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Submitted on 1 Jan 1984
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NUCLEAR ELASTICITY APPROACH TO GIANT
RESONANCES
S. Jang
To cite this version:
NUCLEAR E L A S T I C I T Y APPROACH TO G I A N T RESONANCES
S . Jang
I n s t i t u t des Sciences Nucle'aires, 38026 Grenoble Cedex, France
Resume - La fragmentation des resonances g6antes monopolaire e t quadru-
jj6TZEe i s o s c a l a i r e s dans l e s noyaux d6formGs a 6 t & 6 t u d i e e dans l e cadre de 1 'e l a s t i c i t 6 n u c l e a i r e .
A b s t r a c t
-
The fragmentation o f t h e i s o s c a l a r g i a n t monopole and quadrupole resonances i n deformed n u c l e i has been s t u d i e d w i t h i n the framework o f t h e nuclear e l a s t i c i t y .The present communication i s concerned w i t h t h e i s o s c a l a r g i a n t resonances i n deformed n u c l e i , i n p a r t i c u l a r t h e fragmentation o f the g i a n t monopole and
quadrupole resonances. The model i n which we have worked i s t h e nuclear e l a s t i c i t y
which has been proposed i n i t i a l l y by G. Bertsch /1/ and a l s o by C.Y. Wong /2/.
The conception o f the n u c l e a r e l a s t i c i t y i s s t i l l u n f a m i l i a r w i t h n u c l e a r p h y s i c i s t s b u t I w i l l n o t go i n t o the discussion on the foundation o f t h i s model i n the
p r e s e n t c o n t r i b u t i o n .
Because t h e monopole o p e r a t o r has no d i r e c t i o n a l p r o j e c t i o n , the g i a n t monopole resonance can n o t be s p l i t i n t o f r a ~ m e n t s i n deformed n u c l e i . However,
been observed / 3 / t h a t t h e g i a n t monopole resonance has two components i n
:kehTs4~m nucleus
.
The c o u p l i n g between the monopole and quadrupole o s c i l l a t i o n sby means o f nuclear deformation parameters can p r o v i d e an e x p l a n a t i o n o f t h i s phenomenon.
The nuclear e l a s t i c i t y approach t o nuclear c o l l e c t i v e motions, such as t h e g i a n t resonances, i s e s s e n t i a l l y t o consider t h e n u c l e a r m a t t e r as an e l a s t i c s o l i d body which can v i b r a t e as a whole. As a m a t t e r o f f a c t , the fundamental equation o f motion i n t h i s approach i s t h e s o c a l l e d Lame equation f o r uniform, p e r f e c t l y e l a s t i c medium:
where h and p t h e Lam@ c o e f f i c i e n t s , t h e displacement v e c t o r i n the e l a s t i c medium,
8
the body f o r c e and p t h e d e n s i t y . For t h e n u c l e a r e l a s t i c medium t h e Lam& c o e f f i c i e n t s are shown t o bewhere m i s the e f f e c t i v e nucleon mass, k f t h e Fermi momentum, and K t h e n u c l e a r
c o m p r e s s i b i l i t y . The problem i s now t o solve t h e Lame equation under appropriate
boundary c o n d i t i o n s . For exemple, we can impose a simple boundary c o n d i t i o n which
s t a t e s t h a t t h e s t r e s s components vanish on the n u c l e a r surface. This boundary
c o n d i t i o n t o g e t h e r w i t h t h e assumption o f constant d e n s i t y y i e l d s an eigenvalue equation f o r f r e e v i b r a t i o n s of s p h e r i c a l n u c l e i . This eigenvalue equation takes
t h e form
.
, ,JOURNAL DE PHYSIQUE 1 ( a - l ) ( a t z )
4
-
z a ( a - l ) ( a + z ) ) J a t l ( n )+
[ - 2 +
7+
n 2 ~ ~ ( n )l j L ( ~ )
=O,
n
2 where5
=AR
and i- 2 = m R 0 2 2,
u b e i n g t h e f r e q u e n c y o f v i b r a t i o n .X
+2u 0 !J T h i s e i g e n v a l u e e q u a t i o n may be compared w i t h a c o r r e s p o n d i n g e q u a t i o n w h i c h a r i s e s from t h e w e l l known boundary c o n d i t i o n i n t h e hydrodynamical model, namely, no f l o w s across t h e n u c l e a r s u r f a c e Ro.
T h i s s t a t e m e n t l e a d s t owhere k i s t h e wave number i n t h e hydrodynamical e q u a t i o n . Once t h e e i g e n v a l u e s a r e known, t h e g i a n t resonance e n e r g i e s f o r s p h e r i c a l n u c l e i can be e v a l u a t e d u s i n g t h e n u m e r i c a l y a l ues o f t h e Lam6 c o e f f i c i e n t s f o r w h i c h use o f K = L = 2 2 0 MeV and k =1.3fm- y i e l d a c o r r e c t o r d e r o f magnitude. F o r a more p r e c i s e c a l c u l a t i o n ,
i f
i s necessary t o i n t r o d u c e a r e a l i s t i c p a r a m e t r i z a t i o n o f t h e s e q u a n t i t i e s as w e l l as t h e e f f e c t i v e n u c l e o n mass. M o d i f i c a t i o n o f t h e boundary c o n d i t i o n so as t o i n c l u d e s u r f a c e t e n s i o n and Coulomb i n t e r a c t i o n improves c e r t a i n l y t h e n u m e r i c a l r e s u l t s o f t h e g i a n t resonance e n e r g i e s .F o r deformed n u c l e i , however, t h e problem i s more c o m p l i c a t e d owing t o t h e d i f f i c u l t y o f s o l v i n s t h e Lam6 e q u a t i o n i n s p h e r o i d a l c o o r d i n a t e systems. One method o f o v e r c o m i n ~ t h i s d i f f i c u l t y i s t o a p p l y f i r s t t h e v a r i a t i o n a l p r i n c i p l e t o t h e e q u a t i o n o f m o t i o n and t h e n s o l v e i t by assuming a s m a l l d e v i a t i o n o f n u c l e a r f i g u r e f r o m t h a t o f t h e s p h e r i c a l n u c l e u s . F o r exemple, t h e v a r i a t i o n a l e x p r e s s i o n f o r t h e s c a l a r H e l m h o l t z e q u a t i o n 2 2
?$
t k $ = O , i s s i m p l y k 2 ,< j 1 ? $ l 2 d ~ j l @ 1 2 d ~where k i s t h e wave number. However, t h e procedure o f w r i t i n g down a s i m i l a r v a r i a t i o n a l e x p r e s s i o n f r o m t h e Lame e q u a t i o n i s n o t s t r a i g h t f o r w a r d . A f t e r a r a t h e r l e n g t h y c a l c u l a t i o n , we a r r i v e a t t h e e x p r e s s i o n
where e . a r e t h e s t r a i n t e n s o r s expressed h e r e i n t h e s p h e r i c a l p o l a r c o o r d i n a t e s . F o r exemljple,
1 1 aur "8 1
e12 = eZ1 = erg= ( -+
+
- a r-
-U r e ) 'where
y
and ug a r e r e s p e c t i v e l y t h e r and 8 components o f t h e d i s p l a c e m e n t v e c t o r u':F o r c o n s t a n t d e n s i t y , t h e Lame e q u a t i o n has an a n a l y t i c a l s o l u t i o n . When we i n t r o d u c e t h i s s o l u t i o n as t r i a l f u n c t i o n i n t o t h e v a r i a t i o n a l e q u a t i o n , t h e n t h e r i g h t hand s i d e reduces back t o t h e i n i t i a l frequency f o r s p h e r i c a l n u c l e i . F o r deformed n u c l e i , t h e upper l i m i t o f r a d i a l i n t e g r a l s i n v o l v e d i n t h e v a r i a t i o - n a l e q u a t i o n i s n o t a c o n s t a n t r a d i u s b u t a deformed one which i s a f u n c t i o n o f angles as we1 1 as d e f o r m a t i o n parameters. Therefore, t h e e x p l i 1 c i t e v a l u a t i o n o f t h e r i g h t hand s i d e g i v e s an e x p r e s s i o n w h i c h c o n t a i n s t h e n u c l e a r d e f o r m a t i o n parameters i n a d d i t i o n t o t h e i n i t i a l o s c i l l a t i o n f r e q u e n c y . The n u c l e a r r a d i u s
i s now R = R ( 1 + E . )
,
where E . a r e t h e increments. Assuming a quadrupolewhere fRm-Rh,, and gRm,Q,m, a r e t h e r e s u l t s o f i n t e g r a l s e v a l u a t e d a l s o up t o Having performed a l l a n g u l a r i n t e g r a l s , we g e t a s i m p l e r e s u l t o f w f o r deformed n u c l e i ;
2 'R + d ~ S ~ m
W =
a~ + b ~ S ~ m
3
where aQ,bQ,cR and ddQ a r e t h e r e s u l t i n g r a d i a l i n t e g r a l s e v a l u a t e d up t o f i r s t o r d e r o f t h e c o l l e c t i v e v a r i a b l e s ct
.
ThecRm
i s t h e g e o m e t r i c a l f a c t o r a r i s i n gfrom t h e n u c l e a r d e f o r m a t i o n and "which takes t h e form
2 1
I(
a(a+l)-3m }e3+
Z ~ ( a + l ~ { e3+
2 ~ ~ ( ~ ) } 6 ~ ~ l.
<am = (~R-I)(ZQ+~)
The g e o m e t r i c a l f a c t o r vanishes f o r t h e monopole, t h a t i s f o r R and m equal t o zero. T h e r e f o r e , t h e s t a t i c a l d e f o r m a t i o n a l o n e has no e f f e c t on t h e g i a n t monopole resonance, as was expected.
F i g . 1 shows an exemple o f t h e s p l i t t i n c j o f t h e i s o s c a l a r ~ i a n t quadrupole resonance i n a x i a l l y deformed n u c l e i
.
Here t h e o r d i n a t e i n d i c a t e s t h e r a t i o o f t h e o s c i l l a t i o n frequency o f deformed n u c l e i , W;,
t o t h a t o f SPLITTING OF G Q R s p h e r i c a l n u c l e i , AS we see, A=ISO t h e g i a n t q u a d r u p o l p 2 ' r e s o n a n c e i s 11- s p l i t i n t o t h r e e components a c c o r d i n g t o t h e values o f rn. I t i s n o t e d t h a t i n t h e p o l a r diagram f o r t h e deforma- w:. t i o n parameters B and y, t h e p o i n t s 6 2 64~-"'- l y i n g on t h e axes correspond t o a x i a l l y symmetric shapes and t h e s i x d i f f e r e n t p o i n t s , one i n each s e c t o r ,f i r s t o r d e r o f c o l l e c t i v e v a r i a b l e s . The frequency w a r i s e s from t h e c o u p l i n g and
w i s t h e i n i t i a l frequency b e f o r e c o u p l i n g . We now remark t h a t t h e d i f f e r e n t i a t i o n 'of t h i s e q u a t i o n w i t h r e s p e c t t o t h e v a r i a t i o n a l parameters
r
must v a n i s h i n accordance w i t h t h e r e q u i r e m e n t o f t h e v a r i a t i o n a l p r i n c i p l e . " As a consequence, we o b t a i n t h r e e l i n e a r , homogeneous e q u a t i o n s f o rrim
;which are o b t a i n e d by r e f l e c t i o n i n a9
t h e axes, r e p r e s e n t t h e same shape o f t h e nucleus. T h e r e f o r e , t h e s e c t o r - a y = o b Y =n/3 c y = n 6 (y=O,y=n/3) i s e q u i v a l e n t t o the F ~ Q 1 a1 a2 03 04 s e c t o r (y=-21~/3,y=n) i n t h e p o l a r
diagram. Use o f t h e values y=O and
IT s i m p l i f i e s t h e numerical c a l c u l a t i o n .
The c o u p l i n g o f t h e monopole o s c i l l a t i o n w i t h t h a t o f t h e quadrupole can now be achieved b y i n t r o d u c i n g a t r i a l f u n c t i o n o f t h e t y p e
+ + + +
=
roo uoo
+ r 2 0 u20 + r22 u22'
where a r e t h e s o l u t i o n s o f t h e Lam6 e q u a t i o n p r i o r t o c o u p l i n g and TRm
a r e theQm v a r i a t i o n a l parameters. The f i r s t t e r m describes t h e monopole o s c i l l a -
JOURNAL
DE
PHYSIQUEwhere a . . and b . . are various f a c t o r s r e s u l t i n g from f a c t o r i z a t i o n s o f terms a f t e r d i f f e r e n t d a t i o n J l t h r e s p e c t t o I'Rm and c o n t a i n the square o f w. Here we have used
the N i l s s o n deformation parameter 6 i n stead o f
6
.
This system o f equations hasa non-zero s o l u t i o n i f and o n l y i f t h e determinant c o n s t r u c t e d w i t h t h e f a c t o r s
before the v a r i a t i o n a l parameters, Too,
r20
and dr2 vanishes. Therefore, t h e, 7
determinantal equation leads t o a t h i r d o r d e r equation o f wL;
This cubic equation gives g e n e r a l l y t h r e e r e a l s o l u t i o n s f o r p h y s i c a l l y meaningful
values o f the deformation parameters 6 and y .
Fig.2 shows the r e s u l t o f t h e c o u p l i n g f o r t h r e e values o f y, namely 0"
,
60" and 180". The numerical11-
1.
0/'o0
a9
b e f o r e coupl i n g , whereas t h e lower component i s near 64A-1/3 MeV which i s t h e g i a n t quadrupole resonance energy f o r s p h e r i c a l n u c l e i . I t i s t o be
remarked t h a t t h e c o u p l i n g i s m e a n i n ~ f u l f o r s u f f i c i e n t l y l a r g e values o f 6 and t h i s i s seen from t h e f a c t t h a t t h e c o u p l i n g s t r e n g t h depends m a i n l y on the
deformation parameter 6 . Fig.3 shows t h e
same c a l c u l a t i o n b u t f o r the r e g i o n o f
A=230. The general f e a t u r e i s very s i m i l a r 1 1
t o t h a t o f t h e r e g i o n o f A=150. F i g . 4
d i s p l a y s t h e r e s u l t s o f c o u p l i n g i n energy u n i t s f o r a f i x e d value o f 6, 0.3. The g i a n t monopole resonance energy b e f o r e
c o u p l i n g i s about a t 80A-1/3 MeV. A f t e r 1
coup1 i ng
,
the lower component i s now w,,,a t 63A-1/3 MeV f o r p r o l a t e n u c l e i , whereas
t h e h i g h e r component i s a t 82A-1/3 MeV
0 9 -
which i s n o t very much d i f f e r e n t from the i n i t i a l value b e f o r e coupling. The p o s i - t i o n s o f energies f o r y=60° and 180" a r e a l s o shown.
The c o u p l i n g between the monopole and quadrupole o s c i l l a t i o n s a f f e c t s a l s o the g i a n t quadrupole resonance. F i g.5 shows how t h e g i a n t quadrupole resonance
energies ,which are already s p l i t b e f o r e o I oe S 03 0 4
coup1 i n g , change t h e i r p o s i t i o n s a f t e r FI 9 3
coupling. Contrary t o the g i a n t monopole
resonance, we have now a h i g h e r component
which i s new. Apart from t h e h i g h e r component, the o t h e r components have no
s u b s t a n t i a l d i f f e r e n c e from t h e i n i l t i a l p o s i t i o n s .
c a l c u l a t i o n has been performed using
F R A G M E N T A T I O N OF G M R t h e m o d i f i e d eigenval ue e q u a t i o n which
includes b o t h s u r f a c e t e n s i o n and
Coulomb energy, and t h e parameters f o r
t h e Lame c o e f f i c i e n t s are those o f r e f 2 .
..
- -
pzn
81.2 82 8 1 . ( 1 8 0 69.8
-
64.3-
62.8 v =o y=+ Y = W betare CO"Pl~"0 a l t e r c o u ~ l l o ( i 8a2 7811-
777A detailed description of the nuclear e l a s t i c i t y approach t o g i a n t resonances of deformed nuclei, including both s t a t i c a l deformation and f a s t nuclear r o t a t i o n , will be published elsewhere.
1 ) G.F. Bertsch,
