ( e , e l p ) r e a c t i o n ( s e c t i o n 2 ) and i t s k i n e m a t i c s ( s e c t i o n 3 ) . T e c h n i c a l d e t a i l s a n d d a t a a n a l y s i s p e r t a i n i n g t o t h e NIKHEF-K a p p a r a t u s a r e p r e s e n t e d i n s e c t i o n s 4 and 5. Our p r e l i m i n a r y r e s u l t s f o r a few n u c l e i a r e d i s c u s s e d i n s e c t i o n 6 .
2. THE ( e , e V p ) REACTION
A d i a g r a m o f t h e r e a c t i o n A ( e , e t p ) B i s i l l u s t r a t e d i n f i g . 1 i n P l a n e Wave I m p u l s e Approximation (PWIA). I n t h i s f i r s t - o r d e r a p p r o a c h / 1 8 / ,
which s u f f i c e s t o e n l i g h t e n t h e main c h a r a c t e r i s t i c s o f t h e (e ,e ' p ) S e c h n i q u e , one d e s c r i b e s $he incoming (eo,e) and s c a t t e r e d (e;,e1 ) e l e c t r o n s by p l a n e waves. The v i r t u a l p h o t o n (o,2j) i s exch2nged w i t h a " q u a s i - f r e e " p r o t o n
( p o , p ) a l s o d e s c r i b e d a s a p l a n e wave.
Distortion o f t h e e l e c t r o n waves i n t h e n u c l e a r Coulomb p o t e n t i a l a n d d i s t o r t i o n
( a b s o r p t i o n ) o f t h e knocked-out p r o t o n by t h e r e a l ( i m a g i n a r y ) p a r t o f t h e s t r o n g p o t e n t i a l o f t h e r e s t n u c l e u s
( B ) a r e t h u s n e g l e c t e d i n PWIA. The Fig. I Diagram of the l e , e ' p ) i m p o r t a n t q u a n t i t y t o be e x t r a c t e d from
reaction. t h e measured s i x - f o l d d i f f e r e n t i a l c r o s s s e c t i o n
i s t h e s p e c t r a l f u n c t i o n S which i s a f u n c t i q n o f t h e k i n e m a t i c v a r i a b l e s m i s s i n g e n e r g y (Em) and momentum (p,) d e f i n e d by
I n f o r m u l a ( 2 . 1 ) K i s a k i n e m a t i c a l f a c t o r and a an o f f - s h e l l e l e c t r o n - p r o t o n c r o s s s e c t i o n . eP
The s p e c t r a l f u n c t i o n S c o n t a i n s t h e n u c l e a r - s t r u c t u r e i n f o r m a t i o n . F o r example i n a s i m p l i f i e d s i n g l e - p a r t i c l e s h e l l model i t c a n be e x p r e s s e d a s a sum o v e r s i n g l e - p a r t i c l e energy-momentum d i s t r i b u t i o n s
where P* m e a s u r e s t h e o c c u p a t i o n p r o b a b i l i t y o f t h e s i n g l e - p a r t i c l e s t a t e l a > = I n l j > and @ i s i t s wave f u n c t i o n i n momentum s p a c e
+ -Z
Q ($ m = J. e-iPmr Y (f ) d f . ( 2 . 4 )
F i g . 2 g i v e s a n example of t h e s p e c t r a l f u n c t i o n f o r t h e r e a c t i o n
1 6 O ( e , e ' p ) l S ~ measured a t S a c l a y /38/. ~t c l e a r l y shows p r o t o n knock- o u t from t h e 1 ~ 3 1 2 and l s l l 2 s h e l l s w i t h t h e i r d i f f e r e n t dependence on pm.
Fig. 2 Spectral function for the reaction Fig. 3 ParaZZel (bl and perpendic- 160(e,e ' p ) 1 5 ~ as measured at uZar (a) kinematics for the
SacZay / 3 8 / . le, e 'pl reaction.
3. REACTION KXNEMATICS
I n o r d e r t o c o v e r t h e d e s i r e d r e g i o n i n ( E ,pm) s p a c e one v a r i e s one o r u s u a l l y more o f t h e k i n e m a t i c v a r i a b l e s e' B e t , p, B p ( s e e f i g . 3 ) . The p r o p e r c h o i c e of t h e k i n e m a t i c variables i s i n f l u e n c e d by t h r e e c o n s i d e r a t i o n s : i ) one w i s h e s t o k e e p l p l c o n s t a n t t h r o u g h o u t t h e
e x p e r i m e n t i n o r d e r t o have a c o n s t a n t e f f e c t 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 ; ii) one s h o u l d measure a t q u a s i - e l a s t i c k i n e m a t i c s where t h e o f f - s h e l l e l e c t r o n - p r o t o n c r o s s s e c t i o n o i s w e l l known /14/;
i i i ) one s h o u l d maximize KOee i n o r d e r ep t o have a s h i g h a
c o i n c i d e n c e c o u n t r a t e a s p o > s x b l e . These c o n s i d e r a t i o n s l e a d t o two k i n e m a t i c a r r a n g e m e n t s commonly i n u s e ( s e e f i g . 3 ) : p a r a l l e l kinema-
t i c s , where $m i s p a r a l l e l t o t h e momentum t r a n s f e r 3 t h a t i s v a r i e d by changing 8,:; p e r p e n d i c u Z a r k i n e m a t i c s , w i t h 6, I ?j and t h e v a r i a t i o n i n p 1s a t t a i n e d by changing O p . Each o f b o t h a r r a n g e m e n t s emphasizes d i f ! ? e r e n t a s p e c t s o f t h e f i n a l - s t a t e i n t e r a c t i o n ( d a s h e d l i n e s i n f i g . 2 ) , and i n f a c t a comparison between t h e two methods can b e u s e d t o g a t h e r i n f o r m a t i o n on t h e o p t i c a l p o t e n t i a l and on t h e r e a c t i o n mechanism / 4 / .
TABLE 1 Parameters of ( e , e S p ) f a c i l i t i e s
, eo d.f. *Pm
Laboratory (MeV) ( % 2) (Mev/c) R e f . -'
Frascati
500-780 3 1 0 5-10 ?
1963-1972 /36/
Tokyo
1970-1972 700-750 < 4 - 5 7 16 /37/
Saclay
1971- 500 1 2.5 0.9 8 /2,38/
Amsterdam
1982- 300-500 2.5 0.8 0.2 3 / 3 /
C4-60 JOURNAL DE PHYSIQUE
4. EXPERI2EMTAL TECHNIQUE
T a b l e 1 shows t h a t most ( e , e l p ) e x p e r i m e n t s have been c a r r i e d o u t a t beam e n e r g i e s e o = 300 - 800 MeV. I n o r d e r t o a l l o w discrimination between f i n a l s t a t e s a good r e s o l u t i o n AEm i n t h e m i s s i n g e n e r g y v a r i a b l e i s i m p e r a t i v e . T a b l e 1 shows t h a t t h i s number h a s improved from AE, = 10 MeV t o 200 keV. The l a t t e r v a l u e (see f i g . 4) h a s b e e n o b t a i n e d a t NIKHEF-R by t h e u s e of two h i g h - r e s o l u t i o n m a g n e t i c s p e c t r o m e t e r s / 3 / f o r d e t e r m i n a t i o n of 8' a n d 8 a n d t h e development o f a d i s p e r s i o n n a t c h i n g t e c h n i q u e f o r c o i n c i s e n c e r e a c t i o n s / 2 0 / . I n f i g . 5 we show how AEm i s o p t i m i z e d f o r t h e r e a c t i o n ' ~ ( e , e ' p ) by v a r i a t i o n o f t h e dispersion DB o f t h e e l e c t r o n beam on t h e t a r g e t .
Another improvement t h a t h a s c o n t r i b u t e d v a l u a b l y t o t h e q u a l i t y o f (e , e ' p ) d a t a i s t h e c o n s i d e r a b l y i n c r e a s e d r a t i o R/A o f r e a l - t o - a c c i d e n t a l c o i n c i d e n c e s . I n t h e NIKHEF-K s e t u p t h e s i n g l e s r a t e s i n b o t h s p e c t r o m e t e r s a r e u s u a l l y a
e = 295 MeV/c f a c t o r 100-1000 l a r g e r t h a n t h e
e = 173 MeV/c c o i n c i d e n c e r a t e . C o n s e q u e n t l y a h i g h d u t y f a c t o r ( d . f .)
a c c e l e r a t o r a n d a s m a l l c o i n c i - dence t i m e r e s o l u t i o n ( A t ) o f t h e d e t e c t i o n s y s t e m s a r e n e c e s s a r y t o k e e p R/A, which i s p r o p o r t i o n a l t o d. f . / A t , a t a n a c c e p t a b l e l e v e l . T a b l e 1 shows t h a t t h e q u a n t i t y d. f . / A t h a s improved by an o r d e r o f magnitude. I n f i g . 6 we
i l l u s t r a t e t h a t h i g h - r e s o l u t i o n s p e c t r o m e t e r s w i t h good a n g u l a r r e s o l u t i o n ( < 10 mrad) e n a b l e t o improve t h e h a r d w a r e t i m e
r e s o l u t i o n o f 5 n s t o 0.75 n s by c o r r e c t i o n f o r f l i g h t - t i m e
16 18 20 22 2~ 26 28 d i f f e r e n c e s t h r o u s h t r a i e c t o r v
E, [ M e V 1 + r e c o n s t r u c t i o n . he v a l u e
Fig. 4 M i s s i n g e n s r g IE,J s p e c t r u m of A t = 0.75 n s i s o u r p r e s e n t l o w e r t h e r e a c t i o n y2C(e,e'p)''B. l i m i t s e t by t h e f a s t t r i g g e r
s c i n t i l l a t o r s a n d c o i n c i d e n c e e l e c t r o n i c s .
5. DATA ANALYSIS
W e d i s c u s s t h e ( e , e 1 p ) d a t a a n a l y s i s s h o r t l y o n t h e b a s i s o f f i g . 7 which p r e s e n t s t h e s c h e m a t i c d a t a f l o w . The e v e n t i n f o r m a t i o n o f b o t h
s p e c t r o m e t e r s i s r o u t e d t o t h e program REDUCER, t h a t c h e c k s on t h e v a l i d i t y o f e v e n t t r a c k s , p r o d u c e s a c o i n c i d e n c e - t i m e s p e c t r u m T ( t c ) a f t e r c o r r e c t i o n o f t h e raw c o i n c i d e n c e - t i m i n g d a t a ( t e r , t p ) f o r f l i g h t - t i m e d i f f e r e n c e s a n d b u i l d s a s i x - d i m e n s i o n a l s p e c t r u m c (8' ,$) o f c o i n c i d e n t e v e n t s . I n a l a t e r s t a g e i t i s t r a n s f o r m e d i n t o a two- d i m e n s i o n a l s p e c t r u m C (Em,pm) i n (Em,pm) s p a c e . A s e c o n d program, PHASPA, Uses a Monte C a r l o t e c h n i q u e t 6 c a l c u l a t e t h e p h a s e - s p a c e a c c e p t a n c e V(E;n,pm) from t h e a n g u l a r a n d momentum a c c e p t a n c e s o f b o t h s p e c t r o m e t e r s . I n t h e program ACCIDENT a s p e c t r u m o f a c c i d e n t a l c o i n c i d e n c e s A(Emlpm) = M ( e f ) O M(p) i s c r e a t e d by f o l d i n g ( @ ) t h e s i m u l t a n e o u s l y measured s i n g l e s momentum s p e c t r a (MI o f e a c h s p e c t r o - m e t e r . The o v e r a l l w e i g h t o f A i s d e t e r m i n e d from t h e c o r r e c t e d t i m i n g s p e c t r u m T ( t c ) a n d p h a s e - s p a c e volume V (E,,p,).
-200 0 200 LOO 600 Em [ k e V l -
Fig. 5 K s s i n g energy spectrum of the reaction ' ~ ( e , e ' p ) ; optimum resoZution has been obtained by varying the beam dispersion DB.
'H (e.e'p)
(1
.
R ~ W events -o Flight-time corrected events
FWHM 0.75 ns 3 I
Fig. 6 Coincidence time spectrum before (bZack circZssl and a f t e r (open circZesl correction for f l i g h t - time differences.
Now a n o r m a l i z e d (N) s p e c t r a l f u n c t i o n S (Em,pm) i s formed by sub- f t r a c t i n g t h e a c c i d e n t a l s A , weighting by t h e phase-space volume V and d i v i d i n g by t h e o f f - s h e l l e l e c t r o n - p r o t o n c r o s s s e c t i o n aep / 1 4 / a c c o r d i n g t o sf = N (C-A) /V/Koep. The r e s u l t i n g s p e c t r a l f u n c t i o n sf
h a s t o be c o r r e c t e d f o r v a r i o u s r a d i a t i v e e f f e c t s . I n t h e program, RADCOR, t h e r a d i a t i v e d i s t r i b u t i o n s Ri a r e u n f o l d e d from t h e s p e c t r a l f u n c t i o n sf w i t h a p r o c e d u r e t h a t i s a g e n e r a l i z e d v e r s i o n of a method o r i g i n a l l y p r e s c r i b e d /21/ f o r s i n g l e - a r m measurements. The r a d i a t i v e d i s t r i b u t i o n s Ri a c c o u n t f o r i n t e r n a l /22/ and e x t e r n a l /23/
Bremsstrahlung by t h e e l e c t r o n s and f o r i o n i z a t i o n l o s s /24/ by e l e c t r o n s and p r o t o n s , The u n f o l d i n g p r o c e s s i s r e p e a t e d f o u r t i m e s i n o r d e r t o e n s u r e c o n s e r v a t i o n o f t h e t o t a l number o f c o u n t s /23/.
I n f i g . 8 a r t o f t h e s p e c t r a l f u n c t i o n S(Em,pm) f o r t h e r e a c t i o n
"C(e,e'p)'lB i s shown. I t s h o u l d be compared w i t h t h e "raw" spectrum o f f i g . 4 . S i n c e i t h a s undergone t h e whole c h a i n o f a n a l y s i s programs d e s c r i b e d above it may s e r v e a s a check on t h e v a l i d i t y o f t h e v a r i o u s a p p r o x i m a t i o n s . F i r s t l y we n o t e t h a t w i t h i n t h e e r r o r b a r s S = 0 below Em = 1 6 MeV where t h e f i r s t peak due t o knockout from t h e g. s. o f ' B a p p e a r s . T h i s p r o v e s t h e a c c i d e n t a l c a l c u l a t i o n and s u b t r a c t i o n t o be c o r r e c t . Secondly we a l s o o b s e r v e S = O i n between t h e "B g . s . and t h e f i r s t e x c i t e d s t a t e a t Ex = 2.125 MeV, which d e m o n s t r a t e s t h a t t h e r a d i a t i v e u n f o l d i n g p r o c e s s works s a t i s f a c t o r i l y . Both o b s e r v a t i o n s h o l d t r u e f o r t h e d i f f e r e n t missing-momentum b i n s shown, which i n d i c a t e s t h e c o r r e c t n e s s o f t h e phase-space c a l c u l a t i o n .
A l a s t s t e p i n t h e a n a l y s i s o f t h e ( e , e l p ) d a t a w i l l c o n s i s t o f a c a l c u l a t i o n o f t h e FSI e f f e c t s . A s has been demonstrated / 2 , 4 , 6 / t h e s e e f f e c t s can be r e p r e s e n t e d r o u g h l y by i ) d i s t o r t i o n - a s h i f t i n 6,
s i n c e t h e d i s t o r t e d p r o t o n momentum pD i n s i d e t h e n u c l e u s i s n o t e q u a l t o t h e d e t e c t e d p r o t o n momentum 3; ii) a b s o r p t i o n - an o v e r a l l r e d u c t i o n nCl ($1 which depends on t h e s h e l l quantum numbers ( a ) = { n l ) . The absorption f a c t o r s nc, f o r v a l e n c e s h e l l s a r e i n t h e range 0 . 3 - 0 . 7
f o r n u c l e a r masses A = 1 2 - 5 8 and p r o t o n k i n e t i c e n e r g i e s o f 8 0 - 1 0 0 MeV.
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(e,e'p) ANALYSIS
e transformation 0-, external data
Fig. 7 Schemutic survey o f le,elp) data anaZysis.
6. RESULTS
Below we d i s c u s s t h e r e s u l t s o f o u r f i r s t ( e , e l p ) e x p e r i m e n t s on 12c,
' ' ~ 1 and 'v. They form p a r t o f a l a r g e r program i n which v a r i o u s n u c l e i up t o mass 208 w i l l be s t u d i e d . Because o n l y a s m a l l amount o f beam t i m e h a s been s p e n t on t h e s e e x p e r i m e n t s t h e p r e l i m i n a r y r e s u l t s a r e of r e l a t i v e l y poor s t a t i s t i c a l a c c u r a c y .
The k i n e m a t i c s of t h i s r e a c t i o n h a s been chosen such t h a t we probe t h e maximum of t h e l p - s h e l l momentum d i s t r i b u t i o n . The missing-energy spectrum ( s e e f i g . 4 ) has a r e s o l u t i o n o f about 200 keV. I n t h e
s p e c t r a l f u n c t i o n ( s e e f i g . 8 ) we o b s e r v e t h a t t h e ground s t a t e (3/2-1, 2.125 MeV (1/2-) and 5.021 MeV (3/2-) f i n a l s t a t e s i n "B a r e p r e -
dominantly p o p u l a t e d . These t r a n s i t i o n s a r e known from ( e , e t p ) /2/, ( d , ~ ) /25/ and ( p , 2 p ) /26/ r e a c t i o n s t o have l p - c h a r a c t e r . However, i n t h e
( d , ~ ) and ( p , 2 p ) r e a c t i o n s t h e pick-up f o r b i d d e n 4 . 4 5 MeV (5/2-) s t a t e , i s a l s o s t r o n g l y e x c i t e d , whereas i t i s c l e a r l y a b s e n t i n o u r d a t a .
S i n c e t h e l f 5 / 2 component i n t h e " C g r o u n d s t a t e i s presumably s m a l l , i t h a s been t r i e d /27/ t o d e s c r i b e t h i s t r a n s i t i o n a s a t w o - s t e
p r o c e s s i n which e i t h e r t h e 2+ s t a t e a t 4.44 M e V i n "C o r t h e "B ground s t a t e p l a y an i n t e r m e d i a r y r o l e . Due t o t h e weakness o f t h e e l e c t r o m a g n e t i c i n t e r a c t i o n t h e p r o b a b i l i t y f o r s u c h a t w o - s t e p p r o c e s s i n t h e ( e , e f p ) r e a c t i o n i s much s m a l l e r t h a n i n t h e c a s e o f h a d r o n i c p r o b e s .
Fig. 8 Spectral function of the reaction 12c(e, e ' p l " ~ . A f u r t h e r i n t e r e s t i n g p a r t o f t h e s p e c t r a l f u n c t i o n i s t h e r e g i o n above Em = 2 1 MeV. Here t h e h a d r o n i c e x p e r i m e n t s /25,26/ o b s e r v e up t o 5 p o p u l a t e d s t a t e s , w h e r e a s o u r ( e , e l p ) r e a c t i o n ( a t p m = 120-150 MeV/c) shows none. From t h e ( e , e f p ) s p e c t r a measured e a r l l e r a t S a c l a y / 2 / i t may be deduced t h a t t h i s r e g i o n i s o n l y v e r y weakly e x c i t e d a t low m i s s i n g momentum, which may p o i n t a t 1s w a v e - f u n c t i o n components. I t seems v e r y w o r t h w h i l e t o s t u d y t h i s r e g i o n w i t h t h e h i g h - r e s o l u t i o n a p p a r a t u s a t NIKHEF-K and t o compare t h e d a t a f r o m t h e e l e c t r o m a g n e t i c and h a d r o n i c r e a c t i o n s ,
2 7 ~ ~ ( e , e ' p ) 2 6 ~ g
I n t h e s p e c t r a l f u n c t i o n ( s e e f i g . 9 ) f o r t h e r e a c t i o n 2 7 ~ l ( e , e f p ) 2 6 ~ g f i v e p r o m i n e n t p e a k s due t o knock-out o f I d p r o t o n s a r e o b s e r v e d .
These p e a k s ( g . s . ( 0 + ) , 1 . 8 1 MeV ( 2 + ) , 2.94 MeV (2') 4 , 3 3 MeV ( 2 ' ) and 5.49 MeV (4') a r e a l s o s e e n i n ( d , z ) e x p e r i m e n t s / 2 8 , 2 9 / . T h e i r
momentum d i s t r i b u t i o n s ( s e e f i g . 1 0 a ) have a r e l a t i v e magnitude i n a g r e e m e n t w i t h v a l u e s deduced from p i c k - u p s p e c t r o s c o p i c f a c t o r s ( 0 . 3 , 1 . 0 , 0 . 2 , 2 . 1 , 0 . 3 , a v e r a g e o f r e f s . 28,291. I n f i g . 1 0 a we a l s o show a I d momentum d i s t r i b u t i o n a s c a l c u l a t e d i n a Woods-Saxon
p o t e n t i a l t h a t f i t s m a g n e t i c e l a s t i c e l e c t r o n s c a t t e r i n g from 7 ~ 1 /30/.
As s l i g h t l y d i f f e r e n t momentum dependence i s o b s e r v e d f o r t h e peak a t 4.33 MeV. Although a c c u r a t e v a l u e s a w a i t p e a k - f i t t i n g , i t seems t h a t t h e o b s e r v a t i o n by W i l d e n t h a l e t a l . /29/ o f an Z = 0 t r a n s f e r t o t h i s s t a t e c o u l d p o s s i b l y e x p l a i n t h e d e c r e a s e o f t h e momentum d i s t r i b u t i o n w i t h m i s s i n g momentum.
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F i g . 9 Spectral function for t h e reaction 7 ~ ~ ( e , e 'pl 26 Mg ,
Fig. 10 Momentum d e n s i t i e s for Z = 2 ( a ) and Z = 1 ( b ) proton t r a n s f e r i n the reaction ' 7~~ (e,e lp/ 6 ~ g . The exroes m e a r b i t r a r i l y normalized Woods-Sazon d e n s i t i e s .
Above 6 MeV e x c i t a t i o n e n e r g y knock-out from t h e l p - s h e l l becomes dominant. According t o Wagner e t a l . /28/ t h e peaks a t 7.86 and 9.16 MeV r e p r e s e n t t r a n s f e r from t h e 1p1/2 s u b s h e l l . T h e i r r a t i o o f s p e c t r o s c o p i c f a c t o r s ( 1 . 8 : 1.3) from t h e ( d , ~ ) r e a c t i o n i s i n accordance w i t h t h e peak h e i g h t s i n o u r s p e c t r a ; t h e shape of t h e i r momentum d e n s i t i e s a g r e e s w i t h t h a t of an l p l 2 s t a t e , a s shown i n f i g . lob. The s m a l l peak a t 11.3 MeV i s a s c r i b e d /28/ t o l = 1 t r a n s f e r from t h e 1 ~ 3 1 2 s u b s h e l l b u t i t s s p e c t r o s c o p i c f a c t o r i s s m a l l .
T h e r e f o r e a f u r t h e r h i g h - r e s o l u t i o n ( e , e ' p ) s t u d y i s needed t o l o c a t e t h e d i s t r i b u t i o n o f l p - s t r e n g t h i n t h i s n u c l e u s . Such an i n v e s t i g a t i o n might i n v o l v e e x c i t a t i o n e n e r g i e s up t o 20 MeV/31/ o r even 30 MeV /32/.
T h i s r e a c t i o n h a s been s t u d i e d a t m i s s i n g momenta o f 35 - 190 MeV/c.
The s p e c t r a l f u n c t i o n i s shown i n f i g . 11 where t h e missing-energy s p e c t r a have been a v e r a g e d o v e r 30 MeV/c p b i n s . Assuming p u r e
( 1 f 7 / 2 ) 3 and ( l f 7 / 2 ) c o n f i g u r a t i o n s o u t s i 8 e an i n e r t 1 "ca> c o r e f o r 'V and 4 8 ~ i , r e s p e c t i v e l y , one can e a s i l y c a l c u l a t e t h e s p e c t r o s c o p i c f a c t o r s f o r l = 3 t r a n s f e r t o t h e (0+,2+,4 , 6 + ) q u a d r u p l e t i n "Ti.
The e x p e c t e d r a t i o (9:5:9:13) i s c l o s e l y r e p r o d u c e d by ( d , ~ ) e x p e r i - ments /33,34/ and i s a l s o i n q u a l i t a t i v e agreement w i t h t h e r a t i o o f t h e peak a r e a s i n f i g . 11. The momentum d e n s i t y o f t h e 0' s t a t e shows t h e c h a r a c t e r i s t i c If-shape ( s e e f i g . 1 3 ) . F o r t h e 2+ l e v e l s m a l l 2 p - s h e l l a d m i x t u r e s i n t h e wave f u n c t i o n have been d e t e c t e d i n a
( d , ~ ) r e a c t i o n / 3 5 / a t lower d e u t e r o n e n e r g i e s . We s h a l l i n v e s t i g a t e t h i s f e a t u r e by measuring t h e momentum d i s t r i b u t i o n w i t h b e t t e r s t a t i s t i c s t h a n i n t h e p r e s e n t d a t a .
5' ,' /,'/ / 15 20 1 25
+(+(,;4i,( L
o m u g m t Em [MeV]--+
VI W Q - N ~ O
me%::&
Fig. 11 Spectra2 funcMon for t h e r e a c t i o n ' ~ ( e , e l p l ' O T ~ , n o t coxrected for r a d i a t i v e e f f e c t s .
C4-66 JOURNAL DE PHYSIQUE
+ +
S i n c e t h e ( a f , 2 + , 4 ,6 ) q u a d r u p l e t i s o f r e l a t i v e l y s i m p l e s t r u c t u r e it may s e r v e a s a gauge f o r t h e o p t i c a l - p o t e n t i a l d e s c r i p t i o n o f t h e f i n a l - s t a t e i n t e r a c t i o n . F o r t h i s p u r p o s e we p l a n t o measure t h e s p e c t r a l f u n c t i o n a t a few d i f f e r e n t o u t g o i n g p r o t o n e n e r g i e s . A t t h i s p o i n t w e w i s h t o p o i n t o u t t h a t t h e dependence on b o u n d - s t a t e wave f u n c t i o n s of t h e s p e c t r o s c o p i c f a c t o r s deduced from t h e ( e , e l p ) r e a c t i o n i s c o m p l e t e l y d i f f e r e n t from t h e s e v e r e s e n s i t i v i t y o b s e r v e d
/ l o , 11/ i n one-nucleon t r a n s f e r r e a c t i o n s . I n f a c t i n ( e , e l p ) one maps o u t t h e s i n g l e - p a r t i c l e wave f u n c t i o n and t h e s e n s i t i v i t y o f
C'S o n t h e r a d i a l b o u n d - s t a t e wave f u n c t i o n i s v e r y s m a l l .
I n t h e s p e c t r a o f f i g . 11 t h e 2 = 0 p i c k u p from t h e 4.42 MeV ( 3 - ) and 6.10 MeV ( 4 - ) s t a t e s i s c l e a r l y s e e n . I f t h e s e a r e t h e 2Slh? h o l e s t a t e s t h e i r s t r e n g t h s s h o u l d be p r o p o r t i o n a l t o 2 J f + l . T 1 s
e x p e c t a t i o n i s c o n f i r m e d by t h e ( d , ~ ) d a t a / 3 3 , 3 4 / a s w e l l a s by o u r s p e c t r a . These s t a t e s i n d e e d e x h i b i t a 2s1/2 momentum d i s t r i b u t i o n a s i s shown i n f i g . 13. The s p e c t r a l f u n c t i o n e s s e n t i a l l y p e a k s a t s m a l l m i s s i n g momentum v a l u e s .
Another q u a d r u p l e t i s p r e d i c t e d f o r l d 3 1 2 - h o l e s t a t e s . These a r e t e n t a t i v e l y i d e n t i f i e d / 3 3 , 3 4 / w i t h t h e 2 , 3 - , 4- and 5 - s t a t e s a t 5 . 4 4 , 5.85, 6.46 a n d 7.65 MeV, r e s p e c t i v e l y , which a r e w e l l r e s o l v e d from t h e p r o m i n e n t 6.10 MeV peak ( s e e f i g . 1 2 ) . The momentum d e n s i t y o f t h e s t a t e a t 5.44 MeV i n d e e d e x h i b i t s an I d s h a p e ( s e e f i g , 1 3 ) . A s i s shown i n f i g . 11 t h e r e g i o n between 6 a n d 16 MeV e x c i t a t i o n e n e r g y i s r a t h e r f e a t u r e l e s s w i t h i n t h e p r e s e n t l i m i t e d s t a t i s t i c s . The s t r u c t u r e p e a k s r o u g h l y a t 120 MeV/c m i s s i n g momentum, which may i n d i c a t e t h e p r e s e n c e o f I d h o l e s t a t e s . Whether o r n o t i t w i l l be p o s s i b l e t o d i s e n t a n g l e it from s t r e n g t h a w a i t s t h e r e s u l t o f t h e a n a l y s i s i n p r o g r e s s o f b e t t e r s t a t i s t i c s d a t a .
Fig. 12 Missing energy s p e c t m for t h e ~ e a c t i o n 5 1 ~ ( e , e ' p 1 not corrected for ~ a d i a t i u e e f f e c t s .
F i g , 1 3 Momentum d e m i t i e s ( n o t corrected for r a d i a t i v e e f f e c t s ) for l = 3, l = 0 and 2 = 2 proton t ~ m s f e r i n the reaction " v ( e , e ' p ) ' O T ~ . The curves are a r b i t r a r i l y normal- i z e d Harmonic Osci Z Zator d e n s i t i e s .
7. CONCLUSION
We have d e m o n s t r a t e d t h a t t h e i n s t r u m e n t a t i o n f o r p e r f o r m i n g h i g h - r e s o l u t i o n ( e , e l p ) e x p e r i m e n t s i s now a v a i l a b l e a t NIKHEF-K. I n t h e d a t a a n a l y s i s a c c i d e n t a l s u b t r a c t i o n , phase-space c a l c u l a t i o n a n d r a d i a t i v e u n f o l d i n g have been shown t o y i e l d c o n s i s t e n t r e s u l t s . By a p r o p e r c h o i c e o f t h e k i n e m a t i c a l c o n d i t i o n s a m b i g u i t i e s i n t h e o f f - s h e l l e l e c t r o n - p r o t o n c r o s s s e c t i o n can b e minimized, a n d t h u s an e x p e r i m e n t a l s p e c t r a l f u n c t i o n i s o b t a i n e d w i t h h i g h a c c u r a c y . I n t h e d i s c u s s i o n o f t h e p r e s e n t r e s u l t s we have r e s t r i c t e d o u r s e l v e s t o t h e two o u t e r m o s t s h e l l s , i . e . t h e v a l e n c e s h e l l , a n d one s h e l l d e e p e r . I n a l a t e r s t a g e we s h a l l a l s o e x p l o i t t h e ( e , e l p ) r e a c t i o n f o r t h e s t u d y of d e e p - l y i n g s h e l l s ,
A comparison o f t h e s p e c t r a l f u n c t i o n w i t h s h e l l - m o d e l p r e d i c t i o n s i s somewhat hampered by u n c e r t a i n t i e s i n t h e e f f e c t s o f t h e f i n a l - s t a t e i n t e r a c t i o n . P r o v i d e d t h a t more t h e o r e t i c a l a t t e n t i o n i s g i v e n t o t h i s a s p e c t , t h e ( e , e l p ) r e a c t i o n h a s t h e p o t e n t i a l t o become a n a c c u r a t e method f o r t h e d e t e r m i n a t i o n o f s i n g l e - p a r t i c l e s h e l l s t r u c t u r e . I n a s y n t h e s i z i n g approach w i t h s i n g l e - n u c l e o n t r a n s f e r r e a c t i o n s it w i l l c o n t r i b u t e c o n s i d e r a b l y t o t h e u n d e r s t a n d i n g o f h o l e s t a t e s i n n u c l e i , w e b e l i e v e .
W e w i s h t o t h a n k J.W.A. den H e r d e r , E . J a n s , P.H.M. K e i z e r , L . P r i n s a n d E.M.N. Q u i n t f o r t h e i r v a l u a b l e c o o p e r a t i o n i n t h e ( e , e l p ) e x p e r i m e n t s .
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