Optimum Collective Submanifold in Resonant Cases by the Self-consistent Collective-coordinate Method for Large-amplitude Collective Motion

With 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 op t imum 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 and 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 of 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 motion. I t is 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 ' lmaximal-decoupl ing 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 , and we have 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 The 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 proposed 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 ' loptimum" c o l l e c t i v e submani fo ld i n t h e huge d i m e n s i o n a l t i m e dependent Har t ree-Fock (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 op t imum 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 is 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 op t imum 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 on t h e s u r f a c e . The 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 power 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 method , i t is 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 se 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 mot ion 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 harmonic l i m i t . w i t h t h i s boundary c o n d i t i o n , i t h a s b e e n shown 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 of 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 nonresonant c a s e , t h e f r e q u e n c y of 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 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 of t h e RPA phonon mode and t h e o t h e r a = o 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 ge-ampl i tude c o l l e c t i v e m o t i o n o f n u c l e i , however, 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 C2-80 JOURNAL DE PHYSIQUE power 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 problem of s m a l l denomina tor 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 freedom 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 . Wi th t h e purpose of 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 ropose 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 of t h e SCC method. 52. B a s i c E q u a t i o n s of t h e SCC Method Adopt ing 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 , where t h e t ime-dependent 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 t 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 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 under 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 lways p o s s i b l e / 2 / to 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 which 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 of mot ion i n c l a s s i c a l mechanics ; it a ~ ~ a c * . i d * . = a ~ / a c . U 1 1 ' ,. U l , . U 1 ' The SCC m e t h o d i n t e n d s t o e x t r a c t an 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 is s t a t i o n a r y at e a c h p o i n t on 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 imens ion o f t h e s u r f a c e t o be 2L w h i c h is much s m a l l e r t h a n t h e d imens ion 2MN o f t h e TDHF phase 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 (nn,nE;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* ) on 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 of 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 For any f u n c t i o n K of 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 ne ighborhood of 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 ,. ,. -t [I] 8<$01 i~( ;1~0 , -~ ;0 , ) e ( 2 . 6 ) n where 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: and 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 is d e n o t e d by [I] h e r e a f t e r . The canonical equations of motion f o r the c o l l e c t i v e variables n u , Q:], can be d e r i v e d from [ I I ] , under the condition t h a t the 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 the generators ( 0 ,O,) which have t o s a t i s f y Eq. (2 .9 ) a r e general ly determined through the r e l a t i o n s ; 1 1111 .O J a 0 > = n + S * , and C.C. , (2.10) where S ( n X , n ) is an a r b i t r a r y r e a l funct ion of the variables (na,n:). We may thus express the condition (2.9) i n the following form; Equat ion ( 2 . 1 0 ) i s t h e second of t h e b a s i c equations of the SCC method, which is hereaf te r ca l led the general1 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 is denoted by [ I I ] . ( I n t h e p rev ious expans ion method/ l / , we have f ixed the spec ia l canonical var iables {nfl,n;) from the ou tse t s o as t o s a t i s f y the s implest case with S ( n * , n) = -. 0 i n Eq. ( 2 . 1 0 ) . ) The generalized canonical-variables condition [ I I ] means t h a t we can g e n e r a l l y keep t h e d e g r e e s of freedom t o choose t h e c a n o n i c a l c o l l e c t i v e var iab les {n,,r\;) through the a r b i t r a r y r e a l functfon S ( Q * , ~ ) . With the use of the condition [ I I ] , t h e basic equation [ I ] can be decomposed/l/ i n t o a ) t h e c a n o n i c a l equations of c o l l e c t i v e motion (2.8) Bnd b) the equations Of c o l l e c t i v e submanifold which is denoted by [I] ' he reaf te r . 13. Self-Consistent Solut ions of the SCC Equations -Nonresonant CaseAn e s s e n t i a l idea of the present improved method of solving t h e basic equations of t h e SCC method i s t o choose the canonical c o l l e c t i v e var iab les [n,, Q:) s o as t o put t h e c o l l e c t i v e Hamiltonian i n t o the normal (diagonal) form by a d o p t i n g an a p p r o p r i a t e funct ion S(n* ,n) i n Eq.(2.10). (This representat ion is j u s t the c-number version of the "physical bosonu r e p r e s e n t a t ion / 3 / ,/Q/, i n which the optimum col lec t ive Hamiltonian has a diagonal form with respect t o the number Of p h y s i c a l bosons . ) According t o t h e Birkoff-Gustavson normal-form e x p a n s i o n method/5 / , i t is always p o s s i b l e t o choose such c a n o n i c a l c o l l e c t i v e variables (nu,nL), Provided t h a t the frequencies of t h e R P A normal modes a r e nonresonant . 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 problem 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 can 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 it i n t h e form -t w h e r e [;A,;t. A=0,1 ,2 , + ,MN-l) is t h e comple te s e t of t h e RPA normal 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 expand 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 of 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 ) + 0 , T = l 1 an an* m, l (2m! )<001 [ . . [ , 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 Eqs. ( 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 power 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 can 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 normal modes a r e nonr e s o n a n t . 94. B a s i c E q u a t i o n s i n Resonant 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 problem of t h e a p p e a r a n c e o f lTzero-denomina tor" , l / ( w -n w ) , i n t h e c o e f f i c i e n t s of t h e power1 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 have 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;) , which 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 ' . 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 [II] a r e ( i ) < @ o l [ i A , e i ' { i [ g ) ~ an* an + an an* ,. aX a aX a i G (-)+ (-)-}e ] I @ o > = 0 , A*O, l . an; a n l an l a n l A t 1 a ( i i ) < a O 1 ~ O 1 ~ O > = p * + i ~ ( q * , n ; n ; , n ~ ) and c .c . , a n t 1 = p; + and c.c.. 0 1 0 I n t h e r e s o n a n t c a s e , i t is 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 II] , 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 comple te normal form s u c h a s Eq . (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 ) , however, we can put t h e c o l l e c t i v e Hami l ton ian i n t h e f o l l o w i n g form; which 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 term" xres. The r e s o n a n t t e r m can never be e x p r e s s e d i n t h e normal form and r e s d i s p l a y s an e s s e n t i a l c o u p l i n g between t h e ( Q , q*)-mode and 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 -nou0 2 0 . 1 With t h e boundary 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 ) and ( 4 . 3 ) , can be s o i v e d i n t h e form 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 form 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 of 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 < 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 0 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 C2-84 JOURNAL DE PHYSIQUE The lowest energy s t a t e wi thout i n t e r a c t i o n is The t imedependen t s i n g l e S l a t e r determinant is given a s