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THE QUANTAL PERMEATION CURRENT AND THE DISCREPANT INITIAL STAGE N-AND Z-DRIFTS IN NUCLEAR HEAVY ION COLLISIONS
J. Griffin, M. Dworzecka, A. Lukasiak
To cite this version:
J. Griffin, M. Dworzecka, A. Lukasiak. THE QUANTAL PERMEATION CURRENT AND THE DISCREPANT INITIAL STAGE N-AND Z-DRIFTS IN NUCLEAR HEAVY ION COLLISIONS.
Journal de Physique Colloques, 1987, 48 (C2), pp.C2-259-C2-264. �10.1051/jphyscol:1987239�. �jpa- 00226506�
6 , 48,
THE QUANTAL PERMEATION CURRENT AND THE DISCREPANT INITIAL STAGE N-AND Z-DRIFTS IN NUCLEAR HEAVY ION COLLISIONS
J.J. G R I F F I N , M. DWORZECKA* and A. LUKASIAK*'
Department of P h y s i c s and pstronomy, U n i v e r s i t y o f Maryland, C o l l e g e Park, MLI 20742, U . S . A .
* p h y s i c s Department, George Mason U n i v e r s i t y , F a i r f a x , V A 22030, U . S . A .
'"BRC A s s o c i a t e s I n c . , Bethesda, MD 20814, U . S . A .
A b s t r a c t
A new quanta1 permeation c u r r e n t flowing between i n t e r a c t i n g heavy i o n s i s i d e n t i f i e d and v a l i d a t e d by e x a c t numerical s o l u t i o n s of a model SchrBdinger system. T h i s c u r r e n t i s d r i v e n by t h e d i f f e r e n c e between t h e d e p t h s of t h e p o t e n t i a l w e l l s on t h e two s i d e s of t h e d i n u c l e a r window. It i s not i n c l u d e d i n t h e c o n v e n t i o n a l d e s c r i p t i o n s of heavy i o n p r o c e s s e s , which assume a c u r r e n t dependent only upon t h e d i f f e r e n c e between t h e nucleonic chemical p o t e n t i a l e n e r g i e s .
Accumulating d a t a on t h e N- and Z - d r i f t s observed i n deep i n e l a s t i c c o l l i s i o n s a l s o c o n t r a d i c t t h e c o n v e n t i o n a l d e s c r i p t i o n . By c a r e f u l e x a c t c a l c u l a t i o n w i t h t h e one-dimensional Schrodinger model, t h e permeation c u r r e n t i s h e r e q u a n t i f i e d and t h e n extended t o t h r e e dimensions. The r e s u l t i n g c o r r e c t i o n s a r e q u a l i t a t i v e l y such a s t o a m e l i o r a t e t h e d i s c r e p a n c i e s between t h e o b s e r v a t i o n s and t h e c o n v e n t i o n a l d e s c r i p t i o n .
1. Observed D r i f t s V i o l a t e t h e P o t e n t i a l Energy S u r f a c e ' s P r e d i c t i o n
I n t h e c o n v e n t i o n a l Fokker-Planck d e s c r i p t i c n of _nucleon t r a n s f e r i n deep i n e l a s t i c c o l l i s i o n s [I], t h e time e v o l u t i o n of N and Z f o r t h e p r o j e c t i l e - l i k e fragment -is assumed t o be d r i v e n by t h e g r a d i e n t of t h e d i n u c l e a r energy on t h e N-Z p l a n e [2]. Such a n energy s u r f a c e i s e x h i b i t e d i n F i g . 1 f o r t h e r e a c t i o n [3]
5 6 F e + ' 6 5 ~ o , t o g e t h e r w i t h t h e d r i f t p r e d i c t e d by i t s g r a d i e n t , and t h e e x p e r i m e n t a l d r i f t i n f e r r e d from t h e d a t a a f t e r c o r r e c t i o n f o r n e u t r o n emission.
Although t h e p r e d i c t e d p r o t o n d r i f t i s q u a l i t a t i v e l y c o r r e c t , t h e neutron d r i f t is n e a r l y z e r o , d e s p i t e t h e energy s u r f a c e ' s c l e a r p r e f e r e n c e f o r a n i n c r e a s e i n N,,.
F i g u r e 2 e x h i b i t s a n o t h e r r e a c t i o n [ 4 ] , Fe*, i n which t h e d i r e c t i o n of t h e d r i f t i s even more d r a m a t i c a l l y opposed t o t h e g r a d i e n t of t h e p o t e n t i a l energy s u r f a c e (PES), h e r e a c t u a l l y running up h i l l . Here a l s o t h e c o r r e c t i o n f o r neutron emission ( c i r c l e s v s . c r o s s e s ) i s s o s m a l l a s t o be i r r e l e v a n t . S i m i l a r s i t u a t i o n s a l s o have been observed i n t h e r e a c t i o n s [ 3 , 4 , 5 ] C a w , Cl+Bi and Fe+Bi. On t h e o t h e r hand o b s e r v a t i o n s of d r i f t s i n experiments i n v o l v i n g t h e doubly c l o s e d s h e l l nucleus, 1 3 2 ~ e , have been i n t e r p r e t e d a s conforming b e t t e r [6], although not completely 171, t o t h e p o t e n t i a l energy s u r f a c e .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987239
C2-260 JOURNAL DE PHYSIQUE
LIQUID DROP 30
LOCUS OF KlNET
ZP 20
+ ST6.TISTICAL MODEL PREDICTION
10
20 30 40
NP Fig. 1
'"u + ' ' ~ e Potential Energy Surface
Including The Coulomb Energy
- -
40 44 48 52 56 60 64
MASS NUMBER Fig. 2
2. Exact One-Dimensional Model V a l i d a t e s PES C u r r e n t , But Also E x h i b i t s New Permeation Current
We have s t u d i e d t h e n u c l e o n i c c u r r e n t s by e x a c t c a l c u l a t i o n of t h e simple one-dimensional Schrodinger "Double-Well" model [ 8 ] shown i n Fig. 3 . T h i s model c a l c u l a t e s t h e flow of nucleons between two one-dimensional " n u c l e i " , each of which i s assumed i n i t i a l l y t o comprise N nucleons bound i n a n i n f i n i t e s q u a r e w e l l p o t e n t i a l of l e n g t h L. One imagines t h a t t h e two " n u c l e i " approach one a n o t h e r very slowly ( a non-zero v e l o c i t y of approach can a l s o be included 191, b u t w i l l n o t b e d i s c u s s e d h e r e ) , w i t h t h e i r edges j u s t touching a t t s 0 , when t h e p o t e n t i a l w a l l between them i s removed and t h e subsequent time-dependent flow of nucleons i n both d i r e c t i o n s i s c a l c u l a t e d by t h e e x a c t Schradinger e q u a t i o n .
The problem i s t h u s c h a r a c t e r i z e d by t h e i n i t i a l numbers of nucleons (NR and NL), by t h e volumes (LR and LL), and by t h e d i f f e r e n c e , Vo, between d e p t h s of t h e p o t e n t i a l s i n t h e r i g h t and l e f t n u c l e i , r e s p e c t i v e l y , a s i l l u s t r a t e d i n Fig. 3.
As t h e time i n c r e a s e s , t h e p r o b a b i l i t y amplitudes flow i n both d i r e c t i o n s , a l t e r i n g t h e average number of p a r t i c l e s on t h e r i g h t , s a y , a t a r a t e
I n t h e c o n v e n t i o n a l model [ I ] , t h e s e c u r r e n t s a r e y s t i m a t e d c l a s s i c a l l y using t h e Fermi g a s d i s t r i b u t i o n . Then t h e n e t c u r r e n t A j = N, depends only upon t h e d i f f e r e n c e between t h e chemical p o t e n t i a l s , A , i n t h e l e f t and r i g h t w e l l s :
.L+R - .R+L L R
N ~ ( t ) = A j c l a s s = ' c l a s s ' c l a s s = ( A -A ) / h = ( E ~ V ~ - E : ) / ~ , (2)
L L
where, e . g . , EF i s t h e Fermi k i n e t i c energy on t h e l e f t , EF = ( & n ) * ( ~ ~ + 1 / 2 ) * / 2 ~ ~ : . E a r l i e r , e x a c t q u a n t a l c a l c u l a t i o n s with Double Well [ 8 ] had produced a
c o n t r a d i c t i o n t o Eq. ( 2 ) , a s i l l u s t r a t e d i n Fig. 4. There a s e r i e s of c a l c u l a t i o n s i s d i s p l a y e Q i n which t h e d i f f e r e n c e , AL-AR, i s k e p t p r e c i s e l y c o n s t a n t .
Obviously, N, g i v e n by t h e s l o p e of each curve i n F i g . 4 , depends a l s o upon Vo, and n o t s o l e l y upon A=-AR.
We a r e now a b l e t o e x p l a i n t h i s discrepancy [10,11] i n terms of t h e
"permeation c u r r e n t " a s s o c i a t e d w i t h s t a t e s on t h e r i g h t whose energy Ei i s l e s s t h a n Vo. Although c l a s s i c a l l y such s t a t e s c o n t r i b u t e n o t h i n g t o t h e c u r r e n t a c r o s s t h e window, t h e r e i s a t e a r l y times a q u a n t a l c u r r e n t a s s o c i a t e d w i t h t h e
s i n g l e p a r t i c l e s t a t e s .
DOUBLE WELL PROBLEM and INITIAL CONDITIONS
ElGEN VALUES
N R VS. tlme for Simulated '"Protons"
Vo=O.O MeV V0=2.0 MeV Vo=4.0 MeV Vo = 8.0 MeV
I I I I I I I I I I
f-8 -4 0 4 1 8
-LL ( x ) +LR V,*16.0 MeV
TRANSLATIONAL
ENERGY is Time ( I O - ~ ~ S C C ) characterized by
E/A = C. of Moss F i g . 4
Translational
0 + LR
(XI
F i g . 3 SCHRODINGER CURRENT
n = 6 ; 8.7.2 CLASSICAL CURRENT
n = 5 ; 8.5.1 n = 4 ; 8.3.2 n = 3 ; 8.1.2
- [ t i m e ( t ) ]
-
0'20] SCHR~DINGER CURRENT
, 0.15 PERMEATION CURRENT VS.
-0.05 , , , , , , , , , a , ,
0 1 2 5 6 1
- [ t i m e ( t ) ]
-
F i g . 5
Fig. 6
C2-262 JOURNAL DE PHYSIQUE
F i g u r e 5 e x h i b i t s NL(t) a s c a l c u l a t e d f o r f o u r s t a t e s on t h e r i g h t f o r which 6, = E,,/Vo exceeds t h e v a l u e 1, s o t h a t passage a c r o s s t h e window i s c l a s s i c a l l y allowed. I n each c a s e a s t r a i g h t l i n e w i t h t h e s l o p e implied by t h e c l a s s i c a l assumption i s i n good agreement w i t h t h e e x a c t Schrodinger r e s u l t f o r times l e s s t h a n t h e time r e q u i r e d f o r t h e wave t o r e f l e c t from t h e o p p o s i t e w a l l . Indeed, t h e r a t i o P(n) between t h e b e s t f i t t i n g v a l u e of Nn over t h e range where Nn(t) i s approximately l i n e a r and t h e c l a s s i c a l c u r r e n t , j c l a s s ( n ) = 2En/hn, i s p l o t t e d i n F i g - 7 vs. 6, = En/Vo f o r many c a s e s i n c l u d i n g t h o s e of F i g s . 5 and 6. One s e e s t h a t f o r 6, > 1, Pn 1, i n a c c o r d w i t h t h e c l a s s i c a l assumption.
On t h e o t h e r hand, s t a t e s w i t h 6,<1, from which t h e c l a s s i c a l assumption a l l o w s no c o n t r i b u t i o n t o t h e c u r r e n t , do i n f a c t c o n t r i b u t e i n t h e e x a c t
Schrodinger s o l u t i o n . This i s i l l u s t r a t e d i n Fig. 6 where t h e Schradinger r e s u l t shows a dependence l i n e a r i n time a l s o f o r t h e s e c l a s s i c a l l y excluded s t a t e s . These s l o p e s d e f i n e a n approximately q u a n t a l "permeation c u r r e n t ' ' f o r each such s t a t e , whose v a l u e i s e q u a l t o t h e product of t h e c l a s s i c a l c u r r e n t and a numerical f a c t o r , P ( n ) , which depends only upon 6, = E,/v,, a s i l l u s t r a t e d i n Fig. 7.
Roughly, F i g . 7 can be summarized by t h e r e l a t i o n s h i p ,
P = 0.07 + 0.69 6 , f o r 0 < 6 < 1 . 3 ,
o r , more simply and more approximately ( f 3 0 % ) , by
P = E i / V o = 6 , f o r 6 c 1 .
PERMEATION and CLASSICAL CURRENTS
F i g . 7
Thus, one r e c o g n i z e s t h a t q u a n t a l p a r t i c l e s deeply bound i n t h e right-hand w e l l respond t o t h e r e d u c t i o n i n t h e i r binding p o t e n t i a l (from t o Vo) e f f e c t e d by
t h e approach of t h e d i n u c l e a r p a r t n e r by s p r e a d i n g o u t t h e i r p e n e t r a t i n g t a i l s through t h e window i n t o t h e volume of t h e left-hand n u c l e u s . The r e s u l t i s a n e t c u r r e n t flow a t e a r l y times which moves p a r t i c l e s from t h e s p a t i a l r e g i o n of t h e deeper w e l l i n t o t h a t of t h e shallower well. Most noteworthy i s t h e f a c t t h a t t h i s c u r r e n t does not d i m i n i s h e x p o n e n t i a l l y w i t h i n c r e a s i n g Vo/Ei, but e x h i b i t s i n s t e a d a l i n e a r dependence upon Ei/Vo, such t h a t a t Ei/Vo - 1, i t h a s a v a l u e 1
a p p r o p r i a t e f o r t h e c l a s s i c a l p i c t u r e .
The r e s u l t s imply t h a t t h e c o r r e c t n e t Schrodinger c u r r e n t flowing a c r o s s t h e window t o t h e r i g h t i s i n f a c t approximately e q u a l t o t h e following f u n c t i o n of lL, lR and Vo;
where t h e permeation c u r r e n t , jperm, d e s c r i b e s t h e p u r e l y q u a n t a l p e n e t r a t i o n flow of nucleons, of energy l e s s t h a n Vo, i n t o t h e l e f t - h a n d s i d e , and AjClass i s given i n Eq. (2). Expressions f o r t h e three-dimensional c a s e a r e g i v e n below.
These one-dimensional r e s u l t s can be extended t o t h r e e dimensions by c o n s i d e r i n g two a d j a c e n t volumes i n which t h e p o t e n t i a l d e p t h s d i f f e r by Vo, and which c o n t a i n Fermi g a s e s which f i l l t h e momentum s p h e r e o u t t o &kL, andMkR, on t h e l e f t and r i g h t r e s p e c t i v e l y . T h i s g e n e r a l i z a t i o n t h e n i n c l u d e s t h e a p p r o p r i a t e weighting f o r t h e t r a n s v e r s e s t a t e s a s s o c i a t e d w i t h a wave number k, i n t h e
d i r e c t i o n p e r p e n d i c u l a r t o t h e window, and a l l o w s a c a l c u l a t i o n of t h e permeation c u r r e n t p e r u n i t a r e a a s a f u n c t i o n of Vo.
Then t h e n e t c u r r e n t of p a r t i c l e s from r i g h t t o l e f t through a window of f i n i t e a r e a , ow, i s g i v e n by Eq. (4) w i t h t h e f o l l o w i n g e x p r e s s i o n s
f o r ' j c l a s s and jrm, based upon approximation ( 3 a ) , and a c c u r a t e t o l e a d i n g L R -.
o r d e r i n ( v O / I ) and (A -A )/A.
and
Here j-= P<v>/2 i s t h e average p a r t i c l e f l u x impinging on t h e window from each s i d e , A i s t h e average of t h e two chemical p o t e n t i a l s , 73 i s t h e average of t h e two d e n s i t i e s , and ow i s t h e a r g a of t h e window. Also, f o r a Fermi d i s t r i b u t i o n populated up t o k i n e t i c energy EF, one c a l c u l a t e s <v> = ( 3 / 8 ) ( 2 ~ ~ / ~ ) ~ / ~ .
Note t h a t e q u a t i o n s (5) show t h a t when t h e chemical p o t e n t i a l s k and lR d i f f e r by a n amount comparable t o Vo, t h e n t h e permeation c u r r e n t i s n e a r l y one-half a s l a r g e a s t h e c l a s s i c a l c u r r e n t . Also r e c a l l t h a t t h e conventional PES d e s c r i p t i o n omits e n t i r e l y t h i s Vo-dependent permeation c u r r e n t .
4 . C o r r e l a t i o n w i t h t h e Experimental Data
To make a n i n i t i a l assessement of t h e d i f f e r e n c e s , Vo, which might a r i s e i n a c t u a l heavy i o n r e a c t i o n s , we have u t i l i z e d t h e d r o p l e t model [ 1 2 , 1 3 ] , which p r e d i c t s a c l e a r dependence of t h e n u c l e a r w e l l d e p t h s on t h e n e u t r o n excess, (N-Z) .
( a ) Neutron D r i f t s : Since t h e neutron s h e l l model p o t e n t i a l depth diminishes w i t h i n c r e a s i n g n e u t r o n e x c e s s [12,13], one e x p e c t s Vo t o i n c r e a s e a s (N-Z)
i n c r e a s e s i n t h e nucleus on t h e l e f t i n Fig. 3, r e s u l t i n g i n a n i n c r e a s e d permeation c u r r e n t of n e u t r o n s from t h e nucleus w i t h t h e s m a l l e r neutron excess i n t o t h e n u c l e u s w i t h t h e l a r g e r . T h i s p r e d i c t i o n i s i n q u a l i t a t i v e agreement with
- t h e n e u t r o n d r i f t s observed i n t h e r e a c t i o n s p o r t r a y e d i n F i g s . 1 and 2, and a s w e l l w i t h t h e o t h e r c l a s s i c a l l y anomalous d r i f t s c i t e d above [3-51.
I n each c a s e t h e l i g h t (NnZ) nucleus d e l i v e r s n e u t r o n s t o t h e heavy (N)>Z) n u c l e u s d e s p i t e t h e tendency of t h e p o t e n t i a l energy s u r f a c e t o t h e c o n t r a r y .
( b ) P r o t o n D r i f t s : For p r o t o n s t h e d r o p l e t model p r e d i c t s a n (N-Z) dependence of t h e non-Coulombic p a r t of t h e s h e l l model p o t e n t i a l of t h e same magnitude a s but of o p p o s i t e s i g n from t h e neutron p o t e n t i a l . However, t h e Coulombic p o t e n t i a l s f o r t h e p r o t o n s h e r e become q u i t e important, e s p e c i a l l y t h a t due t o t h e Coulomb
i n t e r a c t i o n between t h e t a r g e t and p r o j e c t i l e - l i k e p a r t s of t h e dinucleus. Upon i n c l u d i n g them, one f i n d s t h a t t h e combined r e s u l t of t h e s e e f f e c t s f o r p r o t o n s i s a p o t e n t i a l d i f f e r e n c e between t h e two s i d e s which i s much s m a l l e r f o r p r o t o n s t h a n f o r n e u t r o n s . Thus a good p r e d i c t i o n of t h e magnitude, o r even of t h e s i g n of t h e p r o t o n permeation c u r r e n t , r e q u i r e s a more c a r e f u l a n a l y s i s t h a n we have done s o f a r .
5. Conclusions
We have d e s c r i b e d c a l c u l a t i o n s f o r a simple one-dimensional Schradinger model which i d e n t i f y a n o n c l a s s i c a l permeation c u r r e n t which flows i n a d i n u c l e u s from
t h e deeper p o t e n t i a l i n t o t h e shallower. This c u r r e n t i s q u a l i t a t i v e l y such a s t o a m e l i o r a t e t h e d i s c r e p a n c i e s between observed d r i f t s and t h e p r e d i c t i o n s of t h e c l a s s i c a l p o t e n t i a l energy s u r f a c e . F u r t h e r r e s e a r c h promises t o t e l l whether t h e improvement can a l s o be made q u a n t i t a t i v e , and u l t i m a t e l y , whether such e f f e c t s can become a t o o l f o r d i r e c t s t u d y of t h e d e t a i l e d n a t u r e of t h e n u c l e a r p o t e n t i a l s i n t h e i r d e e p e s t p a r t s .
JOURNAL DE PHYSIQUE
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