Distorted wave investigation of α-cluster knockout in 6Li

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Nuclear

Physics

A233 (1974) 145- 152; @

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KNOCKOUT

IN 6Li

A. K. JAIN and N. SARMA Nuclear Physics Dicision, Bhabha Atomic

Research

Centre,

Bombay 400 085, India

Received 24 May 1974 (Revised 24 June 1974) Abstract: Experimental results on the 6Li(a, 2a)2H reaction at 70.3 MeV have been compared with a calculation using the distorted wave impulse approximation, taking into account all exchange terms and a proper asymptotic form for the intercluster wave function. The limitation on the number of partial waves imposed by computational facilities is removed by a new prescription. Inconsistencies in the values of the clustering probability as determined by various methods are sought to be explained in terms of the contraction of the deuteron cluster in the 6Li nucleus.

1. Iotroduction Many attempts have been made, both experimental and theoretical to investigate the clustering of nucleons in nuclei. The most direct evidence comes from the analysis of cluster knockout reactions, particularly on the light nuclei 6Li and ‘Li. The analyses carried out so far have, however, raised new problems as the approximations of the earlier theories were reduced le3). For instance, earlier work was carried out in the distorted wave impulse approximation (DWIA) using antisymmetrised cluster model wave functions which fit the binding energy ‘) and charge form factor “) accurately. A major discrepancy which emerged from the analysis of the 6Li data was that the predicted cross sections ‘) were half the measured values “) for the 155 MeV (p, pd) reaction, while being about 80 % higher than the (d, 2d), (d, da) and (z!, 2a) data “) at lower energies (27 to 55 MeV). On the other hand, the ‘Li(p, pt)4He reaction data are in very good agreement with the computed values “). Accurate data are now available at higher energies on the ‘jLi(a, 2a)‘H reaction ‘* ‘), but the analysis of these results has so far been carried out in a plane wave (cut-off) impulse approximation. It would therefore be interesting to apply the full DWIA analysis to these measurements and to inquire into the nature of the earlier discrepancies.

2. Formalism The cross sections were computed using the kinematic coupling approximation ‘) for the three-body final state, consisting of particles 0, 1 and C, with relative coor145

A. K. JAIN AND N. SARMA

146

dinates rot and rrc (fig. 1). In this formulation is written as:

the three-body

final-state

Hamiltonian

In order to separate out the Schriidinger equation as a function of rOc and rrc the (h’/mc)Vcc * V,c term is replaced by its eigenvalue in the plane wave limit. It is however to be remarked that in the plane wave limit the final-state wave function-is Using the knockout approximation exactly separable in the rot, r 1c coordinates. the DWIA matrix element may then be written: Mr,(KO)

= (~YI-‘ltol(rol)l~l+‘>.

(2)

operator for Here Yi-’ is the solution of eq. (1), tol(rol) re p resents the transition the scattering of the incident particle 0 from the struck particle 1. The eigenfunction !I$‘) is the solution of the Schrddinger equation corresponding to the initial state Hamiltonian, Hi :

Hi = To~+Voc+T,,+~,,c+~,+~,+~,,

(3)

where rrc is the interaction between 1 and C corresponding to the bound wave function of the target. If it is assumed that the knockout interaction Y,, is short-ranged and that the change in the effect of distortions is small over this range, the matrix element may then be factorised: Mri =

t,,(P)g,,(Q),

(4)

where gri(Q) = (!P:-‘ld(r, For d- or a-cluster

knockout

reactions

Fig. 1. The coordinate

!f’i+‘).

on 6Li the eigenfunctions

!P!+’ 1 = Nr 4,(1234; Y!-)

-r,)l

56)~b+‘(ke,,

= Nf ~,(1234>~d(56)~~-‘(k,c,

!-Pi’) and !Pi-’ are

r,,,J,

r&-kc,

(5) RI,

(6)

system used in the computations, 0 and 1 refer to the incident and struck particles and C to the residual nucleus.

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KNOCKOUT

where 4*, $, and & are the internal wave functions of the 6Li, a-particle and deuteron respectively. A partial wave treatment of the distorted wave functions xr’ and xi:: yields ‘):

x p:(e,)qe,)+2 i (4 cc,n -14 0>p~(el)p~(~2)l{~~~(Imn) p=1

9

-2~s,(Imn)+~nE(lmn)}dr = N

s

OF(R)dR,

where N contains all the normalisation factors of the internal wave functions, ( .,.,.,.I.,.) is the Clebsch-Gordan coefficient and P:(x) is the associated Legendre polynomial. The terms %r.&1mn), .F&lmn) and F,,E(lmn) are the radial functions corresponding to the no-exchange, single-exchange and double-exchange terms arising from the antisymmetrisation of the target wave function multiplied by the partial waves I, m and n of the scattering functions. As an example, the radial function S,,(lmn) is s,,(lmn)

= {l/&k,,k,ck,c>J1,(1234)ll/,(56)J/,,(1234: x Ur(&A R)f,(k,c

R)f,(k,cR)/R%

56) (10)

where E = (m,-/mA) and f,(kR) are the solutions of the radial Schrodinger equations with appropriate optical potentials. Through a proper choice of coordinate transformations the angular coordinates of R are eliminated from the volume element ds. 3. Calculations in the above formalism as applied to the 6Li(a, 2a) reaction, the scattering matrix element is factorised out of the reaction amplitude. The short-range nature of the a-a interaction justifies this factorisation, while the justification for the replacement of the off-shell z-matrix by an on-shell scattering amplitude obtains from the small binding energy of a and d in 6Li as compared to the incident energy. The spatial wave function for the 6Li nucleus was taken to be ‘)

(11) with +.,x(R) =

R2 exp ( -@R2)

R 5 R,

a exp (- cR)/R

R ZR,,

(12)

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A. K. JAIN AND N. SARMA

with z = 0.514 fm-‘, E = 0.74 fm-’ and fi = 0.33 fm-‘. The decay constant, c = (2~dzEdz/h2)* is determined by the d-z separation energy. The matching radius, R. was determined by matching the logarithmic derivatives of the two forms of +a= (R) in eq. (12). Th is 6Li wave function reproduces the measured charge form factor most satisfactorily ‘) and predicts the correct d-r momentum distribution as measured by the (d, 2d), (d, da) and (a, 2 a ) reactions “) from 27 to 55 MeV and the (p, pd) reaction ‘) at 155 MeV. The nuclear optical potentials used in the calculations were of the form ‘) U,(R)

= UN SW (R,, a, R) - i W, DSW(Ri,

a : R),

(13)

where SW and DSW refer to the Saxon-Woods and derivative Saxon-Woods forms. The Coulomb potential is assumed to be that due to a uniformly charged sphere of radius R, = rcd4+, while UN = U0 - a’E and W, = W, +/I’S The parameters for obtaining the potentials are given in table 1. TABLE

1

The nuclear optical potential parameters

uo 65

r0

I .25

a0

0.592

WO 1.08

10

a0

rc

a'

B’

A

1.25

0.634

1.25

-0.1

0.137

4

Strengths are in MeV and distances in fm.

An accurate calculation requires a large number of partial waves for the evaluation of each of the three distorted waves. However, in practice this number is limited by computer time and memory space. The overlap function from an exact (analytical) plane wave calculation OF(R) = 4q$%,(QR)RZ4(R), where 4(R) is the antisymmetrised intercluster wave function, is shown in fig. 2. Comparison of this quantity with that computed for the plane waves and the distorted waves, using 23,20 and 20 partial waves in the incident and final channels respectively, indicates that the cumulative effect of the higher partial waves is significant, reducing the peak cross section for the 6Li(a, 2a)2H reaction by about 20 % while the width of the momentum distribution is increased. The exact plane wave calculation and the computed plane wave overlap functions are identical up to about 9 fm. Beyond this, the computed plane wave and distorted wave overlap functions are very close to each other but drop off sharply from the exact calculation. Since this drop-off must be due to the neglect of higher partial waves, our prescription for the correction of the computed plane wave or distorted wave matrix element is to add smoothly, using an interpolation procedure, the exact plane wave contribution beyond a certain radius, r,,, , chosen to be the minimum of the /,,l_Orr/k for the three distorted waves.

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149

R (fm)

Fig. 2. Radial dependence of the overlap function for recoil momenta (a) Q = 0 and (b) Q = 50 MeV/c. The real and imaginary parts computed for the DWIA with partial wave cut-off are given by the full and dotted lines respectively. The exact PWIA calculation is represented by the dashed line. The PWIA with plane wave cut-off follows the exact PWIA up to 9 fm, beyond which it overlaps the DWIA with cut-off.

Theoretical justification for this prescription is found from the following arguments. We find the higher partial wave contribution is localised at larger radii. For the high partial waves which contribute beyond 9 fm, the nuclear potential is insignificant compared to the centrifugal barrier. In this reaction, the Coulomb effects are small and so the exact distorted wave and plane wave contributions would be almost identical at such large radii. Because of the kinematics of the experiment “) and the identity of the two alphas in the final state, the reaction is symmetric with respect to zero recoil momentum, Q = 0. The matrix element is therefore calculated only for positive Q. Cross sections [refs. ‘* ‘“)I for different Q were then obtained by multiplying the square of the matrix element by the relevant kinematic factors and the free a-a cross sections ’ ‘). 4. Results and discussion The cross sections computed in the post form approximation “) are compared with the experimental results in fig. 3. The curves are normal&d to the experiment at the peak. It is observed here that exchange effects broaden the spectrum for the distorted wave impulse approximation, but sharpen the spectrum for the plane wave approximation. The distortions reduce the width of the calculated spectrum to bring it in better agreement with experiment. The computed curve, using DWIA and including antisymmetrisation is broader than the experimental data; this is particularly noticeable beyond Q 2 30 MeVlc.

150

A. K. JAIN AND N. SARMA

1 5

0 /Mev/, ) 40 t

30

I 32

20 I

I 34

0 I

20 L 1 3.5

I 1 38

El IMeVi

Fig. 3. Comparison of computed cross sections (normal&d at the peak) for the 6Li(a, 2a)*H reaction with 8, = -Or = 44.25” at El.. = 70.3 MeV. Short dashes: PWIA, and long dashes: DWIA, both without antisymmetrization; dashes and dots: PWIA and continuous line: DWIA, both with antisymmetrization.

The calculated absolute cross sections at the peak position are given in table 2. These results indicate that the antisymmetrisation decreases the peak cross section by a factor of 1.14 for the PWIA while increasing it by I .2 for the DWIA. Inclusion of

Peak

cross

TABLE 2 sections, d3n/dR,dRzdE, (mb/s? . MeV) for the 6Li(a, 2a)ZH reaction at 70.3 MeV for 8, = -& = 44.25’. calculated in the post form approximation

Plane wave Distorted wave

Only no-exchange term included

All exchange terms included

230.76 78.21

202.07 94.72

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151

both effects reduce the plane wave cross section by a factor of 2.13. The computed cross section is 94.7 mb/sr’ - MeV as against the measured value ‘* “) of 43 + 3 mb/sr’ - MeV ( f 32 p/, in the absolute value due to experimental uncertainties). The effective clustering probability’ is then N,,, = 0.45 f 0.18, a value consistent with that obtained from the analysis3) of the 27 to 55 MeV data. On the other hand, Ncfc as derived from a sophisticated analysis’) of the 155 MeV (p, pd) data is the impossible value of 1.75 + 0.5. Other evidence 4*‘) indicates that the ground state is completely clustered, i.e., N,,, should be 1.0. A possible explanation of the discrepancy is in terms of the deformation of the deuteron cluster in 6Li. Such a deformation has been discussed in the study of the 6Li(d, tp)4He reaction I’), of medium energy cluster knockout reactions 3), and binding energy ‘) and charge form factor calculations ‘). Resonating group cluster model calculations [refs. r3*r4)] have also explicitly considered this possibility. In the (p, pd) reaction analysis ‘), it is assumed that the p-d (cluster) cross section is the same as the free p-d cross section. It has been shown I’) that the p-d cross section depends on a sticking factor, S*(p) which increases rapidly as the deuteron contracts. Consequently the increase in the p-d (cluster) cross section may overcome the reduction due to the overlap between the free deuteron and the cluster. Thus, while the deuteron contraction increases the 6Li(p, pd)4He cross section, the 6Li(a, 2a)‘H cross section is reduced. Incorporation of this effect would make the N,,, values more consistent. Following the arguments of ref. “), if the deuteron cluster contracts as it approaches the a-cluster, the overlap between the cluster and the free deuteron is reduced. For Q = 0, a large contribution to the matrix element is from a region of large d-a separation (see fig. 2) while for Q >> 0, the contribution is from smaller separations. If we assume a model where the deuteron cluster contracts at small a-d separations, the matrix element away from Q = 0 will be reduced and the resulting distribution will be sharper, more in agreement with the experimental data. We wish to thank Prof. M. Gusakow, Prof. H. G. Pugh and Dr. B. K. Jain for helpful discussions. One of us (A.K.J.) is grateful to the Institut de Physique Nucltaire de Lyon, where part of this work was done, for its kind hospitality. l

This quantity is defined as the reduced width squared, O2 in ref. a).

References 1) A. K. Jam, N. Sarma and B. Banerjee, Nucl. Phys. Al42 (1970) 330 2) A. K. Jain and N. Sarma, Nucl. Phys. Al% (1972) 566 3) A. K. Jain, J. Y. Grossiord, M. Chevallier, P. Gaillard, A. Guichard, M. Gusakow and J. R. Piui, Nucl. Phys. A216 (1973) 519 4) Y. C. Tang, K. Wildermuth and L. D. Pearlstein, Phys. Rev. 123 (1961) 548 5) A. K. Jain and N. Sarma, Phys. Lett. 338 (1970) 271 6) C. Ruhla, M. Riou, M. Gusakow, J. C. Jacmart, M. Lin and L. Valentin, Phys. Lett. 6 (1963) 282

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7) J. W. Watson, Ph.D. thesis, University of Maryland, 1970, unpublished 8) J. W. Watson, H. G. Pugh, P. G. Roos, D. A. Goldberg, R. A. J. Riddle and D. I. Bonbright, Nucl. Phys. Al72 (1971) 513 9) F. Hinterberger, G. Mairle, U. Schmidt-Rohr, G. J. Wagner and P. Turek, Nucl. Phys. All1 (1968) 265 10) A. K. Jain, report BARC-635 (1972) 11) P. Darriulat, report CEA-R2786 (1965) 12) J. Y. Grossiord, C. Coste, A. Guichard, M. Gusakow, A. K. Jain, J. R. Piui, G. Bagieu and R. de Swiniarski, Phys. Rev. Lett. 32 (1974) 173 13) D. R. Thompson and Y. C. Tang, Phys. Rev. 179 (1969) 971 14) H. Jacobs, K. Wildermuth and E. J. Wurster, Phys. Lett. 298 (1969) 455 15) G. F. Chew, Phys. Rev. 74 (1948) 809