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Transfer Reactions with Deuterons

Transfer Reactions with Deuterons. R C Johnson University of Surrey. Nuclear Physics Group University of Surrey, May 2007. Why study transfer reactions with deuterons?. d +A. p +B. B=A+ n.

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Transfer Reactions with Deuterons

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  1. Transfer Reactions with Deuterons R C Johnson University of Surrey

  2. Nuclear Physics Group University of Surrey, May 2007

  3. Why study transfer reactions with deuterons? d+A p+B B=A+n (i) Angular distribution of p gives orbital angular momentum, ln,, of neutron orbit in B. Polarization may give jn. (ii) Cross-section magnitude reveals single-particle nature of neutron state in B. (iii) Gives insight into the way single particle strength is distributed among states of a given nucleus and over the periodic table. (iv) Modern studies seek to answer these questions for exotic nuclei near the drip lines. This may involve experiments in inverse kinematics. (v) It is essential that we have credible reaction theories if reliable nuclear structure information is to result.

  4. Three Body Model of d+A n r A p R Hamiltonian: ===

  5. Calculating Transfer and Elastic Break-up Amplitudes The 3-body Schroedinger equation: Exact solution of inhomogeneous equation: NB The3-body wavefunction is only needed for n and p within the range of Vnp

  6. Adiabatic, “frozen-halo”, approx. Adiabatic condition:

  7. Implementing the Adiabatic approximation: elastic scattering Adiabatic Equation: Adiabatic 3-body wavefunction: Scattering amplitude d+A: Johnson and Soper PRC1(1970)976, Amakawa, et al, PLB82B(1979)13

  8. Application to transfer and elastic break-up NB Adiabatic wavefunction not valid for r large! Iteration needed! Johnson and Soper, PRC1(1970)976. 3-body Schroedinger equation: 1st iteration: Exact solution of inhomogeneous equation: ADW (see Timofeyuk and Johnson, PRC59(1999)1545 for finite A corrections)

  9. Application to (d,p) V(R,0) = Johnson and Soper PRC1(1970)976); Harvey and Johnson, PRC3(1971)636; Wales and Johnson, NPA274(1976))168. Zero-range n-p interaction: A(d,p)B transition amplitude: Johnson-Soper potential (Zero-range n-p interaction): N.B.

  10. Comparison with DWBA (i) Ignore all components of the 3-body wave function except the elastic A(d, d)A components. (ii) Find deuteron and proton optical potentials which correctly describe elastic d+A data. (iii) Deuteron excitation (break-up) and excitation of A taken into account only in so far as these channels influence elastic d channel. DWBA: DDDDWBA DWBA

  11. Comparison of DWBA and Adiabatic DW implementation Input parameters: DWBA: (a) Deuteron optical potential at incident d energy. (b) Proton optical potential at outgoing energy. (c) Neutron bound state potential (d) Vnp : ADW: (b)+(c)+(d) + nucleon optical potential at half deuteron incident energy. Calculate V(R,0): V(R,0) =

  12. Some applications of the Johnson-Soper ADW model to (d,p) and (p,d) reactions. 1 Cadmus and Haeberli, Nucl Phys A327(1979)419; A349(1980)103. Ed=12.9Mev, Sn target. DWBA fails. ADW works. 2. Liu, et al, Phys.Rev.C69(2004)064313 Ed=12-60MeV, 12C(d,p)13C and 13C(p,d)12C. Consistent spectroscopic factors with ADW. 3.Stephenson, et al, Phys. Rev. C42(1990)2562. Ed=79MeV, 116Sn(d,p)117Sn, ln=0; Ed=88.2MeV, 66Zn(d,p),67Zn(p,d)Zn66. 4. Catford, et al, J.Phys.G.31(2005)S1655. Nucleon Transfer via (d,p) using TIARA with a 24Ne radioactive beam. 5. Guo, et al, J.Phys.G.34(2007)103. 11B(d,p)12B, Ed=12MeV. Asymptotic Normalization Coefficients; Astrophysical S-factor.

  13. Cadmus and Haeberli, Nucl.Phys.A327(1979)419 116Sn(d,p)117Sn, 4.11MeV/A

  14. Spectroscopic Factors(Cadmus and Haeberli, 4.11 MeV/A) Carson and McIntyre, 2.5MeV/A

  15. Application of Johnson-Soper methodto 10Be(d,p)11Be(g.s.) Data: Auton,Nucl.Phys. A157(1970)305 Data: Zwieglinki,et al.Nucl.Phys.A315(1979)124 Timofeyuk and Johnson Phys.Rev. 59(1999)1545

  16. Application of Johnson-Soper method Timofeyuk and Johnson Phys.Rev.C 59(1999)1545 Data:Cooper, Hornyak and Roos, Nucl.Phys.A218(1974)249

  17. The 3-body Wave function inside the range of VnpJohnson and Soper,PRC1(1970)976, Johnson, 2nd RIA Workshop 2005 (AIP Conf Proc 791) JS Zero range Finite Range:

  18. The Sturmian Method of Johnson and Tandy Johnson and Tandy NuclPhys A235(1974)56 Expansion of 3-body w.f.: Complete set: i=0 only: 0 Zero range Vnp:

  19. Sturmian States for Vnp (Hulthen) i i Laid, Tostevin and Johnson, PRC48(1993)1307

  20. Corrections to the Adiabatic approximation at Ed=88.2 MeV 66Zn(d,p)67Zn, 5/2-,l=3, Ed=88.2 MeV Laid, Tostevin and Johnson,Phys.Rev.C48(1993)1307

  21. The CDCC Method Comparison with Faddeev calculations: Deltuva, Moro, Cravo, Nunes and Fonseca, PRC76(2007)064602 Alt, Blokhintsev, Mukhamedzhanov and Sattarov, PRC75(02007)054003.

  22. Many-body theory of A(d,p)B

  23. Nuclear Structure aspects.Overlap functions The Spectroscopic FactorSABis defined by

  24. Effective 3-body potential (1): a challenge to theory. Many-body operator: “Optical model” operators :

  25. Effective 3-body potential (2) where Approximation:

  26. Effective 3-body Hamiltonian(3) 1.Multiple scattering corrections High energy: corrections suppressed by the weak correlation of n and p in the deuteron

  27. Effective 3-body Hamiltonian(4) 1.Multiple scattering corrections 2. Pauli blocking. The Bethe-Goldstone equation. Tostevin, Lopes and Johnson, Nucl.Phys. A465(1987)83. Ioannides and Johnson Phys. Rev. C17 (1978)1331. Binding energy of a d propagating in nuclear matter.

  28. Binding energy of a 50MeV deuteron in 92Zr in the local nuclear matter approximation (Ioannides and Johnson 1978) Ortho: M=0 Para: |M|=1

  29. Johnson-Soper from a modern perspective 1.Johnson-Soper and Harvey-Johnson gave a simple prescription for taking into account coherent , `entangled’, effects of d break-up on (d,p) reactions. 2.They sought to justify their assumptions (`adiabatic’) by applying he same ideas 20 MeV elastic d scattering, with some success. 3. They were able to explain some outstanding discrepancies between old DWBA calculations and experiment. New theory could use existing codes and was simple to implement. Need for deuteron optical potentials disappeared. 4.Johnson and Tandy introduced a new approach which made clear that the adiabatic assumption was not a necessary condition for the validity of the J-S method for transfer. Implemented by Laid, Tostevin and Johnson. 5.For application to low energy the big question is `What is the correct 3-body Hamiltonian?’ 6.Validity of adiabatic approximation for elastic scattering investigated by comparison with CDCC calculations. Generalised to 3-body projectiles.

  30. Deuteron break-up effects 1. In 1970 Johnson-Soper and Harvey-Johnson gave a simple prescription for taking into account coherent , `entangled’, effects of d break-up on (d,p) reactions. Need for Deuteron optical potentials disappears. Only need NUCLEON optical potentials at several energies. No change in the way nuclear structure parameters (overlap functions, spectroscopic factors, ANC’s) appear in the theory. 2.J-S sought to justify their assumptions (`adiabatic’) by applying the same ideas to 20 MeV elastic d scattering, with some success. (See Chau Huu-Tau, Nucl.Phys.A773(2006)56;A776(2006)80 for CDCC developments.) 3. H-J were able to explain some outstanding discrepancies between old DWBA calculations and experiment. New theory could use existing codes and was simple to implement.. 4.Johnson and Tandy introduced a new approach which made clear that the adiabatic assumption was not a necessary condition for the validity of the J-S method for transfer. Implemented by Laid, Tostevin and Johnson. 5.For application to modern low energy experiments (GANIL, MSU) the big question is `What is the correct 3-body Hamiltonian?’ 6.Validity of adiabatic approximation for elastic scattering investigated by comparison with CDCC calculations. Generalised to 3-body projectiles.

  31. Concluding Remarks 1. “Adiabatic” approximation. Must distinguish between (a) Stripping and pick-up. Johnson and Soper. Johnson and Tandy; Laid, Tostevin and Johnson, PRC(1993)1307. (b) Elastic Scattering. Summers, Tostevin and Johnson, PRC66(2002)014614. Johnson and Soper: 2. Implementation needs optical potentials for constituents, not composite system. 3. CDCC is very complicated to implement:. It needs excited continuum state wavefunctions. 4-body CDCC: Matsimoto, et al, PRC70,(2004)061601, Rodriguez-Gallardo, et al, PRC77064609(2008) 4-body adiabatic: Christley, et al, Nucl.Phys.A624(1997)275.

  32. Concluding Remarks • 3-body aspects of (d,p). • DWBA does not work. • 3-body wavefunction needed within Vnp only. • Adiabatic approximation is sufficient but not necessary condition for the • J-S distorting potential to be valid for transfer. • Evidence for validity at low energy(4 MeV/A). • Need to extend work of Laid et al PRC48(1993)1307 on Johnson-Tandy expansion to low energy. • Validity of adiabatic approximation for elastic d scattering is a separate question. • (See Summers, et al, PRC66(2002)014614). • 2. Overlap functions. • What properties of this quantity are measured in any one (d,p) experiment? • See Nunes,et al, PRCC72(2005)017602, C75(2005)024601. • 3. What is the effective 3-body Hamiltonian at low energy? • Multiple scattering effects? • Effective Vnp ? Pauli blocking? • 4. To deduce reliable nuclear structure information we need a co-ordinated programme of transfer and relevant nucleon elastic and inelastic scattering measurements, including polarization variables.

  33. Frozen halo scattering 1.Neutron scattering by a deformed nucleus (Barrett 1960) Frozen body-fixed axis 3.Deuteron-nucleus scattering (Johnson and Soper 1970) Frozen n-p relative co-ordinate.

  34. 2.Coulomb excitation. “Sudden Approximation”.

  35. Instructive Special Case: VnA=0Johnson, Al-Khalili and Tostevin, PRL79(1997)2771 Exact Solution: where

  36. Elastic scattering with VnA =0 d d

  37. 11Be+12C E/A=49.3 MeV. GANIL data (Roussel-Chomaz, 1996) Effect of break-up on elastic scattering

  38. Coulomb break-up of d: VnA=0 n r A p R Other applications to Coulomb break-up : 11Be, 19C: Banerjee, Thompson and Tostevin, PRC58(1998)1042; Zadro, PRC66(2002)034603. 6He :Banerjee, Thompson and Tostevin, PRC58(1998)1337.

  39. Small angle elastic break-up of the deuteron. Tostevin, et al, Phys Lett B424(1998)219, Okamura, et al, RIKEN Data: Theory: Tostevin, Rugmai and Johnson, PRC57(1998)1846. Input: Nuclear Charges, Vnp

  40. Scattering of Halo Nuclei Light nuclei near the neutron drip-line. Scattering and reactions of loosely bound nuclei. The deuteron case. Handling continuum channels. The adiabatic approximation. A special case of the 3-body problem. Application to elastic scattering. Application to transfer and break-up. DWBA and ADW. Concluding remarks. Relation to CDCC method.

  41. Part of the N-Z plane Z=9 Z=8 31F proton drip line neutron drip line N=8 Borromean halo nuclei

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