NONLINEAR EFFECTS AND COLLECTIVE EXCITATIONS

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NONLINEAR EFFECTS AND COLLECTIVE EXCITATIONS

G. Brown

To cite this version:

G. Brown. NONLINEAR EFFECTS AND COLLECTIVE EXCITATIONS. Journal de Physique

Colloques, 1984, 45 (C4), pp.C4-479-C4-487. �10.1051/jphyscol:1984438�. �jpa-00224104�

JOURNAL DE PHYSIQUE

Colloque C 4 , supplkment a u n03, T o m e 45, m a r s 1984 page C4-479

NONLINEAR EFFECTS AND COLLECTIVE EXCITATIONS

G.E. Brown

NORDITA, BZegdamsvej 1 7 , DK-2100 Copenhagen

a,

^Denmark

and

P h y s i c s Department, S t a t e U n i v e r s i t y o f New ~ o r k * , S t o n y Brook, NY 11794, U.S.A.

Resume - Quoique le probleme a

N

corps soit generalement trait6 dans le cadred'une theorie de reponse 1 ineaire, nous montrons que pour calculer la

compressi bil

i

t6, etc . . . on a 1 'habitude d'uti 1 iser des termes non 1 ineaires;

en fait, sans ces termes l'on aboutirait a des absurdites. Dans le cas de la ui.sonance de Gamow-Tel ler, les termes non

1

in6aires etroi tement

1

ies au pro- bleme sont ceux faisant ecran au terme d'echange dans l'interaction nucleon-A.

Pour une fr6quence nulle

(LO = 0)

1 'effet d'ecran rend le terme d'echange petit - ^Pour

^u

^egal

?I

l'energie de la resonance de Gamow-Teller, cet effet d'ecran peut m6me changer le signe du terme d'echange.

A b s t r a c t

Whereas many-body t h e o r y i n g e n e r a l i s o f t e n d i s c u s s e d i n t e r m s o f l i n e a r r e s p o n s e t h e o r y , i t i s shown t h a t i n c a l c u l a t i n g t h e com- p r e s s i b i l i t y , e t c . , we a r e u s e d t o u s i n g n o n l i n e a r t e r m s ; i n d e e d , w i t h o u t them we would g e t n o n s e n s e .

I n t h e c a s e o f t h e g i a n t Gamow T e l l e r Resonance t h e h i g h l y - r e l e v a n t n o n l i n e a r t e r m s a r e t h o s e i n t h e s e l f s c r e e n i n g of t h e ex- change t e r m , i n t h e i n t e r a c t i o n c o u p l i n g n u c l e a r t o A e x c i t a t i o n s . F o r z e r o f r e q u e n c y , w = 0

,

t h i s s c r e e n i n g makes t h e e x c h a n g e t e r m s m a l l ; f o r w = EGTR

,

^{where EGTR} i s t h e e n e r g y o f t h e Gamow-Teller r e s o n a n c e , t h e r e c a n b e o v e r s c r e e n i n g and t h e s i g n o f t h e exchange t e r m c a n b e changed.

1. INTRODUCTION

The c o m p r e s s i o n modulus K i n n u c l e a r m a t t e r h a s t h e e x p r e s s i o n

i n Landau f e r m i l i q u i d t h e o r y . E m p i r i c a l l y , K 220 MeV. F o r m*/m = 1

2

- - kf 2 40 MeV 2m

a n d , t h u s , F o must b e s m a l l , p o s s i b l y s l i g h t l y n e g a t i v e , t o g i v e t h e e m p i r i c a l K O

.

F o r m*/m < 1, F o must b e more n e g a t i v e ; a v a l u e

F o

-

=

-

0.3 ( 1 . 3 )

- -

*work s u p p o r t e d i n p a r t by USDOE u n d e r C o n t r a c t DE-AC02-76ER13001.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984438

C4-W0

seems reasonable in terms of empirical values of K , although there is some uncertainty. Certainly F0 must be > -1 , because at -1 K becomes negative and the system becomes unstable. (Out in the nu- clear surface, in a local density approximation, F0 < -1 , signifying cluster formation .)

In a G-matrix calculation2 F0 = -1.2 for the^Reid potential, -1.43 for the Bonn interaction. (See, also Sjoberg .) The cure for this apparent instability is well known and often used; namely:

the compression modulus is derived by doubly differentiating the system energy:

(1.4) If this differentiation is carried out at an energy minimum, then K is positive, so there is no problem with instability.

In terms of Landau theory, calculating F from the G-matrix corresponds to the term, fig. la. The double differentiation amounts to including also the term, fig. lb, often called a rearrangement

Fig. 1. Particle-hole interaction f . The wavy line here is the G-matrix interaction.

term. The F0 is the L = 0 projection of the particle-hole matrix element

(1.5)

over the Landau angle d , given by cos d = k'ic

Inclusion of b) means that

(1.6)

(1.7) where F is the G-matrix approximation to F , and U(q,0) , the Lindhard function. The relationship between q and d is

(1.8) A reasonable first approximation, to see how things go, is to set

(1.9) and, in general, to keep only the L = 0 Landau parameters. The lat- ter is not quantitatively accurate for the FL's , but should be a very good approximation for the spin-isospin parameter G0' which we

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

Fo = Fo

and one s e e s t h a t Fo > -1 now.

G e n e r a l i z a t i o n must be made, t o i n c l u d e o t h e r c h a n n e l s , which e n t e r i n t o t h e p a r t i c l e - h o l e i n t e r a c t i o n through

where, a g a i n , we have c o n s i d e r e d o n l y L = 0 terms. I n some c a s e s it i s an a d v a n t a g e t o i n c l u d e t h e r e s c a t t e r i n g t o a l l o r d e r s ; i . e . , t o i n c l u d e an a r b i t r a r y number o f b u b b l e s i n f i g . l b ) . F i n a l l y , i t s h o u l d be r e c o g n i z e d 3 r 5 t h a t t h e i n t e r a c t i o n e n t e r i n g i n t o p r o c e s s e s l i k e f i g . l b i s t h e f u l l 3 , n o t t h e G-matrix a p p r o x i m a t i o n . I n t h i s way we a r r i v e a t t h e e q u a t i o n s F = F ( G )

+

Fi

,

e t c . where

0

and t h e r i g h t - h a n d s i d e o f t h e above e q u a t i o n s i s c a l l e d " t h e induced i n t e r a c t i o n " .

Now, from e m p i r i c a l phenomena one can deduce6 : F o g - 0 . 3 , 1

,

^{G o} i s s m a l l ,

G: 1.68

s o one s e e s immediately t h a t t h e t e r m i n 9 G i 2 i n F i w i l l p r e - dominat A major m o d i f i c a t i o n i n t h e formalism must b e made h e r e ,

because t h e assumption was made t h a t t h e v a r i o u s Landau p a r a m e t e r s d i d n o t depend much on 9 ; i . e . , on q

.

However, j u s t t h e G ' c h a n n e l c o n t a i n s t h e one-pion-exchange i n t e r a c t i o n , and G ' i s v e r y q-depeygent. A way t o t a k e t h i s simply i n t o a c c o u n t i s t o m u l t i p l y t h e G o terms i n t h e induced i n t e r a c t i o n by t h e f a c t o r

where G 1 ( q ) i s t h e f u l l

g1-g2 z 1 - ~ 2

i n t e r a c t i o n ( s e e , e . g . , ~ p e t h e t a 1 . 9 ) .

J O U R N A L

DE

PHYSIQUE

w i t h f 2 / 4 n = 0.08

,

fp2/mp2 = 2 f2/mT2

, Fp2

⁼ 0.4 f p 2 where t h e f a c t o r 0.4 t a k e s i n t o a c c o u n t s h o r t - r a n g e c o r r e l a t i o n s . T h i s f a c t o r of K i s roughly 1/12! T h i s f a c t o r o f 1/12 i s s o s m a l l b e c a u s e , w i t h i n c l u s i o n of t h e t e n s o r i n t e r a c t i o n , ~ ' ( q ) becomes v e r y 2 s m a l l f o r q g 2kf

.

On t h e o t h e r hand, s i n c e d ( c o s 9) =

-

qdq/kf

,

be- c a u s e o f t h e phase s p a c e f a c t o r q h e r e , j u s t t h i s r e g i o n i s h i g h l y weighted.

Note t h a t t h e t e n s o r c o n t r i b u t i o n s began from t e n s o r i n v a r i a n t s 8

.

Thus, t e n s o r i n v ~ r i a n t s i n t h e c r o s s e d c h a n n e l c o n t r i b u t e t o t h e d i - r e c t c h a n n e l G o

-

Taking U (qlO) t o be u n i t y ( i n f a c t ,

:2

5 ⁺

^{- 1}

^,

t h e n t h e s i z a b l e c o n t r i b u t i o n s t o Fi a r e

U(q,O) = 1 - -

Even w i t h t h e f a c t o r 12 r e d u c t i o n , which seems e x c e s s i v e , t h e

g1-cs2 ~ ~

c h a n n e l c o n t r i b u t e s a p p r e c i a b l y .

- 2 ~

Adding t h e s e t o o u r o r i g i n a l Fo from Reid o r Bonn g i v e s Fo > -1

,

b u t more n e g a t i v e t h a n found e m p i r i y a l l y . However, from (1.11) one s e e s t h a t a l l terms

,

F O l a r g e r ; t h i s i s a l s o t r u e f o r t h e L

+

0 con- l e f t o u t m k

t r i b u t i o n s

.

Thus, we t e n d towards a r e a s o n a b l e s i t u a t i o n . The p h y s i c a l r e a s o n t h a t e f f e c t s from a l l c h a n n e l s add up, making t h e compression modulus l a r g e r , i s t h e f o l l o w i n g : Each nu- c l e o n c o n s t r u c t s around i t s e l f a c l o u d o f c o l l e c t i v e e x c i t a t i o n s , a r - ranged i n such a way a s t o lower i t s e n e r g y . T h i s c l o u d makes t h e d r e s s e d n u c l e o n , t h e " q u a s i p a r t i c l e ' e f f e c t i v e l y l a r g e r t h a n t h e nu- c l e o n ; t h e r e f o r e it i s h a r d e r t o compress a l i q u i d made up o u t of such l a r g e r o b j e c t s t h a n one c o n t a i n i n g b a r e , u n d r e s s e d n u c l e o n s .

2 . ENERGY DEPENDENCE STRIKES AGAIN

The f o r m u l a t i o n g i v e n above i s , i n t h e Landau s p i r i t , f o r w 'Z 0 ( t h e r e f o r e U ( q , w ) i s t a k e n t o be U (q, 0) b e f o r e b e i n g s e t u n i t y ) , i . e . , phenomena a r e t a k e n on t h e Fermi s u r f a c e . I n r e a l l i f e , i . e . , i n

r&tl

n u c l e i , w i s by no means z e r o . The monopole resonance i n Pb

* ,

c r u c i a l f o r t h e measurement of Fo

,

i s a t 14 MeV, c l o s e t o t h e g i a n t d i p o l e r e s o n a n c e e n e r g y o f 13.5 MeV.

Both o f t h e s e a r e a b o u t 2Kw i n Pb.

,

Now t h e energy denominator 1

+

Fo U(q,O) i n t h e c o n t r i b u - t i o n s from t h e

~ ~ - 2 ,

c h a n n e l t a k e s i n t o accounF t h e push upwards of t h e g i a n t d i p o l e s t a t e t h r o u g h t h e r e p u l s i v e Fo i n t e r a c t i o n . I n t h e c a s e i n hand, t h e r e i s a l a r g e c a n c e l l a t i o n between t h e energy o f t h e g i a n t d i p o l e s t a t e , which o c c u r s t o g e t h e r w i t h t h e u n c o r r e l a t e d p a r t i c l e - h o l e e n e r g y a s t h e v i r t u a l energy of t h e i n t e r m e d i a t e s t a t e

( s e e f i g . 2)

,

and t h e i n i t i a l e n e r g y E ( 0 + )

, .

S i n c e E (O+) E (GDR) i t would b e b e t t e r t o l e a v e o u t t h e 1

+

Fo U(q,O) i n t h e denomina- t o r s o f t h e induced i n t e r a c t i o n . A s i m i l a r argument can b e a p p l i e d t o t h e Gamow-Teller c h a n n e l . One t h e n f i n d s t h e Fi t o b e

now large, but of the right genera12size. Note that the final F o will be small, so that the term F o /4 in the induced interaction will be negligible.

3 . THE GIANT GAMOW-TELLER INTERACTION

I wish now, to discuss work carried out with K. Nakayama, S.

Krewald and J. Speth at Jiilich. Although this work was carried out carefully in a formalism suited to the problem, here I'll give a schematic presentation along the above lines, in order to convey the main idea.

We shall apply a mechanism to the nucleon-isobar transition

po-

tential similar to that suggested by Dickoff et a1. for purely

nu-

clear interactions. In ref. 9 the fact that the contribution

Fig. 2. The energy denominator at the intermediate state shown by the dash-dot line is ( E - E ~ )

+

^{E(GDR) ;} the difference of ini- tial and intermediatg-state energy is E ( O + )

-

[ ( E ~ - E ~ ) + E (GDR) ]

.

from the spin-isospin channel was positive was used to build up Go' and thereby hinder pion condensation. In our estimates this term 1s knocked down a factor of 12 by the averaging over q

,

^{so it is}

smaller than

The calculation in ref. 9 was carried out in finite nuclei, and it is known that q z 0 and q 2 2kf are much more favoured in exchange matrix elements in such calculations than in nuclear matter, so it is likely that our redyction factor of 1/12 is much small. On the other hand, the G o in ~ e f . 2 are only slightly more than half ours.

At first sight, slnce G ' occurs in the induced interaction, ?his would appear to give smaller results. However, the smaller G o means

that their ~ ' ( q ) is substantially more negative than ours at large q

-

^2kf

^,

where most of the phase space is, and this would give them a larger net contribution.

In the coupling of the nuclear GTR to isobars, the transition matrix element shown in fig. 3 is conventionally employed. In order to map the direct matrix element on to particle-particle interactions,

* +

^{direct ( - ) A exchange} Fig. 3 . Transition matrix element.

0 b

C4-484 JOURNAL DE PHYSIQUE

we redraw i t i n f i g . 4 , p r e s e r v i n g t h e l a b e l s a , b , c , d shown i n f i g . 3 . Because t h e i s o b a r h a s T = 3/2

,

S = 3 / 2

,

t h e s t a t e bc

F i g . 4 . The d i r e c t term of f i g . 3 redrawn.

must have T = 1

,

^{S = 1 ( T = 2 o r S =} 2 c o u l d n o t c o n n e c t through t h e s c a l a r i n t e r a c t i o n w i t h t h e lower two-nucleon s t a t e a d ) . So must have ad

.

Thus, o n l y odd r e l a t i v e a n g u l a r momenta ( i n t h e system ad o r i n t h e system b c ) p j e allowed by t h e f ' ~ u l i P r i n c i p l e . T h i s was remarked by Suzuki e t a l . and Arima e t a l .

.

^{Now, a}

major component o f t h e i n t e r a c t i o n comes from p-meson exchange9 ( s e e r e f . 12 f o r a s i m p l e s c h e m a t i c t r e a t m e n t ) , which must n e c e s s a r i l y be o f s h o r t r a n g e , a s i s t h e Lorentz-Lorenz c o r r e c t i o n t o pion exchange.

I n r e f . 11, t h e i n t e r a c t i o n i n t h e exchange c h a n n e l i s r e p r e s e n t e d by*

where V* ( 0 ) = 0 ; V* [ q ) comes from t h e s p i n - s p i n i n t e r a c t i o n . The t h e s i s of r e f . 11 i s t h a t t h e l a r g e G * ' ( o ) c a n c e l s o u t between d i r e c t and exchange i n t e r a c t i o n s , f i g

.

3, l e a v i n g t h e r e l a t i v e l y s m a l l V* ( q )

.

I n comparison w i t h t h e argument of

5

1, t h e t e n s o r i n v a r i a n t i s l e f t o u t h e r e , t h e p o i n t i n r e f G l l l b e i n g how exchange terms i n f l u e n c e t h e Fermi l i q u i d p a r a m e t e r G o

.

Whereas o u r i n t e r e s t i s i n t h e f u l l i n t e r a c t i o n , wherever i t comes from, l e t u s f i r s t make a t r e a t m e n t s t a r t i n g from t h a t o f r e f . 11.

To b e g i n w i t h we s h a l l d r o p

v*!Y)

which, even f o r q r 2kf

,

n e v e r becomes g r e a t e r t h a n 40% of G o ( 0 )

.

(The G * ' ( q ) o f ( 3 . 3 ) d o e s n o t i n c l u d e t e n s o r i n t e r a c t i o n , which gave us o u r extreme q-de- pendence i n

5

1.) We t h u s u s e

Here G ~ * I ( 0 ) i s r e l a t e d t o g o * 1 by m u l t i p l y i n g t h e l a t t e r by t h e d e n s i t y o f s t a t e s a t t h e Fermi s u r f a c e ; S and T a r e t h e t r a n s i t i o n s p i n and i s o s p i n .

With t h e i n t e r a c t i o n ( 3 . 4 ) t h e s e l f s c r e e n i n g of t h e exchange term, f i g . 5, i s e a s y t o c a l c u l a t e ; i t i s :

where w e have a g a i n t a k e n U(q,O) t o be u n i t y . H r e f * / f i s t h e s c a l i n g f a c t o r between NA and NN i n t e r a c t i o n s 1 ' . Note t h a t o n l y t h e s p i n - i s o s p i n component o f t h e f o r c e e n t e r s i n t o t h e s c r e e n i n g now;

t h e o t h e r components c a n n o t f u r n i s h t h e r e q u i s i t e change i n s p i n and i s o s p i n . Note, a l s o , t h a t t h e f a c t o r 1 / 4 of (1.11) i s m i s s i n g h e r e ,

h he

s t a r on G: ( q )

,

e t c . d e n o t e s t r a n s i t i o n p o t e n t i a l , n o t complex con j u g a t e .

F i g . 5. S e l f s c r e e n i n g o f t h e exchange t e r m (induced i n t e r a c t i o n ) . f o r t h e same r e a s o n t h a t it i s n o t p r e s e n t i n t h e exchange term, f i g . 3 .

I n t h e same u n i t s , t h e d i r e c t and exchange t e r m s , f i g . 3, a r e , r e s p e c t i v e l y

{ D i r e c t

1

⁼

fl

^I

Exchange f 0

Thus, f o r G o 1 >> 1

,

t h e s c r e e n i n g p r e c i s e l y removes t h e exchange term, and one i s l e f t w i t h o n l y t h e d i r e c t term. (Note t h a t w i t h i n - c l u s i o n o f The Lindhard f u n c t i o n U(q,O) t h i s i s s t i l l t r u e , r e - q u i r i n g G U(q,O) >> 1 .)

The Z i t u a t i o n h e r e i s s i m i l a r t o t h e w e l l known one i n t h e e l e c - t r o n g a s . The d i r e c t term f o r a Coulomb i n t e r a c t i o n i s

whereas t h e exchange t e r m g e t s s c r e e n e d , becoming

where kFT i s t h e i n v e r s e Fermi-Thomas s c r e e n i n g l e n g t h . Thus, t h e r a t i o o f exchange t o d i r e c t t e r m s ⁺ 0 a s q ^-+ 0

.

I n p r i n c i p l e , t h e d i r e c t t e r m i s n o t s c r e e n e d ; i t e n t e r s a s a k e r n e l i n t o t h e RPA-like i n t e s r a l e q u a t i o n .

-

1ncl;ding now t h e a-dependence o f t h e i n t e r a c t i o n , which w i l l be employed f o r

w = E GTR

we s e e from t h e arguments o f t h e l a s t s e c t i o n t h a t i t would be b e t t e r t o l e a v e o u t t h e denominator i n ( 3 . 2 ) ; i . e . ,

and one s e e s t h a t t h e exchange t e r m , f i g . 3 , i s o v e r s c r e e n e d . T h i s i s p o s s i b l e b e c a u s e we a r e moving up towards t h e p o l e i n t h e p r o c e s s f i g . 5 , t h i s p o l e a r i s i n g from t h e v i r t u a l c o l l e c t i v e e x c i t a t i o n p l u s p a r t i c l e and h o l e . Thus, we come t o t h e c o n c l u s i o n t h a t

Now t h i s i s tremendous. But more i s t o come. Remember t h a t t h e em i r i c a l

r:

( G o v l e f f = 1.68 i n t h e nucleon s e c t o r i s downgraded from G o = 2.34 by s c r e e n i n g due t o i s o b a r s ; namely12

JOURNAL DE PHYSIQUE

F i g . 6 . S c r e e n i n g o f t h e n u c l e a r G o 1

.

t h e g e n e r a l r e s u l t i s

where y i s g i v e n i n r e f . 1 2 , and e s t i m a t e d t o b e y ^s 0.72

.

I n t h e above work, e q . ( 3 . 5 ) , t h e unscreened i n t e r a c t i o n s h o u l d b e used, s i n c e i s o b a r s s h o u l d b e t r e a t e d e x p l i c i t l y . Indeed, i s o b a r - h o l e b u b b l e s s h o u l d a l s o come i n t o ( 3 . 5 ) and t h e y f u r t h e r i n c r e a s e

~ n * '

.

On t h e o t h e r hand, exchange terms p l u s s e l f s c r e e n i n g o f them s h o u l d be added t o t h e n u c l e a r i n t e r a c t i o n , f i g . 6 .

So w e come o u t w i t h t h e c o n c l u s i o n t h a t

i . e . , t h a t t h e r a t i o i s s u b s t a n t i a l l y l a r g e r t h a n would b e g i v e n by

" s c a l i n g " , which would p r e d i c t t h e r a t i o t o b e f * / f = 2

.

R e i n s t i t u t i n g t h e v * ( ~ ) which we dropped i n going from ( 3 . 3 ) t o ( 3 . 4 ) and t h e Lindhard f u n c t i o n U(q,O) w i l l b r i n g t h i s r e s u l t down by a f a c t o r

-

²

^,

s o t h e r e i s no r e a l c o n t r a d i c t i o n w i t h t h e s c a l i n g argument. W e have n o t pursued t h i s q u a n t i t a t i v e l y , b e c a u s e w e b e l i e v e t h a t it i s t a c t i c a l l y p r e f e r a b l e t o i n c l u d e t h e c o n t r i b u t i o n s from t h e t e n s o r i n t e r a c t i o n i n ~ ' ( q )

.

We t h e n p r o j e c t o u t

Thus, r a t h e r l i t t l e o f t h e G:' ( 0 ) from t h e d i r e c t term i s c a n c e l l e d o u t by t h e exchange. From t h e s e l f s c r e e n i n g o f t h e exchange term, f o l l o w i n g t h e arguments of

9

1, one g e t s

+

[ G * ' ( 0 ) 1 2 / 1 2 =

+

^0.46

s o t h a t one i s l e f t e s s e n t i a l l y w i t h j u s t t h e d i r e c t term. I n pro- c e e d i n g i n t h i s way we keep t h e s i z e s of t h e c o r r e c t i o n t e r m s s m a l l e r .

One c o n c l u s i o n i s c l e a r : when t h e Fermi l i q u i d p a r a m e t e r s , such as G o t

,

a r e l a r g e , t h e n h i g h e r - o r d e r e f f e c t s a r e i m p o r t a n t . I t i s d i f f i c u l t t o b e s u r e t h a t t h e a p p r e c i a b l e ones have a l l been i n c l u d e d . T h e r e f o r e , a b e t t e r bookkeeping, such a s t h a t which c o u l d b e p r o v i d e d by a Ward i d e n t i t y , d i s c u s s e d i n t h e p r e v i o u s t a l k by Mannque Rho, would b e h i g h l y d e s i r a b l e .

The importance of t h e s e l f s c r e e n i n g of t h e GTR a r o s e i n d i s - c u s s i o n s w i t h K . Nakayama, S. Krewald and J. S p e t h , J f i l i c h , and I am g r a t e f u l t o them f o r t h e i d e a s developed h e r e ; t h e y s h o u l d n o t b e h e l d

r e s p o n s i b l e f o r my p o s s i b l e o v e r i n f l a t i o n o f them, h o w e v e r . A de- t a i l e d t r e a t m e n t o f t h i s a n d r e l a t e d s u b j e c t s w i l l a p p e a r i n t h e t h e s i s o f K . Nakayama.

R e f e r e n c e s

F . I a c h e l l o a n d A.D. J a c k s o n , P h y s . L e t t . e ( 1 9 8 2 ) 1 5 1 W . H . D i c k o f f , A. F a e s s l e r , H . Miither a n d S . S . Wu, 1 9 8 3

Tiibinqen U n i v e r s i t y P r e p r i n t

0 . S j s b e r g , Ann. P h y s .

2

( 1 9 7 3 ) 39 0 . S j s b e r q , N u c l . P h y s .

A209

( 1 9 7 3 ) 3 6 3

S. Babu a n d G. E . Brown, Ann. P h y s . 78 ( 1 9 7 3 ) 1

G. E . Brown, S.-0. Bzckman a n d J. ~ i s k a n e n , P h y s i c s R e p o r t s , t o b e p u b l i s h e d

E . O s e t a n d P a l a n q u e s - M e s t r e , N u c l . P h y s . ( 1 9 8 1 ) 289 J . D a b r o w s k i a n d P. H a e n s e l , Ann. P h y s . ( N . Y . )

5

( 1 9 7 6 ) 452;

see a l s o G. E . Brown e t a l . , N u c l . P h y s .

A286

( 1 9 7 7 ) 1 9 1 J. S p e t h , V. K l e m t , J . Wambach a n d G. E . Brown, N u c l . P h y s . A343 ( 1 9 8 0 ) 382

T . u z u k i , S . K r e w a l d a n d J . S p e t h , P h y s . L e t t . ( 1 9 8 1 ) 9 A. Arima e t a l . , P h y s . L e t t , ( 1 9 8 3 ) 1 2 6

G. E . Brown a n d Mannque Rho, N u c l . P h y s .

A372

( 1 9 8 1 ) 397