MACROSCOPIC MODELS FOR ISOVECTOR M1 ROTATIONAL STATES

HAL Id: jpa-00224233

https://hal.archives-ouvertes.fr/jpa-00224233

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.

MACROSCOPIC MODELS FOR ISOVECTOR M1 ROTATIONAL STATES

S. Stringari

To cite this version:

S. Stringari. MACROSCOPIC MODELS FOR ISOVECTOR M1 ROTATIONAL STATES. Journal

de Physique Colloques, 1984, 45 (C6), pp.C6-265-C6-267. �10.1051/jphyscol:1984631�. �jpa-00224233�

JOURNAL DE PHYSIQUE

Colloque C 6 , suppl6ment au n06, Tome 45, juin 1984 page C6-265

MACROSCOPIC MODELS FOR ISOVECTOR M1 ROTATIONAL STATES

S. S t r i n g a r i

Universitd degZi Studi d i Trento, Dipartimento d i Fisica,

Povo, Trento, I t a l y

Resume - En u t i l i s a n t l a methode r s g l e de somme, nous derivons une d e s c r i p t i o n fluide-dynamique de 1 ' & t a t M1 de r o t a t i o n i s o v e c t o r i e l l e . La n a t u r e 'elas t i q u e de l ' e x c i t a t i o n emerge du couplage e n t r e l a r o t a t i o n r i g i d e e t l e mouve ment quadrupol a i r e .

A b s t r a c t - S t a r t i n g from a sum r u l e approach, we provide a microscopic basis t o t h e fluid-dynamic d e s c r i p t i o n o f the i s o v e c t o r M1 r o t a t i o n a l s t a t e . The e l a s t i c nature o f such an e x c i t a t i o n i s shown t o emerge from the c o u p l i n g between the r i g i d r o t a t i o n and the quadrupole motion o f i s o v e c t o r nature.

The p o s s i b i l i t y o f d e s c r i b i n g i s o v e c t o r r o t a t i o n s i n deformed n u c l e i u s i n g t h e h y d r o dynamic model was f i r s t explored by Lo I u d i c e and ~alumbo'. The p r e d i c t i o n s o f t h e i r model were based on t h e analogy between such a mode and t h e g i a n t d i p o l e resonance.

The r e s t o r i n g f o r c e was determined u s i n g a Goldhaber-Teller type procedure, w h i l e the c o l l e c t i v e mass parameter was given by the i s o v e c t o r moment o f i n e r t i a . Other attempts t o e x p l o r e such an e x c i t a t i o n have been based on t h e schematic model2 and on t h e i n t e r a c t i n g boson model 3. Recently t h e r e has been some experimental evidence f o r such a c o l l e c t i v e s t a t e . I n p a r t i c u l a r a s i z a b l e MI s t r e n g t h (BMl+= 1.5 - ^{2 pZN)}

has been detected i n e l e c t r o n s c a t t e r i n g experiments a t t h e energy E = 3.075 MeV i n 1 5 6 ~ d 4 . The p r e d i c t i o n s of t h e hydrodynamic model are s i g n i f i c a n t l y l a r g e r than t h e experimental values o f both the s t r e n g t h and t h e e x c i t a t i o n energy.

One can explore t h e problem from a microscopic p o i n t o f view u s i n g the sum r u l e ap- proachfj. The frequency o f t h e e x c i t a t i o n can be estimated through t h e r a t i o o f two d i f f e r e n t sum r u l e s :

where

and

2 - 1 S-l(P)

;l<nlPlo>l (En-Eo) .

The c r u c i a l problem f o r t h e present a n a l y s i s i s the.choice o f t h e e x c i t a t i o n opera- t o r P. While f o r g i a n t resonances o f e l e c t r i c type l o c a l assumptions f o r P provide a n a t u r a l and r a t h e r accurate d e s c r i p t i o n o f t h e e x c i t a t i o n , i n t h e case o f l o w - l y - i n g e x c i t a t i o n s the presence o f non l o c a l components i s c r u c i a l f o r a c o r r e c t de- s c r i p t i o n o f t h e problem. A n a t u r a l choice f o r P i s given by t h e i s o v e c t o r angular

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984631

C6-266 JOURNAL DE PHYSIQUE

momentum ( s p i n components are n o t considered here f o r t h e sake o f s i m p l i c i t y ) :

Operator ( 4 ) e n t e r s e x p l i c i t l y i n t h e transverse e l e c t r o n s c a t t e r i n g cross s e c t i o n i n t h e q-o l i m i t . Choice (4) corresponds t o imagining t h e c o l l e c t i v e motion as a r o t a t i o n aenerated by t h e u n i t a r y t r a n s f o r m a t i o n

x 3

I E > = ~

R.

Ti lo> . ( 5

The sum r u l e S can then be i n t e r p r e t e d as t h e r e s t o r i n g f o r c e associated w i t h t r a n s - formation ( 5 ) w h i l e S-1 1

$ (3,) i s t h e i s o v e c t o r analogous o f the moment o f i n e r t i a 0. The above i n t e r p r e t a t i o n suggests t h a t use o f choice ( 4 ) f o r P y i e l d s t h e hydro- dynamic r e s u l t s f o r E and BMl+. One has:

where E - 34 A - i MeV i s t h e Goldhaber-Teller p r e d i c t i o n f o r t h e i s o v e c t o r d i p o l e energy. Equation ( 6 ) reproduces t h e r e d i c t i o n s o f t h e hydrodynamic model1 and y i e l d s D t o o h i g h values f o r EN, and BMlt i n P56Gd (EM1= 4.7 MeV, BMlte18 p i ) . The main reson f o r such a discrepancy i s the inadequacy o f choice ( 4 ) . I n f a c t t h e o p e r a t o r (4) can a l s o e x c i t e t h e K

1' component o f t h e i s o v e c t o r g i a n t quadrupole resonance occur- r i n g a t much h i g h e r energy. As a consequence t h e average energy ( 1 ) overestimates t h e frequency o f t h e l o w - l y i n g v i b r a t i o n we are i n t e r e s t e d i n .

I n o r d e r t o decouple the low l y i n g mode from t h e g i a n t quadrupole resonance i n the i n t r i n s i c frame one can consider t h e f o l l o w i n g choice f o r p5:

where, w i t h r e s p e c t t o eq.(4), we have added an i s o v e c t o r deformation o p e r a t o r o f quadrupole type. The c o e f f i c i e n t

i s determined by m i n i m i z i n g t h e energy o f t h e s t a t e . One then obtains5

-I

where E i s t h e energy o f the i s o s c a l a r g i a n t quadrupole resonance (E = 65 A 3 MeV) Choice y7) corresponds t o imagining t h e c o l l e c t i v e motion as given by a combination 4

o f a r o t a t i o n and a quadrupole o s c i l l a t i o n (see a l s o r e f . ( 6 ) ) . I n the r e g i o n o f medium-heavy and heavy n u c l e i eq.(8) y i e l d s

1

E M l - ^{56 6 A-7 MeV}

2 2

BMlt - 0.043 6 A ~ / ~ (

~ ~ - ~ ~ )

The c o u p l i n g taken i n t o account i n eq.(7) i s responsible f o r a l o w e r i n g o f both E

and BMlt by a f a c t o r o f - 2 w i t h r e s p e c t t o the hydrodynamic p r e d i c t i o n s ( 6 ) . I n

1 5 6 ~ d one finds E

2.7 MeV BM1+

9 u2. While the frequency i s i n good agreement with experiments, the magnetic strengt! i s s t i l l overestimated. Inclusion of pairing e f f e c t s i s expected t o p a r t i a l l y reduce the discrepancy7.

I t i s i n t e r e s t i n g to i n t e r p r e t the above r e s u l t s i n terms of fluid-dynamics8. The hydrodynamic description of r e f . ( l ) and contained i n assumption ( 4 ) , i s associated w i t h a r i g i d velocity flow given by ( 2 i s the unit vector i n the x d i r e c t i o n )

In a deformed nucleus such a velocity f i e l d does not s a t i s f y the Steinwedel-Jensen boundary condition

which, i n the l i m i t of large systems, y i e l d s t h e stationary conditions of the hydro- dynamic model. This explains why t h e restoring force of r e f . ( l ) a r i s e s a t the sur- face (mode of Goldhaber-Tel l e r type). A velocity f i e l d which s a t i s f i e s condition (1 1) can be constructed by taking a l i n e a r combination of a r i g i d motion and of an i r r o - t a t i o n a l quadrupole flow:

where 6 i s the nuclear deformation. Eq. (1 2 ) provides the macroscopic i n t e r p r e t a t i o n of the coupling contained in e q . ( 7 ) . The velocity f i e l d of eq.(12) i s divergency f r e and consequently no restoring force with respect t o such a motion can a r i s e i n the hydrodynamic model. A restoring force a r i s e s only i f one takes i n t o account the e f f e c t s of the d i s t o r t i o n s induced by the term 6 3 (yz) in the momentum space. Such d i s t o r t i o n s a r e of quadrupole type and coincide with t h e ones occurring i n t h e - i s o - s c a l a r g i a n t quadrupole resonance. This explains why EM1 turns out t o be proportio- nal t o EQ (see eq. ( 8 ) ) r a t h e r than t o E D (see eq. ( 6 ) ) .

In conclusion we find t h a t the isovector M 1 rotational s t a t e rapresent an other beautiful example of e l a s t i c behaviour exhibited by nuclear systems. Further availa- b i l i t y of experimental r e s u l t s will strongly encourage extensive theoretical i n v e s t i gations of such e x c i t a t i o n s .

The main r e s u l t s discussed in the present work have been derived together with E . Lipparini. I t i s a pleasure t o thank h i s collaboration.

REFERENCES

1 ) N . Lo Iudice and F. Palumbo, Phys. Rev. Lett. - 41 (1978) 1532; Nucl. Phys. A _- 326 (1979) 193;

G . De Franceschi, F. Palumbo and N . Lo Iudice, Lett. Nuovo CimentoL7 (1983) 61;

2) T . Suzuki and D. J . Powe, Nucl . ^Phys . A289 (1 977) 461 ; 3) F. Iachello, Nuc1. Phys. A= (1981) 8 E

A . Dieperink, I1 Nuovo Cimento A76 (1983) 377;

4)'A. Richter, in Proceedings of t h e ~ n t e r n a t i o n a l Conference on Nuclear Physics (Florence 1983);

D. Bohle e t a1 . ,Darmstadt preprint;

5) E . Lipparini and S. S t r i n g a r i , Phys. Lett. B - 130 (1983) 139;

6) R . Hilton, i n Proceedings of t h i s Workshop;

7) 0. Bes and R. Broglia, preprint;

8) S. S t r i n g a r i , A n n . of Phys. 151 (1983) 35.