HAL Id: jpa-00224240
https://hal.archives-ouvertes.fr/jpa-00224240
Submitted on 1 Jan 1984
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
THE TIME-DEPENDENT SYMPLECTIC MODEL AS A SEMI-CLASSICAL METHOD FOR NUCLEAR
COLLECTIVE MOTION
P. Arickx, J. Broeckhove, P. van Leuven
To cite this version:
P. Arickx, J. Broeckhove, P. van Leuven. THE TIME-DEPENDENT SYMPLECTIC MODEL AS A SEMI-CLASSICAL METHOD FOR NUCLEAR COLLECTIVE MOTION. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-311-C6-319. �10.1051/jphyscol:1984638�. �jpa-00224240�
THE TIME-DEPENDENT SYMPLECTIC MODEL AS A SEMI-CLASSICAL METHOD FOR NUCLEAR COLLECTIVE MOTION
F. Arickx, J . ~roeckhove* and P. Van Leuven
Dienst Teoretische en Wiskundige Natuurkunde, R i j k s u n i v e r s i t a i r Centrum Antwerpen, Groenenborgerlaan 171, 2020 Antwerpen, Belgium
Rksumg - La vibration monopolaire isoscalaire est ktudike dans le modsle symplectique d6pendant du temps. Les gnergies d'excitation et l'kvolution dans le temps du rayon r.m.s. sont comparQsaux rksultats obtenus dans le mod&le Sp(2,lR) usuel.
Abstract - The time-dependent symplectic model is applied to the isoscalar monopole vibration of spherical nuclei. Results for excitation energies and time-evolution of the r.m.s. radius are compared with the usual Sp(2,lR) model.
1 - INTRODUCTION
In the description of nuclear collective motion using microscopic mo- dels one distinguishes two complementary approaches. The static approach usually considers the diagonalization of the Hamiltonian in the subspace of the many-particle Hilbert space which is believed to contain the more important components for a proper description of a certain collective behaviour. A time dependent approach constrains the dynamics of the system to a collective manifold by prescribing the dependence of the wave function on certain parameters (qi,pi)
The choice of the parameters is made on physical grounds to exhibit the collective motion under study. In this work we wish to present an application of the AympLecLic modex in the time-dependent approach.
The static model, in which the many-particle Hilbert subspace is defined as the space of an infinite dimensional irreducible represen- tation has been reported on extensively elsewhere [I] . This
symplectic model takes into account the monopole and quadrupole vibra- tions as well as rotations. This can be inferred from the generators
*supported by the Interuniversitair Instituut voor Kernwetenschappen
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984638
C6-312 JOURNAL DE PHYSIQUE
o f Sp(6,lR) which may b e w r i t t e n a s f o l l o w s ( p , v i n d i c a t e s p a t i a l d i r e c - t i o n , j i s t h e p a r t i c l e i n d e x , p m d w s t a n d f o r p o s i t i o n and momen- t u m ) .
I n t h i s r e p r e s e n t a t i o n t h e t e n s o r 6 g e n e r a t e s t h e monopole a n d q u a d r u - p o l e v i b r a t i o n . i s t h e k i n e t i c e n e r g y t e n s o r , h c i s t h e r o t a t i o n a l a n g u l a r momentum a n d a r e t h e v i b r a t i o n a l a n g u l a r momenta.
The c o l l e c t i v e m a n i f o l d M i s assumed t o b e g e n e r a t e d by t h e a c t i o n o f t h e s y m p l e c t i c g r o u p o p e r a t o r s on a r e f e r e n c e s t a t e Y O
Y O w i l l b e t a k e n a s t h e l o w e s t w e i g h t s t a t e o f a n i r r e d u c i b l e r e p r e - s e n t a t i o n R o o f S p ( 6 9 7 ) . Because Sp(6,iR) is t h e d y n a m i c a l g r o u p o f t h e 3 - d i m e n s i o n a l harmonic o s c i l l a t o r R o c a n b e r e a l i z e d w i t h i n t h e o s c i l l a t o r s h e l l - m o d e l . I f G i s t h e s t a b i l i t y g r o u p [ 2 ] o f Y O w . r . t . S p ( 6 , R ) t h e f a m i l y o f s t a t e s Y(g) c a n b e l a b e l l e d by t h e c o s e t s o f Sp(G,iR)/G. If ( q . , F i ) form a b a s i s f o r t h e v e c t o r s p a c e complement o f t h e a l g e b r a of G , t h e n Y c a n b e w r i t t e n
The s t a t e s Y d e f i n e d i n t h i s way a r e n o t h i n g b u t t h e g e n e r a l i z e d c o h e t r e n t s t a t e n o f Perelomov 131.
The dynamics i s i n t r o d u c e d by l e t t i n g t h e p a r a m e t e r s q i ( t ) a n d ~ ~ ( t ) depend on t i m e and a p p l y i n g D i r a c ' s t i m e d e p e n d e n t v a r i a t i o n a l p r i n c i - p l e (TDVP)
Kramer and S a r a c e n o [ 2 ] h a v e d i s c u s s e d t h o r o u g h l y t h e c o m b i n a t i o n of g r o u p a c t i o n and TDVP. A s a r e s u l t o f ( 5 ) t h e p a r a m e t e r s q and
i pi s a t i s f y a s e t o f Hamilton e q u a t i o n s
bracket equations. The solution of (6) substituted in (1) yields a Y(t) that traces out a path on the collective manifold. Although this time-dependence of the coherent state is governed by classical eqs. (6) the whole approach must be envisaged as an approximation to the time-dependent Schr6dinger eq. Therefore this approach should be called "pseudo-classical1'.
The Hamiltonian function 2C appearing in (6) is defined by means of the Hamiltonian operator H through
In the application of the symplectic model we intend to use (semi)- realistic nucleon-nucleon interactions and do not take refuge to model Hamiltonians expressed in terms of group operators. There exist at present algorithms [ 4 ] for the calculation of matrix elements of the potential energy operator based on the generating function tech- nique.
The set of eqs. (6) form an initial value problem: the parameter values qi(0), pi(0) at time t = O must be given. To these initial values corresponds an energy which is conserved during the motion.
This energy is not quantized, hence the problem arises of determining the eigenvalues of H. By this we mean the eigenvalues of H in the symplectic-model space Ro. For collective motion characterized by one degree of freedom there exist quantization methods such as the gauge invariant periodic quantization [5] which lead to the old Bohr-Sommer- feld rule. For more degrees of freedom this problem has not yet been settled.
Tn this paper we illustrate the general scheme outlined above on the simplest of collective motions:the breathing mode for sphericalnuclei.
' The m a n i f o l d M i s p r o v i d e d w i t h a s y m p l e c t i c m e t r i c g i v e n by t h e P o i s s o n b r a c k e t . The f a c t t h a t we u s e t h e " s y m p l e c t i c " group t o g e n e r a t e t h e " s y m p l e c t i c " m a n i f o l d is, i n a sense, a c c i d e n t a l .
C6-314 JOURNAL DE PHYSIQUE
2 - T I M E - D E P E N D E N T S Y M P L E C T I C M O D E L F O R T H E " B R E A T H I N G M O D E "
The i s o s c a l a r monopole v i b r a t i o n o r " b r e a t h i n g modeff i s a n i s o t r o p i c c o m p r e s s i o n a l mode g e n e r a t e d by t h e c o r r e l a t e d i n - and o u t w a r d m o t i o n o f a l l p a r t i c l e s . I n s p h e r i c a l n u c l e i t h i s mode i s d e c o u p l e d f r o m b o t h q u a d r u p o l e v i b r a t i o n s and r o t a t i o n s . T h e r e f o r e t h e c o l l e c t i v e m a n i f o l d w i l l b e t a k e n t o c o n s i s t o f t h e c o h e r e n t s t a t e s g e n e r a t e d by t h e a c t i o n o f t h e o p e r a t o r s
T h e s e a r e t h e g e n e r a t o r s o f t h e a l g e b r a s p ( 2 , l R ) and t h e c o r r e s p o n d i n g s y m p l e c t i c g r o u p Sp(2,lR) i s a s u b g r o u p o f S p ( 6 J R ) m e n t i o n e d i n s e c t . 1 . A s t h e r e f e r e n c e s t a t e Y O we t a k e t h e s h e l l - m o d e l ground s t a t e . I t c a n b e shown [I] t h a t Y o i s t h e l o w e s t w e i g h t o f a n i r r e d u c i b l e r e p r e - s e n t a t i o n of Sp(2,lR) c h a r a c t e r i z e d by t h e l a b e l k which i s e i g e n v a l u e o f = t ( a + k ) ; up t o a f a c t o r 2hw t h e o p e r a t o r i s t h e h a r m o n i c o s c i l - l a t o r H a m i l t o n i a n . Hence
T h i s shows t h a t t h e s t a b i l i t y g r o u p o f Y O i s SO(2) g e n e r a t e d by C.
The r e m a i n i n g o p e r a t o r s 6 a n d 6 may b e u s e d t o d e f i n e t h e m a n i f o l d f o r monopole c o l l e c t i v e m o t i o n
The g e n e r a l i z e d monopole c o h e r e n t s t a t e Y ( q , p ) c a n t h u s b e l a b e l l e d by t h e c o s e t s of S p ( 2 , J R ) / S 0 ( 2 ) . I n t e r m s o f t h e p a r a m e t e r s q y p t h e f o r m a l s t r u c t u r e o f t h e TDVP e q u a t i o n s i s now
where
In this representation we can write the coherent state in the form
Y(Q,P) = Q-3A'Zexp [i(p/4kb2~) r.] 2 'Yo(5./Q)
I (15)
I '
A
We see that the action of P results in an isotropic dilation of Y O whereas 6 induces a velocity field. Thus the collective manifold is made up of states describing a scaling type oscillation with both dilational and boost coordinates. It turns out that Q equals the root mean square radius of Y in units JZ?; b whereas P is its canonically conjugate momentum.
The Hamiltonian function X(Q,P) has the form [61
and the equations of motion for Q and P are
3 - N U M E R I C A L R E S U L T S We have applied giant resonance 140Ce and 2 0 8 ~ b .
the above model to the breathing mode or monopole of the doubly closed-shell nuclei 4 0 ~ a , 5 6 ~ i , 'Ozr,
As a semi-realistic nucleon-nucleon interaction we used various Skyrme type forces SkM [71 , Ska (81 , S3 [9] . The collec- tive potential U(Q) is equal to <Y(Q,O) H Y(Q,O)> and can be computed by substituting b Q for the oscillator parameter b in the shell-model matrix element < Y O I ~ I ~ O > .
The solutions of (17) can be understood as the orbits of a fictitious particle with mass 2k/hw in a potential well U . Each orbit is deter- mined by its energy E and forms a closed periodic trajectory with two turning QO(E) and Q1(E). Because of the conservation of energy, along the trajectory with energy E , we have
J O U R N A L DE PHYSIQUE
The p o s i t i v e s i g n a p p l i e s t o t h e f i r s t h a l f p e r i o d ( f r o m Q o t o Q1) a n d t h e n e g a t i v e s i g n t o t h e s e c o n d h a l f p e r i o d . The p e r i o d T ( E ) i s t h e n most e a s i l y w r i t t e n a s
T h e e n e r g y q u a n t i z a t i o n i s o b t a i n e d b y a p ~ l y i n g t h e Bohr-Sommerfeld r u l e
Q1(E)
I ( E ) = $ PdQ = 2 1 P(E,Q)dQ = n ( 2 a h ) ( 2 0 ) Q o ( E )
The bound s t a t e e n e r g i e s c a n now b e f o u n d by t h e i n v e r s i o n o f ( 2 0 )
The l o w e s r e n e r g y s o l u t i o n E 0 ( 4 ) c o r r e s p o n d s t o t h e " o r b i t " P=O, Q = l a n d e q u a l s U(1) = < Y O I ~ I ~ O > . The f i r s t e x c i t e d e n e r g y El i s commonly a s s o c i a t e d w i t h t h e g i a n t r e s o n a n c e e n e r g y . I n f i g . 1 w e show t h e f u n c t i o n ICE) a n d i n t a b l e 1 we l i s t t h e v a l u e s o f El-EO f o r d i f f e r e n t n u c l e i .
F i g . 1: Q u a n t i z a t i o n i n t e g r a l I(E) f o r SkM i n t e r a c t i o n : 1) ' O C ~ , 2 ) 5 6 ~ i , 3 ) ''~r, 4 ) l k O c e , 5 ) 2 0 8 ~ b
From t a b l e 1 we s e e t h a t t h e a g r e e m e n t b e t w e e n b o t h e n e r g i e s i s e x c e l l e n t .
G I P Q 21.64 20.52 1 7 . 8 2 1 5 . 5 4 1 3 . 5 2
SkM
Sp(2,IR) 2 1 . 7 0 20.56 1 7 . 8 4 1 5 . 4 6 1 3 . 5 2
G I P Q 23.38 21.92 1 9 . 2 3 1 6 - 7 0 1 4 . 5 7
S k a
s p ( 2 ,I!?) 23.46 21.97 1 9 . 2 5 1 6 . 7 1 14.58
GIPQ 2 6 . 9 1 25.58 2 2 . 1 9 1 9 . 5 0 1 7 . 0 1
S 3
S p ( 2 ,W) 2 7 . 0 3 25.65 22.23 1 9 . 5 2 1 7 . 0 2
T a b 1 e 1 : citation e n e r g i e s ( E -E ) i n MeV; GIPQ: t i m e d e p e n d e n t model 1 0
see t e x t , S p ( 2 3 ) : r e s u l t s o f r e f . [l] .
The d y n a m i c s o f t h e b r e a t h i n g mode c a n b e s t u d i e d b y l o o k i n g a t t h e t i m e - d e p e n d e n c e o f Q a n d P . I n f i g . 2 we show Q(t)
F i g . 2: r . m . s . r a d i u s Q ( t ) f o r 5 6 ~ i w i t h S 3 - i n t e r a c t i o n f o r e x c i t a t i o n e n e r g i e s 2 5 ( 2 5 ) 1 5 0 MeV.
Q i s e x p r e s s e d i n u n i t s b a n d t i n 21r/w.
C6-318 JOURNAL DE PHYSIQUE
One s t r i k i n g f e a t u r e h e r e i s t h e t y p i c a l asymmetry o f t h e mean s q u a r e r a d i u s : t h e t i m e o f e x p a n s i o n ( Q > l ) i s much l o n g e r t h a n t h e t i m e o f c o n t r a c t i o n ( Q < l ) . T h i s b e h a v i o u r r e f l e c t s t h e f a c t t h a t t h e r e s t o r i n g f o r c e i s much l a r g e r when t h e n u c l e u s i s c o m p r e s s e d t h a n when it i s e x p a n d e d .
I f Q ( t ) a n d P ( t ) a r e s u b s t i t u t e d i n Y ( Q , P ) we o b t a i n t h e t i m e e v o l u - t i o n o f t h e c o h e r e n t s t a t e a c c o r d i n g t o t h e c o n s t r a i n e d dynamics o f TDVP. I t i s i n t e r e s t i n g t o compare t h i s w i t h t h e t i m e - e v o l u t i o n o f t h e same i n i t i a l wave p a c k e t Y ( O ) = Y ( Q ( O ) , P ( O ) ) u n d e r t h e dynamics o f t h e u s u a l Sp(2,IR) m o d e l , i . e .
w h e ~ e $n a n d E n a r e t h e e i g e n s t a t e s a n d e i g e n v a l u e s o f H i n t h e r e p r e - s e n t a t i o n s p a c e k o f S p ( 2 ,lR), a n d c n t h e component o f Y ( 0 ) a l o n g q n . We h a v e compared t h e TDVP a n d S p ( 2 , E ) t i m e e v o l u t i o n s o f t h e r . m . s . r a d i u s
F i g . 3: Difference AR of r . m . s . r a d i i according t o GIPQ and Sp(2,iR) f o r 208~b with SkM interacti6n;AQ is measured in fm and t i n sec.(E-E Q = 1 5 M e V )
Both R a n d R a r e s u b j e c t t o t h e n u m e r i c a l a p p r o x i m a t i o n s u s e d i n - -
d e t e r m i n i n g Q ( t ) r e s p . Y . We h a v e c h e c k e d t h a t t h e a c c u r a c y on t h e s o l u t i o n of e q . ( 1 7 ) r e F p . t h e t r u n c a t i o n o f ( 2 2 ) a t N = l l ( s e e r e f . [ I ] ) a r e s u f f i c i e n t .
4 - C O N C L U S I O N
I n t h i s p a p e r we h a v e i n v e s t i g a t e d a s e m i - c l a s s i c a l method f o r t h e d e s c r i p t i o n o f t h e i s o s c a l a r monopole v i b r a t i o n s by f i r s t c o n s t r u c - t i n g a s e m i - c l a s s i c a l s u b m a n i f o l d , s o l v i n g t h e TDVP e q u a t i o n s a n d t h e n r e q u a n t i z i n g t h e e n e r g i e s by t h e g a u g e - i n v a r i a n t p e r i o d i c quan- t i z a t i o n (GIPQ) method. The m e r i t s o f t h i s method c a n b e d i s c u s s e d by c o m p a r i n g t h e r e s u l t s w i t h t h o s e o b t a i n e d by a d i r e c t d i a g o n a l i - z a t i o n o f t h e H a m i l t o n i a n i n t h e H i l b e r t s p a c e s p a n n e d by t h e i r r e - d u c i b l e r e p r e s e n t a t i o n k o f s p ( 2 , l R ) . T h i s c o m p a r i s o n shows t h a t a l t h o u g h t h e c o h e r e n t s t a t e Y(0) r e m a i n s a s u p e r p o s i t i o n o f s t a t i o n a r y s t a t e s d u r i n g t h e t i m e e v o l u t i o n i t seems t o c o n t a i n t h e n e c e s s a r y i n f o r m a t i o n f o r o b t a i n i n g t h e s t a t i o n a r y e n e r g i e s . The s e t t i n g u p o f t h e TDVP e q s . r e q u i r e s o n l y t h e c o m p u t a t i o n o f d i a g o n a l m a t r i x - e l e m e n t s ( i n f a c t o n l y < Y O I ~ I ~ O > ) a n d i s s i g n i f i c a n t l y l e s s i n v o l v e d t h a n t h e s e t t i n g u p o f t h e c o m p l e t e H a m i l t o n i a n m a t r i x . I t i s a l s o f o u n d t h a t a s f a r a s t h e r m s r a d i u s i s c o n c e r n e d t h e c o n s t r a i n e d dyna- m i c s d i v e r g e s away s l o w l y f r o m t h e u n c o n s t r a i n e d d y n a m i c s .
R E F E R E N C E S
J , Broeckhove a n d P . Van Leuven, P h y s . Rev. C ( t o b e p u b l i s h e d ) P. Kramer and M . S a r a c e n o , Geometry o f t h e t i m e - d e p e n d e n t v a r i a - t i o n a l p r i n c i p l e i n quantum m e c h a n i c s , S p r i n g e r , B e r l i n 1 9 8 1 A . M . P e r e l o m o v , Commun. Math. P h y s . ( 1 9 7 2 ) 222
J . B r o e c k h o v e , J . P h y s . G 71981) L259 K . K . Kan, P h y s . Rev. C24 ( 1 9 8 1 ) 279 -
J . B r o e c k h o v e , M . Buysse a n d P. Van Leuven, P h y s . L e t t . 134B
( 1 9 8 4 ) 379
H . K r i v i n e e t a l . , N u c l . P h y s . A336 ( 1 9 8 0 ) 155 -
H , S . KGhler, N u c l . P h y s . A258 - ( 1 9 7 6 ) 3 0 1 M . B e i n e r e t a l . N u c l . P h y s . A238 ( 1 9 7 5 ) 29
