NUCLEON TRANSFER TO CONTINUUM STATES

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NUCLEON TRANSFER TO CONTINUUM STATES

A. Bonaccorso, D. Brink

To cite this version:

A. Bonaccorso, D. Brink. NUCLEON TRANSFER TO CONTINUUM STATES. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-265-C2-270. �10.1051/jphyscol:1987240�. �jpa-00226507�

JOURNAL DE PHYSIQUE

Colloque C2, supplbment au n o 6, Tome 48, juin 1987

NWCLEON TRANSFER TO CONTINUUM STATES

A. BONfiCCORSO and D.M. BRINK"

~ s t i t u d e Nazionale di Fisica Nucleare, Sezione di Catania, Corso Italia 57, I-95129 Catania, Italy

'~epartment of Theoretical Physics, 1 Keble Road, GB-Oxford OX1 3NP, Great-Britain

ABSTRACT - ^A semiclassical model is presented for the calculation of e-

-- nergy spectra of one nucleon transfer reactions to the continuum states.

It represents a natural extention of theories of transfer between bound states and it is able to explain the main characteristics of experimen- tal spectra.

In this paper we show how the the0 eviously used to describe one nucleon transfer between bound states "-8fcan be extended to the case in which the final state is unbound-Here we describe only the main points of the thgpfy and more details will be available in a forthco- ming publication .Only neutron transfer is discussed.

1. Transfer probability.

Initially the neutron is in a single particle bound state with angu- lar momentum li,mi,and energy Et ( ( 0 ) in the potential representing the projecti1e.A~ the two nuclei pass each other along a classical trajec- tory,the neutron is transfered to an unbound state of the target having angular momentum l6 mb and energy &+ (>O).The transfer happens at the distance of closest approach R=dl+d2 where d l and d 2 are taken along the x-axis and are the positions of the centers of the two nuclei.The relative motion has velocity v at the point of closest approach

(5Lfd it is taken along the z-axis.The transfer amplitude can be wrikten as a surface integral over a surface parallel to the ( Z - Y ) plane between the two nuclei.0n the surfacetthe wave functions for the initial and final states can be approximated by their asymptotic forms which are:

~ ~ ( z ) = ci(-ill yi hi1) (iTir) yLimi(+, q ) (1.1) and

C,gare the asymptotic normalization constants.C, is obtained by mat- ching the numerical solution of the Schradinger equation into Henkel

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987240

C2-266 JOURNAL DE PHYSIQUE

where L i s t h e dimension o f a l a r g e box i n which t h e wave f u n c t i o n eq. (1. 2) i s n o r m a l i z e d .

The t r a n s f e r a m p l i t u d e i s t h e n g i v e n by t h e approximate f o r m u l a

A,, = -

2nmv

where

-

(1. 1) and (1. 2)

t h e q u a n t i t i e s q, k,, k, i n eq. ( 3 ) a r e d e f i n e d by:

( Q +

+

11' = kf + Y: = k: - ^k: ; kl = -

R v

Y

The e x p l i c i t forms o f Qi,f a r e

-

~ l , ( x , k,, k , ) = - ^C, - ^e _YL. ($) ^(1.6)

-

c , ( x , k,, k , ) = - C, e i s ' s i n 6 1 - ^e - y 3 ' x ' y z m ( & )

( 1 . 7 )

7 3

where

A r3 ky k1 ir,

rz = k : + q 2 ; K~ E - - , - ( 1 . 8 )

kf ' k , kf Inserting eqs. (1. 6) and (1. 7) i n eq. (1. 3) one f i n a l l y gets f o r I+ t r a n s - f e r a m p l i t u d e

A(l,m,, &mi) = - ⁴ⁿⁱ R - ^{CiCe K m , - m ( q R ) x}

m v ¹ f

k, k2

w i t h Cos pi = -i - ; s i n pi = 2- ; c o s p, = -

7i 7, k, (1.10)

i q sin g, = -

k ,

and the transfer probability is

Introducing the density of final states

where k , L = n m , the transfer probability for going to a final state with energy E within a range dsf is

f

At this point we would like to comment briefly on some interesting features of the formulae obtained-First one notice that the transfer am- plitude eq. 4-3) is given by the overlap of the two functions 't;. c-Q ?;

eqs.6) and k7) which are in fact the momentum distributions df the neu- tron on the surface with respect to the final and initial nuclei-The other interesting point is that both in eqs. (C7) and U-11) the tern ~ 1 6 s c f ~ takes the place of the asymptotic normalization constant of the final bound state discussed in ref.(l-4).This result is useful because eq.&ll) can deal with the case of an unbound state which is a resonance as well as with a proper continuum state (above the centrifugal barrier).

If the incident energy per nucleon is not too high (Einc/u$lO MeV) transfer will take place manly towards resonance states-In this case one wishes to treat the resonance state in a way very similar to bound sta-

tes and to get a transfer probability for going to the mean energy of the resonance &,,rather than the probability eq.613) for going to an energy

E in a range deres . By integrating eq. (1. 13) over the region of the resonance having taken res

which is the usual one level Breit-Wigner form, one finally gets

JOURNAL DE PHYSIQUE

t h i s formula c a n be o b t a i n e d by a n a l y t i c a l c o n t i n u a t i o n from e formu- l a f o r t h e t r a n s f e r p r o b a b i l i t y between bound s t a t e , p r o v i d e d (tf

where C L was t h e a y m p t o t i c n o r m a l i z a t i o n c o n s t a n t o f t h e f i n a l bound s t a t e wave f u n c t i o n .

When t h e i n c i d e n t energy p e r n u c l e o n i s l a r g e r t h a n a b o u t 10 MeV t r a n s f e r w i l l p o p u l a t e p r o p e r continuum s t a t e and it i s i n t e r e s t i n g t o s t u d y

t h e energy spectrum do/ dcf which can be obtained integrating eq. i l . 13)

i n t h e way w e d i s c u s s i n t h e n e x t s e c t i o n . 2 . Energy s p e c t r a .

Eq- #-?3) b e i n t e g r a t e d o v e r i m p a c t p a r a m e t e r s a c c o r d i n g t o Bonac- c o r s o e t a l C Z P t o l e a d t o t h e c r o s s s e c t i o n

Pel i s t h e p r o b a b i i i t y o f e l a s t i c s c a t t e r i n g g i v e n by t h e s h a r p c u t o f f model w i t h s t r o n g a b s o r p t i o n r a d i u s Rs,ac i s t h e Coulomb l e n g h t parame- t e r .

Now it i s more c o n v e n i e n t t o make t h e s u b s t i t u t i o n

I s i n s l _f

l 2

11

_{l 2}

where S i s t h e e l a s t i c p a r t o f t h e s c a t t e r i n g m a t r i x d e s c r i b i n g t h e s c a t t e r r n g t h a t t h e t r a n s f e r e d n e u t r o n undergoes r e s p e c t t o t h e t a r g e t !h

n u c l e u s .

A f t e r a v e r a g i n g o v e r e n e r g y t h e above term becomes

( 1

where ~ ( l & , l t ) i s an e l e m e n t a r y t r a n s f e r p r o b a b i l i t y d e f i n e d a s

,=- 2 TI%

E q . ( 2 . 6 ) shows t h a t t h e c r o s s s e c t i o n f o r t r a n s f e r t o continuum s t a t e s r e c e i v e s c o n t r i b u t i o n s from two p r o c e s s e s : one is the e l a s t i c s c a t t e r i n g of t h e t r a n s f e r & n u c l e o n on t h e t a r g e t p o t e n t i a 1 , t h e o t h e r i s t h e absorp- t i o n ( c o r r e s p o n d i n g t o compound n u c l e u s f o r m a t i o n ) .

The o p t i c a l model s c a t t e r i n g a m p l i t u d e ^<S, * > can be o b t a i n e d from an o p t i c a l model c o d e , b u t c a n a l s o be p a r a m e t r i z e d w i t h good a c c u r a c y a s

1

( ~ 1 ~= )1 + exp - ^{1, ) / a} 1 ^(2.7)

w i t h lg = KR - - 1 ,where R is t h e dimension o f t h e t a r g e t 2

and K = FK

h

i s t h e a s y m p t o t i c wave number f o r t h e motion o f t h e n u c l e o n w i t h r e - s p e c t t o t h e t a r g e t .

The d i f f u s n e s s p a r a m e t e r a c a n be chosen c o n s t a n t b u t i n o r d e r t o ob- t a i n b e t t e r agreement w i t h e x p e r i m e n t a l r e s u l t s it i s a p p r o p r i a t e t o have a depending on t h e f i n a l e n e r g y Using e q u a n t i o n s ( 2 . 4 ) and ( 2 . 7 ) i n ( 2 . 5 ) , t h e e n e r g y s p e c t r a €& ( o r e q u i v a l e n t l y on Ld ^{) -} d 6 / d e k

c a n be c a l c u l a t e d f o r t r a n s f e r o f a n e u t r o n from a bound s t a t e w i t h e l n e r g y Z; and a n g u l a r momentum 1; i n t h e p r o j e c t i l e t o a f i n a l continuum s t a t e w i t h e n e r g y Eg i n a r a n g e d E y i n t h e t a r g e t . F o r each r a n g e o f t h e f i n a l energy we t a k e an i n c o h e r e n t sum of probabilities f o r t r a n - s f e r t o s t a t e s w i t h d ~ ~ & n i t f i 6 a n g ~ l a ~ 0 y o m e n t u m . F i g s . ( 4 ) and ^{( 2} a r s p e c

r a £ t h e r e a c t i o n P b ( 0 , 0 ) Pb a t 500 MeV and f o r 'BTi(gOAr, ' 9 A r ) B 5 T i a t 1760 MeY.The a b s c i s s a i s d e f i n e d a s

E C i s t h e t o t a l W i n e t i c e n e r g y i n t h e l a b o r a t o r y f r a m e - E i n c i s t h e pro- j e c t i l e i n c i d e n t e n e r g y i n t h e l a b o r a t o r y and Q = &: - ^{&, i s} t h e

Q-value o f t h e r e a c t i o n ( Q < O f o r t r a n s f e r t o unbound s t a t e s ) .

An i n t e r e s t i n g f e a t u r e o f t h e s p e c t r a i s +&at themaxim c o r r e s p o n d t o a f i n a l e n e r g y

EE = E i n c + Q o p t

w i t h 1

l Q o p t I = - ₂ ^mv2

where 1 / 2 -4 i s t h e i n c i d e n t energy p e r . n u c l e o n a t t h e i n s t a n t o f t r a n s f e r . T h i s r e s u l t comes from t h e f a c t t h a t e q . ( 2 . 5 ) h a s a maximum where q, h a s a minimum and t h i s happens when

l a . - a f l = -mv2 1 2

T h i s p r o p r i e t y o f t h e s p e c t 7 y ) o b t a i n e d from e q . ( 2 . 5 ) i s i n agree- ment w i t h e x p e r i m e n t a l r e s u l t s where it h a s been s e e n t h a t t h e maxi- ma do n o t c o r r e s p o n d t o f i n a l e n e r g i e s E&=Einc+Qgg where%s t h e ground s t a t e t o ground Q-value, b u t a r e s h i f t e d towards lower e n e r g i e s - T h e e x p l a n a t i o n o f t h i s phenomenon i s t h a t f o r t h e i n c i d e n t e n e r g i e s p e r n u c l e o n we a r e concerned w i t h ( ~ i n c / A 20-100 MeV) t h e nucleon l e a v e s

1

t h e p r o j e c t i l e w i t h an e n e r g y a i + =; mv2 which i s always l a r g e r

L

t h a n t h e b i n d i n g e n e r g y o f t h e l a s t occupied s h e l l i n t h e t a r g e t and t h e - r e f o r e t r a n s f e r p o p u l a t s s t a t e s i n t h e c o n t i n u u m , s i n c e i n t h i s way t h e b e s t matching c o n d i t i o n s a r e a c h i e v e d .

C2-270 JOURNAL DE PHYSIQUE

In table 1 are shown the peak's values and widths of some spectra cal- culated according to eq.(2.5) using the parameters of ref.5.

Two cases are considered for the diffusness a contained in the para- metrization eq. ( 2 . 7 ) of the scattering amplitude. The widths varie betwee

20 and 60 MeV and they show a llnear increase with the incident energy .Peak's values seem to decrease for very high incident energies-The use of an a depending an energy gives an overall increase in the yields of the spec- tra.This is due to the inclusion of final states with larger l4 which in formula (2.5) s reflected by the term pi

( 5 f

[.

large values

.

Our formula for the cross section . d 6 / d ~ ~ although very simple can give a detailed description of the main characteristics of energy spectra for one nucleon transfer reactions leading to final unbound states.It is interesting than both resonances and proper continuum states can be trea- ted in the same way.

One interesting aspect of transfer reactions to the continuum is that they are in competition with fragmentation in the range of m um to high

(20-80 MeV/u) incident energy.Inclusive experimental spectra 7% contain both contrikutions and our method could help in separating the transfer part.

F I N A L ENERGY C M e V I F I N A L E N E R G Y C M e V l

Table 1 Peak's values and widths of calculated energy spectra

- References

1 ) L.Lo Monaco and D.M.Btink, Jour-Phys. GI 1 (1 985) 935

2) A.Bonaccorso, D.M.Brink and G. Piccolo, Nucl-Phys. A441 (1 985) 555 3) F1.Stancu and D.M.Brink, Phys.Rev C32(1985) 1937

4) A.Bonaccorso, D.M.Brink and L.Lo Monaco, Jour.Phys. G (1987) in press 5) A.Bonaccorso and D.M.Brink, to be published in Phys-Rev C

6) S.Wald et al., Phys-Rev. =(1985)894

7) M.C.Mermaz et al., Z.Phys.A, Atomic Nuclei, =(1986)217