1. The continuum time-dependent Hartree-Fock method for Giant Resonances
C. I Pardi & P. D. Stevenson
Department of Physics University of Surrey Guildford Surrey United Kingdom

2. Background to the problem
Starting from a time-dependent mean field, in the form of the self-consistent TDHF method, is a powerful way to tackle many kinds of collective motion:
Giant resonances
Deep-inelastic collisions
Fusion
Fission (maybe)
For giant resonances, the low-amplitude TDHF limit corresponds to RPA
Giant resonances are in the continuum and require treatment of the continuum no matter the basic theoretical approach
Recent review on TDHF & related methods: Cédric Simenel, Eur. Phys. J. A 48, 152 (2012)

3. Why Giant Monopole Resonances, briefly?
Giant Monopole resonances are radial excitations, with the T=0 oscillation being termed the “breathing mode”
Their properties correlate strongly with the nuclear compressibility, and hence important in determining the Equation of State
More prosaically, their radial nature means that, for a nucleus which is spherical in its ground state, it remains spherical for all time during excitation and decay ⇒ Good simple starting point when developing new theoretical methods.
The spherically-symmetric case can also be calculated by quasi-exact methods, for comparison
IVGDR
Picture: Harakeh and van der Woude, Giant Resonances,
OUP

4. Experimental strength
Y.-W. Lui, H. L. Clark, and D. H. Youngblood, Phys. Rev. C 64, (2001)

5. Reflection

6. Box boundary conditions

7. Strength Functions

8. Absorbing potentials
Can add extra spatial points, and place an imaginary absorbing potential in it.
Can be costly to get a sufficiently exact result.
P.-G. Reinhard, P. D. Stevenson, D. Almehed, J. A. Maruhn and M. R. Strayer, Phys. Rev E 73, (2006)
Much other work on absorbing boundaries in nuclear physics, esp by Nakatsukasa et al (e.g. Nakatsukasa and Yabana, EPJA25, 527 (2005))
<r^2>

9. Strength Functions with absorption
Strength functions are particularly sensitive to the effects of reflection:
P.-G. Reinhard, P. D. Stevenson, D. Almehed, J. A. Maruhn and M. R. Strayer, Phys. Rev E 73, (2006)

10. Laplace Transform Method
We implemented an exact solution to the problem using a Laplace transform of the TDHF equation.
It requires finding analytic solutions for the transformed wave function in a region exterior to the nucleus and transforming back to get the time-dependent wave function
Some lengthy manipulation gives the neutron kernel as:
Full details in Pardi & Stevenson, PRC87, (2013)
Note k=sqrt(2is)
Where R is the spatial coordinate of the edge of the box, and
and

11. Inverse Transform
This kernel is represented in partial fractions, which enables the inverse transform to be performed analytically, giving
In terms of a time-like coordinate 𝜏. This must be integrated from t=0 to the current simulation time to recover the history of all outgoing flux at the boundary.
This exact boundary condition is non-local in time, which incurs a computational cost.
Proton states require a different kernel thanks to the long-range Coulomb potential. We use a numerical approximation to the kernel to enable us to analytically inverse-transform it.

12. Results
Isovector Monopole response in O-16 with simplified Skyrme force (30 fm box)

13. Strengths 16O 40Ca artefact-free strengths
Simple t0-t3 force gives e.g. incorrect ordering of IS vs IV resp: Full Skyrme force needed

14. Summary & Outlook
We have implemented a formally exact boundary condition for time-dependent problems
The application so far is to giant monopole resonances with a simplified Skyrme force
Strength functions can be practically calculated to arbitrary energy resolution.
In the short-term future we will extend to full Skyrme force
Longer term plan is to make multipole expansion of nuclear densities. Implementation in axial or 3D code probably prohibitive at present.

15. Acknowledgements This work funded by the UK STFC
Performed in part in collaboration with K Xu (NAG, Oxford)