Annalen der Physik. 7. Folge, Band 34, Heft 2, 1977, S. 91-98 J. A. Barth, Leipzig Classical Description of Two-Cluster Transfer Reactions in a Coulomb Field By M. EL-NADIand A. OSMAN Cairo University, Physics Department, Faculty of Science, Cairo (Egypt) A b s t r a c t . The theory of two-cluster transfer reactions is reconsidered on the basis of a simple semiclassical approximation. The obtained expression for the differential cross section is in a closed form so as to be easily calculated. A theoretical study with calculations for the angular distrihution for different incoming and outgoing energies, and also for different angular momentum transfer is introduced. Comparing our theoretical calculations with experimental data, satisfactory agreement is obtained especially in the backward direction. Hlassische Beschreibung yon Zwei-,,Cluster-Transfer"-Reaktionen in eineni Coulomb-Feld I n h a l t s u b e r s i c h t . Es wird eine Theorie der Schwerionenreaktionen mit MehrnukleonenTransfer angegeben. Dabei wird das relativ einfache Problem bei Energien unterhalb der Coulombschwelle betrachtet, bei denen die Coulombwechselwirkung den entscheidenden Beitrag liefert. Die Reaktion wird halbklassisch beschrieben, wobei ein geschlossener Ausdruck fur die Winkelverteilung erhalten wird. 1. Introdiiction The theory of two-cluster transfer reactions a t energies well below the Coulomb barrier was considered by EL-NADI [l-31. Due t o the large Coulomb repulsion between the projectile and the target nucleus, the motion of the incoming and outgoing particles is highly distorted. Thus, in such a case the expression of the angular distribution for a given nuclear reaction will largely depend on the Coulomb force. Considering the motion of the particles in the semi-classical approximation, one has t o consider the classical trajectories in the external Coulomb field of the nucleus. The work of ALDERet al. [4] on Coulomb excitation and of LEMMER [ 5 ] on (d, p ) stripping show that in many cases the classical treatment of the Coulomb stripping may lead to simple quantitative expressions for the cross section, which explain the main features of the experimental data. The semi-classical approximation has the advantage of leading t o a closed expression for the angular distribution. I n the quantum mechanical treatment one has t o deal with the complicated Coulomb wave functions which are difficult to compute. I n the present work, we shalluse a classical description for the charged particle motion in the external Coulomb field of the nucleus. The energies of both the incident and outgoing particles are well below the Coulomb barrier; hence following Lemmer's work, one should consider a semi-classical description for the incoming and outgoing particles. 92 M. EL-NADI and A. OSMAN 2. Theory The transition niatrix element for two-cluster transfer reactions is given by [l- 31 : and The transition matrix element for two-cluster transfer reactions is given in ref. [l--31 where we are left withtheintegral i including the Coulomb wave functions f(f)(r) Y f(-)(r)and the bound state wave function G J f M f ( ras: ) f = J dr eiQ.r GhPlff(r)f ( + ) ( r )/*(-)(r) (1) where Expanding Coulomb wave functions in partial wavesfor f ( f ) ( r )and f7-)(r) and introducing a MORINIGO[6] form for the wave function we obtain an expression for with an integral Rn.a refered to as the radial matrix element given by hff R ~= J~dr r+'> e-firFni ~ ~(k,r)~Flf(kfr). (2) I n case when the energies of both the projectile and outgoing particles are well below the Couloinb barrier, i.e. when 7 1, the radial matrix element (2) could be treated by the WKB approximation following the work of ALDERet al. [4] and LEMMER[ 5 ] . The WBK method, which gives good accuracy, gives: with Classical Description of Two-Cluster Transfer Reactions in a Coulomb Field 93 where r,, is the classical turning point given by f(ro) = 0. After a lengthy but straightforward calculation which has no place here, we get for the differential cross-section the following expression : 94 M. EL-NADI and A. OSMAN where q = it - k, - k, f 2k4 +- 2k,, 5 = Vf - Ti, Y = PrT and I n the expression (6) for the differential cross-section the symbols 0 stands for the spectroscopic factors. j a and j b are the total spins of the two clusters in the projectile while Ja and Jb are their spins in the residual nucleus. I , L and 2 are the relative, centre of mass and total angular momentums of the transfered clusters in the projectile, while If, Lfand gf are the corresponding angular momentums in the residual nucleus. 3. Results and Discussion I n some cases of nuclear reactions, the transfered clusters are found in the projectile such that there is no transfer in angular momentum. I n these cases of nuclear reactions, it is found that the transfered relative, I , centre of mass, L,and also the total, 2, angular momentums, are zero. This is very interesting because we find that the very long mathematical expression (6) for the differential cross section reduces t o a very short and closed expression. I n csae of interaction between complex ions where no transfer of angular momentum takes place, i.e. when 1 = L = 2 = 0, we get for the differential cross-section the expression : This forniula is calculated for the nuclear reaction 018(C12,oc) MgZ4with I = L = 2 = 0. It is investigated in case of different C12energies (Fig. 1) and for different excited states of MgZ4(Fig. 2). Similarly, calculations for the differential cross-section for the same reaction were carried out for different angular momentum transfers (Fig. 3). Due to the lack of experimental data a t energies well below the Coulonib barrier, the expression (11) is calculated for the reaction C12(01e,oc) MgZ4which was carried out experimentally a t energies 6.75, 7.393 and 7.929 MeV, all slightly less than the barrier height (= 9.582 MeV), by GROCEand LAWRENCE [7] (Figs. 4, 5 and 6). Cn Figs. 7, 8 and 9 , the expression (6) is applied t o the reaction Bes(Li6,a ) Bll for different excited states of Bll, with Lie energy 1.95 MeV and a Couloinb barrier height of 2.956 MeV. The experimental observations are those of BLAIR[8]. Although the energies in Figs. 4-9 are not sufficiently below the Coulomb barrier height, these curves show significantly a backward rise in the angular distribution which is the characteristic features of Coulomb stripping reactions. The principal result of this work, as may be seen from Figs. 1-3, is that the Coulomb effects become dominating, and the angular distribution changes markedly from the Butler usual forward distributions to a predominantly backward distribution which becomes more peaked as the Coulomb parameter q Land qf increases. Thus, in that typical example one sees that for large q, and qf the angular distribution becomes Gaussian Classical Description of Two-Cluster Transfer Reactions in a Coulomb Field 0 Icm I Fig. 1 95 elcml Fig. 2 Fig. 1 Calculated angular distribution of the ejected a-particles in the nuclear reaction Ola(C12,a ) MgZ4leaving the Mg24 nucleus in its ground state. The calculations are performed at different C12 energies of about one third, half, and near the Coulomb barrier height between the 0 1 % and the C12 nuclei Fig. 2 Calculated angular distribution of the ejected a-particles in the nuclear reaction 016(Cla,a)Mgz4 leaving the Mg24 nucleus in the ground, second excited, and the fourth excited state. The C12energy is taken to be 3 MeV Fig. 3 Calculated angular distribution of the ejected a-particles in the nuclear reaction OlB(C12,a ) Mg24 leaving the MgZ4nucleus in different excited states with different transfered angular momentum L = 0,2, and 4. The Clzenergy is taken to be 4.5 MeV X. EL-NADI and A. OSMAN 96 20.70 - B ($0.07 p- z f -Ti I 0.007 3 ?;, I I I 0 Icm I Fig. 4 Angular distribution of the emitted alpha particles from the reaction C12(016, a)MgZ4 leaving the residual nucleus in its ground state. The energy of the incident 0 l 6 ions is 15.75 MeV. The solid curve in Fig. 4-6 reprzsent our present theoretical calculations. The [7] experimental data are due to GROCEand LAWRENCE 10.0 1 I 0.07 I 0 I I 30 60 1 I 720 150 I 90 0 lcml 1 780 Fig. 6 Angular distribution of the emitted alpha particles from the reaction leaving the residual nucleus in it,s first excited state. The energy of the incident 0 1 6 ions is 17.00 MeV C12(016, a ) Mg24 I 30 I I 60 90 8 icm) I I 120 750 780 Fig. 6 Angular distribution of the emitted alpha particles from the reaction C12(016, a)Mg24leaving the residual nucleus in its ground state. The energy of the incident 0 1 6 ions is 18.60 MeV Classical Description of Two-Cluster Transfer Reactions in a Coulomb Field T 0.700 I -- 97 I I I I I I QO 60 30 . I I I 720 758 mo 8 lcml Fig. 7 Bngular distribution of the emitted alpha particles from the reaction Be9(Li6,a ) B**leaving the residual nucleus in its ground state. The solid curve in Fig. 7-9 is our present theoretical calculations. The experimental data are due t o B ~ a m[8]. ELi6= 3.26 MeV 0' 0 I I I I I 30 60 90 720 750 780 0 lcm 1 Fig. 8 Sngular distribution of the emitted alpha particles from the reaction Be9(Li6,x) B" leaving the residual nucleus in its first excited state 0.700 d I 30 I 60 I go I 720 I 150 1 8 lcml Fig. 9 Angular distribution of the emitted alpha particles from the reaction Be9(Li6,x) Bl' leaving the residual nucleus in the second and third excited state 7 ,\iiti. Pliysik. 7. Folge, Bd. 3'1 M. EL-NADIand A. OSMAN 98 in form about the backward direction, with a width that decreases as q increases. I n the case of L 0, and for low values of q , Butler forward peaks are displaced slightly to larger angles. The same holds for large values of q as is clear from Pig. 3. This shows that the semi-classical treatment of the Coulomb stripping reactions lead to a simple closed forni for the cross-section. + References [l]31.EL-XADI,A. OSMANand T. H. RIHAK,Acta Phys. Hung. 29, 127 (1970). M. EL-NADI,A. OSMANand T. H. RIHAX,Acta Phys. Hung. 29,143 (1970). $1. EL-NADI,A. OSMAN and T. H. RIHAK, U.A.R. J. Physics 5, 45 (1971). K. ALDER,A. BOD, T. Huns, B. MOTTELSOH and A. W’INTHER, Rev. Mod. Phys. 28,432 (1956). R. H. LEYMER, Nucl. Phys. 39, 680 (1962). [GI F. B. MORINIGO, Phys. Rev. 134, B 1243 (1964). [‘i] D. E. GROCE and G. P. LAWRENCE, Nucl. Phys. 65, 277 (1965). [a] J. W.BLAIR,“Proceedings of the Second Conference on Reactions Between Complex Nuclei”, [2] [3] [4] [5] 1960, p. 138. Bei der Redaktion eingegangen am 4. Dezember 1975. Anschr. d. Tierf.: Dr. A. OSMAN Kuwait University, Physics Department, Faculty of Science, P.O. Box 5969, Kuwait

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