1. Some aspects of chaos in TDHF …and a bit of laser-induced fission P. D. Stevenson, University of Surrey, UK …and a bit of laser-induced fission P. D. Stevenson, University of Surrey, UK

2. TDHF workshop, Saclay 2006 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. TDHF workshop, Saclay 2006 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 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. TDHF workshop, Saclay 2006 Details of calculations Spherically symmetric 4 He –allows for more or less arbitrarily large box Zero-range BKN-like force: Solve HF equations: Using Taylor expansion of: Spherically symmetric 4 He –allows for more or less arbitrarily large box Zero-range BKN-like force: Solve HF equations: Using Taylor expansion of:

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

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

7. TDHF workshop, Saclay 2006 time series Timeseries is fluctuation of expectation value of r 2. Top panel calculated in 3.0fm box Lower panel in 38.4fm box Timeseries is fluctuation of expectation value of r 2. Top panel calculated in 3.0fm box Lower panel in 38.4fm box

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

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

11. TDHF workshop, Saclay 2006 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 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. TDHF workshop, Saclay 2006 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:

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

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

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

16. TDHF workshop, Saclay 2006 Level Spacing Large phase-space TDHF calculation with reflections gives a large number of s.p. states Expect Wigner-like distribution for chaotic dynamics: Large phase-space TDHF calculation with reflections gives a large number of s.p. states Expect Wigner-like distribution for chaotic dynamics:

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

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

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

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

22. TDHF workshop, Saclay 2006 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 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