MICROSCOPIC MODEL OF NUCLEUS-NUCLEUS COLLISIONS

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MICROSCOPIC MODEL OF NUCLEUS-NUCLEUS COLLISIONS

B. Harvey

To cite this version:

B. Harvey. MICROSCOPIC MODEL OF NUCLEUS-NUCLEUS COLLISIONS. Journal de Physique Colloques, 1986, 47 (C4), pp.C4-29-C4-34. �10.1051/jphyscol:1986403�. �jpa-00225766�

JOURNAL DE PHYSIQUE

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

MICROSCOPIC MODEL OF NUCLEUS-NUCLEUS COLLISIONS

B.G. HARVEY

Lawrence B e r k e l e y L a b o r a t o r y , B e r k e l e y , C A 94720, U.S.A.

Resume - La c o l l i s i o n de deux noyaux e s t t r a i t e e comme un ensemble de c o l l i s i o n s des nucleons du p r o j e c t i l e avec ceux du noyau c i b l e . Les fragments p r i m a i r e s ne comportent que l e s nucleons q u i n l o n t pas s u b i t une c o l l i s i o n . Les s e c t i o n s e f f i c a c e s i n c l u s i v e s e t l e s coincidences r e s u l t e n t de l a d e s i n t e g r a t i o n des fragments p r i m a i r e s e x c i t e s .

A b s t r a c t - The c o l l i s i o n o f two n u c l e i i s t r e a t e d as a c o l l e c t i o n o f c o l l i - sions between t h e nucleons o f t h e p r o j e c t i l e and those o f t h e t a r g e t nucleus. The primary p r o j e c t i l e fragments c o n t a i n o n l y those nucleons t h a t d i d n o t undergo a c o l l i s i o n . The i n c l u s i v e and coincidence cross s e c t i o n s r e s u l t from t h e decay o f t h e e x c i t e d primary fragments.

I - INTRODUCTION

I t has been known f o r a l o n g time t h a t t h e i n c l u s i v e y i e l d o f n e u t r o n - r i c h fragments from t h e c o l l i s i o n o f a heavy i o n w i t h a n e u t r o n - r i c h t a r g e t i s s u b s t a n t i a l l y l a r g e r a t low beam energies (20-44 MeV/A) than a t h i g h energies [I-31.

The t o t a l cross s e c t i o n f o r n-p s c a t t e r i n g a t low energies i s about t h r e e times l a r g e r than f o r n-n o r p-p s c a t t e r i n g , w h i l e a17 these cross sections a r e n e a r l y equal a t h i g h energies. This suggests t h a t primary p r o j e c t i l e fragments a r e formed as a r e s u l t o f c o l l i s i o n s o f p r o j e c t i l e nucleons w i t h t a r g e t nucleons. When t h e t a r g e t i s n e u t r o n - r i c h and t h e energy i s low, p r o j e c t i l e protons a r e more l i k e l y t o be s c a t t e r e d than p r o j e c t i l e neutrons. I f t h i s s c a t t e r i n g process removes nucleons from t h e p r o j e c t i l e , t h e r e s u l t i n g primary fragments w i l l be p r o t o n - d e f i c i e n t and n e u t r o n - r i c h i n low energy r e a c t i o n s on n e u t r o n - r i c h t a r g e t s .

I n s p i t e of arguments t o t h e c o n t r a r y [4], i t i s now c l e a r t h a t most p r i m a r y frag- ments w i l l be s u f f i c i e n t l y e x c i t e d t o undergo s e q u e n t i a l decay. I n t h e c a l c u l a t i o n of i n c l u s i v e cross sections and e s p e c i a l l y of coincidence cross sections, t h e e f f e c t o f sequential decays o f t h e primary fragments must be included. I n t h e p r e s e n t paper, t h i s i s done i n an approximate way. The f i n a l r e s u l t s a r e i n g e n e r a l l y good agreement w i t h experimental i n c l u s i v e cross s e c t i o n s as w e l l as w i t h t h e l i m i t e d amount o f coincidence data t h a t i s a v a i l a b l e .

I 1 - CALCULATION OF PRIMARY YIELDS

The Monte C a r l o technique t h a t i s used t o c a l c u l a t e t h e p r i m a r y fragment y i e l d s i s described i n d e t a i l i n r e f . 4. B r i e f l y , t h e c a l c u l a t i o n proceeds as f o l l o w s . The d e n s i t y d i s t r i b u t i o n s o f t h e two c o l l i d i n g n u c l e i a r e described by Fermi func- t i o n s w i t h , i n general, d i f f e r e n t r a d i i and d i f f u s i v i t i e s f o r protons and neutrons.

S t a r t i n g a t a given impact parameter, t h e two n u c l e i approach each o t h e r along o r b i t s i n t h e Coulomb p l u s n u c l e a r p o t e n t i a l u n t i l t h e 20% d e n s i t i e s overlap. From t h a t p o i n t on, t h e p r o j e c t i l e nucleons a r e assumed t o f o l l o w s t r a i g h t paths i n t o

and through t h e t a r g e t nucleus.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986403

C4-30 J O U R N A L DE PHYSIQUE

A t a l l e n e r g i e s , t h e N-N c r o s s s e c t i o n s a r e assumed t o have t h e f r e e nucleon v a l u e s f o r c o l l i s i o n s i n t h e s u r f a c e o f t h e t a r g e t . A t e n e r g i e s below t h e Fermi energy, P a u l i b l o c k i n g reduces t h e e f f e c t i v e N-N c r o s s s e c t i o n s i n t h e i n t e r i o r o f t h e t a r g e t . Values a r e chosen t o be c o n s i s t e n t w i t h t h e mean f r e e p a t h o f nucleons i n n u c l e i . They r i s e as t h e d e n s i t y f a l l s u n t i l t h e f r e e v a l u e s a r e reached i n t h e s u r f a c e . A t e n e r g i e s w e l l above t h e Fermi energy, f r e e N-N c r o s s s e c t i o n s a r e assumed a t a l l d e n s i t i e s o f t h e t a r g e t n u c l e u s . The p r i m a r y fragment c r o s s s e c t i o n s and r e a c t i o n c r o s s s e c t i o n s a r e v e r y i n s e n s i t i v e t o assumptions a b o u t N-N c r o s s sec- t i o n s i n t h e i n t e r i o r o f t h e t a r g e t nucleus.

The c o o r d i n a t e s o f a n u c l e o n i n t h e p r o j e c t i l e a r e chosen a t random t o r e f l e c t t h e p r o p e r d e n s i t y d i s t r i b u t i o n s . Each p r o j e c t i l e n u c l e o n i s a1 lowed t o pass t h r o u g h , o r near, t h e t a r g e t . The p r i m a r y fragment c o n s i s t s o f t h o s e p r o j e c t i l e nucleons t h a t d i d n o t s c a t t e r . The s c a t t e r i n g p r o b a b i l i t y f o r a n u c l e o n i s c a l c u l a t e d by n u m e r i c a l e v a l u a t i o n o f t h e i n t e g r a l s : -

Pp = 1 - exp - (

J

J

Pn = 1 - exp - ( / m ( ~ ) d n n ( ~ ) d z + J p p ( ~ ) a n p ( ~ ) d z ) ( 2 ) Here, pn(R), pp(R) a r e t h e d e n s i t i e s o f n e u t r o n s and p r o t o n s a t r a d i u s R i n t h e t a r g e t . The N-N c r o s s s e c t i o n s a r e f u n c t i o n s o f R, as d i s c u s s e d above. A t each i m p a c t parameter, a few thousand p r o j e c t i l e s a r e a l l o w e d t o c o l l i d e w i t h t h e t a r g e t . The i m p a c t parameter b i s t h e n incremented and t h e whole process i s r e p e a t e d u n t i l no more c o l l i s i o n s o c c u r . The c r o s s s e c t i o n f o r a p r i m a r y fragment (Z.A) a t i m p a c t parameter b i s : -

where F i s t h e f r a c t i o n o f i n t e r a c t i o n s a t b t h a t produce (Z.A). The t o t a l c r o s s s e c t i o n f o r a g i v e n fragment i s j u s t t h e sum o v e r a l l i m p a c t parameters, and t h e t o t a l r e a c t i o n c r o s s s e c t i o n er i s t h e sum o f a l l fragment c r o s s s e c t i o n s . Two r e s u l t s emerge a t t h i s p o i n t . F i r s t , t h e p r o d u c t i o n o f a g i v e n p r i m a r y f r a g - ment i s s t r o n g l y l o c a l i z e d i n i m p a c t parameter space. The l i g h t e r fragments come f r o m s m a l l e r impact parameters, and t h e h e a v i e r fragments come f r o m more p e r i p h e r a l c o l l i s i o n s . Second, t h e v a l u e s o f or a r e i n remarkably good agreement w i t h

e x p e r i m e n t a l r e s u l t s a t a l l e n e r g i e s f r o m 20 MeV/A t o 2 GeV/A [4].

The p r i m a r y fragment c r o s s s e c t i o n s do n o t agree w e l l w i t h t h e e x p e r i m e n t a l i n c l u - s i v e values. T h e i r mass d i s t r i b u t i o n s f o r a g i v e n Z-value a r e t o o broad, e s p e c i a l l y on t h e neutron-excess s i d e o f t h e l i n e o f s t a b i l i t y , and t h e y do n o t show t h e char- a c t e r i s t i c peaks f o r fragments t h a t a r e formed by t h e removal f r o m t h e p r o j e c t i l e o f one o r t w o a l p h a p a r t i c l e s . I n r e f . 4, t h e s e d i s c r e p a n c i e s were removed b y t h e use o f t h e f r a g m e n t a t i o n model o f Friedman [5] which assumes t h a t t h e p r o b a b i l i t y f o r removing a g i v e n c l u s t e r o f nucleons f r o m t h e p r o j e c t i l e , and f o r t h e r e m a i n i n g fragment t o escape f u r t h e r i n t e r a c t i o n , depends on t h e s e p a r a t i o n energy o f t h e p r o j e c t i l e i n t o t h e two p a r t s .

Cole [ 6 ] uses an a n a l y t i c a p p r o x i m a t i o n t o t h e c o l l i s i o n geometry. and c o n s i d e r s t h a t d i r e c t N-alpha c o l l i s i o n s can o c c u r as w e l l as N-N c o l l i s i o n s . T h i s t o o has t h e e f f e c t o f i n c r e a s i n g t h e c r o s s s e c t i o n s f o r fragments h a v i n g l o w s e p a r a t i o n e n e r g i e s .

I 1 1 - SEQUENTIAL DECAY

W h i l e t h e r e i s some i n f o r m a t i o n a b o u t t h e e x c i t a t i o n e n e r g i e s and decay modes o f p r i m a r y fragments f r o m l o w beam energy e x p e r i m e n t s [7-81, t h e r e i s v e r y l i t t l e f o r e n e r g i e s above 20 MeV/A. N e v e r t h e l e s s , any model must b e c o n s i s t e n t w i t h t h e f o l l o w i n g e x p e r i m e n t a l o b s e r v a t i o n s : -

1. A t E < 20 MeV/A, t h e number o f charged p a r t i c l e s i n coincidence w i t h a fragment i s small, e i t h e r 0 o r 1 except f o r t h e l i g h t e s t fragments such as L i isotopes from a 160 p r o j e c t i l e [7]. This suggests t h a t t h e primary fragments decay by t h e emission o f no more than one charged p a r t i c l e and/or by t h e emission o f neutrons.

2. A t h i g h energies, a l l fragments, even such l i g h t n u c l e i as 6 ~ i , come from p e r i p h e r a l c o l l i s i o n s [ 9 ] . They must t h e r e f o r e r e s u l t from t h e decay of e x c i t e d p r o j e c t i l e s and from e x c i t e d fragments c l o s e i n mass t o t h e p r o ' e c t i l e .

3. I n t h e c o l l i s i o n o f 12c + 1 % a t 2 GeVIA. about 25% o f t h e fragments a r e i n coincidence w i t h a small-angle beam v e l o c i t y proton. They a r e produced by t h e decay o f e x c i t e d 12c. The o t h e r 75% a r e n o t i n coincidence w i t h such a p r o t o n and a r e presumably s u r v i v i n g primary 118 fragments [ l o ] .

I n t h e same system, t h e channel 1 2 ~ + 6 ~ i + 6Li i s observed w i t h a cross s e c t i o n o f v e r y roughly 1 mb [ l o ] . This shows t h a t t h e temperature o f t h e e x c i t e d I 2 c

n u c l e i i s h i g h enough f o r f i s s i o n processes t o occur [ll]. The cross s e c t i o n f o r t h e decay o f I ~ c i n t o t h r e e a l p h a - p a r t i c l e s i s 9.7 + 5-2.5 mb [ l o ] .

I t has been shown [1,12] t h a t i n c l u s i v e cross s e c t i o n s a t low and h i g h energies can be reproduced by t h e assumption t h a t primary fragments decay i n t o a l a r g e number o f channels and t h a t t h e p r o b a b i l i t y of decay i n t o a s p e c i f i c channel i s : -

Here, Es i s t h e separation energy o f t h e primary fragment i n t o t h e two o r more p a r t s , and T i s a parameter. A very s i m i l a r method has been used t o c a l c u l a t e t h e r e l a t i v e p r o b a b i l i t i e s o f f i s s i o n o f l i g h t and medium n u c l e i [Ill. The cross s e c t i o n f o r t h e f o r m a t i o n o f a given fragment (Z1,A') from p r i m a r y fragment (Z,A) can t h e r e f o r e be w r i t t e n : -

a(Z,A)exp(-ES/T)

u ( z ' , A ' ) = (5)

Z ~ ~ P ( - E ~ / T )

where o(Z,A) i s t h e c a l c u l a t e d primary cross section. The sum includes a l l decay channels o f (Z,A) as w e l l as Es.= 0 t o represent t h e p r o b a b i l i t y t h a t t h e

fragment does n o t decay by p a r t i c l e emission.

A t E 5 20 MeV/A, i t i s e s s e n t i a l t o i n c l u d e a l l primary fragments down t o 6 ~ i i n t h e decay process. F i g . 1 shows a comparison o f t h e f r a c t i o n o f each primary

fragment t h a t survived charged p a r t i c l e decay w i t h an experimental measurement

[a]. The system was 2 0 ~ e + l g 7 ~ u a t 17 MeV/A. Fig. 2 shows a comparison. f o r t h e same system, o f t h e c a l c u l a t e d and experimental r a t i o s o f t h e cross s e c t i o n s of fragments t h a t a r e n o t i n coincidence w i t h a charged p a r t i c l e ( b u t p o s s i b l y i n coincidence w i t h one o r more neutrons) t o t h e i n c l u s i v e cross sections. Both f i g u r e s show q u i t e good agreement between experiment and c a l c u l a t i o n .

F i g . 3 shows a com a r i s o n o f t h e experimental and c a l c u l a t e d i n c l u s i v e cross sec- t i o n s f o r 160 + 20gPb a t 20 MeV/A. The primary y i e l d s are a l s o shown f o r isotopes o f L i , B and C. They a r e broader t h a n t h e experimental values on t h e n e u t r o n - r i c h side. The maximum y i e l d f o r C isotopes f a l l s a t 13c i n s t e a d o f a t 1 2 ~ .

F i g . 4 shows t h e i n c l u s i v e cross s e c t i o n s f o r 12c + 12c a t 2 GeV/A. I n t h i s case, t h e decaying primary fragments were r e s t r i c t e d t o 12c, llc and 118, and decays by t h e emission o f any number o f nucleons o r bound n u c l e i up t o h a l f t h e p r o t o n and neutron number o f t h e primary fragment were included. The 12c was g i v e n a temperature of 8 MeV and t h e A = I 1 n u c l e i , 9 MeV. 21% o f t h e I ~ B comes from I ~ c decay and 79% i s s u r v i v i n g primary 110, i n e x c e l l e n t agreement w i t h

experiment. The channel cross s e c t i o n f o r f i s s i o n o f 12c i n t o two 6 L i n u c l e i i s 1.9 mb, i n agreement w i t h t h e experimental value. The channel cross s e c t i o n f o r decay o f 12c i n t o t h r e e a l p h a - p a r t i c l e s i s 25 mb, somewhat h i g h e r than t h e experimental value o f 9.7 + 5-2.5 mb.

For 12c + I 2 c a t 85 MeV/A, t h e same method and parameters as those used a t 2 GeV/A gave good agreement w i t h i n c l u s i v e cross sections [13]. This suggests

C4-32 JOURNAL DE PHYSIQUE

t h a t , a t t w i c e t h e Fermi energy, t h e r e a c t i o n mechanism has a l r e a d y reached some l i m i t i n g form. Indeed, t h e N-N cross s e c t i o n s a t these two energies a r e n o t very d i f f e r e n t .

Fig. 1 - F r a c t i o n of primary fragments Fig. 2 - R a t i o of c r o s s s e c t i o n t h a t s u r v i v e decay b charged p a r t i c l e Y f o r fragments w i t h o u t a c o i n c i d e n t emission. 2 0 ~ e + l9 Au, 17 MeV/A. charged a r t i c l e t o t h e i n c l u s i v e

y i e l d . + lg7au, I 7 MeV/*.

Hatched areas a r e u n c e r t a i n t y i n experiment, heavy l i n e i s c a l c u l a - t i o n .

1 k l ; e 6 IY;O I; 12;; 1; i3 lh'l3d 14 15 16 A (ejectile)

Fig. 3 - Ex c r i m e n t a l and c a l c u l a t e d (dashed l i n e ) i n c l u s i v e cross s e c t i o n s f o r 160 + 2B8Pb, 20 MeV/A. Primary fragment cross sections f o r L i , B and C a r e d o t t e d l i n e s .

For t h e system 160 + 2 0 8 ~ b , t h e c a l c u l a t e d cross s e c t i o n r a t i o (20 MeV/A)/(2 GeV/A) shows enhancement o f t h e most n e u t r o n - r i c h fragments a t 20 MeV/A by an average f a c t o r of 3.5. For these same fragments, t h e experimental enhancement i s 4.2 + ^.85. Half o f t h e enhancement comes from t h e neutron s k i n o f t h e t a r g e t combined w i t h t h e l a r g e n-p cross s e c t i o n a t 20 MeV. It i s increased, f o r t h e n e u t r o n - r i c h fragments, by a f u r t h e r f a c t o r o f 1.9 i n t h e decay process.

Reactions o f 4 0 ~ r present a s p e c i a l problem a r i s i n g from t h e neutron excess o f t h e p r o j e c t i l e i t s e l f . Any assumptions about t h e e x c i t a t i o n energy and t h e decay

01' ; A A,;,; A A ,;,!,A ; ^{l'o ,'I}

~~

Mass number of fragment

Fig. 4 - Inclusive fragment cross sections, 12c + 12c, 2 GeV/A. Inclusive cross sections for 6 ~ i t o l l c at 85 MeV/A differ by only 3.3% from those at 2 GeV/A.

A (fragment mass)

Fig. 5 - Experimental and calculated isotopic distributions for 4 0 ~ r + l g 7 ~ u at 4 4 MeV/A. Dotted lines are the present calculation, dashed lines are calcu- lation from ref. 14.

JOURNAL DE PHYSIQUE

modes o f t h e primary fragments always lead t o cross s e c t i o n s f o r each Z t h a t have a maximum two mass numbers h i g h e r than observed e x p e r i m e n t a l l y 1141. This can be

"cured" by t h e r a t h e r d r a s t i c assumption t h a t t h e two neutrons o f 4 0 ~ r o u t s i d e t h e N = 20 closed s h e l l a r e always l o s t . The fragment thus produced then decays by the.emission o f l i g h t p a r t i c l e s o r by f i s s i o n . A comparison w i t h experiment o f t h e i s o t o p e d i s t r i b u t i o n s thus obtained i s shown i n Fig. 5. The agreement i s b e t t e r than t h a t obtained w i t h a standard s t a t i s t i c a l evaporation model [14]

(dashed l i n e s ) . A s u b s t a n t i a l f r a c t i o n o f t h e y i e l d s o f l i g h t elements comes from t h e f i s s i o n process.

I V - SUMMARY

Below t h e Fermi energy, primary fragments a r e produced w i t h s u f f i c i e n t l y low e x c i t a t i o n energies t h a t a s u b s t a n t i a l f r a c t i o n s u r v i v e s e q u e n t i a l decays.

e s p e c i a l l y f o r h e a v i e r fragments.

Above t w i c e t h e Fermi energy, t h e primary fragment e x c i t a t i o n s a r e l a r g e enough t h a t decays i n c l u d e t h e f i s s i o n channels. Primary fragments formed by t h e removal of more than a very few nucleons from t h e p r o j e c t i l e do n o t s u r v i v e as bound n u c l e i . They a r e probably d i s i n t e g r a t e d i n t o nucleons and very l i g h t articles bv t h e cas- cade o f s c a t t e r e d nucleons coming from t h e o v e r l a p r e g i o n - o f b r o j e c t i l e and t a r g e t n u c l e i [15].

V - ACKNOWLEDGEMENTS

The author thanks H.J. Crawford, D.E. Greiner and P.J. Lindstrom f o r permission t o quote new experimental r e s u l t s p r i o r t o p u b l i c a t i o n .

T h i s work was supported by t h e D i r e c t o r , O f f i c e o f Energy.Research, D i v i s i o n o f Nuclear Physics o f t h e O f f i c e o f High Energy and Nuclear Physics o f t h e U.S.

Department o f Energy under Contract DE-AC03-76SF00098.

V I - REFERENCES

[I] Gelbke, G.K., e t al., Physics Reports 42 No. 5 (1978) 312.

[2] Homeyer, H., Nuclear Science Research Series, Vol. 6 p. 95.

Nuclear Physics w i t h Heavy Ions. Harwood Academic Publishers, 1984 [ 3 ] Guerreau, D., e t a l . . Phys. L e t t . (1983) 293.

[ 4 ] Harvey, B.G., Nucl. Phys. A444 (1985) 498.

[ 5 ] Friedman, W.A., Phys. Rev. C27 (1983) 569.

[ 6 ] Cole, A.J., Report ISN 84.35, Grenoble, Aug. 1984.

[73 Murphy. M.J.. e t a l . , Phys. L e t t . fZOB (1983) 319.

[8] Wald S., e t a l . , Phys. Rev. C32 (1985) 894.

[ 9 ] Olson D.L.. e t a l . , Phys. Rev. C28 (1983) 1602.

: I 0 1 Crawford, H.J.. Greiner D.E., and Lindstrom, P.J.. P r i v a t e Comnunication.

: I 1 1 Sobotka, L.G., e t al., Phys. Rev. L e t t . 51 (1983) 2187.

: I 2 1 Lukyanov V.K., and T i t o v , A.I., Phys. L e t t . 578 (1975).

1131 Ryde, H., Physica S c r i p t a (1983) 114.

:14] B o r r e l , V., e t al., Orsay Report IPNO-DRE-86/02, 1986 .

:15] Hiifner, J., Schafer K.. and Schiirmann, 0 . . Phys. Rev. C12 (1975) 1888.