OPTIMUM COLLECTIVE SUBMANIFOLD IN RESONANT CASES BY THE SELF-CONSISTENT COLLECTIVE-COORDINATE METHOD FOR LARGE-AMPLITUDE COLLECTIVE MOTION

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OPTIMUM COLLECTIVE SUBMANIFOLD IN RESONANT CASES BY THE SELF-CONSISTENT

COLLECTIVE-COORDINATE METHOD FOR LARGE-AMPLITUDE COLLECTIVE MOTION

Y. Hashimoto, T. Marumori, F. Sakata

To cite this version:

Y. Hashimoto, T. Marumori, F. Sakata. OPTIMUM COLLECTIVE SUBMANIFOLD IN RES- ONANT CASES BY THE SELF-CONSISTENT COLLECTIVE-COORDINATE METHOD FOR LARGE-AMPLITUDE COLLECTIVE MOTION. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-79-C2-85. �10.1051/jphyscol:1987213�. �jpa-00226478�

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JOURNAL DE PHYSIQUE

Colloque C 2 , supplement au n o 6, Tome 48, juin 1987

OPTIMUM COLLECTIVE SUBMANIFOLD IN RESONANT CASES BY THE

SELF-CONSISTENT COLLECTIVE-COORDINATE METHOD FOR LARGE-AMPLITUDE COLLECTIVE MOTION

Y. HASHIMOTO, T. MARUMORI and F. SAKATA*

Institute of Physics, University of Tsukuba, Sakura-Mura, Niihari-Gun, Ibaraki 305, Japan

" ~ n s t i t u t e for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan

ABSTRACT W i t h t h e p u r p o s e o f c l a r i f y i n g c h a r a c t e r i s t i c d i f f e r e n c e o f t h e o p t i m u m c o l l e c t i v e s u b m a n i f o l d s i n n o n r e s o n a n t a n d r e s o n a n t c a s e s , we d e v e l o p a n improved method o f s o l v i n g t h e b a s i c e q u a t i o n s o f t h e s e l f - c o n s i s t e n t c o l l e c t i v e - c o o r d i n a t e (SCC) method f o r l a r g e - a m p l i t u d e c o l l e c t i v e m o t i o n . I t i s shown t h a t , i n t h e r e s o n a n t c a s e s , t h e r e i n e v i t a b l y a r i s e e s s e n t i a l c o u p l i n g t e r m s which b r e a k t h e ' l m a x i m a l - d e c o u p l i n g t f p r o p e r t y o f t h e c o l l e c t i v e m o t i o n , a n d we h a v e t o e x t e n d t h e optimum c o l l e c t i v e s u b m a n i f o l d s o a s t o p r o p e r l y t r e a t t h e d e g r e e s o f f r e e d o m which b r i n g a b o u t t h e r e s o n a n c e s .

5 1 . I n t r o d u c t i o n

T h e s e l f - c o n s i s t e n t c o l l e c t i v e - c o o r d i n a t e method (SCC m e t h o d ) / l / h a s been p r o p o s e d a s a m i c r o s c o p i c t h e o r y t o p r o p e r l y d e f i n e g l o b a l c o l l e c t i v e c o o r d i n a t e s w h i c h s p e c i f y a n ' l o p t i m u m " c o l l e c t i v e s u b m a n i f o l d i n t h e huge d i m e n s i o n a l t i m e d e p e n d e n t H a r t r e e - F o c k (TDHF) m a n i f o l d . The b a s i c p r i n c i p l e o f t h e SCC m e t h o d is t o d e f i n e t h e o p t i m u m c o l l e c t i v e s u b m a n i f o l d ( s u r f a c e ) i n s u c h a way t h a t t h e e x p e c t a t i o n v a l u e o f t h e H a m i l t o n i a n < H > w i t h t h e TDHF wave f u n c t i o n i s s t a t i o n a r y a t e a c h p o i n t o n t h e s u r f a c e w i t h r e s p e c t t o v a r i a t i o n s p e r p e n d i c u l a r t o t h e s u r f a c e . T h u s , t h e g l o b a l c o l l e c t i v e c o o r d i n a t e s a r e d e f i n e d a s c a n o n i c a l v a r i a b l e s t o s p e c i f y t h e o p t i m u m s u r f a c e , a n d t h e c o r r e s p o n d i n g c o l l e c t i v e ,.

H a m i l t o n i a n is s i m p l y g i v e n by t h e e x p e c t a t i o n v a l u e <H> o n t h e s u r f a c e .

T h e b a s i c e q u a t i o n s o f t h e SCC m e t h o d a r e o f t h e s i m p l e f o r m , a n d s e l f - c o n s i s t e n t s o l u t i o n s of t h e set of t h e b a s i c e q u a t i o n s h a s been e a s i l y o b t a i n e d i n t e r m s o f t h e p o w e r s e r i e s e x p a n s i o n o f t h e b a s i c e q u a t i o n s w i t h r e s p e c t t o t h e c o l l e c t i v e v a r i a b l e s / l / . I n t h i s e x p a n s i o n m e t h o d , i t i s n e c e s s a r y t o s e t u p a s p e c i f i c " b o u n d a r y c 6 n a i t i o n " c h a r a c t e r i z i n g t h e c o l l e c t i v e m o t i o n u n d e r c o n s i d e r a t i o n . S u p p o s i n g t h e l a r g e - a m p l i t u d e c o l l e c t i v e v i b r a t i o n i n s o f t n u c l e i , f o r e x a m p l e , we may s e t u p t h e boundary c o n d i t i o n i n s u c h a way t h a t t h e l a r g e - a m p l i t u d e c o l l e c t i v e m o t i o n is c o n n e c t e d w i t h t h e l o w e s t - e n e r g y RPA ( l l p h o n o n f l ) mode i n t h e " s m a l l - a m p 1 i t u d e f f h a r m o n i c l i m i t . w i t h t h i s b o u n d a r y c o n d i t i o n , i t h a s b e e n s h o w n t h a t t h e s e t o f t h e b a s i c e q u a t i o n s c a n be u n i q u e l y s o l v e d , p r o v i d e d t h a t t h e f r e q u e n c y o f t h e RPA phonon mode is i n a n o n r e s o n a n t c a s e . I n t h i s n o n r e s o n a n t c a s e , t h e f r e q u e n c y o f t h e RPA phonon mode d o e s n o t s a t i s f y t h e r e s o n a n c e c o n d i t i o n

W a = o a n d w a (X-1 , 2 , ^{v . )} b e i n g t h e f r e q u e n c i e s o f t h e RPA phonon mode a n d t h e o t h e r RPA normal modes, r e s p e c t i v e l y .

I n t h e r e a l i s t i c l a r g e - a m p l i t u d e c o l l e c t i v e m o t i o n o f n u c l e i , h o w e v e r , we may o f t e n e n c o u n t e r t h e r e s o n a n t cases s a t i s f y i n g E q . ( l . l ) . I n s u c h r e s o n a n t c a s e s , t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987213

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C2-80 JOURNAL DE PHYSIQUE

p o w e r s e r i e s e x p a n s i o n m e t h o d w i t h r e s p e c t t o t h e c o l l e c t i v e v a r i a b l e s e n c o u n t e r s t h e well-known p r o b l e m o f s m a l l d e n o m i n a t o r i n t h e e x p a n s i o n s e r i e s , a n d we h a v e t o p r o p e r 1 y t a k e i n t o a c c o u n t t h e d e g r e e s of f r e e d o m which b r i n g a b o u t t h e r e s o n a n c e s i n t h e power s e r i e s e x p a n s i o n . W i t h t h e p u r p o s e o f i n v e s t i g a t i n g b e h a v i o r o f t h e o p t i m u m c o l l e c t i v e s u b m a n i f o l d i n t h e r e s o n a n t c a s e s , i n t h i s r e p o r t w e p r o p o s e a n i m p r o v e d method o f s o l v i n g t h e b a s i c e q u a t i o n s o f t h e SCC method.

52. B a s i c E q u a t i o n s o f t h e SCC Method

A d o p t i n g t h e c o n v e n t i o n o f u s i n g h - l , we s t a r t w i t h t h e b a s i c e q u a t i o n s o f t h e TDHF t h e o r y ,

w h e r e t h e t i m e - d e p e n d e n t S l a t e r d e t e r m i n a n t I $ ( t ) > is g i v e n by

Here / $ > d e n o t e s t h e H a r t r e e - F o c k g r o u n d s t a t e w i t h e n e r g y Eo, a n d a t a n d b i t 0

r e p r e s e n t t h e p a r t i c l e - and h o l e - c r e a t i o n o p e r a t o r s w i t h r e s p e c t t o I$ 0 >;

M ( N ) b e i n g a number o f s i n g l e - p a r t i c l e ( h o l e ) s t a t e s u n d e r c o n s i d e r a t i o n . Through a v a r i a b l e t r a n s f o r m a t i o n f = f ( C * , C ) , i t is a l w a y s p o s s i b l e / 2 / t o i n t r o d u c e a

U u i

set of c a n o n i c a l v a r i a b l e s {C ,C*. 1 , by w h i c h t h e TDHF e q u a t i o n c a n be e x p r e s s e d a s

U 1

t h e c a n o n i c a l e q u a t i o n s o f m o t i o n i n c l a s s i c a l m e c h a n i c s ;

it - a ~ ~ a c * . i d * . = - a ~ / a c .

U 1 1 ' ,. U l , . U 1 '

T h e SCC m e t h o d i n t e n d s t o e x t r a c t a n optimum c o l l e c t i v e s u r f a c e ( s u b m a n i f o l d ) o u t o f t h e TDHF p h a s e s p a c e ( m a n i f o l d ) c h a r a c t e r i z e d by (C . ,C* 1, i n s u c h a way

U 1 u i

t h a t t h e H a m i l t o n i a n H i s s t a t i o n a r y a t e a c h p o i n t o n t h e s u r f a c e 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 s p e r p e n d i c u l a r t o t h e s u r f a c e . S u p p o s i n g t h e d i m e n s i o n o f t h e s u r f a c e t o be 2L w h i c h i s much s m a l l e r t h a n t h e d i m e n s i o n 2MN o f t h e TDHF p h a s e s p a c e , we may i n t r o d u c e L - p a i r s o f g l o b a l c o l l e c t i v e v a r i a b l e s ( n n , n E ; a - 1 , 2 , . . . , L ) t o s p e c i f y t h e s u r f a c e . The c a n o n i c a l v a r i a b l e s {C ,C* ) o n t h e S u r f a c e a r e t h e n r e g a r d e d a s

u i u i f u n c t i o n s o f t h e c o l l e c t i v e v a r i a b l e s {nn,n;];

CCui] = cUi ( n * , n ) , = c* ( n * , n ) . ( 2 . 5 )

u i

F o r a n y f u n c t i o n K o f t h e c a n o n i c a l v a r i a b l e s { C u i , C ; i ) , w e u s e a s y m b o l [K] t o d e n o t e t h e f u n c t i o n o n t h e s u r f a c e ; [ K ] = k ( n * , n ) . I n t h e n e i g h b o r h o o d o f t h e s u r f a c e , t h u s , t h e TDHF e q u a t i o n ( 2 . 1 ) is r e d u c e d t o

,. ,.

[ I ] 8<$01 i ~ ( ; 1 ~ 0 , - ~ ; 0 , ) -t - e ( 2 . 6 )

n -

w h e r e t h e l o c a l i n f i n i t e s i m a l g e n e ~ a t o r s 6: ^{a n d}in a r e d e f i n e d a s

E q . ( 2 . 6 ) i s t h e f i r s t b a s i c e q u a t i o n o f t h e SCC m e t h o d a n d i s d e n o t e d by [I]

h e r e a f t e r .

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The canonical e q u a t i o n s of motion f o r t h e c o l l e c t i v e v a r i a b l e s nu, Q:],

c a n be d e r i v e d from [ I I ] , under t h e c o n d i t i o n t h a t t h e weak boson-like commutation r e l a t i o n s

- ^{" t} ^{.. ..}

< @ , I ~ 0 ~ , 0 ~ 1 1 + ~ > - ^6,B ^p < @ ~ ~ l [ ~ , ~ o ~ l l @ ~ > = 0 ( 2 . 9 )

A ^ t

have t o be s a t i s f i e d . I t has been proved/l/ t h a t t h e g e n e r a t o r s ( 0 ,O,) which h a v e t o s a t i s f y Eq. ( 2 . 9 ) a r e g e n e r a l l y determined through t h e r e l a t i o n s ;

1111 .O J a 0 > ⁼ 1 n + S * , and C.C. , (2.10)

where S ( n X , n ) i s an a r b i t r a r y r e a l f u n c t i o n of t h e v a r i a b l e s (na,n:). We may t h u s e x p r e s s t h e c o n d i t i o n ( 2 . 9 ) i n t h e f o l l o w i n g form;

E q u a t i o n ( 2 . 1 0 ) i s t h e s e c o n d of t h e b a s i c equations of the SCC method, which i s h e r e a f t e r c a l l e d t h e g e n e r a l 1 zed c a n o n i c a l - v a r i a b l e c o n d i t i o n and i s d e n o t e d by [ I I ] . ( I n t h e p r e v i o u s e x p a n s i o n m e t h o d / l / , we have f i x e d t h e s p e c i a l c a n o n i c a l v a r i a b l e s {nfl,n;) from t h e o u t s e t s o a s t o s a t i s f y t h e s i m p l e s t c a s e with S ( n * , n ) =

-. -

0 i n Eq. ( 2 . 1 0 ) . ) The g e n e r a l i z e d c a n o n i c a l - v a r i a b l e s c o n d i t i o n [ I I ] means t h a t we c a n g e n e r a l l y keep t h e d e g r e e s of freedom t o c h o o s e t h e c a n o n i c a l c o l l e c t i v e v a r i a b l e s {n,,r\;) through t h e a r b i t r a r y r e a l f u n c t f o n S ( Q * , ~ ) .

With t h e use of t h e c o n d i t i o n [ I I ] , t h e b a s i c e q u a t i o n [ I ] can be decomposed/l/

i n t o a ) t h e c a n o n i c a l e q u a t i o n s of c o l l e c t i v e motion (2.8) Bnd b) t h e e q u a t i o n s Of c o l l e c t i v e submanifold

which is denoted by [ I ] ' h e r e a f t e r .

13. S e l f - C o n s i s t e n t S o l u t i o n s of t h e SCC Equations -Nonresonant Case-

An e s s e n t i a l idea of t h e p r e s e n t improved method of s o l v i n g t h e b a s i c e q u a t i o n s of t h e SCC method i s t o choose t h e canonical c o l l e c t i v e v a r i a b l e s [n,, Q:) s o a s t o put t h e c o l l e c t i v e Hamiltonian i n t o t h e normal ( d i a g o n a l ) form

b y a d o p t i n g an a p p r o p r i a t e f u n c t i o n S ( n * , n ) i n Eq.(2.10). ( T h i s r e p r e s e n t a t i o n is j u s t the c-number v e r s i o n of t h e " p h y s i c a l bosonu r e p r e s e n t a t i o n / 3 / , / Q / , i n which t h e optimum c o l l e c t i v e Hamiltonian has a diagonal form with r e s p e c t t o t h e number Of p h y s i c a l b o s o n s . ) A c c o r d i n g t o t h e B i r k o f f - G u s t a v s o n n o r m a l - f o r m e x p a n s i o n m e t h o d / 5 / , i t i s always p o s s i b l e t o choose s u c h c a n o n i c a l c o l l e c t i v e v a r i a b l e s ( n u , n L ) , Provided t h a t t h e f r e q u e n c i e s of t h e R P A normal modes a r e n o n r e s o n a n t .

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C2-82 JOURNAL DE PHY SlQUE

The r e q u i r e m e n t ( 3 . 1 ) is c a l l e d [ I I I ] h e r e a f t e r .

I t is e a s i l y s e e n t h a t t h e p r o b l e m t o s o l v e t h e s e t o f b a s i c e q u a t i o n s [ I ] ,

[I11 a n d [ I I I ] s e l f - c o n s i s t e n t l y c a n be r e d u c e d t o f i n d i n g t h e h e r m i t i a n o p e r a t o r [ F ] s a t i s f y i n g t h e s e t o f e q u a t i o n s . I n o r d e r t o s i m p l i f y t h e p r e s e n t a t i o n , h e r e a f t e r , we r e s t r i c t o u r s e l v e s t o t h e s i m p l e s t c a s e o f L=l w i t h a s i n g l e pair o f c o l l e c t i v e v a r i a b l e s { n , q * ) . S i n c e t h e o p e r a t o r [F] is a o n e b o d y o p e r a t o r , we c a n e x p r e s s i t i n t h e form

- t w h e r e [;A,;t. A = 0 , 1 , 2 , + ,MN-l) is t h e c o m p l e t e s e t o f t h e RPA n o r m a l mode3, X =

t ^.A

l,i($A(ui)a,bi - Y A ( , i ) b . a 1, s a t i s f y i n g t h e RPA e q u a t i o n

1 P

We t h e n e x p a n d t h e c o e f f i c i e n t g A ( q * , n ) a s a power s e r i e s o f ( n , n * ) ;

w i t h t h e b o u n d a r y c o n d i t i o n g ( 1 ) = r76

A A , O . W i t h t h i s b o u n d a r y c o n d i t i o n , t h e q u a d r a t i c p a r t o f x ( n * , n ) i n E q . ( 3 . 1 ) is g i v e n by

wO b e i n g t h e f r e q u e n c y o f t h e RPA phonon mode.

The b a s i c e q u a t i o n s [ I ] ' , [ I I ] ' and [ I I I ] i n t h e p r e s e n t c a s e a r e

aT* aT 1 aic - ^..

( i i ) - + - - _{a n} _an* 0 , T = _m,l - ¹

l (2m! ) < 0 0 1 [ - . . [ - , i ~ ] . . . ] , i ~ 1 ( 0 ~ > a n - - ~ n * (2m)

,.

I n s e r t i n g t h e e x p r e s s i o n ( 3 . 2 ) f o r i C ( n * , n ) i n t o t h e s e t of E q s . ( 3 . 6 1 , ( 3 . 7 ) a n d (3.81, a n d e v a l u a t i n g t h e c o e f f i c i e n t s o f e a c h p o w e r o f ( n , n * ) i n t h e s e e q u a t i o n s s t e p by s t e p , we c a n u n i q u e l y d e t e r m i n e t h e h i g h e r t e r m s g A ( n ) i n E q . ( 3 . 4 ) a s w e l l a s hr i n E q . ( 3 . 8 ) , p r o v i d e d t h a t t h e f r e q u e n c i e s o f t h e RPA n o r m a l modes a r e n o n r e s o n a n t .

94. B a s i c E q u a t i o n s i n R e s o n a n t C a s e

When t h e r e e x i s t s a r e s o n a n c e c o n d i t i o n ( 1 . l ) , wl-nowO 2 0 , w i t h a n i n t e g e r n 0 ( n 0 > 2 ) , t h e p o w e r - s e r i e s e x p a n s i o n method i n 5 3 e n c o u n t e r s t h e w e l l - known p r o b l e m of t h e a p p e a r a n c e o f l T z e r o - d e n o m i n a t o r " , l / ( w - n w ) , i n t h e c o e f f i c i e n t s o f t h e power-

1 0 0

series e x p a n s i o n . I n s u c h a r e s o n a n t c a s e , t h e r e f o r e , we h a v e t o p r o p e r l y t a k e i n t o a c c o u n t a p a i r o f new v a r i a b l e s ( 0 , , n ; ) , w h i c h i s c o n n e c t e d w i t h t h e RPA normal mode w i t h f r e q u e n c y w l i n t h e s m a l l - a m p l i t u d e ( h a r m o n i c ) l i m i t , b y e x t e n d i n g t h e c o l l e c t i v e s u b m a n i f o l d t o { n , q * ; n ,n

1 7'.

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I n t h i s c a s e t h e b a s i c e q u a t i o n s [ I ] ' and [ I I ] a r e ( i ) < @ o l [ i A , e - i ' { i - [ g ) ~ an* a n + an an*

aX a aX a ^{i G},.

- ^(-)- ⁺^(--)-}e ] I @ o > ⁼^{0 ,} A*O, l . an; a n l a n l a n l

A t 1 a

( i i ) < a O 1 ~ O 1 ~ O > ⁼p * ⁺i - ~ ( q * , n ; n ; , n ~ ) _{a n} a n d c . c . ,

t 1

<a ₀10 ₁I Q ₀^>⁼^p;⁺ ^{a n d} ^c.c..

I n t h e r e s o n a n t c a s e , i t i s i m p o s s i b l e t o demand t h e c o n d i t i o n [ I I I ] , which p u t s t h e c o l l e c t i v e H a m i l t o n i a n i n t o t h e c o m p l e t e n o r m a l form s u c h a s E q . ( 3 . 1 ) . By c h o o s i n g a n a p p r o p r i a t e f u n c t i o n S ( n * , n ; n ; , q l ) , h o w e v e r , we c a n p u t t h e c o l l e c t i v e H a m i l t o n i a n i n t h e f o l l o w i n g f o r m ;

w h i c h is of t h e normal ( d i a g o n a l ) form maximally w i t h t h e e x c e p t i o n o f t h e " r e s o n a n t t e r m " xres. The r e s o n a n t t e r m r e s can n e v e r be e x p r e s s e d i n t h e n o r m a l form and d i s p l a y s a n e s s e n t i a l c o u p l i n g b e t w e e n t h e ( Q , q*)-mode a n d t h e ( n l , n;]-mode t h r o u g h t h e r e s o n a n c e w 1 -nou0 2 0 .

W i t h t h e b o u n d a r y c o n d i t i o n i n t h e s m a l l a m p l i t u d e l i m i t , g A _ q ( n * , q ; q ; , q l ) ⁺n,

g A C l ( n * , n ; n y , n l ) ⁺n1 a n d g ^l( n * , q ; q ; , q l ) + 0 , t h e s e t o f t h e b a s i c e q u a t i o n s , ( 4 . 1 ) , ( 4 . 2 ) a n d ( 4 . 3 ) , c a n be s o i v e d i n t h e f o r m o f p o w e r - s e r i e s e x p a n s i o n s w i t h r e s p e c t t o n , n * , q l a n d q;, a n d we c a n d e t e r m i n e g

p o w e r - s e r i e s e x p a n s i o n f o r m a s w e l l a s t h e c o e f f i c i e n t s h i n t h e c o l l e c t i v e H a m i l t o n i a n ( 4 . 3 ) .

55. I l l u s t r a t i v e Example o f S o l u t i o n s

I n s t e a d o f w r i t i n g down t h e g e n e r a l f o r m s o f t h e s o l u t i o n s o f t h e b a s i c e q u a t i o n s , we i l l u s t r a t e t h e s o l u t i o n s o f t h e b a s i c e q u a t i o n s by a d o p t i n g a s i m p l e model.

The model H a m i l t o n i a n is g i v e n a s

T h e r e a r e f o u r l e v e l s w i t h e n e r g i e s E 0 < t z 1 < t z 2 < ~ ? - a n d e a c h l e v e l h a s N - f o l d d e g e n e r a c y . The f e r m i o n p a i r o p e r a t o r s a r e d e f i n e d a s

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C2-84 JOURNAL DE PHYSIQUE

The l o w e s t e n e r g y s t a t e w i t h o u t i n t e r a c t i o n is

The t i m e d e p e n d e n t s i n g l e S l a t e r d e t e r m i n a n t is given a s

A

where t h e b a s i c e x c i t a t i o n modes a r e K l O , K 2 0 and K The e x c i t a t i o n e n e r g i e s c o r r e s p o n d i n g t o ( U } i n t h e p r e v i o u s s e c t i o n s a r e t h e - E ~ - E ~ , = c2-cO and

A

€ 3 0 E E 3 - E ~ v

I n t h e n o n r e s o n a n t c a s e , s t a r t i n g w i t h t h e boundary c o n d i t i o n [ f 1 -, 116 i n

X a , 1

t h e s m a l l - a m p l i t u d e ( h a r m o n i c ) l i m i t , a n d f o l l o w i n g t h e m e t h o d g i v e n i n § 3 by

- t A ^ t .. ^A

r e a d i n g X. a s K 1 0 a n d [XAh0) a s [K20,K301, we can o b t a i n t h e s o l u t i o n of t h e b a s i c e q u a t i o n s ( 3 . 6 ) . (3.7) and (3.8). The diagonal-form H a m i l t o n i a n t h u s o b t a i n e d i s w r i t t e n ( u p t o t h e f o u r t h o r d e r ) a s

When t h e r e s o n a n c e c o n d i t i o n E ~ ~ - Z - E 10 0 i 3 s a t i s f i e d , a c c o r d i n g t o t h e rnethd i n 5 4, we have t o i n t r o d u c e a new s e t of v a r i a b l e s [rl1 ,Q;] which c o r r e s p o n d s t o t h e

^

d e g r e e of freedom of K20. The boundary c o n d i t i o n i n t h i s c a s e i s g i v e n by [f ,. ^l] + q, [f2]. + 9 and [ f 3 ] + 0 i n t h e s m a l l - a m p l i t u d e ( h a r m o n i c ) l i m i t , a n d X i n E q . ( 4 . 1 )

A s h o u l d be r e a d a s K

0 3 ' S o l v i n g t h e b a s i c e q u a t i o n s ( 4 . 1 1 , ( 4 . 2 ) and ( 4 . 3 ) , we o b t a i n t h e H a m i l t o n i a n ( u p t o t h e f o u r t h o r d e r )

C o m p a r i n g t h e H a m l l t o n i a n ( 5 . 6 ) i n t h e r e s o n a n t c a s e w i t h t h e Hamiltonian (5.5) i n t h e n o n r e s o n a n t c a s e , we can s e e t h e f o l l o w i n g f a c t .

I n t h e n o n r e s o n a n t c a s e , t h e c o l l e c t i v e mode ( n , n * ) i s chosen t o be "maximally- decoupled" and t h e t i m e e v o l u t i o n of t h e [n,n*)-mode c a n be w e l l s p e c i f i e d by t h e H a m i l t o n i a n ( 5 . 5 ) . When o n c e t h e r e s o n a n c e c o n d i t i o n happens, t h e r e i n e v i t a b l y a r i s e s t h e e s s e n t i a l c o u p l i n g term Xres which can never be e x p r e s s e d j n t h e n o r m a l

. -

f o r m . B e c a u s e o f t h i s c o u p l i n g , t h e maximal-decoupling p r o p e r t y of t h e c o l l e c t i v e (n.q*)-mode is broken i n t h e r e s o n a n t c a s e . The e n e r g y o f t h e c o l l e c t i v e m o t i o n i n i t i a l l y g i v e n t o t h e c o l l e c t i v e (q,n*]-mode is t h u s a b s o r b e d by t h e (n,,n;)-mode, s o t h a t we have t o t r e a t t h e ( , n l , n f ) - m o d e a s w e l l a s t h e ( n , g * ) - c o l l e c t i v e mode d y n a m i c a l l y .

R e f e r e n c e

1 ) T. Marumori, T. Maskawa, F. S a k a t a and A . Kuriyama, P r o g . Theor. Phys. 64 ( 1 9 8 0 ) 1294. T. Marumori, F. S a k a t a , T. Maskawa, T. Une and Y. Hashimoto, P r o c e e d i n g s o f t h e 1 9 8 2 B r a s o v I n t e r n a t i o n a l S c h o o l (World Scientific P u b l i s h i n g Co. P t e .

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2 ) F. S a k a t a , T , M a r u m o r i , Y . Hashimoto and T. Une, Prog. Theor. Phys. 70 ( 1 9 8 3 ) , 424.

3 ) A . K l e i n , i n Dynamic S t r u c t u e r e of Nuclear S t a t e s , e d . by D.J. Rowe ( U n i v e r s i t y o f Toront P r e s s , 1972).

4) T. Marumori , A . H a y a s h i , T. Tomoda, A . Kuriyama and T. Maskawa, Prog. Theor.

Phys. 63 ( 1 9 8 0 ) , 1576.

5 ) C . D . ~ i r k h o f f , Dynamical Systems ( A m . Math. S o c . , New Yor, 1927), Vol.IX, F.C.

Custavson, Astron. J . 2 ( 1 9 6 6 ) , 670.