- Email: [email protected]
Electroproduction of slow pions on nuclei and giant resonance vibrations
Volume 34B, number 2
PHYSICS LETTERS
ELECTROPRODUCTION O F S L O W P I O N S ON N U C L E I GIANT RESONANCE VIBRATIONS
1 February 1971
AND
J. M. EISENBERG and H. J. WEBER
Department of Physics, University of Virginia, Charlottesville, Virginia, uSA Received 14 December 1970
The (e, e'Y) reaction at high momentum transfer on light nuclei is shown to excite JY = 0- spin-isospin vibrations preferentially, thus allowing for their experimental observation.
The f a m i l y of c o l l e c t i v e giant r e s o n a n c e v i b r a t i o n s in light nuclei with quantum n u m b e r s J n= 0-, 1-, 2" and T = 1 and lying with roughly d e g e n e r a t e e n e r g i e s in the r a n g e of 20MeV to 25MeV, have long been of i n t e r e s t in n u c l e a r s t r u c t u r e s t u d i e s . P h o t o n u c l e a r r e a c t i o n s have b e e n u s e d to e x a m i n e the dipole e x c i t a t i o n s not affecting nucleon spin v a r i a b l e s , and e l e c t r o e x c i t a t i o n h a s s u p p l i e d i n f o r m a t i o n on the J~ = 1-, 2 - s p i n - f l i p v i b r a t i o n s . The 0- m e m b e r s of t h i s f a m i l y have not been o b s e r v e d , h o w e v e r , both b e c a u s e of the i n t r i n s i c difficulty of finding a m e c h a n i s m which will e x c i t e t h e m d i r e c t l y f r o m the g r o u n d s t a t e and h a s a Consequence of t h e i r p r o x i m i t y to the 1- s p i n - f l i p l e v e l s which tend to d o m i n a t e o v e r t h e m T. One method, involving t h e (~, 2~) r e a c t i o n on nuclei, h a s been p r o p o s e d [2] for the e x p e r i m e n t a l study of the 0 - v i b r a t i o n s . The e s s e n t i a l i d e a in question i s to u s e a p r e d o m i n a n t l y s c a l a r i n t e r a c t i o n which i s c a p a b l e of p r o d u c i n g s - w a v e p i o n s (L ~ = 0-); t h i s then s u p p r e s s e s the c o m p e t i n g 1- l e v e l s . In ref. [2], the r e l e v a n t " s c a l a r i n t e r a c t i o n " i s the s - w a v e component of the i n e l a s t i c (~, n') s c a t t e r i n g p r o c e s s . The intent of the p r e s e n t c o m m u n i c a t i o n i s to e x p l o r e the p o s s i b i l i t y of using, for that p u r p o s e , a v i r t u a l photon with a l a r g e s c a l a r component, a s p r o d u c e d in e l e c t r o n s c a t t e r i n g at high m o m e n t u m t r a n s f e r . T h i s s i t u a t i o n will a l s o fulfill the a d d i t i o n a l key r e q u i r e m e n t that the m e c h a n i s m u s e d for e x c i t i n g the 0 - l e v e l s will s u f f i c i e n t l y well u n d e r s t o o d so that the c o r r e s p o n d i n g m e a s u r e m e n t will p r o d u c e d i r e c t i n f o r m a t i o n on the (negative) monopole t r a n s i t i o n s t r e n g t h f r o m the g r o u n d s t a t e . The (e, e'n) r e a c t i o n on n u c l e i h a s been the s u b j e c t of a n u m b e r of e a r l i e r t h e o r e t i c a l i n v e s t i g a t i o n s [3-5]. F o r o u r p r e s e n t p u r p o s e , it i s convenient to f o r m u l a t e the p r o b l e m using s o f t - p i o n techniques; thus r e l a t i n g it to t h e o r i e s of r a d i a t i v e pion a b s o r p t i o n in nuclei [6, 7], but with the photon f i e l d r e p l a c e d by the M ~ l l e r p o t e n t i a l ( s e e f o r e x a m p l e r e f s . [8, 9]). The S - m a t r i x e l e m e n t f o r the p r o c e s s e(p) + N(P) --N'(P ') + e(p') + 7r+(q) ,
(i)
w h e r e the m o m e n t a of the c o r r e s p o n d i n g p a r t i c l e s a r e shown in p a r e n t h e s e s , is
sfi = -in u f dxdy (2q0)-1/2 exp[i(kx - q y ) ] [ - ] x ( % - m
2
) ( N ' ] T ( A g (x)~,+(y)) IN),
(2)
with
a u = ie [u(p')ru
u(P)]/(k2
- k2o) '
(3)
w h e r e k~ = (p - p ' ) ~ , e 2 = 1/137, and the o t h e r notation i s s t a n d a r d [10]. In the s o f t - p i o n l i m i t , t h i s m a t r i x e l e m e n t b e c o m e s [10] Work supported in part by the National Science Foundation. The situation with regard to the study of this family of vibrations is reviewed in refs. [1] and [2], which also supply references to e a r l i e r relevant work. 107
Volume 34B. number 2
Sfi ~
PHYSICS LETTERS
(4nie//n)a u fdx(2qo)-l/2
exp[i(k
-q)x]
1 February 1971
,
(4)
where J ~ is the axial c u r r e n t and fn = 0.83 rnn. The evaluation of the c r o s s - s e c t i o n for the p r o c e s s (1) then p a r a l l e l s ref. [2] v e r y closely, going through the u s u a l steps of choosing a g e n e r a l f o r m for the n u c l e o n axial c u r r e n t , c a r r y i n g out a n o n r e l a t i v i s t i c r e d u c t i o n of this f o r m , and a s s u m i n g that the r e l e v a n t n u c l e a r o p e r a t o r is a sum of the nucleon o p e r a t o r s . F o r p u r p o s e s of obtaining a n u m e r i c a l e s t i m a t e for the c r o s s - s e c t i o n of (1), we use the shell model in the p a r t i c l e - h o l e f o r m a l i s m . The c r o s s section for excitation of a given p a r t i c l e - h o l e configuration (a#), with spin J, in a c l o s e d - s h e l l n u c l e u s is then d c f i / d c o s 8 dq ° = ~ 1
l [
~a ~ p-~ C~p
(gl//~)2 (P'/p)q[4pp'e4/(k 2-k2o)2 ] [']~ ]a(jpj J[½-½0)] 2
~
(-k24k J/k2o~ sin2½8_cosS+jb.]( - C2 O S 22½ 8 +' 2 s i n 2½8) + all,[-~ 2 cos 2132 ~8o k
~,.~ ]
o ko
}
where k = }kl, g l = 1.23 is the axial coupling constant, 8 is the electron scattering angle, A~ ~ and B (7~
a r e c o m b i n a t i o n s of r a d i a l i n t e g r a l s d e f i n e d * in [2], as a r e the q u a n t i t i e s
J all,
and bJl . Since we are not
i n t e r e s t e d in the p r e c i s e energy p a r t i t i o n between the final e l e c t r o n and pion, so long as a given total n u c l e a r excitation energy is m a i n t a i n e d , it is convenient to c o n s i d e r the quantity qo = 170 MeV
f
dafi/d cos
8dq °
dq o dfffi(q ° = 160 + 10 MeV)/d cos8
(6)
P ' + qo = const
qo = 150 MeV
where the r a n g e of i n t e g r a t i o n is chosen to c o v e r those pion e n e r g i e s for which the e s s e n t i a l i n g r e d i e n t s of the p r e s e n t theory a r e valid. We now t u r n to a rough evaluation of the pion e l e c t r o p r o d u c t i o n c r o s s section for the 0-, 1- and 2-, T = 1, giant r e s o n a n c e states, w e use h a r m o n i c o s c i l l a t o r wave functions, which, while c e r t a i n l y not leading to a c c u r a t e r e s u l t s at the high m o m e n t u m t r a n s f e r s we shall r e q u i r e , a r e not g r o s s l y misleadi.ng [11] and in any event p r o v i d e a lower l i m i t for the c r o s s section. The o s c i l l a t o r p a r a m e t e r s ~ for 4He and 160 a r e chosen to be 144MeV and 112 MeV, r e s p e c t i v e l y . R e s u l t s for these two n u c l e i a r e shown in t a b l e s 1 and 2 for m o m e n t u m t r a n s f e r s which a r e s e l e c t e d to be l a r g e enough so that the 0- level g e n e r Note that in the first line of eq. (19) in ref. [2] the radical should be [(2J+ 3)/'(J+ 1)] 1/'2. Table 1 Cross-section for e + 4He ~--~4H*(JY;E) + e'+ ~+. The exitation energy E is the prediction of the particle-hole model [12], and is taken relative to the 4He ground state J~
E (MeV)
Leading configuration
900
90°
60°
450
350
0-
27.7
(1s½)-11P½
2.17
2.96
25.46
129.1
1-
29.0
3.47
2.17
18.76
105.6
1-
25.7
(ls 1) -11pz 2 (ls~)-llp~
1.68
1.07
9.43
53.8
2
24.5
(isl)-llPs
7.86
7.30
62.51
326.2
-
2
108
d(~fi (qo = 160 ~- 10MeV)/d cos8 (I0-36cm2) 700 800
p(MeV/c) =500
PHYSICS
Volume 34B, n u m b e r 2
LETTERS
1 F e b r u a r y 19'71
Table 2 C r o s s - s e c t i o n for e + 160 --~16N*(J~;E) + e' + ~+. The exitation energy E is the prediction of the p a r t i c l e - h o l e model [13] and is taken r e l a t i v e to the 160 ground state JY
0-
E (MeV)
Leading configuration
d(zfi(qo = 160 :~ 10MeV)/d cos 0 (10-36cm 2) p ( M e V / c ) = 500
700
800
900
90 °
60 °
45 °
35o
25.7
(lp3) -1 l d 3
0.165
0.158
2.33
15.25
0-
13.7
~--1 ~ (lP½) 2s½
0.294
0.289
5.14
39.78
1-
25.4
(lps) -1 l d 3
0.194
0.087
1.28
9.02
0.105
0.048
0.723
5.15
0.300
0.264
4.63
37.30
0.140
0.929
16.33
0.269
0.147
2.24
2 -I
12
_
22-
22.7 23.7
(lp3) 2_1
(lp3)
2--1
2
ld s 2
ld3
~
20.2
(lp3)
19.1
~'-1 (lp3) ld_5 2 2
2sl
133.7 15.88
a l l y d o m i n a t e s o v e r t h e 1 - c o m p e t i t o r s . T h e c r o s s - s e c t i o n s g i v e n i n t h e s e t a b l e s do n o t i n c l u d e d i s t o r t i o n s of t h e p i o n w a v e , i n p a r t i c u l a r C o u l o m b e f f e c t s , w h i c h , f o r ~+ m e s o n s i n t h e i n d i c a t e d e n e r g y r a n g e , m a y r e d u c e t h e c r o s s - s e c t i o n s b y a f a c t o r of a b o u t f i v e . W e s u m m a r i z e o u r r e s u l t s by n o t i n g t h a t t h e c r o s s - s e c t i o n s f o r t h e e x c i t a t i o n of n u c l e a r 0 - l e v e l s i n t h e (e, e ' ~ ±) r e a c t i o n s h o u l d b e s u f f i c i e n t l y l a r g e t o b e o b s e r v a b l e w i t h t h e r e q u i s i t e 1 to 2 M e v t o t a l energy resolution provided that modern high intensity electron accelerators are used. It i s a p l e a s u r e t o a c k n o w l e d g e a h i g h l y u s e f u l c o m m e n t d u e to P r o f e s s o r the impetus for this investigation.
P . K. K a b i r w h i c h p r o v i d e d
References [1] J. D. Walecka, in: Proc. W i l l i a m s b u r g Conf. on i n t e r m e d i a t e energy physics, Feb. 1966 (the College of William and Mary, Williamsburg, Virginia, 1966). [2] J . M . E i s e n b e r g , Nucl. Phys. A148 (1970) 135. [3] W. CzyZ and J. D. Waleeka, Nuel. Phys. 51 (1964) 312. [4] V. Devanathan, Nucl. Phys. 87 (1966) 397. [5] E . J . Moniz, Phys. Rev. 184 (1969) 1154. [6] H. P i e t s e h m a n n , L . P . Fu[cher and J. M. Eisenberg, Phys. Rev. L e t t e r s 19 (1967) 1259; L. P. F u l c h e r and J. M. E i s e n b e r g , Nucl. Phys. B 18 (1970) 271. [7] M. E r i e s o n and A. Figureau, Nuel. Phys. B 11 (1969) 621. [8] T. de F o r e s t and J . D . Walecka, Adv. in Phys. 15 (1966) 1. [9] J . M . E i s e n b e r g and W. G r e i n e r , Excitation m e c h a n i s m s of the nucleus (North-Holland, A m s t e r d a m , 1970). [10] S. L. Adler and R. F. Dashen, C u r r e n t a l g e b r a s (W. A. Benjamin, New York, 1968). [11] F. D. Holder and J. M. E i s e n b e r g , Nuel. Phys. A 106 (1968) 261. [12] A. de-Shalit and J. Walecka, Phys. Rev. 147 (1966) 763. [13] V. Gillet and N. Vinh-Mau, Nucl. Phys. 54 (1964) 321; 57 (1964) 698.
109
PHYSICS LETTERS
ELECTROPRODUCTION O F S L O W P I O N S ON N U C L E I GIANT RESONANCE VIBRATIONS
1 February 1971
AND
J. M. EISENBERG and H. J. WEBER
Department of Physics, University of Virginia, Charlottesville, Virginia, uSA Received 14 December 1970
The (e, e'Y) reaction at high momentum transfer on light nuclei is shown to excite JY = 0- spin-isospin vibrations preferentially, thus allowing for their experimental observation.
The f a m i l y of c o l l e c t i v e giant r e s o n a n c e v i b r a t i o n s in light nuclei with quantum n u m b e r s J n= 0-, 1-, 2" and T = 1 and lying with roughly d e g e n e r a t e e n e r g i e s in the r a n g e of 20MeV to 25MeV, have long been of i n t e r e s t in n u c l e a r s t r u c t u r e s t u d i e s . P h o t o n u c l e a r r e a c t i o n s have b e e n u s e d to e x a m i n e the dipole e x c i t a t i o n s not affecting nucleon spin v a r i a b l e s , and e l e c t r o e x c i t a t i o n h a s s u p p l i e d i n f o r m a t i o n on the J~ = 1-, 2 - s p i n - f l i p v i b r a t i o n s . The 0- m e m b e r s of t h i s f a m i l y have not been o b s e r v e d , h o w e v e r , both b e c a u s e of the i n t r i n s i c difficulty of finding a m e c h a n i s m which will e x c i t e t h e m d i r e c t l y f r o m the g r o u n d s t a t e and h a s a Consequence of t h e i r p r o x i m i t y to the 1- s p i n - f l i p l e v e l s which tend to d o m i n a t e o v e r t h e m T. One method, involving t h e (~, 2~) r e a c t i o n on nuclei, h a s been p r o p o s e d [2] for the e x p e r i m e n t a l study of the 0 - v i b r a t i o n s . The e s s e n t i a l i d e a in question i s to u s e a p r e d o m i n a n t l y s c a l a r i n t e r a c t i o n which i s c a p a b l e of p r o d u c i n g s - w a v e p i o n s (L ~ = 0-); t h i s then s u p p r e s s e s the c o m p e t i n g 1- l e v e l s . In ref. [2], the r e l e v a n t " s c a l a r i n t e r a c t i o n " i s the s - w a v e component of the i n e l a s t i c (~, n') s c a t t e r i n g p r o c e s s . The intent of the p r e s e n t c o m m u n i c a t i o n i s to e x p l o r e the p o s s i b i l i t y of using, for that p u r p o s e , a v i r t u a l photon with a l a r g e s c a l a r component, a s p r o d u c e d in e l e c t r o n s c a t t e r i n g at high m o m e n t u m t r a n s f e r . T h i s s i t u a t i o n will a l s o fulfill the a d d i t i o n a l key r e q u i r e m e n t that the m e c h a n i s m u s e d for e x c i t i n g the 0 - l e v e l s will s u f f i c i e n t l y well u n d e r s t o o d so that the c o r r e s p o n d i n g m e a s u r e m e n t will p r o d u c e d i r e c t i n f o r m a t i o n on the (negative) monopole t r a n s i t i o n s t r e n g t h f r o m the g r o u n d s t a t e . The (e, e'n) r e a c t i o n on n u c l e i h a s been the s u b j e c t of a n u m b e r of e a r l i e r t h e o r e t i c a l i n v e s t i g a t i o n s [3-5]. F o r o u r p r e s e n t p u r p o s e , it i s convenient to f o r m u l a t e the p r o b l e m using s o f t - p i o n techniques; thus r e l a t i n g it to t h e o r i e s of r a d i a t i v e pion a b s o r p t i o n in nuclei [6, 7], but with the photon f i e l d r e p l a c e d by the M ~ l l e r p o t e n t i a l ( s e e f o r e x a m p l e r e f s . [8, 9]). The S - m a t r i x e l e m e n t f o r the p r o c e s s e(p) + N(P) --N'(P ') + e(p') + 7r+(q) ,
(i)
w h e r e the m o m e n t a of the c o r r e s p o n d i n g p a r t i c l e s a r e shown in p a r e n t h e s e s , is
sfi = -in u f dxdy (2q0)-1/2 exp[i(kx - q y ) ] [ - ] x ( % - m
2
) ( N ' ] T ( A g (x)~,+(y)) IN),
(2)
with
a u = ie [u(p')ru
u(P)]/(k2
- k2o) '
(3)
w h e r e k~ = (p - p ' ) ~ , e 2 = 1/137, and the o t h e r notation i s s t a n d a r d [10]. In the s o f t - p i o n l i m i t , t h i s m a t r i x e l e m e n t b e c o m e s [10] Work supported in part by the National Science Foundation. The situation with regard to the study of this family of vibrations is reviewed in refs. [1] and [2], which also supply references to e a r l i e r relevant work. 107
Volume 34B. number 2
Sfi ~
PHYSICS LETTERS
(4nie//n)a u fdx(2qo)-l/2
exp[i(k
-q)x]
1 February 1971
(4)
where J ~ is the axial c u r r e n t and fn = 0.83 rnn. The evaluation of the c r o s s - s e c t i o n for the p r o c e s s (1) then p a r a l l e l s ref. [2] v e r y closely, going through the u s u a l steps of choosing a g e n e r a l f o r m for the n u c l e o n axial c u r r e n t , c a r r y i n g out a n o n r e l a t i v i s t i c r e d u c t i o n of this f o r m , and a s s u m i n g that the r e l e v a n t n u c l e a r o p e r a t o r is a sum of the nucleon o p e r a t o r s . F o r p u r p o s e s of obtaining a n u m e r i c a l e s t i m a t e for the c r o s s - s e c t i o n of (1), we use the shell model in the p a r t i c l e - h o l e f o r m a l i s m . The c r o s s section for excitation of a given p a r t i c l e - h o l e configuration (a#), with spin J, in a c l o s e d - s h e l l n u c l e u s is then d c f i / d c o s 8 dq ° = ~ 1
l [
~a ~ p-~ C~p
(gl//~)2 (P'/p)q[4pp'e4/(k 2-k2o)2 ] [']~ ]a(jpj J[½-½0)] 2
~
(-k24k J/k2o~ sin2½8_cosS+jb.]( - C2 O S 22½ 8 +' 2 s i n 2½8) + all,[-~ 2 cos 2132 ~8o k
~,.~ ]
o ko
}
where k = }kl, g l = 1.23 is the axial coupling constant, 8 is the electron scattering angle, A~ ~ and B (7~
a r e c o m b i n a t i o n s of r a d i a l i n t e g r a l s d e f i n e d * in [2], as a r e the q u a n t i t i e s
J all,
and bJl . Since we are not
i n t e r e s t e d in the p r e c i s e energy p a r t i t i o n between the final e l e c t r o n and pion, so long as a given total n u c l e a r excitation energy is m a i n t a i n e d , it is convenient to c o n s i d e r the quantity qo = 170 MeV
f
dafi/d cos
8dq °
dq o dfffi(q ° = 160 + 10 MeV)/d cos8
(6)
P ' + qo = const
qo = 150 MeV
where the r a n g e of i n t e g r a t i o n is chosen to c o v e r those pion e n e r g i e s for which the e s s e n t i a l i n g r e d i e n t s of the p r e s e n t theory a r e valid. We now t u r n to a rough evaluation of the pion e l e c t r o p r o d u c t i o n c r o s s section for the 0-, 1- and 2-, T = 1, giant r e s o n a n c e states, w e use h a r m o n i c o s c i l l a t o r wave functions, which, while c e r t a i n l y not leading to a c c u r a t e r e s u l t s at the high m o m e n t u m t r a n s f e r s we shall r e q u i r e , a r e not g r o s s l y misleadi.ng [11] and in any event p r o v i d e a lower l i m i t for the c r o s s section. The o s c i l l a t o r p a r a m e t e r s ~ for 4He and 160 a r e chosen to be 144MeV and 112 MeV, r e s p e c t i v e l y . R e s u l t s for these two n u c l e i a r e shown in t a b l e s 1 and 2 for m o m e n t u m t r a n s f e r s which a r e s e l e c t e d to be l a r g e enough so that the 0- level g e n e r Note that in the first line of eq. (19) in ref. [2] the radical should be [(2J+ 3)/'(J+ 1)] 1/'2. Table 1 Cross-section for e + 4He ~--~4H*(JY;E) + e'+ ~+. The exitation energy E is the prediction of the particle-hole model [12], and is taken relative to the 4He ground state J~
E (MeV)
Leading configuration
900
90°
60°
450
350
0-
27.7
(1s½)-11P½
2.17
2.96
25.46
129.1
1-
29.0
3.47
2.17
18.76
105.6
1-
25.7
(ls 1) -11pz 2 (ls~)-llp~
1.68
1.07
9.43
53.8
2
24.5
(isl)-llPs
7.86
7.30
62.51
326.2
-
2
108
d(~fi (qo = 160 ~- 10MeV)/d cos8 (I0-36cm2) 700 800
p(MeV/c) =500
PHYSICS
Volume 34B, n u m b e r 2
LETTERS
1 F e b r u a r y 19'71
Table 2 C r o s s - s e c t i o n for e + 160 --~16N*(J~;E) + e' + ~+. The exitation energy E is the prediction of the p a r t i c l e - h o l e model [13] and is taken r e l a t i v e to the 160 ground state JY
0-
E (MeV)
Leading configuration
d(zfi(qo = 160 :~ 10MeV)/d cos 0 (10-36cm 2) p ( M e V / c ) = 500
700
800
900
90 °
60 °
45 °
35o
25.7
(lp3) -1 l d 3
0.165
0.158
2.33
15.25
0-
13.7
~--1 ~ (lP½) 2s½
0.294
0.289
5.14
39.78
1-
25.4
(lps) -1 l d 3
0.194
0.087
1.28
9.02
0.105
0.048
0.723
5.15
0.300
0.264
4.63
37.30
0.140
0.929
16.33
0.269
0.147
2.24
2 -I
12
_
22-
22.7 23.7
(lp3) 2_1
(lp3)
2--1
2
ld s 2
ld3
~
20.2
(lp3)
19.1
~'-1 (lp3) ld_5 2 2
2sl
133.7 15.88
a l l y d o m i n a t e s o v e r t h e 1 - c o m p e t i t o r s . T h e c r o s s - s e c t i o n s g i v e n i n t h e s e t a b l e s do n o t i n c l u d e d i s t o r t i o n s of t h e p i o n w a v e , i n p a r t i c u l a r C o u l o m b e f f e c t s , w h i c h , f o r ~+ m e s o n s i n t h e i n d i c a t e d e n e r g y r a n g e , m a y r e d u c e t h e c r o s s - s e c t i o n s b y a f a c t o r of a b o u t f i v e . W e s u m m a r i z e o u r r e s u l t s by n o t i n g t h a t t h e c r o s s - s e c t i o n s f o r t h e e x c i t a t i o n of n u c l e a r 0 - l e v e l s i n t h e (e, e ' ~ ±) r e a c t i o n s h o u l d b e s u f f i c i e n t l y l a r g e t o b e o b s e r v a b l e w i t h t h e r e q u i s i t e 1 to 2 M e v t o t a l energy resolution provided that modern high intensity electron accelerators are used. It i s a p l e a s u r e t o a c k n o w l e d g e a h i g h l y u s e f u l c o m m e n t d u e to P r o f e s s o r the impetus for this investigation.
P . K. K a b i r w h i c h p r o v i d e d
References [1] J. D. Walecka, in: Proc. W i l l i a m s b u r g Conf. on i n t e r m e d i a t e energy physics, Feb. 1966 (the College of William and Mary, Williamsburg, Virginia, 1966). [2] J . M . E i s e n b e r g , Nucl. Phys. A148 (1970) 135. [3] W. CzyZ and J. D. Waleeka, Nuel. Phys. 51 (1964) 312. [4] V. Devanathan, Nucl. Phys. 87 (1966) 397. [5] E . J . Moniz, Phys. Rev. 184 (1969) 1154. [6] H. P i e t s e h m a n n , L . P . Fu[cher and J. M. Eisenberg, Phys. Rev. L e t t e r s 19 (1967) 1259; L. P. F u l c h e r and J. M. E i s e n b e r g , Nucl. Phys. B 18 (1970) 271. [7] M. E r i e s o n and A. Figureau, Nuel. Phys. B 11 (1969) 621. [8] T. de F o r e s t and J . D . Walecka, Adv. in Phys. 15 (1966) 1. [9] J . M . E i s e n b e r g and W. G r e i n e r , Excitation m e c h a n i s m s of the nucleus (North-Holland, A m s t e r d a m , 1970). [10] S. L. Adler and R. F. Dashen, C u r r e n t a l g e b r a s (W. A. Benjamin, New York, 1968). [11] F. D. Holder and J. M. E i s e n b e r g , Nuel. Phys. A 106 (1968) 261. [12] A. de-Shalit and J. Walecka, Phys. Rev. 147 (1966) 763. [13] V. Gillet and N. Vinh-Mau, Nucl. Phys. 54 (1964) 321; 57 (1964) 698.
109