nuclear deformation in deep inelastic collisions of u + u
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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.
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