1. LLNL-PRES-562100 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC TORUS Annual Meeting June 25, 2012

2. Lawrence Livermore National Laboratory LLNL-PRES-562100 2  Usual formulation of the optical potential: where t z =1/2 for neutrons, -1/2 for protons N and Z are neutron and proton numbers in the target, A=N+Z, V 0 ≈ −52 MeV at center is negative, and V 1 ≈ 26 MeV is positive: (neutrons attract protons more than they do other neutrons)  Define target isospin operator T z = ½( N-Z), so

3. Lawrence Livermore National Laboratory LLNL-PRES-562100 3  Generalize to the full tensor product This has off-diagonal terms: Couples together the neutron and proton channels Get direct (n,p) and (p,n) cross sections: eg. — Fermi transitions in which initial and final orbits are the same — eg. DeVito, Khoa, Austin, et al: arXiv:1202.2660v1. where and Q = energy released in (p,n) = − Coulomb displacement energy Coupled equations!

4. Lawrence Livermore National Laboratory LLNL-PRES-562100 4 Consider 208 Pb(p,n) 208 Bi reaction at low energies Here, Coulomb energies give Q=−18.9 MeV Neutron has much less energy than proton Neutron may be trapped below threshold If near unoccupied bound state, gives resonance: The Isobaric Analog Resonance

5. Lawrence Livermore National Laboratory LLNL-PRES-562100 5 E nF = E p + Q + B n where B n =7.347 MeV and Q = −18.9 MeV Neutron single-particle levels around 208 Pb Can see resonances when the neutron energy is near a bound state. B n =7.347 MeV

6. Lawrence Livermore National Laboratory LLNL-PRES-562100 6  The IAR ‘decay’ to the elastic channel gives resonance phase shifts This is the trapped neutron charge-exchanging back to the elastic proton But it is difficult to measure proton phase shifts accurately at the required energies (14−18 MeV)  Can the IAR decay by other channels? Yes: OTHER neutrons could change to protons! As long as they are in spatial orbital NOT occupied by protons All of the neutrons in the orbitals 1h9/2 to 3p1/2 are thus allowed to charge-exchange back to continuum protons! This leaves nucleus with a weakly bound neutron (eg 4s 1/2 ) and a hole at or below the Fermi level (eg 3p 1/2 ): a particle-hole inelastic excitation Proton has energy reduced by the particle−hole energy difference: inelastic p’ The ph state will eventually gamma-decay.  Experimentally: measure inelastic protons and gamma decays in coincidence ←.. ←

7. Lawrence Livermore National Laboratory LLNL-PRES-562100 7 Left: Resonance at 17 MeV Nearest to 4s 1/2 IAR Decays at 5.292 MeV ≈ E(4s 1/2 ) – E(3p 1/2 ) Right: Resonance at 17.5 MeV Nearest to 2g 7/2 or 3d 3/2 IAR Decays at 5.948 MeV ≈ E(2g 7/2 ) – E(3p 1/2 )

8. Lawrence Livermore National Laboratory LLNL-PRES-562100 8  The (p,p′γ) reaction to the (4s 1/2 ) (3p 1/2 ) −1 particle-hole state can also be modeled as: 1. 208 Pb 3p1/2 (p,p’) 208 Pb 4s1/2 inelastic n* excitation 2. 208 Pb 3p1/2 +p → d+ 207 Pb 1/2- → p’+ 208 Pb 4s1/2 two-step transfer reaction via a deuteron  These are easily modeled in F RESCO  Form a non-resonant background to IAR decay  Note: amplitudes interfere coherently

9. Lawrence Livermore National Laboratory LLNL-PRES-562100 9  Fresco expands in two-body partitions. Here, 4: 1. p + 208 Pb gs KE p =17 MeV n in 3p 1/2 in 208 Pb gs 2. p’ + 208 Pb ph KE p =12 MeV n in 4s 1/2 on 207 Pb 3. n + 208 Bi KE n =−2 MeV n in 4s 1/2 as projectile 4. d + 207 Pb KE d =12 MeV n in deuteron  See the partitions 2. and 3. are NOT orthogonal!  Defined new overlap form in F RESCO for such non-orthogonal bases

10. Lawrence Livermore National Laboratory LLNL-PRES-562100 10 Inelastic cross section from overlap of neutron quasi-bound state (#3) and neutron inelastic state (#2). This calculation used real proton potentials, and a complex deuteron potential

11. Lawrence Livermore National Laboratory LLNL-PRES-562100 11  This work is a ‘valence nucleon’ account of IAR. In the longer-term, full structure-model calculations of widths would be good.  Verification of absolute magnitudes for all peaks.  Choosing the correct energy-averaging interval IARs are too narrow for optical-model averaging! Thus need (p,p′γ) coincidences to see IARs among the compound-nucleus decays  Effects of energy-dependent optical potentials Eg. for transitions from 20 MeV to sub-threshold!

12. Lawrence Livermore National Laboratory LLNL-PRES-562100 12  IAR reactions probe neutron bound states with proton reactions  Should be useful for unstable isotopes!  But: need (p,p′γ) coincidences to see IAR among all the compound-nucleus decays