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Nuclear deformation in deep inelastic collisions of U + U

Nuclear deformation in deep inelastic collisions of U + U. Contents. Introduction Potential between deformed nuclei Multipole expansion of the potential Friction forces Classical dynamical calculations Cross sections Summary and conclusions. Introduction.

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Nuclear deformation in deep inelastic collisions of U + U

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

  2. Contents • Introduction • Potential between deformed nuclei • Multipole expansion of the potential • Friction forces • Classical dynamical calculations • Cross sections • Summary and conclusions

  3. Introduction Motivation: Calculation of sequential fission after deep inelastic collisions of 238U on238U, 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 238U. 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.

  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 238U + 238U 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 238U + 238U, Nucl. Phys. A388 (1982) 381 M. Münchow, W. Scheid Classical treatment of deep inelastic collisions between deformed nuclei and application to 238U + 238U, Phys. Lett. 162B (1985) 265 M. Münchow, W. Scheid Frictional forces for deep inelastic heavy ion collisions of deformed nuclei and application to 238U + 238U, Nucl. Phys. A468 (1987) 59

  7. Expectation that potential of 238U + 238U 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) quasibound states R1+R2 R

  8. 2. Potential between the nuclei Coordinates: ={q1, q2,....q13}=q The potential between deformed nuclei is given by Double-folding model, sudden approximation Condition: analytic calculation

  9. Coordinates

  10. Conditions: • attractive potential • V12(r)=V0exp(-r2/r02) with V0<0 • additional repulsive potential is possible. 2 parameters: V0and r0 • : 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 VN of the potential: with

  14. Average radial density distribution of 238U 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 (Ni=4 ).

  15. spherical deformed, =a20=0.26

  16. Ellipsoidal shapes with eccentricities i (biai): ellipsoidal surface expressed in spherical coordinates ri 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. a20 a40 a00 a40/a202 |a00|/a202 2

  21. with intrinsic deformations al0(1), al0(2) Because of the rotational symmetry about the intrinsic z´-axis we have the transformation: with

  22. with

  23. Choice of potential parameters V0 and r0: as reference potential is taken the Bass potential given by s = distance between nuclear surfaces, fitted with spherical density distributions

  24. U0(R)/a202 U2(R)/a20 U4(R)/a202

  25. W0(R)/a202 W2(R)/a202 W4(R)/a202

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

  27. Taylor expansion -K2 -J2 I2 -R0 dV0 /dR R[fm]

  28. Gaussian M3Y

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

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

  31. friction force acting on center of nucleus 1 with restriction to  (a20) - oscillations

  32. 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

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

  34. 5. Dynamical calculations q={q1,q2,....q13}, p={p1,p2,....p13} 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: 238U + 238U at E=7.42 MeV/amu Experiment: Freiesleben et al. (1979) 

  35. 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:

  36. L=0

  37. L=200

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

  39. final total kinetic energy

  40. 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

  41. 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

  42. 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

  43. exp. calc.

  44. cm

  45. 6. Summary and conclusions We considered classically described, deep inelastic collisions of deformed nuclei and applied the formalism to the collisions of 238U + 238U at Elab=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.

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