1. Lawrence Livermore National Laboratory Two-Step Calculations of Nucleon-Nucleus Optical Potentials LLNL-PRES-414025 Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551 This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 Ian Thompson

2. 2 LLNL-PRES-414025Pack Forest Meeting, June 2009 Lawrence Livermore National Laboratory Nucleon-nucleus Optical Potentials  UNEDF Deliverable: Optical potentials for n-A scattering for energy E n Used for:  Compound-nucleus production – need reaction xsec  R ( ,L)  Entrance/exit channels in direct reactions – need wfs Ψ L (r)  Previous method: Do coupled-channels calculation (CCh) with RPA excited states [T i + V b + e i – E n ] Ψ i (r) + Σ j N V ij (r) Ψ j (r) = 0 Use bare potential V b = V fold that is real. Fit V opt = V b + V DPP by fitting elastic  CCh (  ), after varying parameters of local Woods-Saxon form. Requires ~150,000 RPA channels + ~100 transfer channels Too difficult with present CCh methods (~5000 so far)  UNEDF Deliverable: Optical potentials for n-A scattering for energy E n Used for:  Compound-nucleus production – need reaction xsec  R ( ,L)  Entrance/exit channels in direct reactions – need wfs Ψ L (r)  Previous method: Do coupled-channels calculation (CCh) with RPA excited states [T i + V b + e i – E n ] Ψ i (r) + Σ j N V ij (r) Ψ j (r) = 0 Use bare potential V b = V fold that is real. Fit V opt = V b + V DPP by fitting elastic  CCh (  ), after varying parameters of local Woods-Saxon form. Requires ~150,000 RPA channels + ~100 transfer channels Too difficult with present CCh methods (~5000 so far)

3. 3 LLNL-PRES-414025Pack Forest Meeting, June 2009 Lawrence Livermore National Laboratory Counting RPA excited states (normal parity only) Levels in HO basis N=14RPA levels inside r =16 fm # of Levels E*<10E*<20E*<30E*<40E*<50E*<100 SHO191092795198352297 r = 163929010772602474111763 RPA for 90 Zr

4. 4 LLNL-PRES-414025Pack Forest Meeting, June 2009 Lawrence Livermore National Laboratory How many Reaction Steps are Needed?  Gustavo Nobre’s work tells us we need: All (Q)RPA excited states up to incident energy E n All possible intermediate deuteron channels But not all couplings between the above states.  Jutta Escher’s work tells us we will need: Exchange terms and other non-localities Energy-dependence and/or effective mass and/or nonlocalities in the effective interaction for folding  SO, looks like we need only calculate two-step contributions Couplings from gs to (and from) each excited state. Exactly equivalent to CCh, but more favourable parallelisation  Gustavo Nobre’s work tells us we need: All (Q)RPA excited states up to incident energy E n All possible intermediate deuteron channels But not all couplings between the above states.  Jutta Escher’s work tells us we will need: Exchange terms and other non-localities Energy-dependence and/or effective mass and/or nonlocalities in the effective interaction for folding  SO, looks like we need only calculate two-step contributions Couplings from gs to (and from) each excited state. Exactly equivalent to CCh, but more favourable parallelisation Tried by Coulter & Satchler (1977), but only some inelastic states included

5. 5 LLNL-PRES-414025Pack Forest Meeting, June 2009 Lawrence Livermore National Laboratory Two-Step Approximation  We need only calculate two-step contributions These simply add for all j=1,N inelastic & transfer states: V DPP = Σ j N V 0j G j V j0. G j = [E n - e j – H j ] -1 : channel-j Green’s function V j0 = V 0j : coupling form elastic channel to excited state j Gives V DPP (r,r’,L,E n ): nonlocal, L- and E-dependent. In detail: V DPP (r,r’,L,E n ) = Σ j N V 0j (r) G jL (r,r’) V j0 (r’) = V + i W Quadratic in the effective interactions in the couplings V ij Can be generalised to non-local V ij (r,r’) more easily than CCh. Treat any higher-order couplings as a perturbative correction  We need only calculate two-step contributions These simply add for all j=1,N inelastic & transfer states: V DPP = Σ j N V 0j G j V j0. G j = [E n - e j – H j ] -1 : channel-j Green’s function V j0 = V 0j : coupling form elastic channel to excited state j Gives V DPP (r,r’,L,E n ): nonlocal, L- and E-dependent. In detail: V DPP (r,r’,L,E n ) = Σ j N V 0j (r) G jL (r,r’) V j0 (r’) = V + i W Quadratic in the effective interactions in the couplings V ij Can be generalised to non-local V ij (r,r’) more easily than CCh. Treat any higher-order couplings as a perturbative correction Tried by Coulter & Satchler (1977), but only some inelastic states included

6. 6 LLNL-PRES-414025Pack Forest Meeting, June 2009 Lawrence Livermore National Laboratory Previous examples of Non-local Potentials  Coulter & Satchler NP A293 (1977) 269: Imaginary Part Real Part

7. 7 LLNL-PRES-414025Pack Forest Meeting, June 2009 Lawrence Livermore National Laboratory JIVE: New Project at LLNL Parallel calculation of V DPP (r,r’,L,E n ) now underway  Produce full nonlocality, L- and E-dependence No need for fitting WS parameters  Then calculate V opt, elastic Ψ L (r), and then all nonelastic  j (  ) for comparing with expt. Deliverables: Consider delivering full V DPP (r,r’,L,E n ) ! Look at averages over L, over r-r’, and at parameterising the E n -dependence Up to now, non-locality treated purely ad-hoc, and with several approximations that need to be tested. Methods for non-locality already in the literature. Parallel calculation of V DPP (r,r’,L,E n ) now underway  Produce full nonlocality, L- and E-dependence No need for fitting WS parameters  Then calculate V opt, elastic Ψ L (r), and then all nonelastic  j (  ) for comparing with expt. Deliverables: Consider delivering full V DPP (r,r’,L,E n ) ! Look at averages over L, over r-r’, and at parameterising the E n -dependence Up to now, non-locality treated purely ad-hoc, and with several approximations that need to be tested. Methods for non-locality already in the literature.

8. 8 LLNL-PRES-414025Pack Forest Meeting, June 2009 Lawrence Livermore National Laboratory Target A = (N,Z) UNEDF: V NN, V NNN … V eff for scattering Structure Models Methods: HF, DFT, RPA, CI, CC, … Transition Density [Nobre] Ground state Excited states Continuum states Folding [Escher, Nobre] Transition Densities KEY: UNEDF Ab-initio Input User Inputs/Outputs Exchanged Data Related research E projectile Transition Potentials Coupled Channels or DWBA [Thompson, Summers] Optical Potential [Arbanas] Preequilibrium emission Partial Fusion Theory [Thompson] Hauser- Feshbach decay chains [Ormand] Compound emission Residues (N’,Z’) Elastic S-matrix elements Inelastic production V optical Global optical potentials Deliverables UNEDF Reaction Work Resonance Averaging [Arbanas] Neutron escape [Summers, Thompson] or Two-step Optical Potential Reactions Workflow