Volume 201, number 1 PHYSICS LETTERS B 28 January 1988

LIQUID-DROP EFFECTS IN SUB-BARRIER FUSION REACTIONS ~"

C.E. AGUIAR, V.C. BARBOSA, L.F. CANTO and R. DONANGELO Instituto de Fisica, Universidade Federal do Rio de daneiro, C.P. 68528, 21945 Rio de daneiro, R J, Brazil

Received 5 August 1987

We introduce an operational measure for the enhancement of the fusion cross section at sub-barrier energies in terms of an asymptotic energy shift AE. It is shown that AE has a continuously growing trend with the size of the system. This trend is explained in terms of neck formation using the liquid-drop model. Deviations from this trend are attributed to strong coupling to specific channels.

Tunneling phenomena play a most important role in several areas of sience. In nuclear physics barrier- penetration effects dominate the a-decay, the fission processes and also the low-energy fusion reactions which power energy production in stars. The study of fusion at sub-barrier energies has attracted a lot of interest for many years due to its importance in stel- lar nucleosynthesis, alternative sources of energy, and the production of superheavy elements.

It has been shown that fusion reactions involving light ions can be, in most cases, adequately described as a tunneling process through the one-dimensional barrier generated by the combination of nuclear at- traction and Coulomb and centrifugal repulsions [ 1 ]. It came then as a big surprise that similar attempts to understand heavy-ion fusion cross section data led to predictions which fell short by several orders of magnitude. The observed enhancement in the sub- barrier fusion cross section indicates that the de- scription of this process must involve, in addition to the radial separation, other degrees of freedom. These additional degrees of freedom may be associated to specific details of nuclear structure such as nucleon transfer or vibrational and/or rotational excitations in the colliding nuclei, or to the gross features of nu- clear matter such as the formation of a neck between two liquid drops representing the collision partners.

In this letter we show that the liquid-drop model [2] is able to describe the overal trend of the sub-

Work supported in part by CNPq, FINEP and FUJB.

barrier fusion enhancement. For this purpose we in- troduce an operational measure of the enhancement in terms of an asymptotic energy shift AE which brings the one-dimensional model cross section trof onto the experimental data at energies much below the Coulomb barrier. The cross section aof is para- metrized by Wong's formula [3 ]

O-of(E ) = R 2 ~._..~_~ 2E

× l n [ 1 +exp(~-~ ~ ( E - V ~ ) ] , ( l )

where E is the CM collision energy and liB, RB and ho9 are, respectively, the height, radius and curva- ture of the Coulomb barrier. The three barrier pa- rameters are treated as adjustable quantities. The curvature hm is obtained by imposing that at very low energies the slope of the cross section given by eq. (1) equals that of the data. The parameters VB and RB are determined from a fit of the fusion cross section above the Coulomb barrier (o'f ranging be- tween 100 and 500 mb). These three parameters are not free as they should be related by the constraint

- R----~ Z~--~--~2~,/ J ' (2)

where St is the reduced mass of the system. This re- lation is obtained by requiring that the nuclear po- tential has an exponential behavior at the barrier region. Following this fitting procedure eq. (1) ap-

22 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 201, number 1 PHYSICS LETTERS B 28 January 1988

..El

10 3

10 2

I0

~0 o

161

I 0 - ' I I I 90 I00 105 I10 115

ECM (MeV)

I I I I

58Ni ÷ 58Ni

. f I °°/ :¢

t !

! AE ~j

I

95

Fig. 1. Illustration of the asymptotic shift AE.

proaches the data above the barrier and is displaced by a constant energy shift AE at extreme sub-barrier energies. This is illustrated in fig. 1 for the experi- mental data ofBeckerman et al. [4] on the fusion of two 5SNi nuclei. We applied the above procedure to a variety of systems for which fusion cross sections below 0.1 mb were measured [4-1 1 ]. In fig. 2 we plot the resulting AE values as a function of the sys- tem size, measured by the quantity

( = (Z2/A)ef r = 4Zl Z2 , 4 1 / 3 A I / 3 d A I / 3 _ L A I / 3 % " (3)

• ,'1 I .rx 2 ~., 'a I u ~ n t 2 ]

The error bars indicate the uncertainties in the fits. A different procedure had to be used to find VB for the heaviest systems [10,11 ] included in fig. 2. In such cases the fusion cross section gets large contri- butions from fission channels which were not meas- ured. We carried out a systematic study of the behaviour of the barrier height for the lighter sys- tems, along the lines of ref. [ 1 ], and obtained VB by extrapolating it to larger (-values.

Inspection of fig. 2 shows that the enhancement as measured by AE has a continuously growing trend with the size of the system. The existence of such a global tendency in the data suggests to us that liquid- drop effects are important in order to understand the

enhancement in the fusion cross sections at low energies.

We have recently studied the fusion of two heavy nuclei on the basis of the liquid-drop model and found that at short distances the system becomes un- stable with respect to the formation of a neck be- tween the two nuclei [ 12 ]. This instability appears at distances larger than the radius of the Coulomb barrier and therefore at some energy AB below VB. This situation is illustrated in fig. 3 for the 5SNi+ 58Ni system, where the liquid-drop potential energy is plotted as a function of the radial separation and neck size variables p and n, respectively. The usual Cou- lomb barrier is the peak located at the separation P c - 1.13 and the neck instability occurs at the crit- ical separation p, = 1.27 where there is a sign change in the slope with which the equipotentials rise from the p-axis. This means that the potential-energy bar- rier for fusion along the neck variable vanishes for P<Pn- The quantity AB depends essentially on the size of the system and it may be well parametrized by the expression [ 12]

AE=0.0016( 2"5 MeV. (4)

The neck formation process involves a relatively small fraction of the total mass of the system. The resulting small value of the mass tensor element as- sociated to the neck degree of freedom [ 12] facili- tates the formation of a neck as soon as the barrier disappears. A naive treatment of neck effects in the fusion cross section would then be to simply de- crease the fusion barrier by an amount AB. This quantity should therefore be compared to the asymptotic energy shift AE, taken from the experi- mental data. This comparison is made in fig. 2, where AB is indicated by a solid line. It is immediately clear that AB reproduces very closely the average trend of the AE data. This strongly suggests that neck for- mation is responsible for the overall enhancement of the sub-barrier fusion cross section over the one-di- mensional model predictions. Fig. 2 shows also a few major departures from the average trend. In these cases the discrepancies can be traced to the existence of strong coupling to specific channels. For example the data point indicated by a triangle at (-~ 28, which lies much above the background correspond to the collision of 4°Ar with the highly deformed 1545m iso- tope. In this case there is a very strong coupling to

23

Volume 201, number 1

I...d

I--- I, - r CO

>.- (..9 13/ UJ Z ILl

PHYSICS LETTERS B

2O

- • Si, S+Ni (REF. 7) • S+Mo, Ru, Rh, PcI (REE B)

Ni + Ni (REF. 4,5)

o Ni, Ge + Ge (REE 5,6) 1 5 - o AF + SD (REF. 9)

- • AF +Sm (REE 9)

o Kr +Ge, Mo, Ru (REE I0)

• Zr+Y, Zr~Mo (REF. II)

5 o

÷ >.- - i CO < +

2 8 J a n u a r y 1 9 8 8

15 20 25 50 35 40

SYSTEM SIZE PARAMETER ~;

Fig. 2. Asymptotic energy shift AE (points) as a function of the system size parameters ~. The solid line gives the liquid-drop-model barrier lowering AS, due to the formation of a neck.

O.313 / ~ I / I i t [

v 0.1~

1.0 Pc 1.2 I:)n IA. 1.6 1.8 20

RA[~AL SEPARATN p

Fig. 3. Equipotentials in the liquid-drop model for the 5SNi+ 58Ni system. The numbers on the curves indicate the energies in MeV.

the rotational band which must be treated explicitly. A similar situation occurs with the low lying vibra- tional state in 74Ge which affects the systems indi- cated by open circles in the same ~-region. A detailed

study of the effects of such specific channels in con- junction with the liquid-drop-model background is currently in progress.

References

[ 1 ] L.C. Vaz, J.M. Alexander and G.R. Satchler, Phys. Rep. 69 (1981) 373.

[ 2 ] H.J. Krapl:m, J.R. Nix and A.J. Sierk, Phys. Rex,. C 20 (1979) 992.

[3] C.Y. Wong, Phys. Rev. Lett. 31 (1973) 766. [ 4 ] M. Beckerman et al., Phys. Rev. C 23 (1981 ) 158 I. [5] M. Beckerman et al., Phys. Rev. C 25 (1982) 837. [6] M. Beckerman et al., Phys. Rev. C 28 (1983) 1963. [ 7 ] A.M. Stefanini et al., Nucl. Phys. A 456 (1986) 509. [8] R. Pengo et al., Nucl. Phys. A 411 (1983) 255. [9] W. Reisdorf et al., Nut1. Phys. A 438 (1985) 212.

[ 10] W. Reisdorfet al., Nucl. Phys. A 444 (1985) 154. [ 11 ] J.G. Keller et al., Nucl. Phys. A 452 (1986) 173. [ 12] C.E. Aguiar et al., Phys. Rev. C 31 (1985) 1969; Nucl. Phys.

A, to be published.

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