Breakup Reactions and Spectroscopic Factors: a Theoretical Viewpoint Pierre Capel ULB, Belgium – p.1/31
Breakup reaction Breakup used to study exotic nuclear structures e.g. halo nuclei: large matter radius small S n or S 2n ⇒ seen as dense core with neutron halo Short lived ⇒ studied through reactions like breakup: halo dissociates from core by interaction with target Information sought through reactions: Binding energy (e.g. 19 C) lj of halo neutron(s) (e.g. 31 Ne) SF – p.2/31
Introduction Reaction models rely on single-particle model of a two-body projectile (core c + fragment f ): [ T r + V ( r ) − ǫ ] φ nlj ( r ) = 0 , � ∞ 0 | φ nlj ( r ) | 2 dr = 1 with In reality, there is admixture of configurations: A Y ( J π ) = A − 1 X ( J π c ) ⊗ f ( lj ) + . . . The overlap wave function is ψ lj ( r ) = � A − 1 X ( J π c ) | a lj ( r ) | A Y ( J π ) � � ∞ 0 | ψ lj ( r ) | 2 dr Spectroscopic Factor: S lj = � Single-particle approximation ≡ ψ lj = S lj φ nlj ⇒ usual idea: S lj = σ exp bu /σ th bu – p.3/31
11 Be+Pb → 10 Be+n+Pb @69AMeV Experiment: (our) Theory: 1 [Fukuda et al. PRC 70, 054606 (2004)] [Goldstein et al. PRC 73, 024602 (2006)] 0.1 Dyn. Eik. Dyn. Eik. 0 � � � � 6 � Dyn. Eik. 0 � � � � 1 : 3 � Eik. 0.01 (b/MeV) 0 0.5 1 1.5 2 2.5 3 E (MeV) =dE d� [PC et al. PRC 70, 064605 (2004)] 0.07 Coulomb + Nuclear 0.06 d5/2 Convoluted dσ/dE (b/MeV) 0.05 Experiment 0.04 0.03 0.02 0.01 0 0 0.5 1 1.5 2 2.5 They get S s 1 / 2 = 0 . 72 E (MeV) With S s 1 / 2 =1 for 10 Be( 0 + ) ⊗ n( 2 s 1 / 2 ) – p.4/31
Outline Breakup models: CDCC, Time-Dependent, Dynamical Eikonal Approximation What do we probe in breakup ? Peripherality of breakup reactions (ANC vs SF) Description of the continuum Projectile-target interaction ( V PT ) Influence of couplings upon halo wave function Can we get SF from ANC? Ratio of angular distributions: a new way to remove V PT dependence Conclusion – p.5/31
Framework Projectile ( P ) modelled as a two-body system: core ( c )+loosely bound fragment ( f ) described by H 0 = T r + V cf ( r ) P f r V cf adjusted to reproduce c bound state Φ 0 b and resonances R Target T seen as Z T structureless particle P - T interaction simulated by optical potentials ⇒ breakup reduces to three-body scattering problem: [ T R + H 0 + V cT + V fT ] Ψ( R , r ) = E T Ψ( R , r ) Z →−∞ e iKZ + ··· Φ 0 ( r ) with initial condition Ψ( r , R ) − → – p.6/31
CDCC Solve the three-body scattering problem: [ T R + H 0 + V cT + V fT ] Ψ( r , R ) = E T Ψ( r , R ) by expanding Ψ on eigenstates of H 0 Ψ( r , R ) = � i χ i ( R )Φ i ( r ) with H 0 Φ i = ǫ i Φ i Leads to set of coupled-channel equations (hence CC) [ T R + ǫ i + V ii ] χ i + � j � = i V ij χ j = E T χ i , with V ij = � Φ i | V cT + V fT | Φ j � The continuum has to be discretised (hence CD) [Tostevin, Nunes, Thompson, PRC 63, 024617 (2001)] Fully quantal approximation No approx. on P - T motion, no restriction on energy But expensive computationally (at high energies) – p.7/31
Time-dependent model P - T motion described by classical trajectory R ( t ) [Esbensen, Bertsch and Bertulani, NPA 581, 107 (1995)] [Typel and Wolter, Z. Naturforsch. A54, 63 (1999)] P structure described quantum-mechanically by H 0 Time-dependent potentials simulate P - T interaction Leads to the resolution of time-dependent Schrödinger equation (TD) i � ∂ ∂t Ψ( r , b , t ) = [ H 0 + V cT ( t ) + V fT ( t )]Ψ( r , b , t ) Solved for each b with initial condition Ψ − t →−∞ Φ 0 → Many programs have been written to solve TD Lacks quantum interferences between trajectories – p.8/31
Dynamical Eikonal Approximation Three-body scattering problem: [ T R + H 0 + V cT + V fT ] Ψ( r , R ) = E T Ψ( r , R ) Z →−∞ e iKZ Φ 0 with condition Ψ − → Eikonal approximation: factorise Ψ = e iKZ � Ψ T R Ψ = e iKZ [ T R + vP Z + µ PT 2 v 2 ] � Ψ 2 µ PT v 2 + ǫ 0 Neglecting T R vs P Z and using E T = 1 i � v ∂ Ψ( r , b , Z ) = [ H 0 − ǫ 0 + V cT + V fT ] � � Ψ( r , b , Z ) ∂Z solved for each b with condition � Ψ − Z →−∞ Φ 0 ( r ) → This is the dynamical eikonal approximation (DEA) [Baye, P. C., Goldstein, PRL 95, 082502 (2005)] Same equation as TD with straight line trajectories – p.9/31
15 C + Pb @ 68 A MeV Comparison of CDCC, TD, and DEA [PC, Esbensen, and Nunes, PRC 85, 044604 (2012)] dσ bu /dE dσ bu /d Ω 400 140 cdcc cdcc 120 td dσ bu /dE (mb/MeV) td 300 dea dσ bu /d Ω (b/sr) dea 100 Exp. 80 200 60 40 100 20 0 0 0 1 2 3 4 5 0 1 2 3 4 5 E (MeV) θ (deg) All models agree DEA agrees with CDCC Data: [Nakamura et al. TD reproduces trend PRC 79, 035805 (2009)] but lacks oscillations – p.10/31
ANC vs SF Is S lj = σ exp bu /σ th bu ? Is breakup really sensitive to SF ? i.e. do we probe the whole overlap wave function ? Isn’t breakup peripheral? i.e. sensitive only to asymptotics ? r →∞ C lj e − κr ψ lj ( r ) − → Asymptotic Normalisation Coefficient: C lj Test this with two descriptions of projectile with different interiors but same asymptotics. [PC and Nunes, PRC 75, 054609 (2007)] – p.11/31
SuSy transformations Use 2 V cf with different interior but same asymptotics obtained by SuSy transfo. [ D. Baye PRL 58, 2738 (1987) ] 0.6 20 Deep Deep SuSy SuSy 0.4 0 u p 3 / 2 (fm − 1 / 2 ) V eff (MeV) 0.2 -20 -40 0 -60 -0.2 -80 -0.4 -100 0 2 4 6 8 10 0 2 4 6 8 10 r (fm) r (fm) Deep potential ⇒ spurious deep bound state ⇒ node in physical bound state Remove deep state by SuSy ⇒ remove node but keep same asymptotics (ANC and phase shift) Analyse difference in σ th bu between deep vs SuSy – p.12/31
Peripherality of breakup reactions 8 B+ 58 Ni @ 26MeV 8 B+ 208 Pb @ 44 A MeV 140 0.5 Deep 120 Deep SuSy SuSy 0.4 dσ bu /dE (b/MeV) 100 0.3 80 (d σ /d Ω ) 60 0.2 40 0.1 20 0 0 0.5 1 1.5 2 2.5 3 0 E (MeV) 0 20 40 60 80 100 120 140 θ (degrees) No difference between deep and SuSy potentials at low and intermediate energies, on light and heavy targets, for energy and angular distributions ⇒ breakup probes only ANC ⇒ SF extracted from measurements are questionable? [PC, Nunes, PRC 75, 054609 (2007)] – p.13/31
Similar study Garcia-Camacho et al. NPA 776, 118 (2006) 11 Be+Pb @ 70 A MeV 3000 2500 d σ /d ε (mb/MeV) 2000 1500 1000 500 0 0 0.5 1 1.5 2 ε (MeV) Using either single particle wave function (solid) or its asymptotic expansion (dashed) ⇒ same conclusion with SF � = 1 – p.14/31
Asymptotic version ψ lj and φ nlj exhibit same asymptotics: r →∞ C lj e − κr r →∞ b nlj e − κr ψ lj ( r ) − → φ nlj ( r ) − → ⇒ Asymptotic version of the single-particle approx.: C 2 C lj ψ lj − → b nlj φ nlj ⇒ S lj = lj b 2 r →∞ nlj Since ANC accessible to breakup reactions, can we still extract SF from reaction data? What effects of couplings between configurations ? ψ lj compared to φ nlj SF S lj ANC C lj – p.15/31
c - f system with couplings We use a model where core can be in different states Φ i ( ξ ) described as levels of deformed rotor Ψ J π = � i ψ i ( r ) Y i (Ω)Φ i ( ξ ) The c - f Hamiltonian reads [Nunes NPA 596, 171 (1996)] H 0 = H c + T r + V cf ( r , β, ξ ) � � �� − 1 r − R 0 [1+ βY 0 2 (Ω)] with V cf ( r , β, ξ ) = V 0 1 + exp a ⇒ set of coupled equations [ T r + V ii ( r ) + E i − ǫ ] ψ i ( r ) = − � i ′ � = i V ii ′ ( r ) ψ i ′ ( r ) , with V ii ′ ( r ) = � Φ i ( ξ ) Y i (Ω) | V cf ( r , β, ξ ) | Φ i ′ ( ξ ) Y i ′ (Ω) � We analyse the validity of single-particle approx. for one-neutron halo nucleus 11 Be [PC, Danielewicz, Nunes, PRC 82, 054612 (2010)] – p.16/31
Influence of coupling ( ψ vs. φ ) 11 Be ≡ 10 Be + n has two bound states ε 1 / 2 + = − 0 . 504 MeV ε 1 / 2 − = − 0 . 184 MeV Ψ 1 / 2 + = ψ s 1 / 2 Φ 0 + Ψ 1 / 2 − = ψ p 1 / 2 Φ 0 + + ψ d 3 / 2 Φ 2 + + ψ d 5 / 2 Φ 2 + + ψ p 3 / 2 Φ 2 + + ψ f 5 / 2 Φ 2 + 0.5 β = 0 β = 0 0.4 2 s 1 / 2 (fm − 1 / 2 ) 1 p 1 / 2 (fm − 1 / 2 ) β = 0 . 2 β = 0 . 2 0.4 β = 0 . 4 β = 0 . 4 0.3 β = 0 . 6 β = 0 . 6 0.3 β = 0 . 8 β = 0 . 8 S rot S rot 0.2 � � 0.2 2 s 1 / 2 | / 1 p 1 / 2 | / | rψ rot | rψ rot 0.1 0.1 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 r (fm) r (fm) � ⇒ single-particle approx. fails: ψ lj ( r ) � = S lj φ nlj ( r ) � S s 1 / 2 φ 2 s 1 / 2 But, for the ground state, ψ s 1 / 2 − → ∀ β r →∞ – p.17/31
Comparing S and C 2 /b 2 � S s 1 / 2 φ 2 s 1 / 2 We find ψ s 1 / 2 − → ∀ β r →∞ ⇒ Asymptotic version of single particle approx.? i.e. is C 2 lj /b 2 nlj a good approx. of S lj ? 1 0.9 g.s.: Small admixture, nlj lj /b 2 S 2 s 1 / 2 0.8 approx. OK S nlj or C 2 C 2 s 1 / 2 /b 2 2 s 1 / 2 S 1 p 1 / 2 0.7 e.s.: Large admixture, C 2 p 1 / 2 /b 2 1 p 1 / 2 0.6 approx. fails β > 0 . 2 0.5 0 0.2 0.4 0.6 0.8 1 β ⇒ Approx. breaks at large admixture and/or coupling? – p.18/31
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