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Spectroscopic factors and Asymptotic Normalization Coefficients from the Source Term Approach

Spectroscopic factors and Asymptotic Normalization Coefficients from the Source Term Approach and from ( d,p ) reactions. N.K. Timofeyuk University of Surrey. Can shell model be used to calculated spectroscopic factors (SFs) and Asymptotic Normalization Coefficients (ANCs)?

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Spectroscopic factors and Asymptotic Normalization Coefficients from the Source Term Approach

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  1. Spectroscopic factors and Asymptotic Normalization Coefficients from the Source Term Approach and from (d,p) reactions • N.K. Timofeyuk • University of Surrey

  2. Can shell model be used to calculated spectroscopic factors (SFs) and Asymptotic Normalization Coefficients (ANCs)? • Source Term Approach to SFs and ANCs • Measuring SFs and ANCs: what (d,p) theories are available in the market ? • Non-locality in (d,p) reactions and its influence on SFs and ANCs

  3. Standard shell model works with the wave function Pdefined in a subspace P (HPP – E) P = 0 where HPPis the projection of the many-body Hamiltonian into subspace P. This means that some effective NN interactions should be used. Effective NN interactions are found phenomenologically to reproduce nuclear spectra. If such shell model is to be used to calculate other quantities, the operators that define these quantities should be renormalized. Shell model SFs are calculated using overlaps constructed from wave functionsP(A-1) and P (A) so all the information about missing subspaces is lost.

  4. Is it possible to recover information about missing subspaces for SF calculations staying within the subspace P? One possible way to do it is to use inhomogeneous equation for overlap function with the source term defined in the subspace P.

  5. Overlap function: SF is the norm of the radial part of the overlap function Ilj(r) The asymptotic part of the overlap functions Ilj(r) is given by Ilj(r)  CljW-,l+½ (2r)/r Cljis the asymptotic normalization coefficient (ANC), W is the Whittaker function, = (2με)1/2 ,εis thenucleon separation energy

  6. The overlap integral can be found from the inhomogeneous equation For radial part of the overlap function the inhomogeneous equation gives: source term

  7. Solution of the inhomogeneous equation is (N.K. Timofeyuk, NPA 632 (1998) 19) whereGl(r,r’) is the Green’s function. • Important!!! • Solution of inhomogeneous equation has always correct asymptotic behaviour • Overlap integrals and SFs depend on matrix elements This matrix elements contain contributions from subspace P and from missing subspace Q. N.K. Timofeyuk, Phys. Rev. Lett. 103, 242501 (2009) N.K. Timofeyuk, Phys. Rev. C 81, 064306 (2010)

  8. Effective interactions: Two-body M3YE potential from Bertschet al, Nucl. Phys. A 284 (1977) 399 Where the coefficientsVi,STandai,SThave been found by fitting the matrix elements derived from the NN elastic scattering data (Elliot et al, NPA121 (1968) 241)

  9. 16O • Wave functions: • Independent particle model • Harmonic oscillator s.p. wave functions • Oscillator radius is chosen to reproduce the 16O radius SSTA = 1.45 experiment: Shell model Reduction factor (e,e’p) 1.27 ± 0.13 0ħ (non TI) 2.0 0.64 ± 0.07 p knockout 1.12 ± 0.07 0ħ (TI) 2.13 (p,d) 1.48 ± 0.16 4ħ (non TI) 1.65

  10. A  16 • Model wave functionsare taken from the 0ħ oscillator shell model • Oscillator radius is fixed from electron scattering • Centre-of-mass is explicitly removed

  11. SFs: Comparison between the STA and the experimental values • A  16

  12. Comparison to variational Monte Carlo calculations SSTA 0.600 0.319

  13. A  16 Spectroscopic factors: STA v SM

  14. Double closed shell nuclei A 16 Oscillator IPM wave functions are used with ħ = 41A-1/3 - 25A-2/3 and the M3YE (central + spin-orbit) NN potential Removing one nucleon Adding one nucleon N.K. Timofeyuk, Phys. Rev. C 84, 054313 (2011)

  15. SSTA/SIPM:Comparison to other theoretical calculations and to knockout experiments A A-1 ljSexp/(2j+1)STACBFM SCGFM CCM 16O 15N p1/2 0.64 0.07 0.73 0.89 0.8 0.9 p3/20.56 0.06 0.65 0.89 0.8 0.9 24O 23N p1/20.59 0.61-0.65 24O 23O s1/2 0.87 0.10 0.83 0.92 40Ca 39K d3/2 0.650.05 0.76 0.85 0.8 s1/20.520.04 0.58 0.87 0.8 48Ca 47K s1/20.540.04 0.69 0.84 0.36 d3/2 0.570.04 0.68 0.86 0.59 d5/2 0.110.02 0.71 0.85 57Ni 56Ni p3/2 0.580.11 0.59 0.65 208Pb 207Tl s1/20.490.74 0.74 0.85 d3/2 0.580.06 0.72 0.83 d5/2 0.490.05 0.73 0.83 g7/2 0.260.03 0.61 0.82 h11/2 0.570.06 0.48 0.82

  16. 132Sn(d,p)133Sn, Ed = 9 .46 MeV (N. B. Nguyen et al, Phys. Rev. C84, 044611 (2011)) Adiabatic model + dispersion optical potential Neutron overlap function Standard WS DOM 132Sn+n SDOM = 0.72 SSTA = 0.68 C2DOM = 0.49 C2STA = 0.42

  17. Conclusion over Part 1 The reason why shell model SFs differ from experimental ones is that they are calculated through overlapping bare wave functions in the subspace P. Model spaces that are missing in shell model wave function can be recovered in overlap function calculations by using an inhomogeneous equation with shell model source term. The source term approach gives reasonable agreement between predicted and experimental SFs.

  18. Transfer reaction A(d,p)B d p n B A • (d,p) reactions help • to deduce spin-parities in residual nucleus B • to determine spectroscopic factors (SFs) • to determine asymptotic normalization coefficients (ANCs) • by comparing measured and theoretical angular distributions:

  19. Theory available for  theor( ). Exact amplitude Where (+) is exact many-body wave function B is internal wave function of B p is the proton spin function pB is the distorted wave in the p – B channel obtained from optical model with arbitrary potential UpB

  20. Alternative presentation of exact amplitude Where (+) is exact many-body wave function pB is exact solution of the Schrodinger equation, in which n-p interaction is absent, and has scattering boundary condition in the p – B channel. However, pB is obtained not with the p-B potential but with the p-A potential.

  21. The simplest approximation for Distorted wave Born approximation (DWBA): (+) = dAdA where dA is the distorted wave in the d – A channel obtained from optical model. In this approximation deuteron is not affected by scattering from target A.

  22. Failure of the DWBA: 116Sn(d,p)117Sn Ed=8.22 MeV R.R. Cadmus Jr.,and W. Haeberli, Nucl. Phys. A327, 419 (1979) Deuteron potential:

  23. Including deuteron breakup: is obtained from three-body Schrodinger equation: UnA and UpA are n-A and p-A optical potentials taken at half the deuteron incident energy Ed /2 p rnp d n R A

  24. Solving 3-body Schrödinger equation in the adiabatic approximation. Johnson-Soper model. R.C. Johnson and P.J.R. Soper, Phys. Rev. C1, 976 (1970) Adiabatic assumption: Then the three-body equation becomes Only those part of the wave function, where rnp 0, are needed: The adiabatic d-A potential The adiabatic model takes the deuteron breakup into account as Anp(R,0)includes all deuteron continuum states.

  25. Other methods to solve three-body Schrodinger equation: • Johnson-Tandy (expansion over Weinberg state basis) • R.C. Johnson and P.C. Tandy, Nucl. Phys. A235, 56 (1974) • Continuum-discretized coupled channels (CDCC) • Faddeev equations

  26. Optical potentials are energy-dependent • Optical potentials are non-local because • interactions with complex quantum objects are non-local. • Projectile can disappear from the model space we want to work with and then reappear in some other place • Optical potentials depend on r and r

  27. Non-locality and energy dependence of optical potentials Non-local optical potential: F. Perey and B.Buck, Nucl. Phys. 32, 353 (1962)   0.85 fm is non-locality range ... and equivalent local potential: Link between WFs in local and non-local models:

  28. Three-body Faddeev calculations of (d,p) reactions with non-local potentials A. Deltuva, Phys. Rev. C 79, 021602(R) (2009) 0 60 120 180 c.m. (deg)

  29. Non-locality in DWBA Perey factor distorted waves from local optical model Perey factor reduces the wave function inside nuclear interior.

  30. Adiabatic (d,p) model with non-local n-A and p-A potentials (N.K. Timofeyuk and R.C. Johnson, Phys. Rev. Lett. 110, 112501 (2013) and Phys. Rev. C 87, 064610 (2013)) UdA= UnA + UpA UC is the d-A Coulomb potential M0(0) 0.8 d  0.4 fm is the new deuteron non-locality range d is the d-A reduced mass

  31. For N = Z nuclei a beautiful solution for the effective local d-A potential exists: where ~ 0.86 ~ 1 E0 40 MeV is some additional energy !

  32. Where does the large additional energy E0 40 MeVcome from? E0 to first order is related to the n-p kinetic energy in the deuteron averaged over the range of Vnp Since Vnp has a short range and the distance between n and p is small then according to the Heisenberg uncertainty principle the n-p kinetic energy is large.

  33. Effective local adiabatic d-A potentials obtained with local N-A potentials taken at Ed /2 (Johnson-Soper potentials) and with non-local N-A potentials U0loc

  34. Adiabatic (d,p) calculations with local N-A potentials taken at Ed /2 and with non-local N-A potentials

  35. Ratio of (d,p) cross sections at peak calculated in non-local model and in traditional adiabatic model

  36. All above conclusions were obtained assuming that • non-local potentials are energy-independent • non-local potentials have Perey-Buck form • zero-range approximation to evaluate d-A potential • applications to N=Z nuclei • Work in progress: • To extent the adiabatic model for energy-dependent non- local potentials • Corrections beyond zero-range • To be able to make calculations for N  Z nuclei

  37. Conclusions for Part 2 • Non-local n-A and p-A optical potentials should be used to calculate A(d,p)B cross sections when including deuteron breakup • Effective local d-A potential is a sum of nucleon optical potentials taken at energies shifted from Ed /2 by  40 MeV. • Additional energy comes from Heisenberg principle according to which the n-p kinetic energy in short-range components, most important for (d,p) reaction, is large. • Non-locality can influence both absolute and relative values of SFs

  38. Conclusions: Overlap functions calculations within a chose subspace must include renormalized operators to account for contributions from missing model spaces. Reaction theory used to extract SFs and ANCs must be improved

  39. Double magic 132Sn Fully occupied shells: Neutrons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2,0f7/2, 1p3/2,0f5/2,1p1/2, 0g9/2,0g7/2,1d5/2,1d3/2,2s1/2 ,0h11/2 Protons: 0s1/2, 0p3/2, 0p1/2, 0d5/2, 1s1/2, 0d3/2, 0f7/2, 1p3/2, 0f5/2, 1p1/2, 0g 9/2 Final nucleus J Ex(MeV) SSTA/SIPM 131Sn 3/2+ g.s. 0.80 1/2+ 0.332 0.83 5/2+ 1.655 0.81 7/2+ 2.434 0.75 131In 9/2+ g.s. 0.64 1/2+ 0.30 0.74 3/2+ 1.29 0.74 133Sn 7/2 g.s. 0.68 3/2 0.854 0.72

  40. A  16 nuclei SF of double-closed shell nuclei obtained from STA calculations: Oscillator IPM wave functions are used with ħ = 41A-1/3 - 25A-2/3 and the M3YE (central + spin-orbit) NN potential A A-1 lj SIPMSexp(e,e’p) SSTA SSTA/SIPM 16O 15N p1/2 2.0 1.27(13) 1.45 0.73 p3/2 4.0 2.25(22) 2.61 0.65 40Ca 39K d3/2 4.0 2.58(19) 2.90 0.73 s1/2 2.0 1.03(7) 1.15 0.58 48Ca 47K s1/2 2.0 1.07(7) 1.38 0.69 d3/2 4.0 2.26(16) 2.70 0.68 d5/2 6.0 0.683(49) 4.21 0.71 208Pb 207Tl s1/2 2.0 0.98(9) 1.48 0.74 d3/2 4.0 2.31(22) 2.88 0.72 d5/2 6.0 2.93(28) 4.38 0.73 g7/2 8.0 2.06(20) 4.88 0.61

  41. DWBA calculations of the 10Be(d, p)11Be Reaction at 25 MeV B. Zwieglinski et al, NPA 315, 124 (1979) N.K. Timofeyuk and R.C. Johnson, PRC 59, 1545 (1999) When optical potentials reproduce elastic scattering in entrance and exit channels we obtain: Optical potentials used don’t reproduce elastic scattering in entrance and exit channels!

  42. Spectroscopic factors obtained using • Global systematic of nucleon optical potentials CH89 • DOM DOM used for neutron bound state Woods-Saxon potential used for neutron bound state STA 0.52 0.67 0.68 0.64

  43. SSTA/SIPM 0.62 0.61 0.75 1.10 0.63 0.77 N.K. Timofeyuk, Phys. Rev. C84, 054313 (2011)

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