Fusion excitation function of the 40Ca + 40Ca system close to the threshold

Nuclear Physics A373 (1982) 341-348 ® North-Holland Publishing Company

FUSION EXCITATION FUNCTION OF THE 4°Ca+"Ca SYSTEM CLOSE TO THE THRESHOLD E . TOMAS], D . ARDOUIN', J . BARRETO, V . BERNARD, B . CAUVIN, C. MAGNAGO, C. MAZUR, C . NGA, E . PIASECKI and M . RIBRAG DPh.N/MF, CEN Saclay, 91191 Gif 3wr-Yretle Cedex, France Received 20 July 1981 (Revised 15 September 1981) Abstract : We have measured the fusion excitation function of the "Ca +` °Ca system between 112 .5 and 165 MeV . The results are in agreement with those of Barreto et al., but not with those of Doubre et al. The fusion threshold which is deduced agrees with the one calculated using an interaction potential calculated by the energy density formalism. Furthermore this potential allows us to reproduce rather well the excitation function.

1. Introdaction The investigation of fusion excitation functions at low bombarding energies above the interaction barrier is useful to calculate the fusion threshold and the fusion radius . For light systems and at these energies, the compound nucleus which is formed decays almost entirely by light-particle and y-ray emission. Consequently the fusion cross section turns out to be equal to the evaporation residues cross section. The investigation of the "Ca +"Ca system is of some interest for the following reasons : being a doubly magic system, we could expect to observe the influence of shell effects on the fusion threshold and on the fusion radius compared to nonmagic systems. On the other hand, because we have to deal with magic nuclei we expect that they will remain spherical at the stage where the fusion process is decided. Therefore this allows a more direct comparison with model calculations where the interaction potential is usually calculated within the sudden approximation assuming two interacting spherical frozen densities. The fusion excitation function of the "Ca + "Ca system has already been measured by Doubre et al. ') and Barveto et al. Z). However they have obtained quite different results, especially just above the fusion threshold. They therefore reached different conclusions concerning the possible structure effects and dissipative processes which are involved in the fusion reaction . '

Université de Nantes, Institut de Physique, 44072 Nantes Cedex, Frana. 341

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2. Experimental procedure The "Ca beam was provided by the Tandem accelerator of Orsay. The target of 4'Ca (50 2 pg/cm ), evaporated on a carbon backing (15 pg/CM2), was bombarded with the "Ca beam with an energy ranging between 112.5 and 165 MeV . The reaction products were detected using a large-area two-dimensional positionsensitive ionization chamber which was developed by Sann et al . '). The details of this set-up can be found in ref. ') but we shall recall here a few of its characteristics. It is a twin ionization chamber but only one part of it was used in our experiment . This part covers 22° in the horizontal plane and 3° in the vertical plane. It was filled with methane at pressures of 30 to 50 Torr, depending on the incident energy . The length of the ionization chamber was 34 cm and consisted of two AE (2 cm and 3 cm) detectors and a longer part for the residual energy E(29 cm) detector ; the bias given to the electrodes was such that the electric field in V/cm divided by the pressure in Torr was equal to 1 . The position in the horizontal plane ofthe incident particle was measured by means of a grid read by a delay line. The position in the vertical direction was measured by the drift time of the electrons from the initial track to the anode. The resolution of the position, measured with fission fragments, was 0.3° in the horizontal plane and 0.15° in the vertical plane when the entrance of the ionization chamber was located at 1 m from the source . The energy resolution obtained in this experiment was around 3 %. A drawing of the entrance window is shown in fig. 1 . It differs from the one used in ref. 3). It has 13 small apertures, each one covering 3° in the horizontal plane and 0.2° in the vertical plane. 100 pg/cm' stretched polypropylene foils were glued on each aperture . The choice of this entrance window, compared to the one of ref. '), was done for two reasons (i) Because the ionization chamber was located at forward angles it is necessary to reduce the solid angle of ref. ') (25 msr). (ii) The evaporation residues have a relatively low kinetic energy, and consequently the window should be very thin. This can only be done if the area of the foil is not

Fig. l . Scheme of the entrance window of the ionization chamber (see text) .

E . Tomasi et al. / Fusion excitation

343

40Co + 40Ca -_ 135 MeV Elab 3! < 0 < 27! from Car

ton

t

Evaporation residues Elastic scatterinq_

200(-

100

E (Channels)

200

Fig . 2 . Typical AE -E plot observed in the experiment .

very large, which is the case for these small apertures, but not the case of the window of ref. 3). The ionization chamber was located in such a way that we could detect the products which were emitted between 3° and 25° in the horizontal plane. Five bombarding energies (112 .5, 120, 135, 150 and 165 MeV) were investigated . Due to the small value of the pressure in the ionization chamber, it was not possible to completely stop the elastic scattering products. Nevertheless, in the E-AE plane, the evaporation residues, the elastic scattering events and the evaporation residues coming from the carbon backing could be, at all bombarding energies, well separated from each other. This is illustrated in fig. 2 for a typical example where we see that the evaporation residues are well separated from the elastic products and from the reaction products arising from the carbon backing or the oxygen impurities in the target . The great advantages of the experimental set-up used here are the following: (i) It allows us to cover a large angular range and consequently covers at the same time all the angles usually experinientally available. (ii) The elastic scattering is measured at the same time and this allows an easy normalization of the residual nuclei cross section to the Rutherford cross section. (iii) In addition, due to the large angular range covered, a determination of the exact position of the chamber with respect to the beam axis can be done by means of the variation of the elastic cross section as a function of the angle. 3. Results In fig. 3 is shown the angular distribution of the residual nuclei at a bombarding energy of 150 MeV. We see that the evaporation residues are emitted between 0°

E. Tomasi et al. / Fusion excitation

344

10

f

0

"Ca

fil

. "C'

150 MeV 10 E lob -

00-

i-

S!

10"

ii 15 " 0 tab' (degrees)

Fig. 3 . Angular distribution of the evaporation residues for E,,b = 150 MeV .

and 15°. Therefore all the points of this angular distribution could be measured at the same time. For each dot, the statistical error is indicated. Although the minimum detection angle with the ionization chamber was equal to 3° we were not able to measure the maximum of the angular distribution . In fig. 4 is shown the evaporation residue cross section for the "Ca +"Ca system as a function of the c.m . energy . It includes the results of Doubre et al., Barreto et al., and the results of our measurement . Typical errors are of the order of 25 %. These errors include the statistical errors and the uncertainty in the maximum of the angular distribution ; that is, on the extrapolation of it at small angles . We see that we are in agreement with Barreto et al. If we fit the plot of the evaporation residue cross section (which is for this system equal to the fusion cross section) as a function of 1/E, ., by a straight line, we ob. tain the following value for the fusion barrier : VFP = 55.56 MeV. From this value we can deduce a reduced interaction radius defined as V«P Z1Zze z F r , (A, / s+A2 / 3)'

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345

Fig. 4. Evaporation residues cross section for the "Ca+' °Ca system . Our results are reported together with the ones of Barreto et al. and Doubre et al. It also shows (full curve) the result of a calculation according to ref.') (see text). The error bars include statistical errors together with the errors due to the uncertainty of the angular distribution at small angles .

where A,, A2, Z, and Z2 are the masses and the atomic numbers of the projectile and of the target . We obtain r, = 1 .52 fm. Eq. (1) assumes that the nuclear potential is zero at the fusion barrier, which is of course wrong, but this definition of r, is often used in the literature to quickly compute VF . We shall now compare the experimental results with the interaction potential calculated using the energy density formalism of ref. `). The theoretical fusion barrier VF which is calculated is equal to 55.9 MeV and the position of the top of the barrier is equal to R F = 9.4 fm. R F and YF are connected to each other by VF = Vc(RF) + VN(RF),

(2)

where we have VN(R F) = - 5.2 MeV for the nuclear part and Vc(RF) = 61 .1 MeV for the Coulomb part. Ibis latter value is close to the Coulomb interaction energy between two point charges Z,Z2e2 RF

= 61 .3 MeV.

(3) .

For the same system the proximity potential ') gives a fusion barrier which is a little bit higher (see table 1 where the results are summarized) . We now calculate the fusion cross section OF using the energy density interaction

346

E. Tomasi et al. / Fusion excitation TABLE I

Interaction barrier VF and fusion radius RF obtained from the experiment and from calculation using : (i) the energy density `) and (ii) the proximity formulation') System 4 0Ca+ 4

0Ca

Experiment

Energy density

Proximity

VF (MeV)

RF (fm)

VF (MeV)

VN (MeV)

RF (fm)

VF (MeV)

VN (MeV)

RF (fm)

55 .6±0 .8

9.1 t0 .6

55 .9

-5 .2

9.4

57.7

-4 .9

9.2

potential of ref. '). It is given by it °` QF = z Il (21+ =0

1)T,,

(4)

where k is the wave number and T, the transmission coefficient for the wave l. We approximate the transmission coefficients T, by those of a parabolic barrier. This parabolic barrier is osculating the total interaction potential at RF,,, the position of the barrier for the wave l. In such a case T, is given by the Hill-Wheeler formula : T,

_

1 l +exp (2$( VF., - E, .m ./hw,) '

(5)

where VF., is the fusion barrier for the wave l and to, the frequency associated to the inverted oscillator defined by the parabolic barrier: (d2 V it tor = . _ -AD 2 h where u is the reduced mass, V, the total interaction potential for the wave 1. The stiffness coefficient d2 V,/dR2 is calculated at R = RF,,, the position of the fusion barrier for the wave 1. The T, have been calculated for several bombarding energies and uF has been calculated using eq . (4). The results are shown in fig. 4 by the full line which reproduces the experimental results of Barreto et al., and our results, rather well. In calculating the interaction potential according to ref. 4) there are no adjustable parameters . It should be noted that, except for the lowest bombarding energy, the transmission coefficients go from a value close to 1 to a value close to zero in a domain of 3 !values . This means that we have an almost sharp cut-off behaviour of the transmission coefficients . Assuming the sharp cut-off approximation and RF,, = RF, 0 we would obtain for QF QF = ?rRF .O(1 - vF .OIEa.m .) "

E. Tomasi et al . / Fusion excitation TABLE

347

2

For the "Ca +"Ca system at the investigated bombarding energies are indicated the experimental cross section, the calculated cross section (see text), 1. deduced from the experimental value of aF according to formula (8) and the 1« corresponding to a value of the transmission coefficients T, = 0.5, using the energy density formalism System "Ca +"Ca

(Mev)

E,.,

E, ... (Mev)

112.5 120 135 150 165

56.25 60.00 67.50 75 .00 82.50

(mb) (exp)

(calc)

OF

1« (exp)

1« (calc)

49± 7 160± 24 444± 67 828± 87 852+128

37 202 477 680 833

8±1 16±2 29±4 42+6 45±7

7 18 30 38 45

aF

Even if this simple formula is applied we obtain a rather good agreement with the experimental results 6). We report in table 2 the experimental and theoretical values for the fusion cross section. We report also the calculated value of the critical angular momentum l, for the different bombarding energies using the energy density potential and the relation OF Q

-

n 1)2, k2(lcr+

which is derived within the sharp cut-off approximation. 4. Conclusion

Our measurement of the ""Ca +"°Ca fusion excitation function is in agreement with the experiment performed by Barreto et al. but disagrees with the measurement of Doubre et al. The experimental results obtained here are rather well reproduced by the interaction potential calculated using the energy density formalism within the sudden approximation . We would like to acknowledge the staff of the Orsay Tandem accelerator for providing us a very good beam during this experiment. We would also like to express our gratitude to Hubert Doubre who helped us a lot during the, experiment . We also thank R. Marquette and D. Sznajderman for preparing the target .

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E. Tomasi et al. / Fusion excitation

References 1) H . Doubre, A . Gamp, J . C . Jacmart, N . Poffé, J . C. Roynette and J . Wilczynski, Phys . Lett . 739 (1978) 135 2) J . Barreto, G . Auger, H . Doubrc, M . Langevin and E. Plagnol, Annual Report IPN Orsay (1980), to be published ; J . Barreto, thesis, Orsay (1980) 3) H . Sann, H . Damjantschitsch, D. Hebbard, J . Junge, D. Pelte, B . Povh, D. Schwalm and D. B. Tran Thoai, Nucl . Instr . 124 (1975) 509 4) H . Ng6 and C. Ng6, Nucl . Phys. A348 (1980) 140 5) J . Blocki, J . Randrup, W. J . Swiatecki and C . F. Tsang, Ann . of Phys. 105 (1977) 427 6) E . Tomasi, D. Ardouin, J . Barreto, V . Bernard, H . Doubre, C. Ng6, C. Mazur, E. Piasecki and M . Ribrag, 3rd Adriatic Europhys. Study Conf. on the dynamics of heavy-ion collisions, Hvar, Yugoslavia May, 1981, Fizika 13(1981)6