FLUID DYNAMICAL DESCRIPTION OF GIANT RESONANCES ON HIGH SPIN STATES

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FLUID DYNAMICAL DESCRIPTION OF GIANT RESONANCES ON HIGH SPIN STATES

M. Di Toro, G. Giansiracusa, H. Lombardo, G. Russo

To cite this version:

M. Di Toro, G. Giansiracusa, H. Lombardo, G. Russo. FLUID DYNAMICAL DESCRIPTION OF

GIANT RESONANCES ON HIGH SPIN STATES. Journal de Physique Colloques, 1984, 45 (C4),

pp.C4-297-C4-301. �10.1051/jphyscol:1984422�. �jpa-00224088�

JOURNAL DE PHYSIQUE

Colloque C4, suppl6ment au n03, Tome 45, mars 1984 page C4-297

F L U I D DYNAMICAL DESCRIPTION OF GIANT RESONANCES ON H I G H S P I N STATES

M. Di Toro, G. Giansiracusa, U. Lombardo and G. Russo I s t i t u t o Dipartimentale di Fisica, Universita' di Catania, 951 29 Catania, I t a l y

R6sum6 - Les r6sonances geantes construites sur les 6tats de haut spin sont d6crites comme des solutions de 'scaling' d'une kquation lin6arisde de Vlasov dans un r6fGrentiel en rotation.

On decrit ici les effets des dgformations dynamiques,de la force centrifuge et des diffgrents couplages cr6Bs par la force de Coriolis.On arrive ainsi 2 reproduire les principales carac- tgristiques des donn6es experimentales.

Abstract - Giant resonances built on high spin states are descri bed as scaling solutions of a linearized Vlasov equation in a rotating frame.We discuss the relative effects of dynamical de- formations,centrifugal force and Coriolis Coupling.We can repro- duce the main features of the experimental data.

In this contribution we are able to show the effects of dynamical deformations and Coriolis coupling on the frequencies and strengths of Giant Resonances-built on high spin states.In particular we obtain two main features that have been observed in the first available experimeg tal data:a shift of the centroid of the resonance to lower energies with higher angular momenta and a larger overall width,mainly due to a splitting of the giant level/l,2/.

Our approach is fluid-dynamical in the sense that we are working in a phase space but it is also fully microscopic since our dynamics can be easily derived from a semiclassical limit of the self-consistent TDHF equations.Consequently our results,which are quite easy to work out,can be obtained using realistic interactions and can be directly compared with full cranked RPA calculations.

The key point of our approach is to assume that for a giant colle- ctive state all the strength is concentrated on only one 1evel.This ansatz is largely justified from RPA calculationsas well as from vari- ational fluid-dynamical approaches,and it corresponds to take into account only the lowest multipole distortions of the momentum distri- bution (l?=0,1,2) during the vibration,which is described as a scaling mode.0ur general philosophy is to use this simplifying assumption to study giant resonances in a quite wider context,like in a rotating nucleus or as doorway states in particular reactions.

The dynamical equations are obtained from the Vlasov equation in a rotating frame

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984422

JOURNAL DE PHYSIQUE

with L

r ^{x p} , ^being

^{and 2}

m ( ~ + ~ x ~ ) canonical conjugate variables.

We remark that this equation can be obtained as a semiclassical limit of the cranked TDHF equation for the Wigner lransform f(r,p,t) of the one-body density matrix/3/,with h(~,p,t)

p /2m

W(r,p,t) transform of the HF harniltonian.For a scaling mode we can exactly close the fluid dynamical chain of equations derived from eq.(l) at the lowest two v-moments,continuity and Euler equations/4,5/.

-

For isoscalar modes in a spin-saturated system,with a general Gali- lei invariant non local potentia1,the equation for the zeroth moment

(continuity) assumes the form

with J

f(g,p,t) d p ,and,for a Skyrme-type interaction,the first 3 moment equation(Eu1er) is

3 *

where r..

m j v . v . f (r,p, t) d p is the kinetic energy tensor and m is the ilfective ka4s.u is the local part of the potential and B is the usual Skyrme non locality parameter.

In the rotating frame Giant Resonances are described as small oscil- lations Gf(r,p,t) of the distribution function around the stationary value f which is solution of the cranked eqnation {h - w * L , f } = O .

st st -

Assuming the nucleus to undergo rigid rotations, the generallzeatscaling generator takes the form K(r,p, t)=X(r,t)+ (p - moxr) .s(r, t) .It involves second order distortions o f the momentum distribution while taking into account the shift of the Fermi Sphere due to the rotation/6/.

In the limit of irrotational flow,one obtains from the continuity equation the simple relation

which allows us to express the 2-moments involved in the fluid-dynami-

cal chain in terms o f only the scaling field s(r,t):

Finally,we are left with the following linearized Euler equation

It describes the dynamics of the collective motion and,in principle, can be solved by imposing suitable boundary conditi0ns.A~ far as we are interested in gross properties of the giant modes such as centroid energies and transition strengths as a function of the angular frequen- cy,we follow a simplified procedure.It corresponds to the m3/ml esti- mates calctilated in RPA within the sum rules approach.

Let us consider the case of Giant Quadrupole Resonances and assume for the corresponding scaling field the usual Tassie-Bohr form 5

Taking the scalar product of eq.(4) with each component of the scaling field and integrating over the space coordinates,we reduce it to a set of coupled equations

. .

nK a K ( k ) + BK d j - (k) t ei( b y ( k ) = O

where for rotation around the x -axis one has 3

(collective mass) ( Z )

(a ( ~ 1

= - B - . 2 w Z 1- ( A R -Q1: h*;

; II.

(Coriolis coupling

K

*

parameter)

C', = c'~ ⁼ !! 4 _{- 2} a- 6 _~ ';~fi>'z~)(restoring parameter)

The coefficients 1 = ( ) d3r are the inertial parameters.

We remark that,as : n the static case/4,7/,the only details of the in- teraction are in the effective mass.

This coupling only between the k and components of a given angu-

lar momentum projection,is actually valid if the nucleus undergoes an

axially symmetrical deformation with the rotation.In the general case

of a triaxial shape we get a much more complicated set of five coupled

equations.We will restrict our analysis to dynamical axial symmetric

C4-300 JOURNAL DE PHYSIQUE

deformations.Therefore we expect to seepsplitting in three levels due to the effect of the deformations and a further splitting in five le- vels due to the Coriolis coupling.

Assuming a harmonic time evolution,we can easily get the eigenfre-

quencies ,

8 ; +

- ^- - ^{K = 0, 1 , 2}

K

4 r1,2 2MK

where for k=O,B =O.Everything can be evaluated once we know the cranked selfconsistent stationary solutions

and

(w).In this con-

St. St

tribution we show some first,not selfcons~stent,results obtained from a stationary Thomas-Fermi distribution in a oblate cranked shape/8,4/.

20 50 80 100 200

I

14

Fig. L

Fig. 2

In Fig.1 we report the GQR energies for 1 6 8 ~ r as a function of angg lar frequency and classical angular momentum L=7w,1=1(~) being the ri- gid moment of inertia along the deformation axis.The pattern of the energy branches results from the competition between dynamical defor- mation and centrifugal force.This latter prevails at high angular mo- menta leading to a overall shift of the energies to lower values in agreement with the experimental data/l,Z/fV.Metag,private communication).

The Coriolis splitting is shown in Fig.2 where the k=1,2 frequencies are compared with those calculated dropping out the Coriolis doupling t?rm.~o* the k = l case the Coriolis splitting is strongly reduced for high angular velocity.This is related to the oblate deformation we are only allowing in our calculation.The scaling velocity fields

$5, I

become more a n d more a l i g n e d t o t h e r o t a t i o n a x i s and t h e C o r i o l i s term o f t h e E u l e r e q . ( 4 ) h a s no e f f e c t .

A l l t o g h e t e r t h e f i v e c u r v e s s p r e a d o u t w i t h i n c r e a s i n g a n g u l a r momentum and t h i s c a n p a r t i a l l y e x p l a i n t h e o b s e r v e d i n c r e a s i n g o f t h e w i d t h 0 6 t h e r e s o n a n c e / l , Z , /.

I n o u r model c a l c u l a t i o n t h e r e d u - c e d e.m. t r a n s i t i o n s t r e n g t h c a n a l s o b e e a s i l y c a l c u l a t e d / 4 / . The r e s u l t s a r e r e p o r t e d i n F i g . 3 . The mode w i t h l o w e r e n e r g y g e t s t h e l a r g e r s t r e n g t h and t h i s f u r t h e r s u p p o r t s t h e e x p e c t a t i o n o f a n o b s e r ved s h i f t o f t h e r e s o n a n c e c e n t r o i d t o l o w e r v a l u e s w i t h i n c r e a s i n g a n -

/

^,

^, I g u l a r m o m e n t u m .

0

2 hw(MeV)

F u l l y s e l f c o n s i s t e n t c a l c u l a t i o n s w i t h r e a l i s t i c Skyrme f o r c e s a r e

u n d e r way a l s o w i t h i n c l u s i o n o f i s o s p i n d e g r e e s o f f r e e d o m .

F i g . 3

R e f e r e n c e s

1 ) Newton J . O . e t a l . , P h y s . R e v . L e t t . 46 (1981) 1 3 8 3 . 2 ) D r a p e r J . E . e t a l . , P h y s . R e v . L e t t . c(1982) 434.

3 ) B r i n k D . M . and D i Toro M.,Nucl.Phys. A372 (1981) 1 5 1 . 4) D i Nardo M.et a l . , P h y s . R e v . E28 (1983) 929 and

P h y s . L e t t . (1983) 240.

5) D i Toro M.,"Microscopic F o u n d a t i o n s o f N u c l e a r F l u i d Dynamics" i n

" N u c l e a r C o l l e c t i v e Dynamics" Ed.V.Ceausescu,World S c i e n c e P u b l i - s h i n g , i n p r e s s .

6 ) D i Nardo M.et a l . , " C o r i o l i s c o u p l i n g e f f e c t s i n G i a n t R e s o n a n c e s o n r o t a t i n g n u c l e i " , P h y s . L e t t . ~ i n p r e s s .

7 ) D i Nardo M . . e t a l . , N u o v o Cimento L e t t . 35 (1982) 1 1 3 .

8) I g n a t y u k A . V . and M i k h a i l o v I . N . , S o v . ~ . ~ c l . ~ h ~ s . 33 (1981) 483 .