NEW DEVELOPMENTS IN THE STUDY OF CONTINUUM SPECTRA

HAL Id: jpa-00224066

https://hal.archives-ouvertes.fr/jpa-00224066

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.

NEW DEVELOPMENTS IN THE STUDY OF CONTINUUM SPECTRA

H . Machner

To cite this version:

H . Machner. NEW DEVELOPMENTS IN THE STUDY OF CONTINUUM SPECTRA. Journal de

Physique Colloques, 1984, 45 (C4), pp.C4-3-C4-12. �10.1051/jphyscol:1984401�. �jpa-00224066�

JOURNAL DE PHYSIQUE

Colloque C*, supplément au n°3, Tome 45, mars 198*f page C4-3

NEW D E V E L O P M E N T S IN T H E STUDY OF C O N T I N U U M SPECTRA

H. Machner

Institut fur Kernphysik, Kernforschungsanlage Jillieh, D-5170 Julioh, F.R.G.

Résumé - Les sections e f f i c a c e s i n c l u s i v e s e t exclusives de r é a c t i o n s nuclé- a i r e s à moyenne énergie sont analysées dans l e cadre d'un modèle

" e x c i t o n " é l a b o r é . Le concept d ' e x c i t a t i o n l o c a l e s ' a v è r e ê t r e u t i l e à l ' i n t e r p r é t a t i o n des données des expériences de coincidence.

Abstract - Inclusive and exclusive cross sections in medium energy nuclear reactions are discussed and analyzed in terms of an extended exciton model.

I t i s shown t h a t the concept of local e x c i t a t i o n s might by useful in e x p l a i - ning coincident data.

I - INTRODUCTION

The study of continuous energy spectra from medium energy nuclear reactions is of im- portance due to d i f f e r e n t reasons. One aspect i s t h a t by knowing the underlying reac- tion mechanism one is able to subtract a physical background below s t r u c t u r e s of i n t e r e s t . For the case of giant resonances the importance of t h i s knowledge has been shown by Osterfeld in his contribution

) to t h i s conference. Examples for break-up studies are given in r e f s . 2 and 3. However, the study of the reaction mechanisms i s important by i t s e l f : the continuous part of the spectra contains the l a r g e s t fraction of the cross s e c t i o n .

In the experiments recently performed new aspects of the reaction mechanism have been attacked. These are for example:

a) large range of bombarding energies,

b) variation of the projectiles from nucleons to light heavy ions, c) use of polarized beams,

d) exclusive or semiexclusive measurements of particle spectra.

The models developped originally to understand light ion induced reactions at mode- rate energies got their theoretical foundations

*"

) recently. They seem to work in an unexpected large region of excitation energies

). Because they give reliable results in the case of ion induced reactions they can be used to analyze continuous spectra from more exotic probes like pion-s

'

), muons

) or Y ' S

) to study the vertex of probenucleus interaction.

In this contribution we would like to present first the phenomena occuring at medium energies. We then describe the generalized exciton model extensions followed by a com- parison between model predictions and data. In this comparison coincident data ana- lysis is included. Finally we will discuss the concept of local excitations.

II - PHENOMENA AT MEDIUM ENERGIES

It was only recently that physicists have started to perform experiments in the ener- gy region above 100 MeV and produce data "with anything like the completeness and precision that nuclear physicists have become used to when working at lower energies"

(ref. 12). While low energy protons and complex particles with the same amount of energy/nucleon probe only the surface of target nuclei high energy protons are

"looking" deeper into the nuclear volume

). It is therefore very interesting to prove models designed originally for much smaller energies. Here we will mainly con- centrate on the generalized exciton m o d e l

) .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984401

JOURNAL DE PHYSIQUE

PROTONS 200 MeV

I n f i g u r e 1 we p r e s e n t p r o t o n s p e c t r a f r o m 200 MeV p r o t o n s on 2 7 A l a t 200 MeV. The expe- r i m e n t s have been performed a t t h e Synchro- C y c l o t r o n o f t h e INP by a J i i l i c h - O r s a y c o l l a - b o r a t i o n 1 5 ) . To r e a l l y produce d a t a o f h i g h q u a l i t y we have used a s p e c t r o m e t e r c o n s i s t i n g o f s e v e r a l HPGe diodes. T h i s d e v i c e i s i l l u s - t r a t e d i n f i g u r e 2. I t a l l o w s us t o s p e c t r o s - copy p r o t o n s up t o 200 MeV w i t h an energy r e - s o l u t i o n o f lo-,. A p r e l i m i n a r y v e r s i o n o f t h e s p e c t r o m e t e r has been p u b l i s h e d i n r e f . 16.

The f i r s t Ge-counter c o n s i s t s o f an i n n e r c i r c l e a c t i n g as AE c o u n t e r and an o u t e r r i n g used as a c t i v e c o l l i m a t o r . T h i s has been essen- t i a l i n p r o d u c i n g c l e a n r e l i a b l e s p e c t r a . The f o r w a r d a n g l e s p e c t r a i n f i g u r e 1 show a d i s t i n c t peak which has t h e s i g n s o f q u a s i - f r e e nucleon-nucleon s c a t t e r i n g . F o r heavy t a r g e t s l i k e g o l d we see t h i s bump o n l y a t v e r y f o r w a r d angles (14O). The peak shows s t r u c t u r e n o t h a v i n g been r e p o r t e d so f a r . Besides t h e knock-out bump t h e s p e c t r a a r e dominated by t h e so c a l l e d continuum. T h i s continuum i s f o r w a r d peaked b u t n o t t h a t s t r o n g l y than t h e knock-out y i e l d . T h i s i s a c l e a r i n d i c a t i o n o f t h e mu1 t i s t e p n a t u r e o f t h e continuum. The changes i n t h e s p e c t r a l shapes f o r d i f f e r e n t angles i s i l l u s t r a t e d i n f i g u r e 3.

I 1 1 - EXCITON MODEL EXTENSIONS

I n p r i n c i p l e one would l i k e t o d e s c r i b e s c a t - t e r i n g processes w i t h t h e h e l p o f t h e Schro- d i n g e r e q u a t i o n . However, t h i s i s by f a r t o c o m p l i c a t e d . T h e r e f o r e one has t o deal w i t h more o r l e s s s u c c e s s f u l models. One o f these i s t h e e x c i t o n model which has i n a h e u r i s t i c

1 ,

4 I way been i n t r o d u c e d by G r i f f i n 1 7 ) . I t s t h e o r e -

0 50 100 150 c ( M e V )

t i c a l f o u n d a t i o n has been g i v e n by Agassi e t

Figure Forward angle spectra from a1.

Here we w i l l s t a r t w i t h t h e master the indicated reaction. The distinct e q u a t i o n l a ) . T h i s e q u a t i o n can be deduced peak i n t h e s p e c t r a stems f r o m q u a s i - from t h e Schrodi n g e r e q u a t i o n by coarse g r a i - f r e e nucleon-nucleon s c a t t e r i n g . n i n g i .e. by lumping a s m a l l number o f e i g e n -

s t a t e s o f t h e u n p e r t u b a t e d h a m i l t o n i a n l a b e l e d

ION PUMP

- -

PARTICLE BEAM

by rl, r,, r,, . . . i n t o a group l a - b e l d r. L e t P ( r , t ) be t h e p r o b a b i l i - t y o f f i n d i n g t h e system i n s t a t e r a t t i m e t and x ( r + r t ) t h e t r a n s i t i o n p r o b a b i l i t y p e r u n i t t i m e between t h e groups o f s t a t e s r and r ' . Be- cause n u c l e i a r e s m a l l and f i n i t e systems one has t o t a k e e m i s s i o n i n t o account by

t h e t r a n s i t i o n r a t e i n t o t h e continuum. Then t h e P a u l i master e q u a t i o n reads

F i g u r e 2: The h i g h r e s o l u t i o n s o l i d

s t a t e s p e c t r o m e t e r i n s i d e t h e cryos-

t a t . The f i r s t Ge-diode c o n s i s t s o f

an a c t i v e c o l l i m a t o r .

where t h e summation c o n t a i n s i n t h e case o f c o n t i n u o u s v a r i a b l e s a l s o i n t e g r a t i o n . T h i s e q u a t i o n i s i r r e v e r s i b l e w h i l e t h e S c h r o d i n g e r e q u a t i o n i s r e v e r s i b l e . The i r r e v e r s i b i 1 i t y i s o b t a i n e d as can be seen i n van Kampen's d e r i v a - t i o n o f t h e master e q u a t i o n l g ) by i n v o k i n g a r e p e a t e d random phase a p p r o x i m a t i on, i . ^{e. by}

n e g l e c t i n g o r s u p p r e s s i n g any dynamical b u ~ l d - up o f phases as t i m e e v o l v e s . A s u f f i c i e n t c o n d i t i o n i s t h a t t h e l i f e t i m e o f t h e s t a t e r i s l o n g i n comparison t o t h e response t i m e o f t h e r e s i d u a l i n t e r a c t i ~ n ~ ~ ~ ~ ) . T h i s c o n d i t i o n i s f u l l f i l l e d i n t h e e x c i t o n model f o r a l o n g mean f r e e p a t h 2 0 ) . As "good" quantum numbers one choses i n a s h e l l model r e p r e s e n t a t i o n t h e number o f e x c i t e d p a r t i c l e s p and h o l e s h w i t h r e s p e c t t o t h e Fermi energy and n

(g,p) t h e d i r e c t i o n o f a p a r t i c l e 1 4 ) . The sum n=p+h i s c a l l e d t h e e x c i t o n number. The t r a n s i t i o n r a t e s i ( r + r l ) may be o b t a i n e d f r o m nucleon-nucleon s c a t t e r i n g i n n u c l e a r m a t t e r . I n p r a c t i c e we use

w i t h /A(n,n)dn

1 . F i g u r e 3: I n c l u s i v e p r o t o n s p e c t r a ob- ( 3 ' 3 ) t a i n e d a t t h e i n d i c a t e d angles.

Then eq. (3.1) reduces t o an a n g l e independent e q u a t i o n . A(n,n) i s t h e p r o b a b i l i t y o f f i n d i n g a p a r t i c l e w i t h v e l o c i t y i n t o d i r e c t i o n n i n an n - e x c i t o n s t a t e . Because one can n o t d i s t i n g u i s h between t h e s t r u c k and t h e s c a t t e r e d n u c l e o n one has two p a r - t i c l e s c a r r y i n g memory of t h e i n c i d e n t d i r e c t i o n 2 1 ) . I f t h e energy o f a system i s s u f f i c i e n t l y h i g h one can t r e a t t h e e q u i l i b r a t i o n o f e v e r y r e s i d u a l system by a s e t o f master e q u a t i o n s . Eq. (3.1) i s s o l v e d f o r t h e i n i t i a l c o n d i d a t i o n

P(n,n,t=O)

s A(no,n) oo

n ,no (3.4)

w i t h a . t h e r e a c t i o n c r o s s s e c t i o n . The m a s t e r e q u a t i o n s f o r t h e r e s i d u a l systems have t o be s o l v e d f o r

P(m,n,t=O)

6 m,n-n.

B(n,ei ,E,<) ( 3 . 5 )

w i t h B t h e c r o s s s e c t i o n f o r p r o d u c i n g a r e s i d u a l system when a p a r t i c l e w i t h n i e x c i - t o n s and energy

has been e m i t t e d i n t o t h e d i r e c t i o n T i . Then t h e cross s e c t i o n f o r e m i s s i o n o f a p a r t i c l e w i t h energy sj i n t o t h e d i r e c t i o n 3 f r o m such a r e s i d u a l system i s 2 1 )

w i t h f an i s o s p i n m i x i n g f u n c t i o n , W t h e e m i s s i o n p r o b a b i l i t y which i s c a l c u l a t e d as i n t h e compound nucleus t h e o r y . F o r complex p a r t i c l e s a f i n a l s t a t e i n t e r a c t i o n (coalescence) i s i n c l ~ d e d ~ ~ ) .

The c o i n c i d e n t cross s e c t i o n i s t h e n g i v e n as

C4-6 JOURNAL DE PHYSIQUE

. . dri dni d ~ . dn. d ~ ~ doi a dc . dn. d c . d n . oo dri dni

J J O J J J J

The second f a c t o r r e p r e s e n t s t h e n a c o n d i t i o n a l p r o b a b i 1 i t y . I V - - THE TRANSITION RATES

As mentioned above t h e t r a n s i t i o n r a t e s may be c a l c u l a t e d f r o m c o r r e c t e d f r e e nuc- leon-nucleon cross s e c t i o n s ann. We t h e n o b t a i n

ann(al-n) w i t h W(n'-n)

/ann(n1--n)dn

The t r a n s i t i o n r a t e s A(n-tnt2)

~ + ( n ) and ~ ( n - t n - 2 )

k ( n ) a r e d e r i v e d b y energy a v e r a g i n g and w e i g h t i n g w i t h t h e s t a t i s t i c a l w e i g h t o f t h e d i f f e r e n t s t a t e s . The ma- t r i x element f o r t h e r e s i d u a l i n t e r a c t i o n i s d e r i v e d f r o m t h e mean f r e e p a t h o f nucleons i n n u c l e a r m a t t e r

2 v

i(lp-tZp+lh,U)

2 IMI2 p ( 2 , l , u )

-

Ti ii ( 4 . 2 )

w i t h

t h e l e v e l d e n s i t y , U t h e energy o f t h e p a r t i c l e undergoing t h e c o l l i s i o n and v t h e v e l o c i t y c o n t a i n i n g t h e r e a l p o t e n t i a l depth. We may c a l c u l a t e A f r o m t h e f r e e nucleon-nucleon c r o s s s e c t i o n and make t h e f o l l o w i n g a n s a t z :

A(E,T)

( < o ( ~ , ~ ) > p K ) - l ( 4 . 3 )

with

0.17 fm-3 b e i n g t h e n u c l e a r d e n s i t y and <a> t h e averaged nucleon-nucleon cross s e c t i o n

Z N -AT

<o(E,T)>

(A opp(E)+x anp(E)) ( l - f ( Z ) e . ( 4 . 4 )

The f u n c t i o n f ( Z ' ~ i r ~ ) EF i s t h e P a u l i b l o c k i n g f a c t o r and t a k e n f r o m t h e work o f K i k u c h i and Kawai

fl

F o r f i n i t e temperatures t h e P a u l i b l o c k i n g l o s e s i t s e f f e c t i n p a r t . T h i s i s taken i n t o account by t h e e x p o n e n t i a l i n eq. ( 4 . 4 ) . We can reproduce t h e f i n d i n g s o f C o l l i n s and G r i f f i n 2 4 ) by a c o n s t a n t

0.02 ( n o t t o be mixed w i t h t h e t r a n s i t i o n r a t e ) . I n f i g u r e 4 we compare t h e energy dependence o f A m u l t i p l i e d by k

4 t o t a k e e f f e c t i v e mass e f f e c t s and reduced d e n s i t y i n t h e n u c l e a r s u r f a c e i n t o account w i t h one o b t a i n e d f r o m

( 4 . 6 ) F i g u r e 4: The mean f r e e path m n s i n n u c l e a r m a t t e r as f u n c t i o n o f t h e a s y m p t o t i c energy. The s o l i d l i n e shows t h e r e s u l t o f eq.(4.3) w i t h E ~ = 4 0 MeV and a b i n d i n g ener- gy B=10 MeV. The r e s u l t s a r e m u l t i p l i e d by f o u r as d i s - cussed i n t h e t e x t . The o p t i - c a l p o t e n t i a l r e s u l t ( e q . ( 4 . 6 ) ) i s p l o t t e d as dashed l i n e .

-

T = o MeV

10

- ---_____ -

with o p t i c a l model parameters obtained by Bohr and ~ o t t e l s o n ~ ~ ) by assuming only vo- lume geometries. Figure 5 shows the e f f e c t of f i n i t e temperatures. A t T=10 MeV the

mean f r e e path i s nearly c o n s t a n t .

For the Ericson form of the p a r t i c l e - h o l e level d e n s i t y the averaged t r a n s i t i o n r a t e s a r e g i - ven by polynomials

, 6 a . j ! ~ j x+(n,E)

"-

j L ( n - j + ~ ) ! (4.7)

A . 0 0 2

and ( n - l ) ! p h ( n - 2 ) 6 a j j ! ~ j

1-(n,E)

1 ( 4 . 8 )

~ . ( s E ) ~ j = o (n-3+j)!

with parameters a . given i n r e f . ₃ 20 .

I I I I I I L I

20 LO 50

bo

V - COMPARISONS BETWEEN DATA AND MODEL CALCU-

Figure 5: The mean f r e e path of nuc- LATIONS

leons i n "'Iear matter as In the following we w i l l t e s t t h e exciton model

O f

the asymptotic energy a n d d i f f e - p r e d i c t i o n s a g a i n s t data. In a l l c a l c u l a t i o n s rent temperatures as i n d i c a t e d i n the t h e modelparameter k=4 has been kept f i x e d . In f i g u r e . f i g u r e 6 angle i n t e g r a t e d ( p , p l ) d a t a z 6 ) from

90 MeV proton bombardment of nuclei spanning the p e r i o d i c t a b l e a r e compared with model pre- I

d i c t i o n s . S p e c t r a l shapes and absolute heights

a r e well reproduced. We can a l s o look t o o t h e r e x c i t channels. In f i g u r e 7 which i s taken from

I

101

- >

27AI

(p.n)X

r Ep= 90 MeV

0

P V)

v - i _T

-

-

0 20 LO 60 E,(M~V)

X 2

(D

Figure 7: Inclusive angle i n t e g r a t e d

7'

neutron cross s e c t i o n s a r e compared with two d i f f e r e n t e x c i t o n model c a l - c u l a t i o n s ( s e e t e x t ) . This f i g u r e i s taken from r e f . 27.

r e f . 27 (p ,n) d a t a f o r the same bombarding ener- gy a r e shown. In a d d i t i o n a c a l c u l a t i o n labeled Chang i s shown which used d i f f e r e n t t r a n s i t i o n -

I

r a t e s z 8 ) not a p p l i c a b l e f o r high e n e r g i e s . In

2b

LO

^QO

f i g u r e 8 the dependence of t h e bombarding ener- g i s s t u d i e d . Here data from 2 7 A l ( p , p ' ) a t

4. 50° f o r e n e r g i e s spanning

60 t o 600 MeV Figure 6 : I n c l u s i v e angle i n t e g r a -

bombarding energies

a r e shown with c a l - ted proton c u l a t i o n s . In the case of t h e s m a l l e s t energy

bombardment

t h e i n d i c a t e d an evaporative p a r t has been added. A t t h i s

target '"lei are

w i t h ex- p o i n t we would l i k e t o s t r e s s t h a t a l l calcula-

c i ton model c a l c u l a t i o n s . t i o n s a r e a b s o l u t e and t h a t t h e r e has been no

C4-8 JOURNAL DE PHYSIQUE

+ i n d i v i d u a l adjustment. Even a t t h e v e r y h i g h energy o f 558 MeV t h e gross f e a t u r e s o f t h e d a t a a r e reproduced. F i g u r e 9 g i v e s t h e angu- l a r d i s t r i b u t i o n s o f t h e same d a t a f o r t h r e e e x c i t channel e n e r g i e s t o g e t h e r w i t h two s e t s o f c a l c u l a t i o n s . The dashed l i n e drawn uses an i n i t i a l a n g u l a r d i s t r i b u t i o n

A(n,.n)

a e x p ( - 7 . 2 8 ( 5 . 1 )

u

- d h i c h has been found t o reproduce d a t a w e l l

when t h e p r o j e c t i l e i s a composite system32).

The parameter a i s determined by f u l l f i l l i n g 1u2-

2OOMeV eq.(3.3). The c a l c u l a t i o n w h i c h i s s o l i d l i n e

drawn uses

A(no,n)

a C O S ( ! ~ ~ ) Q ( ~ / ~ - B 8

-

( 5 . 2 )

; I I I I I I I I I

- w l t h B = & ^, Gax

( Z n ) / ( m ) and a g a i n

20 LO 60 80

cp~lEp(%) a n o r m a l i z a t i o n c o n s t a n t . T h i s r e l a t i o n i s ob- t a i n e d f r o m t h e i s o t r o p y o f nucleon-nucleon v o t l i cross s e c t i o n s and t a k i n g i n t o account t h a t for 7 ; goo and three bombarding ener- t h e a n g u l a r d i s t r i b u t i o n s i n f i n i t e n u c l e i g i e s . The e j e c t i l e e n e r g i e s a r e g i v e n cannot have peaks i n t h e i n t e r v a l below in fractions of the projectile &ax1'+). F o r v e r y h i g h bombarding e n e r g i e s

f o r w a r d angles a r e o v e r e s t i m a t e d b y t h e c a l - c u l a t i o n s .

We now want t o s w i t c h o v e r t o c o i n c i d e n t data. So f a r o n l y t h e Maryland group33) ( c o n t r i b u t i o n t o t h i s c o n f e r e n c e ) and t h e J u l i c h group3'+) have p u b l i s h e d c o i n c i d e n t cross s e c t i o n s . Here we want t o c o n c e n t r a t e on t h e l a t e r . These d a t a a r e i n p l a n e c o i n c i d e n t c r o s s s e c t i o n s from 5 8 N i ( a , a ' p ) r e a c t i o n s a t E,

140 MeV. The a - p a r t i c l e s have been s p e c t r o s c o p i e d i n t h e range f r o m 50 t o 140 MeV and p r o t o n s f r o m 2 t o 28 MeV.

I n a f i r s t s t e p we a d j u s t (a,,') c a l c u l a t i o n s t o i n c l u s i v e d a t a . We t h e n c a l c u l a t e s p e c t r a u s i n g eq. ( 3 . 7 ) . Because t h e d a t a a r e i n t e g r a t e d o v e r energy b i n s i n t h e a ' - s p e c t r a we do t h e same f o r t h e c a l c u l a t i o n s . A comparison between d a t a and c a l c u l a - t i o n i s shown i n f i g u r e 10. The h i g h energy re-equilibrium p a r t i s accounted f o r by t h e c a l c u l a t i o n . Aqain we would l i k e t o s t r e s s t h a t t h e r e i s n o i n d i v i d u a l a d j u s t - ment. The same agreement between d a t a and c a l c u l a t i o n s i s a l s o o b t a i n e d f o r o t h e r energy i n t e r v a l l s 2 1 ) .

V I - THE CONCEPT OF LOCAL EXCITATIONS

A r a t h e r d i f f e r e n t approach t o understand p r e - e q u i 1 i b r i um phenomena than t h e one d i s - cussed above i s t h e concept of l o c a l e x c i t a t i o n s . I t has been o r i g i n a l l y proposed by B e t h e 3 5 ) who p o i n t e d o u t t h a t e m i s s i o n f r o m h o t s p o t s w i l l l e a d t o h a r d e r s p e c t r a t h a n e v a p o r a t i o n f r o m a f u l l y e q u i l i b r a t e d compound nucleus. I n a s e r i e s o f papers t h e o r i s t s have r e c e n t l y developped models on t h e b a s i s o f h o t s p o t s 3 6 ) . A l t h o u g h i t has become v e r y p o p u l a r t o p a r a m e t r i z e s p e c t r a i n terms o f moving s o u r c e s 3 7 ) t h e ex- p e r i m e n t a l e v i d e n c e f o r h o t s p o t s i s scant.y.

F o r t h e d e t e c t i o n o f h o t spots - t h e f o l l o w i n g r e l a t i o n s have t o be f u l f i l le d

T c < T d < T

S

( 6 . 1 )

w i t h Tc b e i n g t h e t i m e a h o t s p o t i s b u i l d - u p , ~ d i s i t s p a r t i a l l i f e t i m e due t o p a r t i c l e decay and

i s t h e s p r e a d i n g t i m e . I t has been shown i n a h a r d sphere s c a t t e r i n g mode138) % h a t o n l y t h r e e c o l l i s i o n s p e r sphere a r e s u f f i c i e n t t o e s t a b l i s h an e q u i l i b r i u m v e l o c i t y d e s t r i b u t i o n . The H f u n c t i o n a l r e a d y assumes t h e e q u i l i b r i u m v a l u e r a p i d l y i n about two mean c o l l i s i o n t i m e s 3 8 ) . T h i s i s s u p p o r t e d by t h e p r e - e q u i l i b r i u m model b y Chiang and H ~ f n e r ~ ~ ) . I f we assume 3.10-23 s as mean c o l l i s i o n t i m e we g e t

= 3.3.10-23 s = 9.10-23

I f we c a l c u l a t e i n t h e above d e s c r i b e s e x c i t o n model t h e t i m e necessary t o b u i l d up an 8p+4h s t a t e i n a 58Ni + a r e a c t i o n a t E,

140 MeV we a r e a g a i n l e f t w i t h 8 - 1 0 - 2 3

From t h i s we may conclude t h a t

T C %

10-22 s.

2 7 ~ 1

ip,p')x

Ep=62MeV ..>

\;;. f = ~ ~ , / E p

\ ^\,..

^'. _{--_}

_-:-- I

t

\ 1=050\,

I \\

'\

' "

'

'

'

'

'

' 4b

I n o r d e r t o have a s p a t i a l we1 1 d e f i n e d e x c i t a t i o n p r o j e c t i l e

t h e mean f r e e s h o u l d p a t h be o f v e r y t h e s m a l l . I t has been p o i n t e d i o u t t h a t t h e optimum regime

i f o r l o w e s t mean f r e e p a t h f o r f i n i t e - n u c l e i i n t e r a c - t i o n s i s l i g h t n u c l e i i n c i - d e n t on medium o r heavy n u c l e i a t o r around t h e Fermi energy40). To reduce

-

(p,p'lX

\Iev \

\

-

f=050

^.

t h e e f f e c t o f t h e Coulomb b a r r i e r we have chosen 58Ni as

as d i s t o r t i o n s much t a r g e t as nucleus. p o s s i b l e f r o m t h e To Coulomb s c a t - a v o i d 2 t e r e d p r o j e c t i l e a - p a r t i c l e s

a t 35 MeV/nucleon have been

: s e l e c t e d . T h e i r mean f r e e p a t h d e r i v e d f r o m eq. ( 4 . 6 )

-

i s 1.8 fm.

We w i l l now d i s c u s s

and

~ d . Both q u a n t i t i e s depend

zb

'

l;oZ;o ' 60 ido ' g on t h e temperature T. I n a r e c e n t paper K ~ h l e r ~ ~ ) has

1 2 7 ~ 1

( p , p ' ~ x

\

\

I I , , , , , 1 I I I , , I I , I ,

20 60 100 " 20 60 1 0 0 ~ 20 M)

d e r i v e d values f o r

f r o m TDHF c a l c u l a t i o n s w i t h and w i t h o u t

A ~ d comparison has been two body made between by c o l l K i n d

i s i o n s . and and P a t e r g n a n i 4 1 ) . However, these a u t h o r s do n o t s p e c i -

7 f y t h e way t h e y g e t t h e i r r e s u 1 t s . T h e d e c a y t i m e i s t h e i n v e r s e o f t h e t o t a l

I:

p r o b a b i l i t y o f a p a r t i c l e b e i n g e m i t t e d and has been d e r i v e d b y W e i s s k ~ p f ~ ~ ) b y 6 i n t e g r a t i o n o f t h e e m i s s i o n

r a t e o v e r t h e r e s i d u a l ex- c i t a t i o n energy

F i g u r e 9: A n g u l a r d i s t r i b u t i o n s f o r p r o t o n s a t t h r e e rTi3

d i f f e r e n t f r a c t i o n s o f t h e bombarding e n e r g i e s a r e corn- ~ d = e B / T ( E ) p a r e d w i t h e x c i t o n model c a l c u l a t i o n s . The dashed l i n e (2s+l)R2pT2(E-B)

drawn c a l c u l a t i o n i s p e r f o r m e d w i t h eq. ( 5 . 1 ) t h e s o l i d

l i n e drawn c a l c u l a t i o n w i t h eq.(5.2) f o r t h e i n i t i a l ( 6 . 2 )

a n g u l a r d i s t r i b u t i o n . Here s denotes t h e s p i n ,

t h e reduced mass and B t h e b i n d i n g energy o f t h e p a r t i c l e b e i n g e m i t t e d . E i s t h e t o t a l e x c i t a t i o n energy and R t h e r a d i u s o f t h e r e s i d u a l n u c l e u s .

The s p r e a d i n g t i m e may be d e r i v e d i n a rough e s t i m a t e . T ~ m o n a g a ~ ~ ) g i v e s an expres- s i o n f o r t h e h a l f - l i f e t i m e t h a t i n i n f i n i t e n u c l e a r m a t t e r w i t h temperature 0 i n one h a l f and T i n t h e o t h e r h a l f a t a p o i n t R f r o m t h e o r i g i n t h e temperature changes by 1 / 4 T. T h i s e x p r e s s i o n reads

1 m.pcR2

T S =

--.

0.92

('5.3)

Here c a r e t h e s p e c i f i c h e a t and

t h e h e a t c o n d u c t i v i t y o f n u c l e a r m a t t e r . The spe- c i f i c h e a t i s k 4 )

c =

2 / ~ ) ( T / ~ E F ) ( 6 . 4 )

JOURNAL DE PHYSIQUE

d a t a shown i n f i g u r e 10 i n F i g u r e 10: C o i n c i d e n t p r o t o n cross s e c t i o n s f r o m

5 8 N i ( a , n 1 p ) r e a c t i o n s a t E,

140 MeV a r e compared an energy range f r o m 6 t o w i t h model p r e d i c t i o n s f o r

(64k9.4) MeV. 12 MeV. The o b t a i n e d values

a r e shown i n f i a . 12 f o r

L

and t h e h e a t c o n d ~ c t i v i t y ~ ~ )

t h e t h r e e energy b i n s i n t h e a ' - s p e c t r a w e l l above t h e GQR. I n a l l cases we found h i g h e r T-parameters ( i .e.

d e v i a t i o n s f r o m compound n u c l e u s temperatures which a r e shown as c i r c l e s ) f o r t h e f o r w a r d hemisphere. From these f i n d i n g s we can r u l e o u t t h e e x i s t e n c e o f a s t a t i c h o t s p o t . However, we propose a s i m p l e p i c t u r e as g i v e n i n f i g u r e 12d. A s m a l l number o f nucleons i s moving t h r o u g h t h e f o r w a r d hemisphere c a r r y i n g t h e t r a n s f e r r e d energy and l i n e a r momentum. Because o f t h e h i g h e r energy d e n s i t y i n t h i s p a r t o f t h e nucleus we o b t a i n h i g h e r T-values than i n t h e backward d i r e c t i o n where t h e T-values a r e con- s i s t e n t w i t h compound n u c l e a r temperatures.

58Ni

(a.a'p) Ea=lLOMeV 10-1

eb=65MeV

EF3/2

I< ⁷

---

4 8 0 7 1 V'-?<G>T ( 6 -51

3a*=-300 ^: 10-1

t t

I t

I

t + i , t t w i t h a l l n o t a t i o n s as above.

T h i s l e a d s t o

-rs=163.4

~ ' . ( 6 . 6 )

.I ^{A ( T )}

I", 0

110 The decay t i m e (eq. 6 . 2 ) as f u n c t i o n o f t h e temperature and f o r two d i f f e r e n t nuc-

+ + + 4

-10-1 l e o n numbers N i s shown i n

t

3, = 60" f i g u r e 11. I t i s obvious

t h a t f o r temperatures above 3 MeV t h e d i f f e r e n c e between a h o t s p o t and t h e f u l l y e q u i l i b r a t e d compound nuc- l e u s i s s m a l l . The s p r e a d i n g t i m e i s c a l c u l a t e d u s i n g t h e l o n g mean f r e e p a t h as has been done i n t h e e x c i t o n model c a l c u l a t i o n s . The mean f r e e p a t h f o r t h e

\

s m a l l e s t and l a r q e s t

values

0

P)

(eq.4.4) has been c a l c u l a t e d

-0

10% f o r T/2, t h e one f o r t h e

most r e a l i s t i c v a l u e ~ = 0 . 0 2 f o r T. The temperature depen- dence o f t h e s p r e a d i n g t i m e 10-lr

i s a l s o shown i n f i g u r e 11.

F o r temperatures above 5 MeV t h e r i g h t hand p a r t o f t h e r e l a t i o n ( 6 . 1 ) h o l d s . I t i s j u s t t h i s temperature where a l l t h e times a r e more o r l e s s e q u a l .

0 10 20 30 0 10 20 To deduce T-values we have

~ ~ ( M e v ) f i t t e d e x p o n e n t i a l s t o t h e

10-l9 - F i g u r e 11: The dependence o f t h e s p r e a d i n g t i m e

and t h e decay t i m e r d as a func- t i o n o f t h e temperature.The decay t i m e i s shown f o r the f u l l e q u i l i b r a t e d A=58 com- pound nucleus and a sub-

- system c o n s i s t i n g o f 10

nucleons. A b i n d i n g energy

lo-2o- B=10 MeV has been assumed

- as b e f o r e . The s p r e a d i n g

t i m e i s shown f o r t h r e e

2 d i f f e r e n t values o f 1 (eq.

- _m ^4.4).

- %

P 4

f B

10-2' - -

2 L 6 8 10 TlMeV)

F i g u r e 12: a-c: A n g u l a r dependence o f t h e e x t r a c t e d l o c a l temperature l i k e T para- meters f o r d i f f e r e n t e n e r g i e s o f t h e i n - e l a s t i c s c a t t e r e d a - p a r t i c l e . A l s o shown a r e t h e compound n u c l e u s t e m p e r a t u r e as c i r c l e and l i n e a r momenta o f t h e incoming and o u t g o i n g a - p a r t i c l e pa and pal and t h e t r a n s f e r r e d momentum q.

d: Schematic p i c t u r e o f an e a r l y phase o f t h e r e a c t i o n .

c d

References :

1 ) OSTERFELD, F., Phys. Rev. C26 (1982) 762 and c o n t r i b u t i o n t o t h i s conference 2 ) BECHSTEDT, u., MACHNER, A U R , , H . G., SHYAM, R., ALDERLIESTEN, c., BOUSSHID, o.,

DJALOEIS, A., JAHN,P., MAYER-BORICKE, C., ROSEL, F. and TRAUTMANN, D., Nucl.Phys.

A343 (1980) 221

3) ANN, B., REBEL, H., GILS, H.J., PLANETA, R., BUSCHMANN, J., KLEWE-NEBENIUS, H., ZAGROMSKI, S., SHYAM, R. and MACHNER, H., N u c l . Phys. A382 (1982) 296 4 ) AGASSI, D., WEIDENMOLLER, H.A. and MANTZDURANIS, G., ~ h y s . 5 . 22C (1975) 145 5 ) FESHBACH, H., KERMAN, A., KOONIN, S., Ann. Phys. (N.Y.) 125 ( 1 9 8 v 4 2 9

6 ) MC VOY, K.W. and TANG, X.T., Phys. Rep. 94C (1983) 139

7) MACHNER, H., Proc. I n t . N u c l e a r P h y s i c s X n f . , F l o r e n c e 1983

8 ) CHIANG, H.C. and HOFNER, J., Nucl . Phys. - A352 (1981) 442

C4-12 J O U R N A L DE PHYSIQUE

9 ) MACHNER, H., N u c l . Phys. A395 (1983) 457

10) LIFSHETZ, M. and SINGER, T P h y s . Rev. C22 (1980) 2135 11) WU, J.R. and CHANG, C.C., Phys. Rev. C16 v 9 7 7 ) 1812

12) SEGEL, R.E., CHEN,T., RUTLEDGE, L.L. , m H E R , J.V., WIGGINS, J . , SINGH, P.P., DEBEVEC, P.T., Phys. Rev. C26 (1982) 2424

13) FRIEDMAN, E., GILS, H.J. REBEL, H., Phys. Rev. C25 (1982) 1551

14) MANTZOURANIS, G., WEIDENMOLLER, H.A. and AGASSI, 0.7. Phys. A276 (1976) 145 15) MACHNER, H., PRO TI^, D., RIEPE, G., MORSCH, H.P., DIDELEZ, J.P?RASCARIA, N.,

GERLIC, E., HOURANI, E. and MORLET, M., s u b m i t t e d f o r p u b l i c a t i o n

16) RIEPE, G., PROTIC, D., SOKOSD, C., DIDELEZ, J.P., FRASCARIA, N., GERLIC, E., HOURANI, E. and MORLET, M., Nucl. I n s t r . Methods 117 (1980) 361

17) GRIFFIN, J.J., Phys. Rev. L e t t . 17 (1966) 478 -

18) PAULI, W., i n P. Debeye ( E d . ) : PFibleme d e r Modernen P h y s i k , S. H i r z e l Verlag, L e i p z i g 1928

19) VAN KAMPEN, N.G., F o r t s c h r . P h y s i k 4 (1956) 405 20) MACHNER, H., Z. Phys. A302 (1981) 125

21) MACHNER, H., s u b m i t t e m p u b l i c a t i o n

22) MACHNER, H., Phys. L e t t . 868 (1979) 129 and Phys. Rev. C21 (1980) 2695

23) KIKUCHI, K. and KAWAI, M.,uclear M a t t e r and N u c l e a r Reactions, N o r t h - H o l l a n d P u b l . Comp. Amsterdam 1968, p . 37

24) COLLINS, M.T. and GRIFFIN, J.J., N u c l . Phvs. A348 (1980) 63

25) BOHR, A. and MOTTELSON, B.R., N u c l . S t r u c t u r e I , W.A. Benjamin I n c . , New York 1969, p. 237

26) WU, J.R., CHANG, C.C. and HOLMGREN, H.D., Phys. Rev. C19 (1979) 659

27) KALEND, M., ANDERSON, B.D., BALDWIN, A.R., MADEY, R. ,XTSON, J.W., CHANG, C.C., HOLMGREN, H.D., KOONTZ, R.W., WU, J.R. and MACHNER, H., Phys. Rev. C28 (1983) 105 28) KALBACH-CLINE, C . , N u c l . Phys. A210 (1973) 590

29) BERTRAND, F.E. and PEELLE, R.W. , m L 4455 (1969)

30) DIDELEZ, J.P., FRASCARIA, N., HOURANI, E., GERLIC, E., MORLET, M., MACHNER, H., RIEPE, G., PROTIC, D. and SOK'OSD, C . , i n E. G a d i o l i (Ed.), Proc. 3 r d I n t . Conf.

on N u c l e a r R e a c t i o n Mechanisms, Varenna, 1982, R i c . S c i . e d Educ. Perm., Suppl.

28 (1982) 237

31) BFCK, M.SH. and POWELL, C.A., NASA TN D-8119 (1976)

32) MACHNER, H., BECHSTEDT, U., DJALOEIS, A. and JAHN, P., Phys. Rev. 5% (1982) 411 33) CIANGARU, C . , CHANG, C . C . , HOLMGREN, H.D., NADASEN, A. and ROOS, P.G., Phys. Rev.

C27 (1983) 1360 and c o n t r i b u t i o n t o t h i s conference

34) BECHSTEDT, U., MACHNER, H., BUDZANOWSKI, A., JAHN, P. and MAYER-BORICKE, C., Phys. Rev. C25 (1982) 3221

35) BETHE, H . A . 7 h y s . Rev. 53 (1938) 675

36) KOHLER, H.S., N u c l . P h y s T A 3 7 8 (1982) 159 and r e f e r e n c e s t h e r e i n

37) SCOTT, D.K., i n N. C i n d r o 5 1 . ( E d t s . ) : Dynamics o f Heavy-Ion C o l l i s i o n s , N o r t h - H o l l a n d P u b l . Comp., Amsterdam 1981, p. 241

38) ALDER, B.J. and WAINWRIGHT, T., i n I . P r i g o g i n e (Ed.) T r a n s p o r t processes i n s t a t i s t i c a l mechanics, I n t e r s c i e n c e (1958) p . 97

39) CHIANG, H.C. and HUFNER, J . , N u c l . Phys. A349 (1980) 466 40) SINHA, B., Phys. Rev. L e t t . 50 (1983) 9 1

41) KIND, S. and PATERGNANI, G.;-Nuovo Cim. 10 (1953) 1375 42) WEISSKOPF, V.F., Phys. Rev. 52 (1937) 295-

43) TOMONAGA, S., Z . Phys. 110 ( l m 8 ) 573

44) BETHE, H.A., Rev. Mod. Phys. 9 (1937) 7 1