Nuclear Physics A396(1983) 171 c- 180c. o N orth-HoUand Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher.
NUCLEAR
FLUID
DYNAMICS
G. H O L Z W A R T H
FOR GIANT
171c
RESONANCES
a n d G. E C K A R T
Siegen University, Fachbereich 7 - Physik A. R e i c h w e i n s t r . , 59 S i e g e n 21 W.-Germany A b s t r a c t : N u c l e a r f l u i d d y n a m i c s is an a t t e m p t to u n d e r s t a n d n u c l e a r c o l l e c t i v e m o t i o n in t e r m s of one l o c a l m a c r o s c o p i c v e l o c i t y f i e l d c o m m o n to all n u c l e o n s of a g i v e n type. B a s e d on a g e n e r a l i z e d s c a l i n g a s s u m p t i o n for the s i n g l e - p a r t i c l e d e n s i t y m a t r i x it i n c o r p o r a t e s d y n a m i c a l d i s t o r t i o n s of the l o c a l F e r m i s u r f a c e . S o u n d p r o p a g a t i o n m a y thus be c o n s i d e r e d as an a p p r o x i m a t i o n to L a n d a u ' s zero s o u n d m o d e s in F e r m i f l u i d s . F o r f i n i t e s p h e r i c a l n u c l e i the r e s u l t i n g d i f f e r e n t i a l e q u a t i o n s are solv e d for i s o s c a l a r a n d i s o v e c t o r m o d e s of d i f f e r e n t s p i n - p a r i ties. B o u n d a r y c o n d i t i o n s a l l o w for b o u n d d i s c r e t e s o l u t i o n s and for u n b o u n d m o d e s e m b e d d e d in the p a r t i c l e c o n t i n u u m . E x c i t a t i o n e n e r g i e s and t h e i r d e p e n d e n c e on p a r t i c l e n u m b e r A and on L a n d a u p a r a m e t e r s , t r a n s i t i o n d e n s i t i e s , f l o w p a t t e r n s of t r a n s i t i o n c u r r e n t s a n d B - v a l u e s are o b t a i n e d . F o r u n b o u n d sol u t i o n s s t r e n g t h f u n c t i o n s for the e x c i t a t i o n w i t h e x t e r n a l f i e l d s m a y be c a l c u l a t e d .
F l u i d d y n a m i c a l m e t h o d s in a p p l i c a t i o n to g i a n t m u l t i p o l e r e s o n a n c e s of a t o m i c n u c l e i a i m at u n d e r s t a n d i n g s a l i e n t f e a t u r e s of t h e s e c o l l e c t i v e m o d e s w i t h o u t e n t e r i n g into the c o m p l e x i t y of detailed microscopic descriptions. They have found renewed attenL i o n 1 - 1 5 ) f o l l o w i n g an a t t e m p t of B e r t s c h I ) to r e f o r m u l a t e the R P A e q u a t i o n s in t e r m s of a m a c r o s c o p i c s c a l i n g f i e l d w h i c h d e s c r i b e s c o l l e c t i v e d i s p l a c e m e n t s for a l l the n u c l e o n s . W i t h o u t r e f e r r i n g to the RPA, f l u i d d y n a m i c a l e q u a t i o n s h a v e b e e n s u g g e s t e d on the b a s i s of a g e n e r a l i z e d s c a l i n g a s s u m p t i o n for the s i n g l e - p a r t i c l e d e n s i t y m a t r i x 2) s u p p l e m e n t e d b y the r e q u i r e m e n t t h a t the t i m e - d e r i v a t i v e of the s c a l i n g f i e l d ~ s h o u l d c o i n c i d e w i t h the v e l o c i t y f i e l d ~/p c h a r a c t e r i z i n g the flow. O n e t h u s a r r i v e s at a c l o s e d set of e q u a tions which couple macroscopic time-dependent variables like+the d e n s i t y p(~,t) , the c u r r e n t 5(~,t) , the p r e s s u r e t e n s o r Pij (x,t) and p r o v i d e a t r a n s p a r e n t p i c t u r e of the c o l l e c t i v e m o d e s . It h a s b e e n n o t i c e d 3) t h a t the s c a l i n g a s s u m p t i o n i m p l i e s l o w m u l t i p o l a r i t y d i s t o r t i o n s for the l o c a l F e r m i s u r f a c e thus ~ i e l d i n g e s s e n t i a l p a r t s of the r e s t o r i n g f o r c e s for c o l l e c t i v e m o t i o n . F o r an i n f i n i t e F e r m i f l u i d in the c l a s s i c a l l i m i t s c a l i n g c o r r e s p o n d s to a t r u n c a t i o n 4) of the c o u p l e d e q u a t i o n s for the m o m e n t s of the d i s t r i b u t i o n f u n c t i o n at i<2. T h e same set of e q u a t i o n s t h e r e f o r e m a y a l s o be o b t a i n e d b y t r u n c a t i n g the c o u p l e d m o m e n t s of the L a n d a u - V l a s o v e q u a t i o n a f t e r the s e c o n d m o m e n t s'6) w i t h o u t e x p l i c i t u s e of the s c a l i n g p r e s c r i p t i o n . A partial justification of nuclear fluid dynamics through a variational approach based on a single determinant with explicit s p e c i f i c a t i o n of its t i m e - o d d p a r t has b e e n g i v e n r e c e n t l y - ) . It s h o w s the r e m a r k a b l e f e a t u r e t h a t a s i z a b l e c h a n g e in the t r a n s v e r s e s o u n d s p e e d c a u s e d b y i n c l u d i n g o c t u p o l e d i s t o r t i o n s in m o m e n t u m s p a c e is e f f e c t i v e l y c o m p e n s a t e d by the b o u n d a r y c o n d i t i o n s for the f i n i t e s y s t e m so t h a t the s i m p l e r e l a t i o n s of n u c l e a r f l u i d d y n a m i c s are essentially reestablished.
172c
G. HOLZWARTH, G. ECKART
W e s h a l l t h e r e f o r e p u t at the b e g i n n i n g nuclear fluiddynamics ~ ~ m j2 + L = S m s'j d 3 r - / ~ -p- d ~ r - E[s]
the L a g r a n g i a n
of
(I)
T h e s c a l i n g f i e l d s d e t e r m i n e s the d e v i a t i o n of the t i m e - e v e n p a r t of the s i n g l e - p a r t i c l e d e n s i t y m a t r i x ~ f r o m its g r o u n d - s t a t e v a l u e Po by ~(x,x',t)
= exp E(s(x,t).V
+ V-s(x,t))
× (2)
Remembering
that
T = - 2~
x exp ~I( s+( x÷' , t ) "~'
+ ~' .S(X + + ' ,t) ) PO (X,X+ ÷ ') .
the k i n e t i c
density
energy
[A+, + ~ ( x , x ' , t ) ] + X
--X
is g i v e n
by
+
X:X
(3) i
the t o t a l i n t r i n s i c e n e r g y E m a y t h e n be e x p r e s s e d in a l o c a l d e n s i t y a p p r o x i m a t i o n as a f u n c t i o n a l E[~] of the l o c a l s c a l i n g f i e l d and its d e r i v a t i v e s . T h e g r o u n d - s t a t e d e n s i t y po(~) and the g r o u n d s t a t e k i n e t i c e n e r g y T0(~) e n t e r as an i n p u t into E[~] and m a y be t a k e n f r o m a n y f e a s i b l e d e s c r i p t i o n of the g r o u n d - s t a t e ( H a r t r e e Fock, T h o m a s - F e r m i , or ~ v e n a s i m p l e s q u a r e d e n s i t y ) . It s h o u l d be n o t e d that ±n (2) P 0 ( X , X ) g e n e r a l l y zs net r e s t r l c t e d to a d e t e r m i n a n t a l form. W h i l e the f i r s t t e r m in the L a g r a n g i a n (I) m a y be d e r i v e d f r o m the q u a n t u m - m e c h a n i c a l L a g r a n g i a n <% | i 3 t - H l % > for a g e n e r a l t i m e - d e p e n d e n t m a n y - b o d y w a v e f u n c t i o n i} >2) the s e c o n d t e r m r e p r e s e n t i n g the c o l l e c t i v e k i n e t i c e n e r g y has a v e r y s p e c i a l f o r m w h i c h w i l l not be v a l i d in g e n e r a l . H o w e v e r , in the e x p l i c i t c a s e d i s c u s s e d in ref. 7, w h e r e the c o l l e c t i v e e n e r g y l o o k s f o r m a l l y q u i t e d i f f e r e n t from / ( m / 2 ) j 2 / p d 3 r the n u m e r i c a l r e s u l t s still are r a t h e r c l o s e to t h o s e o b t a i n e d f r o m (I). T o p o s t u l a t e the c o l l e c t i v e k i n e t i c e n e r g y in the f o r m c o n t a i n e d in (I) i m p l i e s the v e r y n a t u r a l r e l a t i o n (as is e v i d e n t f r o m v a r i a t i o n w i t h r e s p e c t to ]) •
^
+
s
t
'
+
p
=
O
•
,
(4)
i.e. the t i m e d e r i v a t i v e of -s e q u a l s the f l o w v e l o c i t y ]/p. V a r i a t i o n of (I) w i t h r e s p e c t to ~ l e a d s to the E u l e r - t y p e e q u a t i o n ~ _ I @E[s] m ~
(5)
Eqs. (4) and (5) c o n s t i t u t e the c l o s e d e q u a t i o n s of m o t i o n for the n u c l e a r fluid. F r o m the l o c a l (x'=x) p a r t of (2) the t i m e d e p e n d e n c e of the local d e n s i t y p(x,t) is o b t a i n e d as +
p(x,t)
+
= exp
(~-s(x, ÷ t))
t h e r e f o r e (4) o b v i o u s l y the c o n t i n u i t y e q u a t i o n
P0 (x)
is s u f f i c i e n t
+
v.]
=
o
.
(6) to g u a r a n t e e
the v a l i d i t y
of
(7)
As w e h a v e s t r e s s e d a l r e a d y it is not c l e a r w h e t h e r it is p o s s i b l e to d e r i v e (I) f r o m the q u a n t u m - m e c h a n i c a l L a g r a n g i a n u n d e r s u i t a b l e r e s t r i c t i o n of the v a r i a t i o n a l d e g r e e s of f r e e d o m . F o r a g e n u i n e l y v a r i a t i o n a l a p p r o a c h it w o u l d be n e c e s s a r y to s p e c i f y a l s o e x p l i c i t l y the t i m e - o d d p a r t of the s i n g l e - p a r t i c l e d e n s i t y m a t r i x . T h e p r e s e n t a t i o n we h a v e g i v e n here, h o w e v e r , l e a v e s the t i m e - o d d p a r t o p e n e x c e p t for the r e l a t i o n (4) w h i c h c o n n e c t s the s c a l i n g field, i.e. a q u a n t i t y s p e c i f y i n g the t i m e - e v e n p a r t w i t h the f i r s t m o m e n t
NUCLEAR FLUID DYNAMICS FOR GIANT RESONANCES
173c
of the t i m e - o d d p a r t (namely the c u r r e n t 5)- A l t h o u g h there is thus no c l e a r c o n n e c t i o n w i t h the RPA a c o m p a r i s o n of c o r r e s p o n d i n g results seems a p p r o p r i a t e e s p e c i a l l y after f i n d i n g that the v a r i a t i o n a l d e r i v a t i o n 7) s u p p o r t s the s i m p l e and t r a n s p a r e n t a p p r o a c h p r e s e n t e d here. E l i m i n a t i n g j from (4) and (5) we o b t a i n for s m a l l - a m p l i t u d e h a r m o n i c o s c i l l a t i o n s the d i f f e r e n t i a l e q u a t i o n for the space part of the s c a l i n g field m~200(A)s(x)~
=
[s(x)]
(8)
w h e r e the h e r m i t i a n linear d i f f e r e n t i a l o p e r a t o r ~ is d e r i v e d from l i n e a r i z i n g the v a r i a t i o n of the i n t r i n s i c e n e r g y E w i t h r e s p e c t to the s c a l i n g field ~. For i s o s c a l a r and i s o v e c t o r m o d e s e x p l i c i t exp r e s s i o n s for ~ have b e e n d i s c u s s e d in refs. 8 and 9. D i f f e r e n t m u l t i p o l e s and p a r i t i e s s e p a r a t e for s p h e r i c a l g r o u n d - s t a t e d e n s i ties P0 and the e l e c t r i c (E) and m a g n e t i c (M) type s o l u t i o n s of (8) m a y be o b t a i n e d in the form SE = ~(~ll(r)Y~m (~))
+ ~x ( ~ L ( r ) ~ m ( ~ ) )
,
(9a)
SM = ~ ( r ) Y ~ m ( ~ )
(95)
A n i n t e r e s t i n g f e a t u r e of (8) w i t h Po t a k e n from a H a r t r e e - F o c k c a l c u l a t i o n is that for e n e r g i e s he
d3x
= 6 nm
(10)
w h i l e for here F (8) shows a c o n t i n u u m of s o l u t i o n s w h i c h (when m u l t i p l i e d by /po) b e h a v e like s p h e r i c a l w a v e s for large v a l u e s of I~I and m a y be n o r m a l i z e d to the ~ - f u n c t i o n / Po s ~ ' s e ' d 3 x From these solutions transition m a y be o b t a i n e d t h r o u g h 6Q =
(P0S)
W i t h the d e f i n i t i o n f i e l d ~(x)
,
densities
] = ieoos
= ~ h / ( 2 m ~ n)
~p and t r a n s i t i o n
currents
(12) amplitude
$ 60n(X)~(x) states
(11)
•
of the t r a n s i t i o n
B(E£) v a l u e s for d i s c r e t e tinuous solutions S~(e)
= 6(e-e') .
and
for a g i v e n
local
d3x
strength
(13) functions
for
the con-
= I ~ ( e - e ' ) l < e ' l~lO>I2de
(14)
m a y be c a l c u l a t e d . For p o s i t i v e - o d d - p o w e r c l o s e d e x p r e s s i o n m a y be d e r i v e d ~°) m 2 K + 1 ~ / ( h a ~ ) 2 K + I s ~ ( e ) d e = ½ ~ ,h2,K+1 --)
energy-weighted
÷ + IOo?~" (F ~)
sums
a
K [ ~ ] d3x (15) ~o w h i c h shows that for m I and m 3 n u c l e a r fluid d y n a m i c s r e p r o d u c e s the exact quantum-mechanical sum rulesl 9) . The fact that two q u a n t u m - m e c h a n i c a l sum r u l e s are f u l f i l l e d is a r a t h e r s a t i s f a c t o r y f e a t u r e of the f l u i d - d y n a m i c a l f o r m a l i s m , a l t h o u g h it m a y also be the o r i g i n of p r o b l e m s in c a s e s w h e r e l o w - l y i n g states c o n t r i b u t e a p p r e c i a b l y to
174c
G. HOLZWARTH, G. ECKART
6
g
~
A =1o00
\\
~fm)
Fig. I. T r a n s i t i o n d e n s i t i e s for the f i r s t and s e c o n d e x c i t e d isos c a l a r m o n o p o l e m o d e s for s m o o t h s u r f a c e (A=IOOO) . D a s h e d lines are B e s s e l f u n c t i o n s Jo(kr) w i t h k o b t a i n e d f r o m the c a l c u l a t e d f r e q u e n cies and the l o n g i t u d i n a l s o u n d speed in the n u c l e a r i n t e r i o r .
\/~,r)
T=0
/
/ I'0
I'5
2'0
r(fm)
Fig. 2. C o m p a r i s o n b e t w e e n the RPA r e s u l t (full line) and the f l u i d d y n a m i c a l r e s u l t (dashed line) for the r a d i a l v e l o c i t y of the f i r s t e x c i t e d i s o s c a l a r m o n o p o l e r e s o n a n c e (from ref.15) . the t o t a l sum v a l u e s . It s e e m s t h a t the s c a l i n g a p p r o a c h is too limited to d e s c r i b e a l s o l o w - l y i n g q u a d r u p o l e and o c t u p o l e s t a t e s : t h e r e fore in o r d e r to o b t a i n the full sum r u l e v a l u e the g i a n t s t a t e s m a y be a f f e c t e d in an u n p h y s i c a l way. In such c a s e s it m i g h t be n e c e s s a r y to g o b e y o n d the s c a l i n g a p p r o x i m a t i o n ' S ) . L e t us in the f o l l o w i n g d i s c u s s a f e w t y p i c a l r e s u l t s of the f o r m a l i s m o u t l i n e d above. M o r e d e t a i l e d i n v e s t i g a t i o n s m a y be f o u n d in refs. 6, 8, 9, 12, 13 and 14. Fig. I s h o w s t r a n s i t i o n d e n s i t i e s for the f i r s t two e x c i t e d i s o s c a l a r m o n o p o l e states. E v i d e n t l y t h e y r e p r e s e n t t y p i c a l c o m p r e s s i v e m o d e s w h i c h are in the n u c l e a r i n t e r i o r w e l l a p p r o x i m a t e d by B e s s e l f u n c t i o n s Jo(kr), w i t h f r e q u e n c i e s conn e c t e d to k t h r o u g h the l o n g i t u d i n a l s o u n d s p e e d c L = ( 3 / 5 + F o / 3 ) 1 / 2 v F, w h e r e F o is the (£=O) L a n d a u p a r a m e t e r of the u n d e r l y i n g e n e r g y f u n c tional. T h e r e s u l t i n g r a d i a l v e l o c i t y is in e x c e l l e n t a g r e e m e n t w i t h
NUCLEAR FLUID DYNAMICS FOR GIANT RESONANCES
T
500
I
t=0
I
400 4-3()0
200
~
I
175c
A:80
\ \
\~-A = 40 \
\\
100'
\ \ k
J
15
2O
hLu
Fig. 3. S t r e n g t h nuclei with A=40
\
~ 25
[ MeV]
f u n c t i o n S(e) for m o n o p o l e e x c i t a t i o n s for two N=Z a n d A=80. G r o u n d - s t a t e p r o p e r t i e s are t a k e n f r o m a Hartree-Fock calculation.
a c o r r e s p o n d i n g R P A - r e s u l t , as d e m o n s t r a t e d in fig. 2 (from ref. 15) for a v e r y l a r g e s y s t e m (A=4096) w h e r e the e n e r g y is w e l l b e l o w the F e r m i e n e r g y E F and t h e b r e a t h i n g m o d e is o b t a i n e d as a d i s c r e t e b o u n d state. F o r s m a l l e r s y s t e m s the m o n o p o l e s t r e n g t h is d i s t r i b u t e d c o n t i n u o u s l y a b o v e EF; the s t r e n g t h f u n c t i o n (14) for ~ = r 2 Y o o is p l o t t e d in fig. 3 for A = 4 0 a n d A=80. T h e r e s u l t i n g s h a p e s a g r e e v e r y c l o s e l y w i t h L o r e n t z i a n s a l t h o u g h at the l o w - e n e r g y side S(~) g o e s to zero at e = £ F a n d the d e c r e a s e for h i g h v a l u e s of ~ is s l i g h t l y f a s t e r than for a L o r e n t z i a n . Figs. 4 a n d 5 s h o w r e s u l t s for i s o s c a l a r q u a d r u p o l e m o d e s . A m a j o r a c h i e v e m e n t for the f l u i d - d y n a m i c a l a p p r o a c h w a s t h a t it c o r r e c t l y o b t a i n e d the A - I / 3 d e p e n d e n c e I-3) for the e n e r g y of an a l m o s t -pure s u r f a c e o s c i l l a t i o n . Fig. 4 s h o w s t h a t the n u m e r i c a l l y o b t a i n e d 6p for the f i r s t e x c i t e d s t a t e is in v e r y g o o d a g r e e m e n t w i t h the s t a n d a r d a n s a t z for a s u r f a c e mode. A b o v e t h i s strong, a l m o s t inc o m p r e s s i b l e m o d e t h e r e o c c u r s a t y p i c a l c o m p r e s s i v e q u a d r u p o l e state w i t h a m u c h r e d u c e d B(E2) s t r e n g t h (~10-20% of the s u r f a c e mode) w h i c h s h o w s a v e r y i n t e r e s t i n g f l o w p a t t e r n g i v e n in fig. 5(b) w h e r e the f l o w l i n e s of the v e l o c i t y f i e l d ~/p are p l o t t e d in the x-z plane. W h i l e the g i a n t s u r f a c e m o d e s h o w s a f l o w p a t t e r n (fig. 5(a) s i m i l a r to i r r o t a t i o n a l flow, the s e c o n d e x c i t e d s t a t e d i s p l a y s a v o r t e x i n s i d e t h e n u c l e u s w i t h the f l o w l i n e s b e i n g a l m o s t p a r a l l e l to the n u c l e a r s u r f a c e . It s h o u l d be noted, h o w e v e r , t h a t a l s o the g i a n t m o d e c o n t a i n s s t r o n g t r a n s v e r s e c o m p o n e n t s , w h i c h m i g h t be an a r t i f a c t Is) d u e p e r h a p s to the m i s s i n g l o w - l y i n g q u a d r u p o l e states. A v e r y s i m i l a r p a t t e r n e m e r g e s for the i s o v e c t o r d i p o l e case. In this c a s e the s t a t e w i t h the v o r t e x f l o w p a t t e r n (fig. 6(a)) o c c u r s at an e n e r g y b e l o w the s t r o n g s u r f a c e m o d e (fig. 6(b)) d u e to a r e d u c t i o n of the s u r f a c e s y m m e t r y e n e r g y in a s i t u a t i o n w h e r e m a n y f l o w l i n e s a r e p a r a l l e l to the s u r f a c e . T h u s n u c l e a r f l u i d d y n a m i c s r e p r o d u c e s the s p l i t t i n g of the g i a n t i s o v e c t o r d i p o l e w h i c h is k n o w n f r o m R P A c a l c u l a t i o n s 1 6 ) . F o r a c o m p a r i s o n w i t h the f l o w p a t t e r n s r e s u l t i n g f r o m R P A fig. 7 s h o w s v e l o c i t y f i e l d s for the two s t r o n g e s t T=I I=I s t a t e s in a R P A c a l c u l a t i o n w i t h a r a t h e r s o p h i s t i c a t e d c h o i c e of the
176c
G. HOLZWARTH,
G. ECKART
69
/ /
/
A = 1000
X
5
\
10
15
r(fm)
Fig. 4. T r a n s i t i o n d e n s i t i e s for the f i r s t a n d s e c o n d e x c i t e d isos c a l a r q u a d r u p o l e states. T h e d o t t e d line i n d i c a t e s the T a s s i e d e n sity @p[ = r Z - 1 D p o / D r ; the d a s h e d line is the B e s s e l f u n c t i o n J2(kr) •
z(frn)
/_<.o... L=2
7"
A=
z(frn) L = 2
208
,_v
7-
A = 208
T = 0
1%w =
11.8MeV
\
1
2
3
4
5
6
7
x(fm)
Fig. 5. F l o w p a t t e r n s for the s e l f - c o n s i s t e n t s m o o t h s u r f a c e (A=208) for the f i r s t two e x c i t e d T=O, I=2 s t a t e s w i t h (a) E = 9 . 4 MeV, B ( E 2 ) = 1430 e 2 f m 4 and (b) E = 1 1 . 8 MeV, B ( E 2 ) = 2 8 3 e 2 f m 4. t w o - b o d y forcelV) . As an e x a m p l e for p u r e l y t r a n s v e r s e , m a g n e t i c m o d e s let us fin a l l y d i s c u s s the c o n v e c t i o n a l (S=O T=0) m a g n e t i ~ q u a d r u p o l e state. In t h i s c a s e the e x c i t a t i o n o p e r a t o r B + E ~ . ~ = ~ ( r ) Y 2 2 0 - ~ = ( ~ ( r ) / r 2 ) Z L z d e s c r i b e s a r o t a t i o n a r o u n d the z - a x i s by an a n g l e w h i c h is p r o p o r t i o n a l to z, i.e. it r e p r e s e n t s a t w i s t i n g m o t i o n of the n u c l e u s w h e r e the top is r o t a t e d in o p p o s i t e p h a s e to the b o t t o m . Fig. 8 s h o w s t h a t the n u m e r i c a l l y o b t a i n e d Po~ is F a t h e r c l o s e to p0 r2 t h e r e f o r e the m o d e c a n s i m p l y be t a k e n as zL z. D e p e n d i n g on the surface p r o f i l e the e n e r g y of this t w i s t i n g m o d e lies a r o u n d 4 5 A - 1 / 3 M e V in c l o s e c o r r e s p o n d e n c e to R P A r e s u l t s for 2- states1?) . M i c r o s c o p i c a l l y zL z i n v o l v e s I ~ e 0 e x c i t a t i o n s so we e x p e c t t h a t this s t a t e m i g h t be m i x e d w i t h the I ~ e 0 s p i n - f l i p (S=I) m o d e and the c o r r e s p o n d i n g i s o v e c t o r s t a t e s d u e to s p i n - o r b i t and e x c h a n g e f o r c e s in
NUCLEAR
FLUID
DYNAMICS
FOR GIANT
RESONANCES
177c
z(frn)
z(fm)
6
6-
5
5.
/
4 3
2
1
2
3
4
4
I,
5
6
/
__
3
/
2
1
4
x(fm)
5
x(fm)
6
Fig. 6. Flow patterns of the first two T=I, ~=I states of 40Ca for s e l f - c o n s i s t e n t smooth surface: (a) E=14 ~eVL B(EI)=O.3 e2fm 2, (b) E=20.I MeV, B(E1)=2.2 e2fm 2 z(fm
z(fm)
2
1
m
1
2
3
4
5
6
x(fm,
1
2
3
4
6
x(frn)
~ig. 7. F l o w patterns of T=I, £=I states of 40Ca resulting from a m i c r o s c o p i c RPA-type calculation: (a) E=16.9 MeV, B(EI)=O,2 e2fm 2, (b) E=19 MeV, B(E1)=2.71 e2fm 2 the r e s i d u a l interactionl 8). Fig. 9 shows the DWBA formfactors for inelastic electron scattering into these four 2- components (without any q u e n c h i n g for the g-factors). A l t h o u g h the m a x i m u m of the S=O, T=O f o r m f a c t o r occurs in the v i c i n i t y of the S=I, T=I m i n i m u m the mixing among these components and admixtures of other m u l t i p o l a r i t i e s will make a clear o b s e r v a t i o n of the c o n v e c t i o n a l 2- difficult. To conclude this brief survey we remark that, naturally, the extent to w h i c h the simple a p p r o a c h d e s c r i b e d here applies to reality is not clear from the outset. Its range of v a l i d i t y can, however, be checked against d e t a i l e d m i c r o s c o p i c c a l c u l a t i o n s and w i t h i n that range it p r o v i d e s a very t r a n s p a r e n t picture of the main physical i ng r e d i e n t s w h i c h d e t e r m i n e essential p r o p e r t i e s of giant r e s o n a n c e s in nuclei. By comparing with p r e v i o u s h y d r o d y n a m i c a l ("liquid drop '~)
178c
G. HOLZWARTH, G. ECKART
9~ d~q.2
/ i
//?F
ii ;,....................._/i ............. .
i
r cA)
5
Fig. 8. R a d i a l d e p e n d e n c e of the c u r r e n t d e n s i t y for the f i r s t e x c i ted 2- (twist) m o d e for A = 2 0 8 . I n c l u d e d for c o m p a r i s o n is por 2 (dashd o t t e d line) and the s q u a r e - d e n s i t y r e s u l t J2(kr)@ (Ro-r) (dashed line)
IF~qJl ~
~Zr
40MO -6 / , i
20~I0 -~
~ !~
DWB4 --S=O
i
/
~
8=~5" T=O
....
S=O
T=I
....
S=I
r=!
i
20~10 -6
f0~10 6
0
I
Fig.
9. F o r m
factors
Js
,.'o
,.'s
q r,-"l
for the four d i f f e r e n t c o m p o n e n t s c a l c u l a t e d by D W B A for 90Zr.
S=O,I;
T=O,I
m o d e l s we m a y u n d e r s t a n d w h y t h e s e m o d e l s w e r e a b l e to d e s c r i b e special s t a t e s like the b r e a t h i n g m o d e or the i s o v e c t o r d i p o l e r e s o n a n c e w i t h g o o d s u c c e s s a l t h o u g h t h e y w e r e b a s e d on the a s s u m p t i o n of l o c a l e q u i l i b r i u m w h i c h c o n t r a d i c t s our k n o w l e d g e a b o u t the m e a n free p a t h of n u c l e o n s in n u c l e i . O n the o t h e r h a n d the p r e s e n t a p p r o a c h w h i c h m a y be c o n s i d e r e d as an a p p r o x i m a t i o n to L a n d a u ' s zero s o u n d a p p l i e s o n l y for s i t u a t i o n s in w h i c h all r e l e v a n t r e l a x a t i o n t i m e s are l a r g e as c o m p a r e d to the t y p i c a l o s c i l l a t i o n times.
NUCLEAR FLUID DYNAMICS FOR GIANT RESONANCES
179c
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