Some aspects of chaos in TDHF
…and a bit of laser-induced fission
P. D. Stevenson, University of Surrey, UK
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
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
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
box-size dependence
• as space increases,
density of eigenmodes
increases
• corresponding timeseries
look very different
• strength function
converges as space
increases
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
TDHF workshop, Saclay 2006
Strength functions
Continuum strength regained by smoothing
procedure
TDHF workshop, Saclay 2006
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
Phase plots
• Again; 3fm and 38fm
boxes. Compare with
previous IS vs IV:
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?
TDHF workshop, Saclay 2006
“Continuum” calculation
Continuum calculation is quickly damped
“chaotic” region occurs later
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
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:
TDHF workshop, Saclay 2006
Dependence on initial conditions
• At large time, similar initial conditions become
large differences
TDHF workshop, Saclay 2006
Duffing Oscillator
• Oscillator with linear + cubic force, damping
and driving term.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
TDHF workshop, Saclay 2006
Analogy
kinetic
energy
Nonlinear potential
Driving: reflected
flux impinging on
nucleus
Damping - particle escape
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:
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
TDHF workshop, Saclay 2006
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
t=1475 fm/c
t=1503 fm/c
t=1512 fm/c
TDHF workshop, Saclay 2006
t=1528 fm/c
t=1536 fm/c
t=1552 fm/c
TDHF workshop, Saclay 2006
t=1556 fm/c
t=1560 fm/c
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
TDHF workshop, Saclay 2006