1. Non-locality in nuclear reactions
What to do about it, and does it matter?
Filomena Nunes
Michigan State University
ECT*, Feb 2017
Supported by: NNSA, NSF, DOE

2. Reaction theory for heavy exotic nuclei
6Li(d,p)7Li
132Sn(d,p)133Sn
59Cu(d,ng)60Zn*
95Mo(d,pg) 96Mo* …?
PRC93, (2016)

3. Outline 0. Our starting point Reduction to a few-body problem
Solving the few-body problem
Our effective interactions
Non-locality in reactions

4. Our starting point A complex many-body problem
Scattering boundary conditions
Importance of thresholds
Large Coulomb interactions
Specific clustering
MeV/u

5. 1. reduction to few-body
Reducing the many-body problem to a few-body problem introduces effective interactions.
How does the original many-body Hamiltonian relate to the few-body Hamiltonian?
We assume that
What are these UNA? R.C. Johnson, ECT*, November 2014

6. 1. reduction to few-body
The role of core excitation? (Summers, Nunes, Moro, Deltuva, etc)
Seems to be important for (d,p) on loosely bound nuclei…
Deltuva, PRC91, (2015)

7. 2. solving the few-body Let’s say we know how to do this!
Faddeev Formalism
CDCC, ADWA, etc, etc…
Let’s say we know how to do this!
(this is another talk…)

8. 3. determining Veff Currently our bipolar thinking:
Veff is self energy of N+A system and can be extracted from many-body theories (microscopic optical potential)
Veff is effective interaction between N-A and should describe elastic scattering (global optical potential)

9. 3. microscopic Veff
Veff is the self-energy extracted from coupled-cluster CCSD
Ab-initio Hamiltonian: NNopt
Basis: HO and Breggren
Larger model space for convergence of potential.
n + 10 MeV
Rotureau et al., PRC in press (2017)

10. 3. microscopic Veff The effective interaction is non-local!
n + 10 MeV
n + 10 MeV
s1/2
d3/2
The effective interaction is non-local!
Rotureau et al., PRC in press (2017)

11. 3. microscopic Veff Non-locality is large and non-Gaussian!
n + 10 MeV
Non-locality is large and non-Gaussian!
Rotureau et al., PRC in press (2017)

12. 3. microscopic Veff There remains an energy dependence!
n + 16O
There remains an energy dependence!
Absorption is small from E=0-10 MeV.
Rotureau et al., PRC in press (2017)

13. 3. determining Veff Currently our bipolar thinking:
Veff is self energy of N+A system and can be extracted from many-body theories (microscopic optical potential)
Veff is effective interaction between N-A and should describe elastic scattering (global optical potential)

14. 3. phenomenological Veff
Perey and Buck (1962): only surface absorption
Tian, Pang and Ma (2015): volume and surface imaginary

15. 3. phenomenological Veff
Perey and Buck: best for E<20 MeV
Tian, Pang, Ma: best for E>20 MeV
(volume absorption important)
Joint analysis of low energy and high energy data indicates, for both PB and TPM, residual energy dependence is needed! PB
a = 0.33±0.02
b = 21.7±2.1
c = 9.5±0.5
TPM
a = 0.30±0.02
b = 15.0±2.1
c = 9.3±0.5
Ws = aE + b(N-Z)/A + c
Bacq et al., in preparation (2017)

16. 4. non-locality in reactions
How to deal with non-locality?
What is the effect of non-locality?
How to pin down non-locality?
Is this a relevant question?

17. non-locality effect in transfer reactions
Systematic study of effect of nonlocality in (p,d) with PB
Titus et al., PRC89, (2014)
Similar study with DOM interaction
Ross et al., PRC92, (2015)
Inclusion of non-locality in adiabatic theories
Titus et al. PRC 93, (2016)
New reaction code NLAT
Titus et al., CPC 207, 499 (2016)
Systematic study of effect of nonlocality in (d,n)
Ross et al., PRC 94, (2016)

18. non-locality effect on wavefunctions
BOUND STATES
Fitted separation energy
Reduction of strength in interior
Increase of magnitude in asymptotics
SCATTERING STATES
Fitted nucleon elastic scattering
Reduction of strength in interior
p+48Ca at 50 MeV

19. non-locality effect in transfer reactions
48Ca(p,d)49Ca at 20 MeV
DWBA
Nonlocality only added to proton channel.
Bound state nonlocality enhances cross section.
Scattering state nonlocality decreases cross section.
Perey correction factor not sufficient.

20. non-locality effect on wavefunctions
d+48Ca at 50 MeV
BOUND STATES
Fitted separation energy
Reduction of strength in interior
Increase of magnitude in asymptotics
SCATTERING STATES
Fitted nucleon elastic scattering
Reduction of strength in interior
d+208Pb at 50 MeV
THREE-BODY DEUTERON SCATTERING STATES (ADWA)
Fitted nucleon elastic scattering
Reduction of strength in interior
Deuteron elastic no longer reproduced

21. non-locality effect in (d,p) with ADWA
48Ca(d,p) at 10 MeV
208Pb(d,p) at 20 MeV

22. non-locality effect in (d,p) with ADWA
132Sn(d,p) at 50 MeV
208Pb(d,p) at 50 MeV

23. non-locality effect in (d,p) with ADWA
Transfer cross sections: Nonlocal relative to local at first peak
Low Energy
General enhancement of cross section
Proton channel most important
Deuteron channel had a modest impact
High Energy
Deuteron channel more important, specially for heavy targets
Competition between effects of bound and scattering effects in proton channel.

24. non-locality effect in transfer reactions
20 MeV
50 MeV
In general there are very few examples of (d,n) data out there
Non-locality in optical potential can produce large differences in the angular distribution
Neutron angular distributions can provide constrains
Important to get the most forward angles!!!
Ross, Titus and Nunes, PRC 94, (2016)

25. non-locality effect: energy shift
MeV
MeV
Energy shift does not provide a quantitative description of the effect of nonlocality:
neither shape nor magnitude
MeV

26. Concluding remarks Solving the few-body problem
A lot of progress has been made and more developments are ongoing for (d,p) on heavy targets.
Determining the effective interactions
Revival of microscopic interactions from ab-initio calculations
All unsatisfying in describing accurately elastic scattering
Need for both non-locality and energy dependence
Including non-locality in transfer reactions
We understand non-locality affects transfer observables and know how to include it. How do we constrain it? Need guidance from microscopic theory…

27. Thank you for your attention
My group: Luke Titus, Alaina Ross, Amy Lovell, Terri Poxon-Pearson, Jimmy Rotureau and Gregory Potel
Collaboration with Pierre-Loic Bacq and Pierre Capel (ULB)

28. 1. reduction to few-body The role of core excitation?
Ross et al., PRC 94, (2016)

29. Benchmarking few-body methods
4N bound state
H. Kamada, et al, PRC 64, (2001)

30. Benchmarking few-body methods
n-3He scattering
p-3He scattering (N3LO)
Viviani et al., Phys. Rev. C 84, (2011)

31. Benchmarking few-body methods
10Be(d,p)11Be
12C(d,p)13C
48Ca(d,p)49Ca
Nunes and Deltuva, PRC 84, (2011)
Upadhyay, Deltuva and Nunes, PRC 85,

32. Faddeev in the Coulomb distorted basis
Faddeev AGS with screened Coulomb (Deltuva et al., PRC71, )
equations written in the plane wave basis
screening radius increases with increasing Z target
larger number of partial waves needed for convergence
integral equation solvers break down
Faddeev AGS including unscreened Coulomb
equations written in the momentum space Coulomb distorted basis
Mukhamedzhanov et al., PRC86, (2012)
assumes interactions are separable
Hlophe et al., PRC88, 06408(2013); PRC 90, (2014)
no screening of interactions – Coulomb included in the basis
Eremenko et al., CPC 187, 195 (2015)
challenge to calculate the Coulomb distorted nuclear form factors
Upadhyay et al. PRC 90, (2014)
implementation of the equations ongoing

33. 2. solving the few-body The problem for a few nucleons (weak Coulomb!)
Scattering is harder than bound states: there are small discrepancies…
When the problem involves intermediate mass systems:
Approximate methods are often used
Depending on the observables, different energy region of validity
When the problem involves heavy mass systems:
No exact methods currently available
Cannot determine whether approximate methods are suitable
The goal: to benchmark various methods for reactions with heavy nuclei and at low energy!!
Transfer to the continuum: recent revival but still approximate methods. Need to benchmark…

34. What about transfer to the continuum?

35. (d,pg) as surrogate for (n,g)
Carlson, Moro and Potel will report on this

36. FRIB in the near horizon
Reach into the r-process nuclei:
masses and detailed spectroscopy
of the r-process path nuclei
W. Nazarewicz