MEDIUM-ENERGY SCATTERING OF NUCLEONS BY NUCLEI

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MEDIUM-ENERGY SCATTERING OF NUCLEONS BY NUCLEI

H. Machner

To cite this version:

H. Machner. MEDIUM-ENERGY SCATTERING OF NUCLEONS BY NUCLEI. Journal de Physique

Colloques, 1987, 48 (C2), pp.C2-271-C2-274. �10.1051/jphyscol:1987241�. �jpa-00226509�

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MEDIUM-ENERGY SCATTERING OF NUCLEONS BY NUCLEI

H. MACHNER

Institut fur Kernphysik, Kernforschungsanlage Julich GmbH.

Postfach 1913, 0-5170 Julich, F.R.G.

A b s t r a c t

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The e x c i t o n model f o r nuclear r e a c t i o n s i s reformulated t o includeenergy-momentum coupling. A new d e f i n i t i o n f o r t h e cross s e c t i o n i s g i v e n s i m i l a r t o t h e Glauber model. The model p r e d i c t i o n s a r e compared w i t h medium energy data.

I

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I n t r o d u c t i o n

The e x c i t o n model f o r nuclear r e a c t i o n s as was invented o r i g i n a l l y by

riffi in')

a1 lowed only t o c a l c u l a t e t h e s p e c t r a l shape o f energy d i f f e r e n t i a l cross sections.

The model was developed f u r t h e r by d i f f e r e n t groups w i t h t h e f o l l o w i n g main goals:

t c p r e d i c t cross s e c t i o n s on an absolute basis, i n c l u d e composite p a r t i c l e s i n t h e entrance and t h e e x i t channels and account f o r t h e forward peaked angular d i s t r i - butions. A recent review i s given i n Ref. 2. I n t h i s c o n t r i b u t i o n we would l i k e t o focus on t h e l i n e a r momentum d i s s i p a t i o n associated w i t h t h e angular d i s t r i b u t i o n problem. A r e f o r m u l a t i o n of e x i s t i n g models i n a u n i f i e d approach should make d i f - f e r e n t model assumptions r a t h e r transparent. F i n a l l y , we w i l l compare t h e model p r e - d i c t i o n s w i t h medium energy data.

I I

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The model

Suppose an e n e r g e t i c nucleon e n t e r s t h e nuclear p o t e n t i a1 we1 1. A nucl eon-nucl eon i n t e r a c t i o n w i l l lead t o nuclear e x c i t a t i o n s which can be d i s c r i b e d i n terms o f e x c i t e d p a r t i c l e s p above and holes h below t h e Fermi energy. The sum o f these e x c i t o n s i s c a l l e d m. More complex s t a t e s can be formed through p a r t i c l e - p a r t i c l e and hole-hole i n t e r a c t i o n s . A l l i n t e r m e d i a t e s t a t e s a r e grouped i n t o classes w i t h

(m=p+h, Q=($,~),E) where i s t h e d i r e c t i o n o f t h e momentum v e c t o r of a p a r t i c l e w i t h energy ^Eabove t h e Fermi energy. The energymomentum d i s s i p a t i o n can be d i s c r i b e d by t h e P a u l i master equation f o r t h e occupation p r o b a b i l i t y P(m,~i,~,t) a t t i m e t:

The t a s k i s now t o d e r i v e t h e spreading r a t e s A, t h e decay r a t e s 'A and t o s o l v e t h e system Eq.'s (1) which w i l l be done f o r t h e i n i t i a l c o n d i t i o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987241

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C2-272 JOURNAL DE PHYSIQUE

w i t h mo being t h e i n i t i a l e x c i t o n number, (o,o) t h e beam d i r e c t i o n , E t h e p r o j e c t i l e energy and a t h e absorption cross section.

abs

The decay r a t e i s c a l c u l a t e d from t h e time-reversed r e a c t i o n 2 ) and t h e spreading r a t e from nucleon-nucleon c o l l i s i o n s i n t h e Fermi gas. As previous authors we assume a f a c t o r i s a t i o n

From t h i s ansatz i t f o l l o w s t h a t t h e occupation p r o b a b i l i t y a l s o f a c t o r i s e s

The q u a n t i t y A i s t h e n t h e t i m e independent occupation p r o b a b i l i t y . From Eq. ( 1 ) t h e f o l l o w i n g recurrence r e l a t i o n f o r A i s obtained ( i f t r a n s i t i o n s t o s t a t e s w i t h small e r e x c i t o n numbers are neglected) :

The s c a t t e r i n g p r o b a b i l i t y gm i s d e r i v e d from t h e s i n g l e s c a t t e r i n g kernel g by

Here a,,, i s t h e p r o b a b i l i t y t h a t t h e p a r t i c l e w i t h (a,€) i s scattered3). I n t h i s con- t r i b u t l o n we have chosen two values: e i t h e r am=l which i s t h e f a s t p a r t i c l e assumption o r t h e o t h e r extreme t h a t a l l p a r t i c l e s a r e being scattered w i t h t h e same p r o b a b i l i t y . The cross s e c t i o n f o r a p a r t i c l e o f t y p e x may then be c a l c u l a t e d by

T h i s d e f i n i t i o n i s d i f f e r e n t from previous ones because i t c o n t a i n s an energy averaged decay r a t e w h i l e t h e p a r t i c l e energy- and momentumdependence i s i n t h e occupation p r o b a b i l i t y .

I 1 1

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Comparison w i t h Data and Conclusions

I n F i g u r e 1 t h e angle i n t e g r a t e d cross s e c t i o n s 4 ) a r e shown t o g e t h e r w i t h t h e model p r e d i c t i o n s . The c o n t r i b u t i o n s of i n d i v i d u a l steps are shown separately. Fig. 2 shows t h e double d i f f e r e n t i a l cross s e c t i o n s f o r t h r e e d i f f e r e n t e x i t channel energies. Again t h e c o n t r i b u t i o n s from d i f f e r e n t steps are separately shown. At t h e

s m a l l e s t energy t r a n s f e r (f=0.7) only t h e f i r s t i n t e r a c t i o n i s e f f e c t i v e . On t h e c o n t r a r y f o r l a r g e energy t r a n s f e r s and backward angle emission up t o f o u r r e s i d u a l i n t e r a c t i o n s a r e necessary, t o reproduce experimental data. The uriderestimation o f the data a t small angles i s due t o t h e r i g o r o u s way t h e P a u l i p r i n c i p l e i s t r e a t e d i n t h e nucl eon-nucl eon s c a t t e r i n g . A1 so d i s t o r t i o n s n o t present i n t h e s e m i c l a s s i c a l approach seemed t o be important. In conclusion we have presented a new f o r m u l a t i o n o f t h e e x c i t o n model o f n u c l e a r reactions. T h i s model accounts w e l l f o r experimental data a t medium energies on an absolute scale except f o r v e r y small emission angles.

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=

1

~ ( ~ , x ) < ( E ) * A ( ~ , E , o ) d ~ d a

t eq w i t h O;(E) = h i (m,E)

I

^P(m,t)dt.

0

Fig. 1: Energy d i f f e r e n t i a l c r o s s s e c t i o n s f o r t h e i n d i c a t e d r e a c t i o n . Data a r e f r o m Ref. 5. C a l c u l a t i o n s employing Eq. ( 7 ) f o r a m = l a r e shown as s o l i d curves. The t h i n s o l i d c u r v e s a r e t h e c o n t r i b u t i o n s f r o m d i f f e r e n t i n t e r a c t i o n s , t h e i r sum i s t h e t h i c k s o l i d curve.

Fig. 2: A n g u l a r d i s t r i b u t i o n s a r e shown f o r t h e i n d i c a t e d r e a c t i o n (see Fig. 1).

D i s t r i b u t i o n s a r e shown f o r t h r e e d i f f e r e n t energy b i n s g i v e n i n terms o f f r a c t i o n s o f t h e i n c i d e n t energy.

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C2-274 JOURNAL DE PHYSIQUE

I n t h i s f o r m u l a t i o n t h e s e r i e s Eq. (f) i s a mu1 t i p l e s c a t t e r i n g expansion s i m i l a r t o the approach by Glauber and Matthiae ) o r i g i n a l l y derived f o r much h i g h e r energies.

Both approaches take t h e a t t e n u a t i o n o f t h e i n c i d e n t beam e x p l i c i t y i n t o account.

However, t h e Glauber model t r e a t s only c o l l i s i o n s o f t h e beam p a r t i c l e s w i t h nucleons from t h e Fermi sea, which i s a good approximation a t h i g h energies, and w i l l t h e r e f o r e never lead t o an e q u i l i b r a t e d system. This f e a t u r e i s on t h e c o n t r a r y included i n t h e present approach. This can be seen i n Fig. 3 where Glauber model c a l c u l a t i o n s and e x c i t o n model c a l c u l a t i o n s are compared w i t h each other. Shown a r e t h e c o n t r i b u t i o n s o o f d i f f e r e n t numbers o f i n t e r a c t i o n s n = (m-m ) / 2 f o r two d i f f e r e n t nuclei. ~o"r t h e f i r s t i n t e r a c t i o n s t h e two models y i e l d n#arly i d e n t i c a l r e s u l t s where as f o r l a r g e i n t e r a c t i o n numbers t h e e x c i t o n model leads t o t h e compound nucleus. T h i s i s n o t t h e case f o r t h e Glauber model due t o i t s " t h e y never cane back" approximation. However, c o n t r i b u t i o n s of these steps t o t h e t o t a l y i e l d a t t h e energies under c o n s i d e r a t i o n (aNN-3hb) i s marginal, as already s t a t e d above.

Fig.

0,001 ( +-x exciton model

1 1 ~ ' ' 1 ' 1 1 ' 1 1 ' ' ' 1 1

0 5 10 0 10 20 30

interaction number

3: P a r t i a l cross s e c t i o n f o r i n c i d e n t protons of some hundreds o f MeV. The curves are o n l y t o guide t h e eyes. The Glauber model r e s u l t s are from Ref. 6.

References :

1) G r i f f i n , J.J., Phys. Rev. L e t t . 17 (1966) 478 2) Machner, H., Phys. Rep. 127 (1985) 309

3 Iwamoto, A., Harada, K., Nucl. Phys. A419 (1984) 472 41 Machner, H.; e t a1

.,

Phys. Lett. 1388 (1984) 39

5 ) Glauber, R.J., Matthiae, G., Nucl. Phys. 821 (1970) 135

6) Abul-Magd, A.Y., Friedman, W.A., Hiifner, J., Phys. Rev. C34, 113 (1986)