A Distorted-Wave Born-Approximation Theory of ... - Science Direct

ANNALS

OF PHYSICS

163, 265-287

(1985)

A Distorted-Wave Heavy-Ion-Induced

Born-Approximation Projectile Breakup

Theory of Reactions

R. SHYAM GSI,

Poslfach

110541,

Planckstrasse

I, D-6100

Darmstadt

II,

West Germany

AND

M. A. NAGARAJAN Science

and Engineering Research Council, Daresbury Warrington, United Kingdom Received

April

Laboratory,

24, 1984

A distorted-wave Born-approximation (DWBA) formalism is developed to describe the “elastic” and “inelastic” breakup of the projectile in the heavy-ion-induced reasctions. Using a “local momentum approximation” and the surface localization property of these reactions expressions for the DWBA transition amplitude are derived. One of the attractive features of the present formulation is the fact that the radial integrals involved in it are the same as those contained in the transition amplitude evaluated with a zero range approximation. The validity of the approximations used in these derivations has been tested and it is shown that these approximations are well fulfilled. 0 1985 Academic Press. Inc

1, INTRODUCTION During recent years, there has been an accumulation of a large amount of experimental data on the breakup of light and heavy ion projectiles in the Coulomb and nuclear fields of the target nuclei [l-7]. These reactions are characterised by a peak near the beam velocity in the spectra of the observed fragments. Also, several attempts have been made to understand these kinds of reactions theoretically. For the breakup reactions induced by light projectiles (mass number 64) considerable success has been achieved in describing all the aspects of the measured inclusive as well as coincidence spectra [l, 8-101 in terms of a theory which is formulated in the framework of the post-form distorted-wave Born approximation (DWBA) [11-121. In this theory, which is also known as the “spectator-participant model,” one of the constiuents of the projectile is assumed to interact strongly with the target and is knocked out of the beam leaving the remaining piece essentially undisturbed. For the heavy-ion-induced reactions, though some attempts have been 265 0003-4916/85

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Copyright 0 1985 by Academic Press, Inc. All rights of reproduction m any Iorm reserved.

266

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made [13-161 to describe some gross features of the spectra (like the widths of the breakup bumps and the total cross sections), a quantitative theoretical understanding of the details of the data (inclusive as well as coincidence measurements) is still lacking. Our main effort in this paper is to reformulate the spectator-participant model of the breakup reactions so that it can be used for a quantitative theoretical analysis of the experimental data for the heavy-ion-induced breakup reactions. There is one major problem in a straightforward application of the spectatorparticipant model in its present form to the heavy projectiles. Presently calculations based on this model use the simplifying approximation of a zero range interaction between the constituents of the projectile [17]. One of the consequences of this approximation is that the details of the internal structure of the projectile do not appear in the description of the amplitude except through an overall normalization constant. Whereas this approximation may be reasonable in the case of the deuteron and to a lesser extent for 3He and a-particles, it does not seem right to apply it to heavier projectiles. This approximation will obviously be unsuitable for the projectile whose constituents have a relative orbital angular momentum different from zero. It is therefore necessary to extend the formalism to include the finite range effects of the interaction. This would automatically incorporate more of the details of the projectile structure in the breakup amplitude. However, the computation of the distorted-wave integrals is already a major problem even in a zero range approximation because one needs to evaluate integrals of a product of three scattering wavefunctions. The inclusion of the finite range effects exactly is likely to make the calculation of these integrals much too complicated. In order to avoid this, we present a simple approximation based on a method similar to that of Braun-Munzinger and Harney [18]. This “local momentum approximation” results in two simplifying features, First, it factorizes the dynamics of reaction from the structure effects of the projectile and, second, it results in an amplitude which can utilize all the integrals already evaluated in the zero range approximation. In Section 2, we derive expressions for the finite range DWBA amplitudes for the “elastic” and “inelastic” breakup. By elastic breakup we mean that kind of reaction in which the target and one of the fragments remain in their ground states during the breakup process. However, for the calculation of the inclusive spectra in which only one of the constituents of the projectile is observed in the final channel, one has to integrate over all the possible channels that the rest of the projectile and the target nucleus could go into. This kind of process will be called as inelastic breakup. For example, if we consider the breakup of I60 by a target nucleus A and wish to evaluate the coincidence cross section for the fragments l*C and a-particle as well as the inclusive cross section for observing the ‘*C fragment in a given state of excitation, we shall define reactions of the type 160+A+

‘*C+a+A

(1.1) (1.2)

A DWBA

THEORY

OF HEAVY-ION-INDUCED

BREAKUP

REACTIONS

267

as elastic breakup reactions, because a and A are in their ground states. In the case of inclusive reactions 160 + A + 12C + anything + 12C:+ + anything

(1.3) (1.4)

the (a + A) system is composed of all possible open channels which contribute to the inclusive cross sections. These will be referred to as the inelastic breakup channels. It must be mentioned at this point that there is also another simple mechanism of breakup that may contribute to the inclusive spectra: the projectile is inelastically scattered to some excited state above the particle emission threshold, which will then decay into the fragents after the projectile has left the region of interaction with the target nucleus (see Ref. [ 193). This “sequential” mechanism of breakup has not been considered in this paper. Our derivations have been checked by extracting from them the corresponding zero range expressions. The calculation of the geometrical part of the amplitude and the estimates of the additional angular momenta brought in by the finite range effects are also presented in Section 2. In Section 3, the validity of various approximations used in our derivations is checked. Finally, discussions and conclusions are presented in Section 4.

2.

FORMALISM

(2.1) Elastic Breakup

In this section we derive expressions for the elastic breakup cross section for the case where the target nucleus and both of the fragments remain in their respective ground states during the breakup process (i.e., reactions of the type (1.1)). However, cross sections for reactions of the type (1.2) where one of the fragments can be in one of its excited states can be calculated analogously. Consider a reaction A+a+A+b+x

(2.1)

where the projectile a (= b + x) disintegrates into the constituents b and x in the Coulomb and nuclear fields of the target nucleus A. Both constituents b and x are supposed to be detected and these as well as the target remain in their respective ground states. The T-matrix for this reaction in the post-form DWBA is written as

where qa, qb and qX are the momenta of the particles in the initial and final channels. x’s denote the scattering wavefunctions of the nuclei a, b and x, which are

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generated by appropriate optical potentials, and Q’s are the wavefunctions of the internal coordinates. V,, represents the interaction between the constituents b and x (which acts as the breakup interaction). In fact the breakup interaction has additional terms of the type VbA - iJbA, where V,,, is the microscopic interaction between b and A and UbA is the optical potential for the elastic scattering of b by A [20]. However, these terms are usually neglected in DWBA. Since, the projectile a could be composed of the nuclei b and x in different states of excitation with different probabilities, the transition amplitude (Eq. (2.2)) should in general be multiplied by the corresponding spectroscopic factors. These are not specifically shown in our derivations. The system of coordinates is shown in Fig. 1, where ri = (M,/M,)r,

+r

rf = WCJWA

+ KJ)r,

(2.3) + (MA/W,

+ KJr

(2.4)

rx = rl + r

(2.5)

with M,, M, and M, being the masses of the target, the projectile and the fragment x, respectively. The integration over internal coordinates can be performed as follows:

(2.6)

where 1 and j are the respective quantum numbers of the relative b + x system, and Y,m, are the spherical harmonics. Thus the T-matrix (Eq. (2.2)) can be written as

=,z,

(jbpb jp I j,pa)(lm,jxpx

FIG.

1.

System

I jp> i’T,~AST’C(qbq,;

of coordinates.

qa)

(2.7)

A DWBA

THEORY

OF HEAVY-ION-INDUCED

BREAKUP

269

REACTIONS

where

X G&

1 %(rl)

YI,,(PI

Iv= (2N+

1y.

1 X!’

)(q,,

(2.8)

ri)

with

The triple differential cross section for the reaction described by Eq. (2.1) is written as (d3aELASr*Cfdf2, dQ, dE,)

(2.9) Substituting Eqs. (2.7) into (2.9) and carrying magnetic quantum numbers, we get

out the summations

over the

(d3aELAST’C/dSZ, dL2, dE,)

(2.10) In the next two subsections we shall derive expressions for the amplitude B

ELASTIC Im,

.

(2.1.1) Evaluation of the Amplitude /lEASTIC (a) Zero range approximation. The evaluation of the DWBA amplitude (Eq. (2.8)) in a zero range approximation is discussed in detail in Ref. [ 111. Therefore, we give here only a brief description of this. In the zero approximation one assumes K&

(2.11)

1 udrl I= Do Ql 1

where D, is the usual zero range normalization constant. Approximation (2.11) reduces the six-dimensional integral in Eq. (2.8) to a three-dimensional integral. It should be noted that the approximation introduced by Eq. (2.11) necessarily implies that the relative motion between b and x in the projectile a has s state only. This restricts its applicability to only s-wave projectiles. Introducing partial wave expansions for the distorted waves and performing the angular integral one finally gets for the zero range amplitude

x

%$qbY

qX)(LboLXoI

&‘)

&,L,L,(&,qX;

4.3)

(2.12)

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SHYAM

AND

NAGARAJAN

where (2.13) and

The quantities ( ) in Eq. (2.12) are the standard Clebsch-Gordan coefficients. Due to the appearance of (L,OL,OI L,O) in this equation the values of L, (orbital angular momentum associated with the motion of the particle x with respect to the target) are restricted by the following condition: lL,-Lbl 1 L,O)(IOLO

I Zm> W(L,Z-lZbA; YLAMAvu

qm,&,.

1Z,O)

LZ) (‘4.8)

A DWBA

THEORY

OF HEAVY-ION-INDUCED

BREAKUP

In Eq. (A.6) P, represents the Legendre polynomials. I,(q)

1 l+Af1b)((2LA+ L.dJdbmb x 1))1/2(1-~oLo

1lo) -‘w(LJ-&A;

1 L,O)(/1OLO

1lb0)

LI)

i.bwiJ,L(h&)

(A.lO)

where W represents the Racah coefficients as defined in Ref. [24]. Now to evaluate G~,,X,r,(K,K,K,) we define in Eq. (2.25)

K, = K. + (M,/MJK,

(A.1 I)

and use (A.9) a second time. The final result is G:,,,~WLJ~KJ

(21-2/I,+ x ((21+ 1)/(2L,+ x (I-l,OL,O x (&OL,O x W(f,l-&I&,;

l)“2(-)L’+Lz

1)(21,+

I1,0)(1,OL,O 1 Z,O)(L,Ol,O L,l)

1)(21,+

l)p2

I Lo)-‘(1,oLo W(I,L-f&1,&;

x((M,IM,)~,)'-~'((M,I(M,+M,))~,)L-"2(~,)"2~,L:,'~,

I&O)

I LO)(l-&OL,O I 10)-l LZL)

04.12)

where we have defined p1

bL2

(A.13)

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SHYAM

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NAGARAJAN

with Z=K;

(A.14)

K,.

One can easily show that Eq. (A.lO) follows from (A.12) in the limit K, -+ 0. Equation (A.12) is our final result which relates the integral G:b,X, to the Legendre expansion of the Fourier transform of V(rl) uI(rl).

ACKNOWLEDGMENTS One of the authors (R.S.) is thankful to Professor W. Nijrenberg and Dr. J. Knoll for the kind hospitality of GSI Darmstadt and their ineterst in the present work. He is also thankful to Drs. G. Baur, F. Rose1 and D. Trautmann for their constant encouragements and several useful and stimulating discussions.

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