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Nuclear Reaction Theory Nuclear Reaction Theory Antonio M. Moro Universidad de Sevilla, Spain UK Nuclear Physics Summer School Belfast, 18th-21st August 2017 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Table of contents I 1 Introduction 2 Weakly-bound vs. “normal” nuclei in reaction observables 3 Modelling reactions Feshbach formalism: P and Q spaces Defining the modelspace 4 Single-channel scattering: the optical model Optical model formalism Elastic scattering of weakly-bound nuclei 5 Inelastic scattering General features of inelastic scattering Collective excitations Inelastic scattering within a few-body model 6 Breakup The CDCC method Exploring the continuum with breakup reactions Structures in the continuum 7 Transfer reactions General considerations 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 2 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Table of contents II Formal treatment of transfer reactions Transfer reactions with weakly bound nuclei Accessing continuum structures via transfer reactions 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 3 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Unstable nuclei and the limits of stability Note that: Not all unstable nuclei are weakly-bound. There are weakly-bound nuclei which are not unstable (eg. deuteron). 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 4 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Light exotic nuclei: halo nuclei and Borromean systems 71 Ne 1 7 9 F 16 8 31 41 15 O 17 O O O O 7 12 13 14 15 16 17 18 19 20 21 N N N N N N N N N N 6 9C 10 11 12 13 14C 15C 16 17C 18C 19 20C 22C C C C C C C 8 10 11 21 31 41 51 17 19 5 B B B B B B B B B 7 9 12 41 4 Be Be 10 11 Be Be BeBe 6 7 8 9 11 Stable 3 Li Li Li Li Li 8 Unstable 2 3 4 6 He He He He Proton number Neutron halo 1 H D T n Borromean 12 3 4 5 6 7 8 9 10 11 12 13 14 Neutron number 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 5 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Light exotic nuclei: halo nuclei and Borromean systems Radioactive nuclei: they typically decay by β emission. β 6 − 6 E.g.: He Li (τ / 807 ms) −−→ 1 2 ≃ Weakly bound: typical separation energies are around 1 MeV or less. Spatially extended Halo structure: one or two weakly bound nucleons (typically neutrons) with a large probability of presence beyond the range of the potential. Borromean nuclei: Three-body systems with no bound binary sub-systems. 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 6 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Signatures of weakly-bound nuclei in reaction observables 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 7 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Elastic scattering: Rutherford experiment...100 years later 1,5 4,6 208 4,6 208 He+ Pb @ 22 MeV He+ Pb at 19 MeV Rutherford 4 208 He + Pb (experimental) 1 4 1 He R R σ / σ / σ σ 6 0.5 He 0,5 6 208 He + Pb (experimental) 0 0 0 50 100 150 0 50 100 150 θ θ (deg) c.m. (deg.) c.m. 4He follows Rutherford formula at 19 MeV but not at 22 MeV. 6He drastically departs from Rutherford formula at both energies! 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 8 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Inclusive breakup cross sections 6He +208 Pb α + X → 2 10 6 208 He + Pb @ 22 MeV 1 10 yield 0 10 He 6 / yield -1 10 He 4 PH189 experiment -2 10 PH215 experiment -3 10 0 30 60 90 120 150 180 θ LAB (deg) ➩At large angles, there are more α’s than 6He (elastic) ! ➩What are the mechanisms behind the α producion and how can we compute it? 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 9 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering High-energy interaction cross sections with light targets Interaction cross sections of nuclei on light targets and high energies are proportional to the size of the colliding nuclei. 2 σI π(Rp + Rt) ≃ From I. Tanihata 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 10 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering High-energy interaction cross sections with light targets Interaction cross sections of nuclei on light targets and high energies (hundreds MeV/nucleon) are proportional to the size of the colliding nuclei. 2 σI π(Rp + Rt) ≃ Tanihata et al, PRL55, 2676 (1985) 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 10 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering High-energy interaction cross sections with light targets What do momentum distributions tell us about the size of the nucleus? ☞ A narrow momentum distribution is a signature of an extended spatial distribuion 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 10 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering High-energy interaction cross sections with light targets 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 10 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Modelling nuclear reactions 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 11 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Why reaction theory is important? Reaction theory provides the necessary framework to extract meaningful structure information from measured cross sections and also permits the understanding of the dynamics of nuclear collisions. The many-body scattering problem is not solvable in general, so specific models tailored to specific types of reactions are used (elastic, breakup, transfer, knockout...) each of them emphasizing some particular degrees of freedom. In particular, exotic nuclei close to driplines are usually weakly-bound and breakup (coupling to the continuum) is important and must be taken into account in the reaction model. Few-body models provide an appealing simplification of this complicated problem. 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 12 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Direct and compound nucleus processes 2H Elastic 10Be Transfer p DIRECT 11Be 2 REACTIONS H p Breakup 10Be n 10Be α COMPOUND Fusion NUCLEUS + n evaporation t 12B* 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 13 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Direct versus compound reactions Direct: elastic, inelastic, transfer,. 21 “fast” collisions (10− s). only a few modes (degrees of freedom) involved small momentum transfer angular distribution asymmetric about π/2 (peaked forward) Compound: complete, incomplete fusion. many degrees of freedom involved large amount of momentum transfer “loss of memory” almost symmetric distributions forward/backward ⇒ 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 14 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Linking theory with experiments: the cross section THEORY EXPERIMENT (HΨ= EΨ) 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 15 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Linking theory with experiments: the cross section THEORY EXPERIMENT (HΨ= EΨ) CROSS SECTIONS dσ dσ , , etc dΩ dE 19thUK Nuclear PhysicsSummer School,2017 A. M.Moro Universidad de Sevilla 15 / 129 Introduction Weakly-bound vs. “normal” nuclei in reaction observables Modelling reactions Single-channel scattering: the optical model Inelastic scattering Experimental cross section Detector dσ ∆I = I n ∆Ω θ 0 t dΩ Target Source ∆I: detected particles per unit time in
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