STATISTICAL ASPECTS OF HIGHLY EXCITED AND NON-EQUILIBRIUM NUCLEAR SYSTEMS

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STATISTICAL ASPECTS OF HIGHLY EXCITED AND NON-EQUILIBRIUM NUCLEAR SYSTEMS

W. Lynch

To cite this version:

W. Lynch. STATISTICAL ASPECTS OF HIGHLY EXCITED AND NON-EQUILIBRIUM NUCLEAR SYSTEMS. Journal de Physique Colloques, 1986, 47 (C4), pp.C4-51-C4-61.

�10.1051/jphyscol:1986405�. �jpa-00225768�

JOURNAL D E PHYSIQUE

Colloque C4, supplement au n o 8, Tome 47, aoGt 1986

STATISTICAL ASPECTS OF HIGHLY EXCITED AND NON-EQUILIBRIUM NUCLEAR SYSTEMS

W.G. LYNCH

National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, M I 48824, U . S . A .

Abstract:One and two l i g h t p a r t i c l e i n c l u s i v e measurements can provide -ion about t h e nonequi l i b r i u m i n i t i a l stages o f heavy i o n induced

r e a c t i o n s a t i n t e r m e d i a t e energies. C o r r e l a t i o n f u n c t i o n s between c o i n c i d e n t l i g h t p a r t i c l e s become more pronounced w i t h i n c r e a s i n g k i n e t i c energy o f t h e detected p a r t i c l e s i n d i c a t i n g emission from a r e g i o n o f t h e combined system which i s more l o c a l i z e d i n space time. The r e l a t i v e populations of p a r t i c l e unbound s t a t e s o f e m i t t e d fragments are c o n s i s t e n t w i t h temperatures o f about 5 MeV. Low temperatures d e r i v e d from t h e measurement o f low energy

t r a n s i t i o n s occur as a r e s u l t o f sequential decay processes which render these measurements i n s e n s i t i v e t o t h e temperatures o f h i g h l y e x c i t e d s y s t e i : ~ .

One c h a l l e n g i n g q u e s t i o n f o r i n t e r m e d i a t e energy heavy i o n r e a c t i o n s concerns whether one can d e t e r m i n e t h e r m o d y n a m i c p r o p e r t i e s of n u c l e a r m a t t e r f r o m e x p e r i m e n t s . C a l c u l a t i o n s i n d i c a t e t h a t a l i q u i d gas phase t r a n s i t i o n e x i s t s f o r n u c l e a r m a t t e r a t m o d e r a t e t e m p e r a t u r e s of T<20 MeV, t e m p e r a t u r e s w h i c h , i n p r i n c i p l e , c a n be a c h i e v e d i n heavy i o n r e a c t i o n s . / l - 3 / D i f f i c u l t i e s i n m e e t i n g t h i s challenge stem from the f $ s t t h a t systems formed i n t h e s e c o l l i s i o n s a r e s m a l l (A<500) and s h o r t l i v e d (T=10 s e c ) . Experimental evidence f o r s h o r t r e a c t i o n times i s i n d i c a t e d by measurements which show t h a t p a r t i c l e e m i s s i o n a t t h e s e e n e r g i e s o c c u r s p r i o r t o t h e a t t a i n m e n t of s t a t i s t i c a l e q u i l i b r i u m of t h e c o m p o s i t e system./4-5/

Models t o d e s c r i b e t h e r e l a t i v e p r o d u c t i o n o f d i f f e r e n t p a r t i c l e s p e c i e s f r e q u e n t l y employ t h e assumption of s t a t i s t i c a l p a r t i c l e e m i s s i o n from s u b s e t s of nucleons/5-8/- subsets which a r e c h a r a c t e r i z e d by t h e i r average v e l o c i t y , space- time e x t e n t , and e x c i t a t i o n energy d e n s i t y o r "temperature". Experimental d e t e r m i n a t i o n of t h e s t a t i s t i c a l and s p a c e - t i m e a s p e c t s of t h i s emission mechanism i s c l e a r l y important. Information a b u t t h e space-time l o c a l i z a t i o n o f t h e s e s u b s e t s can be o b t a i n e d from measurements of c o r r e l a t i o n f u n c t i o n s between c o i n c i d e n t l i g h t p a r t i c l e s . / 9 - 1 3 1 I t i s a l s o important t o determine whether s t a t i s t i c a l concepts such a s t e m p e r a t u r e a r e r e l e v a n t t o t h e s e processes. U n t i l r e c e n t l y , most attempts t o e x t r a c t temperatures have been based on analyses of t h e k i n e t i c e n e r g y s p e c t r a of t h e e m i t t e d p a r t i c l e s . The i n t e r p r e t a t i o n o f s u c h s p e c t r a , however, c a n be complicated by s e n s i t i v i t i e s t o t h e c o l l e c t i v e motion/l5/ and t h e temporal evolution o f t h e e m i t t i n g s y s t e m . / 5 , 1 6 / An a l t e r n a t i v e d e t e r m i n a t i o n of t h e " e m i s s i o n temperaturef1, i . e . t h e temperature a t t h e p o i n t a t which t h e p a r t i c l e s l e a v e t h e e q u i l i b r a t e d subsystem, i s based on t h e r e l a t i v e population of s t a t e s . / l 7 - 2 1 /

The populations of p a r t i c l e unbound s t a t e s a r e o b t a i n e d from the measurement of t h e i r c o r r e l a t e d decay products. The same coincidence measurements, when combined w i t h s i n g l e p a r t i c l e i n c l u s i v e d a t a t o c o n s t r u c t c o r r e l a t i o n f u n c t i o n s , p r o v i d e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986405

C4-52 JOURNAL

DE

PHYSIQUE

i n f o r m a t i o n about t h e s p a c e - t i m e e x t e n t of t h e e m i t t i n g s u b s y s t e m . / 9 - 1 3/ The connection between t h e c o r r e l a t i o n f u n c t i o n and t h e r e l a t i v e populations of p a r t i c l e u n s t a b l e e x c i t e d s t a t e s c a n be i l l u s t r a t e d i n t h e l i m i t of e q u i l i b r i u m thermodynamics. I n t h i s l i m i t , t h e t w o - p a r t i c l e c o r r e l a t i o n f u n c t i o n and t h e r e l a t i v e decay y i e l d s a r e independent f u n c t i o n s of one v a r i a b l e each,/l 3/ the source volune and temperature, r e s p e c t i v e l y . For two non-identical p a r t i c l e s of s p i n s s , and s,, t h e c o r r e l a t i o n f u n c t i o n , R ( q ) , can be approximated as/13/

where q d e n o t e s t h e momentum of r e l a t i v e motion and r denotes the source r a d i u s . This formula has r e c e n t l y been shown/l3/ t o be c o n s i s t e n t w i t h t h e o r i g i n a l model proposed by Koonin./9/ The r e l a t i v e populations of exci t e d s t a t e s can be expressed as:

where T, denotes t h e t e m p e r a t u r e o f t h e e m i t t i n g s o u r c e . Thus, i n t h i s l i m i t , measurements of t h e t w o - p a r t i c l e c o r r e l a t i o n f u n c t i o n and t h e r e l a t i v e populations oP s t a t e s p r o v i d e i n d e p e n d e n t i n f o r m a t i o n c o n c e r n i n g t h e t e m p e r a t u r e t h e space-time e v o l u t i o n of the e m i t t i n g system.

The r e s u l t s w i l l be organized i n t h e f o l l o w i n g manner. I n f o r m a t i o n from two p a r t i c l e c o r r e l a t i o n measurements about t h e space-time localization of the r e a c t i o n w i l l be p r e s e n t e d f i r s t . The d i s c u s s i o n o f e m i s s i o n t e m p e r a t u r e m e a s u r e m e n t s f o l l o w s . New measurements of t h e r e l a t i v e populations of Y-ray e m i t t i n g p a r t i c l e s t a b l e e x c i t e d s t a t e s a r e presented and t h e d i f f i c u l t i e s of e x t r a c t i n g t e m p e r a t u r e s from t h e s e measurements a r e discussed. Emission temperature measuranents f rom t h e populations of s h o r t - l i v e d p a r t i c l e unstable e x c i t e d s t a t e s a r e t h e n p r e s e n t e d f o r ' O A r induced r e a c t i o n s on 19'Au a t E/A=60 MeV and f o r "N induced r e a c t i o n s on '"Au a t E/A=35 MeV.

Two P a r t i c l e C o r r e l a t i o n Measurements

To i l l u s t r a t e t h e b a s i c f e a t u r e s of the two p a r t i c l e c o r r e l a t i o n Measurements, we b r i e f 1 y d i s c u s s r e s u l t s of r e c e n t measurements performed a t Michigan S t a t e U n i v e r s i t y f o r "N i n d u c e d r e a c t i o n s on I9'Au a t E / A = 35 MeV. C o r r e l a t i o n f u n c t i o n s were constructed from one and two l i g h t p a r t i c l e ( p , d . t , a ) i n c l u s i v e d a t a . The c o r r e l a t e d l i g h t p a r t i c l e s (222) were d e t e c t e d w i t h a c l o s e - p a c k e d hexagonal a r r a y o f 1 3 AE-E t e l e s c o p e s . I n o r d e r t o r e d u c e c o n t r i b u t i o n s from p r o j e c t i l e f r a g m e n t a t i o n r e a c t i o n s , measurements were performed w i t h t h e c e n t e r of t h e hodoscope p o s i t i o n e d a t l a b o r a t o r y a n g l e s of O =35O and 50°. These angle. a r e s i g n i f i c a n t l y l a r g e r than t h e grazing a n g l e , O ,,==l&'?

The d a t a a r e presented i n terms of the t w % p a r t i c l e $ o r r e l a t i o n f y c t i o n , R ( q ) , which i s d e f i n e d i n t e r m s of t h e s i n g l e s y i e l d s , Y , ( p , ) and Y , ( p 3 ) , a n d t h e

+ + . .

- -

coincidence y i e l d , Y1 2(p1 , p 2 ) , of p a r t i c l e s 1 and 2:

-t 4

H e r e , p l a n d p2 a r e t h e l a b o r a t o r y momenta of p a r t i c l e s 1 a n d 2, a n d q i s the manentun of r e l a t i v e motion. For each p a r t i c l e p a i r , t h e normalization constant C12 was c h o s e n s u c h t h a t R ( q ) = O a t s u f f i c i e n t l y l a r g e r e l a t i v e momenta, w h e r e c o r r e l a t i o n s due t o f i n a l s t a t e i n t e r a c t i o n s s h o u l d v a n i s h ; was c h o s e n i n d e p n d e n t l y of the c o n s t r a i n t s uhich were placed on t h e e n e r g i e s

:J 5

he coincident p a r t i c l e s . For a given energy c o n s t r a i n t , t h e e x p e r i m e n t a l c o r r e l a t i o n f u n c t i o n s were o b t a i n e d by i n s e r t i n g t h e measured y i e l d s i n t o eq. 3 and sunming t n t h s i d e s Of

t h e equation over a l l e n e r g i e s and a n g l e s c o r r e s p o n d i n g t o a g i v e n v a l u e o f q . F i g u r e 1 s h o w s t h e c o r r e l a t i o n s f o r p r o t o n s

,

d e u t e r o n s

,

and t r i t o n s t h a t were measured a t eav=350. I n o r d e r t o e x p l o r e the dependence of the c o r r e l a t i o n f u n c t i o n s on t h e e n e r g y of t h e o u t g o i n g p a r t i c l e s . t h e c o r r e l a t i o n f u n c t i o n s were e v a l m t e d f o r d i f f e r e n t c o n s t r a i n t s on t h e sum

e n e r g y , E 1 + E 2 , o f t h e t w o c o i n c i d e n t p a r t i c l e s . These c o n s t r a i n t s a r e i n d i c a t e d o n t h e f i g u r e s . For t h e t w o - p r o t o n c o r r e l a t i o n f u n c t i o n , shown i n t o p p a r t of f i g u r e 1, t h e a t t r a c t i v e s i n g l e t S-wave i n t e r a c t i o n gives r i s e t o a maximun a t q=20 M e V / c . T h i s maximum h a s a l s o b e e n interpreted/22-24/ i n terms of the emission o f p a r t i c l e u n b o u n d 'He n u c l e i . The c o r r e l a t i o n f u n c t i o n s f o r c o i n c i d e n t d e u t e r o n s and t r i t o n s , shown i n t h e center and bottan part of F i g u r e 1, r e s p e c t i v e l y , d o n o t e x h i b i t m a x i m a s i n c e t h e i n t e r a c t i o n s between t h e s e two p a i r s of p a r t i c l e s a r e not resonant a t low r e l a t i v e mcmenta./l 1 , 2 5 /

The d o t - d a s h e d , s o l i d , d a s h e d a n d d o t t e d c u r v e s s h o w n i n t h e f i g u r e s represent t h e o r e t i c a l c o r r e l a t i o n f u n c t i o n s p r e d i c t e d by a g e n e r a l i z a t i o n o f t h e f i n a l - s t a t e i n t e r a c t i o n model of Koonin.

I n t h e c a l c u l a t i o n o f t h e c o r r e l a t i o n f u n c t i o n , t h e t w o - p a r t i c l e wave f u n c t i o n i s f o l d e d w i t h a G a u s s i a n s p 5 t i a j s o u r c e d i s t r i b u t i o n , p ( r ) = p o . e - r / r o

.

F i n i t e l i f e t i m e e f f e c t s were n e g l e c t e d . I n t h i s approximati on, t h e model becomes equivalent t o t h e t h e r m a l model/l 3/ of e q u a t i o n 1 . Q u a l i t a t i v e l y , t h e i n c l u s i o n o f t h e temporal evolution of t h e e m i t t i n g s y s t e m i s e x p e c t e d t o r e d u c e t h e c a l c u l a t e d t w o - p a r t i c l e c o r r e l a t i o n s . T h e r e f o r e , source r a d i i e x t r a c t e d under t h e assunption of n e g l i g i b l e l i f e t i m e r e p r e s e n t u p p e r l i m i t s f o r t h e s p a t i a l e x t e n t of t h e e m i t t i n g svstem./26/

24- 50 M e V

0 30- 80 M e V

I I I I I

120-200 M e V .

he

dependence o f t h e two c a l c u l a t e d

corr ati on un c t u p o n t h e s o u r ce

re

^1: C o r r e l a t i o n f u n c t i o n s f o r 14N a d i us

=,

^{at e d} t o a t t r a c t i ve or induced r e a c t i o n s on lg7Au a t E/A=35 MeV.

r e p u 1 s i v " e n a t u r e o f t h e t w o b o d y i n t e r a c t i o n . C o r r e l a t i o n f u n c t i o n s with a t t r a c t i v e ( r e p u l s i v e ) i n t e r a c t i o n s

g e n e r a l l y i n c r e a s e ( d e c r e a s e ) a s t h e s o u r c e r a d i u s i s decreased. For a l l cases i n v e s t i g a t e d t o d a t e (p+p, d+d, t + t , p + a , d+a) t h e measured c o r r e l a t i o n s become more pronounced w i t h i n c r e a s i n g e n e r g y of the two coincident p a r t i c l e s i n d i c a t i n g t h a t more e n e r g e t i c l i g h t p a r t i c l e s a r e emitted from s u b s e t s of nucleons which a r e more l o c a l i z e d i n s p a c e - t i m e . This f e a t u r e i s q u a n t i f i e d by t h e e s t i m a t e d s o u r c e r a d i i sunmari zed i n Table 1. Also included i n t h e t a b l e a r e sou-ce r a d i i e x t r a c t e d from the measurements with t h e center of the hodoscope p s i t i o n e d a t 50°. No s i g n i f i c a n t dependence of the c o r r e l a t i o n f u n c t i o n s on s c a t t e r i n g a n g l e i s o b s e r v e d . Source r a d i i determined i n * O A r induced r e a c t i o n s on 19'Au a t E/A=60 MeV (shown i n F i g . 2).

a n d i n

'

⁶⁰ i n d u c e d r e a c t i o n o n 19'Au a t E/A=25 MeV/1 0 / , a l s o d i s p l a y s i m i l a r

C4-54 JOURNAL DE PHYSIQUE

d e p e n d e n c e s on t h e k i n e t i c e n e r g y of the

detected p a r t i c l e s . For t h e ' O A r i n d u c e d 2.0

,

^.

,

.

,

. , . , . , , , r e a c t i ons

,

t h e a v e r age c o r r e l a t i ons a r e

m n s i d e r a b l y s m a l l e r

,

c o n s i s t e n t w i t h a n

e m i t t i n g s u b s ys tem of l a r g e r s p a c e - t i m e @,.=30° 4 frn

extent

.

_{5 fm}

The two-proton c o r r e l a t i o n s f o r "N induced

r e a c t i o n s a r e more comparable i n magnitude . . . . . . t o t h e o n e s measured c l o s e t o t h e grazing

a n g l e (0=15O) f o r ''0 induced r e a c t i o n s on

1.2

" ' A u a t E / A = 2 5 MeV. The f a c t t h a t s i g n i f i c a n t c o r r e l a t i o n s a r e o b s e r v e d a t

a n g l e s s i g n i f i c a n t l y l a r g e r t h a n t h e -

grazing a n g l e renders i n t e r p r e t a t i o n s / 2 3 / of t h e o b s e r v e d c o r r e l a t i o n s f o r t h e 160

i n d u c e d r e a c t i o n s i n t e r m s of t h e

+

u n l i k e l y .

s e q u e n t i a l decay o f p r o j e c t i l e fragments

,

1.0

6

V

The s o u r c e s i z e s e x t r a c t e d f o r t h e emission of e n e r g e t i c p-p and a-d p a i r s a r e smaller than t h e s i z e of the t a r g e t nucleus o t h e r s o u r c e dimensions a r e l a r g e r . (We

[ r ,(Au) = /2/3 .rrms!Au) " 4 . 3 f m l . The 1.0 m u s t , however, r e - e m p h a s i z e t h a t s o u r c e 0.8 r a d i i e x t r a c t e d i n t h e l i m i t of n e g l i g i b l e

source l i f e t i m e must be r e g a r d e d a s upper l i m i t s f o r t h e s p a t i a l e x t e n t of t h e e m i t t i n g s o u r c e .) The e x t r a c t e d s o u r c e r a d i i may be o r d e r e d a p p r o x i m a t e l y a s f o l l o w s : r o ( c t - d ) 5 r o ( p - p ) 5 r o ( a - p ) 5 r , ( t - t ) 5 r , ( d - d ) . T h i s sequence of

o r d e r i n g cannot be u n d e r s t o o d i n t e r m s of 0.6

t r i v i a l geometrical considerations based on 0 20 40 60 80 100 120

t h e s i z e o f t h e e m i t t e d l i g h t p a r t i c l e s . q (MeV/c) F u r t h e r experiment a1 i n v e s t i g a t i o n s a r e

c l e a r l y n e e d e d . I n p a r t i c u l a r

,

o n e would

1 i ke t o know whet her t h i s o r d e r i n g can be Figure 2: Correlation functions for 40Ar r e l a t e d t o d i f f e r en c e s i n t h e i m p a c t induced r e a c t i o n s on "'AU a t E/A=60 M ~ V . p a r a m e t e r s s a m p l e d by t h e v a r i o u s

c o r r e l a t i o n s . Table 1 :

Source r a d i i , r , f o r a source of n e g l i g i b l e l i f e t i m e and Gaussian s p e c i a l d e n s i t y . c o r r e l a t i o n r 0 ( 3 5 " ) [ f m l r O ( 5 O 0 ) [ f m l

5.2 ^+_ 0.5

Emi ssi on Ten per a t u r e Meas ur enents D u r i n g t h e n o n - e q u i l i b r i u m i n i t i a l s t a g e of a n u c l e a r r e a c t i o n , t h e i n d i v i d u a l degrees of freedom of t h e combined s y s t e m become e x c i t e d a t d i f f e r e n t r a t e s . W i t h i n t h e s i m p l i f y i n g a s s u n p t i o n of l o c a l thermal

e q u i l i b r i u m , t h e r e l a t i v e p o p u l a t i o n s of d i f f e r e n t p h a s e - s p a c e co nf i gur a t i ons a r e d e s c r i b e d by a t e m p e r a t u r e . S i n c e t e m p e r a t u r e s d e r i v e d f r o m t h e s l o p e s of

e n e r g y s p e c t r a m a y b e a f f e c t e d by s e n s i t i v i t i e s of the s p e c t r a t o c o l l e c t i v e motion and t o t h e t e n p o r a l e v o l u t i o n of t h e e m i t t i n g s ys tem, e n s u i n g d i s c u s s i o n w i l l c o n c e n t r a t e o n t h e e x t r a c t i o n of " e m i s s i o n t e n p r a t u r e s " frcm the r e l a t i v e p o p u l a t i o n s o f g r o u n d a n d e x c i t e d s t a t e s of e m i t t e d fragments ./l 7-21 /

R e c e n t m e a s u r e m e n t s of e m i s s i o n t e m p e r a t u r e s have p r o v i d e d c o n t r a d i c t o r y r e s u l t s . / l 7 - 2 1 / The r e l a t i v e p p u l a t i o n s of p a r t i c l e u n s t a b l e s t a t e s of 6 L i , ' L i , a n d 'Be n u c l e i p r o d u c e d i n i n d u c e d r e a c t i o n s on lg7Au a t E/A=60 MeV and i n "N

i n d u c e d r e a c t i o n s on 1.97Au a t E/A=35 MeV a r e c o n s i s t e n t w i t h a v e r a g e e m i s s i o n t e m p e r a t u r e s of a b o u t 4-5 MeV. /I 9-21 / I n c o n t r a s t , measurements of low e n e r g y Y-ray t r a n s i t i o n i n ' L i , 'Li, a n d 7Be f r a g m e n t s g o d u c e d i n "Ii induced r e a c t i o n s on Ag a t E/A=35 MeV y i e l d e d e m i s s i o n t e n p e r a t u r e s of l e s s t h a n 1 M e V . / l 7 , 1 8 / T h e s e l o w t e m p e r a t u r e s w e r e a t t r i b u t e d t o t h e breakdown o f t h e a p p r o x i m a t i on o f 1 o c d l t h e r m a l e q u i l i b r i u m . / l 7 , 1 8 / I n o r d e r t o a d d r e s s t h e e f f e c t s of s e q u e n t i a l d e c a y o n t e m p e r a t u r e m e a s u r e m e n t s u s i n g low energy Y-ray t r a n s i t i o n s , we have m e a s u r e d l o w e n e r g y Y-ray t r a n s i t i o n s i n ' L i , 'Be, 'OB, I 2 B , a n d 13C f r a g m e n t s f o r 3 Z S i n d u c e d r e a c t i o n s on Ag a t 71 5 MeV and have a p p l i e d t h e quant un s t a t i s t i c a l model t o e s t i m a t e t h e e f f e c t of s e q u e n t i a l d e c a y . T h e s e measurements w i l l be d i s c u s s e d f i r s t . Then w e w i l l t u r n t o t h e d e t e r m i n a t i o n o f m i s s i o n t e m p e r a t u r e measurements f r o m t h e r e l a t i v e p o p u l a t i o n s of p a r t i c l e u n s t a b l e

e x c i t e d s t a t e s

.

. . . . I . . I . . . . I _ . . . I . . . . I .

A~(''s,'OB) , E/A=22.3 MeV

-*. B

-

' '

I . .

0 50 100 150 200 250

E

(MeV)

Figure 3: Energy spectrum for 'OB fragments for 3 2 ~ induced reactions at E/A=22.3 MeV.

ZW

LOO ZOO 0 Z 4 e 8 10

E (keV) T (MeV)

a .) _Tem_pe_~aLuye-M_eas_ur_em_en_t_s_-Cram_

P o p u l a t i o n s of P a r t i c l e S t a b 1 e S t a t e s :

Figure 4: Left side: gamma ray energy spectra; Right side: theoretical and T h e e x p e r i m e n t was p e r f o r m e d a t t h e experimental gamma ray fractions.

H o l i f i e l d Heavy I o n R e s e a r c h F a c i l i t y o f 3 a k R i d g e N a t i o n a l L a b o r a t o r y . I s o t o p e

r e s o l v e d complex fragments w i t h 3 ( 2 ( 8 were measured i n c o i n c i d e n c e w i t h Y-rays d e t e c t e d i n Compton s h i e l d e d gerrnaniun d e t e c t o r modules of t h e S p i n S p e c t r m e t e r . F i g . 3 sbws t h e measured e n e r g y s p e c t r a of 1°B i s o t o p e s . Coincident Y-ray s p e c t r a f o r t h e s e f r a g m e n t a t i o n p r o d u c t s a r e s h o w n i n t h e l e f t h a n d s i d e of f i g u r e 4. The f r a c t i o n of d e t e c t e d f r a p p e n t s which e m i t t h e s e c o i n c i d e n t Y-rays a r e b u n d e d by t h e

JOURNAL

DE

PHYSIQUE

r e d u c e d e m i s s i o n p r o b a b i l i t i e s a r e d i r e c t l y

e l a t q u a l d e c a Figure 5: d-alpha c o r r e l a t i o n function f o r 4 0 ~ r induced r e a c t i o n s on At-

c o r r e c t i o n s . Not o n l v a r e t h e e m i s s i o n d o t t e d l i n e s o n t h e r i g h t s i d e o f t h e f i g u r e . P r e d i c t i o n s f o r t h e s e Y - r a y e m i s s i o n p r o b a b i l i t i e s

,

when s e q u e n t i a 1 d e c a y c o r r e c t i o n s a r e n e g l e c t e d , a r e i n d i c a t e d by t h e d a s h e d c u r v e s . I f o n e f o l l o w s t h e p r o c e d u r e of R e f s . 17 a n d 1 8 i n w h i c h s e q u e n t i a 1 d e c a y i s n e g l e c t e d , t e m p e r a t u r e s of -55.1, 0. 9c. 3, 2. 4 c . 8 a n d

1 . 6 c . 2 MeV w o u l d be e x t r a c t e d from Y-ray

2

^{6 -}

.

t r a n s i t i o n s , ' ~ e ( . 4 2 9 M e V + g . s ) , ' L i ( . 9 8 1

2 ^-

^*

MeV + g . s ) , ' ' ~ ( 2 . 154 M e V + 1.740 MeV), a n d

1 3 C ( 3 . 8 5 4 + 3 . 6 8 4 M e V ) , r e s p e c t i v e l y . I . . . . I . . . .

p r o b a b i l i t i e s r e d u c e d , t h e c a l c u l a t i o n s

p r e d i c t t h a t t h e s e Y-ray measurements are YYI-LI'.IO?

E;,, (MeV)

r e l a t i v e l y i n s e n s i t i v e t o t h e s t a t i s t i c a l p r o p e r t i e s o f t h e e m i t t i n g s y s t e m f o r

t e m p e r a t u r e s g e a t e r t h a n 2 MeV. C l e a r 1 y , 195.( 4 0 ~ r , ( I ~ I X , E/A=60MeV

m o r e a c c u r a t e p r o b e s of t h e f r a g n e n t a t i o n p r o c e s s a r e n e c e s s a r y .

T h e s e t e m p e r a t u r e s i n c r e a s e w i t h t h e e x c i t a t i o n e n e r g y o f t h e e x c i t e d s t a t e a s

b . ) Tern P e r a t u r e-M_eas_ur e m e n t 3 CrOE

$

^,04

P o p u l a t i o n s of P a r t i c l e U n s t a b l e S t a t e s . 4

8

0 50 100 150 -

-

2)

: : 14yzMev

I n o r d e r t o r e d u c e t h e i n f l u e n c e of -

s e q u e n t i a l decay, we have i n v e s t i g a t e d t h e 1 0 MCV

p o p u l a t i o n s o f s h o r t - l i v e d p a r t i c l e - u n s t a b l e s t a t e s of 6 L i , ' L i , a n d

'Be f r a m e n t s f o r " A r i n d u c e d r e a c t i o m on ' 0 2 ~ 2 4 6

e x p e c t e d i f s e q u e n t i a l d e c a y i s

dominant ./l 9/ The s o l i d curves c o r r e s p o n d I . . . . I . . . . ~ . . . . I . . . .

0 100 200 300

t o t h e e m i s s i o n p r o b a b i l i t i e s p r e d i c t e d by q [MeV/cl

t h e quantun s t a t i s t i c a l mode1/27-28/; t h e

l g 7 A u a t E/A

-

= 6 0 MeV a n d "N i n d u c e d Tc.,!MeV)

r e a c t i o n s on l S 7 A u a t E / A = 35 MeV. A s a

f i r s t e x a m p l e , w e d i s c u s s e m i s s i o n F i g u r e 6 : E x c i t a t i n e n e r g y S p e c t r w n f o r

z

t em p e r a t U r e m e a s u r em e n t s f r o m t h e p a r t i c l e unstable Li fragments

p p u l a t i o n of p a r t i c l e u n s t a b l e 6 L i n u c l e i i n ' O A r i n d u c e d r e a c t i o n s on 19'Au a t E/A

-

60 MeV. /19/

F i g u r e 5 s h o w s t h e measured a-d c o r r e l a t i o n f u n c t i o n , R ( q ) , d e f i n e d i n e u a t i o l )

a

3 . The c o r r e l a t i o n f u n c t i o n e x h i b i t s two maxima c o r r e s p n d i n g t o t h e T=O, J = 3 s t a t e i n 6 L i a t 2.186 MeV and two o v e r l a p p i n g T=O s t a t e s a t 4. 31 MeV ( J" = 2+,

rtot

= 1 . 3 ,

r$rtot

⁼ 0 . 9 7 ) , a n d a t 5.65 MeV (J" = I + , r t o t = 1 . 9 MeV,

r a / r t o t

= 0 . 7 4 ) . The c o i n c i d e n c e y i e l d r e s u l t i n g f r a n t h e decay of e x c i t e d 6 L i n u c l e i was o b t a i n e d by a s s u n i n g t h a t t h e a-d c o i n c i d e n c e y i e l d i s g i v e n by Yad=Y6Li+ Yb, where Y d e n o t e s t h e y i e l d f r o m d e c a y i n g 6 L i n u c l e i a n d Y b d e n o t e s t h e "background" y i e l d . The background y i e l d was a s s u n e d t o k given by Yb= YaYd[l+Rc(q)], w h e r e Y d a n d Y a are t h e s i n g l e s y i e l d s a n d Rc(q) c o r r e s p o n d s t o t h e c o r r e l a t i o n f u n c t i o n e x p e c t e d frcm t h e Coulanb r e p u l s i o n o f t h e two c o i n c i d 6 n t p a r t i c l e s . T h i s background i s i n d i c a t e d by t h e s o l i d c u r v e s i n F i g u r e 5. Y i e l d s of p a r t i c l e u n s t a b l e e x c i t e d 6 L i n u c l e i , s b w n i n F i g u r e 6, w e r e e x t r a c t e d by b i n n i n g t h e e x p e r i m e n t a l y i e l d w i t h r e s p e c t t o t h e k i n e t i c e n e r g y i n t h e 'Li r e s t f r a m e and s u b t r a c t i n g t h e background y i e l d shown i n F i g u r e 5.

The y i e l d YeLi(E) i s r e l a t e d t o t h e e n e r g y s p e c t r u m , dn (E)/dE, i n t h e 6 L i center-of-mass frame by t h e equation:

where E ( E , E ' ) i s the e f f i c i e n c y f u n c t i o n f o r t h e r e s p o n s e of t h e hodoscope t o a-d p a i r s a r i s i n g f r m the decay of e x c i t e d ' L i nuclei; E and E ' denote t h e measured and a c t u a l e x c i t a t i o n e n e r g i e s , r e s p e c t i v e l y .

The e f f i c i e n c y f u n c t i o n f o r t h e hodoscope was c a l c u l a t e d f o r t h e p r e c i s e geometry, l i g h t p a r t i c l e d e t e c t i o n t h r e s h o l d s (Ed' 15 MeV, E t 40 MeV) a n d d e t e c t o r energy r e s o l u t i o n t h a t $as determined during t h e e ~ p e r i m e n t . ~ I n t h e s e c a l c u l a t i o n s , the parent nucleus, ' L i

,

w a s assuned t o decay i s o t r o p i c a l l y i n i t s r e s t frame. The l a b o r a t o r y e n e r g y s p e c t r a and a n g u l a r d i s t r i b u t i o n s of e x c i t e d 6 L i nuclei were constrained t o be i d e n t i c a l t o t h e s p e c t r a f o r t h e emission of p a r t i c l e - s t a b l e 6 L i n u c l e i .

The e x c i t a t i o n energy s p e c t r u n , dn/dE for t h e r m a l l y e m i t t e d 'Li nuclei i s given by :

where N i s a n o r m a l i z a t i o n c o n s t a n t and t h e sum i n c l u d e s t h e t k e e T=O e x c i t e d s t a t e s of ' L i b e l w 10 MeV e x c i t a t i o n energy. Equation 5 c o r r e s p o n d s t o t h e p h a s e s p a c e m o d i f i c a t i o n s g i v e n by e q u a t i o n 2 f o r t h e a-d i n t e r a c t i o n when t h e energy dependence of the phase s h i f t s i s dominated by a s e r i e s of resonances.

The measured y i e l d s a r e c o n s i s t e n t with a mean temperature a t emission of T 1 5 MeV. This emission temperature i s l w e r t b n t h e temperature p a r a m e t e r T ^.;20 MeV which c h v a c t e r i zes t h e e n e r g y s p e c t r a of complex n u c l e i emi t t e d a t i n t e r m e d i a t e r a p i d i t y . ⁾

To a s s e s s whether t h i s discrepancy may be c a m e d by secondary processes such a s s e q u e n t i a l decay, we have analyzed t h e r e l a t i v e populations of e x c i t e d s t a t e s of ' L i and 8 B e . / 2 0 / T h e s e s t a t e s a r e widely s e p a r a t e d i n e x c i t a t i o n energy (AE=15 MeV).

Figure 7shows the c o r r e l a t i o n f u n c t i o n s f o r t h e g r o u n d s t a t e ' L i +a+p a n d t h e 090

e x c i t e d s t a t e ' ~ i * ~ ~ . 7+ d+*He. F i g u r e 8 shows the c o r r e l a t i o n f m c t i o n s f o r t h e e x c i t e d s t a t e 'Be 3. 0,,+2a and t he e x c i t e d s t a t e 'Be 7. 6+p+7Li.

The r e l a t i v e w p u l a t i o n s of the f o u r p a r t i c l e u n s t a b l e resonances, ' L i O e O + a+p, 'Lil 6.7+ d+He, 'Be3. 04+ 2a, 'Be1 7. 6+ p+'Li d e r e d e t e r m i n e d by i n t e g r a t i n g t h e c a l c u l a t e d a n d measured decay y i e l d s over t h e range of e x c i t a t i o n energy f o r which t h e coincidence y i e l d s were dominated by t h e s e r e s p e c t i v e d e c a y s . The i n t e g r a l s o v e r t h e two s t a t e s a t low and h i g h e x c i t a t i o n e n e r g i e s f o r a g i v e n p a r t i c l e u n s t a b l e n u c l e u s a r e d e n o t e d by NL(T) a n d N H ( T ) , r e s p e c t i v e l y . The f u n c t i o n a l dependence of t h e c a l c u l a t e d y i e l d r a t i a N /N L H f o r p a r t i c l e u n s t a b l e ' L i and 'Be nuclei i s shown by t h e s o l i d l i n e s i n F i g u r e 9. The h a t c h e d r e g i o n s i n t h e f i g u r e i n d i c a t e t h e r a n g e of y i e l d r a t i c a and emission temperatures which a r e c o n s i s t e n t with t h e e x t r e m e background a s s u m p t i o n s shown i n F i g s . 7 a n d 8. The r e l a t i v e p o p u l a t i o n s of s t a t e s i n 'Li and 'Be correspond t o emission t m p e r a t w e s of T = 4.6

+

0.7 MeV and T = 4.2

+

0.5 MeV, r e s p e c t i v e l y . (The u n c e r t a i n t i e s shown i n F i g u r e 9 a r e

not

s t a t i s t i c a l ; they correspond t o conservative e s t i m a t e s of the u n c e r t a i n t i e s of the background.) These t e m p e r a t u r e s a r e c o n s i s t e n t w i t h t h e r e s u l t s of t h e a n a l y s i s of t h e a-d c o i n c i d e n c e s p e c t r u m . A s w i t h t h e 6 L i a n a l y s i s , t h e y a r e c o n s i d e r a b l y lower t h a n t h e t e m p e r a t u r e p a r a m e t e r s of a b o u t 20 MeV w h i c h c h a r a c t e r i z e t h e k i n e t i c energy s p e c t r a of the e m i t t e d p a r t i c l e s . The l a r g e energy s p a c i n g between t h e s t a t e s a t low and high e x c i t a t i o n e n e r g y e n s u r e s t h a t t h e e x t r a c t e d t em p e r a t u r e s a r e r e l a t i v e l y i n s e n s i ti ve t o t h e e f f e c t s of s e q u e n t i a l decay. Thus b t h s i g n i f i c a n t l y higher and l w e r emission temperatures a r e excluded.

JOURNAL DE PHYSIQUE

W - 8 5 - P O

E/A = 60 MeV

Figure 7: Correlation function displaying particle unstable states of 5 ~ i . S i m i l a r a n a l y s i s of " N - i n d u c e d r e a c t i o n s on lg7Au a t E/A = 35 MeV provide

e m i s s i o n t e m p e r a t u r e s of T = 4 . 4 & 0 . 4 MeV. / 2 1 / The s i m i l a r i t y o f e m i s s i o n t m p e r a t u r e s f w t h e two r e a c t i o n s i s q u i t e u n e x p e c t e d . Perhaps i t i s p o s s i b l e t h a t s e c o n d a r y p r o c e s s e s , s u c h a s s e q u e n t i a 1 d e c a y , l i m i t t h e e m i s s i o n t e m p e r a t u r e s m e a s u r a b l e i n a r e a c t i o n . B e f o r e t h a t conclusion i s mandated, however, one should e x t r a c t e m i s s i o n t e m p e r a t u r e s s u b j e c t t o c o n s t r a i n t s w h i c h p r o v i d e a m o r e r e s t r i c t i v e event s e l e c t i o n . T e m p e r a t u r e measurements w i t h t r i g g e r c o n d i t i o n s on a s s o c i a t e d p a r t i c l e m u l t i p l i c i t y o r momentum t r a n s f e r

,

b o t h of which may place s a n e c o n s t r a i n t s on i m p a c t p a r a m e t e r

,

a r e

c u r r e n t l y u n d e r a n a l y s i s a n d w i l l be a v a i l a b l e soon. Sane i n s i g h t may be gained by e x a m i n i n g t h e emission temperature as a f u n c t i o n of the t o t a l k i n e t i c energy of the p a r t i c l e unbound f r a g a e n t . One may expect t h a t t h e more e n e r g e t i c f r a g m e n t w i l l r e f l e c t t h e s t a t i s t i c a l p-operties of t h e e a r l i e r s t a g e of the c o l l i s i o n .

Figure 8: Correlation function displaying particle unstable states of 8 ~ e .

T (MeV)

Figure 9: Cross hatched region designates the range of temperatures consistent with the relative populations of states of 8 ~ e and 5 ~ i nuclei.

K i n e t i c e n e r g y s e l e c t e d , e x c i t a t i o n energy s p e c t r a f o r p a r t i c l e u n b u n d 'Li n u c l e i a r e shown i n F i g u r e 10 f o r "N i n d u c e d r e a c t i o n s on "'Au a t E / A = 3 5 MeV./21/ I n t h e upper p a r t of the f i g u r e s p e c t r a m e a s u r e d w i t h t h e center of the hodoscope p s i t i o n e d a t 35O a r e shown; the lower part shows s p e c t r a o b t a i n e d a t 50°.

t h e f o l l o w i n g c o n s t r a i n t s were applied: E 2 40 MeV, Ed? 15 MeV

and:

55 MeV

<

^{E + E} I 100 MeV ( l e f t Mnd p a r t ) , 100 MeV

<

E +Ed

8

150 MeV ( c e n t e r m r t ) . 150 MeV

<

220 MeV ( r i g h t hand p a r t ) .

197 A U ( ' ~ N , ~ ~ ) X , E/A = 35 MeV USU-85-531

5

V

t

100-150 MeV 150-220 MeV

1

.

55-100 MeV

Figure 10: Energy gated c o r r e l a t i o n f u n c t i o n s for 1 4 ~ induced r e a c t i o n s on 197Au a t E/A=35 MeV.

Calculations based on eq. 4 and 5 shown i n F i g u r e 2 f o r e m i s s i o n t e m p e r a t u r e s of T = 1 , 2 . 5 , 5 , a n d 1 0 MeV. The c u r v e s a r e n o r m a l i z e d t o r e p r o d u c e t h e experimental y i e l d i n t e w a t e d over t h e e n e r g y r a n g e o f T A = 0 . 3

-

1 . 2 MeV. I n order t o e x t r a c t m i s s i o n t m p r a t u r e s , we have integratecft'Fe decay y i e l d s over t h e e n e r g y r a n g e s o f T = 0 . 2 5 - 1 . 4 5 a n d 1 . 5 - 6 . 2 5 M e V a n d c o m p a r e d t h e r a t i o o f t h e s e y i e l d s t o t h e c o r r & k n d i n g r a t i o c a l c u l a t e d from eq. 3. The r e s u l t s a r e sunmarized i n Table 2. Higher m i s s i o n temperatures a r e e x t r a c t e d f o r higher k i n e t i c e n e r g i e s , EdEd, of the emitted p a r t i c l e s .

Table 2:

c o n s t r a i n t on

I

^{0 = 3S0}

1

^{8 =} 50"

I

E l + E,

I

T(MeV) [ r,(frn) ( T(MeV) ( r,(fm)[

5 5 - 2 2 0 M e V 1 4 1 3 . 4 4 1 3 . 6 1 55

-

^{100 MeV}

1

⁴

1

3.8

1

3

1

3 . 9

1

100

-

150 MeV

1

4

1

3 . 0

1

5

1

^{2 . 8}

1

1 5 0 - 2 2 0 M e V 1 7

1

3 . 0

1

⁹

1

^{2 . 7}

1

J,OURNAL DE PHYSIQUE

Refer en ces

When e m i s s i o n t e m p e r a t u r e i n f o r m a t i o n i s combined w i t h s o u r c e r a d i i o b t a i n e d from c o r r e l a t i o n f u n c t i o n measurements ( a l s o s u n m a r i z e d i n T a b l e 2 ) , o n e o b s e r v e s t h a t h i g h e r m i s s i o n t e m p e r a t u r e s and s m a l l e r s o u r c e r a d i i a r e e x t r a c t e d f o r higher k i n e t i c e n e r g i e s of t h e e m i t t e d p a r t i c l e s . T h e s e f i n d i n g s a r e c o n s i s t e n t w i t h p a r t i c L e emission from a subsystem which i s i n t h e process of c o o l i n g a n d expanding.

C o o l i n g a n d e x p a n d i n g s u b s y s t e m s of h i g h e x c i t a t i o n c o u l d a r i s e f r o m t h e e q u i l i b r a t i o n o f p a r t i c i p a n t m a t t e r w i t h t h e s u r r o u n d i n g c o l d t a r g e t n u c l e a r n a t t e r / 5 / o r from an i s e n t r o p i c e x p a n s i o n a s e x p e c t e d f r o m i n t r a n u c l e a r c a s c a d e c a l c u l a t i o n s ./29/ I t i s a l s o c o n C e i ~ b l f 2 t h a t t h e m e a s w e d e n e r g y dependence o f t h e emission t e m p e r a t u r e s a r i s e s from an energy dependence o f t h e f e e d i n g f r o m h i g h e r 1 y i n g p a r t i c l e - uns t a b l e s t a t e s . T h i s p o s s i b i l i t y c a n be i n v e s t i g a t e d by measuring che r e l a t i v e p o p u l a t i o n of s t a t e s w i t h l e v e l s e p a r a t i s l a r g e r t h a n &he e m i s s i o n t e m p e r a t u r e . T h i s l i m i t i s s a t i s f i e d by t h e decays:

Ti

⁺ a + p , 'Li 16.7* d+'He,

g . s .

f o r which s t a t i s t i c a l c a l c u l a t i o n s i n d i c a t e n e g l i g i b l e f e e d i n g f r o m h i g h e r l y i n g s t a t e s . / 2 7 / U n f o r t u n a t e l y , t h e a n g u l a r s e p a r a t i o n ( 6 . 1 ° ) b e t w e e n n e i g h b r i n g t e l e s c o p e s was t o o l a r g e t o p e r m i t an e x p l o r a t i o n o f t h e e n e r g y d e p e n d e n c e o f t h e r e l a t i v e p o p u l a t i o n of t h e s e s t a t e s i n 'Li f o r t h i s r e a c t i o n .

The d e p e n d e n c e o f t h e e m i s s i o n

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t e m p e r a t u r e s f o r ' L i a n d 'Be n u c l e i upon 14- t h e k i n e t i c e n e r g y o f t h e e m i t t e d f r a g m e n t

h a v e b e e n e x p l o r e d f o r ' O A r i n d u c e d r e a c t i o n s o n 1 9 ' ~ u a t E/A = 6 0 MeV. The r e s u l t s a r e shown i n F i g . 11. The widths

B

of t h e g a t e s o n k i n e t i c e n e r g y o f t h e

e m i t t e d f r a g m e n t a r e shown by t h e dashed

-

l i n e s . There i s o n l y a s l i g h t i n c r e a s e o f t h e e m i s s i o n t e m p e r a t u r e w i t h t h e k i n e t i c e n e r g y o f the em1 t t e d fragment

.

$ j

+

%inmar y

Emission t e m p e r a t u r e measurements from t h e r e l a t i v e p o p u l a t i o n s of p a r t i c l e

, I , I ,

,

^,

LMSUiss;547

-

^{GOMeV/u * ~ r}

+ "hu

12- -

- e , = ~

lo: '~i*. 'Bc.

weM ^I _-

:1

^a-

⁺

.. .---

^....-. + \...::;:+::- -..

^-..-.

^-{

-

..- .f

....- -

2-

-

u n s t a b l e s t a t e s of 'Li, 'Li, and 'Be n u c l e i ^OO ' l k ' & ' & ' ~ o l & ' e o

h a v e b e e n m e a s u r e d f o r ' O A r i n d u c e d (EIA),,, (MeV) r e a c t i o n s on 19'Au a t E/A

-

60 MeV and

'

*N

induced r e a c t i o n s on lg7Au a t E/A = 3 5 MeV.

Aver a g e e m i s s i o n t e m p e r a t u r e s of 4 5 MeV Figure 11: Energy gated correlation source

ar b t ai ne ,j f or bo t h a ct i

.

^N~ radii for 4 0 ~ r induced reactions on 1 9 7 ~ "

e x p l a n a t i o n f o r t h e i n s e n s i t i v i t y o f arE'A=60MeV' m i s s i o n t e m p e r a t u r e s on bombarding e n e r g y

i s e v i d e n t . More e x c l u s i v e measurenents w i t h impact parameter or manentun t r a n s f e r s e l e c t i o n a r e c l e a r l y r e q u i r e d . For

'

*N i n d u c e d r e a c t i o n s , b o t h e m i s s i o n t e m p e r a t u r e s a n d s o u r c e r a d i i d e p e n d s i g n i f i c a n t l y o n t h e k i n e t i c e n e r g y of t h e d e t e c t e d p a r t i c l e u n s t a b l e f r a g m e n t

.

For " A r i n d u c e d r e a c t i o n s , t h i s e n e r g y d e p e n d e n c e i s l e s s p r o n o u n c e d . R e s u l t s of emission t e m p e r a t u r e measurements from low energy Y-ray t r a n s i t i o n f o r 32S induced r e a c t i o n s a t E/A = 2 3 MeV a r e s e e n t o be s t r o n g l y i n f l u e n c e d by s e q u e n t i a l d e c a y p r o c e s s e s . Thus none o f t h e Y-ray t r a n s i t i o n s w h c h have been s t u d i e d t o d a t e a r e s u i t a b l e f o r e m i s s i o n t e m p e r a t u r e measurements.

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