1. Nuclear deformation in deep inelastic collisions of U + U

2. Contents 1.Introduction 2.Potential between deformed nuclei 3.Multipole expansion of the potential 4.Friction forces 5.Classical dynamical calculations 6.Cross sections 7.Summary and conclusions

3. Motivation: Calculation of sequential fission after deep inelastic collisions of 238 U on 238 U, Exp.:Glässel,von Harrach, Specht et al.(1979) Needed: Excitation energy and angular momentum of primary fragments. These quantities depend strongly on deformation and initial orientation of 238 U. Siwek-Wilczynska and Wilczynski (1976) showed that the distribution of final kinetic energy versus scattering angle depends on deformation. Modification of potential in exit channel. 1.Introduction

4. Schmidt,Toneev,Wolschin (1978): extension of this model by taking into account the dependence of deformation energy on angular momentum. Deubler and Dietrich (1977), Gross et al. (1981), Fröbrich et al. (1983): Classical models applied to deep inelastic collisions and fusion processes with deformed nuclei. Dasso et al. (1982): Double differential cross sections as functions of angular momentum and scattering angle for collision of a spherical projectile on a deformed target.

5. Here: Complete classical dynamical treatment of orientation and deformation degrees of freedom of deep inelastic collisions of 238 U + 238 U by Münchow (1985) (before not fully taken into account). Model: double-folding model for potential; extended model of Tsang for friction forces; classical treatment of relative motion, orientation and deformation of the nuclei.

6. Publications: M. Münchow, D. Hahn, W. Scheid Heavy-ion potentials for ellipsoidally deformed nuclei and application to the system 238 U + 238 U, Nucl. Phys. A388 (1982) 381 M. Münchow, W. Scheid Classical treatment of deep inelastic collisions between deformed nuclei and application to 238 U + 238 U, Phys. Lett. 162B (1985) 265 M. Münchow, W. Scheid Frictional forces for deep inelastic heavy ion collisions of deformed nuclei and application to 238 U + 238 U, Nucl. Phys. A468 (1987) 59

7. Expectation that potential of 238 U + 238 U has minimum at touching distance. Study of molecular configurations in the minimum in connection with electron- positron pair production by Hess and Greiner (1984) V(R) R 1 +R 2 R quasibound states

8. 2. Potential between the nuclei Coordinates: ={q 1, q 2,....q 13 }=q The potential between deformed nuclei is given by Condition: analytic calculation Double-folding model, sudden approximation

9. Coordinates

10. Conditions: (i)attractive potential V 12 (r)=V 0 exp(-r 2 /r 0 2 ) with V 0 <0 additional repulsive potential is possible. 2 parameters: V 0 and r 0 (ii) : equidensity surfaces have ellipsoidal shapes.

11. equidensity surfaces are given by: with deformation parameters transformation to principle axes with coordinates :

12. Conservation of mass between two equidensity surfaces when deformation is changed during collision (iii) expansion of This yields the nuclear part of the potential

13. Nuclear part V N of the potential: with

14. Average radial density distribution of 238 U can be expressed in the form of a Fermi distribution: (r)= 0 /(1 + exp((r –c)/a) The parameters are c=6.8054 fm, a=0.6049 fm and  0 =0.167 fm -3. Fitted by Gaussian expansion, only 5 terms are needed (N i =4 ).

15. spherical deformed, =a 20 =0.26

16. Ellipsoidal shapes with eccentricities  i (b i a i ): ellipsoidal surface expressed in spherical coordinates r i and  i : Expansion into a multipole series

17. Axial deformation of the nuclei

18. 3.Multipole expansion of the potential The ellipsoidal shapes can be related to multipole deformations of even order, defined by  lm (1) and  lm (2) with l=0,2,4. General expansion:

19. Leading deformation of ellipsoidal shapes is the quadrupole deformation and is taken into account up to quadratic terms. Monopole and hexadecupole terms can be expressed as Inserted in the potential yields 8 potentials

20. a 20 a 40 a 00 a 40 /a 20 2 |a 00 |/a 20 2 22

21. with intrinsic deformations a l0 (1), a l0 (2) Because of the rotational symmetry about the intrinsic z´-axis we have the transformation: with

23. Choice of potential parameters V 0 and r 0 : as reference potential is taken the Bass potential given by s = distance between nuclear surfaces, fitted with spherical density distributions

25. U 0 (R)/a 20 2 U 2 (R)/a 20 U 4 (R)/a 20 2

26. W 0 (R)/a 20 2 W 2 (R)/a 20 2 W 4 (R)/a 20 2

27. The Taylor expansion method yields the following approximations for the potentials: This formula gives the same result for

28. R[fm] I 2 -R 0 dV 0 /dR -K 2 -J 2 Taylor expansion

31. Gaussian M3Y

32. 4. Friction forces extended model of Tsang infinitesimal force with 2 parameters: k and 

34. relative velocity: relative motion rotation vibration liquid drop model, incompressible and vortex-free liquid: with

35. friction force acting on center of nucleus 1 with restriction to  (a 20 ) - oscillations

36. moment of force acting on nucleus 1 with Comparison with radial friction force of Bondorf et al. k = 5 x 10 -20 MeV fm s for  = 2.3 fm

37. k=5x10 -20 MeVfms Bondorf et al. R[fm]

38. 5. Dynamical calculations q={q 1,q 2,....q 13 }, p={p 1,p 2,....p 13 } Hamiltonian H=T(p,q)+V(p,q), friction forces Q classical equations of motion =1,....13 : dq /dt=dH/dp dp /dt=-dH/dq + Q We considered: 238 U + 238 U at E=7.42 MeV/amu Experiment: Freiesleben et al. (1979) 

39. Assumption: rotationally symmetrical shapes,  i =0 Excitation energy of nucleus i: with and friction coefficient  j for  j – vibration Spin of nucleus i after collision:

40. L=0 

41. L=200 

42. final excitation energy of projectile final total angular momentum of projectile

43. final total kinetic energy

44. 6. Cross sections Classical double differential cross section integration over impact parameter b E = Total Kinetic Energy (TKE) after collision  cm is scattering angle. P = distribution function obtained by averaging over the initial orientations

45. Distribution function (E = final TKE): obtained by solving the classical equations of motion Initial orientation of intrinsic axes: isotropically distributed No events with energy loss >200 MeV. Neglected: statistical fluctuations Single differential cross section d/ cm

46. In the reconstruction of the primary distribution and in the calculation the events with energy losses TKEL < 25 MeV were excluded. Cross section for deep inelastic reaction: d/d is integrated over 50° cm 130 ° It resulted:  DIR cal = 970 mb,  DIR “exp“ = (80050) mb

48. exp. calc.

49.  cm

50. 6. Summary and conclusions We considered classically described, deep inelastic collisions of deformed nuclei and applied the formalism to the collisions of 238 U + 238 U at E lab =7.42 MeV/ amu. The internuclear potential, the densities of nuclei and the friction forces are written by using Gaussian functions and can be solved for arbitrary directed and deformed nuclei.

51. Quantum mechanical studies lead to coupled channel calculations. Such calculations are only practically possible for light nuclei, for example 12 C + 12 C. Shell effects for arbitrary oriented nuclei can be calculated with the new two– center shell model of A. Diaz Torres. But for heavy nuclei this model needs large numerical computations.

52. Here, we only studied ellipsoidally deformed nuclei. Also important is the extension of the theory to odd deformation degrees of freedom of the nuclear densities (octupole deformation of the nuclei). We propose to use shifted quadratic surfaces with middle points at. D.G. The same formalism is possible. Also the treatment of the neck degree of freedom is needed.