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Virtual nuclear excitation corrections to elastic electron scattering at low energies
Volume 33B, number 7
PHYSICS
VIRTUAL NUCLEAR ELASTIC ELECTRON
LETTERS
EXCITATION SCATTERING
7 December 1970
CORRECTIONS TO AT LOW ENERGIES*
G. H. R A W I T S C H E R
University of Connecticut, Storrs, Connecticut 06268, USA Received 23 October 1970
It is suggested that the energy region where virtual nuclear excitation c o r r e c t i o n s to the elastic e l e c tron s c a t t e r i n g c r o s s sections should be looked for experimentally is where the f i r s t diffraction minimum occurs at large angles. F o r Ca, Nd and Pb this energy interval is in the vicinity of 120, 60 and 50 MeV. r e s p e c t i v e l y . F o r the nucleus of Ca the c o r r e c t i o n to the c r o s s section should be in the range of 5 to 15~. T h e p u r p o s e of t h i s n o t e i s to p o i n t out t h a t t h e e f f e c t s of v i r t u a l n u c l e a r e x c i t a t i o n s (VNE) on t h e e l a s t i c e l e c t r o n n u c l e u s s c a t t e r i n g c r o s s s e c t i o n a r e m o r e l i k e l y to b e i d e n t i f i a b l e e x p e r i m e n t a l l y at low e n e r g i e s ( b e t w e e n 100 a n d 15 M e V f o r 4 0 C a , f o r e x a m p l e ) t h a n at t h e h i g h e r e n e r g i e s (250 M e V a n d a b o v e ) . T h e v i r t u a l e x c i t a t i o n of n u c l e a r l e v e l s d u r i n g a 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 is t h e q u a n t u m m e c h a n i c a l d e s c r i p t i o n of n u c l e a r p o l a r i z a t i o n ( i , e . , c h a n g e of the n u c l e a r s h a p e ) w h i c h o c c u r s d u r i n g the i n t e r a c t i o n of t h e i n c i d e n t p r o j e c t i l e w i t h t h e n u c l e o n s in t h e t a r g e t n u c l e u s . T h i s effect for example lowers the energy eigenvalue of a t o m i c electror~$ o r m u o n s w h i c h a r e in b o u n d s t a t e s s u r r o u n d i n g t h e n u c l e u s , a n d it a p p e a r s t o h a v e b e e n m e a s u r e d in t h e c a s e of m u - m e s i c X - r a y s f o r t h e n u c l e i of 2 0 6 p b [1] a n d 142Nd [2,3]. In 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 t h e e f f e c t a p p e a r s not to h a v e b e e n i d e n t i f i e d , a n d a g o o d d e a l of u n c e r t a i n t y e x i s t s r e g a r d i n g a s to w h a t to e x p e c t f o r t h e m a g n i t u d e of t h e e f f e c t d e s p i t e t h e f a c t t h a t a g o o d d e a l of t h e o r e t i c a l w o r k h a s l a t e l y b e e n d o n e on t h i s s u b j e c t [4-6]. T h e r e a s o n why VNE e f f e c t s a r e e x p e c t e d to m a n i f e s t t h e m s e l v e s at low e n e r g i e s i s c o n n e c t e d w i t h t h e p r o p e r t i e s of t h e d i f f r a c t i o n m i n i m a of the scattering cross section. These are the a n g u l a r r e g i o n s w h e r e t h e i m a g i n a r y p a r t of t h e s c a t t e r i n g amplitude is l a r g e r than the r e a l part, n e a r t h e z e r o ' s of t h e r e a l p a r t . In t h e r e g i o n s of t h e d i f f r a c t i o n m i n i m a VNE e f f e c t s c a u s e r e l a t i v e l y l a r g e p e r c e n t c o r r e c t i o n s to the c r o s s s e c t i o n f o r two r e a s o n s . O n e , t h e c r o s s s e c t i o n i s u s u a l l y s m a l l , a n d h e n c e s e n s i t i v e to c o r r e c * R e s e a r c h supported by Grant GP-9227 of the National Science Foundation.
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Fig. 1. Electron s c a t t e r i n g from the nucleus of 142Nd. All c r o s s sections are in units of a theoretical static c r o s s section based on the F e r m i charge distribution e = 5.75 fm and t = 2.38 fm derived f r o m m u - m e s i c X-ray e n e r g i e s c o r r e c t e d for VNE effects [3]. The c i r c l e s with e r r o r flags r e p r e s e n t Yale's experimental r e s u l t s . The solid curves r e p r e s e n t c r o s s sections c o r r e c t e d by the VNE effects as r e p r e s e n t e d by the monopole model with the total inelastic electron c r o s s section, Cqn, taken equal to 0.60 fm2, which is about five t i m e s l a r g e r than a theoretical estimate, as d i s c u s s e d in the text. The dashed curves show the r e s u l t s after the real part of the virtual nuclear monopole c o r r e c t i o n to the partial wave phase shifts is set artificially to z e r o . 445
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Fig. 2. P e r c e n t c o r r e c t i o n to the elastic c r o s s section for the nucleus of Ca, for various incident electron e n e r g i e s . The solid lines use a value for the shape p a r a m e t e r Zin for the monopole t r a n s i t i o n charge density of 2 fm, the dashed lines use 0.5 fin. The cut off radius R, defined in ref.[9], is 8.5 fm in the f i r s t case and 8.3 fm in the second. The value of ain is taken equal to 0.027 fm 2. The static charge d i s t r i b u tion is of a F e r m i f o r m , with p a r a m e t e r s c = 3.673 and t = 2 . 7 0 4 f r o . t i o n s . T w o , w h e n t h e VNE c o r r e c t i o n i s s m a l l , w h i c h i s t h e c a s e a t low i n c i d e n t e n e r g i e s , t h e c o r r e c t i o n to t h e c r o s s s e c t i o n a r i s e s f r o m t h e interference between the static and the "dynamic" c o r r e c t i o n to t h e s c a t t e r i n g a m p l i t u d e s . T h e VNE c o r r e c t i o n to t h e a m p l i t u d e i s m a i n l y i m a g inary, and hence the effect is large where the static amplitude is also mainly imaginary, which i s in t h e d i f f r a c t i o n r e g i o n . At h i g h e n e r g i e s t h e d i f f r a c t i o n r e g i o n s a r e narrow and the cross section has deep minima. A t low e n e r g i e s t h e c r o s s s e c t i o n m a y s h o w o n l y a slight shoulder, but the diffraction region extends over a large angular interval and the imag446
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i n a r y p a r t of t h e s c a t t e r i n g a m p l i t u d e i s q u i t e large. F o r e x a m p l e , f o r 60 M e V e l e c t r o n s i n c i d e n t oi1 Nd, t h e z e r o in t h e r e a l s c a t t e r i n g a m p l i t u d e o c c u r s a t 135 ° a n d t h e d i f f r a c t i o n r e g i o n e x t e n d s f r o m 90 ° to 170 °. It i s p r e c i s e l y f o r Nd t h a t a n a n o m a l y in t h e 40 to 60 M e V s c a t t e r i n g c r o s s s e c t i o n h a s b e e n d e t e c t e d [7]. 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 i s up to 15% l a r g e r t h a n t h e t h e o r e t i c a l v a l u e b a s e d on m u - m e s i c X - r a y c h a r g e d i s t r i b u t i o n p a r a m e t e r s [3] ( d e r i v e d a f t e r VNE e f f e c t s a r e a l l o w e d for). T h e r a t i o of t h e e x p e r i m e n t a l v a l u e s [7] to t h e t h e o r e t i c a l v a l u e s i s i l l u s t r a t e d in fig. 1 b y t h e p o i n t s w i t h t h e e r r o r f l a g s . By c o n t r a s t , a t h i g h e n e r g i e s n o t o n l y d o e s the diffraction region being very narrow weigh l i t t l e in t h e d e t e r m i n a t i o n of c h a r g e d i s t r i b u t i o n p a r a m e t e r s b u t a l s o t h e r e a r e r e a s o n s to b e l i e v e [8] t h a t t h e c o r r e c t i o n to t h e f o r m f a c t o r in t h e r e g i o n s o u t s i d e of t h e d i f f r a c t i o n m i n i m a a r e e n e r g y i n d e p e n d e n t (only m o m e n t u m t r a n s f e r d e p e n d e n t ) a n d h e n c e will b e h a r d to d i s t i n g u i s h from equivalent static charge distribution modifications. For the above mentioned reasons the energy region which is probably well suited to observe t h e VNE c o r r e c t i o n i s t h a t r e g i o n f o r w h i c h t h e first diffraction minimum appears at backward a n g l e s . F o r C a t h i s r e g i o n i s b e t w e e n 100 a n d 150 M e V , f o r Nd a n d P b it i s b e t w e e n 40 a n d 100 MeV. In t h e s e r e g i o n s t h e s t a t i c c h a r g e d i s t r i b u t i o n f i t t e d to t h e d a t a s h o u l d s h o w a n energy dependence. Calculations based on a m o n o p o l e m o d e l d i s c u s s e d b e l o w s h o w t h a t low v a l u e s of Z ( l i k e 20) a r e to b e p r e f e r r e d o v e r h i g h v a l u e s ( l i k e 82) b e c a u s e in t h e d i f f r a c t i o n r e g i o n t h e p e r c e n t c o r r e c t i o n to t h e c r o s s s e c t i o n i s l a r g e r in t h e f o r m e r c a s e . Q u a n t i t a t i v e s u p p o r t of t h e a b o v e - m e n t i o n e d ideas is based on a simple nuclear monopole exc i t a t i o n m o d e l , d i s c u s s e d p r e v i o u s l y [9]. T h e m o d e l c o n s i s t s in s o l v i n g e x a c t l y t h e D i r a e coupled channel equations which describe the e x c i t a t i o n of a n u c l e a r m o n o p o l e s t a t e . T h e e x c i t a t i o n e n e r g y of t h e n u c l e a r s t a t e i s s e t e q u a l to z e r o . T h e c h a r g e d e n s i t y w h i c h in t h i s m o d e l i s r e s p o n s i b l e f o r t h e t r a n s i t i o n f r o m t h e g r o u n d to the excited state is normalized by the requirement that the corresponding inelastic electron cross s e c t i o n i s e q u a l to the e x p e r i m e n t a l t o t a l i n e l a s t i c electron nucleus cross section. A l t h o u g h q u i t e c r u d e , t h e m a i n a d v a n t a g e of t h i s m o d e l i s t h a t it p r o v i d e s a m e c h a n i s m f o r making the elastic partial wave phase shifts c o m p l e x (due to t h e p r e s e n c e of a n o p e n c h a n n e l ) . F u r t h e r , in t h i s m o d e l it i s p o s s i b l e to c a l c u l a t e
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t h e c o r r e c t i o n to t h e e l a s t i c c r o s s s e c t i o n in a " d i s t o r t e d w a v e m a n n e r " w h i c h is q u i t e s i m p l e a n d d o e s not u s e f i r s t o r s e c o n d o r d e r B o r n a p p r o x i m a t i o n , w h i c h i s a n i m p o r t a n t p o i n t 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 . A n o b v i o u s d i s a d v a n t a g e of t h e m o d e l i s t h a t it u s e s a m o n o pole excitation rather than a physically more acceptable dipole or quadrupole excitation for the i n e l a s t i c l e v e l . H o w e v e r , s i n c e t h e s i g n of t h e i m a g i n a r y p a r t s of t h e c o r r e c t i o n to t h e p h a s e s h i f t i s ( d e f i n i t e n e g a t i v e ) i n d e p e n d e n t of t h e m u l t i p o l a r i t y of the e x c i t a t i o n , a n d s i n c e , to l o w e s t o r d e r in t h e t r a n s i t i o n p o t e n t i a l , t h e c o n t r i b u t i o n to t h e i m a g i n a r y p a r t of t h e p h a s e s h i f t i s a d d i t i v e f o r a l l e x c i t a t i o n s , it i s e x p e c t e d t h a t a t t h e m a g n i t u d e of t h e c o r r e c t i o n to t h e c r o s s section is given reasonably well by this model for angles near the diffraction minima. For other a n g l e s t h e r e a l p a r t of t h e c o r r e c t i o n to t h e p h a s e s h i f t s b e c o m e s i m p o r t a n t a n d t h e m o d e l i s n o t to b e t r u s t e d . T h e s i g n of t h e c o r r e c t i o n to t h e c r o s s s e c t i o n in t h e d i f f r a c t i o n m i n i m a i s n o t p r e d i c t e d b y t h i s m o d e l , s i n c e it d e p e n d s o n t h e c h o i c e of t h e r a d i a l d e p e n d e n c e of the t r a n s i t i o n c h a r g e density. These various points are illustrated by t h e c u r v e s i l l u s t r a t e d in fig. 1. T h e two v a l u e s of Zin l i s t e d i n fig. 1 d e s c r i b e [9] two q u i t e d i f f e r e n t r a d i a l d e p e n d e n c e s of t h e t r a n s i t i o n c h a r g e d e n s i t y . T h e d o t t e d l i n e s a r e o b t a i n e d if t h e r e a l p a r t of t h e p h a s e s h i f t c o r r e c t i o n i s a r t i f i c a l l y s e t e q u a l to z e r o . T h e e f f e c t on t h e c o r r e c t i o n to t h e c r o s s s e c t i o n i s s e e n to b e small. F o r C a , t h e e f f e c t of a v i r t u a l d i p o l e t r a n s i t i o n h a s b e e n c a l c u l a t e d b y O n l e y [5] a t v a r i o u s " h i g h " e n e r g i e s , a n d c o m p a r i s o n w i t h the m o n o p o l e m o d e l s h o w s q u a l i t a t i v e a g r e e m e n t f o r t h e two c a l c u l a t i o n s f o r a v a l u e of t h e m o n o p o l e s h a p e p a r a m e t e r Z i n of 2 f m . It i s i n t e r e s t i n g t h a t , w h i l e the two calculations give comparable magnitudes f o r t h e c o r r e c t i o n to t h e e l a s t i c c r o s s s e c t i o n i n t h e r e g i o n of t h e s e c o n d d i f f r a c t i o n m i n i m u m , in t h e f i r s t m i n i m u m t h e d i p o l e m o d e l g i v e s an e f f e c t a b o u t t h r e e t i m e s l a r g e r . T h i s is n o t s u r prising since the dipole transition charge density has a long tail, while the monopole one vanishes o u t s i d e t h e n u c l e a r r e g i o n . Fig. 2 i l l u s t r a t e s t h e p e r c e n t c o r r e c t i o n f o r C a in t h e i n t e r e s t i n g " l o w " e n e r g y r a n g e f o r two c h o i c e s of Zin. T h e v a l u e of t h e t o t a l i n e l a s t i c c r o s s s e c t i o n i s t a k e n e q u a l to Crin = 0.027 f m 2. T h i s v a l u e i s b a s e d o n a n i n d e p e n d e n t p a r t i c l e s h e l l m o d e l of t h e n u c l e a r e x cited states, which assumes plane waves for the incident and outgoing electrons, but which does n o t m a k e u s e of c l o s u r e in t h e s u m o v e r n u c l e a r s t a t e s [10]. T h e d i p o l e t r a n s i t i o n s g i v e t h e l a r g e s t
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c o n t r i b u t i o n s to t h e c r o s s s e c t i o n , a n d in o r d e r to i n c l u d e e f f e c t s of c e n t e r of m a s s m o t i o n , t h e e f f e c t i v e c h a r g e of t h e p r o t o n s a n d n e u t r o n s i s t a k e n r e s p e c t i v e l y e q u a l to ALIA a n d - Z i " A *. In v i e w of t h e d i s c u s s i o n a b o v e r e g a r d i n g t h e c o m p a r i s o n w i t h O n l e y ' s c a l c u l a t i o n , it i s b e l i e v e d that the curves for Zin = 2fm are more realistic t h a n f o r Z i n = 0.5 f m in t h e c a s e of C a , a n d t h a t t h e s i z e of t h e c o r r e c t i o n s h o w n v e r y l i k e l y r e presents an underestimate. T h e c a s e of Nd, w h i c h a c t u a l l y p r o v i d e d t h e m o t i v a t i o n f o r t h e p r e s e n t n o t e [3,7], a p p e a r s to b e d i f f e r e n t f r o m t h a t of Ca. T h e v a l u e of Zin n e e d s to l i e n e a r 0.5 fro, r a t h e r t h a n 2 f m , if t h e a n o m a l y i s to b e i n t e r p r e t e d in t e r m s of VNE c o r r e c t i o n s . F u r t h e r , t h e v a l u e of Crin r e q u i r e d to produce qualitative agreement with experiment ( s o l i d l i n e in fig. 1) h a s t h e v a l u e of 0.6 f m 2, w h i c h i s a b o u t 4.5 t i m e s l a r g e r t h a n t h e v a l u e of 0.133 fro2 e x p e c t e d on t h e b a s i s of t h e s h e l l m o d e l calculation. These discrepancies could quite well b e due to t h e l a r g e c o l l e c t i v e n a t u r e of t h e f i r s t low l y i n g e x c i t e d s t a t e s in Nd, of m u l t i p o l a r i t y 2 + a n d 3-. T h i s b e l i e f i s s t r e n g t h e n e d b y t h e r e s u l t t h a t t h e s c a t t e r i n g a n o m a l y i s f o u n d to v a r y s t r o n g l y f r o m o n e i s o t o p e to t h e n e x t [11], a r e s u l t w h i c h w o u l d not b e e x p e c t e d if t h e g i a n t dipole excitations were mainly responsible for t h e VNE e f f e c t s , s i n c e t h e e x c i t a t i o n of d i p o l e s t a t e s s h o u l d b e r e l a t i v e l y i n d e p e n d e n t of t h e p r o p e r t i e s of t h e low l y i n g s t a t e s , F i n a l l y , it i s i n t e r e s t i n g to n o t e t h a t " l o w " e n e r g y e l e c t r o n s c a t t e r i n g o n i s o t o p e s of T i s h o w a n o m a l i e s of t h e t y p e a n d m a g n i t u d e d i s c u s s e d h e r e [12]. It is a p l e a s u r e to a c k n o w l e d g e a v e r y s t i m u l a t ing contact with the Yale group, particularly Professor Bockelman and Messers. Madsen and Cardman, as well as several very useful convers a t i o n s w i t h P r o f e s s o r S. W a l l . M a n y of t h e c o m p u t a t i o n s h a v e b e e n c a r r i e d o u t at t h e U n i v e r s i t y of C o n n e c t i c u t C o m p u t i n g C e n t e r , s u p p o r t e d b y NSF Grant GT-9. * The r e s u l t given for Ca in ref. [10] is l a r g e r by a factor of two than the value Grin = 0.027 fm 2) used here because the effective charge in ref. [10] is taken as 1 and 0 for protons and neutrons, r e s p e c t i v e l y .
Reference s [1] H . L . A n d e r s o n et al., Phys. Rev. L e t t e r s 22 (1969) 221. [2] T. T. Bardin et al., Phys. Rev. L e t t e r s 16 (1966) 718; E . R . M a c a g n o et al., Phys. Rev. Cl (1970) 1202. 447
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[3] Min-Yi Chen. P h y s . R e v . C1 (1970) 1167, and private communication to the Yale Group, quoted in ref.[7]. [4] C. "I oepffer and W. Greiner, Ann. P h y s . N. Y. 47 (1968) 285; Phys. Rev.186 (1969) 1044; C. Toepffer and D. Drechsel, Phys. Rev. L e t t e r s 24 (1970) 1131. [5] D.S.Onley, Nucl. Phys. A l l 8 (1968) 436. [6] A. Bottino, G. Ciocchetti and A. Molinari, Nucl. Phys.89 (1966) 192; E.Kujawski, Phys. L e t t e r s 32B (1970) 75; W. D. Brown and E. Kujawski, P r e p r i n t , University of Maryland; 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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[7] D. W. Madsen et al.. Phys. Rev. L e t t e r s 23 (1969) 1122. [8] G. H. Rawitscher, unpublished. [9] G . H . R a w i t s c h e r . Phys. Rev.151 (1966) 846. [10] G. H. Rawitscher, Bull. Am. Phys. Soc. 15 (1970) 595 and to be published. [11] D.W.Madsen, L.S. Cardman, G . R . L e g g a n d C . K . Bockelman, to be submitted for publication. [12] E . R o m b e r g et al., Bull.Am. Phys. Soc.14 (1969) 1242; E. Romberg, Pd. D. T h e s i s , University of Maryland (1970); N. S. Wail, to be published.
PHYSICS
VIRTUAL NUCLEAR ELASTIC ELECTRON
LETTERS
EXCITATION SCATTERING
7 December 1970
CORRECTIONS TO AT LOW ENERGIES*
G. H. R A W I T S C H E R
University of Connecticut, Storrs, Connecticut 06268, USA Received 23 October 1970
It is suggested that the energy region where virtual nuclear excitation c o r r e c t i o n s to the elastic e l e c tron s c a t t e r i n g c r o s s sections should be looked for experimentally is where the f i r s t diffraction minimum occurs at large angles. F o r Ca, Nd and Pb this energy interval is in the vicinity of 120, 60 and 50 MeV. r e s p e c t i v e l y . F o r the nucleus of Ca the c o r r e c t i o n to the c r o s s section should be in the range of 5 to 15~. T h e p u r p o s e of t h i s n o t e i s to p o i n t out t h a t t h e e f f e c t s of v i r t u a l n u c l e a r e x c i t a t i o n s (VNE) on t h e e l a s t i c e l e c t r o n n u c l e u s s c a t t e r i n g c r o s s s e c t i o n a r e m o r e l i k e l y to b e i d e n t i f i a b l e e x p e r i m e n t a l l y at low e n e r g i e s ( b e t w e e n 100 a n d 15 M e V f o r 4 0 C a , f o r e x a m p l e ) t h a n at t h e h i g h e r e n e r g i e s (250 M e V a n d a b o v e ) . T h e v i r t u a l e x c i t a t i o n of n u c l e a r l e v e l s d u r i n g a 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 is t h e q u a n t u m m e c h a n i c a l d e s c r i p t i o n of n u c l e a r p o l a r i z a t i o n ( i , e . , c h a n g e of the n u c l e a r s h a p e ) w h i c h o c c u r s d u r i n g the i n t e r a c t i o n of t h e i n c i d e n t p r o j e c t i l e w i t h t h e n u c l e o n s in t h e t a r g e t n u c l e u s . T h i s effect for example lowers the energy eigenvalue of a t o m i c electror~$ o r m u o n s w h i c h a r e in b o u n d s t a t e s s u r r o u n d i n g t h e n u c l e u s , a n d it a p p e a r s t o h a v e b e e n m e a s u r e d in t h e c a s e of m u - m e s i c X - r a y s f o r t h e n u c l e i of 2 0 6 p b [1] a n d 142Nd [2,3]. In 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 t h e e f f e c t a p p e a r s not to h a v e b e e n i d e n t i f i e d , a n d a g o o d d e a l of u n c e r t a i n t y e x i s t s r e g a r d i n g a s to w h a t to e x p e c t f o r t h e m a g n i t u d e of t h e e f f e c t d e s p i t e t h e f a c t t h a t a g o o d d e a l of t h e o r e t i c a l w o r k h a s l a t e l y b e e n d o n e on t h i s s u b j e c t [4-6]. T h e r e a s o n why VNE e f f e c t s a r e e x p e c t e d to m a n i f e s t t h e m s e l v e s at low e n e r g i e s i s c o n n e c t e d w i t h t h e p r o p e r t i e s of t h e d i f f r a c t i o n m i n i m a of the scattering cross section. These are the a n g u l a r r e g i o n s w h e r e t h e i m a g i n a r y p a r t of t h e s c a t t e r i n g amplitude is l a r g e r than the r e a l part, n e a r t h e z e r o ' s of t h e r e a l p a r t . In t h e r e g i o n s of t h e d i f f r a c t i o n m i n i m a VNE e f f e c t s c a u s e r e l a t i v e l y l a r g e p e r c e n t c o r r e c t i o n s to the c r o s s s e c t i o n f o r two r e a s o n s . O n e , t h e c r o s s s e c t i o n i s u s u a l l y s m a l l , a n d h e n c e s e n s i t i v e to c o r r e c * R e s e a r c h supported by Grant GP-9227 of the National Science Foundation.
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Fig. 1. Electron s c a t t e r i n g from the nucleus of 142Nd. All c r o s s sections are in units of a theoretical static c r o s s section based on the F e r m i charge distribution e = 5.75 fm and t = 2.38 fm derived f r o m m u - m e s i c X-ray e n e r g i e s c o r r e c t e d for VNE effects [3]. The c i r c l e s with e r r o r flags r e p r e s e n t Yale's experimental r e s u l t s . The solid curves r e p r e s e n t c r o s s sections c o r r e c t e d by the VNE effects as r e p r e s e n t e d by the monopole model with the total inelastic electron c r o s s section, Cqn, taken equal to 0.60 fm2, which is about five t i m e s l a r g e r than a theoretical estimate, as d i s c u s s e d in the text. The dashed curves show the r e s u l t s after the real part of the virtual nuclear monopole c o r r e c t i o n to the partial wave phase shifts is set artificially to z e r o . 445
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Fig. 2. P e r c e n t c o r r e c t i o n to the elastic c r o s s section for the nucleus of Ca, for various incident electron e n e r g i e s . The solid lines use a value for the shape p a r a m e t e r Zin for the monopole t r a n s i t i o n charge density of 2 fm, the dashed lines use 0.5 fin. The cut off radius R, defined in ref.[9], is 8.5 fm in the f i r s t case and 8.3 fm in the second. The value of ain is taken equal to 0.027 fm 2. The static charge d i s t r i b u tion is of a F e r m i f o r m , with p a r a m e t e r s c = 3.673 and t = 2 . 7 0 4 f r o . t i o n s . T w o , w h e n t h e VNE c o r r e c t i o n i s s m a l l , w h i c h i s t h e c a s e a t low i n c i d e n t e n e r g i e s , t h e c o r r e c t i o n to t h e c r o s s s e c t i o n a r i s e s f r o m t h e interference between the static and the "dynamic" c o r r e c t i o n to t h e s c a t t e r i n g a m p l i t u d e s . T h e VNE c o r r e c t i o n to t h e a m p l i t u d e i s m a i n l y i m a g inary, and hence the effect is large where the static amplitude is also mainly imaginary, which i s in t h e d i f f r a c t i o n r e g i o n . At h i g h e n e r g i e s t h e d i f f r a c t i o n r e g i o n s a r e narrow and the cross section has deep minima. A t low e n e r g i e s t h e c r o s s s e c t i o n m a y s h o w o n l y a slight shoulder, but the diffraction region extends over a large angular interval and the imag446
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i n a r y p a r t of t h e s c a t t e r i n g a m p l i t u d e i s q u i t e large. F o r e x a m p l e , f o r 60 M e V e l e c t r o n s i n c i d e n t oi1 Nd, t h e z e r o in t h e r e a l s c a t t e r i n g a m p l i t u d e o c c u r s a t 135 ° a n d t h e d i f f r a c t i o n r e g i o n e x t e n d s f r o m 90 ° to 170 °. It i s p r e c i s e l y f o r Nd t h a t a n a n o m a l y in t h e 40 to 60 M e V s c a t t e r i n g c r o s s s e c t i o n h a s b e e n d e t e c t e d [7]. 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 i s up to 15% l a r g e r t h a n t h e t h e o r e t i c a l v a l u e b a s e d on m u - m e s i c X - r a y c h a r g e d i s t r i b u t i o n p a r a m e t e r s [3] ( d e r i v e d a f t e r VNE e f f e c t s a r e a l l o w e d for). T h e r a t i o of t h e e x p e r i m e n t a l v a l u e s [7] to t h e t h e o r e t i c a l v a l u e s i s i l l u s t r a t e d in fig. 1 b y t h e p o i n t s w i t h t h e e r r o r f l a g s . By c o n t r a s t , a t h i g h e n e r g i e s n o t o n l y d o e s the diffraction region being very narrow weigh l i t t l e in t h e d e t e r m i n a t i o n of c h a r g e d i s t r i b u t i o n p a r a m e t e r s b u t a l s o t h e r e a r e r e a s o n s to b e l i e v e [8] t h a t t h e c o r r e c t i o n to t h e f o r m f a c t o r in t h e r e g i o n s o u t s i d e of t h e d i f f r a c t i o n m i n i m a a r e e n e r g y i n d e p e n d e n t (only m o m e n t u m t r a n s f e r d e p e n d e n t ) a n d h e n c e will b e h a r d to d i s t i n g u i s h from equivalent static charge distribution modifications. For the above mentioned reasons the energy region which is probably well suited to observe t h e VNE c o r r e c t i o n i s t h a t r e g i o n f o r w h i c h t h e first diffraction minimum appears at backward a n g l e s . F o r C a t h i s r e g i o n i s b e t w e e n 100 a n d 150 M e V , f o r Nd a n d P b it i s b e t w e e n 40 a n d 100 MeV. In t h e s e r e g i o n s t h e s t a t i c c h a r g e d i s t r i b u t i o n f i t t e d to t h e d a t a s h o u l d s h o w a n energy dependence. Calculations based on a m o n o p o l e m o d e l d i s c u s s e d b e l o w s h o w t h a t low v a l u e s of Z ( l i k e 20) a r e to b e p r e f e r r e d o v e r h i g h v a l u e s ( l i k e 82) b e c a u s e in t h e d i f f r a c t i o n r e g i o n t h e p e r c e n t c o r r e c t i o n to t h e c r o s s s e c t i o n i s l a r g e r in t h e f o r m e r c a s e . Q u a n t i t a t i v e s u p p o r t of t h e a b o v e - m e n t i o n e d ideas is based on a simple nuclear monopole exc i t a t i o n m o d e l , d i s c u s s e d p r e v i o u s l y [9]. T h e m o d e l c o n s i s t s in s o l v i n g e x a c t l y t h e D i r a e coupled channel equations which describe the e x c i t a t i o n of a n u c l e a r m o n o p o l e s t a t e . T h e e x c i t a t i o n e n e r g y of t h e n u c l e a r s t a t e i s s e t e q u a l to z e r o . T h e c h a r g e d e n s i t y w h i c h in t h i s m o d e l i s r e s p o n s i b l e f o r t h e t r a n s i t i o n f r o m t h e g r o u n d to the excited state is normalized by the requirement that the corresponding inelastic electron cross s e c t i o n i s e q u a l to the e x p e r i m e n t a l t o t a l i n e l a s t i c electron nucleus cross section. A l t h o u g h q u i t e c r u d e , t h e m a i n a d v a n t a g e of t h i s m o d e l i s t h a t it p r o v i d e s a m e c h a n i s m f o r making the elastic partial wave phase shifts c o m p l e x (due to t h e p r e s e n c e of a n o p e n c h a n n e l ) . F u r t h e r , in t h i s m o d e l it i s p o s s i b l e to c a l c u l a t e
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t h e c o r r e c t i o n to t h e e l a s t i c c r o s s s e c t i o n in a " d i s t o r t e d w a v e m a n n e r " w h i c h is q u i t e s i m p l e a n d d o e s not u s e f i r s t o r s e c o n d o r d e r B o r n a p p r o x i m a t i o n , w h i c h i s a n i m p o r t a n t p o i n t 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 . A n o b v i o u s d i s a d v a n t a g e of t h e m o d e l i s t h a t it u s e s a m o n o pole excitation rather than a physically more acceptable dipole or quadrupole excitation for the i n e l a s t i c l e v e l . H o w e v e r , s i n c e t h e s i g n of t h e i m a g i n a r y p a r t s of t h e c o r r e c t i o n to t h e p h a s e s h i f t i s ( d e f i n i t e n e g a t i v e ) i n d e p e n d e n t of t h e m u l t i p o l a r i t y of the e x c i t a t i o n , a n d s i n c e , to l o w e s t o r d e r in t h e t r a n s i t i o n p o t e n t i a l , t h e c o n t r i b u t i o n to t h e i m a g i n a r y p a r t of t h e p h a s e s h i f t i s a d d i t i v e f o r a l l e x c i t a t i o n s , it i s e x p e c t e d t h a t a t t h e m a g n i t u d e of t h e c o r r e c t i o n to t h e c r o s s section is given reasonably well by this model for angles near the diffraction minima. For other a n g l e s t h e r e a l p a r t of t h e c o r r e c t i o n to t h e p h a s e s h i f t s b e c o m e s i m p o r t a n t a n d t h e m o d e l i s n o t to b e t r u s t e d . T h e s i g n of t h e c o r r e c t i o n to t h e c r o s s s e c t i o n in t h e d i f f r a c t i o n m i n i m a i s n o t p r e d i c t e d b y t h i s m o d e l , s i n c e it d e p e n d s o n t h e c h o i c e of t h e r a d i a l d e p e n d e n c e of the t r a n s i t i o n c h a r g e density. These various points are illustrated by t h e c u r v e s i l l u s t r a t e d in fig. 1. T h e two v a l u e s of Zin l i s t e d i n fig. 1 d e s c r i b e [9] two q u i t e d i f f e r e n t r a d i a l d e p e n d e n c e s of t h e t r a n s i t i o n c h a r g e d e n s i t y . T h e d o t t e d l i n e s a r e o b t a i n e d if t h e r e a l p a r t of t h e p h a s e s h i f t c o r r e c t i o n i s a r t i f i c a l l y s e t e q u a l to z e r o . T h e e f f e c t on t h e c o r r e c t i o n to t h e c r o s s s e c t i o n i s s e e n to b e small. F o r C a , t h e e f f e c t of a v i r t u a l d i p o l e t r a n s i t i o n h a s b e e n c a l c u l a t e d b y O n l e y [5] a t v a r i o u s " h i g h " e n e r g i e s , a n d c o m p a r i s o n w i t h the m o n o p o l e m o d e l s h o w s q u a l i t a t i v e a g r e e m e n t f o r t h e two c a l c u l a t i o n s f o r a v a l u e of t h e m o n o p o l e s h a p e p a r a m e t e r Z i n of 2 f m . It i s i n t e r e s t i n g t h a t , w h i l e the two calculations give comparable magnitudes f o r t h e c o r r e c t i o n to t h e e l a s t i c c r o s s s e c t i o n i n t h e r e g i o n of t h e s e c o n d d i f f r a c t i o n m i n i m u m , in t h e f i r s t m i n i m u m t h e d i p o l e m o d e l g i v e s an e f f e c t a b o u t t h r e e t i m e s l a r g e r . T h i s is n o t s u r prising since the dipole transition charge density has a long tail, while the monopole one vanishes o u t s i d e t h e n u c l e a r r e g i o n . Fig. 2 i l l u s t r a t e s t h e p e r c e n t c o r r e c t i o n f o r C a in t h e i n t e r e s t i n g " l o w " e n e r g y r a n g e f o r two c h o i c e s of Zin. T h e v a l u e of t h e t o t a l i n e l a s t i c c r o s s s e c t i o n i s t a k e n e q u a l to Crin = 0.027 f m 2. T h i s v a l u e i s b a s e d o n a n i n d e p e n d e n t p a r t i c l e s h e l l m o d e l of t h e n u c l e a r e x cited states, which assumes plane waves for the incident and outgoing electrons, but which does n o t m a k e u s e of c l o s u r e in t h e s u m o v e r n u c l e a r s t a t e s [10]. T h e d i p o l e t r a n s i t i o n s g i v e t h e l a r g e s t
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c o n t r i b u t i o n s to t h e c r o s s s e c t i o n , a n d in o r d e r to i n c l u d e e f f e c t s of c e n t e r of m a s s m o t i o n , t h e e f f e c t i v e c h a r g e of t h e p r o t o n s a n d n e u t r o n s i s t a k e n r e s p e c t i v e l y e q u a l to ALIA a n d - Z i " A *. In v i e w of t h e d i s c u s s i o n a b o v e r e g a r d i n g t h e c o m p a r i s o n w i t h O n l e y ' s c a l c u l a t i o n , it i s b e l i e v e d that the curves for Zin = 2fm are more realistic t h a n f o r Z i n = 0.5 f m in t h e c a s e of C a , a n d t h a t t h e s i z e of t h e c o r r e c t i o n s h o w n v e r y l i k e l y r e presents an underestimate. T h e c a s e of Nd, w h i c h a c t u a l l y p r o v i d e d t h e m o t i v a t i o n f o r t h e p r e s e n t n o t e [3,7], a p p e a r s to b e d i f f e r e n t f r o m t h a t of Ca. T h e v a l u e of Zin n e e d s to l i e n e a r 0.5 fro, r a t h e r t h a n 2 f m , if t h e a n o m a l y i s to b e i n t e r p r e t e d in t e r m s of VNE c o r r e c t i o n s . F u r t h e r , t h e v a l u e of Crin r e q u i r e d to produce qualitative agreement with experiment ( s o l i d l i n e in fig. 1) h a s t h e v a l u e of 0.6 f m 2, w h i c h i s a b o u t 4.5 t i m e s l a r g e r t h a n t h e v a l u e of 0.133 fro2 e x p e c t e d on t h e b a s i s of t h e s h e l l m o d e l calculation. These discrepancies could quite well b e due to t h e l a r g e c o l l e c t i v e n a t u r e of t h e f i r s t low l y i n g e x c i t e d s t a t e s in Nd, of m u l t i p o l a r i t y 2 + a n d 3-. T h i s b e l i e f i s s t r e n g t h e n e d b y t h e r e s u l t t h a t t h e s c a t t e r i n g a n o m a l y i s f o u n d to v a r y s t r o n g l y f r o m o n e i s o t o p e to t h e n e x t [11], a r e s u l t w h i c h w o u l d not b e e x p e c t e d if t h e g i a n t dipole excitations were mainly responsible for t h e VNE e f f e c t s , s i n c e t h e e x c i t a t i o n of d i p o l e s t a t e s s h o u l d b e r e l a t i v e l y i n d e p e n d e n t of t h e p r o p e r t i e s of t h e low l y i n g s t a t e s , F i n a l l y , it i s i n t e r e s t i n g to n o t e t h a t " l o w " e n e r g y e l e c t r o n s c a t t e r i n g o n i s o t o p e s of T i s h o w a n o m a l i e s of t h e t y p e a n d m a g n i t u d e d i s c u s s e d h e r e [12]. It is a p l e a s u r e to a c k n o w l e d g e a v e r y s t i m u l a t ing contact with the Yale group, particularly Professor Bockelman and Messers. Madsen and Cardman, as well as several very useful convers a t i o n s w i t h P r o f e s s o r S. W a l l . M a n y of t h e c o m p u t a t i o n s h a v e b e e n c a r r i e d o u t at t h e U n i v e r s i t y of C o n n e c t i c u t C o m p u t i n g C e n t e r , s u p p o r t e d b y NSF Grant GT-9. * The r e s u l t given for Ca in ref. [10] is l a r g e r by a factor of two than the value Grin = 0.027 fm 2) used here because the effective charge in ref. [10] is taken as 1 and 0 for protons and neutrons, r e s p e c t i v e l y .
Reference s [1] H . L . A n d e r s o n et al., Phys. Rev. L e t t e r s 22 (1969) 221. [2] T. T. Bardin et al., Phys. Rev. L e t t e r s 16 (1966) 718; E . R . M a c a g n o et al., Phys. Rev. Cl (1970) 1202. 447
Volume 33B. number 7
PHYSICS
[3] Min-Yi Chen. P h y s . R e v . C1 (1970) 1167, and private communication to the Yale Group, quoted in ref.[7]. [4] C. "I oepffer and W. Greiner, Ann. P h y s . N. Y. 47 (1968) 285; Phys. Rev.186 (1969) 1044; C. Toepffer and D. Drechsel, Phys. Rev. L e t t e r s 24 (1970) 1131. [5] D.S.Onley, Nucl. Phys. A l l 8 (1968) 436. [6] A. Bottino, G. Ciocchetti and A. Molinari, Nucl. Phys.89 (1966) 192; E.Kujawski, Phys. L e t t e r s 32B (1970) 75; W. D. Brown and E. Kujawski, P r e p r i n t , University of Maryland; 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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LETTERS
7 December 1970
[7] D. W. Madsen et al.. Phys. Rev. L e t t e r s 23 (1969) 1122. [8] G. H. Rawitscher, unpublished. [9] G . H . R a w i t s c h e r . Phys. Rev.151 (1966) 846. [10] G. H. Rawitscher, Bull. Am. Phys. Soc. 15 (1970) 595 and to be published. [11] D.W.Madsen, L.S. Cardman, G . R . L e g g a n d C . K . Bockelman, to be submitted for publication. [12] E . R o m b e r g et al., Bull.Am. Phys. Soc.14 (1969) 1242; E. Romberg, Pd. D. T h e s i s , University of Maryland (1970); N. S. Wail, to be published.