WIGNER TRANSFORM METHODS IN INCLUSIVE ELECTRON SCATTERING FROM NUCLEI

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WIGNER TRANSFORM METHODS IN INCLUSIVE ELECTRON SCATTERING FROM NUCLEI

R. Rosenfelder

To cite this version:

R. Rosenfelder. WIGNER TRANSFORM METHODS IN INCLUSIVE ELECTRON SCAT- TERING FROM NUCLEI. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-3-C6-10.

�10.1051/jphyscol:1984601�. �jpa-00224202�

JOURNAL DE PHYSIQUE

Coiioque C6, supplement au n°6, Tome 45, juin 1984 page C6-3

WIGNER TRANSFORM METHODS IN INCLUSIVE ELECTRON SCATTERING FROM NUCLEI

R. Rosenfelder

Institut fur Physik, University of Mains, D-6S00 Mainz, F.R.G.

Abstract - A multiple scattering series for deep inelastic lepton- induced reactions is derived by using semiclassical Wigner transform methods. In contrast to the usual Glauber theory there is no limi- tation for the energy loss since a time-dependent formulation is used throughout. A simple parametrization of the generalized profile function yields a closed analytical expression for the longitudinal and transverse response function of p-shell nuclei. Comparison is made with the Saclay data for -'•^C.

I - Introduction

It is common knowledge that geometrical optics is valid if the wavelength of the scattering wave is small compared to the dimensions of the scatterer.

Under these conditions the phase-space description of the scattering process by means of the Wigner transform is best suited to derive semiclassical approximations /I/. As a simple example consider scattering of a nonrela- tivistic particle by a potential V(f). The scattering matrix is given by U(oo, -co) where the time evolution operator U(t,t

) obeys (Ti = 1)

(1)

and V

(t) = exp(i H t)U exp(-i H t) is the potential in the interaction picture. The scattering matrix element for the transition f<.-? I<p can be expressed as Jd^r exp(-iq^-f) U..(r,p;oo, -oo) where q = f<

- 6. is the momen- tum transfer, "p = (k. + k*,,)/2 the mean momentum and U. XT,~p) the Wigner trans-

i ~ 1 T" W

form of U. To get this quantity one performs a Wigner^transformation of eq.(l). In lowest order semiclassical approximation (\/.(t) U(t,t )),, ^

V

i,(t) U,,(t,t ) and one obtains

IW w o

(2)

Eq.(2) shows the familiar straight-line propagation of high-energy particles.

The scattering matrix then takes the familiar impact parameter form Résumé - Une série de diffusion multiple pour des réactions profon- dément inélastiques induites par des leptons est dérivée à l'aide des méthodes de transformation demi-classiques de Wigner. Par oppo- sition à la théorie conventionnelle de Glauber, il n'y a pas de limitation de perte d'énergie, vu qu'une formule dépendant du temps est utilisée constamment. Un simple paramètre du fonction-profil généralisé fournit une expression analytique fermée pour la fonction- réponse longitudinale et transversale des noyaux de la couche p.

Comparaison est faite avec les valeurs de Saclay pour '^C.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984601

'36-4 JOURNAL DE PHYSIQUE

where

Y ( ~ l , ~ , ^{= -}

i s t h e e i k o n a l p h a s e . H i g h e r o r d e r c o r r e c t i o n s c a n be c a l c u l a t e d s y s t e m a t i - c a l l y /2/ and t h e g e n e r a l i z a t i o n t o n o n l o c a l p o t e n t i a l s is s t r a i g h t f o r w a r d / 3 / . Cqs. ( 2 ) - ( 4 ) a r e t h e b a s i s f o r t h e d e s c r i p t i o n o f h i g h - e n e r g y s c a t t e r i n g o f p a r t i c l e s by cnmplcx t . a r q e t s ( e . g . n u c l e i ) i n G l n u b e r t h e o r y / 4 / . Assuming

A

an l r l t e r a c t i o n o f t h e f.orm X v(;-Ti) b e t w e e n p r o j e c t i l e a n d t a r q e t n u c l e o r i s ,

1-1

t h e p o t e n l i a l i n t h e i n L e r a c t s u -

ir e x p ( i ^ H N t ) 6 ( 9 + $ t / ~ - i ~ ) e x ~ ( - i ~ d ) i = l

wrtic:re tiN i s t h e n u c l e a r I l a m i l t u n i a n . I f , i n a d d i t i o n , t h e n u c l e a r m o t i o n i s assumed t o be s l o w compared witti t h e veloc:it;y o f t h e p r o j e c t i l e ( " f i x e d s c n t t e r e r " o r " f r o z e n n u c l e u s " n p p r o x l m a t i o n ) H N c a n be n e g l e c t e d a l t o g e t h e r

analy!;is; f o l l o v ~ s ttie

l i n e s a s h n f o r c : I n lowest. o r d e r s e m i c l a s - s l c a l a p p r o x i m a t i o n o n e qct.s f o r t h e t r a n s i t i o n

I ) 3 ( f , F )

1 n t r u d ~ : c i r l q t h e p r o f i l e f u n c t i o n

a ( f i n i t e ) m u l t i p l e s c a t t e r i n y e x p a n s i o n is o b t a i n e d by w r i t i n g

; F X ( r 1 - TI)

e

- ^q = T c ~ ~ ~ - 7 i ~ ) 4,

;:1

3 - ' ^{( 7 )}

11 - G l a u b e r I h e o r y f o r I n c l u s i v e P r o c e s s e s

Due t o t h e s t r a i g t - l l n e p r o p a g a t i o n t h e l o n q i t u d l n a l momcrltum t r a r i s f e r q ,, ^{= ( 2} f ^-I? 1 ^{) .} $ ⁽² f ^+i? 1 ^{)/p =} M(k:/>M-k 2 ; /2M)/p i s c o n s e r v e d i n e q . ( 5 ) . T h i s means t h a t . G l a u b a r t h e o r y c a n u s u a l l y o n l y be a p p l i e d t o e l a s t i c s c a t t e r i n g o r e x c 1 t a t : i o n o f l o w - l y i n g s t - : ~ t . e s . A p r a c t i c a b l e m u l t i p l e s c a t t e r i n g forma- l i s m f o r d e c p i n e l a s t i c r e a c t i o r i s i s s t i l l u n d e r d i s c u s s i o n /5-7/. Lept.on- i n d u c e d r e a c t i u n s h a v c t h e a d v a n t a g e t h a t t h e weak o r e l e c t r o m a g n e t i c i n t e r - a c t l o r 1 o f t h e p r o j e c t i l e w i t h t h e t a r g e t r a n b c t r e a t e d i n f i r s t o r d e r . F-or ~ n s t . a n c e , t t i e d o u b l e d i f f e r e n t i a l c r o s s s c c t . i o n f o r i r l c l u s i v e e l e c t r o r i s c o t . t e r i n g

i s p r o p o r t i o n a l t o t h e s l r u c t u r e ( o r r e s p o n s e ) f u n c t i o n o f t h e t a r g e t

As an example c o n s i d e r l o n g i t u d i n a l e x c i t a t i o n o f

n u c l e u s where t h e e x c i - t a t i o n o p e r a t o r i s g i v e n by

Here&,< a r e t.he energy- and momentum t r a n s f e r t o t h e n u c l e u s , e i i s t h e c h a r g e o f t h e i ' t h n u c l e o n and t h e sum i n e q . ( 9 ) r u n s over a l l unobserved f i n a l s t a t e s I n > w i t h e n e r g y E . A f t e r b e i n g s t r u c k by t h e v i r t u a l photon, n u c l e o n // i p r o p a g a t e s t h r o u g h n t h e n u c l e u s d i s t r i b u t i n g i t s energy and momen- tum g a i n by subsequent c o l l i s i o n s w i t h o t h e r n u c l e o n s . I t i s t h e aim o f t h e p r e s e n t work t o d e s c r i b e t h i s p a r t o f t h e r e a c t i o n by a G l a u b e r - t y p e m u l t i p l e s c a t t e r i n g f o r m a l i s m . As compared t o i n c l u s i v e r e a c t i o n s i n d u c e d b y s t r o n g l y i n t e r a c t i n g p a r t i c l e s we n o t e t h e f o l l o w i n g d i f f e r e n c e s and c o m p l i c a t i o n s : F i r s t , t h e r e a c t i o n s t a r t s i n s i d e t h e n u c l e u s so t h a t o f f - s h e l l i n f o r m a t i o n about t h e n u c l e o n - n u c l e o n i n t e r a c t i o n i s needed. Second, i m p o r t a n t c o n s t r a i n t s from v a r i o u s sum r u l e s have t o be obeyed. F o r example, t h e non-energy-weighted sum r u l e

c o u n t s t h e numbers o f s c a t t e r e r s i r r e s p e c t i v e o f t h e i r i n t e r a c t i o n .

The s t a r t i n g p o i n t i s t h e time-dependent f o r m u l a t . i o n o f i n c l u s i v e s c a t - t e r i n g /8/

where t h e " c h a r a c t e r i s t i c f u n c t i o n " $ ( t ) i s t h e g e n e r a t i n g f u n c t i o n o f t h e moments

. . * I

and H = T + V t h e n u c l e a r tiamilt~cm~an. W i t h t h e e x p l i c i t form ( 1 0 ) o f t h e e x c i t a t i o n o p e r a t o r t h e p r o d u c t Q W c o n t a i n s an i n c o h e r e n t ( i = j ) and a co- h e r e n t p a r t ( i f j ) . The l a t t e r one c a n be n e g l e c t e d a t h i g h momentum t r a n s - f e r . Remembering t h a t e x p ( i 4 . l ) i s a momentum s h i f t o p e r a t o r one can w r i t e

Here GOi+, i s t h e n u c l e a r W a m i l t o n i a n w i t h momentum o p e r a t o r Fi r e p l a c e d

rA 7

by pi+:, ^{i . e .} i t describes n n u c l e u s wherc p a r t i c l e i/ i has absorbed momentum

2

q from t h e v i r t u a l photon. The " l o c a l Fermlgns" model o f r e f . / 3 / i s d i r e c t l y o b t a i n e d from e q . ( 1 4 ) b y t r e a t i n g t h e H a m i l t o n o p e r a t o r s as c l a s s i c a l , Winner t r a n s f o r m e d q u a n t i t i e s . tiowever, t h e e f f e c t s due t o t h e n u c l e a r i n t e r a c t i o n

CI

V car1 be made more e x p l i c i t by u s i n g

C6-6 J O U R N A L DE PHYSIQUE

Eq. ( 1 4 ) t t i e n becomes f o r

l o c a l l s o s p i r i - i r i d c p c : r ~ d c r ~ t p o t . c r ~ t . i a l V

Z V ( $ ; - ? ; )

w h ~ c h 1 s s t 1 1 1 an exact r e s u l t . Here P,(I , p ) denotes Lhe Wlgner t r a n s 1 o r m o f t h e A-body d e n s l t y m a t r l x . Clslng t h e semiclassical a p p ~ o x ~ r n a t ~ o n ( 2 ) arid n e g l e c t ~ n g p c i t e n t l a l d e r ~ v a t ~ v e s t h e l a s t f a c t o r I n e q . ( 1 5 ) car1 h?

w r l t t e n as

where t h e g e n e r a l l r e d p r o f i l e f u n c t i o n

? = e x p ( i X > - 1 w i t h

Expanding t q . ( 1 6 ) i n powers o f ' r * q e n e r n t e s a f i n i t e m u l t l p l e s c a t t e r i n g s e r l e s f o r t h e c h a r a c t c r i s t ~ c f u n r t l o n as shown i n riq.1.

I 1 1 - E v a l u a t i o n o f Che M u l t i p l e S c a t - t e r i n g S e r i e s

At v e r y h i q h momsntum t r a n s f e r t h e n u c l e a r momenta i n eq. ( 1 7 ) can he ncq- l c c t e d ( f i x e d s c a L t e r e r a p p r o x i m a t i o n ) . T h i s has t h e udvar~tayle th a t f o r l a r g e i n t e r a c t i o n t i m e s t h e u s u a l p r o f i l e f u n c t i o n ( 6 ) 1s o b t a i n e d . Another i m p o r t a n t feature: ~ n a p p l y ~ n g G l a u h e r ' s t h e o r y t o t ~ a d r o r i - r ~ u c ~ l e u r ; s c a t t c r i n q i s t h e e l i m i n a t i o n o f Lhe p o t e n t i a l by u s i n q p r o f i l e furict.ioris o b t a i n e d from t h e e l e m e r ~ t : ~ r y hac1ror1-nucleon s c : a t t o r l r ~ g amplitude. T h i s l e a d s t o a d e s c r i p t i o n o f t h e s c a t t e r i n q prc)ress whicti i s f a r s u p e r l o r 1.0 t h e o r i g i r i a l one based or1 t h e r ~ o t i o r ~ o f potcint-

a r ~ d t.he e i k o n a l a p p r o x ~ r n a t i o n . 1r1 t h o somc s p i r i t I pl.opost: t o rr:plec:e e q . ( 1 7 ) by a p a r a m e l . r i 7 a t i o r i whic:t~

n c c o t r r ~ t s f o r t h e e x p e c t e t i .short- and lorig-t.lme b e h a v i o u r , v i 7 .

2 2

Here f ( t ) = e x p ( - t / t ) qovcrris t l i e ; ~ p p r o a c h t o t h e asympl.otlc form and t h e sccurid t e r m d e s r r y b c s i n a rough rnarirlcr exchanqe e f f e c t s i n i n c l i r s i v r : r e a c : t i o r ~ s . l t i e l i n e a r t-df:pr:ritJeric:e o f T' f o r s m a l l I. l e a d s 1.0 {.he observed s h i f t o f t h e q u a s i e l a s t i c peak /[1,9/.

To e x h i t l l t most r l e a r l y t h e : i t r u r t t r ~ . ( - ' o i ' t h e m u l t . i p l c s c n t t c i r i r i g s e r i e s , I use t h e c o r ~ v c n t i o r i a l Gauss

on poromct r i / a t i n n o f t h e e l e m e n t a r y p r o f i l c

A 2 2

f u r i c t i o r i / l o - 1 2 / and a s l m p l c p r o c l t ~ c t tu;jvc f u n c t l o n r( f i ' b ) - ' / Z e X p ( - r /2h )

1 - 1 J

f o r t h e q r o o n d st.i~l.c!. I hc

z ; ~ t i o l i f o r h a r m o n i c - 0 x 1 1 l a t o r s h e l l - m o d e l

wave f-i~nctlor?:; i s q l v e r l I n t t i ~ A(:lxlr~d

r 1 q . l : I n c l u s i v e e l e c t r o n s c a t t e r i r i r j from n u c l e i and m t ~ l t l p l e s c a t t e r i n g expanslori. I n t h e upper p a r t , e q s . ( l 2 ) and ( 1 3 ) :ire d e p l c t e d g r a - p h i c a l l y : c u t t i n g t h e diagram i s e q u i v a l c r i t t o L a k i r ~ g t h e i m a q i n ~ r y p a r t o f t h e c t ) a r a c t . c r i s t i c f u n c t i o n , t h e l n c o h e r e n L p a r t o f w h i c h

IS

i n d i c a t e d by t t i c two p t i o t o n l i n e s c n d i n q on t h e same n u c l e o n .

I n t h e l o w e r p a r t t h e m u l t l p l c s c a t t e r i n g e x p a n s i o n o f e q s . ( l S j

and ( 1 6 ) i s shor.rn: t h e f i r s t . t c r m c o r r e s p o n d s Lo t h e Fermigas ap-

p r o x l m a t i o n , t h e second d t ! s c r l b e s t h e i n t e r a c t i o n o f t h e s t r u c k

n t l c l e o n # i w i t h n u r l e o r i # j e t c .

C6-8 J O U R N A L DE PHYSIQUE

The r e s u l t f o r t h e s t r u c t u r e f u n c t i o n 1s

i . e . a s u p e r p o s i t i o n o f Gaussians w i t h d i f f e r e n t w i d t h s

and peak p o s ~ t l o n s = q 2 /2M + m4. The qeomet.rjca1 p a r a m e t e r s

a r e f a m i l a r from s i m i l a r work i n h a d r o n - n u c l e u s s c a t t e r i n q /13/. The sum r u l e c o n s t r a i n t s a r e checked by c a l c u l a t i n g

A t h i g h $ t h e e r r o r f u n c t i o n t e n d s t o one. U s i n q t h e i n t e g r a l r e p r e s e n t a - t i o n bp,r ~ p ( . ( + y ) ( a + t ~ c 4 - * 2 ) ' t h e summation can t h e n be p e r - formed orld one f i n d s m -Z i n accordance w i L h eq. ( 1 1 ) . Thus, t h e non- e n e r g y - w e l g h t e d sum r u y e i s p r e s e r v e d i n d e p e n d e n t o f t h e (complex) i n t e r a c - t i o n parameters. Note t h a t t h i s o n l y h o l d s i f t h e f u l l m u l t i p l e s c a t t e r i n g s e r i e s i s summed up. S i m i l a r l y , one f l n d s f o r a l l <

w i t h J1 = ( A - l ) o ( . F i n a l l y , t h e second moment t a k e s a t h i g h < t h e g e n e r a l form /8/

2 2

where i n t h e p r e s e n t model < T > = 3/4Mb and A 2 ⁼ (A-1) Re ( x ( l + y i / ( l + y ) / 2 ) / t ~ . A c c o r d i n g t o e q s . ( 2 4 ) , ( 2 5 ) t h e e f f e c t o f t h e n u c l e a r i n t e r a c t i o n i s a s h i f t and a b r o a d e n i r i y o f t h e q u a s ~ e l a s t i c peak w i t h o u t c:har~ying t h e t o t a l e x c i t a t i o n strengt.h. The b r o a d e n i n g i s m a i n l y due t o t h e f i n i t e l i f e t i m e o f l p - l h s t a t e s w h i c h decay i n t o more c o m p l i c a t e d s t a t e s /14/. T h i s can

be seen most e a s i l y i n t h e p r e s e n t f o r m u l a t . i o n i n t h e l l m ~ t y << 1 ( i n t e r a c t i o n range s m a l l compared t o mean n u c l e o n d l s t a n c c ) . Then R e x = G /2*b2 = % / 2 ( ~ - 1 ) 2

t o t

is i n v e r s e l y p r o p o r t i o n a l t o t h e mean f r e e p a t h A o r p r o p o r t i o n a l t o t h e i m a g i n a r y p a r t o f t h e o p t i c a l p o t e n t i a l f o r t h e s t r u c k n u c l e o n i n t h e n u c l e a r i n t e r i o r .

F i g . 2 shows a c o m p a r i s o n o f e q . ( 1 9 ) w i t h t h e measured l o n g i t u d i n a l s t r u c t u r e f u n c t i o n f o r "C /15/. R e l a t i v i s t i c c o r r e c t i o n s and f i n i t e n u c l e o n s i z e a r e t a k e n i n t o a c c o u n t by t h e Darwin-Foldy t e r m and a d i p o l e form f a c t o r , r e s p e c t i v e l y . Compared w i t h a Fermi g a s d e s c r i p t i o n t h e a g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a i s much b e t t e r , e s p e c i a l l y a t h l g h e n e r g y t r a n s f e r . C a l - c u l a t i o n s w i t h more r e a l i s t i c n u c l e a r wave f u n c t i o n s a n d form f a c t o r s a r e i n p r o g r e s s .

F i g . 2 : L o n g i t u d i n a l s t r u c t u r e f u n c t i o n f o r "C a t two d i f f e r e n t momentum t r a n s f e r s . The d a s h e d l i n e is t h e F e r m i y a s r e s u l t /16/.

IV - Summary

A G l a u b e r - t y p e m u l t i p l e s c a t t e r i n g s e r i e s f o r i n c l u s i v e r e a c t i o n s h a s b e e n

d e v e l o p e d which s h o u l d b e v a l i d a t high-momentum t r a n s f e r . The u s u a l d i f f i -

c u l t i e s f o r l a r g e e n e r g y l o s s o f t h e p r o j e c t i l e a r e a v o i d e d by u s i n g a t i m e -

d e p e n d e n t f o r m u l a t i o n o f i n c l u s i v e s c a t t e r i n g . The p r i c e t o b e p a i d is t h e

a p p e a r e n c e o f g e n e r a l i z e d , t i m e - d e p e n d e n t p r o f i l e f u n c t i o n s which o n l y f o r

l a r g e i n t e r a c t i o n t i m e s ( i . e . s m a l l e x c i t a t i o n e n e r g i e s ) make c o n t a c t w i t h

t h e u s u a l o n - s h e l l a m p l i t u d e s . I n a s i m p l i f i e d model i t h a s b e e n shown t h a t

t h e t o t a l s t r e n g t h is p r e s e r v e d and t h a t t h e n u c l e a r i n t e r a c t i o n m e r e l y

r e d i s t r i b u t e s t h e e x c i t a t i o n s t r e n g t h . Due t o t h e s e m i c l a s s i c a l a p p r o x i m a -

t i o n and t h e a s s u m p t i o n o f a " f r o z e n " n u c l e u s , o n l y t h e s t r u c k n u c l e o n u n d e r -

g o e s m u l t i p l e s c a t t e r i n g and t h e c a s c a d e o f low-momentum s e c o n d a r y n u c l e o n s

i s n o t c o n s i d e r e d e x p l i c i t l y . I n t h i s r e s p e c t t h e p r e s e n t a p p r o a c h r e s e m b l e s

t h e work o f r e f . / l 4 / w h e r e t h e a b s o r p t i v e p a r t o f t h e o p t i c a l p o t e n t i a l

h a s b e e n t a k e n i n t o a c c o u n t i n a t i m e - i n d e p e n d e n t f o r m u l a t . i o n . The a p p e a l i n g

s i m p 1 i c i t . y o f G l a u b e r t h e o r y , however, may a l l o w f u r t h e r e x t e n s i o n and i m -

p r o v e m e n t ( c o h e r e n t s c a t t e r i n g , h i g h e r - o r d e r c o r r e c t i o n s , r e s o n a n c e p r o d u c -

t i o n , e t c . ) o f t h e p r e s e n t a p p r o a c h t o i n c l u s i v e s c a t t e r i n g .

J O U R N A L DE PHYSIQUE

V - Appendix

4 1Zc 16 I-lere we q i v e t h e q e n e r a l i ~ a t i o n o f eq.(19) f o r p - s h e l l n u c l e i ( He, , 0 ) and longitudinal ( a z l ) arid t r a n s v e r s e e x c i t a t i o n s ( a r 2 : c o n v e c t i o n c u r r e n t , a13 : m a g n e t i z a t i o n c u r r e n t c o r ~ t r i b u t i o n ) . I f exchange m a t r i x c l c m e n t s a r e n e g l e c t e d , t h e c o e f f i c i e r ~ t fiA-l-rn ^{i r I} eq. ( 1 9 ) i s t.o be r e p l a c e d by

\ ^{9"' - 7}

9

^{( ? -} ~ ( W - P ? ? M - ~ ~ I ~ / ~ : ) ~ ~ ( "

i 2 b ( i ! , - w

where ?I = 2(Z-21/32 and g ( 1 9 3 ) = 1 - A , g"' = 1. The new c o e f f i c i e n t s a r e q i v e n b y

I n a d d i t i o n , r e l a t i v i s t i c c o r r e c t i o n s and f i n i t e n u c l e o n s i r e e f f e c t s r e q u i r e t t i c structure f u n c t - i o n s t o be m u l t i p l i e d b y

f z l c t o r C where

( 1 ) 2

F1/2(Q ) a r e t h e Pau.Li-Dirac n u c l e o n form f a c t . o r s f o r p r o t o n and n e u l r o n 2 2

c v a l u a t c d a t Q' = W - d .

V T - References

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/11/ F. Schcck: "Leptons, Hadrons and N u c l e i " , N o r t h H o l l a n d , Amsterdam ( 1 9 8 j ) , p. 10-22

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