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Dispersion corrections to elastic electron scattering by 16O and 12C
Volume 39B, number 5
DISPERSION
PHYSICS LETTERS
CORRECTIONS
TO
ELASTIC
BY 1 6 0
AND
29 May 19~2
ELECTRON
SCATTERING
12C
J. L. FRIAR
Center for Theoretical Physics, M.I.T., Cambridge, Mass.02139, USA and
M. ROSEN
Center for Theoretical Physics, M.I. T., Cambridge, Mass. O2139,USA and Naval Research Laboratory j Washington, D.C.20390, USA** Received 24 March 1972 Longitudinal dispersion corrections to elastic electron scattering from 160 and 12C are calculated using oscillator wave functions and closure approximation. Near the first diffraction minimum, the contributions from the Pauli, COM and self-correlations almost cancel. The correction increases away from the minimum.
As the a c c u r a c y and e n e r g y of 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 e x p e r i m e n t s have i n c r e a s e s so has i n t e r e s t in the c a l c u l a t i o n of c o n t r i b u t i o n s c o m i n g f r o m inter.mediate i n e l a s t i c s c a t t e r i n g p r o c e s s e s - the so c a l l e d d i s p e r s i o n c o r r e c t i o n s [1-3]. T h e s e a r e events in which the nucleus is f i r s t v i r t u a l l y e x c i t e d and subsequently d e - e x c i t e d by the incident e l e c t r o n and which t h e r e f o r e c o n t r i b u t e to the o v e r a l l e l a s tic amplitude. In this note we r e p o r t on a c a l c u l a t i o n of the longitudinal d i s p e r s i o n c o r r e c t i o n to the 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 f r o m 12C and 160 at 374.5 MeV b o m b a r d i n g e n e r g y (the l o w e r of the two e n e r g i e s of the Stanford e x p e r i m e n t of Sick and McCarthy [4]). We e x p l i c i t l y exhibit the s e p a r a t e c o n t r i b u t i o n s made to the d i s p e r s i o n a m p l it u d e by the t h r e e k i n e m a t i c ground state c o r r e l a t i o n s - the P a u l i , c e n t e r of m a s s (COM) and self c o r r e l a t i o n s [5] - and show why a l e s s than c a r e f u l c a l c u l a t i o n done c o n s i s t e n t l y within one model is a l m o s t s u r e to lead to e r r o n e o u s r e s u l t s . We w r i t e the total e l a s t i c a m p l it u d e as the sum of static and d i s p e r s i o n a m p l i t u d e s ; we c a l c u l a t e the l a t t e r in second B o r n a p p r o x i m a t i o n , but obtain the static am p l i t u d e f r o m a p h ase shift p r o g r a m using the b e s t fit c h a r g e d i s t r i b u t i o n s of Sick and M c C a r t h y [4]. T h i s allows us to obtain d i s p e r s i o n c o r r e c tions throughout the r e g i o n of the d i f f r a c t i o n mtnimun unlike m o s t p r e v i o u s c a l c u l a t i o n s . The two photon exchange c o n t r i b u t i o n to the d i s p e r s i o n am p l i t u d e (including both d i r e c t and c r o s s e d graphs) is g i v en by~
G(2)(q)
~2 1 ~ ½PnS2+Ptp" S (PIP" S/P-½ $2) :~2-g-e S n¢O fd3p q ~ L ~ - En (P+Pn)(P+Pn+2En) I x
(i) Z
Z (fli__~1 exp('iql" ri)ln>
615
Volume 39B, number 5
PHYSICS
LETTERS
29 May 1972
H e r e PL(Pf, P) is the i n i t i a l (final, i n t e r m e d i a t e ) e l e c t r o n momentum,q=pi - P f , q l = P-Pf, q 2 = PL-P, S = pi~Pf, Pn = Pi-En and the n u c l e a r s t a t e s a r e w r i t t e n as the p r o d u c t of a n u c l e a r C O M p l a n e w a v e f u n c t i o n and an i n t r i n s i c t r a n s l a t t o n - i n v a r t a n t w a v e f u n c t i o n ; i n t e g r a t i o n o v e r i n t e r m e d i a t e COM m o m e n t a is i n c l u d e d in the s u m o v e r i n t e r m e d i a t e s t a t e s (we w i l l r e p r e s e n t the i n t r i n s i c g r o u n d s t a t e by the s y m b o l 10)). T h e i n t e r m e d i a t e e n e r g y of the n u c l e u s is d e n o t e d by En; it i n c l u d e s the r e c o i l e n e r g y of the COM, i . e . , p2
En =
1
p2 + (%z - Wo)
(2)
2M
w h e r e (w u - w o) is the i n t r i n s i c e x c i t a t i o n e n e r g y . R e l a t i v e to the high e n e r g y of the i n c i d e n t e l e c t r o n s , the s t r u c t u r e of the u n i m p o r t a n t . To f a c i l i t a t e c a r r y i n g out the s u m o v e r s t a t e s , we w r i t e E n = the i n t e g r a n d of eq. (1) about an e f f e c t i v e e x c i t a t i o n e n e r g y (3 - Wo). In the e x i c t e d - s t a t e d e p e n d e n c e o c c u r s only in the m a t r i x e l e m e n t s ; we c a n i n v o k e m a t r i x e l e m e n t s u m a s the g r o u n d s t a t e t w o - p r o t o n c o r r e l a t i o n f u n c t i o n
e x c i t e d s t a t e s p e c t r u m is (~ - ¢Oo) + A n and e x p a n d l o w e s t o r d e r t e r m the c l o s u r e and e x p r e s s the
Z
C(ql,q 2) = (OI
exp ( i q l "
r i + iq2" rj)10} - Z2Fo(ql)Fo(q 2)
(3)
i,j=l
w h e r e Fo(q) is the g r o u n d s t a t e c h a r g e f o r m f a c t o r . T h i s in t u r n c a n be w r i t t e n a s the s u m of the t h r e e kinematical correlations (4)
C(ql,q2) : Cs(ql,q2 ) + Cptq l, q 2) ~-CcM(ql,q 2) w h e r e CS, Cp and C C M a r e the s e l f c o r r e l a t i o n , r e s p e c t i v e l y , and
P a u l t c o r r e l a t i o n and c e n t e r of m a s s c o r r e l a t i o n ,
C S : Z[Fo(q) - F o ( q l ) Fo(q2)]
(5a)
Cp - 2AF(CAM- 1( )q ) ~ 4 ~ 1
(5b)
CCM =
-)(01 zcj ~ e x p ( i q l . r i + iq2" rj) 0 - Fo(ql)Fo(q2) 1
(Z2-Z)Fo(ql)F~(q2) I FcM(ql)FcM(q2) Fc~q)
- 1 ].
(5c)
H e r e FCM(q) is the g r o u n d s t a t e c e n t e r of m a s s f o r m f a c t o r and the p r i m e i n d i c a t e s f o r m f a c t o r s c a l c u l a t e d with r e l a t i v e c o o r d i n a t e s , i . e . , with the c e n t e r of m a s s m o t i o n r e m o v e d . T h u s to l o w e s t o r d e r , a l l we n e e d to know about the e x c i t e d s t a t e s is t h e e f f e c t i v e e x c i t a t i o n e n e r g y (~ - ¢Oo). I n d e e d t h i s is a l s o t r u e of the f i r s t o r d e r t e r m in the e x p a n s i o n . F o r o u r i m m e d i a t e p u r p o s e s , we a s s u m e a v a n i s h i n g e f f e c t i v e a v e r a g e e x c i t a t i o n e n e r g y and u s e s h e l l m o d e l w a v e f u n c t i o n s f o r the g r o u n d s t a t e . T h e d e p e n d e n c e of the d i s p e r s i o n c o r r e c t i o n s on ~ t o g e t h e r with d e t a i l e d r e s u l t s of the f i r s t o r d e r c o r r e c t i o n to the c l o s u r e a p p r o x i m a t i o n w i l l be g i v e n in a ia6!er p u b l i c a t i o n . We a s s u m e the lp.~/2 l e v e l is f i l l e d f o r 12C and t a k e the O g r o u n d s t a t e to be a c l o s e d l p s h e l l . In e a c h c a s e we c h o o s e - f h a t o s c i l l a t o r p a r a m e t e r that g i v e s a b e s t fit to the S t a n f o r d e l a s t i c s c a t t e r i n g data. T h e h a r m o n i c o s c i l l a t o r f o r m f a c t o r is
Fo(q) : (1
616
-
~flo q2) e x p [ - f l o q 2 ( 1 - 1 / A )];
(6)
V o l u m e 3 9 B , number 5
PHYSICS Itc
LETTERS
29 May 1972
DISPERSION CORRECTIONS
.f
t
-8 -12 ........ . . . . . .
-16
SELF SELF+PAULI SELF +PAULI+ COM
-20
-24 -28
2
3
4
5
6
7
8
SQUARE OF MOMENTUM TRANSFER (fm -2)
Fig.1. Contribution of the self correlation, Pauli c o r r e lation and center of m a s s correlation to the percentage dispersion corrections to the 12C elastic c r o s s section. f o r 12C, c ~ = 4 a n d w e f t n d f l of m a s s f o r m f a c t o r to be
F C M(q )
=
o =0.6769fm
-2while
for 160
c~=land/3 o =0.7859fm
-2. W e t a k e the c e n t e r
exp[ -/3oq 2 / A ]
(7)
and w h e n f i t t i n g the data w e t a k e the f i n i t e n u c l e o n s i z e into a c c o u n t by m u l t i p l y i n g the s h e l l m o d e l f o r m f a c t o r by the J a n s s e n s s c a l a r e l e c t r i c n u c l e o n f o r m f a c t o r [6]. 12C DISPERSION CORRECTIONS
10-3 12C DISPERSION CORRECTIONS
10-4
\
i<
"~.. "~--~..
e
IO-S
ffl
-~ IO-6
~
.......
SELF
.....
,,AUU
.... •_
IO "7
/ /
COM
~
TOTAL IMAGINARYAMPLITUDE
/
ID - 6
........
/
. . . . . . . . .
~~
10-7
SELF PAULI COM TOTAL REAL AMPLITUDE
\ /i
/
'
/
/
/
i p
/ I0
I
2
I
3
I
4
~_____
5
6
I
7
SOUARE OF MOMENTUM TRANFER (fm "z)
Fig.2. The absolute value of the contribution of each of the correlations to the imaginary dispersion amplitude. The abcissa shows the imaginary amplitudes multiplied by S.
10-8
I
I
I
SQUARE OF MOMENTUM TRANSFER (fin- t )
Fig. 3. The abso|ute value of the contribution of each of the correlations to the real dispersion amplitude. The abcissa shows the real amplitudes multiplied by S.
617
Volume 39B, n u m b e r 5 ~,
PHYSICS
LETTERS
IgC ELASTIC FORM FACTOR SQUARED
29 May 1972
IC
=2C DISPERSION CORRECTIONS
x SICK- MCCARTHY DATA
to-,~
\
~ -tO
(9 ......... "~
#~'
"~X~//
g
/
SELF
~-20
........ .....
SELF SELF + PAULI SELF+ PAULI + COM
SELF+PAULI ÷COM
//
STATIC AMPLITUDE
/
/
-30 //
I
3.0
I
L
5.2 5.4 3.6 SQUARE OF MOMENTUM TRANSFER (fro-=:)
I
I
5.8
4.0
Fig.4. C o r r e c t i o n s to the best Sick and McCarthy fit to the elastic form factor in the region of the f i r s t d i f f r a c tion m i n i m u m .
L, / 200
J I 400
I 600
I 800
I
IO00
LAB ENERGY (MeV)
Fig.5. Energy dependence of the percentage d i s p e r s i o n c o r r e c t i o n s at the f i r s t diffraction mini'mum.
T h e c o r r e l a t i o n C ( q l , q2 ) i s n o w p u t i n t o eq. (1), t h e a n g u l a r i n t e g r a l s d o n e a n a l y t i c a l l y a n d t h e r a d i a l i n t e g r a t i o n p e r f o r m e d n u m e r i c a l l y . T h e r e s u l t s a r e g i v e n in f i g s . 1 - 5 f o r 12C. In fig.1 a r e i n d i c a t e d t h e p e r c e n t a g e d i s p e r s i o n c o r r e c t i o n s to t h e e l a s t i c c r o s s s e c t i o n f o r a n i n c i d e n t e l e c t r o n e n e r g y of 374.5 MeV. It i s c l e a r t h a t a l t h o u g h e a c h of t h e c o r r e l a t i o n s c o n t r i b u t e s s i g n i f i c a n t l y to t h e d i s p e r s i o n c o r r e c t i o n s , e x p e c i a l l y in t h e r e g i o n of t h e d i f f r a c t i o n m i n i m u m , t h e n e t c o r r e c t i o n t s s m a l l a n d e s p e c i a l l y s o j u s t i n t h a t r e g i o n . F i g s . 2 a n d 3 s h o w c l e a r l y w h a t i s h a p p e n i n g . In t h e r e g i o n of t h e m i n i m u m , b o t h r e a l a n d i m a g i n a r y p a r t s of t h e s e l f c o r r e l a t i o n a m p l i t u d e a r e l a r g e (and n e g a t i v e ) . T h e y a r e h o w e v e r a l m o s t e x a c t l y c o m p e n s a t e d b y t h e P a u l t a n d c e n t e r of m a s s c o r r e l a t i o n a m p l i t u d e s , so t h a t t h e t o t a l a m p l i t u d e s a r e m o r e t h a n two o r d e r s of m a g n i t u d e s m a l l e r t h e r e . T h i s p o i n t s up t h e i m p o r t a n c e of t h e r e l a t i v e l y n e g l e c t e d c e n t e r of m a s s c o r r e l a t i o n s a n d a l s o t h e n e c e s s i t y of c a l c u l a t i n g t h e c o r r e l a t i o n s c o n s i s t e n t l y w i t h i n t h e f r a m e w o r k of a s i n g l e m o d e l . If o n e u s e s d i f f e r e n t m o d e l s to c a l c u l a t e t h e r e s p e c t i v e c o r r e l a t i o n s , a n y s i g n i f i c a n t r e s u l t (in t h e r e g i o n of t h e f i r s t m i n i m u m c e r t a i n l y ) i s m o r e l i k e l y to b e a m e a s u r e of d i f f e r e n c e s i n t h e m o d e l s t h a n of d i s p e r s i o n c o r r e c t i o n s . A l t h o u g h less pronounced elsewhere than near the diffraction minimum, there is a clear destructive interference a m o n g t h e c o r r e l a t i o n a m p l i t u d e s o v e r m o s t of t h e r a n g e of m o m e n t u m t r a n s f e r s c o n s i d e r e d . F i g . 4 s h o w s e x p l i c i t l y how t h e d i s p e r s i o n c o r r e c t i o n s a f f e c t t h e e l a s t i c f o r m f a c t o r n e a r m i n i m u m - t h e r e i s c l e a r l y no s i g n i f i c a n t c h a n g e h e r e . N o r do t h e p e r c e n t a g e d i s p e r s i o n c o r r e c t i o n s a t t h e m i n i m u m c h a n g e m u c h w i t h e n e r g y . I n f i g . 5 it m a y b e s e e n t h a t a b o v e 250 MeV t h e c o r r e c t i o n i s a l m o s t e n e r g y i n d e p e n d e n t . F u r t h e r m o r e , w h e n o n e e x a m i n e s t h e f i r s t o r d e r c o r r e c t i o n s to t h e c l o s u r e a p p r o x i m a t i o n , o n e f i n d s t h a t t h e d o m i n a n t c o n t r i b u t i o n i s p r o p o r t i o n a l to Fo(q) , t h e B o r n e l a s t i c f o r m f a c t o r , w h i c h v a n i s h e s a t t h e d i f f r a c t i o n m i n i m u m . T h a t i s we f i n d t h a t c l o s u r e i s a g o o d a p p r o x i m a t i o n a t t h e m i n i m u m . T h i s i s in c o n t r a s t w i t h t h e w o r k of D e F o r e s t [7]. H i s r e s u l t m u s t a r i s e f r o m t h e s t r o n g e n e r g y d e p e n d e n c e of t h e p o t e n t i a l h e u s e d . G u r c o n c l u s i o n i s t h e r e s u l t of u s i n g a s u m r u l e w h i c h a s s u m e s t h a t s u c h a d e p e n d e n c e d o e s n o t o c c u r in t h e n u c l e a r H a m i l t o n i a n . T h e d i s p e r s i o n a m p l i t u d e s f o r 1 6 0 b e h a v e t n v e r y m u c h t h e s a m e way a s t h o s e of 12C - t h e d e s t r u c t i v e i n t e r f e r e n c e b e i n g if a n y t h i n g g r e a t e r , s o t h a t t h e p e r c e n t a g e d i s p e r s i o n c o r r e c t i o n s a r e s m a l l e r t h a n t h o s e of 12C. T h e d e t a i l s of t h e o x y g e n r e s u l t s w i l l a l s o b e g i v e n l a t e r . Not a l l s t a t e s c o n t r i b u t e to o u r s u m o v e r e x c i t e d s t a t e s . O u r m o d e l i s a s i n g l e p a r t i c l e m o d e l a n d so the summation goes over all allowed lp-lh states (thus implicitly including contributions from interm e d i a t e q u a s i e l a s t i c s c a t t e r i n g a n d g i a n t d i p o l e r e s o n a n c e e x c i t a t i o n ) , b u t d o e s not i n c l u d e c o l l e c t i v e m u l t i p a r t i c l e - m u l t i h o l e s t a t e s . N o r d o e s it i n c l u d e s t a t e s t h a t c a n b e r e a c h e d only by m a g n e t i c t r a n s i t i o n s . A n d of c o u r s e , o u r s h e l l m o d e l g r o u n d s t a t e d o e s not c o n t a i n s h o r t r a n g e d y n a m i c a l c o r r e l a t i o n s . 618
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LETTERS
29 May 1972
O n e of u s ( M . R . ) w o u l d l i k e to t h a n k P r o f . H e r m a n F e s h b a c h f o r t h e w a r m h o s p i t a l i t y s h o w n h i m d u r i n g h i s s t a y a t t h e C e n t e r f o r T h e o r e t i c a l P h y s i c s a t M . I . T . , a n d b o t h w o u l d l i k e to t h a n k P r o f . J . H e i s e n b e r g of M . I . T . f o r t h e u s e of h i s e l e c t r o n s c a t t e r i n g p h a s e s h i f t c o d e .
References [1] [2] [31 [4] [5] [6] [7]
A. Bottino, G. Ciocehetti and A. Molinari, Nucl. Phys. 89 (1966) 192 and r e f e r e n c e s cited therein. A. Bottino and G. Ciocchetti, Nuel. Phys. A178 (1972) 593 and r e f e r e n c e s cited therein. H. A. Bethe and A. Molinari, Ann. Phys. (N.Y.) 63 (1971) 393. I. Sick and J. S. McCarthy, Nucl. Phys. A150 (1970) 631. H. Feshbach, A. Gal and J. Hiifner, Ann. Phys. (N.Y.) 66 (1971) 20. T. J a n s s e n s , R. Hofstadter, E. B. Hughes and M. R. Yearion, Phys. Rev. 142 (1966) 922. T. D e F o r e s t J r . , Phys. L e t t e r s 32B (1970) 12. * * * * *
619
DISPERSION
PHYSICS LETTERS
CORRECTIONS
TO
ELASTIC
BY 1 6 0
AND
29 May 19~2
ELECTRON
SCATTERING
12C
J. L. FRIAR
Center for Theoretical Physics, M.I.T., Cambridge, Mass.02139, USA and
M. ROSEN
Center for Theoretical Physics, M.I. T., Cambridge, Mass. O2139,USA and Naval Research Laboratory j Washington, D.C.20390, USA** Received 24 March 1972 Longitudinal dispersion corrections to elastic electron scattering from 160 and 12C are calculated using oscillator wave functions and closure approximation. Near the first diffraction minimum, the contributions from the Pauli, COM and self-correlations almost cancel. The correction increases away from the minimum.
As the a c c u r a c y and e n e r g y of 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 e x p e r i m e n t s have i n c r e a s e s so has i n t e r e s t in the c a l c u l a t i o n of c o n t r i b u t i o n s c o m i n g f r o m inter.mediate i n e l a s t i c s c a t t e r i n g p r o c e s s e s - the so c a l l e d d i s p e r s i o n c o r r e c t i o n s [1-3]. T h e s e a r e events in which the nucleus is f i r s t v i r t u a l l y e x c i t e d and subsequently d e - e x c i t e d by the incident e l e c t r o n and which t h e r e f o r e c o n t r i b u t e to the o v e r a l l e l a s tic amplitude. In this note we r e p o r t on a c a l c u l a t i o n of the longitudinal d i s p e r s i o n c o r r e c t i o n to the 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 f r o m 12C and 160 at 374.5 MeV b o m b a r d i n g e n e r g y (the l o w e r of the two e n e r g i e s of the Stanford e x p e r i m e n t of Sick and McCarthy [4]). We e x p l i c i t l y exhibit the s e p a r a t e c o n t r i b u t i o n s made to the d i s p e r s i o n a m p l it u d e by the t h r e e k i n e m a t i c ground state c o r r e l a t i o n s - the P a u l i , c e n t e r of m a s s (COM) and self c o r r e l a t i o n s [5] - and show why a l e s s than c a r e f u l c a l c u l a t i o n done c o n s i s t e n t l y within one model is a l m o s t s u r e to lead to e r r o n e o u s r e s u l t s . We w r i t e the total e l a s t i c a m p l it u d e as the sum of static and d i s p e r s i o n a m p l i t u d e s ; we c a l c u l a t e the l a t t e r in second B o r n a p p r o x i m a t i o n , but obtain the static am p l i t u d e f r o m a p h ase shift p r o g r a m using the b e s t fit c h a r g e d i s t r i b u t i o n s of Sick and M c C a r t h y [4]. T h i s allows us to obtain d i s p e r s i o n c o r r e c tions throughout the r e g i o n of the d i f f r a c t i o n mtnimun unlike m o s t p r e v i o u s c a l c u l a t i o n s . The two photon exchange c o n t r i b u t i o n to the d i s p e r s i o n am p l i t u d e (including both d i r e c t and c r o s s e d graphs) is g i v en by~
G(2)(q)
~2 1 ~ ½PnS2+Ptp" S (PIP" S/P-½ $2) :~2-g-e S n¢O fd3p q ~ L ~ - En (P+Pn)(P+Pn+2En) I x
(i) Z
Z (fli__~1 exp('iql" ri)ln>
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H e r e PL(Pf, P) is the i n i t i a l (final, i n t e r m e d i a t e ) e l e c t r o n momentum,q=pi - P f , q l = P-Pf, q 2 = PL-P, S = pi~Pf, Pn = Pi-En and the n u c l e a r s t a t e s a r e w r i t t e n as the p r o d u c t of a n u c l e a r C O M p l a n e w a v e f u n c t i o n and an i n t r i n s i c t r a n s l a t t o n - i n v a r t a n t w a v e f u n c t i o n ; i n t e g r a t i o n o v e r i n t e r m e d i a t e COM m o m e n t a is i n c l u d e d in the s u m o v e r i n t e r m e d i a t e s t a t e s (we w i l l r e p r e s e n t the i n t r i n s i c g r o u n d s t a t e by the s y m b o l 10)). T h e i n t e r m e d i a t e e n e r g y of the n u c l e u s is d e n o t e d by En; it i n c l u d e s the r e c o i l e n e r g y of the COM, i . e . , p2
En =
1
p2 + (%z - Wo)
(2)
2M
w h e r e (w u - w o) is the i n t r i n s i c e x c i t a t i o n e n e r g y . R e l a t i v e to the high e n e r g y of the i n c i d e n t e l e c t r o n s , the s t r u c t u r e of the u n i m p o r t a n t . To f a c i l i t a t e c a r r y i n g out the s u m o v e r s t a t e s , we w r i t e E n = the i n t e g r a n d of eq. (1) about an e f f e c t i v e e x c i t a t i o n e n e r g y (3 - Wo). In the e x i c t e d - s t a t e d e p e n d e n c e o c c u r s only in the m a t r i x e l e m e n t s ; we c a n i n v o k e m a t r i x e l e m e n t s u m a s the g r o u n d s t a t e t w o - p r o t o n c o r r e l a t i o n f u n c t i o n
e x c i t e d s t a t e s p e c t r u m is (~ - ¢Oo) + A n and e x p a n d l o w e s t o r d e r t e r m the c l o s u r e and e x p r e s s the
Z
C(ql,q 2) = (OI
exp ( i q l "
r i + iq2" rj)10} - Z2Fo(ql)Fo(q 2)
(3)
i,j=l
w h e r e Fo(q) is the g r o u n d s t a t e c h a r g e f o r m f a c t o r . T h i s in t u r n c a n be w r i t t e n a s the s u m of the t h r e e kinematical correlations (4)
C(ql,q2) : Cs(ql,q2 ) + Cptq l, q 2) ~-CcM(ql,q 2) w h e r e CS, Cp and C C M a r e the s e l f c o r r e l a t i o n , r e s p e c t i v e l y , and
P a u l t c o r r e l a t i o n and c e n t e r of m a s s c o r r e l a t i o n ,
C S : Z[Fo(q) - F o ( q l ) Fo(q2)]
(5a)
Cp - 2AF(CAM- 1( )q ) ~ 4 ~ 1
(5b)
CCM =
-)(01 zcj ~ e x p ( i q l . r i + iq2" rj) 0 - Fo(ql)Fo(q2) 1
(Z2-Z)Fo(ql)F~(q2) I FcM(ql)FcM(q2) Fc~q)
- 1 ].
(5c)
H e r e FCM(q) is the g r o u n d s t a t e c e n t e r of m a s s f o r m f a c t o r and the p r i m e i n d i c a t e s f o r m f a c t o r s c a l c u l a t e d with r e l a t i v e c o o r d i n a t e s , i . e . , with the c e n t e r of m a s s m o t i o n r e m o v e d . T h u s to l o w e s t o r d e r , a l l we n e e d to know about the e x c i t e d s t a t e s is t h e e f f e c t i v e e x c i t a t i o n e n e r g y (~ - ¢Oo). I n d e e d t h i s is a l s o t r u e of the f i r s t o r d e r t e r m in the e x p a n s i o n . F o r o u r i m m e d i a t e p u r p o s e s , we a s s u m e a v a n i s h i n g e f f e c t i v e a v e r a g e e x c i t a t i o n e n e r g y and u s e s h e l l m o d e l w a v e f u n c t i o n s f o r the g r o u n d s t a t e . T h e d e p e n d e n c e of the d i s p e r s i o n c o r r e c t i o n s on ~ t o g e t h e r with d e t a i l e d r e s u l t s of the f i r s t o r d e r c o r r e c t i o n to the c l o s u r e a p p r o x i m a t i o n w i l l be g i v e n in a ia6!er p u b l i c a t i o n . We a s s u m e the lp.~/2 l e v e l is f i l l e d f o r 12C and t a k e the O g r o u n d s t a t e to be a c l o s e d l p s h e l l . In e a c h c a s e we c h o o s e - f h a t o s c i l l a t o r p a r a m e t e r that g i v e s a b e s t fit to the S t a n f o r d e l a s t i c s c a t t e r i n g data. T h e h a r m o n i c o s c i l l a t o r f o r m f a c t o r is
Fo(q) : (1
616
-
~flo q2) e x p [ - f l o q 2 ( 1 - 1 / A )];
(6)
V o l u m e 3 9 B , number 5
PHYSICS Itc
LETTERS
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DISPERSION CORRECTIONS
.f
t
-8 -12 ........ . . . . . .
-16
SELF SELF+PAULI SELF +PAULI+ COM
-20
-24 -28
2
3
4
5
6
7
8
SQUARE OF MOMENTUM TRANSFER (fm -2)
Fig.1. Contribution of the self correlation, Pauli c o r r e lation and center of m a s s correlation to the percentage dispersion corrections to the 12C elastic c r o s s section. f o r 12C, c ~ = 4 a n d w e f t n d f l of m a s s f o r m f a c t o r to be
F C M(q )
=
o =0.6769fm
-2while
for 160
c~=land/3 o =0.7859fm
-2. W e t a k e the c e n t e r
exp[ -/3oq 2 / A ]
(7)
and w h e n f i t t i n g the data w e t a k e the f i n i t e n u c l e o n s i z e into a c c o u n t by m u l t i p l y i n g the s h e l l m o d e l f o r m f a c t o r by the J a n s s e n s s c a l a r e l e c t r i c n u c l e o n f o r m f a c t o r [6]. 12C DISPERSION CORRECTIONS
10-3 12C DISPERSION CORRECTIONS
10-4
\
i<
"~.. "~--~..
e
IO-S
ffl
-~ IO-6
~
.......
SELF
.....
,,AUU
.... •_
IO "7
/ /
COM
~
TOTAL IMAGINARYAMPLITUDE
/
ID - 6
........
/
. . . . . . . . .
~~
10-7
SELF PAULI COM TOTAL REAL AMPLITUDE
\ /i
/
'
/
/
/
i p
/ I0
I
2
I
3
I
4
~_____
5
6
I
7
SOUARE OF MOMENTUM TRANFER (fm "z)
Fig.2. The absolute value of the contribution of each of the correlations to the imaginary dispersion amplitude. The abcissa shows the imaginary amplitudes multiplied by S.
10-8
I
I
I
SQUARE OF MOMENTUM TRANSFER (fin- t )
Fig. 3. The abso|ute value of the contribution of each of the correlations to the real dispersion amplitude. The abcissa shows the real amplitudes multiplied by S.
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29 May 1972
IC
=2C DISPERSION CORRECTIONS
x SICK- MCCARTHY DATA
to-,~
\
~ -tO
(9 ......... "~
#~'
"~X~//
g
/
SELF
~-20
........ .....
SELF SELF + PAULI SELF+ PAULI + COM
SELF+PAULI ÷COM
//
STATIC AMPLITUDE
/
/
-30 //
I
3.0
I
L
5.2 5.4 3.6 SQUARE OF MOMENTUM TRANSFER (fro-=:)
I
I
5.8
4.0
Fig.4. C o r r e c t i o n s to the best Sick and McCarthy fit to the elastic form factor in the region of the f i r s t d i f f r a c tion m i n i m u m .
L, / 200
J I 400
I 600
I 800
I
IO00
LAB ENERGY (MeV)
Fig.5. Energy dependence of the percentage d i s p e r s i o n c o r r e c t i o n s at the f i r s t diffraction mini'mum.
T h e c o r r e l a t i o n C ( q l , q2 ) i s n o w p u t i n t o eq. (1), t h e a n g u l a r i n t e g r a l s d o n e a n a l y t i c a l l y a n d t h e r a d i a l i n t e g r a t i o n p e r f o r m e d n u m e r i c a l l y . T h e r e s u l t s a r e g i v e n in f i g s . 1 - 5 f o r 12C. In fig.1 a r e i n d i c a t e d t h e p e r c e n t a g e d i s p e r s i o n c o r r e c t i o n s to t h e e l a s t i c c r o s s s e c t i o n f o r a n i n c i d e n t e l e c t r o n e n e r g y of 374.5 MeV. It i s c l e a r t h a t a l t h o u g h e a c h of t h e c o r r e l a t i o n s c o n t r i b u t e s s i g n i f i c a n t l y to t h e d i s p e r s i o n c o r r e c t i o n s , e x p e c i a l l y in t h e r e g i o n of t h e d i f f r a c t i o n m i n i m u m , t h e n e t c o r r e c t i o n t s s m a l l a n d e s p e c i a l l y s o j u s t i n t h a t r e g i o n . F i g s . 2 a n d 3 s h o w c l e a r l y w h a t i s h a p p e n i n g . In t h e r e g i o n of t h e m i n i m u m , b o t h r e a l a n d i m a g i n a r y p a r t s of t h e s e l f c o r r e l a t i o n a m p l i t u d e a r e l a r g e (and n e g a t i v e ) . T h e y a r e h o w e v e r a l m o s t e x a c t l y c o m p e n s a t e d b y t h e P a u l t a n d c e n t e r of m a s s c o r r e l a t i o n a m p l i t u d e s , so t h a t t h e t o t a l a m p l i t u d e s a r e m o r e t h a n two o r d e r s of m a g n i t u d e s m a l l e r t h e r e . T h i s p o i n t s up t h e i m p o r t a n c e of t h e r e l a t i v e l y n e g l e c t e d c e n t e r of m a s s c o r r e l a t i o n s a n d a l s o t h e n e c e s s i t y of c a l c u l a t i n g t h e c o r r e l a t i o n s c o n s i s t e n t l y w i t h i n t h e f r a m e w o r k of a s i n g l e m o d e l . If o n e u s e s d i f f e r e n t m o d e l s to c a l c u l a t e t h e r e s p e c t i v e c o r r e l a t i o n s , a n y s i g n i f i c a n t r e s u l t (in t h e r e g i o n of t h e f i r s t m i n i m u m c e r t a i n l y ) i s m o r e l i k e l y to b e a m e a s u r e of d i f f e r e n c e s i n t h e m o d e l s t h a n of d i s p e r s i o n c o r r e c t i o n s . A l t h o u g h less pronounced elsewhere than near the diffraction minimum, there is a clear destructive interference a m o n g t h e c o r r e l a t i o n a m p l i t u d e s o v e r m o s t of t h e r a n g e of m o m e n t u m t r a n s f e r s c o n s i d e r e d . F i g . 4 s h o w s e x p l i c i t l y how t h e d i s p e r s i o n c o r r e c t i o n s a f f e c t t h e e l a s t i c f o r m f a c t o r n e a r m i n i m u m - t h e r e i s c l e a r l y no s i g n i f i c a n t c h a n g e h e r e . N o r do t h e p e r c e n t a g e d i s p e r s i o n c o r r e c t i o n s a t t h e m i n i m u m c h a n g e m u c h w i t h e n e r g y . I n f i g . 5 it m a y b e s e e n t h a t a b o v e 250 MeV t h e c o r r e c t i o n i s a l m o s t e n e r g y i n d e p e n d e n t . F u r t h e r m o r e , w h e n o n e e x a m i n e s t h e f i r s t o r d e r c o r r e c t i o n s to t h e c l o s u r e a p p r o x i m a t i o n , o n e f i n d s t h a t t h e d o m i n a n t c o n t r i b u t i o n i s p r o p o r t i o n a l to Fo(q) , t h e B o r n e l a s t i c f o r m f a c t o r , w h i c h v a n i s h e s a t t h e d i f f r a c t i o n m i n i m u m . T h a t i s we f i n d t h a t c l o s u r e i s a g o o d a p p r o x i m a t i o n a t t h e m i n i m u m . T h i s i s in c o n t r a s t w i t h t h e w o r k of D e F o r e s t [7]. H i s r e s u l t m u s t a r i s e f r o m t h e s t r o n g e n e r g y d e p e n d e n c e of t h e p o t e n t i a l h e u s e d . G u r c o n c l u s i o n i s t h e r e s u l t of u s i n g a s u m r u l e w h i c h a s s u m e s t h a t s u c h a d e p e n d e n c e d o e s n o t o c c u r in t h e n u c l e a r H a m i l t o n i a n . T h e d i s p e r s i o n a m p l i t u d e s f o r 1 6 0 b e h a v e t n v e r y m u c h t h e s a m e way a s t h o s e of 12C - t h e d e s t r u c t i v e i n t e r f e r e n c e b e i n g if a n y t h i n g g r e a t e r , s o t h a t t h e p e r c e n t a g e d i s p e r s i o n c o r r e c t i o n s a r e s m a l l e r t h a n t h o s e of 12C. T h e d e t a i l s of t h e o x y g e n r e s u l t s w i l l a l s o b e g i v e n l a t e r . Not a l l s t a t e s c o n t r i b u t e to o u r s u m o v e r e x c i t e d s t a t e s . O u r m o d e l i s a s i n g l e p a r t i c l e m o d e l a n d so the summation goes over all allowed lp-lh states (thus implicitly including contributions from interm e d i a t e q u a s i e l a s t i c s c a t t e r i n g a n d g i a n t d i p o l e r e s o n a n c e e x c i t a t i o n ) , b u t d o e s not i n c l u d e c o l l e c t i v e m u l t i p a r t i c l e - m u l t i h o l e s t a t e s . N o r d o e s it i n c l u d e s t a t e s t h a t c a n b e r e a c h e d only by m a g n e t i c t r a n s i t i o n s . A n d of c o u r s e , o u r s h e l l m o d e l g r o u n d s t a t e d o e s not c o n t a i n s h o r t r a n g e d y n a m i c a l c o r r e l a t i o n s . 618
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O n e of u s ( M . R . ) w o u l d l i k e to t h a n k P r o f . H e r m a n F e s h b a c h f o r t h e w a r m h o s p i t a l i t y s h o w n h i m d u r i n g h i s s t a y a t t h e C e n t e r f o r T h e o r e t i c a l P h y s i c s a t M . I . T . , a n d b o t h w o u l d l i k e to t h a n k P r o f . J . H e i s e n b e r g of M . I . T . f o r t h e u s e of h i s e l e c t r o n s c a t t e r i n g p h a s e s h i f t c o d e .
References [1] [2] [31 [4] [5] [6] [7]
A. Bottino, G. Ciocehetti and A. Molinari, Nucl. Phys. 89 (1966) 192 and r e f e r e n c e s cited therein. A. Bottino and G. Ciocchetti, Nuel. Phys. A178 (1972) 593 and r e f e r e n c e s cited therein. H. A. Bethe and A. Molinari, Ann. Phys. (N.Y.) 63 (1971) 393. I. Sick and J. S. McCarthy, Nucl. Phys. A150 (1970) 631. H. Feshbach, A. Gal and J. Hiifner, Ann. Phys. (N.Y.) 66 (1971) 20. T. J a n s s e n s , R. Hofstadter, E. B. Hughes and M. R. Yearion, Phys. Rev. 142 (1966) 922. T. D e F o r e s t J r . , Phys. L e t t e r s 32B (1970) 12. * * * * *
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