Breakup models Pierre Capel 17 July 2015 1 / 28
Summary of Lecture 1 In a quantum collision various reactions can take place each corresponds to one channel that can be open or closed Elastic scattering corresponds to a + b → a + b always open ; described by stationary scattering states � � e iKZ + ... + f ( θ ) e iKR + ... R →∞ (2 π ) − 3 / 2 Ψ Z ( R ) −→ K ˆ R Cross section obtained from scattering amplitude f ( θ ) d σ | f ( θ ) | 2 = d Ω Other channels affect elastic scattering ⇒ use of optical potential U opt ( R ) = V ( R ) + iW ( R ) imaginary part W simulates absorption from elastic channel We now see how to include the breakup channel 2 / 28
Halo nuclei Halo nuclei Light, neutron-rich nuclei small S n or S 2n low- ℓ orbital 13O 14O 15O 16O 17O 18O 19O 20O 21O 22O 23O 24O One-neutron halo 12N 13N 14N 15N 16N 17N 18N 19N 20N 21N 22N 23N 11 Be ≡ 10 Be + n 9C 10C 11C 12C 13C 14C 15C 16C 17C 18C 19C 20C 22C ✻ Z 8B 10B 11B 12B 13B 14B 15B 17B 19B 15 C ≡ 14 C + n 7Be 9Be 10Be 11Be 12Be 14Be 6Li 7Li 8Li 9Li 11Li Noyau stable Two-neutron halo Noyau riche en neutrons 3He 4He 6He 8He Noyau riche en protons 6 He ≡ 4 He + n + n 1H 2H 3H Noyau halo d’un neutron Noyau halo de deux neutrons N ✲ 11 Li ≡ 9 Li + n + n n Noyau halo d’un proton Proton halœs are possible but less probable : 8 B, 17 F Two-neutron halo nuclei are Borromean. . . c +n+n is bound but not two-body subsystems e.g. 6 He bound but not 5 He or 2 n 3 / 28
Halo nuclei Borromean nuclei Named after the Borromean rings. . . [M. V. Zhukov et al. Phys. Rep. 231, 151 (1993)] 4 / 28
Breakup reaction Breakup reaction Breakup ≡ dissociation of projectile in constituent clusters by interaction with target 11 Be + 12 C 10 Be + n + 12 C → 8 B + 208 Pb 7 Be + p + 208 Pb → Used to study cluster structure in nuclei e.g. halo nuclei infer reaction rates of astrophysical interest Need a good understanding of the reaction mechanism i.e. an accurate theoretical description of reaction coupled to a realistic model of projectile Elastic breakup ≡ all clusters measured in coincidence target not excited 5 / 28
Breakup reaction Framework Projectile ( P ) modelled as a two-body system : core ( c )+loosely bound fragment ( f ) described by H 0 = T r + V c f ( r ) P f r V c f adjusted to reproduce c P spectrum R Target T seen as 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 ) 6 / 28
Breakup reaction Projectile Hamiltonian H 0 H 0 = − � 2 ∆ r + V c f ( r ) 2 µ c f V c f has usually a Woods-Saxon form factor V 0 V c f ( r ) = 1 + e ( r − R 0 ) / a c - f relative motion described by H 0 eigenstates ǫ nl < 0 : discrete set of bound states H 0 φ nlm ( r ) = ǫ nl φ nlm ( r ) ǫ > 0 : c - f continuum ≡ broken up projectile H 0 φ klm ( r ) = ǫ φ klm ( r ) where ǫ = � 2 k 2 / 2 µ c f Breakup ≡ transition from bound state to continuum through interaction with target (Coulomb and nuclear) Breakup can take place in one or more steps will be sensitive to both bound and continuum states (for simplicity, spin will be ignored) 7 / 28
Breakup reaction Example : 11 Be 11 Be ≡ 10 Be(0 + ) + n 10 Be cluster assumed in 0 + ground state (extreme shell model) ⇒ spin and parity of 11 Be states fixed by angular momenta l and j of n : 1 / 2 + ground state in s 1 / 2 5/2 + d 5 / 2 1.274 1 / 2 − excited state in p 1 / 2 10 Be + n 5 / 2 + resonance in d 5 / 2 1/2 − -0.184 0 p 1 / 2 ⇒ fit V 0 in s 1 / 2 , p 1 / 2 and d 5 / 2 waves 1/2 + -0.504 1 s 1 / 2 but not in p 3 / 2 . . . 11 Be spectrum 8 / 28
Breakup reaction Projectile-target interaction : V cT and V fT The breakup channel is now included in the collision description However other channels not included : absorption of the fragment by the target breakup of the core . . . c - T and f -T interactions described by optical potentials V cT and V fT Their imaginary parts simulate the other channels Usually chosen in the literature V cT : problematic if c - T scattering not measured ⇒ extrapolate what exists V fT : many N - T global potentials exist [Becchetti and Greenlees, Phys. Rev. 182, 1190 (1969)] [Koning and Delaroche NPA 713, 231 (2003) ] 9 / 28
Reaction models Three-body Scattering Problem Within this framework breakup reduces to three-body problem � � P f T R + H 0 + V cT + V fT Ψ ( r , R ) = E T Ψ ( r , R ) r c with the initial condition Z →−∞ e iKZ + ··· φ l 0 m 0 ( r ) R Ψ ( r , R ) −→ T ⇔ P in its ground state φ l 0 m 0 impinging on T Various methods developed to solve that equation Coupled-channel method with discretised continuum (CDCC) Time-dependent approach (TD) (semiclassical) Eikonal approximation 10 / 28
Reaction models CDCC Coupled-Channel method The eigenstates of H 0 {| φ i �} are a basis in r : H 0 | φ i � = ǫ i | φ i � Ψ ( r , R ) = � Idea : expand Ψ on that basis : i χ i ( R ) | φ i � � � T R + H 0 + V cT + V fT Ψ ( r , R ) = E T Ψ ( r , R ) � T R χ i ( R ) | φ i � + χ i ( R ) H 0 | φ i � + ( V cT + V fT ) χ i ( R ) | φ i � ⇔ i � = E T χ i ( R ) | φ i � i � φ j | ⇓ � T R χ j ( R ) + ǫ j χ j ( R ) + � φ j | V cT + V fT | φ i � χ i ( R ) = E T χ j ( R ) i This is a set of coupled equations in χ i ( R ) where the coupling terms are � φ j | V cT + V fT | φ i � i.e. connect the various projectile states through the P - T interaction Problem : continuum states φ klm are not discrete. . . 11 / 28
Reaction models CDCC Discretising the Continuum Model of breakup requires description of continuum must be tractable in computations, i.e. discrete [Rawitscher, PRC 9, 2210 (1974)] with k ∈ R + → φ ilm φ klm with i ∈ N Various methods exist : mid-point : divide continuum in bins [ ǫ i − ∆ ǫ i / 2 , ǫ i + ∆ ǫ i / 2] and choose φ ilm ( r ) = φ k i lm ( r ) to describe bin i average the wave function over the bin � ǫ i + ∆ ǫ i � ǫ i + ∆ ǫ i φ ilm ( r ) = 1 2 2 f i ( ǫ ) φ klm ( r ) d ǫ with W i = f i ( ǫ ) d ǫ W i ǫ i − ∆ ǫ i ǫ i − ∆ ǫ i 2 2 ⇒ square-integrable wave functions φ ilm pseudo-states : solve H 0 φ ilm = ǫ φ ilm on a finite basis ⇒ square-integrable wave functions φ ilm but ǫ i not chosen 12 / 28
Reaction models CDCC CDCC Continuum Discretised Coupled-Channel : CDCC [Austern et al. , Phys. Rep. 154, 125 (1987)] [Tostevin, Nunes, Thompson, PRC 63, 024617 (2001)] Recent review : [Yahiro et al. , Prog. Th. Phys. Supp. 196, 87 (2012)] Fully quantal approximation No approximation on P - T motion, nor restriction on energy But expensive computationally (at high energies) Various codes have been written to solve these coupled equations fresco written by Ian Thompson is free on www.fresco.org.uk 13 / 28
Reaction models CDCC CDCC breakup cross sections Expanding χ into spherical harmonics � χ j ( R ) = 1 i L u jL ( R ) Y 0 L ( Ω ) R L . . . and coupling l and L into J , the coupled equations read � d 2 � � − � 2 dR 2 − L ( L + 1) u J V J cc ′ ( R ) u J ( E T − ǫ j ) u J c ( R ) + c ′ ( R ) = c ( R ) 2 µ PT R 2 c ′ with c ≡ { j , L } and J T = L + l These equations are solved assuming the asymptotic behaviour � � u J δ c 0 I L ( η, KR ) − S J c ( R ) −→ c 0 O L ( η, KR ) R →∞ where I L = G L − iF L incoming Coulomb function O ∗ = outgoing Coulomb function L The S matrix is used to compute the breakup cross sections 14 / 28
Reaction models CDCC Example : 8 B breakup 8 B + 58 Ni → 7 Be + p + 58 Ni @26MeV Exp. :[V. Guimar˜ aes et al. PRL 84, 1862 (2000)] 120 100 d σ /d Ω c (mb/sr) 80 60 40 20 0 0 20 40 60 80 7 Be) (degrees) θ lab ( Th. :[Tostevin et al. PRC 63, 024617 (2001)] 15 / 28
Reaction models Time-dependent approach Time-dependent model P - T motion described by classical trajectory R ( t ) defined by V traj ( R ) T R fT ( t ) b R ( t ) f R cT ( t ) r c P P structure described quantum-mechanically by H 0 Time-dependent potentials simulate P - T interaction ⇒ time-dependent (TD) Schr¨ odinger equation i � ∂ ∂ t Ψ ( r , b , t ) = [ H 0 + V cT ( t ) + V fT ( t ) − V traj ( t )] Ψ ( r , b , t ) Solved for each b with initial condition Ψ ( m 0 ) −→ t →−∞ φ l 0 m 0 16 / 28
Reaction models Time-dependent approach Numerical resolution of the TD Schr¨ odinger equation Time-step evolution approximating the evolution operator Ψ ( m 0 ) ( r , b , t + ∆ t ) U ( t + ∆ t , t ) Ψ ( m 0 ) ( r , b , t ) = � t ′ with U ( t ′ , t ) = exp[ i t H ( τ ) d τ ] and Ψ ( m 0 ) ( r , b , t → −∞ ) = φ l 0 m 0 ( r ) � Faster computation compared to CDCC because each trajectory treated separately Lacks quantum interferences between trajectories Many programs developed to solve TD Partial-wave expansion of Ψ : [Kido, Yabana, and Suzuki, PRC 50, R1276 (1994)] [Esbensen, Bertsch and Bertulani, NPA 581, 107 (1995)] [Typel and Wolter, Z. Naturforsch.A 54, 63 (1999)] Expansion on a 3D spherical mesh : [P . C., Melezhik and Baye, PRC 68, 014612 (2003)] Expansion on 3D cubic lattice : [Fallot et al. NPA700, 70 (2002)] 17 / 28
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