The Nuclear Velocity Field with and without the Vortex Spin

We a r e i n t e r e s t e d i n t h e e x t r a c t i o n of t h e c o l l e c t i v e q u a n t i t i e s , such as t h e c o l l e c t i v e v e l o c i t y f i e l d , o u t o f t h e many p a r t i c l e t h e o r y through some c o o r d i n a t e t r a n s f o r m a t i o n f o r t h e system of A-1 : n Jacobi v e c t o r s such t h a t t h e c o l l e c t i v e c o o r d i n a t e s belong t o t h e s e t of t h e new 3n c o o r d i n a t e s . For t h i s i d e a s s e e r e f s . / 1 , 2 , 3 / . F i r s t we d e f i n e a s t h e c o l l e c t i v e c o o r d i n a t e s t h r e e r o t a t i o n a l coord i n a t e s , a n g l e s y,, )I = 1 , 2 , 3 and t h r e e v i b r a t i o n a l c o o r d i n a t e s al 1 = 1 , 2 , 3 which one o b t a i n s through t h e d i a g o n a l i z a t i o n o f t h e coord i n a t e quadrupole t e n s o r , n w i t h i,j= 1 , 2 , 3 t h e space index and s = 1 , . . , n t h e J a c o b i v e c t o r index. Such a c o o r d i n a t e t r a n s f o r m a t i o n had been done i n r e f s . /4,5/ i n o r d e r t o s t u d y t h e problem of s e p a r a t i o n of t h e c o l l e c t i v e k i n e t i c energy. We d i d t h e c o o r d i n a t e t r a n s f o r m a t i o n u s i n g t h e t echn ique o f an o r b i t a n a l y s i s w . r . t . t h e SO(3)xSO(n) group / 6 , 7 / . The t y p e o f c o l l e c t i v i t y r e l a t e d t o t h i s o r b i t a n a l y s i s we c a l l SO(n) c o l l e c t i v i t y . The group SO(3)xSO(n) a c t on t h e 3n-dimensional Euc l idean manifold paramet r ized by t h e C a r t e s i a n c o o r d i n a t e s x i s such t h a t t h e " c o l l e c t i v e " group SO(3) a c t s on t h e index i and t h e " i n t r i n s i c " group SO(n) on t h e index s. Through t h e o r b i t a n a l y s i s one g e t s r i d o f t h e redundant coord ina tes w i t h o u t i n t r o d u c i n g c o n s t r a i n t s / 6 , 7 / , and a s t h e r e s u l t one w r i t e s the p o i n t o f t h e c o n f i g r u a t i o n space a s a f u n c t i o n o f t h e new c o o r d i n a t e s i n t h e unique f a s h i o n ( t h e r e s t k i c t i o n from SO(3) t o SO(3)/M i s n o t iApor tan t i n f u r t h e r * Work s u p p o r t e d by Deutsche Forschungsgemeinschaft ** Permanent address : I n s t i t u t e o f Phys ics , Belgrade, J u g o s l a v i a Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984636 C6-298 JOURNAL DE PHYSIQUE a n a l y s i s because M i s a d i s c r e t e g roup) ; R o'(B) E S O ( ~ ) / S O ( ~ 3 ) I C t h e r i g h t c o s e t space . Both v 1 s and B's a r e e x p o n e n t i a l pa ramete rs . The v i b r a t i o n a l variables gl 1 = 1 , 2 , 3 one o b t a i n s through t h e d i a g o n a l i z a t i o n o f Q . . 1 3 0 2 to Q= 1 oi l (v lx l lJ 1 l j (TI LJOut o f t h i s r e l a t i o n oge s e e s t h a t t h e x ' s must be chosen a3 an ordered s e t , f o r example xi 2 x2 -> g3. We r e s t r i c t o u r a n a l y s i s t o xl > ?i2 > g3x, r e l a t e d w i t h t h e group M. Through t h i s r e s t r i c t i o n , unimportant f o r the quantum system, f o r t h e c l a s s i c a l system we a r e bound t o one type of o r b i t s / 7 / . The same r e s t r i c t i o n was used i n r e f . / 8 / . The c a n o n i c a l l y con juga te momenta r v v = 1 , ..., 3n t o t h e new coord i n a t e s a r e r e l a t e d t o t h e C a r t e s i a n momenta pis by t h e i n v e r s e J a c o b i m a t r i x J -1 , 3n P i s = 1 JT' v= 1 l S , V nv ' where 1 ' is ,v a x i s The J a c o b i m a t r i x J i s g iven by i t s m a t r i x e lements where R ? ~ l p ) and L?(p) a r e e s s e n t i a l l y g iven by t h e e x p r e s s i o n s f o r any SO(j) group, compare r e f . / 7 / f o r t h e d e t a i l s . We s t a t e t h e fo l lowing p r o p o s i t i o n wi thou t p r o o f : 1 P r o p o s i t i o n : The i n v e r s e J a c o b i m a t r i x J I i s g iven by i t s m a t r i x e lements L where RO(q) i s t h e i n v e r s e m a t r i x t o R~ (q) and O(B) i s r e l a t e d t o L~ (6) (no te t h a t t h e B T T = 1 , . . . ,3n-6 a r e t h e c o s e t pa ramete rs and c p b P = 1 , 2 , 3 t h e group paramete rs up t o t h e d i s c r e t e group M) a s L L L I L r 5 T , ( 6 ) o f R = o,,, (Rl Y T I T (Bl = 6 G T T T 1 The e x p r e s s i o n one can use bo th t o c a l c u l a t e t h e k i n e t i c energy / 7 / and t h e v e l o c i t y f i e l d . For t h e v e l o c i t y f i e l d we proceed a s i n r e f . / 8 / . The C a r t e s i a n momenta become where t h e new c l a s s i c a l momenta a r e d e f i n e d a s f o l l o w s : 311-6 I, I : = I Og,v7 'T g = 1 , 2 , 3 v = 4 , ..., n a n d gv T=l rw a r e t h e momenta c a n o n i c a l l y con juga te t o t h e c o o r d i n a t e gW, C6-300 JOURNAL DE PHYSIQUE w = 1 ,2 ,3 . The aomenta Lgv a r e a n g u l a r momenta " r e f e r r e d " t o t h e body i s t h e v o r t e x s p i n / l o / . : ~ / t ~ ~ d c ~ f ' ) ; e c t i v e p a r t i n C a r t e s i a n momenta we mean t h e p a r t t h a t can be expressed a s 3 c o l l = P i s j=1 1 X i j x j s wi th c o e f f i c i e n t s X depending on ly on c o l l e c t i v e c o o r d i n a t e s and new momenta r e l a t e d t o t h e c o l l e c t i v e manifold. For l o c a l hami l ton ians we have m v . = 1 s P i s We e x t r a c t vE:l1 a s