Nuclear deformation in deep inelastic collisions of U + U

Nuclear deformation in deep inelastic collisions of U + U

Contents1. 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

Motivation: Calculation of sequential fission after deep inelastic collisions of 238U on 238U, 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.

1.Introduction

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.

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.

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

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)

R1+R2 R

quasibound states

2. Potential between the nuclei

Coordinates:

={q1, q2,....q13}=q

The potential between deformed nuclei is given by

Condition: analytic calculation

Double-folding model, sudden approximation

Coordinates

Conditions:

(i) attractive potential

V12(r)=V0exp(-r2/r02) with V0<0

additional repulsive potential is possible. 2 parameters: V0 and r0

(ii) : equidensity surfaces have ellipsoidal shapes.

equidensity surfaces are given by:

with

deformation parameters

transformation to principle axes

with coordinates :

Conservation of mass between two equidensity surfaces when deformation is changed during collision

(iii) expansion of

This yields the nuclear part of the potential

Nuclear part VN of the potential:

with

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 ).

spherical

deformed, =a20=0.26

Ellipsoidal shapes with eccentricities i (biai):

ellipsoidal surface expressed in spherical coordinates ri and i:

Expansion into a multipole series

Axial deformation of the nuclei

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:

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

a20

a40

a00

a40/a202 |

a00|/a202

2

with

intrinsic deformations al0(1), al0(2)

Because of the rotational symmetry about the intrinsic z´-axis we have the transformation:

with

with

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

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

2

W0(R)/a202 W2(R)/a20

2 W4(R)/a202

The Taylor expansion method yields the following approximations for the potentials:

This formula gives the same result for

R[fm]

I2 -R0 dV0 /dR

-K2

-J2

Taylor expansion

Gaussian M3Y

4. Friction forces

extended model of Tsang

infinitesimal force

with

2 parameters: k and

relative velocity:

relative motion

rotation vibration

liquid drop model, incompressible and vortex-free liquid:

with

friction force acting on center of nucleus 1

with

restriction to (a20) - oscillations

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

k=5x10-20 MeVfms

Bondorf et al.

R[fm]

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)

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:

L=0

L=200

final excitation energy of projectile

final total angular momentum of projectile

final total kinetic energy

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

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

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°cm130 °

It resulted:

DIR cal = 970 mb, DIR “exp“ = (80050) mb

exp.

calc.

cm

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.

Quantum mechanical studies lead to coupled channel calculations. Such calculations are only practically possible for light nuclei, for example 12C + 12C.

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.

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.