A SEMICLASSICAL THEORY FOR MASS AND CHARGE TRANSPORT COEFFICIENTS IN HEAVY-ION REACTIONS

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A SEMICLASSICAL THEORY FOR MASS AND CHARGE TRANSPORT COEFFICIENTS IN

HEAVY-ION REACTIONS

K. Hartmann

To cite this version:

K. Hartmann. A SEMICLASSICAL THEORY FOR MASS AND CHARGE TRANSPORT COEFFI-

CIENTS IN HEAVY-ION REACTIONS. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-401-

C6-406. �10.1051/jphyscol:1984648�. �jpa-00224250�

JOURNAL DE PHYSIQUE

Colloque C6, suppl6ment au n06, Tome 45, juin 1984 page C6-401

A SEMICLASSICAL THEORY FOR MASS AND CHARGE TRANSPORT COEFFICIENTS I N HEAVY-ION REACTIONS

K.M. Hartmann

Sektion Physik der Universitat &chen, 8046 Garching, F. R. G.

RBsum6 - On d g v e l o p p e un modele m i c r o s c o p i q u e p o u r l e s c o e f f i c i e n t s de t r a n s p o r t d e s p r o t o n s e t n e u t r o n s , qui p e u t e x p l i q u e r d e s c a r a c t Q r i s t i q u e s o b s e r v e e s dans l e c o u r s d ' e x p e r i e n c e s de & a c t i o n s i n d u i t e s q u a s i - e ' l a s t i q u e s p a r d e s i o n s l o u r d s .

A b s t r a c t - We d e v e l o p a m i c r o s c o p i c model f o r n e u t r o n and p r o t o n t r a n s p o r t c o e f f i c i e n t s , which i s a b l e t o a c c o u n t f o r t h e e x p e r i m e n t a l l y observed f e a t u r e s i n q u a s i - e l a s t i c heavy- i o n induced r e a c t i o n s .

I - I N T R O D U C T I O N

Over t h e p a s t t e n y e a r s e x p e r i m e n t a l i s t s have been a b l e t o e x t r a c t i n - f o r m a t i o n a b o u t an i n c r e a s i n g v a r i e t y o f d e g r e e s o f freedom t h a t a r e i n v o l v e d i n q u a s i - e l a s t i c and d e e p - i n e l a s t i c c o l l i s i o n s between heavy n u c l e i . T h i s has prompted t h e o r i s t s t o d e v e l o p more r e f i n e d models in o r d e r t o a c c o u n t f o r t h e more complete measurements.

C l a s s i c a l f r i c t i o n / I / and t r a n s p o r t / 2 / models d e s c r i b e d a d e q u a t e l y t h e i n i t i a l d a t a on e n e r g y l o s s , c h a r g e t r a n s f e r and s c a t t e r i n g a n g l e . A s d a t a c o n c e r n i n g c h a r g e and mass t r a n s f e r became a v a i l a b l e , it was found n e c e s s a r y t o d i s t i n g u i s h c a r e f u l l y between n e u t r o n s and p r o t o n s i n a t r a n s p o r t t h e o r e t i c a l d e s c r i p t i o n o f n u c l e o n t r a n s f e r / 3 , 4 / . I n t h i s c o n t r i b u t i o n we review t h e s a l i e n t f e a t u r e s a p p e a r i n g i n t h e n u c l e o n t r a n s p o r t c o e f f i c i e n t s of Gross and Partmann / 4 / . I n o r d e r t o e s t a b l i s h a t what p o i n t i n t h e dynamics o f heavy-ion s c a t t e r i n g t h e s e n e u t r o n and p r o t o n t r a n s p o r t c o e f f i c i e n t s e n t e r , we f i r s t r e c a l l o u r d e s c r i p t i o n o f t h e s c a t t e r i n g p r o c e s s . F i n a l l y , some o f t h e more d r a m a t i c p r e d i c t i o n s o f t h e model, confirmed by e x p e r i m e n t , a r e d i s c u s s e d .

I1 - ^HEAVY-ION SCATTERING W I T H FRICTION AbJD NUCLEON TRANSPORT

The p h y s i c a l p i c t u r e o f t h e s c a t t e r i n g p r o c e s s i s d e p i c t e d i n f i g . 1 . The two n u c l e i move ( i n t h e c . m. frame) on c l a s s i c a l t r a j e c t o r i e s determined by t h e Coulomb and n u c l e a r i n t e r a c t i o n s and by a f r i c t i o n f o r c e / 1 , 5 / . T h i s f r i c t i o n f o r c e g i v e s r i s e t o e n e r g y l o s s from t h e r e l a t i v e motion i n t o i n t r i n s i c e x c i t a t i o n o f t h e n u c l e i . A t each p o i n t a l o n g t h e t r a j e c t o r y , t h e n u c l e i a r e assumed t o be a t a t e m p e r a t u r e T

(same f o r b o t h n u c l e i ) determined by t h e e x c i t a t i o n e n e r g y a t t h a t p o i n t on t h e t r a j e c t o r y . The p r o t o n s and n e u t r o n s i n e a c h n u c l e u s move i n t h e i r r e s p e c t i v e s i n g l e - p a r t i c l e p o t e n t i a l s (whose p a r a m e t e r s a r e t a k e n from / 6 / ) . These p o t e n t i a l s d e v e l o p a b a r r i e r a s t h e n u c l e i p a s s e a c h o t h e r , and n u c l e o n s may be exchanged between p r o j e c t i l e and t a r g e t / 7 / , a s shown i n f i g . 1. The nucleon exchange i s assumed t o be

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984648

JOURNAL DE PHYSIQUE

TARGET PROJECTILE

FIG 1 : S i n g l e p a r t i c l e p r o t o n and n e u t r o n p o t e n t i a l s f o r t h e combined p r o j e c t i l e - p l u s - t a r g e t system.

d e s c r i b e d by a 2-dimensional Fokker-Planck e q u a t i o n e . g . t h e p r o b a b i - l i t y P ( N , Z , t ) t h a t a t t i m e t t h e p r o j e c t i l e ( o r t a r g e t ) c o n t a i n N n e u t r o n s and Z p r o t o n s s a t i s f i e s

The n e u t r o n and p r o t o n d r i f t and d i f f u s i o n c o e f f i c i e n t s a r e denoted by V N , v Z I DNN and D Z Z r e s p e c t i v e l y . Eq. ( 1 ) i s coupled t o t h e c l a s s i c a l e q u a t i o n s of motion and t h e r e b y t h e t r a n s p o r t c o e f f i c i e n t s c a r r y t h e dynamical i n f o r m a t i o n a b o u t t h e s c a t t e r i n g p r o c e s s , e . g . t h e opening and c l o s i n g o f a window, t h r o u g h which nucleons can p a s s , between t h e n u c l e i ( l o w e r i n g and h e i g h t e n i n g o f t h e s i n g l e p a r t i c l e p o t e n t i a l b a r r i e r s ) and t h e changes i n t h e s t a t i c Fermi e n e r g i e s o f n e u t r o n s and p r o t o n s i n one n u c l e u s caused by t h e mean f i e l d of t h e o t h e r n u c l e u s . These a r e d i s c u s s e d i n more d e t a i l below.

I n i t i a l l y , t h e d i s t r i b u t i o n f u n c t i o n P i s a d e l t a f u n c t i o n peaked a t t h e p r o j e c t i l e n e u t r o n and p r o t o n numbers. We assume t h a t a t l a t e r t i m e s P d e v e l o p s a s a two-dimensional Gaussian d e t e r m i n e d by t h e mean n e u t r o n a n d p r o t o n numbers < N > and < Z > and t h e v a r i a n c e s

o$(%<N2 > - < N > 2 ) , ~i ^{and ahZ} ( E < M Z > - < N > < Z > ) . I t t h e n f o l l o w s from (1 ) t h a t t h e s e f i v e q u a n t i t i e s s a t i s f y f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s i n t h e t i m e /4/, e . g.

The s o l u t i o n o f t h e s e e q u a t i o n s f o r l a r g e t a r e compared w i t h t h e e x p e r i m e n t a l mean v a l u e s and v a r i a n c e s .

111 - MICROSCOPIC THEORY FOR TEE TRANSPORT COEFFICIENTS

We c o n s i d e r t h e t r a n s f e r o f a phase s p a c e c e l l o f s i z e d 3 r d 3 p

( q u a n t i z e d i n u n i t s o f h 3 ) l o c a t e d a t (?,s) i n t h e t a r g e t t o t h e

p r o j e c t i l e , s e e f i g . 1 . The p r o b a b i l i t y U T , ~ f o r s u c h a t r a n s f e r

c o n s i s t s o f t h e p r o d u c t o f t h r e e t e r m s :

( i ) t h e p r o b a b i l i t y t h a t t h e phase s p a c e c e l l i s o c c u p i e d i n t h e t a r g e t . Assuming a Fermi gas w i t h Fermi energy fiT ( d i f f e r e n t f o r n e u t r o n s and p r o t o n s ) and t e r m p e r a t u r e T , t h e o c c u p a t i o n proba- b i l i t y i s

w i t h F = p2/2m and m t h e nucleon mass. Note t h a t b o t h T and yT depend on t h e n u c l e a r s e p a r a t i o n R. Thus T = 0 f o r R - ^{- m , and}

i n c r e a s e s a s t h e f r i c t i o n f o r c e s e t s i n . Also BT, i n a d d i t i o n t o t h e ' s t a t i c ' Fermi e n e r g y , c o n t a i n s t h e e f f e c t of t h e mean f i e l d o f t h e p r o j e c t i l e ( e . g. t h e Coulomb f i e l d o f t h e p r o j e c - t i l e p r o t o n s ' r a i s e s ' t h e energy o f e a c h p r o t o n i n t h e t a r g e t by an amount Z e 2 / R ) . W e have i n a d d i t i o n allowed s h e l l e f f e c t s p r e s e n t i n i t i a l l y i n t h e ' s t a t i c ' Fermi e n e r g i e s , t o ' d i e o u t ' w i t h i n c r e a s i n g t e m p e r a t u r e .

( i i ) t h e p r o b a b i l i t y t h a t t h e c o r r e s p o n d i n g phase s p a c e c e l l i s un- o c c u p i e d i n t h e p r o j e c t i l e ,

w i t h f i p and p f h e Fermi energy and momentum o f N o r Z i n t h e p r o j e c t i l e ?n$ R t h e p r o j e c t i l e - t a r g e t r e l a t i v e v e l o c i t y . The q u a n t i t y ppR i n ( 4 ) t a k e s i n t o account t h e e f f e c t i v e l o y e r i n g o f t h e Fermi' l e v e l of N and Z i n t h e e n t r a n c e c h a n n e l ( R < 0 ) and t h e e f f e c t i v e i n c r e a s i n g of t h e Fermi l e v e l i n t h e e x i t c h a n n e l ( R > O ) , due t o t h e r e l a t i v e motion o f t h e n u c l e i / 4 / . The r e l a t i v e motion enhances p a r t i c l e t r a n s f e r i n t h e e n t r a n c e c h a n n e l and h i n d e s i t i n t h e e x i t c h a n n e l .

( i i i ) t h e p r o b a b i l i t y t h a t t h e phase s p a c e c e l l t u n n e l s t h r o u g h t h e b a r r i e r

Here V b a r r i g r i s t h e b a r r i e r h e i g h t a t s e p a r a t i o n R , ( E - V O ) i s t h e separation e n e r g y o f t h e phase s p a c e c e l l and w i s r e l a t e d t o t h e b a r r i e r c u r v a t u r e a t s e p a r a t i o n F. L e t u s n o t e t h a t f o r g r a z i n g c o l l i s i o n s , where t h e s i n g l e p a r t i c l e p o t e n t i a l s o v e r l a p o n l y a l i t t l e , t h e quantum mechanical formula ( 5 ) i s i m p o r t a n t t o account f o r s l i g h t d i f f e r e n c e s i n t h e n e u t r o n and p r o t o n t u n n e l i n g p r o b a b i l i t i e s . I n a c l a s s i c a l window t r e a t m e n t , £ ( F , R ) would be e i t h e r 0 o r I , s e e below.

The s i n g l e phase s p a c e t r a n s f e r p r o b a b i l i t y i s t h u s

I t i s shown i n / 4 / how t o c o n s t r u c t t h e p r o b a b i l i t y W , t h a t d e s c r i b e s t h e t r a n s f e r o f a g i v e n f r a c t i o n o f t h e tot$TPnumber o f phase s p a c e c e l l s , u s i n g t h e s i n g l e phase s p a c e t r a n s f e r p r o b a b i l i t i e s w ~ + p . The d r i f t and d i f f u s i o n c o e f f i c i e n t s a p p e a r i n g i n ( 1 ) a r e t h e t i m e d e r i v a t i v e s o f t h e f i r s t and second moments o f WT,p and W ~ , T .

1

Under t h e s i m p l i f y i n g assumption t h a t R = 0 i n ( 4 ) , we o b t a i n / 4 / t h e

foll.owing r e l a t i o n s between t h e d r i f t and d i f f u s i o n c o e f f i c i e n t s :

C6-404 JOURNAL DE PHYSIQUE

(i) i n t h e low t e m p e r a t u r e l i m i t , T = 0 ,

D~~ = + ^{1 <n> - n} i n i t i a l I ^{( 7}

where n = N o r Z . The d i f f u s i o n c o e f f i c i e n t e q u a l s t h e number o f t r a n s f e r r e d p a r t i c l e s , a s i n a c l a s s i c a l random walk problem.

( i i ) i n t h e h i g h t e m p e r a t u r e l i m i t ,

Dnn

v n ~

( 8

which i s - t h e s o - c a l l e d E i n s t e i n r e l a t i o n .

L

F i g . 2 s k e t c h e s t h e b e h a v i o u r o f t h e r e l a t i v e v e l o c i t y R, t h e

t u n n e l i n g p r o b a b i l i t y f Z and t h e d i f f u s i o n c o e f f i c i e n t D Z Z a l o n g one t r a j e c t o r y i n t h e 4 0 ~ r + l o 0 ~ o r e a c t i o n a t Elab = 270 MeV. The f r i c - t i o n f o r c e used i s t h a t o f r e f . / I / . The dashed l i n e shows t h e 'clas- s i c a l window' approximation o b t a i n e d by s e t t i n g f & l a s s = 0 i f f Z ( e q . ( 5 ) ) < 1 / 2 and f21ass = 1 i f f Z (eq. ( 5 ) ) > 1/2. The asymme- t r y i n D Z Z , r e f l e y t i n g enhanced p r o t o n flow i n t h e e n t r a n c e c h a n n e l , i s c a u s e d by t h e R t e r m i n ( 4 ) .

FIG 2: Dynamical b e h a v i o u r o f v a r i o u s q u a n t i t i e s de- s c r i b e d i n t h e t e x t f o r t h e r e a c t i o n " ~ r + "'MO

a t Elab = 270 MeV and i n i t i a l a n g u l a r momentum R = 100.

I n summary, we r e s t a t e t h e i m p o r t a n t dynamical e f f e c t s t h a t have been i n c l u d e d i n t h e e v a l u a t i o n o f t h e t r a n s p o r t c o e f f i c i e n t s :

(i) S o f t e n i n g o f t h e p a r t i c l e and h o l e Fermi d i s t r i b u t i o n s by t h e

R-dependent t e m p e r a t u r e T .

( l i ) D i s a p p e a r e n c e o f s h e l l e f f e c t s i n t h e b i n d i n g e n e r g i e s w i t h i n - c r e a s i n g T .

(iii) D i s t o r t i o n o f t h e b i n d i n g e n e r g i e s by t h e mean f i e l d o f t h e p a s s i n g n u c l e u s .

( i v ) E f f e c t o f t h e r e l a t i v e motion of t h e n u c l e i on t h e t r a n s f e r pro- b a b i l i t i e s .

( v ) The d i f f e r e n c e s between n e u t r o n s and p r o t o n s a r e p r o p e r l y t a k e n i n t o account i n t h e b i n d i n g e n e r g i e s and t h e s i n g l e p a r t i c l e p o t e n t i a l s .

N e v e r t h e l e s s o u r t r e a t m e n t o f t h e t u n n e l i n g (3-dimensional t u n n e l i n g reduced t o I-dimension; n e g l e c t o f t h e change i n t h e b a r r i e r s h a p e d u r i n g t u n n e l i n g ) i s c r u d e , and s h o u l d b e improved.

IV - COMPARISON WITH EXPERIMENT

We show r e s u l t s which h i g h l i g h t some o f t h e e f f e c t s t h a t t h e dynamics c a n have on t h e v a r i a n c e s .

Fig. 3 g i v e s t h e r a t i o o h / o i a s a f u n c t i o n o f e n e r g y l o s s f o r two symmetric and two asymmetric s y s t e m s / 8 , 4 / . The s t e e p r i s e o f t h i s

F I G 3 : R a t i o o f n e u t r o n t o p r o t o n v a r i a n c e a s f u n c t i o n o f e x c i t a - t i o n energy f o r two near-symme- t r i c and two asymmetric systems.

Data from r e f . / 8 / .

O : ' l'o ' 2'0 ' 3; ' Lo ' ;o ' ;o ' ;o

E X l T A T l O N E N E R G Y { M e V 1

r a t i o a t low energy l o s s e s i n t h e Xe + Sn r e a c t i o n s i s a consequence

of t h e h i g h e r p r o t o n s i n g l e p a r t i c l e b a r r i e r r e s u l t i n g i n a quenching

C6-406 JOURNAL DE PHYSIQUE

of oi, a s s t r e s s e d by Brosa and Gross / 7 / . I n c o n t r a s t t h e Xe + Fe r e a c t i o n s do n o t show t h i s r i s e . This i s due t o t h e d i s t o r t i o n o f t h e p r o t o n b i n d i n g e n e r g i e s by t h e Coulomb f i e l d . S i n c e t h e n u c l e i a r e asymmetric, t h e p r o t o n s i n t h e l i g h t e r n u c l e u s ( F e ) w i l l be l i f t e d o u t of t h e i r s i n g l e p a r t i c l e p o t e n t i a l by t h e Coulomb f i e l d o f t h e h e a v i e r n u c l e u s (Xe) much more t h a n t h e p r o t o n s i n Xe by t h e Coulomb f i e l d o f Fe. A l a r g e p o t e n t i a l head d e v e l o p s which enhances 0 5 .

I n f i g . 4 t h e " ~ r + l o O ~ o and 3 6 ~ r + "MO r e a c t i o n s / 9 , 1 0 / a r e compared.

7 - ^{F I G 4 :} Energy l o s s d e endence o f

c r 2 K f o r t h e *OP.r

'"Mo and - 3 W ~ r +

r e a c t i o n s a t E

⁼

- 270 MeV. Data from r e f . /9)?b

- LOAr

-100

Mo -

1 c'-t-t-t-" + + -

-

1 - -

I l l I I , I , , , I

0 50 100

T K E L ( M e V )

The d i f f e r e n c e i n t h e r a t i o a&/02 a t s m a l l e n e r y l o s s e s i s due, on t h e one hand, t o t h e n e u t r o n r i c H n e s s o f b o t h ''Ar and

and, on t h e o t h e r hand, t o t h e c l o s e d n e u t r o n s h e l l i n "MO.

We b e l i e v e t h a t t h e p r e s e n t model i s e s p e c i a l l y v a l u a b l e i n t h e q u a s i - e l a s t i c regime o f heavy i o n s c a t t e r i n g , where t h e more s u b t l e e f f e c t s o f ( N , Z ) p o t e n t i a l energy s u r f a c e s , quantum t u n n e l i n g e t c . a r e n o t

'washed o u t ' by t h e s t a t i s t i c s .

T h i s work was performed t o g e t h e r w i t h D . Gross. I s h o u l d a l s o l i k e t o t h a n k W. Bohne, U . Brosa and P. FrGbrich f o r v a l u a b l e d i s c u s s i o n s .

REFERENCES

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/ 2 / NbRENBERG, W . , Phys. L e t t . 52B (1974) 289.

/ 3 / SCHRbDER, W. U . , EUIZENGA, J.R. and RRNDRUP, J . Phys. L e t t . 98B (1981) 355.

/ 4 / GROSS, ^D. H. ^E. and HARTMANN, K. M . , Phys. Rev. C24 (1981) 2526.

/ 5 / GROSS, D . E. E., NAYAK R. C . and SATPATHY, L., Z . Physik A299

(1981) 63.

/ 6 / BOHR, A. and MOTTELSON, B . , N u c l e a r S t r u c t u r e (Benjamin, New York, 1 9 6 9 ) , v o l . 1 , p. 236.

/ 7 / BROSA, U . and GROSS, D. E. E . , Z . Physik g , ( 1 980) 91.

/ 8 / SCHULL D. e t a l . , phys. L e t t . 2 (19F1) 116 and GSI S c i e n t i f i c Report 1980, p. 18.

/ 9 / BOHNE, W . e t a l . , Z . Physik A313 (1983) 19.

/ l o / GROSS, D . H. E. and PARTMANM, K. M., L e c t u r e Notes i n P h y s i c s 168 (1982) 175.

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