Some aspects of chaos in TDHF

1
Some aspects of chaos in TDHF

• and a bit of laser-induced fission
• P. D. Stevenson, University of Surrey, UK

2
Revisiting chaos in GRs

• Previous study of chaos in GR (Vretenar et al.
  PRE56, 6418 (1997)

• TDHF gives time series solution to equations of
  motion
• ISGMR showed regular motion with a single strong
  peak in Fourier spectrum
• ISVGR showed more complicated motion

3
Discrete Continuum RPA

• With reflecting boundary conditions, outgoing
  spherical wave is reflected back causing resonant
  standing waves
• structure of spectrum, and timeseries is highly
  dependent on available space
• Choosing a small enough space should allow
  excitation of a single mode

4
Details of calculations

• Spherically symmetric 4He
• allows for more or less arbitrarily large box
• Zero-range BKN-like force
• Solve HF equations
• Using Taylor expansion of

5
box-size dependence

• as space increases, density of eigenmodes
  increases
• corresponding timeseries look very different
• strength function converges as space increases

6
Strength functions

• Continuum strength regained by smoothing procedure

7
time series

• Timeseries is fluctuation of expectation value of
  r2.
• Top panel calculated in 3.0fm box
• Lower panel in 38.4fm box

8
Phase plots

• Again 3fm and 38fm boxes. Compare with previous
  IS vs IV

9
RPA

• There is a strong dependence in (phase) space for
  the timeseries.
• These are small amplitude calculations so the
  Fourier transforms give the RPA amplitudes.
• Motion is bound to be made of superposition of
  harmonic RPA eigenmodes.
• What happens when the degrees of freedom become
  infinite?

10
Continuum calculation

• Continuum calculation is quickly damped
• chaotic region occurs later

11
Reflected flux

• The chaotic region is caused by the reflected
  flux
• Unphysical in the sense that nuclei do not
  usually sit in reflecting boxes
• Physical in the sense of a plausible thought
  experiment
• Only input is nuclear effective interaction and
  TDHF

12
Control Parameter

• Because of the box size dependence can use it as
  a control parameter to see the onset of chaos -gt
  a bifurcation-like plot

13
Dependence on initial conditions

• At large time, similar initial conditions become
  large differences

14
Duffing Oscillator

• Oscillator with linear cubic force, damping and
  driving term.

15
Analogy

• kinetic
• energy

Nonlinear potential
Driving reflected flux impinging on nucleus
Damping - particle escape
16
Level Spacing

• Large phase-space TDHF calculation with
  reflections gives a large number of s.p. states
• Expect Wigner-like distribution for chaotic
  dynamics

17
Laser-induced fission

• Recent (1999 2000) experiments have
  demonstrated laser-induced fission
• Motivated by application to waste transmutation
• Intense laser pulse creates plasma
• Fission then induced by Bremsstrahlung

18
Demonstration

• Real experiment on 238U
• For demonstration purposes use a light deformed
  nuclide 12C
• No detailed analysis yet but some ASCII density
  plots

t0
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t1475 fm/c
t1503 fm/c
t1512 fm/c
20
t1528 fm/c
t1536 fm/c
t1552 fm/c
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t1556 fm/c
t1560 fm/c
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acknowledgements

• In collaboration with
• D. Almehed, C. Goddard, University of Surrey
• J. A. Maruhn, Universität Frankfurt
• P.-G. Reinhard, Universität Erlangen
• M. R. Strayer, Oak Ridge National Laboratory
• J. Rikovska Stone, University of Surrey