collision of almost identical nuclei: fusion cross sections and barrier distributions

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NUCLEAR PHYSICS A ELSEVIER Nuclear Physics A 591 (1995) 341-348 Collision of almost identical nuclei: fusion cross sections and barrier distributions J.A. Christley a, M.A. Nagarajan h, A. Vitturi c a Department of Physics, University of Surrey, Guildford GU2 5XH, UK b Laboratori Nazionali di Legnaro, INFN, Italy e Dipartimento di Fisica and INFN, Padova, Italy Received 17 March 1995 Abstract The effect of elastic transfer on the fusion of almost identical nuclei is investigated. It is shown that even in cases where no oscillatory structure is visible in the fusion excitation function, the signature of the elastic transfer process is clear in the partial fusion cross sections and in the barrier distributions. These features are illustrated with calculations for 160-+-180 and 28Si+3°Si systems. 1. Introduction The importance of elastic transfer in collisions of almost identical nuclei was pointed out by von Oertzen and Ntrenberg [ 1 ] and detailed analyses of the elastic and inelastic scattering in such systems have been performed and reviewed by von Oertzen and Bohlen [2]. It was suggested by Frahn and Hussein [3] that the effect of elastic transfer on the elastic scattering can be easily taken into account by the introduction of a L-dependent (parity dependent) term in the optical potential. This was, for example, shown very clearly by Dasso and Vitturi [4] in their analysis of the elastic scattering of 160 by 180 where the strength of the parity dependent term was determined by the coupling of a neutron pair to the 160 core. In some of the systems comprised of either identical or almost identical nuclei, the total fusion cross sections have been observed to possess an oscillatory structure as a function of the projectile energy for values just above the Coulomb barrier. Such fusion oscillations have been observed in systems like 12C +12C [5] and 160 + 160 [6] as well as in 12C +13C [7] and 160 +12C [8]. Poff6 et al. [9] showed that the fusion oscillations observed in the 12C +12C system are a consequence of a sharp cut-off in the 0375-9474/95/$09.50 t~) 1995 ElsevierScience B.V. All rights reserved SSDI 0375-9474(95)00186-7

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Page 1: Collision of almost identical nuclei: fusion cross sections and barrier distributions

N U C L E A R P H Y S I C S A

ELSEVIER Nuclear Physics A 591 (1995) 341-348

Collision of almost identical nuclei: fusion cross sections and barrier distributions J.A. Christley a, M.A. Nagarajan h, A. Vit tur i c

a Department of Physics, University of Surrey, Guildford GU2 5XH, UK b Laboratori Nazionali di Legnaro, INFN, Italy

e Dipartimento di Fisica and INFN, Padova, Italy

Received 17 March 1995

Abstract

The effect of elastic transfer on the fusion of almost identical nuclei is investigated. It is shown that even in cases where no oscillatory structure is visible in the fusion excitation function, the signature of the elastic transfer process is clear in the partial fusion cross sections and in the barrier distributions. These features are illustrated with calculations for 160-+-180 and 28Si+3°Si systems.

1. Introduction

The importance of elastic transfer in collisions of almost identical nuclei was pointed

out by von Oertzen and Ntrenberg [ 1 ] and detailed analyses of the elastic and inelastic

scattering in such systems have been performed and reviewed by von Oertzen and Bohlen [2]. It was suggested by Frahn and Hussein [3] that the effect of elastic transfer on the elastic scattering can be easily taken into account by the introduction of

a L-dependent (parity dependent) term in the optical potential. This was, for example, shown very clearly by Dasso and Vitturi [4] in their analysis of the elastic scattering of 160 by 180 where the strength of the parity dependent term was determined by the

coupling of a neutron pair to the 160 core.

In some of the systems comprised of either identical or almost identical nuclei, the total fusion cross sections have been observed to possess an oscillatory structure as a function of the projectile energy for values just above the Coulomb barrier. Such fusion oscillations have been observed in systems like 12C +12C [5] and 160 + 160 [6] as well as in 12C +13C [7] and 160 +12C [8]. Poff6 et al. [9] showed that the fusion

oscillations observed in the 12C +12C system are a consequence of a sharp cut-off in the

0375-9474/95/$09.50 t~) 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(95)00186-7

Page 2: Collision of almost identical nuclei: fusion cross sections and barrier distributions

342 J.A. Christley et al./Nuclear Physics A 591 (1995) 341-348

barrier transmission coefficients, magnified by the occurrence of only even partial waves due to the identity of the colliding nuclei. They further showed that the sharp cut-off

in the barrier transmission coefficients was related to the large diffuseness of the real nucleus-nucleus potential. This idea was extended by Kabir et al. [ 10] to the 160 + 12C system where they suggested that the sharp cut-off features of the barrier transmission

coefficients coupled with the parity-dependence of the optical potential could account

for the observed fusion oscillations. The interpretation of fusion oscillations in the light identical or almost identical

systems hinges according to Poff6 et al. [9] and Kabir et al. [ 10] on the large diffuseness

of the real nucleus-nucleus potential. This could imply that one should observe them

only in reactions involving very light nuclei. In this paper, we investigate the question if

there may be other signatures of elastic transfer also in cases where the fusion excitation function seems structureless. We specifically consider two systems, namely 160 +180

and 28Si + 3°Si and study the effect of the diffuseness (or more specifically the barrier

transmission coefficients) and the strength of the transfer coupling on the characteristics

of fusion.

2. Fusion of ZSsi + 3°Si system

As an illustration of the ideas of Kabir et al. [ 10], we use a simple model to study the implications of the diffuseness of the real potential and the strength of the elastic

transfer on the characteristics of fusion for the above system. The total cross section can be expressed in terms of the barrier transmission coeffi-

cients as

o'f(Ecm ) = ~ Z(2L+ 1) TL(Ecm), (1) L

where TL(Ecm) is the transmission coefficient for the L-th partial wave. It is determined

by the barrier height, the barrier radius and the width of the barrier in the simple model where effective potential is represented as an inverted oscillator potential. If one uses the universal Akyiiz-Winther [ 11 ] parametrization of the real potential, one obtains for the 28Si + 3°Si system a barrier height V = 29.4 MeV, a barrier radius RB = 8.86 fm

and a barrier width hw = 3.44 MeV. The effect of the elastic transfer (which has Q = 0) will be to introduce a parity-dependent term in the optical potential thus causing an

opposite shift in the barrier heights for positive (even L) and negative-parity (odd L) barriers. This energy shift is a measure of the strength of the transfer coupling. We shall

designate this shift by AV. In Fig. la, we show the predicted fusion excitation function as a function of the

projectile energy for AV = 0(no elastic transfer) and ~V = 0.3 MeV. There is little if no effect of the parity dependence on the fusion excitation function. We then calculate

the barrier distribution function [ 12] defined by

Page 3: Collision of almost identical nuclei: fusion cross sections and barrier distributions

J.A. Christley et al./Nuclear Physics A 591 (1995) 341-348 343

10 ~

=

b

lO 1

2 8 S i + 3 ° S i

26

I I I I I I _

a) i

I

28 30 3e 34 36 38 40 E (MeV)

0 . 6 i 1 i i I i

b)

0.4

0.2

0.0

-0.2 I I I I I I 26 28 30 32 34 36 38 40

E (MeV)

Fig. 1. (a) Fusion excitation function for the reaction 28Si+3°Si as a function of the c.m. energy. The barrier height was taken to be 29.4 MeV, its radius RR = 8.86 fm and the width h~o = 3.44 MeV. The dashed line correspond to the case where there is no parity-dependent potential. The solid line corresponds to the case where the even and odd L barriers are shifted by 4-0.3 MeV. (b) The barrier distribution for the system 28Si+30Si.

1 d 2 n(Ecm ) = qTR2 d E 2 (Ecmorf(Ecm)), (2)

normalized with respect to the geometrical cross section ~R~. This quantity is shown

in Fig. lb. The barrier distribution clearly shows the evidence of the parity dependence

of the potential through oscillations at energies above the Coulomb barrier. We then

investigated the effect of the diffuseness of the real nuclear potential, or equivalently of

the barrier width. (Larger diffuseness is in fact related to smaller barrier width [ 10] ).

The resulting fusion excitation function and the corresponding barrier distribution are

shown in Figs. 2a and b. One can now begin to see the fusion oscillations as well as

greater ampli tude variations in the barrier distribution. This clearly illustrates the fact

that fusion oscil lations need small barrier width, as pointed out by Kabir et al. [ 10]. It

can be noticed, however, that even in the absence of parity dependence (dashed line)

one observes oscil lations in the barrier distribution. This implies that it is an extremely

narrow potential where the introduction of each discrete partial wave can be observed.

This is possibly a very unrealistic barrier.

To better interpret the characteristic maxima and minima in the barrier distribution

above the Coulomb barrier, we calculate the barrier transmission coefficients at an energy

corresponding to the mid point between a maximum and a minimum, e.g. Ecm = 34 MeV

(Fig. 3). It can be seen that at this energy, the transmission coefficient for an even partial

wave coincides with the one for the next adjacent partial wave. For example, in our case,

TL=15 = TL=16. In fact, it was this feature that Kabir et al. pointed out for the occurrence

of fusion oscillations. Here, it is seen to correspond to the structure in the barrier

Page 4: Collision of almost identical nuclei: fusion cross sections and barrier distributions

344 J.A. Christley et al./Nuclear Physics A 591 (1995) 341-348

E IO

b

28Si..F3°Si

103 3 I I 1 1 1 1 _

1 0 1 I I I I I I

26 28 30 32 34 36 38 40

E (MeV)

I I I I I I

b)

2

1

0

-1

-2 26 28 30 32 34 36 38 40

E (MeV)

Fig. 2. (a) Fusion excitation function for 28Si+3°Si, but now the barrier width has been reduced to 1.72 MeV. The inset shows an expanded version of the fusion cross section in the range of energies 34-38 MeV and shows the onset of oscillations clearly. (b) Fusion barrier distribution for the same parameters as in (a).

28Si _t_3°Si

E = 34 MeV 1.2 ~ ~ ~ ~ ; ~ = , ,

0.8 \

S 0 . 6 \\~_~ 0.4 \ \ \ ~ k \ \

0.2 \ ' ~ \ X=

0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . 7 7 . ; 7 ~ . ~

- o . 2 ' ' ' ' ' ' ' ' ' 10 11 12 13 14 1~ 16 17 18 19 2 0

L Fig. 3. The barrier transmission functions TL(E) shown for E = 34 MeV. It can be seen that even L and the adjacent odd L transmission coefficients become equal beyond a characteristic L.

Page 5: Collision of almost identical nuclei: fusion cross sections and barrier distributions

J.A. Christley et al./Nuclear Physics A 591 (1995) 341-348

-28Si+3°Si

345

0.6

0.~

0.2

0 . 0

-0.2 26.

f . . . . . . . . . . . . . . . . . . . . ili .... r, Strong cou ng t

t

28. 30. 32. 3~. 36. 38. ~0.

E (MeV) Fig. 4. Effect of the very strong parity dependence on the barrier distribution (AV = 1.0 MeV). The splitting of the barrier at subbarrier energies is clearly seen.

distribution. Next, we study the effect of the coupling strength on the barrier distribution. This

is shown in Fig. 4, where one can see the splitting of the barrier at energies below the Coulomb barrier and the oscillations at energies above the barrier. Thus, the elastic

transfer plays two roles. I f it is very strong, it is clearly observed at subbarrier energies by allowing the single barrier to split into two, and at the same time due to the increased parity dependence also affects the barrier distribution at above barrier energies.

In the next section, we consider a more microscopic description of fusion and barrier distribution for the system 160 + 180.

3. The fusion of 160 + 180 system

The elastic scattering of 160 by 180 was investigated by Dasso and Vitturi [4]. They treated the elastic transfer of a pair of neutrons between the identical 160 cores and

showed that its effect can be incorporated in terms of a parity-dependent term in the optical potential of the form

( _ 1 ) L flpR d x / ~ 3 A dr Uopt(r), (3)

where Uopt(r) is the optical potential, A is the mass number of the target and tip is a "pair deformation parameter". They were able to fit the elastic scattering at Elab = 24 MeV

Page 6: Collision of almost identical nuclei: fusion cross sections and barrier distributions

346 J.A. Christley et al./Nuclear Physics A 591 (1995) 341-348

16 0 +18 0

E = 24 M e V 1

10 , , . . . . . .

100

& 10-'

b

10 .2

10 .3 I I I I I I I I

0 20 40 60 80 100 120 140 160 180

ecru (deg)

Fig. 5. The elastic scattering cross section for 160 q- 180 at Elab = 24 MeV. The pairing deformation parameter flt~ was chosen to be 15.11. The experimental data is from Gelbke et al. [15].

with a tip of 16. (The parameter fie was found to decrease with increasing projectile

energy and this may have to do with the decrease in collision time at higher energies

thus diminishing the importance of elastic transfer). The optical potential Uopt(r) was defined according to the Akyiiz-Winther parametrization. The resulting elastic cross

section is shown in Fig. 5 in comparison with the data.

In order to study the behaviour of the fusion cross section as a function of the projectile

energy, we fixed the strength of the coupling to that corresponding to Elab = 24 MeV.

The fusion cross section as a function of the projectile energy is shown in Fig. 6a. It is seen to be smooth and structureless. We then evaluated the fusion barrier distribution,

displayed in Fig. 6b. In contrast to the fusion excitation function, one clearly observes

oscillations in the barrier distribution. In Fig. 7, we show the partial fusion cross sections at a few projectile energies. It

can be observed that there is an onset of oscillations for each of these energies beyond

a partial L. This is again related to the coincidence of the transmission coefficients for even and the adjacent odd partial wave which progressively move to larger L's with

increasing energy. The above results show that the signature of the elastic transfer (and the consequent

parity-dependence of the optical potential) is clearly visible in the barrier distributions

and in the partial fusion cross sections even when they are not manifestly seen in the

fusion excitation function.

Page 7: Collision of almost identical nuclei: fusion cross sections and barrier distributions

1(¢

lO ~

10 2 E

# 1 o 1

lO 0

10 -~ 10 15 20

E (MeV)

16 0 +18 0

. . . . I . . . . I . . . .

a)

J.A. Christley et al./Nuclear Physics A 591 (1995) 341-348 347

0.6

0.4

r,n 0.2

0.0

-0.2 5

. . . . I . . . . I ' ' ' '

b)

, , , , , , , , I , , , ,

10 15 20

E (MeV)

Fig. 6. (a) Fusion excitation function for 160 -I- 180 system. The real potential was chosen to be the one used to calculate the elastic scattering cross section displayed in Fig. 5. (b) The corresponding barrier distribution for 160 + 180 system.

4. Summary and conclusions

We investigated the effect of elastic transfer on the fusion of almost identical nuclei.

It was possible to show that even in cases where the effect of elastic transfer is barely

16 0 _1_18 0

Partial cross sec t ions 100 I / 90 30 MeV

8O

7O

60

~ so . . .1

b 4O

30

2O

10

o '

0 5 10 15

L

Fig. 7. The partial fusion cross sections for 160 + 180 system at different projectile energies.

Page 8: Collision of almost identical nuclei: fusion cross sections and barrier distributions

348 J.A. Christley et al./Nuclear Physics A 591 (1995) 341-348

observed in the fus ion excitat ion funct ion, it is more clearly observed in the barrier dis-

t r ibut ion and partial fus ion cross sections. The barrier distr ibution exhibits an oscil latory

structure at energies above the Co u l o mb barrier. One thus has to use the technique of

ut i l iz ing large-angle quasielastic data to extract this, as done by Leigh et al. [ 13].

An alternative method recently proposed by Ackermann [ 14] allows one to directly

measure the partial fusion cross sections. These also carry the knowledge of elastic

transfer as clearly as the barrier dis t r ibut ion itself.

Acknowledgements

One of the authors (J .A.C.) is supported by EPSRC grants G R / J / 9 5 8 6 7 and

G R / K / 3 3 0 2 6 . This work was also supported by the EEC Hum an Capital and Mobi l i ty

program E R B C H R X - C T 9 2 0 0 7 5 .

References

I l l W. von Oertzen and W. N6renberg, Nucl. Phys. A 207 (1973) 113. [21 W. yon Oertzen and H.G. Bohlen, Phys. Reports 19 (1975) 1. [3] W. Frahn and M.S. Hussein, Nucl. Phys. A 346 (1980) 287. [4] C.H. Dasso and A. Vitturi, Nucl. Phys. A 458 (1986) 157. [51 E Sperr et al., Phys. Rev. Lett. 37 (1976) 321. [61 B. Fernandez et al., Nucl. Phys. A 306 (1978) 259. [7[ D.G. Kovar et al., Phys. Rev. C 20 (1979) 1305. [8] J.J. Kolata et al., Phys. Lett. B 65 (1976) 333. [9] N. Poff6, N. Rowley and R. Lindsay, Nucl. Phys. A 410 (1983) 498.

110] A. Kabir, M.W. Kermode and N. Rowley, Nucl. Phys. A 481 (1988) 94. [1 ll O. Akyiiz and A. Winther, in Proc. Enrico Fermi Intern. School of Physics, 1979 (eds. R.A. Broglia,

C.H. Dasso and R.A. Ricci). [121 N. Rowley, G.R. Satchler and P.H. Stelson, Phys. Lett. B 245 (1991) 25. 113] J.R. Leigh et al., in Heavy-ion fusion: exploring the variety of nuclear properties, eds. A.M. Stefanini

et al. (World Scientific, Singapore, 1994) p. 15. 1141 D. Ackermann, Proc. Summer School on Nuclear Physics, Zakopane (1994), and private communication. [151 C.K. Gelbke et al., Phys. Rev. Lett. 29 (1972) 1683.