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GIANT RESONANCES IN HEAVY ION COLLISIONS
A. Bonaccorso, M. Di Toro, U. Lombardo, G. Russo
To cite this version:
A. Bonaccorso, M. Di Toro, U. Lombardo, G. Russo. GIANT RESONANCES IN HEAVY ION COLLI- SIONS. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-269-C6-279. �10.1051/jphyscol:1984632�.
�jpa-00224234�
JOURNAL DE PHYSIQUE
Colloque C6, suppldment a u n06, Tome 45, juin 1984 page C6-269
GIANT RESONANCES I N HEAVY ION COLLISIONS
A. Bonaccorso, M. Di Toro, U. Lombardo and G. Russo
I s t i t u t o DipartimentaZe d i Fisica, Uniuersita' d i Catania, I s t i t u t o NazionaZe d i Fisica NucZeare, Sez. d i Catania 5 7 , Corso ItaZia, 1-95229 Catania, I t a l y
Resume - Nous t r a i t o n s l e s resonances g6antes de d i p o l e i s o v e c t o r i e l l e s c o n s t r u i t e s s u r l e s Gtats de haut s p i n observees dans l e s r e a c t i o n s de fusion. Nous reproduisons l e s p r i n c i p a u x aspects des donnees experimen- t a l e s par l ' a n a l y s e des s o l u t i o n s de " s c a l i n g " des p e t i t e s o s c i l l a t i o n s hors de phase de deux f l u i d e s de Vlasov dans un r e f e r e n t i e l en r o t a t i o n . Nous considerons aussi l e problsme de l a p r o d u c t i o n de II sous l e s e u i l dans l e s c o l l i s i o n s e n t r e i o n s lourds. Nous montrons comment o b t e n i r une s e n s i b l e hausse dans l e rendement seulement en c o n s i d e r a n t des de- formations dynamiques c o l l e c t i v e s , du t y p e quadrupole ggant, dans l a pha- se d'approche.
A b s t r a c t - We discuss i s o v e c t o r g i a n t d i p o l e resonances b u i l t on h i g h s p i n s t a t e s observed i n f u s i o n r e a c t i o n s . We reproduce t h e main f e a t u r e o f t h e experimental data from t h e a n a l y s i s of s c a l i n g s o l u t i o n s f o r out-of-phase small o s c i l l a t i o n s o f two Vlasov f l u i d s i n a r o t a t i n g frame. We consider a l s o t h e problem o f subthreshold T-production i n heavy i o n c o l l i s i o n s . We show how t o g e t a n o t i c e a b l e increase o f t h e y i e l d j u s t c o n s i d e r i n g c o l - l e c t i v e dynamical deformations, o f g i a n t quadrupole type, i n t h e approaching phase.
Giant resonances are extremely important dynamical p r o p e r t i e s o f n u c l e i and we should expect t o see r e l a t e d e f f e c t s i n i o n - i o n c o l l i s i o n s . I n t h i s c o n t r i b u t i o n we a r e m a i n l y going t o discuss a t h e o r e t i c a l i n t e r p r e t a t i o n o f t h e main f e a t u r e s o f g i a n t modes b u i l t o n h i g h s p i n s t a t e s . I n t h e l a s t p a r t we a l s o show how a g i a n t quadrupo l e resonance c o u l d a c t as doorway s t a t e i n order t o enhance t h e r a t e o f subthreshofd a-production .
I - ISOVECTOR GIANT DIPOLE RESONANCES ON HIGH SPIN STATES
A g r e a t deal o f progress has been made t h e l a s t years i n t h e i n v e s t i g a t i o n o f GDR's b u i l t on h i g h s p i n s t a t e s observed i n h i g h energy y - r a y spectra from t h e deexcita- t i o n o f medium and heavy n u c l e i produced i n f u s i o n and deep i n e l a s t i c r e a c t i o n s /I/.
The appealing p h y s i c a l aspects o f such a study l i e i n t h e p r o p e r t i e s o f t h e GDR s t r e n g t h f u n c t i o n i n systems f a r from t h e ground s t a t e and i t s dependence on proper- t i e s o f hyghly e x c i t e d n u c l e i such as e x c i t a t i o n energy, s p i n and nuclear deforma- t i o n .
Although some experimental discrepancies s t i l l e x i s t , t h r e e main f e a t u r e s come' o u t from t h e a v a i l a b l e experimental data: 1) s t r e n g t h f u n c t i o n n o t much depending on the g.s. deformation, 2 ) s h i f t of t h e c e n t r o i d o f t h e resonance t o lower energies w i t h h i g h e r angular momenta (aER/AL *.05+.1 MeV/%), 3) GDR o v e r a l l widths much broader
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984632
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than the typical ground s t a t e GDR widths.
Several theoretical attempts t o t h i s problem have been proposed recently /2/, ranging from simple solvable models t o q u i t e huge cranked RPA calculations.
The approach proposed in t h i s paper i s fluid-dynamical in t h a t the c o l l e c t i v e nuc- l e a r dynamics i s developed in a phase space as a semiclassical l i m i t of the s e l f - consistent TDHF equations /3/. Consequently our r e s u l t s , which are q u i t e easy t o work out, can be obtained using r e a l i s t i c interactions and can be d i r e c t l y compared with f u l l cranked RPA calculations.
The key point of our approach i s t o assume t h a t f o r a Giant Collective s t a t e a l l t h e strength i s concentrated on only one level. This ansatz i s largely j u s t i f i e d from RPA calculations a s well a s from variational f l u i d-dynami cal approaches and i t corrg sponds t o take i n t o account only t h e lowest multipole d i s t o r t i o n s of the momentum d i s t r i b u t i o n during the vibration, which i s described as a scaling mode.
O u r general philosophy i s t o use t h i s simplifying assumption t o study Giant Resonan- ces in a q u i t e wider context, l i k e in a r o t a t i n g nucleus o r as doorway s t a t e s in par t i c u l a r reactions.
In t h i s paper we a r e a b l e t o show the e f f e c t s of dynamical deformations and Coriolis coupling on the frequencies and strengths of GDR's b u i l t on high spin s t a t e s . In p a r t i c u l a r an agreement with t h e above mentioned experimental features i s found.
I1 - CRANKED VLASOV EQUATION
The dynamics of a two-fluid system will derived from t h e Vlasov equation i n a frame rotating w i t h angular frequency (assumed t o be along the z-axis)
where h' = h-:.t i s t h e Wigner transform of t h e s e l f c o n s i s t e n t cranked Hamiltonian with a Skyrme interaction. The label q i s the isospin coordinate; quantities without any isospin labels a r e understood t o denote t o t a l values.
E q . (2.1) can be obtained+az a semiclassical 1 imit of t h e cranked TDHF equation f o r the Wigner transform f ( r , p ; t ) of the one-body density matrix.
q
Transforming i n the i n t r i n s i c coordinate (?'=?,%=~-m;x?) t h e equation of motion becomes
~ ~ , 1 ? , ~ ~ , e ) = { ~ ; ( t , ~ , t ) , P ~ ( 6 ~ , t ) ] + z m i : - ( ~ ~ x ~ # ~ ) ( 2 . 2 ) Q t
For Galinei invariant Skvrme forces the transformed cranked Hamiltonian assumes the form
K' + $(1)-i? + A g ( 4 ) - f .m ~ S X P I '
c; = tx=
1 *
where t h e e x p l i c i t expressions f o r m , A and B a r e given in r e f . /4/.
In the following we shall focus our attention on the SKM force which reproduces very well the nuclear compressibilities and t h e symmetry energy near t h e nuclear sur- face 151.
I t i s worth noticing t h a t , l i k e i n the c l a s s i c a l case, the centrifugal and the Coriolis forces appear e x p l i c i t l y i n the ~ a m i l t o n i a n and in the equation of motion, respectively, as an e f f e c t of the rotation.
From eq.(2.2) we can generate an i n f i n i t e chain o f fluid-dynamical equations f o r t h e -+ k-moments o f t h e d i s t r i b u t i o n f u n c t i g n . I f we a r e able, w i t h some physical anaatz, t o t r u n c a t e t h e chain a t t h e lowest k-moments l e v e l , i n s t e a d o f t h e camplicated Cran ked Vlasov equation,+we should solve a s e t o f d i f f e r e n t i a l coupled equations f o r r e l a - t i v e l y few unknown ( r , t ) - f u n c t i o n s , as d e n s i t y , c u r r e n t , k i n e t i c energy t e n s o r and so on. These equations are completely c l a s s i c a l , quantum e f f e c t s being i n t h e i n i - t i a l c o n d i t i o n s and i n t h e t r u n c a t i o n procedure.
For s c a l i n g modes we can e x a c t l y c l o s e t h e fluid-dynamical chain a t t h e lowest two z-moments, c o n t i n u i t y and Eul e r equations / 3 / .
The equation f o r t h e z e r o t h 2-moment assumes t h e form
w i t h
and
As i t i s w e l l known t h e neutron-proton exchange o f a given i s o s p i n L y p e
disappears, being Jce
served i n any case.
c u r r e n t i s n o t conserved l o c a l l y due t o t h e presence o f the term i n t h e nun-local p o t e n t i a l . The t o t a l number o f nucleons i s g f c o u r s e , conserved. For i s o s c a l a r modes t h e source term
. Obviously, t h e t o t a l d e n s i t y i s l o c a l l y con For t h e f i r s t 2-moment we g e t t h e E u l e r equation
L a s t f o u r t e r m s i n t h e s e c o n d l i n e do n o t g i v e a n y c o n t r i b u t i o n a f t e r The k i n e t i c energy t e n s o r i s given by l i n e a r i z a t i o n
I 1 1 - ISOVECTOR DIPOLE MODE
I s o v e c t o r g i a n t d i p o l e rezo$ances a r e $e%cri bed as out-of-phase small amp1 i t u d e c o l - l e c t i v e o s c i l l a t i o n s s f ( r , k , t ) = - s f ,(r,k,t) o f t h e neutron/proton d i s t r i b u t i o n f u n c t i o n around t h e s t a t i o n a r y value 4 4fst(:,t,t). S c a l i n g deformations correspond t o a f l o w p a t t e r n given by an i r r o t a t i o n a l d i p o l e v e l o c i t y f i e l d . q
Using a g e n e r a l i z e d s c a l i n g generator /3/
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we have
I t i n v o l v e s second o r d e r d i s t o r t i o n s o f t h e momentum d i s t r i b u t i o n s w h i l e t a k i n g i n t o account t h e s h i f t o f t h e Fermi sphere t o t h e r o t a t i o n .
Assuming t h e nucleus t o undergo r i g i d ro_fations ( ?ft= 0 ) and i n t h e l i m i t of i r r o t a t i o n a l displacement f i e l d Jgav ,one ob b i n s from t h e c o n t i n u i t y equa- t i o n t h e simple r e l a t i o n 'C 9
which a l l o w s us 40 close t h e fluid-dynamical c h a i n a t t h e lowest two z-moments. I n f a c t , t h e f i r s t k-moments o f t h e t r a n s i t i o n d i s t r i b u t i o n f u n c t i o n can be expressed o n l y i n terms o f t h e s c a l i n g f i e l d 3 ( r , t )
9
The l i n e a r i z e d E u l e r equation becomes t h e equation of motion f o r t h e s c a l i n g f i e l d
where D i s t h e l o c a l p a r t o f t h e Skyrme HF p o t e n t i a l and p=-t-/4h2,t-=t,-t2.
9
I t describes t h e dynamics o f t h e c o l l e c t i v e motion and, i n p r i n c i p l e , can be solved by imposing s u i t a b l e boundary c o n d i t i o n s .
As f a r as we are i n t e r e s t e d i n gross p r o p e r t i e s o f t h e g i a n t modes such as c e n t r o i d energies and t r a n s i t i o n strengths i n terms o f t h e angular frequency, we f o l l o w a sim- p l i f i e d procedure.
L e t us assume f o r t h e s c a l a r f i e l d t h e usual Tassie-Bohr form
By t a k i n g t h e s c a l a r product o f eq.(3.7) w i t h each component o f t h e r e a l s c a l i n g f i e l d (3.8) and by i n t e g r a t i n g over t h e s p a t i a l coordinates we end up w i t h t h r e e d i f f e r e n t i a l coupled equations f o r t h e dqx,y ,r (t) unknown f u n c t i o n s .
As Vi(sS); = o V;,j t h e g r a d i e n t and divergence terms i n t h e l i n e a r i z e d E u l e r equation g i v e no c o n t r i b u t i o n a f t e r p r o j e c t i o n .
A c t u a l l y , from a d i r e c t e v a l u a t i o n o f t h e c o l l e c t i v e k i n e t i c energy, i t i s p o s s i b l e t o show how t h i s procedure gives t h e r i g h t c o l l e c t i v e mass. For r o t a t i o n s about t h e z-axis,the z-mode i s n o t a f f e c t e d by t h e presence o f c e t r i f u g a l and C o r i o l i s f o r ces b u t o n l y i n d i r e c t l y " f e e l s " t h e r o t a t i o n through dynamical deformations i n the s t a t i o n a r y d e n s i t y .
For a s t a t i o n a r y d i s t r i b u t i o n f u n c t i o n which i s i n v a r i a n t w i t h respect t o r o t a t i o n o f a about each o f t h r e e s p a t i a l orthogonal axes, t h e equation f o r t h e z-mode i s
w h i l e the x,y-modes, coupled o n l y through t h e C o r i o l i s term, s a t i s f y
where
d3;
c o l l e c t i v e i n e r t i a l parameter, w i t h Z = 9
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k i n e t i c energy and i n t e r a c t i o n c o n t r i b u t i o n s , r e s p e c t i v e l y . I n t h e harmonic approximation eq. (3.9) g i v e s
w h i l e f o r t h e x,y-modes we g e t
w i t h c , = ~ ( c , ~ c ~ )
I t should be n o t i c e d t h a t t h e major c o n t r i b u t i o n s t o t h e i n t e g r a l s of eq.(3.11) come from t h e n u c l e a r surface. Thus, i t i s n o t t h e d e t a i l e d behaviour of t h e l o c a l densi t y i n s i d e t h e nucleus which i s important here, b u t r a t h e r t h e shape a t t h e surface.- The expressions f o r t h e d i p o l e frequencies, obtained so f a r a r e v e r y general and a l l o w f u l l 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 a f i n i t e temperature cranked HF code t o g e t t h e s t a t i o n a r y s o l u t i o n /6/.
As p r e l i m i n a r y c a l c u l a t i o n we s p e c i a l i z e o u r method f o r a c l a s s i c a l r o t a t i n g nucleus about t h e z-symmetry a x i s , which corresponds t o a maximum c o l l e c t i v i t y c o n d i t i o n f o r t h e r o t a t i o n .
This i s shown i n t h e n e x t s e c t i o n .
I V - EVALUATION OF THE DIPOLE ENERGIES
Although we have developed t h e formalism f o r a general case, i n order t o s i m p l i f y t h e d i s c u s s i o n and t o emphasize t h e main e f f e c t s o f t h e r o t a t i o n on t h e d i p o l e f r e - quencies, we s h a l l assume t h a t
and we n e g l e c t t h e Coulomb f i e l d . I n t h e Thomas-Fermi approximation f o r t h e k i n e t i c energy d e n s i t ~ ~ i n t e g r a t i n g by p a r t s eq.(3.11) we g e t C;=Z9E/A where
w i t h
f o r a x i a l l y symmetric dynamical deformations along z-axis (F = F ) we g e t x Y
w i t h
F o r t h e cranked s t a t i o n a r y d e n s i t y we assume an o b l a t e Wood-Saxon shape
ySt(*, p) = Po
i + exp T - Rte)
a d
w i t h po =0.145 f n r 3 , a=0.5 fm and R0(B) f i x e d by t h e c o n d i t i o n
p5 ( * t ~ ~ . ) ) = A
The e q u i l i b r i u m deformation B ( W ) i s constructed by t h e v i r i a l t e n s o r method /7/ which c o n s i s t s t o impose a balance among t h e f o r c e s a c t i n g o n t h e nucleus, namely pressure, surface, c e n t r i f u g a l , Coulomb and n u c l e a r forces. We s o l v e a n a l y t i c a l l y t h e i n t e - g r a l s (4.2), assuming w i t h a very good approximation
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In t h i s way, e x p l i c i t expressions f o r the dipole frequencies are obtained
where
i s t h e dipole centroid energy f o r spherical nuclei in t h e non-rotating case, which provides a good estimate of t h e experimental values /8/.
Trasforming t o t h e laboratory system, we get four frequencies Q+t w and Q-A w , i n addition t o the unaffected n Z . ( ~ n i n t h e f i g . s )
The expected e f f e c t s are shown in fig.1 f o r 40Ca and in f i g . 2 f o r 1 6 8 ~ r . W predict e
a c l e a r increasing of the width mainly due t o a s p l i t t i n g of t h e giant level. The dipole absorption cross sections, e a s i l y computed in our approach /3/, are used as weights i n order t o get t h e average dipole energy (*Q),
1 2 hw (MeV)
We obtain a small s h i f t of the centroid of the resonance t o lower energies. This e f f e c t i s enhanced in the laboratory frame, b u t s t i l l the maximum s h i f t i s of the order of 1.5 MeV f o r angular v e l o c i t i e s which can be reached in r e a l i s t i c heavy ion reactions (%LIZ 1 MeV). Actually t h e rotation discussed here i s f u l l y classical and then the corresponding angular momentum completely c o l l e c t i v e . I f we take i n t o ac- count a l s o quantum alignement e f f e c t s of single p a r t i c l e spins we get a much lower c o l l e c t i v e angular momentum and a corresponding substantial decrease of the angular velocity and of the related e f f e c t s . In conclusion while we s t i l l predict a noti- ceable s p l i t t i n g i t i s hard t o say t h a t we should a l s o expect a systematic decrease of the peak energy t o lower values.
V - COLLECTIVE DYNAMICAL EFFECTS ON SUBTHRESHOLD II-PRODUCTION IN HEAVY ION COLLISIONS The production of pions in heavy ion reactions f a r below the f r e e N N s c a t t e r i n g thre- shold i s maybe the f i r s t experimental evidence of new phenomena we could observe in medium energy heavy-ion c o l l i s i o n s . Of course one should take i n t o account the mo- mentum d i s t r i b u t i o n of nucleons in the two c o l l i d i n g ions, but a l l the calculations performed with Fermi gas models or more r e a l i s t i c shell model wave functions, give production r a t e s well below (of about a f a c t o r 100) t h e experimental values observed a t CERN i n the 12C+12C a-inclusive reaction a t the energy region 60-85 MeV/A /9,10/.
These r e s u l t s a r e inducing a search f o r production mechanisms d i f f e r e n t from the NN c o l l i s i o n (cooperative process, pionic fusion, nuclear bremsstrahlung e t c ...).
In t h i s contribution we would l i k e t o show how a c o l l e c t i v e d i s t o r t i o n in t h e momell turn d i s t r i b u t i o n s of the nucleons inside the approaching two ions can account f o r a large part of t h e observed discrepancy. We are assuming a diabatic polarization of the two ions along the l i n e which connects the two centres, which corresponds t o t h e
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s e t up o f a g i a n t quadrupole o s c i l l a t i o n , whose d i s t r i b u t i o n f u n c t i o n i n phase space can be e a s i l y w r i t t e n i n t h e s c a l i n g approximation. I n o u r c a l c u l a t i o n we c o n s i d e r an harmonic o s c i l l a t o r s h e l l model t o d e s c r i b e 1 2 C /11/ and then we s c a l e c o o r d i n a - t e s and momenta i n a quadrupole way a l o n g z ( d i r e c t i o n between t h e two c e n t r e s )
where a i s t h e a m p l i t u d e o f t h e deformation. The p r o d u c t i o n c r o s s - s e c t i o n has t h e s t r u c t u r e
ua - (Glauber f a c t o r ) . (Phase space f r a c t i o n ) . ~ Y ( E ~ ~ ) - ( P a u l i b l o c k i n g ) . ( a a b s o r p t i o n ) where: t h e Glauber f a c t o r g i v e s t h e average number o f f i r s t NN c o l l i s i o n s ;
t h e phase space f r a c t i o n g i v e s t h e p a r t o f phase space a v a i l a b l e t o produce p i o n s i n t h e cm o f t h e t w c o l l i d i n q nucleons: -
a:N i s t h e p i o n p r o d u c t i o n c r o s s s e c t i o n f o r f r e e NN s c a t t e r i n g ;
P a u l i b l o c k i n g and a - a b s o r p t i o n a r e e f f e c t s t o t a k e i n t o account i n t h e e x i t channel.
A l l t h e s e terms a r e a f f e c t e d by t h e c o l l e c t i v e d e f o r m a t i o n i n t h e approaching phase.
We have focused o u r a t t e n t i o n on t h e phase space f r a c t i o n . F i g . 3 shows t h e b e h a v i o u r o f t h i s q u a n t i t y as a f u n c t i o n o f t h e d e f o r m a t i o n parameter a f o r a 12C+12C c o l l i s i o n
a t 85 MeV/A. It i s v e r y i n t e r e s t i n g t h e i n i t i a l i n c r e a s e o f t h e curve ( a l a r g e r t h a n .25 have n o p h y s i c a l meaning). We can go f r o m l e s s t h a n 1% o f phase space avaL l a b l e t o about a 30%.
We can c o n s i d e r as a minimum v a l u e o f a t h e z e r o p o i n t m o t i o n a m p l i t u d e f o r a sca- l i n g quadrupole mode o b t a i n e d f r o m t h e e q u a t i o n
w i t h
which l e a d s t o aZp .08.
T h e r e f o r e we c o u l d e a s i l y g a i n a f a c t o r between 10 and 20 j u s t t a k i n g i n t o account c o l l e c t i v e d e f o r m a t i o n s i n t h e i n i t i a l stage o f t h e r e a c t i o n . More p r e c i s e c a l c u l a - t i o n s a r e under way a l o n g t h i s l i n e /12/. The i n t e r p r e t a t i o n o f t h e enhancement as due t o a g i a n t quadrupole resonance a c t i n g as doorway s t a t e i s perhaps t o o n a i v e . However t h i s r e s u l t c e r t a i n l y shows t h a t dynamical d e f o r m a t i o n s i n the approaching phase can be e x t r e m e l y i m p o r t a n t t o e x p l a i n t h e h i g h r a t e o f produced s u b t h r e s h o l d p i o n s .
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