Probing nuclear surfaces by heavy ion reactions

Volume 47B, number 3

PHYSICS LETrERS

12 November 1973

P R O B I N G N U C L E A R S U R F A C E S BY H E A V Y I O N R E A C T I O N S B. NILSSON** Nordita, Copenhagen, Denmark

R.A. BROGLIA, S. LANDOWNE*, R. LIOTTA and A. WINTHER The Niels Bohr Institute, University of Copenhagen, Denmark

Received 2 October 1973 An analysis of the recently measured 160 + 26Mg one- and two-proton transfer reactions is given. It is shown that these reactions probe progressively deeper regions of the surface of the two colliding nuclei. In connection with a discussion of the recent experiment [1] 160 + 26Mg we draw attention to the fact that in this case, as in many others, the two nucleon transfer cross section is of the same order of magnitude as the cross section for one particle transfer. In fig. 1 we illustrate, on the same scale, the form factors for elastic, inelastic and transfer reaction for the above case. Provided that the one- and two-nucleon transfer reactions can be described by the first order perturbation theory it is a consequence of this figure that similar cross sections for the reactions 26Mg(160,15N)27A1 and 26Mg(160,14C)28Si can only be obtained if distances of the order of 7 fm (or smaller) be probed. This distance should be compared to the half density radius of 6 f m = 1.1 (A ~/3 +A1/3) fm where reflection or total absorption is expected to occur. In fig. 2 we display DWBA calculations showing the dependence of the cross sections as functions of the effective radius of the absorptive potential for a fixed real potential, fitting elastic scattering. The expected predominant sensitivity of the two-particle transfer cross section is evident. Note also the correlation between the small angle rise of the one-particle transfer cross section with the overall (renormalization type) increase of the two-particle transfer cross section. The elastic cross section is rather insensitive to the changes in the absorption.

* Present address: Hahn Meitner Institute, Sektion Kernphysik, Berlin, West Germany. ** Present address: The Niels Bohr Institute, University of Copenhagen, Denmark.

From total cross section measurements [ 12] one may estimate the effective radius of total absorption consistent with the real potential used above to be of the order of 7 - 7 . 5 fm. The discrepancy between this number and the absorption radius needed to explain the two-nucleon transfer may be due to the effects of higher order processes. Thus pilot calculations on the effect of inelastic processes indicate that in the present case of strongly deformed nuclei the two particle transfer cross section could be changed by a factor of about two, while the expected effect of successive transfer [3] leaves an uncertainty of a factor of about four [4]. While the total cross section is mainly (for given form factors) related to the range of the absorption, the angular distribution is sensitive to the real potential, the different reactions testing the potential at closer and closer distances. Because of the uncertainties mentioned above no attempt has been made to vary the real potential in order to obtain a fit to the data. The angular distribution in fig. 2 were calculated on the basis of a real standard folded potential [5], which gives rise to the deflection function illustrated in fig. 3(A). The distances of closest approach for the different trajectories are indicated on the curve. Thus the distances 6 - 7 fm correspond to l ~ 18-19. The partial reaction cross section u I for the different partial waves in the exit channel is indicated on fig. 3(B). The maximum for the two-nucleon transfer reaction is located in that region. The fact that the peak in the proton transfer reaction is shifted to l = 17 is associated with the fact that the angular momentum transfer in 189

Volume 47B, number 3

PHYSICS LETTERS

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26Mg+160

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Fig. 1. Inelastic (real part) and one- and two-proton transfer form factors associated with the reactions

26Mg(160,160')26Mg(2+), 26Mg(160, 15N)27AI(gs) and

26Mg(160, 14C)28Si(gs). The real part of the optical potential in the entrance channel is also plotted. The nuclear part of this potential is given by a Saxon-Woods potential of depth V= 35 MeV, rv = 1.14 fm and av = 0.69 fro. This potential has a similar asymptotic behaviour as the folded potential I5] V(r) = 0.8754 r 2 exp (-[r-1.14(161/3 +261/3)]/0.59} MeV. In the calculation of the one-particle form factor it was assumed that the proton moves in the lPl/2 orbital in 160 and in the lds/2 orbital in 27A1.The corresponding amplitudes were obtained using the spectroscopic factors So,,, = 2 (i.e. assuming the Pl/2 shell full) and [ 11 ] Sds/2 (26Mg~(~",~I)27AI)= 2.0. For the 26Mg(160, 14C)28Si reaction the following spectroscopic amplitudes were utilized: (16OI[Cnl] Cnl]] 01140 = 0.88 (lp]/2); -0.08 (ld5/2); -0.06 (ld3/2); -0.11 (lSl/2) and (28Sil[Cnl]Cnlf] o 126Mg)+_-1.03 (ld5/2); 0.51 (2Sl/2); 0.25 (ld3/2). The operator Cnlj creates a proton in the singleparticle orbital nlj. Utilizing an enlarged configuration space which comprises the s, d, f and p shells, one obtains a form factor of the same shape, but larger by a factor 1.2. The interaction responsible for the transfer process was equal to the sum of the single-particle potentials acting on each transferred proton and generated by the 14C core. 190

12 November 1973

the reaction is k = 3 and that the Q-value for the reaction ( 3.85 MeV) is such [5] that it requires the exit partial wave angular m o m e n t u m If to be three units less than the partial wave angular m o m e n t u m in the entrance channel l i given in fig. 3(A). The partial wave cross sections also give a qualitative understanding of the angular distributions. Thus the two particle transfer received contribution in the region l i = 2 1 - 1 6 corresponding classically to forward scattering. Because the deflection function crosses 0 = 0 one expects oscillations from the glory effect• However, since by far the most significant contribution comes from very few partial waves around l 0 = 19, the angular distribution is dominated by a superposition of a few Legendre polynomials with a period of oscillation of 180°/(/0 + 21-). The one particle transfer receives contributions from the range l i = 2 7 - 1 7 which gives rise to an angular distribution characteristic of rainbow scattering at grazing collisions [2]. Thus the drop off at angles beyond the peak is rather insensitive to absorption. The shoulder in the angular distribution at small angles can be ascribed to the deep penetrating orbits, as is also evident from the sensitivity of this part of the cross section to the absorption. The smoothness of the angular distribution is not only due to the moderately large number of partial waves contributing to it, but is also associated with the fact that several magnetic substates ILXMI<_ 3 contribute to the reaction. The theoretical inelastic scattering is typical for the interference between Coulomb and nuclear excitation in a rainbow situation and displays the phase-rule [6] when compared to elastic scattering. Both are relatively insensitive to the changes in the absorption. Finally we point out that the present data strongly support an attractive nuclear potential as used above. Thus the grazing angle can be determined from the one particle transfer reaction to be 3 0 0 - 3 5 ° . Without a nuclear attraction this corresponds to distances of closest approach between 10 and 12 fm (see fig. 3(A)) where the form factors are so small that the absolute cross sections for the one-particle transfer reaction could never be achieved. Also the period in the two nucleon transfer angular distribution of 9 ° implies angular momenta of order If = 19.5. In a Coulomb field this corresponds to a distance of 8.3 fm, which would lead to the same difficulty concerning the two-particle absolute cross section, and would predict [7] a grazing peak at 0 = 50 °.

Volume 47B, number 3 i

PHYSICS LETTERS

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12 November 1973 r

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Fig. 2. Angular distributions associated with the elastic, inelastic, one- and two-proton transfer reactions induced by 160 ions impinging on 26Mg with a laboratory energy of 45 MeV. The two angular distributions displayed in each case were calculated with the same real potential plotted in fig. 1. Two imaginary potentials with Saxon-Woods parametrization were utilized. In both cases the depth was kept fixed at 35 MeV, while the corresponding radius and diffuseness are explicitly displayed in the figures. The one corresponding to r w = 1.04 fm and a w = 0.61 fm has a similar asymptotic behaviour as the foldedpotential [ 5 ]. W(r) = - 1.217 r 2 exp ~ - [ r - 1.04 ( 1 6 1 / 3 + 2 6 1 / 3 ) ] / 0 . 5 4 } MeV. The form factors plotted in fig. 1 were utilized in conjunction with the code DWUCK [ 8] to calculate the angular distributions displayed in (B) and (C). Seventy partial waves were included in this calculation. For the inelastic angular distribution, a version of the code DWUCK modified [9] according to ref. [10] was utilized. In this case 4 0 0 partial waves were included. The experimental data displayed in (A), (B) and (C) are from ref. [1]

OCM

191

Volume 47B, number 3 "--'T

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PHYSICS LETTERS

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Discussions w i t h O. H a n s e n a n d D. Sinclair are acknowledged.

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References

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Fig. 3. Classical deflection function and partial cross sections as functions of l. The classical deflection function 0(l) is obtained from the real potential plotted in fig. 1, and is a function of the angular momentum in the entrance channel. The part of the deflection function corresponding to positive angles has been plotted as a continuous line, while the part corresponding to negative deflection angles has been plotted as dashed-point line. The numbers by the side of each box (or crossed circle for the case of negative angles) indicates the corresponding distance of closest approach. The dashed line is the deflection function when only the Coulomb interaction between 160 and 26Mg is considered. Note that the partial angular momentum cross section for the one- and two-proton transfer are given as functions of the exit channel angular momentum.

192

12 November 1973

[1] P.R. Christensen et al., Phys. Lett. 45B (1973) 107; J.B. Ball, D. Sinclair, J.S. Larsen, O. Hansen and F. Videbaek, Phys. Rev. Lett., to be published. [2] R.A. Broglia, S. Landowne, R.A. Malfliet, V. Rostokin and A. Winther, Phys. Rep., to be published. [3] R.A. Broglia et al., Phys. Lett. 45B (1973) 23. [4] U. G6tz and M. Ichimura, private communication. [5] R.A. Broglia and Aage Winther, Phys. Rep. 4C (1972) 153. [6] R.A. Malfliet, S. Landowne and V. Rostokin, Phys. Lett. 44B (1973) 238. [7] W.E. Frahn and R.H. Venter, Nucl. Phys. 59 (1964) 651. [ 8] P.D. Kunz, private communication. [9] F. Videbaek, private communication. [10] M. Samuel and U. Smilansky, Comp. Phys. Comm. 2 (1971) 455. [11] W. Bohne et al.,Nucl. Phys. A131 (1969) 273. [12] B. Wilkins and G. Igo, Proc. Third Conf. on Reactions between complex nuclei (Univ. of California Press, 1963) p. 241; L. Kowalski et al., Phys. Rev. 169 (1968) 894.