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

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 Elsevier Science B.V. All rights reserved SSDI 0375-9474(95)00186-7

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

342

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 coefficients as o'f(Ecm ) =

~ Z ( 2 L + 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

343

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

28Si+3°Si 10 ~

I

I

I

I

I

I

a)

_

0.6

i

1

i

i

I

i

i

b) 0.4

0.2 =

b

0.0

lO 1

I

26 28 30 3e 34 36 38 40 E (MeV)

-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 d2 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 o f the parity dependence o f the potential through oscillations at energies above the Coulomb barrier. We then investigated the effect o f the diffuseness o f 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 amplitude variations in the barrier distribution. This clearly illustrates the fact that fusion oscillations 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 oscillations 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

344

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

28Si..F3°Si 103

I

I

1

1

1

1

3

_

I

I

I

I

I

I

b) 2

1

E IO b

0

-1

101

I

I

I

I

I

I

-2

26 28 30 32 34 36 38 40

26 28 30 32 34 36 38 40

E (MeV)

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

\'~

0

........................................................

-o.2 10

' 11

' 12

' 13

' 14

' 1~

' 16

\ X= 7.77.;7~.~

' 17

' 18

' 19

20

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.

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

345

-28Si+3°Si 0.6

0.~

ng f . . . . . . . r,. . . . . . . . .Strong . . . . couili .... t

t

0.2

0.0

-0.2 26.

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. If 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

x/~3A

d

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

346

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

160 +18 0 10

1

,

,

E = 24 M e V . . . . . .

100

& 10-' b

10.2

10.3 0

I

I

I

I

I

I

I

I

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 o f 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.

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

347

160 +18 0

1(¢

.

.

.

.

I

.

.

.

.

I

.

.

.

0.6

.

.

.

.

.

I

.

.

.

.

I

'

a)

'

'

'

b)

lO ~

0.4

10 2 E

r,n

0.2

#1o 1 0.0

lO 0

-0.2

10-~ 10

15

20

,

,

,

,

,

10

5

E (MeV)

,

,

,

I

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

100

_1_180

Partial cross s e c t i o n s

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.

348

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

observed in the fusion excitation function, it is more clearly observed in the barrier distribution and partial f u s i o n cross sections. The barrier distribution exhibits an oscillatory structure at energies above the C o u l o m b barrier. O n e thus has to use the technique o f utilizing large-angle quasielastic data to extract this, as d o n e by Leigh et al. [ 13]. A n alternative method recently proposed by A c k e r m a n n [ 14] allows one to directly measure the partial fusion cross sections. These also carry the knowledge o f elastic transfer as clearly as the barrier distribution itself.

Acknowledgements O n e o f the authors (J.A.C.) is supported by E P S R C 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 H u m a n Capital and M o b i l i t y program ERBCHRX-CT920075.

References Ill [21 [3] [4] [51 [61 [7[ [8] [9] 110] [1 ll

W. von Oertzen and W. N6renberg, Nucl. Phys. A 207 (1973) 113. W. yon Oertzen and H.G. Bohlen, Phys. Reports 19 (1975) 1. W. Frahn and M.S. Hussein, Nucl. Phys. A 346 (1980) 287. C.H. Dasso and A. Vitturi, Nucl. Phys. A 458 (1986) 157. E Sperr et al., Phys. Rev. Lett. 37 (1976) 321. B. Fernandez et al., Nucl. Phys. A 306 (1978) 259. D.G. Kovar et al., Phys. Rev. C 20 (1979) 1305. J.J. Kolata et al., Phys. Lett. B 65 (1976) 333. N. Poff6, N. Rowley and R. Lindsay, Nucl. Phys. A 410 (1983) 498. A. Kabir, M.W. Kermode and N. Rowley, Nucl. Phys. A 481 (1988) 94. 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.