DIRECT REACTION MECHANISMS : ONE-STEP AND MULTI-STEP PROCESSES

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DIRECT REACTION MECHANISMS : ONE-STEP

AND MULTI-STEP PROCESSES

K. Low

To cite this version:

JOURNAL DE PHYSIQUE Colloque C5., supplément au n° \\, Tome 37., Novembre 1976., page C5-15

DIRECT REACTION MECHANISMS : ONE-STEP AND MULTI-STEP PROCESSES K.S. Low

Departement de Physique Nucleaire, CEN Saclay, BP2, 91190, Gif-sur-Yvette, France

Résumé.- ',es principes de base de la théorie des réactions directes sont revus dans le cadre de leur" application aux réactions induites par ions lourds. Il a été tenté de présenter une description cohérente de tous les mécanismes de réaction impliqués dans la diffusion inélastique et les réactions de transfert d'un et de quelques nucléons. Les causes de l'échec de la théorie au premier ordre sont discutées. Les désaccords observés sont généralement supprimés en incluant les termes d'ordre supérieur. La convergence rapide des termes corrobore l'application des théories des réactions directes à cette classe de réactions induites par ions lourds.

Abstract.- The basic framework of direct reaction theory in the description of heavy ion induced reactions will be reviewed. Attempts will be made to obtain a coherent picture of all the reaction mechanisms at work in inelastic scatterings and the transfer of one and a few nucleons. The nature of the failure of the first order theory will be discus-sed. The inclusion of higher order effects generally resolve the discrepancies reported so far. The rapid convergence of such higher order effects tends to support the applica-tion of direct reacapplica-tion theories to this class of reacapplica-tions induced by heavy ions.

1. INTRODUCTION

The title of this talk coincides with the stan-dard technique used to obtain solutions to a set of coupled integro-differential equations in direct reaction theories for inelastic or transfer reactions induced by the collision of two heavy ions. One would hope that the inter-actions responsible for the coupling between the different solutions, or channels, will be small and thus a single step perturbative treatment will be sufficient. In the past few years [l] , however, various experimental data have been accumulated which seemingly indicates that such channel couplings are rather strong. In particu-lar, transitions to low lying collective states are strongly affected by inelastic excitations. Thus, one step Distorted Wave Born Approximation (DWBA) is extended to include such inelastic excitations and treated to all orders in the coupled channel formalism. This results in the Coupled Channel Born Approximation (CCBA) which still treats the transfer process as a single step. Such asymmetric treatment of the inelastic and the transfer processes causes suspicion again if this is not the culprit of part of the

"malaise" observed. One has particularly in mind various cases of strongly populated one nucleon transfer reactions, which one would expect to occur in a single transition but which has not

been satisfactorily described by the DWBA. Whether this failure of the DWBA can be remedied by CCBA or whether one must consider the transfer coupling matrix elements as strong and thus taken to all orders will be the theme of the present talk. In the process, we will also exa-mine a vital question in the basic framework of direct reaction theories. It has never been determined directly whether the usage of non-orthogonal channel basis can give rise to erro-neous informations. Therefore, the basic coupled reaction channel (CRC) theory will be first reviewed and the origin and consequences of the non-orthogonality terms examined. The validity of the DWBA together with other practical genera-lizations of the DWBA will then be discussed. In particular, for coherence, we will concentrate on the comparison of various aspects of these

48 theories applied mainly to the reactions of Ca induced by 0. Therefore, inelastic scatterings, transfer of one, two and multi-nucleon reaction data and analyses will be discussed. Various deductions and general features of such reactions will be drawn.

2. DIRECT REACTION THEORY

The basic premise that we will adopt here is the theory of direct reactions which has been

applied rather successfully in the past two decades such that for light ion induced reactions. The transition

from the analyses of light ion to heavy ion reac- tion data has produced a number of numerical diffi- culties which are slowly but surely overcome in recent years. The formalism of' the direct reaction theories are rather well known [2], but to facili- tate our discussions on the nature of the DWBA as well as the origin of the non-orthogonality term,

it will still be useful to review the basic equations involved.

2.1. Coupled Reaction Channel Equations.- For an incoming channel h = a + A, we select for discus- sion a few outgoing channels B = b + B, such as inelastic scattering and the transfer of a few nucleons. These channels have strong overlaps with the original elastic channel and presumably are direct in nature. Thus, we seek a solution to the time independent Schroedinger equation

with the total wave function expanded in these few channels

where the standard notations for the reaction A(a,b)B is adopted [2,3]. For simplicity, we write down explicitly only one extra outgoing channel B.

In terms of the different partitions, we write the total hamiltonian as follows

- -

whereby

5

aaY

0 and O

-

are eigenfunctions of

B b

the intrinsic hamiltonians h A' ha, hB and hb respectively. A very important assumption is now made that one can replace the interaction between the two heavy ions by a local effective optical potential. One can think of extracting from a time dependent Hartree Fock solutions such an effective potential, although its procedure is neither obvious nor trivial at the moment especially if an individual channel has to be identified. On the other hand, folding procedures based on nuclear matter distribution have been attempted in recent years. We will not go into the details of any of these as they might be covered by Prof. Brink in his talk. We adopt a phenomenological approach requiring that these potentials reproduce the elastic scattering data in the zeroth order. Thus, we define effective optical model potentials

xa(ra) U a ( r a > r c4

1

°A@a> Y and a ( 4 ) -.,

Subsequent discussions will indicate how a modi- fied effective optical potential must be used in order to improve on the description of the data. With a local effective potential, one can now derive the following coupled reaction channel

(CRC) equations xB(rB) <@ O

I

1

Vij -UBl @B@b> A T ij *g

.

and " (5\

where E = E-E -E and E = E-E -E and E

A a

B

B b A' 'a'

cB, eb are the eigenvalues of h A, ha, hB and h b respectively.

The first terms on the right hand sides in the above equations contain the usual residual inter- actions responsible for the transitions from channel a to 8 and from $ back to a. The second terms containing <O @

I@

@ > obviously arise due

A a B b

to the non-orthogonality of the basis states. For inelastic scattering, the basis states are rear- rangements of the intrinsic wave functions of one particular nucleus and can therefore be chosen to be orthogonal and thus these terms do not exist. For transfer reactions, these non-

orthogonality terms are usually neglected and the importance of these correction terms have so far not been directly evaluated [ 3 , 4 ] . It should also be mentioned that only a few attempts to solve the above coupled equations exactly including transfer reactions have been reported for light ion reactions [5]. In all cases reported, non- orthogonality correction terms are neglected.

DIRECT REACTION MECHANISMS C5-17

from the boundary conditions which one imposes on the coupled equations. Asymptotically, there is an incoming wave (and an outgoing wave also) in channel a only with the outgoing waves in all the other channels fed by the couplings to this elas- tic channel. Thus, one expects that the elastic cross sections defined by the diagonal potential U is large and one can neglect the second,order feed back from the non-diagonal term. Thus, from the solution'of the homogeneous equation

with the boundary condition that

-. -

one obtains the scattering amplitude for the elastic channel which occurs due to both the nuclear and Coulomb parts of the diagonal poten- tial.

On the other hand, for the inelastic or transfer channels 6, the solution can now be obtained from the uncoupled inhomogeneous equations, knowing

x

of the elastic channel

with the asymptotic condition that

T o a n

CQB can also be expressed in the more familiar integral form proportional to

where xO* (fB) is the time reversed regular solu- tion of the homogeneous equation

x;(rB)

[T +U -E

]

-

=

B

B

B

r B 0

satisfying the asymptotic boundary condition that there is an incoming wave in channel

B.

The transition from a to

B

occurs solely from the residual interaction (C V

-

U), which by

ij

its name implies that its effects are smaller than the diagonal potential U. The propagation of the elastic channel wave function has been governed by the diagonal potential U and the channel wave function is just given by x (r ); On the other

a a

hand, the propagation of the channel

B

is also governed by the distorting potential UB after the transition form a to

B

induced by the residual interaction. But it must not be confused that the

channel wave function

x

is equal to Xi, which is

B

the regular solution of the diagonal potential U 6' Thus, the physical basis for DWBA as a perturba- tion method will be satisfied if the residual interaction causing transitions between channels are small, and in particular, when second or higher order feed back from B to a can be neglec- ted. In that case, the residual interaction acts once only causing the transition from a to 6.

2.3. Exact Coupled Reaction Channel Solutions.- If the residual interaction is strong, the full coupled channel solutions must be obtained. In the standard method of solving these coupled equations, one obtains as many independent solu- tions as the number of coupled equations [6]. This is a rather time consuming procedure and a well known iterative procedure is preferred [7].

In this method, the N coupled equations are solved as uncoupled successively with the inhomogeneous terms calculated with solutions of other channels obtained in previous steps until convergence is obtained. Thus in this method, DWBA solutions are obtained at the first iteration and higher order solutions from the other iterations. To speed up this procedure and also to eliminate the possibi- lity of divergence, the method of the Pad6 approximant can be used

171.

It is clear that if the coupling is strong, the angular distribution of the elastic channel will be drastically different between DWBA and CRC calculations if the same effective optical poten- tial is used in both cases. The simple procedure of a

x2

fit to the elastic scattering data to obtain an optical model potential will have to be generalized therefore to include a fit to the inelastic or transfer data in the CRC approach. However, unless all reaction channels are coupled at the same time, which is unfeasible, different optical model potentials will be generated when different outgoing channels are coupled. This is rather unsatisfactory as we can no longer identify an effective interaction between two given nuclei scattering at a given energy.

C5-18 K.S. cular channel. This is then compared to the results obtained from an exact solution of the coupled reaction channel equation using the same optical model potential. The difference observed between

the two gives us a measure of the breakdown of the simple DWBA procedure. On the other hand, we can selectively pick out a few strong channels and treated in CRC in order to extract a general optical potential. Which reaction channels are to be included in this procedure will therefore be also an important part of the present review.

3. HEAVY ION REACTION

-

DWBA AND CRC COMPARISON 3.1. Overview of Experimental Data : 160

+

4 8 ~ a reactions.- To get ourselves oriented to the kind of discussion that we intend to pursue further, it will be helpful to look first at Fig. 1 for the reactions of 4 8 ~ a induced by 56 MeV 160

181.

This figure shows the angular distributions for cross sections of individual fragments which.have been classified as quasi-elastic, apparently due to the close similarity in kinematics as well as strong overlap with the intrinsic states of the original elastic channel. Each fragment consists of a sum of all the states, mostly not resolved experimentally, up to about 10 MeV in excitation energy. Except for the very strong 1 5 ~ and 170

channels, which have "bell-shaped" angular distri- butions, most of the angular distributions are forward peaked.

Assuming that these reactions are peripheral in nature, Henning et aZ. [8] extracted a mean par- tial reaction cross section distribution from various DTJBA analyses of several strong transitions and the results are shown in Fig.2. The area under the curve is therefore adjusted to be equal to the experimental sum of 136 mb. This is then compared to the absorption or reaction cross section extracted from an optical model fit to the elastic scattering data. It is seen that direct reaction cross section accounts for a significant portion of the total reaction cross section at the nuclear surface. The question is then whether one can still treat these large cross sections in a one step Born approximation.

Furthermore, DWBA analyses also fail to describe the data of a few of the individual channels. Thus, some weakly excited states in the one nucleon transfer reactions to 1 5 ~ and "0 are not described by the DWBA. Similarly, the rather

F I I , ~ , I , I , I ~

0" 10" 2d 30° 40° 50° 60'

'lob

Fig.1

-

Angular distributions of individual channels integrated over energy for the 160+48~a induced reactions at 56 MeV incident energy [ B ]

.

DIRECT REACTION MECHANISMS

strongly populated "C channels are also not pre- dicted well by DWBA. In the two nucleon transfer reactions to 180 and 14c, 'DWBA also fails to account for the absolute magnitudes of these cross sections as well as for the shapes of the angular distributions. Further, one can also expect that the transfer of three of five nucleons and in ~articular, "exotic" reaction like those of "C channels, may not occur in a single step. In all cases, therefore, one would like to find out the reaction mechanisms that cause the transitions to these channels.

In the following, we will therefore confine our discussion mainly to these reactions induced by

160 on 4 8 ~ a taken at Argonne [8], being the most complete set of experimental data for such quasi- elastic reactions to date. O n the other hand, we will supplement our discussions here and there with results from other reactions but still

drawing inferences to be applied to this particular set of reactions. In the comparison of the DWBA and CRC results, only the two strongest reaction channels will be discussed. These are the inelastic and the transfer of one nucleon reaction. The channels involved are shown in Fig.3. Note that for uniformity, we have also identified the analy- ses of inelastic scatterings as CRC analyses rather than the more commonly used name of coupled channel analyses.

Fig.3

-

Channels investigated in the comparison of DWBA and CRC analyses. The inelastic and the transfer channels are treated separately.

3.2. Inelastic scattering.- For inelastic scattering, the macroscopic model is used [6]. The collective form factor is defined by the defor- mation parameter BE, a measure of the density perturbation from the spherical shape. Thus,

au

V i j - U = B 2 R - Y

ar Em (14)

which is generally taken to be complex, together with a Coulomb term. There is no individual

experimental angular distributions reported in comparing DWBA and.CRC results, it will not be inappropriate to use deformation values averaged from those extracted in light ion analyses.

In Fig.4 are shown the results of DWBA and CRC calculations for the first excited 2' and 3- of 4 8 ~ a as well as the elastic scattering. The 2+ showed a more distinctive interference dip near the grazing angle than the 3- state. This is due to the opposite signs of the Coulomb and nuclear form factors which cancels more strongly for the

2' than the 3-, the Coulomb form factor being -9.- I

r in radial form. This effect can be seen

more clearly in Fig.5 where we show the partial reaction cross section for one magnetic substate of the 2+ state. Thus, the nuclear part is short ranged and peak at the surface. The Coulomb part is also peaked at the surface but is longer in range. Thus, integration to as large a distance as 50 fm and up to 300 or more partial waves are needed to take into proper account the Coulomb

t

56

MeV

I

K.S. LOW

excitation effects. However, this is not much of a problem due to the extremely smooth nature of the matrix elements and interpolation techniques can be applied.

-

I 6 'I6

48

,",

4 8 ~ o (

0,

01

ca

8 '

\

:

; 1 1 I *

2'

CRC

----

.NUCLEAR

...--

COU

LOM

B

Fig.5

-

Destructive cancellation of Coulomb and nuclear amplitudes as evidenced by the dip at k=3I in the summed amplitude for magnetic sub- state X=O.

Looking next at the comparison of DWBA and CRC results, it is seen that the angular distributions are not significantly different except for the tail at the larger angles. However, the overall cross sections are increased by about 35 to 40 %

in the strsng coupling limit for both states. Thus, even at these relatively small deformation parameters of

B2

= 0.15 and

B3

= 0.20, DWBA could give information in difference by as much as 40 %.

To further investigate the differences between

DWBA and CRC, we show in Fig.6 the partial reac- tion cross sections for individual incident partial wave Ra. It is seen that in the interior, where the nuclear part of the form factor domina- tes, both the elastic and inelastic reaction cross sections are increased due to multiple scattering compared to the one step DWBA results. In the outer region, the Coulomb excitation has the opposite effect. However, the net effect in the interior is more important with partial cross sec- tion almost doubled at .9 = 26. This is also evident in Fig. 4 where the outer tail of the angular distributions, which is the most sensitive to the smaller partial waves, are drastically different between the DWBA and CRC results. This drastic modification of even the elastic channel indicates that when explicit coupling of inelastic

channels are made, the optical potential must now be obtained with a procedure different from the one we mentioned earlier. The method of

x2

fit of the data for extracting a potential must now be made not only with the elastic but also with the inelastic scattering data, as mentioned earlier.

1

(incident)

Fig.6-Comparison of DWBA and CRC partial reaction cross sections of elastic, 2' and 3- inelastic scatterings as a function of incident channel R values. The inset shows the total and elastic

scatter;-ng cross sections.

3.3. One nucleon transfer reactions.- We now proceed to examine the differences between DWBA and CRC for one nucleon transfer reactions, which are the next strongest group of populated channels. It will, however, be appropriate to summarize first some of the important refinements found necessary in making the transition from analysing light ion to heavy ion reaction data.

3.3.1.

Form

factor : nuclear and Coulomb terms.- For transfer reactions to the channel B, the Hamiltonian is partitioned in the notations of Fig.7 into two different forms

Post :

1

_Vij

_-

_uB

_{= VXb + VbA}

-

_UB

iab

Prior :

1

Vij

-

Ua =

vxA

v b ~

-

U a iaa

j c A

DIRECT REACTION MECHANISMS C5-2 1

and t h a t t h e n u c l e a r p a r t of VbA = U = U This a 6'

l e a v e s o n l y t h e n u c l e a r p a r t of Vbx o r VxA which can t h e n be assumed t o be 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 of t h e t r a n s f e r r e d nucleon x i n t h e c o r e s b and A r e s p e c t i v e l y . However, i t h a s s i n c e been shown by DeVries, Kramctr and S a t c h l e r [9]

t h a t a d i s c r e p a n c y a s l a r g e as 40 % c a n e x i s t between t h e two d i f f e r e n t forms of t h e i n t e r a c t i o n f o r one nucleon t r a n s f e r r e a c t i o n s . I n o r d e r t o minimize t h e d i s c r e p a n c y between t h e s e two forms, t h e y show t h a t one must r e t a i n e x p l i c i t l y t h e Coulomb p a r t of each of t h e s e t:hree terms. Thus, we w r i t e where t h e s u p e r s c r i p t s N and C r e f e r t o t h e n u c l e a r and Coulomb p a r t s of t h e i n t e r a c t i o n r e s - p e c t i v e l y . Note t h a t i n t h e coupled r e a c t i o n c h a n n e l e q u a t i o n s ( 5 ) , we have chosen t o u s e t h e been a t t e m p t e d w i t h v a r y i n g d e g r e e of s u c c e s s t o i.nclude t h e s e r e c o i l e f f e c t s [16]. However,

i t h a s a l s o been shown t h a t an e x a c t transforma- t i o n of t h e c o o r d i n a t e s can be performed f a i r l y e a s i l y i f one r e a l i z e s t h e smooth n a t u r e of t h e q u a n t i t i e s b e i n g transformed [I 71. Numerical i n t , e r p o l a t i o n t e c h n i q u e s can b e i n c o r p o r a t e d , r e d u c i n g t h e problem of huge c o r e s t o r a g e r e q u i r e d f o r such a multi-dimensional c o o r d i n a t e t r a n s f o r m a t i o n . 3 - 3 - 3 -

EWBA2-GEC-and-zon:of:thog!!nf1it~-corf

ea-

tion:;.- The r e s u l t s of DWBA and CRC c a l c u a l t i o n s

---

using; f i n i t e range form f a c t o r s which i n c l u d e r e c o i l e f f e c t s e x a c t l y a s w e l l a s t h e Coulomb terms e x p l i c i t l y a r e shown i n F i g . 8 [18]. A l l t h e ca1cu:lations a r e performed i n t h e p r i o r form which i s 2.1 Z l a r g e r t h a n t h e corresponding p o s t f o r m i n DWBA. Without t h e Coulomb terms i n c l u d e d e x p l i c i t l y , a p o s t - p r i o r d i s c r e p a n c y o f 1 3 . 5 % w i l l be i n c u r r e d . p r i o r form of t h e i n t e r a c t i o n f o r b o t h terms s o The dashed c u r v e s a r e t h e r e s u l t s of t h e f u l l t h a t t h e non-orthogonality c o r r e c t i o n h a s t h e CRC an.nlyses w i t h o u t t h e n o n - o r t h o g o n a l i t y c o r r e c - same formal s t r u c t u r e a s t h e main d i a g o n a l term

t i o n (IJOC) terms. Thus, i t g i v e s r i s e t o a n f o r numerical convenience.

3.3.2. R e c o i l e f f e c t s . - The coupled e q u a t i o n s i i v o l v e two independent s e t s of c . o o r d i n a t e s ( r a , r ) and ( r l , r 2 ) . The f i r s t a r e t h e channel

6

c o o r d i n a t e s whereas t h e second s e t a r i s e s from t h e s t r u c t u r e form f a c t o r i n terms of bound s t a t e c o o r d i n a t e s . I n heavy i o n r e a c t i o n s , t h e e x a c t t r a n s f o r m a t i o n of t h e c o o r d i n a t e s must be p e r f o r - med and normally goes w i t h t h e name ,of e x a c t f i n i t e range a n a l y s e s . An approxinlate t r e a t m e n t of t h i s c o o r d i n a t e t r a n s f o r m a t i o n which i s p h y s i c a l l y understood a s t h e no-retcoil a p p r o x i - mation [lo] normally y i e l d s an e r r o r a s l a r g e a s a f a c t o r of f o u r f o r b o t h one [I

11

and two

[12,13,14] nucleon t r a n s f e r r e a c t i o n s . This d r a s t i c d i f f e r e n c e is r e l a t e d t o t h e s t r o n g p e r i p h e r a l n a t u r e of t h e r e a c t i o n and t h u s any approximation made f o r t h e coordinai:es w i l l r e s u l t i n a s i g n i f i c a n t displacement of t h e r e g i o n of i n t e r a c t i o n . For one nucleon t r a n s f e r r e a c t i o n a t s u f f i c i e n t l y h i g h energy above t h e Coulomb b a r r i e r , DeVries and Kubo [I 51 has shown t h a t

e r r o n e o u s a n g u l a r d i s t r i b u t i o n s can even r e s u l t due t o t h e l i m i t a t i o n s of a n g u l a r momentum s e l e c t i o n r u l e s imposed by t h e n o - r e c o i l approxi- mation. Various h i g h e r o r d e r c o r r e c t : i o n s have

a n g u l a l - d i s t r i b u t i o n more forward peaked t h a n t h e cox-responding DWBA r e s u l t s . Note t h a t we have n c ~ t included t h e experimental d a t a p o i n t s

i n t h e s ; e f i g u r e s f o r y h i c h t h e DWBA reproduces w e l l tile G.S. 7/2- b u t i s s h i f t e d by about 5' t o t h e backward a n g l e s f o r t h e 3.09 MeV 312- a n g u l a r d i s t r i b u t i o n s . Thus, t h e 712- CRC r e s u l t s w i t h o u t t h e NOC no l o n g e r reproduces t h e e x p e r i - mental a n g u l a r d i s t r i b u t i o n a l though t h e 312- i s now s l i g h t l y b e t t e r d e s c r i b e d by CRC w i t h o u t

NOC. However, when f u l l CRC with NOC i s performed, t h e r e s u l t s a s shown by t h e s o l i d curve i s l e s s forward peaked and l a r g e r a t t h e backward a n g l e s t h a n t h ~ ? dashed CRC w i t h o u t NOC c u r v e s . Thus, t h e a n g u l a r d i s t r S b u t , i o n s a r e now n o t s i g n i f i c a n - t l y d i f f e r e n t from t h o s e of t h e DWBA a l t h o u g h t h e o v e r a l l magnitude i s i n c r e a s e d by 18 and 10 %

K.S. LOW

NOC, but reversed when NOC is included. To examine this further, one must look at the individual partiar wave contributions to the cross section. This is defined from the total reaction cross section as follows

In Fig.9 is shown the partial Q cross sections of the G.S. 7/2- state for the magnetic substate m = 4. The top part of the figure shows the

comparison between the CRC with NOC and those of the DWBA. A comparison is also made for the ratio of these partial R cross sections for the CRC with and without NOC to those of the DWBA at the

lower part of the figure. It is seen that for the partial waves smaLler than the grazing, CRC with NOC is larger than those without NOC. Though these cross sections are small, being removed by the imaginary part of the optical potential, it nevertheless implies that at small impact para- meter, non-orthogonality of the basis is very

large. At the larger impact parameters, CRC without NOC overestimates the transfer cross

sections. Thus, the two cross sections are equal

at an R very close to the grazing. This is also evident from the angular distributions of the CRC results with and without NOC which has a crossing point at the grazing peak. At this stage, the physical origin of this particular feature of the non-orthogonality term is not well understood.

Fig.7--Vector diagram of the coordinates and, the notations for reaction A(a,b)B.

56

MeV

Fig.8

-

Comparison of DWBA and CRC results with and without non-orthogonality correc- tion terms. The amplitudes c2s1c2s are respectively 1.5 and 0.9 for the 7?2- and 312- states giving peak cross sections in close agreement with experimental data of Ref. 8. f? [q(CRC)

/ y

(DWBAII x DWBA O CRC(no NOC) o CRC (with NOCI 25 , 30 35 1

DIRECT REACTION MECHANISMS

3.4. Validity of the DWBA.- Sunrmarizing our results in Table 1, we find that for the inelastic scattering, even a not too strong inelastic tran- sitions with total cross section of about 12 mb for the 2+ state, a difference of 36 % can be incurred if DWBA is used. Since most nuclei have deformation of this order, one can no longer easily extract the deformation values by simply normalizing the calculated results to the experi- mental data for inelastic scattering. Note also that the absorption cross sections of the elastic channel obrained from optical model or the coupled reaction channel is only different by 4.4 mb from the original of 1286.5 mb. However, we have alrea- dy mentioned that the shape of the elastic cross section is rather drastically affected. Thus, when coupling is made with inelastic channels, a simultaneous fitting to both the elastic and inelastic scattering data must be made.

On the other hand for the one nucleon transfer reactions, where the strong transitions are normally-less than or of the order of 10 mb as the G.S. 712

,

DWBA would incur a somewhat smaller error of about 20 % when spectroscopic infoma- tions are extracted. No difference in the shape of the angular distributions is incurred by

the UbBA. Similarly, the shape of the elastic cross sectionsare not significantly affected although the absorption cross sections are affected rather strongly.

To further understand the differences in the importance of strong coupling of the inelastic and the transfer of one nucleon channels, we show in Fig.10 the partial reaction cross sections of the

the transfer of a nucleon, which one expects to have been transferred to the outer surface. Note that the Q-value matching condition to the 712- state is already optimum and even if transfer to other Q-valued states may peak at a smaller R, the absolute magnitudes of these cross sections will be much smaller. Further realizing the strong absorption of the elastic channel as required by the fit to the elastic scattering data, we find that the partial inelastic cross sections is as large as the partial elastic cross sections at the smaller impact parameter. One step DWBA leads to an error as large as a factor 2 which has already been shown in Fig.6. At the larger partial waves, elastic cross sections beco- mes Rutherford and thus large and DWBA begins to be more reliable. In the case of transfer of one nucleon, it is at the worst only one-fifth of the elastic cross sections and perturbation theory. becomes rather appropriate. We have thus shown a rather strong dependence in the validity of the Born approximation for both inelastic scatterings and transfer reactions.

160

+

4 8 ~ a

56

MeV

DWBA

-

ELASTIC

cn

&."...A

'60

2

+

Fig.10

-

Partial DWBA reaction cross sections compared to the optical model partial elastic reaction cross section as a function of the incident channel 9. values, showing the R dependence of the validity of the DWBA. different channels compared to the partial elastic reaction cross sections of the different channels compared to the partial elastic reaction cross section. It is seen that inelastic channels involving the rearrangement of the intrinsic wave function peak at a smaller grazing 9. and thus occurs more inside the nuclear surface than

4. PRACTICAL GENERALIZATIONS OF DWBA

It has already been mentioned that the 312- state of 4 9 ~ c is not well predicted by DWBA, neither by CRC with or without non-orthogonality corrections. This brings us to an entire class of one nucleon transfer reactions, which cannot be described by DWBA, and by inference neither by CRC with coupling to the transfer channels. These cases fall into two main categories [20] :

i) transitions which have its experimental phase of oscillations different from DWBA predictions ; ii) transitions which have its grazing peak positions shifted with respect to the DWBA predic-

t ions.

In this section, we will therefore review various cases of one and multi-nucleon transfer reactions which are not described by DWBA. These require the inclusion of higher order effects in terms of simple generalizations of the DWBA. However, in all these cases reported, the methods used fall short of an exact solution of the coupled reaction channel equations.

K.S. LOW

examine three cases and we draw the conclusion that inelastic excitations are responsible for all the failures of the DWBA. In the past, such multi-step processes have been treated in the CCBA approach, with the inelastic transitions treated to all orders but the transfer step treated by the Born approximation. The transition amplitudes can then be shown to be of the form [21]:

This has the same formal structure as the DWBA transfer amplitudes given by Eq.(12) with the

xa

and

x0

replaced by the generalized xaa,and Xo

B

si3"

obtained from an exact solution of the coupled reaction channel equation for inelastic scatte- ring. In this form, the numerical extension from DWBA to CCBA can be obtained rather easily. The method for the solution of the coupled reaction channel equations for inelastic excitations are rather well developed [6,7] due to the simpler nature of the non-diagonal coupling potentials involved compared to the complicated coordinate transformations needed to generate the transfer coupling matrix elements. The numerical evaluation of the multi-dimensional integrals of the transi- tion amplitudes above is then a straight forward extension of those of the DWBA.

On the other hand, one has always been doubtful whether the asymmetric treatment of the two different types of transitions is valid [5].

Our discussion in the last section, however, provides the first solid indication that CCBA is in fact a rather good alternative to an exact solution of the CRC equations. The transfer step is shown to be at most 20 % in error but if full coupling must be made, the non-orthogonality correction term must also be included if correct angular distributions are to be obtained. Thus, Born approximation for this transfer step is a rather appropriate substitute. In contrast, the coupling between inelastic channels are rather strong but the exact solution of these coupled equations are not difficult. In the following, we will present various cases of one-nucleon transfer reactions analysed with CCBA. It should be mentioned, however, that in the iterative scheme we have outlined earlier, an exact solution of the CRC equation including transfer

consuming compared to those of the CCBA.

4.1 . I . _~o_rPjddgq__tr_a_n~i_t~o-n-

5%

!9_Fc_60_,!5N)-20r\re reaction.- In the reaction "~('~0, 15~)20Ne [22],

3

it is found that the 4 state is populated as

+

strongly as the G.S. and 2 of

ON^.

Since the direct population of this 4' state is "forbidden", having negligible admixture of g-shell component, it must have been excited inelastically through other states as multi-step processes. The results of EFR-CCBA calcualtions [23] with couplings of both G.S. rotational bands of "F and 'ONe are shown in Fig. I I. The relative magnitudes as well as well as the shape of the angular distributions are rather well predicted. Of particular interest is the phase change observed between the DWBA and CCBA results for both the G.S. and the 2' of 20~e. This foretells part of the solutions to the

discrepancies in other one-nucleon transfer reaction data observed in DWBA analyses.

ec.,.

(el)

Fig.11

-

Comparison of DWBA and CCBA results for 19~(160, 15N)20Ne data of Refs. 22 and 23.

DIRECT REACTION MECHANISMS

4.1.2. F'jlc-up and-gt~ippin~ reaction diffe- z~?g_cu?s.- A Brookhaven group [241 has recently reported that in the one nucleon transfer reac- tions induced by

I3c on 40'42944~a isotopes, the

angular distributions are out of phase between the pick-up and stripping reactions and could not be explained by DWBA analyses. In particular, the range of Q-values spanned by the different isotopes seems to indicate that Q-effect does not play a direct role in this discrepancy which occurs even for the very well matched reactions.

We have however performed CCBA analyses 1251 by including inelastic excitations of 4 0 ~ a to the one-particle one-hole 3- and 5- states. The results are shown in Fig.12. It is seen that in

40ca

+I3c

-

41

ca

I - ' 2 ~ .

,

3 9 ~

-t-

1 4 ~ .

ELab

=

68

MeV

1

%a

G.S.

712-

1

I

--

EFR- CCBA (INDIRECT)

'-'\I

Fig.12

-

Comparison of DWBA and CCBA analyses for the stripping and pick-up reactions of Refs. 24 and 25.

the case of the stripping reaction, the inelastic routes are about 3 orders of magnitude smaller and thus does not affect significantly the cross section compared to those of the direct route.

On the other hand, the indirect routes of the pick-up reaction are about a factor 10 smaller only giving rise to the strong difference as shown in the summed cross section. In particular, the oscillation of the angular distributions are distinctly out of phase with those of the DWBA.

The different behaviour of the same inelastic excitations in the same target nucleus on the pick-up and stripping reactions can be attributed to the differences in the kinematical matching conditions as well as geometrical properties of the single particle states. They are summarized in Fig.13 :

(i) Better Q-value matching conditions of the indirect routes in the pick-up reaction. In the stripping reaction, it is poorer matched.

(ii) Less bound particles in the indirect routes in the pick-up reaction compared to those of 'the direct. The reverse is true for the stripping reaction.

(iii) Stronger jz(R+1/2) transitions in the ground

state to ground state transition. The opposite is true for the stripping reaction.

The above three factors all served to enhance the importance of the indirect routefor the pick-

.. .

up reaction. However, despite the rather satis- factory result obtained by the CCBA at the more backward angles, the fit at the more forward angles are not completely adequate. This may be due to contributions from even higher excited states which we have not included in our CCBA analyses.

On the other hand, for the stripping reaction, we find that both CCBA and DWBA give equally good fit to the experimental data. Good agreement of the DWBA with data is in fact normally the case for strongly populated one-nucleon transfer reactions as is the case here. The striking ano- maly of the strongly populated pick-up reaction due to a fortuitous combination of various factors is we believe, the exception rather than the rule for strongly populated one-nucleon transfer reac- tions. It is also no small consolation to find that the absolute magnitude in the case of the anomalous pick-up reaction is not drastically changed between DWBA and CCBA analyses !

4.1.3. Regctio_zg-with different ~r_o_igctiles.- Another case of interest is the one-proton trans- fer reactions on 6 2 ~ i induced by 160 and I2c [26]. The results of DWBA and CCBA analyses are shown in Fig. 14. In the case of 62~i(12~, 'B) 6 3 ~ u reaction, inelastic excitation to the 2+ of 12c is very important affecting the peak position of thecross section. There is an enhancement of the forward angle cross section with a shift of the peak position by 5 O

to the forward angles. The absolute magnitude can also be different by as much as a factor two for these not so strongly populated one-proton trans- fer reactions. Note also the out of phase nature at the small angles between the direct and the summed CCBA angular distributions.

On the other hand, inelastic excitation to the 3- of 160 has relatively smaller effect on the 62~i(160, 15~)63~u reaction. Despite the fact that their deformation values are only a factor two different (-0.60 and 0.31 for the 2+ of "C and 3- of 160 respectively), a strong difference in the relative importance of the inelastic excitation in the projectiles is again due to a

combination of kinematical and geometrical fac- tors. These results provided a clue. to the problem encountered in getting consistent

soectrosco~ic values of the various excited states of '09gi from the (160,15~), ( 1 2 ~ , 1 1 ~ ) and

1 1 10

( B, Be) reactions [l l]

.

.The discrepancy observed is most likely due to the different importance of projectile inelastic excitations.

Lar\\

2

02

t 1

:.

,,,A E

-

I ,-I

',-\.,-.,

s

0, v

:"

--.

0

:

'k. kvm€CT ',, ', oc6

,.,

,

.

' OOL '.'

=,,

'

it, '\

',

om

'.

001

t

w m 3 0 L 0 5 0 w 7 0 8 0

Fig.14

-

Comparison of the relative importance of the indirect routes of the one-proton stripping ( 1 2 ~ , 1 1 ~ ) and (160,15~) reactions on 6 2 ~ i target [26].

DIRECT REACTION MECHANISMS

same intrinsic rotational band are coupled stron- gly and is exhibited rather "unambiguously" in the

4-

19~(160,15~)20~e reaction to the 4 state. For vibrational nuclei involving one particle, one hole type collective states, the nature of the particle and hole states involved and various kinematical effects govern the relative importance of multi-step processes. Thus, inelastic excita- tions in the same target nucleus can produce different effects in the stripping and the pick-up reactions. On the other hand, inelastic excitations in different projectiles can also have different degree of importance on the transfer reactions with the same target-residual nuclei pair, resulting in different spectroscopic informations extracted for the same final states.

The effects on the angular distributions are two fold. There could be a unilateral shift of the angular distributions to the forward angles for those with a bell shape. However, at the higher energies, with forward peaked angular distributions, this is transformed to become a change in the slope of the angular distributions. Also, the phase of the angular distributions can be altered and is more pronounced for the forward peaked angular distributions. All these phenomena are due to the different scattering phases of the indirect routes compared to those of the direct routes. We will not discuss the physical origin of this difference except to mention that Udagawa and Tamura [27] have made qualitative estimates obtaining satisfactory understanding of such a mechanism.

It is clear from these discussions that multi- step processes will be of particular importance when the direct transitions are weak, either due to nuclear structrual properties or dynamical matching conditions. In Fig.15 is shown the one- proton transfer (160, l5N) reactions [8,20] on various Ca isotopes taken at Argonne. It is seen thzt the fit by DWBA becomes the worst when the cross sections decreases with Q-value mismatch in the case of the G.S. 712- states. Thus, inelastic excitations of Ca or 160 could play a role on these reactions. On the other hand, it is known that the excited 312- states of Sc isotopes contain a significant admixture of multi-particle, multi-hole configurations. In particular, the first excited 312- state of 4 3 ~ c isotopes

contain a significant admixture of multi-particle,

multi-hole configurations. For example the

first excited 312- state of 4 3 ~ c is known to contain mainly 5p-2h configuration [28] and must come mainly through the parent 4p-2h configuration in 42~a. This explains why the angular distribu- tion is not correctly predicted by the DWBA. It must come mainly as a higher order process

through the 2+ of 42~a. In this respect, it should be mentioned that the shapes of the angular distributions of multi-step processes observed in the few cases studied earlier are more forward peaked than the corresponding direct D ~ A transitions. One is rather optimistic that analy- ses in terms of CCBA could resolve the anomalies observed in these reactions in DWBA analyses.

co ( ' 6 ~ . ' s ~ ) ~ c E,-,..56 MeV

16

Fig.15

-

Comparison of DWBA analyses of ( 0,

15?l) reactions on different Ca isotopes [~ef .20]

.

simplicity of DWBA, one can ask for more structu- ral. informations regarding the multi-particle, multi-hole configurations in these states in the more elaborate framework of CCBA.

4.2. Two-nucleon transfer reactions.- Theoreti- cal efforts in the study of channel coupling effects have in the past being concentrated to reactions that involve the transfer of two nucleons [29,30]. This is numerically simpler due to the spin zero nature of the interacting nuclei involved. Also, since these transfer cross sec- tions are normally a factor five or more smaller than one-nucleon transfer reactions, one hopes that the transfer step can be treated by the Born approximation retaining the strong coupling nature of the inelastic scattering. Such CCBA analyses have been made for nuclei in the deformed s-d shell for 2 0 ~ e [29] and Mg [31,32] isotopes, vibrational nuclei for Ge [33]. Ni and Mo [34], Sn [30,35] and Te [36] isotopes as well as for the heavier deformed rare earth nuclei [37,38]

.

Some of these analyses are carried out in the no-recoil approximation. It should not affect the basic conclusions derived from these analyses regarding the interference phenomena of multi-step processes. However, absolute magnitudes predicted by no-recoil calculations may be in error by as much as a factor four as shown by various groups [12,13,14] in the comparison to the exact finite range calculations.

We will in the following review two basic aspects of such two-nucleon transfer reactions. The first relates to interference phenomenon due to diffe- rent mechanisms and the second, the absolute magnitudes of these cross sections and the role played by the sequential transfer reaction mechanism.

4.2.1. Coulomb=nxlgeay,-direct-indirect interfe- g~c_e_-phenonte~~n.- For strongly collective states like the first 2+ of vibrational nuclei which are only weakly populated by a direct transition, interference with the indirect routes via the G.S. produces distinctive angular distributions. Since the indirect routes contains a nuclear and

interfere destructively with the nuclear part and constructively with the Coulomb part of the tran- sition amplitudes or vice versa. This aspect of the interference phenomenon has been discussed rather clearly in the talk of Ascuito and Vaagen at Nashville [39]. Of particular interest is the question of how the different terms would interfere in the pick-up and stripping reac- tions [35,38,40,41,42] involving a pair of vibrational nuclei.

In the following, we will discuss the analyses of ,the first definitive data [42] for the two- neutron pick-up and stripping reactions

16 18 74

76~e( 0% 0) Ge measured at Brookhaven. The exact finite range CCBA calculations are made with coupling of not only the target-residual nuclei

2' states, but includes also the much stronger transition to the 2' of 180. Such simultaneous coupling of both target-residual, projectile- ejectile has never been reported before but has been pointed out to be necessary by Glendenning and Wolschin [35] in order to reproduce the angular distributions of the G.S. to G.S. transitions.

In Fig.16 is shown the coupling scheme and the results for pick-up reaction in Fig.17. Of particular interest is the presence of an inter-

+

ference dip for the 2 state of 74Ge but not for the 2+ of 180 at the grazing angles. The population of the 2' of 180 is four of five times stronger than those to the G.S. and both comes mainly from the direct transitions. On the other hand, the direct transition to the 2+ of 7 4 ~ e is negli- gible. This transition corresponds to the removal o£ two neutrons from the G.S. BCS-vacuum of 7 6 ~ e

a Coulomb terms with opposite signs, these two

Fig. 6

-

ou lin scheme of the stripping, pick- parts will interfere differently with the direct _{up 71Ge( 1g & 7 6 0) $6 Ge reactions. Siaul taneous}

route. Thus whatever the relative sign may exist coupling is made for the 2+ states of the target-

DIRECT REACTION MECHANISMS

leaving 7 4 ~ e in an excited 2+ state, which is mainly a one-particle, one-hole type quadrupole vibrational ~onfi.~uration.' The population of this state must come mainly as a two-step process with first the removal of a neutron pair to the G.S. of 7 4 ~ e followed by the creation of a quasi-particle pair of the 2'. This fact is expressed in the incoherent addition of the f and g-shells components resulting in small direct cross section. Thus, the 7 4 ~ e 2+ cross section, being an indirect process through inelastic excitation, exhibits a characteristic Coulomb-nuclear inter- ference dip at the grazing angle much like those of the inelastic scattering discussed earlier. There is no constructive interference observed in this particular reaction between the direct and the indirect routes.

Fig.17

-

Results of EFR-CCBA analyses [~ef .42]. The 7 4 ~ e 2+ cross section comes mainly from the indirect trans<- tion via the G.S. and exhibits a characteristic Coulomb-nuclear inter- ference dip at the grazing angle.

For the stripping reaction as shown in Fig.18, the direct transition to the 2+ of 7 6 ~ e is signi- ficant since it can occur by the direct addition of two particles to the core of 74~e. This fact is expressed in the dominant contribution of the g-shell components to the direct transition amplitudes of the 2' state. This rather strong direct amplitude interferes destructively with the nuclear amplitude but constructively with the Coulomb part of the indirect routes. The result is a further enhancement of the interference dip at the grazing angle. Such a sensitive dependence of the shape of the angular distributions on the strength of the direct route gives rise to increasing interest of these two-nucleon transfer reactions as a spectroscopic tool for the study of pairing vibrations.

--

K.S. LOW

It should be mentioned here that without the simultaneous inclusion of the coupling of the 180

2

' state, the angular distributions of the ground

states of 74Ge and 7 6 ~ e will be shifted by about

+

4' more backwards and the 2 states will be less

~*

forward peaked. This will produce poorer agreement the experimental data. The enhancement of the forward cross sections as discussed in Ref. [35] occurs due to the rather large cross section of the

180 2+ state. The magnitude of the de-excitation amplitude is therefore of significance to the G.S. cross sections.

4.2.2. Absolute magnitudes of two-nucleon transfer reactions.- A common normalization

---

factor of 2.3 has been reported [42] in fitting the calculated results to the experimental data in the above two neutron transfer reactions. This underestimation of the calculated cross sections is a common problem encountered in such

reactions [12,14]

.

It is particularly serious in the case of two-proton transfer reactions, where a factor as large as 100 to 500 are commonly obtained [33,34,43]

.

One must therefore question the basic assumption we made that the two-nuclons are transferred together in a single step. This is particularly relevant when one realizes the relatively weak two-nucleon correlation, typically 1.5 MeV compared to the 7 or 8 MeV required to separate each of the two nucleons. One must therefore consider the possibility of a successive or the sequential transfer of the two nucleons in two steps. 4.2.3. Sequential transfer reaction process in second order DWBA.- For this purpose, the coupled

---

reaction channel equations are generalized to include an intermediate channel y. The solution of these equations with three coupled states have so far been attempted in second order DWBA for heavy ion reactions

[I

4,43,44,45]

.

Thus, from the elastic channel

x

of the intermediate one-nucleon transfer channel in the DWBA procedure described earlier. This

x

is then used to feed to the final

Y

two-nucleon transfer channel. Various comments have been made in the past regarding the possible

importance of the non-orthogonality terms [3,4] in this second order DWBA procedure. Our discussions earlier indicate that these non-orthogonality correction terms should be small.

A transition scheme is shown in Fig.19 for the 48~a(180, 160) 5 0 ~ a reaction including the sequen- tial transfer process through the intermediate

170 channel. In Fig.20 are shown the results in comparison with the experimental data of Petersen

et aZ. [46]. The sequential transfer cross section. [14,45] given by curves 3 is as large as

Fig.19

-

Successive and simultaneous transfer schemes for the 48~a(180, 160)50ca reaction.

the simultaneous transfer cross sections given by curves 1, which alone would underestimate the expe- rimental cross sections by factors of 8. The simul- taneous transfer cross sections given by curves 2 are obtained using a procedure[47] taking into account the residual interaction between the two transferred nucleons. The use of such extended shell model wave functions enhances the simulta- neous transfer cross sections by factors of two. The coherent addition of these two amplitudes, simultaneous and sequential, then gives rise to cross sections in rather good agreement in absolute magnitude with the experimental cross

sections.

Similar results showing the importance of the sequential transfer reactions are all predicted within a factor of two [45]. On the other hand, they reported that serious discrepancies still exist for various two proton transfer reactions and Fig.21 illustrates the problems encountered

[8,45]. Both the G.S. to G.S. transitions of the 4 2 ~ a 60,

14c)

4 4 ~ i and 4 8 ~ a (

'

60, 4 ~ ) 5 0 ~ i reactions are underestimated by large factors of

DIRECT REACTION MECHANISMS

On the other hand, the 48~a(160,14~)50~i reaction going to the excited '0 state at 7.19 MeV is better matched in Q-value and the predicted cross section is in rather good agreement in absolute magnitude with the experimental results. This provides an important hint that there may be other mechanisms at work for these poorly Q-matched reactions. We are rather optimistic that more extensive coupling of the channels, especially highly excited inelastic channels may eventually provide an answer to this problem of the absolute magnitudes of two-proton transfer reactions.

2.00 Elob = 50 MeV

Fig.20

-

Curves marked 1 and 2 are respectively first order DWBA results for simultaneous transfer of two neu- trons using simple or more sophisticated wave functions. Curves marked 3 are results of second order DWBA calcula- tions for the'sequential transfer processes. The sum of these first and

second order-results, curves 2 and 3, are

shown as curves 4 [14].

Fig.21

-

Results of analyses of two- proton transfer reactions (simultaneous as sequential processes included) illustrating the problems encountered in the absolute magnitudes of these reactions.

5. OTHER REACTION PROCESSES.

The study of two-nucleon transfer reactions indicated that sequential transfer process is significantly large and can be treated by second order DWBA. A logical question that arises is how large is the third and higher order processes. In this section, we would like to examine the reaction processes that leads to the transfer of three or more nucleons.

5.1. (I60,l5c) reaction as a third order

C5-32 K.S. LOW

vast number of possible combinations of transi- tions that can lead to the

15c

channel. In parti- cular, this reaction could occur first by a one-proton stripping to 1 5 ~ , followed by a charge exchange 1 5 ~ + 1 5 ~ type transition. However, it is well known in light ion reactions that the stripping, pick-up double-step mechanism is a more favourable transition [48) compared to those of the simpler charge exchange reactions. One

16 15 14 15 must consider also three steps O-t N+ C+ C

16 17 16 15

(and also O+ O+ N+ C) for this reaction. Since our previous section indicated that simultaneous and sequential processes occur equally probable in two-nucleon transfer reac- tions and also since it has been shown [45] that the angular distribution for the two different processes are not significantly different, it is justified to perform a simplified calculation where the two-protons are assumed to be transfer- red simultaneously. This is with the understanding that the amplitudes have to be adjusted to include the contribution from a second order sequential

16 15c) process. In this way, we can treat the ( 0, reaction in second order DWBA calculation [49]. The strength of the two transitions 160+14~ and

370;-15~ are fitted to the experimental cross 16 14

sections of ( O+ C), assuming that the neutron in 170 in the latter is only a spectator. Similarly, the strength of the transitioris

l6O+I70 and

I4c+l5c are fitted to the

160+170

cross sections.

The results of the second order DWBA calcula- tions are shown in Fig.23. Qualitative agreements in shape and magnitude of the cross sections are obtained. The cross sections, however, are rather small indicating taht such third order processes must be small. On the other hand, the qualitative agreement of this analysis indicates that the other neglected reaction processes like charge exchange type reactions must aLso be negligible. The overall smallness of this type of third order processes enhances our earlier conjecture regarding the convergence of the reaction series.

Fig.22

-

All possible transition schemes, one-step, two-step, three-step and general nucleon exchange processes in the . 48ca(I60, 15~)49Ti reaction.

Fig.23

-

Second order DWBA analyses of the 48~a(1 60, 1 5 ~ ) 49Ti reaction through intermediate

DIRECT REACTION MECHANISMS

5.2. (160,13c) , ( 1 6 ~ , 1 3 ~ ) and (160,12c) reactions. -We mentioned that experimentally (160,

13c)

integra- ted cross section [8] is almost as large as those of the (IGO,

14c) reaction. This implies that

I4c is not an

intermediate channel to 13c as it was for the much weaker I5c channels. On the other hand, it is also remarked that (l60,I2c) integrated cross section is only a factor three smaller than the strongest (160, 1 5 ~ ) cross section and is of the same

160 17

magnitude as the (

,

0) cross section; This suggests to us the interesting possibility that

13c occurs mainly via the intermediate 12c

channels. This does not exclude the much weaker transition via the intermediate

14c

channels although it can be expected that its contributions must be of the same order of those of 15c channeis, which is two orders of magnitude smaller.

16 1 1

A similar observation is made that the ( 0, B) reaction which has about the same magnitude as those of the (160,180) reaction. Thus, we expect

16 12 1 1

that the reaction to occur as O-t C+ B much 16 17 18

like the O-t Ft 0 reaction.

The above two observations could be used as argu- ments in favour of considering (160,

12c)

reaction as a direct one-step process. Efforts in the past in the study of (160,32~) reactions tends to support this picture, especially in the close similarity for the selective populations of states

6 in this reaction compared to those of the ( Li,d)

7

or ( Li, t) reaction [50]. The strong population of the 40~a(160, 12~)44~i reaction performed at Argonne [8] and similarly 58~i(160,

12c)

6 2 ~ n at Saclay [51] as well as few other reactions also confirm the nature of these reactions. It did not occur through intermediate one or two-nucleon transfer channels, which are only weakly populated due to poor Q-value matching conditions. Thus, it is becoming clear that (360,32~) reactions must have occurred as the simultaneous transfer of four nucleons coupled to internal spin equal to zero,

i.e., as an a-cluster. Our present discussion indicated also that it is highly possible that a sequential transfer of four nucleons followed by another four nucleons can occur, when it is favoured by kinematic matching conditions. The magnitude of such eight nucleon transfer reactions as secondorder processes may not be small.

5.3. Pick-up, stripping mechanism in inelastic scattering.- From the earlier discussions, it is

clear tbat the second order process involving the pick-up followed by the stripping of a particle or vice versa can lead to large contributions to inelastic cross sections. This causes us to reexamine the standard macroscopic model for inelastic scattering there it is known that complex form factor must be used in order to obtain satisfactory fit td the data. However, a microscopic calculation like the RPA procedure

only predicts a real coupling form factor, which normally agrees well with those of the macroscopic model. It has therefore been assumed that the complex part of the form~factor takes into account in a macroscopic way the effects of other neglected higher order processes, much like the imaginary part of the optical potential for elastic scatte- ring. However, due to the rather strong dependence of heavy ion reactions on kinematics as well as on structural properties of the particular nuclei involved, it may happen that the use of such complex form factor can be inadequate. In parti- cular, in the event that only a few intermediate channels are available in the double pick-up, stripping mechanism feeding to the inelastic channel, then it can no longer be described by an average complex form factor. This may indeed be the source of discrepancy in the various cases reported where the macroscopic model failed to account for projectile inelastic excitations

[~ef .52]. It is therefore necessary to include explicitly such higher order processes to verify the origin as well as the range of validity of the use of the macroscopic complex form factor for inelastic scattering.

6. STJMMARY OF 160 ON 4 8 ~ a REACTIONS.

K.S. LOW

This picture can be further illustrated by looking at a summary of the quasi-elastic cross sections observed in 160 on 4 8 ~ a scattering in Fig.24. Here, we sum the two observed one-nucleon transf er channels of and j70 in the first step of a sequence and for the second step, the addi- tion or subtraction of another nucleon. On the other hand, l2c channel is observed to b e very large. We thus ascribe another sequence to it. The next step of this sequence is formed by the addition or subtraction of another nucleon. In this way, we find that more than 99 % of all the transfer cross sections is exhausted. The limit is defined by the ratio of the two sequences to the total cross section of 96 f 10 mb.

1 6 0

r4$ca

56

MeV

I

I

,

1

I

1

0

1

2

STEP

I f

(1) Z ' ~ N , ' ~ O

Fig.24

-

Summary of 160 on 4 8 ~ a reaction. The first step cross sections (1) and (a) make up 80 % of the total cross section observed. The second step is the addition or subtraction of another nucleon accounting most of the remai-

ning 20 %.

440

2

b

20

It is interestingto.know that for the one- nucleon sequence that we have identified, the first step is 86 % of the limit. It is no mere coincidence that this corresponds closely to the percentage attained by DWBA for the two cases of one-nucleon transfer reactions that we have studied earlier. This is shown in Fig.25. Here also, the sequences converge very rapidly and the first steps accounts for 83 and 87 % respec- tively for the cross sections of the two states. On the other hand, the first term of the four nucleon transfer sequence that we have identified is only 68 % of the

-,

1 (1 2 1 ) r 14 14 18

- r /

/

LIMIT

( H ) z12c C, N, 0

-,'I

,f. 11 -13 11

( o c + l )

=

C,

B

limit. This is a mixed sequence with possibly different strength in the first and the second step, It may also not be completely justified to classify this reaction as a pure four nucleon plus another nucleon transfer reaction. It can have a not too small contribution from a direct three nucleon stripping and also from the sequence

14C

to 13c. Nevertheless, the experi- mental data interpreted in this way seems to

suggest that both one and four nucleon transfer occurs in a single step accounting for 80 % of the total cross section. The second step is the transfer of another nucleon making up most of the remaining 20 %. Third order processes is an order of magnitude smaller.

I'

,

'w)

I----

4

ITERATION

Fig.25

-

Graph showing the rapid convergence of the iterated sequence in the CRC analyses of the one-proton transfer reaction. The first step isequivalent to the results of first order DWBA analyses.

However, the inability to separate out distinct two or three nucleon reaction sequence has also important implications. In this particular reaction, the transfer of two or three nucleons are kinematically disfavoured and the observed cross sections of these reactions are tacked to higher order successive type processes. On the other hand, one can ask if this is not a

DIRECT REACTION MECHANISMS

four nucleons. This conjecture is, however, not easily substantiated due to the strong kinematical and other matching conditions. More systematic study in this direction is required. In particular, by careful choice, one may be able to perform a reaction where only a few of the governing factors are present allowing a more definite conclusion be drawn regarding such selective reaction events through favoured channels.

7. THE OUTLOOK OF HEAVY ION INDUCED REACTIONS. One must now review what we have gained in the study of these heavy ion induced reactions. A

foremost question in our minds is whether we are not just repeating the experiments and analyses of those made using light ion projec- tiles. However, in this paper, we show a rather severe dependence on higher order effects for both the shape and magnitude of the angular distributions for heavy ion reactions. Such effects are less striking in light ion induced reactions

[s],

presumably due to the less peri- pheral nature of these reactions. Since these higher order effects are correlated to the more complicated multi-particle, multi-hole configu- rations in nuclei, heavy ion induced reactions therefore provide a more severe measures for the relative magnitudes of these components compared to those of the simpler single particle normally obtained by DWBA. Also, heavy ion reacti.ons are expected to provide easier access to information,, regarding the "clustering" aspects of nucleons in nuclei. Through the availability of a larger range of projectiles, multi-nucleon transfer reactions can be more easily manipulated with heavier projec- tiles. However, such analyses are necessarily more complicated but are beginning to be amassed. Much has still to be done in such spectroscopic studies of nuclei with heavy ions reactions beyond those obtained with light ions.

Studies in heavy ion reactions have also more acutely sharpened our awareness of the physical processes in the scattering of nuclei and the approximations made in direct reaction theories. Out of necessity,'we are made to understand the meaning of recoil effects in transfer reactions. Thus, zero range approximation gave way to exact finite range theory. Similarly, how the residual interaction between nuclei are to be chosen have

also been reexamined. This led to Coulomb and core corrections in order to establish post- prior form equivalence. We have also seen a'surge of interest in the effects of higher order scatterings. Thus, multi-step processes are pursued in fervour and angular distributions are being discussed in terms of Coulomb-nuclear, direct-indirect interference patterns with respect to specific nuclear properties. And due to the more peripheral nature of these heavy ion reac- tions, angular distributions are simpler and normally more regular. This led to the usage of more physical terminologies like diffraction

?'

patterns, rainbow scattering in common with those of optical scattering. We thus find that there is an all round bioadening of our understanding of direct reaction processes, induced by our study of heavy ion induced reactions !

We can also view the results of direct reaction theories from another angle. Much attention has been devoted in the last few years to certain class of macroscopic phenomenon like deep- inelastic scattering observed in heavy ion reactions. Theoretical efforts in the study of these reaction events are based either on a microscopic approach in the framework of the

time-dependent Hartree-Fock (TDHF) or in a macroscopic approach using classical concepts of frictional forces. These theories purportedly strive to account for the macroscopic features like mass, charge and energy distributions of the outgoing channels, and has been contrasted to direct reaction theories where only a few

outgoing channels are being considered at a time. However, we believe that these different models are complimentary, The study of average properties must come together with a detailed understanding of the basic reaction mechanism in the interaction between two complex nuclei. By restricting our-