1. Nuclear Structure Tools for Continuum Spectroscopy
H. Lenske
Institut für Theoretische Physik
Justus-Liebig-Universität Giessen

2. Continuum Tools, H. Lenske, ECT* 2016
Agenda:
Many-body theory in weakly bound nuclei
Sampling the continuum
„Pairing“ resonances in 10Li
Bound states in the continuum (BiC) and Fano-dynamics
Summary
Continuum Tools, H. Lenske, ECT* 2016

3. Nuclear Many-Body Approaches
Continuum Tools, H. Lenske, ECT* 2016

4. Strategies for Nuclear Spectroscopy
Multi-configuration Shell Model
Truncation to a few valence shells
Complete np-nh treatment in the valence sector
Multi-configuration Mean-Field Model
Truncation in ph-number
Limited, but unrestricted configuration space

5. Density Functional Theory and Multi-Phonon QRPA Theory
Single Particle Self-Energy:
Landau-Migdal Residual Interaction :
Choice of wave function:

6. Binding Energies of Sn-Isotopes
DFT and HFB:
Binding Energies of Sn-Isotopes
N. Tsoneva, H.L., PRC 2008…2016

7. Giessen DFT & QPM: „(D)QPM“ 1-,1+,2+… new Modes of Excitation
PhysRevC (2015)
PhysRevC (2014)

8. Dynamical Core Polarization in 11Be
Single Neutron Spectral Distributions
[0+ × 1/2+]: 0.79
[2+ × 5/2+]: 0.18
[0+ × 1/2-]: 0.58
[2+ × 3/2-]: 0.28

9. Discretization of the Single Particle Continuum
Continuum Dynamics:
Discretization of the Single Particle Continuum

10. Frequently used discretization methods
Expansion into a discrete basis {fnℓ}:
eigenfunctions of the 3-D quantum Harmonic oscillator
Spherical Bessel functions with infinite well boundary condition at R>>Rturn : jℓ (kR)=0
Binning of the continuum (Austern, CDCC…):
Direct integration by e.g. Numerov method:
Continuum Tools, H. Lenske, ECT* 2016

11. Discretization of the Single Particle Continuum
Continuum Tools, H. Lenske, ECT* 2016

12. Numerical approach by expansion into a basis
Expand uℓj into a (complete) basis {fnℓ}
Choice of (orthonormal) basis:
spherical Bessel functions {fnℓ =Anℓxnℓjℓ(xnℓ)}
with xnℓ = qnℓr and fnℓ(R)=0 ↔ jℓ(qnℓR)=0
Obtain the knℓj values from the eigenvalue problem
The scattering phase shifts are obtained by
(c/o the famous Luescher formula of LQCD!)
Continuum Tools, H. Lenske, ECT* 2016

13. ½+ proton Scattering Phase Shifts on C Isotopes
Continuum Tools, H. Lenske, ECT* 2016

14. ½+ Proton Partial Wave Cross Section on C Isotopes
Continuum Tools, H. Lenske, ECT* 2016

15. Density of States in the Continuum: the Speed Plot
Continuum Tools, H. Lenske, ECT* 2016

16. ½+ Proton Density of States in C isotopes
Continuum Tools, H. Lenske, ECT* 2016

17. Sampling the 5/2+ resonance in p+12C
Continuum Tools, H. Lenske, ECT* 2016

18. Evolution of Neutron Single Particle Continuum Strength
Density of States in the Continuum  Speed plot
Jp=5/2-

19. QRPA 1- and 2+ Multipole-Response 128Sn
Microscopic
DD-QRPA

20. Pairing across the Dripline
Continuum Dynamics:
Pairing across the Dripline

21. Pairing in Infinite Nuclear Matter
Free Space SE (S=0,T=1) Interaction:
(Bonn-B NN Potential)
Pairing is a LOW DENSITY Phenomenon

22. Pairing Theory as Coupled Channels Problem: The Gorkov-Equations

23. Pairing in the Continuum
S. Orrigo, H.L., PLB 677 (2009)

24. Mapping to Single Particle Pairing Self-Energies
Energy Shifts and Widths
Spectral Functions for particles and holes

25. Spectrum of the Gorkov Equation:

26. Neutron Spectral Functions in 9Li(3/2-)

27. Neutron Spectral Functions in 9Li(3/2-):
Continuum Admixtures into the g.s.
Continuum Admixture!

28. Pairing Resonances in Dripline Nuclei: 9Li+n 10Li
S. Orrigo, H.L., PLB 677 (2009) & ISOLDE newsletter Spring 2010

29. n+9Li Scattering Phase Shifts
Jp=½+
Jp=½-
n+9Li s-wave
Mean-Field:
as : [fm]
rs : [fm]
Pairing:
as : 1.76[fm]
rs : [fm]
Continuum Tools, H. Lenske, ECT* 2016

30. Effective Range Expansion and Pole Structure
n+9Li s-wave virtual state:
Mean-Field:
as : fm
rs : fm
 E=-1.38 MeV
Pairing:
as : fm
rs : fm
 E=-3.63 MeV
Continuum Tools, H. Lenske, ECT* 2016

31.  talk of Manuela Cavallaro
Continuum Spectroscopy of 10Li=9Li+n via
Numerical challenge for „standard (d,p) stripping theory“:
How to evaluate a 3-D integral with only oscillatory wave functions?
New TRIUMF data and analysis:
 talk of Manuela Cavallaro
Data: H. Jeppesen et al., REX-ISOLDE Collaboration, NPA 738 (2004) 511,;NPA 748 (2005) 374.

32. Bound States in the Continuum
Continuum Tools, H. Lenske, ECT* 2016

33. 1/2+ Particle and Hole Strength Functions in 14C
Particle strength function

34. Interference of Closed Channels (BiC) and Open Channels:
Fano-Dynamics of Asymmetric Resonance Line Shapes
Science 340,716 (2013): Stimulated Ionization of double-excited He

35. Reaction Amplitude and Formation Cross Section
The Fano-Formula:

36. Resonance Production Scenarios in Nuclear Physics
The Fano-Wave Function:

37. Sonja Orrigo, H.L., et al. Phys.Lett. B633 (2006)
Correlation Dynamics in an Open Quantum System: 5/2+ Fano-Resonances in 15C by Core Polarization
G~60…140keV
Sonja Orrigo, H.L., et al. Phys.Lett. B633 (2006)

38. Bound States by Correlation Dynamics through Continuum Coupling
The DCP picture: Binding by Virtual Continuum Coupling
The s.p. shell model picture
H.L.,J. Prog.Part.Nucl. 561 (2004)

39. Supported by DFG, BMBF, HIC for FAIR, and GSI
Summary
Discretization of the single particle continuum
Sampling continuum observables
Correlations across the particle threshold in Li-isotopes
Bound States in the Continuum
Interference of open and closed channels
…with special ontributions by Sonja Orrigo and Nadia Tsoneva
Supported by DFG, BMBF, HIC for FAIR, and GSI
Continuum Tools, H. Lenske, ECT* 2016