Lawrence Livermore National Laboratory 1 PLS Directorate, Physics Division – LLNL, Livermore, CA 2 CEA, DAM, DIF, Arpajon, France 3 University of North Carolina, Chapel Hill, NC Accurate Reaction Cross-section Predictions for Nucleon-Induced Reactions 06/24/2010 G. P. A. Nobre 1,*, I. J. Thompson 1, J. E. Escher 1, F. S. Dietrich 1, M. Dupuis 2, J. Terasaki 3 and J. Engel 3 Prepared by LLNL under Contract DE-AC52-07NA27344 Performance Measures x.x, x.x, and x.x *

2 LLNL-PRES PLS Directorate - Physics Division - NPS

3 LLNL-PRES : UNEDF project: a national 5-year SciDAC collaboration 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 [Thompson, Summers] Optical Potentials [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 PLS Directorate - Physics Division - NPS

4 LLNL-PRES Nuclear Excited States from Mean-field Models  Mean-field HFB calculations using SLy4 Skryme functional  Use (Q)RPA to find all levels E*, with transition densities from the g.s. Uncorrelated particle-hole states Correlated p-h states in HO basis Correlated p-h states in 15 fm box Neutron separation energy is 9.5 MeV. Above this we have discretized continuum. Collaboration with Chapel Hill: Engel & Terasaki PLS Directorate - Physics Division - NPS

5 LLNL-PRES Cross Sections for Excited States Uncorrelated p-h RPA Correlated states PLS Directorate - Physics Division - NPS

6 LLNL-PRES Diagonal Density PLS Directorate - Physics Division - NPS Example of diagonal Density for 90 Zr Example of diagonal Density for 90 Zr RPA Folding of densities with n-n interaction  Transition potentials

7 LLNL-PRES Transition densities to Transition potentials Diagonal folded potentialOff-diagonal couplings Natural parity states only: no spin-flip, so no spin-orbit forces generated. No energy or density dependence. Exchange contributions included implicitly. All potentials real-valued (So far) PLS Directorate - Physics Division - NPS

8 LLNL-PRES PLS Directorate - Physics Division - NPS Reaction Cross Sections with Inelastic Couplings  (Q)RPA Structure Calculations for n,p + 40,48 Ca, 58 Ni, 90 Zr and 144 Sm  Couple to all excited states, E* < 10, 20, 30, 40 MeV  Find what fraction of σ R corresponds to inelastic couplings Not Converged yet!  E* < 50, 60, 70, …?

9 LLNL-PRES Inelastic Convergence  Coupling to more states gives larger effect  Convergence appears when all open channels are coupled PLS Directorate - Physics Division - NPS For reactions with protons as projectile, inelastic convergence is achieved with less couplings due to the Coulomb barrier Protons as projectile

10 LLNL-PRES Coupling Between Excited States PLS Directorate - Physics Division - NPS g.s. At not too low energies:  Individual cross-sections change very little, except for some few states: up to 20%  Overall sum of reaction over states remains the same  Supports the concept of “doorway states” At not too low energies:  Individual cross-sections change very little, except for some few states: up to 20%  Overall sum of reaction over states remains the same  Supports the concept of “doorway states” Details in paper being prepared for submission to PRC

11 LLNL-PRES Pick-up Channel: Deuteron Formation PLS Directorate - Physics Division - NPS N. Keeley and R. S. Mackintosh * showed the importance of including pick-up channels in coupled reaction channel (CRC) calculations. 40 Ca(d,d) elastic scattering * Physical Review C 76, (2007) Physical Review C 77, (2008) d

12 LLNL-PRES Contribution of Transfer Channels There are many nucleons in the target that can be picked out to make a deuteron. Effect depends on binding energy and size of bound state wave functions: Given by the mean-field model  Large contribution to σ R : closer to OM!  Significant non-orthogonality effects

13 LLNL-PRES Non-Orthogonality and Fraction of σ R PLS Directorate - Physics Division - NPS Behaviour of non-orthogonality is sensitive to changes of the deuteron potential:  Coupling to 90 Zr(n,d,n) channel gives a large increment, approaching to the optical model calculation.  Non-Orthogonality has an additional effect. Better definition needed! Using Johnson-Soper * prescription: V d (R)=V n (r)+V p (R) α CC < α CC+CRC and α CC+CRC+NO Using the Daehnick et al. § potential for the deuteron. * Physical Review C 1, 976 (1970) § Physical Review C 21, 2253 (1980)

14 LLNL-PRES Comparison with Experimental Data PLS Directorate - Physics Division - NPS Good description of experimental data! There is still possibility for improvements. Inelastic convergence when coupling up to all open channels

15 LLNL-PRES Comparison with Experimental Data PLS Directorate - Physics Division - NPS Good description of experimental data! Inelastic and pick-up channels account for all reaction cross sections arXiv: Submitted to PRL arXiv: Submitted to PRL

16 LLNL-PRES Summary of Results at E lab = 30 MeV PLS Directorate - Physics Division - NPS Inelastic + Transfer with non-orthogonality Inelastic couplings only Inelastic + Transfer Phenomenological Optical Model Targets 40 Ca, 48 Ca, 58 Ni, 90 Zr, 144 Sm With all couplings, calculations agree with experimental data arXiv: Submitted to PRL

17 LLNL-PRES Elastic Angular Distributions PLS Directorate - Physics Division - NPS Provide complementary information on reaction mechanisms Are sensitive to the effective interaction used Density-dependent effective interaction: Resulting coupling potentials improve large-angle behavior, still need improvements for small angles. Work in progress to treat and then test UNEDF Skyrme functionals. Our approach predicts a variety of reaction observables. Data provides constraints on the ingredients. Our approach predicts a variety of reaction observables. Data provides constraints on the ingredients. Results will be shown in paper being prepared for submission to PRC

18 LLNL-PRES Conclusions  Inelastic (Q)RPA couplings account for a fraction of reaction cross- section  To achieve convergence, couplings to (at least) all open channels is necessary  Coupling to pick-up channel is very important Deuteron potential Non-orthogonality  Concept of “doorway states” is a good approximation: simplifies the problem, saves computational time  Coupling to inelastic and deuteron channels accounts for (almost) all reaction cross sections  Angular distributions are sensitive to the effective interaction PLS Directorate - Physics Division - NPS arXiv: Submitted to PRL

19 LLNL-PRES PLS Directorate - Physics Division - NPS Future Work - Next Steps  Incorporate UNEDF functionals into folding potentials and examine effects, in particular density-dependence (Jutta Escher)  Use densities from deformed QRPA code (Terasaki & Engel, Chapel Hill, NC)*  Analyze reactions on a range of nuclei, using spherical and deformed QRPA transition densities and functionals from UNEDF*  Two-step approach (Ian Thompson)  Couple to even higher states to achieve convergence*  Solve consistency issues about deuteron potential, break-up, triton coupling  Draw conclusions about the most important ingredients for a predictive reaction calculation. *Computational Challenges Future Publications: Detailed paper to be submitted to PRC Proceeding for INPC2010, Vancouver

20 LLNL-PRES Folding potentials and effective interactions PLS Directorate - Physics Division - NPS Status: Folding code for Gogny interaction is written, results are being incorporated in reaction calculations and tested. Focus: study effects of density dependence on angular distributions. Formalism for Skyrme functionals in development. Folding code to be extended to accommodate UNEDF functionals. Science questions (below) to be addressed. Brief outline of approach and status summary… Details of approach: Exchange potentials are non-local and contain non-diagonal density matrices. Slater LDA is used for the density matrix:  (r,r’) =  SL (r) 3j 1 (k F s)/(k F s) with s = |r-r’| and k F (r) = (3  2  0 (r)/2) 1/3 Alternative DMEs can be considered: Negele-Vautherin, Campi- Bouyssy, UNEDF-developed versions Slater-Exchange Approximation (SEA): The non-locality is treated by using a Fourier transform of the product of the exchange term of the interaction, multliplied by 3j 1 (k F s)/(k F s), evaluated at the local momentum, (W.G. Love, 1978) The treatment goes beyond the local energy approximation (LEA), gives larger exchange contributions, and is expected be better at lower energies, while reducing to the LEA at higher energies. Different prescriptions can be used for the effective momentum (require extensions). Equations are solved self-consistently. Inputs for folding code are: Transition densities from QRPA (Terasaki and Engel), RPA, other structure model. Effective interaction: Gogny-type, Skyrme-type Science questions to address: How sensitive are reaction observables to details of the effective interaction (e.g. density dependence)? What are the best interactions for reaction applications?

21 LLNL-PRES Appendix: Elastic and Inelastic Scattering PLS Directorate - Physics Division - NPS g. s. E*0E*0 E*1E*1 E*nE*n Elastic Inelastic Excitations Vibrational Rotational Particle-hole d

22 LLNL-PRES Appendix: Coupled Channels Equations PLS Directorate - Physics Division - NPS Schrödinger Equation Optical Model: