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 TDHF workshop, Saclay 2006

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 TDHF workshop, Saclay 2006

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: TDHF workshop, Saclay 2006

5. box-size dependence • as space increases, density of eigenmodes increases • corresponding timeseries look very different • strength function converges as space increases TDHF workshop, Saclay 2006

6. Strength functions Continuum strength regained by smoothing procedure TDHF workshop, Saclay 2006

7. time series • Timeseries is fluctuation of expectation value of r2. • Top panel calculated in 3.0fm box • Lower panel in 38.4fm box TDHF workshop, Saclay 2006

8. Phase plots • Again; 3fm and 38fm boxes. Compare with previous IS vs IV: TDHF workshop, Saclay 2006

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? TDHF workshop, Saclay 2006

10. “Continuum” calculation Continuum calculation is quickly damped “chaotic” region occurs later TDHF workshop, Saclay 2006

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 TDHF workshop, Saclay 2006

12. Control Parameter • Because of the box size dependence can use it as a control parameter to see the onset of chaos -> a bifurcation-like plot: TDHF workshop, Saclay 2006

13. Dependence on initial conditions • At large time, similar initial conditions become large differences TDHF workshop, Saclay 2006

14. Duffing Oscillator • Oscillator with linear + cubic force, damping and driving term. TDHF workshop, Saclay 2006

15. Analogy kinetic energy Nonlinear potential Driving: reflected flux impinging on nucleus Damping - particle escape TDHF workshop, Saclay 2006

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: TDHF workshop, Saclay 2006

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 TDHF workshop, Saclay 2006

18. Demonstration • Real experiment on 238U • For demonstration purposes use a light deformed nuclide: 12C • No detailed analysis yet… but some ASCII density plots: t=0 TDHF workshop, Saclay 2006

19. t=1475 fm/c t=1503 fm/c t=1512 fm/c TDHF workshop, Saclay 2006

20. t=1528 fm/c t=1536 fm/c t=1552 fm/c TDHF workshop, Saclay 2006

21. t=1556 fm/c t=1560 fm/c TDHF workshop, Saclay 2006

22. 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 TDHF workshop, Saclay 2006