Nuclear Reaction Theory

Transcription

1 Nuclear Reaction Theory Antonio M. Moro Universidad de Sevilla, Spain UK Nuclear Physics Summer School Belfast, 18th-21st August 2017

2 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 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 2/ 129

3 Table of contents II Formal treatment of transfer reactions Transfer reactions with weakly bound nuclei Accessing continuum structures via transfer reactions 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 3/ 129

4 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). 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 4/ 129

5 Light exotic nuclei: halo nuclei and Borromean systems Proton number He He H D T n Ne F O 14 O 15 O 16 O 17O 12N 13 N 14 N 15 N 16 N 17 N 18 N 19 N 20 N 21 N 9 C 10 C 11C 12 C 13C 14 C 15 C 1 C 6 17 C 18 C 19 C 20C C B 10 B 11 B 12 B 13 B B B B 19 B 7Be Be Be Be Be Be Li 7 Li 8 Li 9 Stable Li Li He He Unstable Neutron halo Borromean Neutron number 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 5/ 129

6 Light exotic nuclei: halo nuclei and Borromean systems Radioactive nuclei: they typically decay by β emission. E.g.: 6 He β 6 Li (τ 1/2 807 ms) 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. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 6/ 129

7 Signatures of weakly-bound nuclei in reaction observables 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 7/ 129

8 Elastic scattering: Rutherford experiment years later 1,5 4,6 He+ 208 Pb at 19 MeV Rutherford 4,6 He MeV 1 4 He Pb (experimental) 1 4 He σ/σ R 0,5 σ/σ R He 6 He Pb (experimental) θ c.m. (deg.) θ c.m. (deg) 4 He follows Rutherford formula at 19 MeV but not at 22 MeV. 6 He drastically departs from Rutherford formula at both energies! 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 8/ 129

9 Inclusive breakup cross sections 6 He+ 208 Pb α+x He MeV 4 Heyield / 6 He yield PH189 experiment PH215 experiment θ LAB (deg) At large angles, there are moreα s than 6 He (elastic)! What are the mechanisms behind theαproducion and how can we compute it? 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 9/ 129

10 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. σ I π(r p + R t ) 2 From I. Tanihata 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 10/ 129

11 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. σ I π(r p + R t ) 2 Tanihata et al, PRL55, 2676 (1985) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 10/ 129

12 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 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 10/ 129

13 High-energy interaction cross sections with light targets 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 10/ 129

14 Modelling nuclear reactions 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 11/ 129

15 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. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 12/ 129

16 Direct and compound nucleus processes 2 H Elastic 10 Be 2 H 11 Be p p Transfer Breakup DIRECT REACTIONS 10 Be 10 Be n α n t Fusion + evaporation COMPOUND NUCLEUS 12 B* 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 13/ 129

17 Direct versus compound reactions Direct: elastic, inelastic, transfer,... fast collisions (10 21 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 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 14/ 129

18 Linking theory with experiments: the cross section EXPERIMENT THEORY (HΨ=EΨ) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 15/ 129

19 Linking theory with experiments: the cross section EXPERIMENT THEORY (HΨ=EΨ) CROSS SECTIONS dσ dω, dσ de, etc 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 15/ 129

20 Experimental cross section Detector θ I= I 0 n t dσ dω Ω Source Target I: detected particles per unit time in Ω I 0 : incident particles per unit time n t : number of target nuclei per unit surface Ω: solid angle of detector dσ/dω: differential cross section dσ dω = flux of scattered particles through da=r2 dω incident flux 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 16/ 129

21 Projectile and target internal Hamiltonians Mass partitions: α, β,... Internal proj.+ target Hamiltonians: H α (ξ α ) H p (ξ p )+H t (ξ t ) Internal states: [H α (ξ α ) ε α ]Φ α (ξ α )=0 {ε α }=excitation energies Different mass partitions have different Hamiltonians: H α (ξ α ),H β (ξ β ), etc 10 Be 2 H Elastic Inelastic H α (ξ α ) 2 H p Transfer H β (ξ β ) 11 Be 10 Be α channel p Breakup H γ (ξ γ ) 10 Be n 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 17/ 129

22 Model Hamiltonian and model wavefunction Full Hamiltonian H= ˆT R + H p (ξ p )+H t (ξ t )+V(R,ξ p,ξ t ) ˆT R : proj. target kinetic energy H p (ξ p ): projectile Hamiltonian H t (ξ t ): target Hamiltonian V(R,ξ p,ξ t ): projectile target interaction Time-independent Schrodinger equation: [H E]Ψ(R,ξ p,ξ t )=0 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 18/ 129

23 Scattering wavefunction Source K i Target Kf Detector θ K f θ K i q Among the many mathematical solutions of [H E]Ψ=0 we are interested in those behaving asymptotically as: Ψ (+) K α Φ α (ξ α )e ik α R α + (outgoing spherical waves inα,β,...) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 19/ 129

24 Scattering amplitude and cross sections Ψ (+) K α R α Φ α (ξ α )e ik α R α +Φ α (ξ α )f α,α (θ) eik αr α + Φ α (ξ α )f α,α(θ) eikα R α α α Ψ (+) R β K α Φ β (ξ β )f β,α (θ) eik βr β β R β R α R α (elastic) (inelastic) (transfer) Cross sections: ( ) dσ = K β fβ,α (θ) 2 dω α β K α E= 2 K 2 α 2µ α +ε α = 2 K 2 β 2µ β +ε β f β,α is called scattering amplitude 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 20/ 129

25 Ideally, the strategy would be: 1 Choose structure model for H α (ξ) 2 ComputeΨ (+) by solving [H E]Ψ (+) = 0 3 Consider the limit R ofψ (+) 4 Project it on the desired final state to extract the scattering amplitude: (Φ α (ξ α ) Ψ (+) =f α,α(θ) eik α R α R α 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 21/ 129

26 Ideally, the strategy would be: 1 Choose structure model for H α (ξ) 2 ComputeΨ (+) by solving [H E]Ψ (+) = 0 3 Consider the limit R ofψ (+) 4 Project it on the desired final state to extract the scattering amplitude: (Φ α (ξ α ) Ψ (+) =f α,α(θ) eik α R α But... Ψ is a solution of a complicated many-body problem, not solvable in most cases. The number of accesible channels and states can be huge. R α So, in practice, we will be happy with an approximation ofψ(or f (θ)) in a restricted modelspace 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 21/ 129

27 Defining our model space: Feshbach formalism Divide the full space into two groups: P and Q P: channels of interest Q: remaining channels WriteΨ=Ψ P +Ψ Q (E H PP )Ψ P = H PQ Ψ Q (E H QQ )Ψ Q = H QP Ψ P ( H PP = PHP, H PQ = PHQ, etc ) Eliminate (formally)ψ Q : [ ] 1 H PP + H PQ E H QQ + iǫ H QP Ψ P = EΨ P } {{ } H eff H eff too complicated (complex, energy dependent, non-local) needs to be replaced by a simpler Hamiltonian: H eff H model (complex, energy dependent) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 22/ 129

28 Strategy for reaction calculaions We need to make a choice for: 1 Modelspace: what channels are to be included? 2 Structure model: for projectile and target (Microscopic, collective, cluster...) 3 Reaction formalism 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 23/ 129

29 Choice of the modespace: the d+ 10 Be example p + n + Be d+ Be* p + n + Be d+ Be* 10 d+ Be 11 p+ Be (a) 1 channel (elastic) 10 d+ Be 11 p+ Be (b) 2 channels (elastic + inelastic) p + n + Be d+ Be* p + n + Be d+ Be* 10 d+ Be 11 p+ Be 10 d+ Be 11 p+ Be (c) elastic + inelastic + transfer (d) elastic + inelastic + transfer + breakup 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 24/ 129

30 Choice of structure model: from the many-body to the few-body problem Microscopic models 11 Be n 10 Be* Fragments described microscopically Realistic NN interactions (Pauli properly accounted for) Numerically demanding/not simple interpretation. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 25/ 129

31 Choice of structure model: from the many-body to the few-body problem Microscopic models 11 Be n 10 Be* Fragments described microscopically Realistic NN interactions (Pauli properly accounted for) Numerically demanding/not simple interpretation. Many-body Inert cluster models Few-body 11 Be 10 Be n Ignores cluster excitations (only few-body d.o.f). Phenomenological inter-cluster interactions (aprox. Pauli). Exactly solvable (in some cases). Achieved for 3-body and 4-body (eg. coupled-channels, Faddeev). 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 25/ 129

32 Choice of structure model: from the many-body to the few-body problem Microscopic models 11 Be n 10 Be* Fragments described microscopically Realistic NN interactions (Pauli properly accounted for) Numerically demanding/not simple interpretation. Many-body Non-inert-core few-body models 11 Be n 10 Be* Few-body+some relevant collective d.o.f. Pauli approximately accounted for. Achieved for 3-body problems (coupled-channels, Faddeev). Inert cluster models Few-body 11 Be 10 Be n Ignores cluster excitations (only few-body d.o.f). Phenomenological inter-cluster interactions (aprox. Pauli). Exactly solvable (in some cases). Achieved for 3-body and 4-body (eg. coupled-channels, Faddeev). 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 25/ 129

33 Single-channel scattering: optical model potential P space represents just the ground state of projectile and target Wavefunction: Ψ= Ψ P Schrodinger equation in modelspace: }{{} elastic + Ψ Q }{{} non-elastic [ T+ Hα (ξ α )+V ] Ψ P = EΨ P V= V PP }{{} Bare interaction + V PQ 1 E H QQ + iǫ V QP } {{ } Polarization potential V bare + V pol V too complicated usually replaced by some phenomenological (complex) potential U(R) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 26/ 129

34 Microscopic folding model for V PP Start from some (effective) nucleon-nucleon potential v NN (JLM, M3Y, etc): 1 Single-folding potential: V(R)= ρ t (s t )v NN ( R s )ds s t R s t ρ t (s t )=target g.s. density. R 2 Double-folding potential: V(R)= ρ p (s p )ρ t (s t )v NN ( R+ s p s t )ds p ds t s p R + s p s t R s t Ifρ p andρ t are g.s. densities, V(R) accounts only for the bare potential (V PP ) (P-space part) and ignores the effect of non-elastic channels. A model for V pol must be supplied. If v NN is real, V(R) is also real. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 27/ 129

35 Phenomenological optical model Effective potential:v U(R)=U nuc (R)+U coul (R) Coulomb potential: charge sphere distribution U coul (R)= ( ) 2R c 3 R 2 Z 1 Z 2 e 2 Z 1 Z 2 e 2 R R 2 c if R R c if R R c Nuclear potential (complex): Eg. Woods-Saxon parametrization U nuc (R)=V(r)+iW(r)= V 0 1+exp ( R R 0 a 0 ) i W 0 1+exp ( ) R R i a i Popular parametrization: R 0 = r 0 (A 1/3 p + A 1/3 t ) (r 0 =reduced radius) For normal nuclei: r 0 r fm a 0 a i fm 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 28/ 129

36 Elastic scattering within the optical model Effective Hamiltonian: U(R) independent of{ξ α } Schrödinger equation: Boundary condition: H= T R + H α (ξ α )+U(R) (U(R) complex!) Ψ (+) K (ξ α, R)=Φ 0 (ξ α )χ (+) 0 (K, R) [T R + U(R) E α ]χ (+) 0 (K, R)=0 (E α= E ε α = 2 K 2 2µ ) χ (+) 0 (K, R) eik R + f (θ) eikr R 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 29/ 129

37 Partial wave decomposition For a central potential (U(R)=U(R)): χ (+) 1 0 (K, R)= KR i l (2l+1)χ l (K, R)P l (cosθ) lm (θ= ˆR ˆK) χ l (K, R) obtained from: [ 2 d 2 2µ dr µ ] l(l+1) + U(R) E R 2 0 χ l (K, R)=0. For U(R)=0,χ (+) 0 (K, R) must reduce to the plane wave: e ik R = 1 KR i l (2l+1)F l (KR)P l (cosθ) l So, for U= 0 χ l (K, R)=F l (KR)=(KR)j l (KR) sin(kr lπ/2) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 30/ 129

38 Asymptotic solution for the case U(R) 0 For R U(R)=0 χ l (K, R) will be a combination of F l and G l F l (KR) sin(kr lπ/2); G l (KR) cos(kr lπ/2) or their outgoing/ingoing combinations: H (±) (KR) G l (KR)±iF l (KR) e ±i(kr lπ/2) The physical solution is determined by the known boundary conditions: χ (+) 0 (KR) eik R + f (θ) eikr R U= 0 χ l (KR) F l (KR) + 0 U 0 χ l (KR) F l (KR) + T l H (+) (KR) The coefficients T l are called transition matrix elements. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 31/ 129

39 Numerical procedure 1 Fix a matching radius, R m, such that U(R m ) 0 2 Integrateχ l (R) from R=0 up to R m, starting with the condition: lim χ l(k, R)=0 R 0 3 At R=R m impose the boundary condition: S l =reflection coefficient (S-matrix) 4 Phase-shifts: χ l (K, R) F l (KR)+T l H (+) l (KR) = i 2 [H( ) l (KR) S l H (+) l (KR)] S l = 1+2iT l e i2δ l T l = e iδ l sin(δ l ) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 32/ 129

40 The S-matrix and phase-shifts S l =coefficient of the outgoing wave for partial wavel. S l 2 is the survival probability for the partial wavel: U real S l =1 δ l real U complex S l <1 δ l complex Sign of Re[δ]: δ>0 attractive potential δ<0 repulsive potential δ=0(s l = 1) no potential (U(R)=0) Forl S l 1 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 33/ 129

41 The S-matrix and phase-shifts χ l (R) 1 H (-) l (R) -S l H (+) l (R) 20 1 H (-) l (R) -S l H (+) l (R) E (MeV) 0-20 W(R) S l =e 2 i δ l -40 V(R) R (fm) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 33/ 129

42 The scattering amplitude Replace the asymptoticχ l (K, R) in the general expansion: χ (+) 1 0 (K, R) KR l = e ik R + 1 K i l (2l+1) { F l (KR)+T l H (+) l (KR) } P l (cosθ) l (2l+1)e iδ l sinδ l P l (cosθ) eikr R The scattering amplitude is the coefficient of e ikr /R: f (θ)= 1 (2l+1)e iδ l sinδ l P l (cosθ) K l = 1 (2l+1)(S l 1)P l (cosθ). 2iK Elastic cross section: l dσ dω = f (θ) 2. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 34/ 129

43 Coulomb plus nuclear case Radial equation: [ d 2 dr 2+ K2 2ηK R + 2µ U(R)+l(l+1) 2 R 2 Asymptotic condition: ] χ l (K, R)=0 η= Z pz t e 2 χ (+) (K, R) e i[k R+η log(kr K R)] log 2KR) ei(kr η + f (θ) R v = Z pz t e 2 µ 2 K (Sommerfeld parameter) [ χ l (K, R) e iσ l Fl (η, KR)+T l H (+) l (η, KR) ] [ = (i/2)e iσ l H ( ) l (η, KR) S l H (+) l (η, KR) ] σ l (η)=coulomb phase shift F l (η, KR)=regular Coulomb wave H (±) l (η, KR)=outgoing/ingoing Coulomb wave 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 35/ 129

44 Coulomb plus nuclear case: scattering amplitude Total scattering amplitude: f (θ)=f C (θ)+ 1 2iK (2l+1)e 2iσ l (S l 1)P l (cosθ) l f C (θ) is the amplitude for pure Coulomb: dσ R dω = f C(θ) 2 = η 2 4K 2 sin 4 ( 1 2 θ)= ( Zp Z t e 2 ) 2 1 4E sin 4 ( 1 2 θ) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 36/ 129

45 Integrated cross sections Total elastic cross section (uncharged particles!) σ el = dω dσ dω = π (2l+1) 1 S K 2 l 2 Total reaction cross section (loss of flux from elastic channel) σ reac = π (2l+1)(1 S K 2 l 2 )= π (2l+1) T K 2 l 2 l l l 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 37/ 129

46 4 He+ 58 Ni example Effective potential: U(R)=U nuc (R)+U coul (R) Effective potential (MeV) W(R) Z 1 Z 2 e 2 /R V nuc (R) α+ 58 Ni E=25 MeV E=10 MeV Real (nuclear + Coulomb) Imaginary Nuclear (Real) Coulomb V=191.5 MeV, W=23.5 MeV, =1.37 r fm, a=0.56 fm R (fm) The maximum of V nuc (R)+ V C (R) defines the Coulomb barrier. Approximately: R b 1.44(A 1/3 p + A 1/3 t ) fm E b = Z pz t e 2 R b Z p Z t p + A 1/3 ) MeV (A 1/3 t 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 38/ 129

47 Effect of indicent energy Depending on the bombarding energy E and the charges of the interacting nuclei, we observe different patterns of elastic scattering. For medium/heavy systems, this can be characterized in terms of the Coulomb (or Sommerfeld) parameter: η= Z pz t e 2 4πǫ 0 v E well above the Coulomb barrier (η 1) Fraunhofer scattering E around the Coulomb barrier (η 1) Fresnel scattering E well below the Coulomb barrier (η 1) Rutherford scattering 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 39/ 129

48 Elastic scattering: energy dependence 4 He+ 58 E=5 MeV He+ 58 E=10.7 MeV 4 He+ 58 E=25 MeV 10 5 dσ/dω (mb/sr) dσ/dω (mb/sr) Coulomb + Nuclear potential Rutherford formula dσ/dω (mb/sr) θ c.m. (deg) θ c.m. (deg) θ c.m. (deg) 4 He+ 58 E=5 MeV 4 He+ 58 E=10.7 MeV He+ 58 E=25 MeV σ/σ R 0.5 σ/σ R 0.5 σ/σ R θ c.m. (deg) θ c.m. (deg) θ c.m. (deg) Rutherford scattering Fresnel Fraunhöfer 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 40/ 129

49 Effect of indicent energy (cont.) Example: 4 He+ 58 Ni at E=5, 10.7, 25 and 50 MeV Coulomb barrier: R b 7.8 fm ; V b 10.2 MeV E lab η K Ż=1/K 2a 0 (*) (MeV) (fm 1 ) (fm) (fm) (*) classical distance of closest approach in head-on collision. η 1: Rutherford scattering:σ(θ) 1/ sin 4 (θ/2) η 1: Fresnel scattering (rainbow) η 1: Fraunhofer scattering (oscillatory behaviour): 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 41/ 129

50 Rutherford scattering Near side waves He+ 58 E=5 MeV Far side waves θ dσ/dω (mb/sr) Bombarding energy well below the Coulomb barrier Purely Coulomb potential (η 1) Obeys Rutherford law:. dσ dω = Z pz t e 2 4E 1 sin 4 (θ/2) θ c.m. (deg) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 42/ 129

51 Fresnel scattering Strongly absorbed Near side waves 4 He+ 58 E=10.7 MeV 1 Far side waves θ σ/σ R θ c.m. (deg) Bombarding energy around or near the Coulomb barrier Coulomb strong (η 1) Illuminated region Coulomb + nuclear trajectories Shadow region strong absorption 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 43/ 129

52 Fraunhofer scattering 4 He+ 58 E=25 MeV Strongly absorbed waves Near side waves θ σ/σ R 10-2 Far side waves θ c.m. (deg) Bombarding energy well above Coulomb barrier Coulomb weak (η 1) Nearside/farside interference pattern (difracction) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 44/ 129

53 Fraunhofer scattering Strongly absorbed waves Near side waves He+ 58 E=25 MeV θ σ/σ R 10-2 total far-side near-side Far side waves θ c.m. (deg) Bombarding energy well above Coulomb barrier Coulomb weak (η 1) Nearside/farside interference pattern (difracction) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 44/ 129

54 Normal versus halo nuclei How does the halo structure affect the elastic scatterig? 4 He+ 208 E=22 MeV 6 He+ 208 E=22 MeV σ/σ R 0.6 σ/σ R 0.6 V 0 =5.89 MeV r 0 =1.33 fm, a=1.15 fm W 0 =9.84 MeV r i =1.33 fm a i =1.7 fm 0.2 V 0 =96.4 MeV r 0 =1.376 fm a 0 =0.63 fm W 0 =32 MeV r i =1.216 fm a i =0.42 fm θ c.m. (deg) θ c.m. (deg) 4 He+ 208 Pb shows typical Fresnel pattern and standard optical model parameters 6 He+ 208 Pb shows a prominent reduction in the elastic cross section, suggesting that part of the incident flux goes to non-elastic channels (eg. breakup) Understanding and disentangling these non-elastic channels requires going beyond the optical model (eg. coupled-channels method next lectures) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 45/ 129

55 Origin of the long-range absorption in 6 He 0.5 db(e1)/de (e 2 fm 2 /MeV) Aumann et al 3-body calculation E (MeV) The Coulomb force on the core induces a tidal force which may eventually break 6 He. From the structure point of view, this translates into a large B(E1) strength near breakup threshold. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 46/ 129

56 Second order perturbative amplitude ( ) 2 i + c (2) n = z t dt z V 1 (t ) 0 exp { i dt n V 1 (t) z exp { i (E z E 0 )t } (E n E z )t } z 0 z z V 1(t) V 1(t) 0 0 z 0 0> V 0(t) V 0(t) 0> V 1(t) V 1(t) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 47/ 129

57 How does a weakly bound nucleus behaves in the field of a heavy target? 1 The strong Coulomb field will produce a polarization ( stretching ) of the projectile, giving rise to a dipole contribution on the real potential: V(R) Z 1Z 2 e 2 R α Z 1Z 2 e 2 2R 4 2 The weakly bound nucleus can eventually break up, leading to a loss of flux of the elastic channel imaginary polarization potential. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 48/ 129

58 The effect of E1 on elastic scattering of weakly-bound nuclei σ/σ R PH189 data PH215 data U bare ( 6 Li+ 208 Pb potential) U bare + (Coulomb) U pol 6 He MeV θ c.m. (deg) Real (MeV) Imag (MeV) Coulomb polarization potential V bare V pol R(fm) E1 Coulomb couplings produces a sizable effect on the elastic cross section of neutron-halo nuclei (we have learnt something!) but...some additional physics is still missing. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 49/ 129

59 Eg: deuteron polarizability from d+ 208 Pb: Adiabatic limit (E x ): Vpol= α dip Z 1Z 2 e 2 2R 4 E= g.s. E1 E1 d p n E1 E1 Rodning et al, PRL49, 909 (1982) α=0.70±0.05 fm 3 Pb 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 50/ 129

60 Exercices Exercices 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 51/ 129

61 Exercises Problem 1 Complete the following table and predict the type of elastic scattering: System/E lab R b V b η 2a 0 (fm) (MeV) (fm) 4 He MeV 4 He MeV 4 He MeV 10 Be MeV Useful constants: amu=931.5 MeV, c=197.3 MeV.fm, e 2 =1.44 MeV.fm 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 52/ 129

62 Exercises: solution to problem 1 2a 0 = Z p Z t e 2 /E cm R b 1.44 (A 1/3 p + A 1/3 t ); V b Z p Z t e 2 /R b k= 2µE/ = 2(µc 2 )/( c) η=z p Z t e 2 / v=z p Z t e 2 /[ ( k/µ)]= Z p Z t e 2 (µc 2 )/[( c) 2 k] System/E lab R b V b η 2a 0 Type (fm) (MeV) (fm) 4 He MeV Rutherford 4 He MeV Fresnel 4 He MeV Fraunhofer 10 Be MeV Fresnel 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 53/ 129

63 Exercises: solution to problem 1 4 He MeV 10 1 σ/σ R 4 He MeV 10 1 σ/σ R 4 He MeV 10 1 σ/σ R θ cm, deg (Rutherford) θ cm, deg (Fresnel) θ cm, deg (Fraunhofer) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 53/ 129

64 Exercises: solution to problem Be+ 64 Ec.m. =24.5MeV 11 Be+ 64 Ec.m. =24.5 MeV Ratio to Rutherford σ/σ R c.m. angle (deg) (Fresnel) θ c.m. (deg) (?) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 53/ 129

65 Exercises Problem 2: Scattering amplitude and cross sections. We have defined the differential cross section as the flux of scattered particles through the area da=r 2 dr in the directionθ, per unit incident flux. Recalling the definition of flux (current) in quantum mechanics, j= 2mi (Ψ Ψ Ψ Ψ ) and the asymptotic form of a scattering wavefunction for a spinless case, Ψ( R) R e ikiz + f (θ,φ) eik f R R show that the differential cross section is given by ( ) dσ = K f f (θ,φ) 2 dω K i Hint: Use spherical coordinates, and recall that: = e r if r + e θ r θ + e φ r sinθ φ, (1) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 54/ 129

66 Inelastic scattering 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 55/ 129

67 Inelastic scattering Nuclei are not inert or frozen objects; they do have an internal structure of protons and neutrons that can be modified (excited) during the collision. Quantum systems exhibit, in general, an energy spectrum with bound and unbound levels. 11 Be 11 Be* Pb 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 56/ 129

68 Models for inelastic excitations 1 COLLECTIVE: Involve a collective motion of several nucleons which can be interpreted macroscopically as rotations or surface vibrations of the nucleus. 2 FEW-BODY/SIGLE-PARTICLE: Involve the excitation of a nucleon or cluster. 11 Be 11 Be* Pb 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 57/ 129

69 Types of collective excitations The nucleons can move inside the nucleus in a coherent (collective) way. 1 Vibrations (spherical nuclei): small surface oscillations in shape. 2 Rotations (non-spherical nuclei): permanent deformation. 3 Monopole (breathing) mode: oscillations in the size (radius). 4 Isovector excitations (protons and neutrons move out of phase) (eg. giant dipole resonance) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 58/ 129

70 Types of collective excitations The type of collective motion is closely related to the kind of energy spectrum. Rotor: E J J(J+ 1) Vibrator: E J n ω 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 58/ 129

71 Microscopic description in the IPM: the 11 Be case Ground state (1/2 + ) First excited state (1/2 ) 1d 5/2 1p 1/2 n p 2s 1/2 1d 5/2 1p 1/2 n p 2s 1/2 2s 1/2 1p 1/2 2s 1/2 1p 1/2 1p 3/2 1s 1/2 1p 1s 1/2 3/2 core 1p 3/2 1s 1/2 1p 1s 1/2 3/2 core 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 59/ 129

72 Models for inelastic excitations Microscopically, what we describe in both cases are quantum transitions between discrete or continuum states: Collective excitations can be regarded as a coherent superposition of many single-particle excitations. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 60/ 129

73 By doing inelastic scattering experiments we measure the response of the nucleus to an external field (Coulomb, nuclear). This response is related to some structure property of the nucleus. Example: for a Coulomb field: B(Eλ; i f )= 1 2I i + 1 Ψ f M(Eλ) Ψ i 2 wherem(eλ,µ) is the electric multipole operator: Z p M(Eλ,µ) e ri λ Y λµ (ˆr i) The structureψ i,f can be described in a collective, few-body or microscopic model. i 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 61/ 129

74 The coupled-channels method for inelastic scattering We need to incorporate explicitly in the Hamiltonian the internal structure of the nucleus being excited (eg. target). H= T R + h(ξ)+v(r,ξ) T R : Kinetic energy for projectile-target relative motion. {ξ}: Internal degrees of freedom of the target (depend on the model). h(ξ): Internal Hamiltonian of the target. h(ξ)φ n (ξ)=ε n φ n (ξ) V(R, ξ): Projectile-target interaction. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 62/ 129

75 Defining the modelspace: d+ 10 Be d+ 10 Be* example p + n + Be d+ Be* 10 d+ Be 11 p+ Be P space composed by ground states (elastic channel) and some excited states (inelastic scattering) Boundary conditions: Cross sections: Ψ (+) K 0 (R,ξ) R e ik0 R φ 0 (ξ) + f 0,0 (θ) eik 0R R φ 0(ξ) + f n,0 (θ) eik nr R φ n(ξ) n>0 } {{ }} {{ }} {{ } incident elastic inelastic ( ) dσ(θ) dω 0 n = K n K 0 f n,0 (θ) 2 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 63/ 129

76 CC model wavefunction (target excitation) We expand the total wave function in a subset of internal states (the P space): Ψ model (R,ξ)=φ 0 (ξ)χ 0 (K 0, R)+ φ n (ξ)χ n (K n, R) Boundary conditions for theχ n (R) (unknowns): n>0 χ (+) 0 (K 0, R) e ik0 R + f 0,0 (θ) eik 0R R χ (+) n (K n, R) f n,0 (θ) eik nr R for n=0 (elastic) for n>0 (non-elastic) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 64/ 129

77 Calculation ofχ (+) n (R): the coupled equations The model wavefunction must satisfy the Schrödinger equation: [H E]Ψ (+) model (R,ξ)=0 Multiply on the left by eachφ n(ξ), and integrate overξ coupled channels equations for{χ n (R)}: [ E εn T R V n,n (R) ] χ n (R)= V n,n (R)χ n (R) n n The structure information is embedded in the coupling potentials: V n,n (R)= dξφ n (ξ)v(r,ξ)φ n(ξ) φ n (ξ) will depend on the assumed structure model (collective, few-body, etc). 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 65/ 129

78 Optical Model vs. Coupled-Channels method Optical Model The Hamiltonian: H= T R + V(R) Internal states: Justφ 0 (ξ) Model wavefunction: Ψ mod (R,ξ) χ 0 (K, R)φ 0 (ξ) Schrödinger equation: [H E]χ 0 (K, R)=0 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 66/ 129

79 Optical Model vs. Coupled-Channels method Optical Model Coupled-channels method The Hamiltonian: H= T R + V(R) Internal states: Justφ 0 (ξ) Model wavefunction: Ψ mod (R,ξ) χ 0 (K, R)φ 0 (ξ) Schrödinger equation: [H E]χ 0 (K, R)=0 The Hamiltonian: H= T R + h(ξ)+v(r,ξ) Internal states: h(ξ)φ n (ξ)=ε n φ n (ξ) Model wavefunction: Ψ model (R,ξ)=φ 0 (ξ)χ 0 (K, R)+ n>0φ n (ξ)χ n (K, R) Schrödinger equation: [H E]Ψ model (R,ξ)=0 [ E εn T R V n,n (R) ] χ n (K, R)= V n,n (R)χ n (K, R) n n 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 66/ 129

80 A first-order formula for f (θ): the DWBA approximation Assume that we can write the p-t interaction as: V(R,ξ)=V 0 (R)+ V(R,ξ) Use central V 0 (R) part to calculate the (distorted) waves for p-t relative motion: [ ˆT R + V 0 (R) E i ] χ (+) i (K i, R)=0 (E i = E ε i ) [ ˆT R + V 0 (R) E f ] χ (+) f (K f, R)=0 (E f = E ε f ) In first order of V(R,ξ) (DWBA) : fi f DWBA (θ)= µ 2π 2 χ ( ) f (K f, R) V if (R)χ (+) (K i, R) dr i with the transition potential: V if (R) φ f (ξ) V(R,ξ)φ i(ξ) dξ 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 67/ 129

81 Physical interpretation of the DWBA method DWBA can be interpreted as a first-order approximation of a full coupled-channels calculation: a a DWBA a a CC The auxiliary potential U β generating the entrance and exit distorted waves is usually chosen in order to reproduce the elastic scattering at the corresponding c.m. energy. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 68/ 129

82 Collective excitations: example Physical example: 16 O+ 208 Pb 16 O+ 208 Pb(3,2 + ) Outgoing 16 O energy: i=3 i=2 i= g.s. 208 Pb Nucl. Phys. A517 (1990) th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 69/ 129

83 Collective excitations: example Coulomb barrier: Z p Z t e 2 V barrier 78 MeV 1.44(A 1/3 p + A 1/3 ) t 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 70/ 129

84 208 Pb( 16 O, 16 O) 208 Pb inelastic scattering Coulomb and Nuclear excitations can produce constructive or destructive interference: Below the barrier, the Coulomb excitation is dominant, and the interference is smaller 208 Pb( 16 O, 16 O) 208 Pb*(3 - ) 208 Pb( 16 O, 16 O) 208 Pb*(3 - ) 10 DWBA: Coulomb + nuclear DWBA: Nuclear DWBA: Coulomb 1 DWBA: Coulomb+Nuclear DWBA: nuclear DWBA: Coulomb dσ inel /dω (mb/sr) dσ inel /dω (mb/sr) E lab =78 MeV θ c.m. (deg) 0.01 E lab =69 MeV θ c.m. (deg) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 71/ 129

85 Inelastic scattering within a few-body model 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 72/ 129

86 Many-body to few-body reduction 11 Be 10 Be* 11 Be 10 Be n n V pt = V ij (r ij ) ij V pt = U ct (r ct )+U nt (r nt ) Effective three-body Hamiltonian: H= T R + h r (r)+u ct (r ct )+U nt (r nt ) U ct (r ct ), U nt (r nt ) are optical potentials describing fragment-target elastic scattering (eg. target excitation is treated effectively, through absorption) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 73/ 129

87 Inelastic scattering in a few-body model Some nuclei allow a description in terms of two or more clusters: d=p+n, 6 Li=α+d, 7 Li=α+ 3 H. Projectile-target interaction: V(R,ξ) V(R, r)=u 1 (r 1 )+U 2 (r 2 ) Transition potentials: V n,n (R)= drφ n(r) [U 1 (r 1 )+U 2 (r 2 )]φ n (r) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 74/ 129

88 Inelastic scattering in a few-body model Some nuclei allow a description in terms of two or more clusters: d=p+n, 6 Li=α+d, 7 Li=α+ 3 H. Projectile-target interaction: V(R,ξ) V(R, r)=u 1 (r 1 )+U 2 (r 2 ) Transition potentials: V n,n (R)= drφ n(r) [U 1 (r 1 )+U 2 (r 2 )]φ n (r) Example: 7 Li=α+t r α = R m t m α r; r t = R+ r m α + m t m α + m t 7 Li r t r t Internal states: (two-body cluster model) [T r + V α t (r) ε n ]φ n (r)=0 α R r α T 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 74/ 129

89 Example: 7 Li(α+t)+ 208 Pb at 68 MeV CC calculation with 2 channels (3/2, 1/2 ) (Phys. Lett. 139B (1984) 150) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 75/ 129

90 Application of the CC method to weakly-bound systems Example: Three-body calculation (p+n+ 58 Ni) with Watanabe potential: V dt (R)= drφ gs(r) { V pt (r pt )+V nt (r nt ) } φ gs (r) 10 1 d + 58 Ni at E=80 MeV (dσ/dω)/(dσ R /dω) Stephenson et al (1982) No continuum (Watanabe potential) θ c.m. Three-body calculations omitting breakup channels fail to describe the experimental data. 2 H n r R p T 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 76/ 129

91 Inclusion of breakup channels: the CDCC method 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 77/ 129

92 Breakup modelspace p + n + Be d+ Be We want to include explicitly in the modelspace the breakup channels of the projectile or target. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 78/ 129

93 Bound versus scattering states φ ε (r) Continuum wavefunctions: ε 0 φ gs (r) 8 B ϕ k,ljm (r)= u k,lj(r) [Y l (ˆr) χ s ] jm r V(r) [MeV] -50 V(r) ε= 2 k 2 2µ th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 79/ 129

94 Bound versus scattering states Continuum wavefunctions: φ ε (r) ε 0 φ gs (r) 8 B ϕ k,ljm (r)= u k,lj(r) [Y l (ˆr) χ s ] jm r V(r) [MeV] V(r) ε= 2 k 2 2µ Unbound states are not suitable for CC calculations: They have a continuous (infinite) distribution in energy. Non-normalizable: u k,lsj (r) u k,lsj(r) δ(k k ) SOLUTION continuum discretization 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 79/ 129

95 The role of the continuum in the scattering of weakly bound nuclei Continuum discretization method proposed by G.H. Rawitscher [ PRC9, 2210 (1974)] and Farrell, Vincent and Austern [Ann.Phys.(New York) 96, 333 (1976)]. Full numerical implementation by Kyushu group (Sakuragi, Yahiro, Kamimura, and co.): Prog. Theor. Phys.(Kyoto) 68, 322 (1982) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 80/ 129

96 Continuum discretization for deuteron scattering φ l,2 φ l,2 φ l,1 n=3 n=2 n=1 l=0 l=1 l= s waves p waves Breakup threshold d waves ε max ε=0 ε = 2.22 MeV ground state Select a number of angular momenta (l=0,...,l max ). For eachl, set a maximum excitation energyε max. Divide the intervalε=0 ε max in a set of sub-intervals (bins). For each bin, calculate a representative wavefunction. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 81/ 129

97 CDCC formalism: construction of the bin wavefunctions Bin wavefunction: ϕ [k u[k1,k2] 1,k 2 ] ljm (r)= lj (r) [Y l (ˆr) χ s ] jm [k 1, k 2 ]=bin interval r u [k 1,k 2 ] lsjm (r)= 2 πn k2 k 1 w(k)u k,lsj (r)dk k: linear momentum u k,lsj (r): scattering states (radial part) w(k): weight function u(r) Continuum WF at ε x =2 MeV Bin WF: ε x =2 MeV, Γ=1 MeV r (fm) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 82/ 129

98 CDCC formalism for deuteron scattering Hamiltonian: H= T R + h r (r)+ V pt (r pt )+V nt (r nt ) Model wavefunction: N Ψ (+) (R, r)=φ gs (r)χ 0 (R)+ φ n (r)χ n (R) n>0 2 H p r R n Coupled equations: [H E]Ψ(R, r)=0 [ E εn T R V n,n (R) ] χ n (R)= V n,n (R)χ n (R) n n T Transition potentials: V n;n (R)= drφ n(r) [V pt (R+ r 2 )+V nt(r r ] 2 ) φ n (r) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 83/ 129

99 Application of the CDCC formalism: d+ 58 Ni Coupled-Channels + Continuum discretization Continuum-Discretized Coupled-Channels (CDCC) 10 1 l=0 l=2 ε max (dσ/dω)/(dσ R /dω) Exp. (80.0 MeV) Exp. (79.0 MeV) CDCC: No continuum CDCC θ c.m. φ l,n n=3 n=2 n=1 s waves d waves ε = 2.22 MeV ground state ε min Coupling to breakup channels has a important effect on the reaction dynamics 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 84/ 129

100 Application of the CDCC method: 6 Li and 6 He scattering The CDCC has been also applied to nuclei with a cluster structure: 6 Li=α+d (S α,d =1.47 MeV) 11 Be= 10 Be+n(S n =0.504 MeV) Li+ 40 Elab =156 MeV No continuum σ/σ R σ/σ R Including continuum 10-4 Experiment No continuum CDCC θ c.m. (deg) Be + 49 MeV/A θ (degrees) c.m. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 85/ 129

101 Extension to 3-body projectiles projectile y x r r2 R r3 target To extend the CDCC formalism, one needs to evaluate the new coupling potentials: V n;n (R)= drφ n(x, y){v nt (r 1 )+V nt (r 2 )+V αt (r 3 )}φ n (x, y) φ n (x, y) three-body WFs for bound and continuum states: hyperspherical coordinates, Faddeev, etc (difficult to calculate!) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 86/ 129

102 Four-body CDCC calculations for 6 He scattering 6 He+ 208 Pb: PH189 data 1 6 He+ 208 Pb: PH215 data 1 channel (no continuum) 3-body CDCC: di-neutron model 4-body CDCC σ/σ R He+ 22 MeV θ c.m. (deg) N.b.: 1-channel potential considers only g.s. g.s. coupling potential: V 00 (R)= drφ g.s. (x, y){v nt(r 1 )+V nt (r 2 )+V ct (r 3 )}φ g.s. (x, y) Data (LLN): Sánchez-Benítez et al, NPA 803, 30 (2008) L. Acosta et al, PRC 84, (2011) Calculations: Rodríguez-Gallardo et al, PRC 80, (2009) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 87/ 129

103 Polarization potential from CDCC calculations 1 Coulomb polarization potential 1 Nuclear polarization potential Re[U pol (R)](MeV) U bare U pol Re[V TELP (R)](MeV) U bare U pol Im[U pol (R)](MeV) Im[W TELP (R)](MeV) R(fm) Polarization potentials are long-ranged. Both nuclear and Coulomb couplings are important R(fm) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 88/ 129

104 Exploring the continuum with breakup reactions 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 89/ 129

105 Exclusive breakup measurements of halo nuclei Example: 11 Be+ 208 Pb 10 Be+ n+ 208 Pb measured at RIKEN (69 MeV/u). Fukuda et al, PRC70, (2004)) 11 Be SF3 BDC Target Neutron VETO NEUT FDC HOD 10 Be Dipole Magnet 11 Be excitation energy can be reconstructed from core-neutron coincidences (invariant mass method) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 90/ 129

106 Dominance of E1 coupling 11 Be+ 208 Pb 10 Be+ n+ 208 Pb measured at RIKEN (69 MeV/u). CDCC calculations include nuclear and Coulomb couplings to all orders. dσ/dω c.m. (mb/sr) Be MeV/u 1/2 + 3/2-1/2 - Fukuda et al. XCDCC: j=1/ /2 - XCDCC: j=1/ /2 - (convoluted) (E rel =0-5 MeV) 5/2 + 3/ θ c.m. (deg) Forθ c.m., the breakup is dominated by g.s. to continuum E1 transitions (1/2 + 1/2, 3/2 ). For pure E1 transitions, we can resort to the simpler semiclassical theory of Coulomb excitation. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 91/ 129

107 Semiclassical 1st order Eλ excitation (Alder & Winther) For Eλ excitation to bound states (0 n) ( ) dσ = dω 0 n ( Zt e 2 ) 2 B(Eλ, 0 n) v e 2 a 2λ 2 0 f λ (θ,ξ) ξ 0 n = (E n E 0 ) a 0 v 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 92/ 129

108 Semiclassical 1st order Eλ excitation (Alder & Winther) For Eλ excitation to bound states (0 n) ( ) dσ = dω 0 n ( Zt e 2 ) 2 B(Eλ, 0 n) v For continuum states (breakup): e 2 a 2λ 2 0 f λ (θ,ξ) ξ 0 n = (E n E 0 ) a 0 v dσ(eλ) dωde = ( Zt e 2 v ) 2 1 e 2 a 2λ 2 0 db(eλ) df λ (θ,ξ) de dω db(eλ)/de can be extracted from small-angle Coulomb dissociation data. dσ θmax de (θ<θ max)= 0 dσ(eλ) db(eλ) dω dωde de 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 92/ 129

109 Extracting B(E1) of 11 Be from 11 Be+ 208 Pb Coulomb dissociation Be MeV/u dσ/dω (mb/sr) RIKEN data Semiclassical (convoluted) dσ(eλ) dωde = ( Zt e 2 v ) 2 1 e 2 a 2λ 2 0 db(eλ) df λ (θ,ξ) de dω 10 3 (E rel =0-5 MeV) (db(e1)/de from a two-body model, 10 Be+n) θ c.m. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 93/ 129

110 Extracting B(E1) of 11 Be from 11 Be+ 208 Pb Coulomb dissociation Be MeV/u dσ/dω (mb/sr) RIKEN data Semiclassical (convoluted) dσ(eλ) dωde = ( Zt e 2 v ) 2 1 e 2 a 2λ 2 0 db(eλ) df λ (θ,ξ) de dω 10 3 (E rel =0-5 MeV) (db(e1)/de from a two-body model, 10 Be+n) θ c.m. dσ/de rel (b/mev) Be MeV/u RIKEN data: 0 o < θ < 6 o RIKEN data: 0 < θ < 1.3 o Semiclassical Semiclassical db(eλ) de dσ de db(e1)/de (e 2 fm 2 /MeV) Fukuda 04 Two-body model ( 10 Be+n ) E rel (MeV) See Fukuda et al, PRC70, (2004)) E rel 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 93/ 129

111 Why heavy targets and why low energies? 11 Be+ 197 Au 10 Be+n+ 197 Au (V b 40 MeV) dσ bu /dω (mb/sr) a) E lab = 39.6 MeV Semiclassical (E1) XCDCC Exp dσ/ bu dω (mb/sr) b) E lab = 31.9 MeV θ lab (deg) In slow collisions, multistep couplings are more likely to occur 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 94/ 129

112 Why heavy targets and why low energies? DWBA calculations Continuum structures (resonances, virtual states) can be accessed via breakup, but also with transfer reactions 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 94/ 129

113 Exploring structures in the continuum The continuum spectrum is not homogeneous ; it contains in general energy regions with special structures, such as resonances and virtual states 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 95/ 129

114 What is a resonance? It is a pole of the S-matrix in the complex energy plane. It is a structure on the continuum which may, or may not, produce a maximum in the cross section, depending on the reaction mechanism and the phase space available. The resonance occurs in the range of energies for which the phase shift is close toπ/2. In this range of energies, continuum wavefunctions have a large probability of being in the radial range of the potential. The continuum wavefunctions are not square normalizable. For practical reasons, a normalized wave-packet (or bin ) can be constructed to represent the resonance. 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 96/ 129

115 Distinctive features of a resonance In the energy range of the resonance, the continuum wavefunctions have a large probability of being within the range of the potential. Potential and WFs Bound state Resonant state Non-resonant state potential (Courtesy of C. Dasso) 0 r 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 97/ 129

116 Distinctive features of a resonance The decay of the resonance is also behind theα-decay phenomenon: 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 98/ 129

117 Resonances and phase-shifts χ l (R) 1 H (-) l (R) -S l H (+) l (R) 20 1 H (-) l (R) -S l H (+) l (R) E (MeV) 0-20 W(R) S l =e 2 i δ l -40 V(R) R (fm) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 99/ 129

118 Resonances and phase-shifts (borrowed from J. Tostevin) 19th UK Nuclear Physics Summer School, 2017 A. M. Moro Universidad de Sevilla 99/ 129