Classical Dynamics of Damping Effects and of Sub-coulomb Transfer in Collisions of Deformed Heavy Ions : 238u + 238u Reaction

A classical model with dynamical deformations and orientations degrees of freedom is presented for collisions of deformed nuclei. Applications of this model are made to the damping effects and to sub-coulomb transfer in 'U+ 38U reactions at Elab'7. 42 and 5.05-6.07 MeVlu, respectively. I INTRODUCTION The 'U+ 'U reaction is well studied at beam energies of both above and below the Coulomb barrier. For the incident energy above the Coulomb barrier, namely ElabZ7. 42 MeVIu , the characteristic damping effects were observed 11 / and at Elab=5 .7-6.1 MeVIu a new effect of peaked structure in positron energy spectra was found 121. More recently, there have been the measurements 131 of excitation function and angular distributions for the one-neutron-transfer product 2 3 9 U at Elab= 5.05-6.07 MeVlu. The deformation and orientation degrees of freedom affect strongly the dissipation of energy and angular momentum in deep inelastic collisions 141. Inclusion of deformation and orientation effects of the colliding nuclei also indicated I51 the possibility of a minimum or "pocket" in the interaction potential of 38U+ 8U. Such a pocket in the heavy-ion potential is shown to have significant bearing on the cross sections for sub-Coulomb transfer of neutrons 161. For the sub-coulomb transfer in 238U+ 238U the interpretation of the data on the semi-classical theory for spherical nuclei show large deviations, particularly for central collisions 131. The deformation and orientation effects, not yet included, might also be important for these data. In this paper, we describe a classical dynamical model for collisions of two deformed nuclei and consider its application to the data on damping effects and sub-coulomb transfer in 8U+ 8U reactions. I1 THE MODEL We use the classical Hamilton equations of motion for the collective coordinates q v and their canonically conjugate momenta p,, with frictional forces €&included;: with H = T(p,q)+V(q) , v = 1 , 2 , . . . . ,13. (2) ('I Work supported by BM3T and GSI (Darmstadt) ( ' ) Permanent address : Physics Department. Panjab University. Chandigarh-160014. India Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987238 C2-256 JOURNAL DE PHYSIQUE The coordinates are: the relative vector %(R, 0 ,$I) between the nuclear centres of mass, the Euler angles ni=( $Ii, 0i, $i) defining the orientation of the intrinsic principal axes of the two nuclei (i=1,2) with respect to the laboratory system and the intrinsic quadrupole deformations pi and yi. We assume yi=O here. The potential V in (2) consists of Coulomb, nuclear, and deformation energies that are calculated in Ref./7/.The mass parameters, defining the kinetic energy T of the rotation and vibration of the nuclei in (2), are determined by the rotation-vibration model. For the calculation of frictional forces Q,, the Tsang model 181 is extended to the case of two arbitrarily oriented deformed nuclei by allowing a finite range po of the frictional force between the matter elements of two nuclei moving with certain velocity fields. For details, we refer to Ref. 141. The time evolution of the process is studied for 238U+ 2 3 8 U by solving the coupled equations (1) for various incident energies and orbital angular momenta. For incident energies below the barrier, the nuclear interaction potential and the frictional forces are zero, such that only the Coulomb potential acts. Since the deformed nuclei initially can have various orientations, we have solved the equations of motion for arbitrary initial orientations of 8U nuclei. I11 DISSIPATIVE EFFECTS IN 7.42 MEVIU COLLISION OF 238U+ 2 3 8 U The trajectory calculations are first used to calculate the dissipation of energy AE and angular momentum that depend on the choice of the frictional force parameters pO and k. Then, the total kinetic energy (TKE) after the collision, defined as 2 TKE = Ecm AE C Erot,i ( 3 ) i= 1 is calculated as a function of the scattering angle . Fig. 1 shows the results of this calculation for various orientations (solid lines) as well as that obtained by averaging over all possible initial orientations (dashed line). We notice in Fig.1, a spreading of TKE about the mean value, which is largest for central collisions ( 1 3 ~ ~ = 1 8 0 ~ ) and arises mainly due to the nose-to-nose configuration. We have then calculated the double differential cross section d20/(dTKE d 0,,), the differential cross section d olda and the total cross section. Fig. 2 illustrates our Fig.1 The final total kinetic energy vs. scattering angle for 8U+ Z 3 8 U . -Histogram. present colcul. Calc. R.Schmidt et 01.1978 Wolschin 1977 Fig.2 The differential cross section vs. scattering angle for DIC of, 2 3 8 UC 238U at 7.42 MeVlu. calculated doldo (integrated over TKE>25 MeV) in comparison with transport model calculations of Schmidt et al.191 and of Wolschin 1101 and the experimental data 191. The comparison is satisfactory. Integrating over a for 50°<8cm S130°, and multiplying by 0.5 because of the identity of the projectile and target nuclei, the total calculated cross section is 972mb compared with the experimental (800 f50)mb. The qualitative agreement is once again obtained. IV SUB-COULOMB TRANSFER IN 38U+ 238U AT 5.05-6.07 MEVIU For deformed nuclei, apparently the trajectory is no more the true Rutherford trajectory. However, the variation of angular momentum L as a function of does not change much in going from spherical to deformed nuclei, with a very weak dependence on the orientations of nuclei. Also, for larger impact parameters the distance of closest approach for deformed nuclei Rmin coincides with that for spherical nuclei,and at zero and smaller impact parameters the variation of Rmin with 8, is such that for, say, the nose-to-nose configurations the deformed nuclei behave as spherical nuclei of larger radii and for the belly-to-belly configurations as spherical nuclei with smaller radii. According to the semi-classical theory of neutron tunneling, first developed by Breit and his collaborators 1111, and later deduced from DWBA expressions by Buttle and Goldfarb 1121 the integrand of the transfer amplitude is well localized near the distance of closest approach D( gem) for spherical nuclei. The differential cross section can be written in terms of the dimensionle?~ factor CAB containing spectroscopic factors and the wave number a = ( ~ ~ ~ ~ / k ~ ) ? associated with the appropriate neutron binding energy EB, as given in terms of classical Rutherford scattering cross section and the Sommerfeld parameter q. Allowing for the deformation of colliding nuclei in this formalism would mean that the corresponding distance of closest approach Rmin now depend's on the orientations of the nuclei. The averaging over the various possible initial orientations 8i (i=1,2) is carried out analytically by using the following parametrized form: 1 Rmin( 015$1,82 ,@2 ,gem) = ~ ~ ~ ~ f 8 = 9 0 ~ , gem)+ 5 AR(COS ei+coS 92 1 with AR = Rmin(8=00,8 )-R ( 8 = 9 0 ~ , 8 ~ ~ ) cm min where 8=8,=8,=90~ or 0' refer, respectively, to belly-to-belly and nose-to-nose configurations. Eq. (5) is found to fit the actual trajectory calculations made for 238U+ 38U at 5.65 MeVIu. The averaged differential cross section is then given by d o <-> d o nci 8cm d fi 0 [=Ic CAB s i n ( 7 ) ' However, for transfer between deformed wclei one may argue that the correzponding distance between the surfaces, dmin, may be more relevant rather than the centre-to-centre distance Rmin. 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