Coupled-channels Neutron Reactions on Nuclei Ian Thompson with: Gustavo Nobre, Frank Dietrich, Jutta Escher (LLNL) and: Toshiko Kawano (LANL), Goran Arbanas (ORNL), P. O. Box, Livermore, CA! This work performed under the auspices of the U.S. Department of Energy by under Contract DE-AC-NA LLNL- PRES-
Channel Couplings in Neutron-nucleus Collisions Neutrons incident on Spherical Nuclei Scidac Project UNEDF Use mean-field models with RPA excited states Use real effective interactions Calculate inelastic cross sections to all RPA states Calculate transfer cross sections to all one-nucleon-transfer states Predict Reaction cross sections Predict Optical Potentials: Nonlocal & Local-equivalent
: UNEDF project: a national -year SciDAC collaboration Target A = (N,Z) UNEDF: V NN, V NNN V eff for scattering E projectile Structure Models Methods: HF, DFT, RPA, CI, CC, Ground state Excited states Continuum states Transition Densities Transition Density [Nobre] Folding [Escher, Nobre] KEY: UNEDF Ab-initio Input User Inputs/Outputs Exchanged Data Related research UNEDF Reaction Work Deliverables Residues (N,Z ) Hauser- Feshbach decay chains [Ormand] Partial Fusion Theory [Thompson] Transition Potentials Inelastic production Coupled Channels [Thompson, Summers] Compound emission Preequilibrium emission Neutron escape [Summers, Thompson] V optical Two-step Optical Potential or Elastic S-matrix elements Resonance Averaging [Arbanas] Global optical potentials Optical Potentials [Arbanas]
Diagonal Density. From M. Dupuis calculations density [fm - ].........! Total! n -! p! p! n Example of diagonal Density for Zr RPA r [fm] Folding of densities with n-n interaction Transition potentials
Nuclear Excited States from Mean-field Models Excitation energy (MeV) Mean-field HFB calculations using SLy Skryme functional Use (Q)RPA to find all levels E*, with transition densities from the g.s. Particle!hole levels in Zr Spin of state Excitation energy (MeV) RPA levels in Zr Spin of state QRPA states in Zr Collaboration with Chapel Hill: Engel & Terasaki Uncorrelated particle-hole states Correlated p-h states in HO basis Correlated p-h states in fm box Neutron separation energy is. MeV. Above this we have discretized continuum.
Transition densities to Transition potentials Diagonal folded potential Off-diagonal couplings RPA transition potentials from the gs to states E* < MeV Real central potential V(r)!! HFB diagonal folded potential KD optical potential n + Zr at MeV V f (r) (MeV)!! Radius (fm) All potentials real-valued! Radius r (fm) Natural parity states only: no spin-flip, so no spin-orbit forces generated. No density dependence. Direct terms only: no exchange contributions. (Yet.)
Cross Sections for Excited States!!!! Inelastic cross section (mb)! + Inelastic cross section (mb)! + + + + + Excitation energy (MeV) Uncorrelated p-h Excitation energy (MeV) RPA Correlated states
Reaction Cross Sections with Inelastic Couplings (Q)RPA Structure Calculations for n,p +, Ca, Ni, Zr and Sm Couple to all excited states, E* <,,, MeV Find what fraction of σ R corresponds to inelastic couplings Not Converged yet! E* <,,,?
Summed inelastics => Reaction Cross Section Use doorway model : these inelastic cross sections are sum of escape and compound-nucleus production rates.! R (mb) Couplings to/from g. s. only L =, L =,, E lab = MeV! R (mb) CC; QRPA E* < MeV CC; QRPA E* < MeV CC; QRPA E* < MeV E lab = MeV σ R (L) Effects of Couplings between States! R (mb) E lab = MeV Partial Wave Partial Wave Convergence with E* But: long way from reaction cross section from optical model! R (mb) Optical Model CC; QRPA E* < MeV CC; QRPA E* < MeV CC; QRPA E* < MeV E lab (MeV)
Pick-up Channel: Deuteron Formation Ca(d,d) elastic scattering N. Keeley and R. S. Mackintosh * showed the importance of including pick-up channels in coupled reaction channel (CRC) calculations. * Physical Review C, () Physical Review C, () d
Many Transfer Channels! N d N There are many nucleons in the target that can be picked out to make a deuteron. These give large contributions to the reaction cross sections. Harmonic Oscillator Finite Well N=n+L With spin-orbit force n occ (j) Sum Closed shells BG optical potl QRPA inelastic Inel+transfers crcn!da.reac Neutrons Protons!"!" g p f d s g / d / p / g / p / f / f / d / s / d / () () () () () () () () Reaction cross section (mb) n + Zr MeV!" p p / p / () ()!" s s / N!" nl nl j j + j + () " Partial Wave Effect depends on binding energy and Size of bound state wave functions. These are given by the mean-field model.
Comparison with Experimental Data Good description of experimental data! There is still possibility for improvements. Inelastic convergence when coupling up to all open channels
Several Projectiles, Targets and Energies More Proton data exists for reaction cross sections Multiple targets (normalised to nuclear area) E lab = MeV r =. fm.! R (mb) p+ Ca J. F. Turner et al., R. F. Carlson et al., J. F. Dicello et al., " R /!(r A / )..! R (mb) p+ Ca R. F. Carlson et al.,.. neutron as projectile proton as projectile. transfers! R (mb) p+ Ni J. J. Menet et al., J. F. Turner et al., T. Eliyakut!Roshko et al., E lab (MeV) Inel: dash +transfer: dash-dotted +tr+nono: short-dash optical model: solid " R /!(r A / ).. inelastic optical potential.!(r A / )
Non-Orthogonality and Fraction of σ R Behaviour of non-orthogonality is sensitive to changes of the deuteron potential: Better definition needed! Using the Daehnick et al. potential for the deuteron. Using Johnson-Soper * prescription: V d (R)=V n (r)+v p (R) Coupling to Zr(n,d,n) channel gives a large increment, approaching to the optical model calculation. Non-Orthogonality has an additional effect. α CC < α CC+CRC and α CC+CRC+NO * Physical Review C, () Physical Review C, ()
Elastic Angular Distributions #!, #! + Provide complementary information on reaction mechanisms Are sensitive to the effective interaction used,! /D'@E-E@)F& GDB &H&+! #!, #! +,! ;'?E-E?)B&F A &G&+!D -".-# #! * #! # -".-# #! * #! # #!! //&&&&+!& // //&:&;< //&:&=>?@!(AB?C //&:&;<&:&=>?@!(AB?C #!! /&$&(-&'.&:) ;<;&=&> ;<;&=&>&=&:?!(@A ;<;&=&>&=&:B&/&C&D A@ ;<;&=&>&=&:B&/&C&D A@ &?-&D -! "! #!! #"!! $% &' ( )! "! #!! #"!! $% &' ( ) Our approach predicts a variety of reaction observables. Data provides constraints on the ingredients. 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.
Optical Potentials Define: The one-channel effective interaction to generate all the previous reaction cross sections Needed for direct reactions: use to give elastic wave function Hauser-Feshbach: use to generate reaction cross sections = Compound Nucleus production cross sec. In general, the exact optical potential is Energy-dependent L-dependent, parity-dependent Non-local Empirical: local, L-independent, slow E-dependence fitted to experimental elastic data
Two-Step Approximation We found we need only two-step contributions These simply add for all j=,n inelastic & transfer states: V DPP = Σ N j V j G j V j. G j = [E n - e j H j ] - : channel-j Green s function V j = V j : 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 j (r) G jl (r,r ) V j (r ) = V + iw 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 (), but only some inelastic states included
Previous examples of Non-local Potentials Coulter & Satchler NP A () : Real Part Imaginary Part
Calculated Nonlocal Potentials V(r,r ) now Real Imaginary - - - -. -. - -. - -. - -. L= - - - - - - - - - - - - - - - -. -. - -. - -. - -. L= - - - - - - LLNL- PRES- and!
Low-energy Equivalents: V low-e (r) = V(r,r ) dr Real Imaginary Increasing L KD optical potential V lowe (MeV) Imag V low!e (MeV)!! Increasing L! Radius (fm)! Radius (fm) See strong L-dependence that is missing in empirical optical potentials.
Comparison of (complex) S-matrix elements Imag(S L ).!. Koning!Delaroche optical potential Becchetti!Greenless optical potential CRC + NONO!!!.. Real(S L ) Labeled by partial wave L Comparison of CRC+NONO results with Empirical optical potls (central part). See more rotation (phase shift). Room for improvements!
Exact equivalents: fitted to S-matrix elements Fit real and imaginary shapes of an optical potential to the S-matrix elements. Real part of fitted optical potential (MeV)!!!! MeV MeV MeV MeV KD optical potential Radius (fm) Imag part of fitted optical potential (MeV)!!!!! MeV MeV MeV MeV KD optical potential Radius (fm) Again: too much attraction at short distances
Further Research on Optical Potentials. Compare coupled-channels cross sections with data. Reexamine treatment of low partial waves: improve fit?. Effect of different mean-field calculations from UNEDF.. Improve effective interactions: Spin-orbit parts spin-orbit part of optical potential Exchange terms in effective interaction small nonlocality. Density dependence (improve central depth).. Examine effect of new optical potentials: Are non-localities important? Is L-dependence significant?. Use also ab-initio deuteron potential.. Do all this for deformed nuclei (Chapel Hill is developing a deformed-qrpa code)