Programa de doctorado de Fsica Nuclear. Reacciones Nucleares

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1 Programa de doctorado de Fsica Nuclear Reacciones Nucleares Antonio M. Moro Universidad de Sevilla Santiago de Compostela de marzo de 29 1

2 Elastic scattering: optical model calculations 2 /?? Direct versus compound reactions DIRECT: elastic, inelastic, transfer,... 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 lose of memory almost symmetric distributions forward/backward Programa de doctorado en Fsica Nuclear Reacciones Nucleares 3 /?? Elastic scattering 1 1 d+ 58 Ni at E=8 MeV 4 He+ 28 E=22 MeV 6 He MeV (dσ/dω)/(dσ R /dω) σ/σ R σ/σ R θ c.m θ c.m. (deg) θ c.m. (deg) What can we learn from an optical model analysis of the elastic cross section? Programa de doctorado en Fsica Nuclear Reacciones Nucleares 4 /?? 2

3 Elastic scattering: phenomenology EFFECTIVE PROJECTILE-TARGET INTERACTION: Optical potential (MeV) W(r) Z 1 Z 2 e 2 /R V(r) Nuclear + Coulomb Nuclear (imaginary) Nuclear (Real) E=25 MeV E=1 MeV V=191.5 MeV, W=23.5 MeV, r =1.37 fm, a=.56 fm R (fm) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 5 /?? Elastic scattering: phenomenology Depending on the bombarding energy E and the charges of the interacting nuclei, we observe different types of elastic scattering. This can be characterized in terms of the Coulomb (or Sommerfeld) parameter: η = Z1Z2e2 4πǫ 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 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 6 /?? 3

4 Elastic scattering: phenomenology RUTHERFORD SCATTERING Near side waves He+ 58 E=5 MeV Far side waves θ dσ/dω (mb/sr) θ c.m. (deg) Purely Coulomb potential (η 1) Bombarding energy well below the Coulomb barrier Obeys Rutherford law:. dσ dω = zze2 1 4E sin 4 (θ/2) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 7 /?? Elastic scattering: phenomenology FRAUNHOFER SCATTERING: Strongly absorbed waves Near side waves 1 4 He+ 58 E=25 MeV θ 1-1 σ/σ R 1-2 Far side waves θ c.m. (deg) Bombarding energy well above Coulomb barrier Coulomb weak (η 1) Nearside/farside interference pattern (difracction) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 8 /?? 4

5 Elastic scattering: phenomenology FRESNEL SCATTERING: Strongly absorbed Near side waves 4 He+ 58 E=1.7 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 interference pattern (near-side/far-side) Shadow region strong absorption Programa de doctorado en Fsica Nuclear Reacciones Nucleares 9 /?? Elastic scattering: optical model How does one describe the motion of a particle in quantum mechanics? Hamiltonian: H = T R + U(R) U(R): optical model effective projectile-target interaction Schrodinger equation: [H E]Ψ(R) = Partial wave expansion of the model wavefunction: Ψ(R) = X LM C LM f L (R) YLM( ˆR) R f L (R) obtained as solution of:» 2 2µ d 2 dr L(L + 1) + U(R) E f L (R) = 2µR 2 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 1 /?? 5

6 Elastic scattering: optical model Numerical procedure: 1. Fix a matching radius, R m, such that V nuc(r m) 2. Integrate f(r) from R = up to R m, starting with the condition: 3. At R = R m impose the boundary condition: S L=scattering matrix lim R fl (R) = f L (R) I L(R) S LO L(R) I L and O L are the so called incoming and outgoing waves: I L(R) = O L(R) = 1 (KR)h i(kr η log 2KR) L(KR) e v 1 i(kr η log 2KR) (KR)h L(KR) e v Programa de doctorado en Fsica Nuclear Reacciones Nucleares 11 /?? The S-matrix S L =coefficient of the outgoing wave for partial wave L. Phase-shifts: S L = e 2iδ L U(R) = No scattering S L = 1 δ L = U real S L = 1 δ L real U complex S L < 1 δ L complex For L S L 1 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 12 /?? 6

7 Elastic scattering: the scattering amplitude Which one of the many solutions of Schrödinger equation is the one that correspond to a scattering experiment? Detector θ Source Target Ψ Ki (R) = e ik i R + χ (+) K i (R) Scattering amplitude: A(θ) Partial wave decomposition: Incident plane wave: Outgoing spherical waves: Ψ Ki (R) = e ik i R + χ (+) K i (R) e ik i R + A(θ) eik ir R Ψ Ki (R) = 1 X C LM f L (R)Y LM( R ˆR) 1 X C LM [I L(R) S LO L(R)] Y LM( R ˆR) LM e ik i R = X LM = X LM LM 4πY LM( ˆK i)i L Y LM( ˆR)j L(K R) 2πi v KR Y LM( ˆK i)i L Y LM( ˆR) [I L(R) O L(R)] χ (+) K i (R) X 2πi v KR Y LM( ˆK i)(1 S L )Y LM( ˆR)O L(R) LM Programa de doctorado en Fsica Nuclear Reacciones Nucleares 13 /?? 7

8 Scattering amplitude and cross sections Scattering amplitude: Nuclear potential alone: A(θ) = i 2K Nuclear+Coulomb: A(θ) = A C(θ) + A (θ) A C(θ) = i 2K A (θ) = i 2K X (2L + 1)P L(cos θ)(1 S L ) L X (2L + 1)(1 e 2iσ L )P L(cos θ) L X (2L + 1)e 2iσ L (1 S L )P L(cos θ) L Differential cross section: dσ dω = A(θ) 2 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 14 /?? 8

9 Extracting structure information from elastic scattering measurements Eg: deuteron polarizability from d+ 28 Pb: Deuteron polarizability: P = αe For E < V b, the main deviation from Rutherford scattering comes from dipole polarizability. In the adiabatic limit (E x ): V dip = α Z 1Z 2 e 2 2R 4 Rodning, Knutson, Lynch and Tsang, PRL49, 99 (1982) α =.7 ±.5 fm 3 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 15 /?? Halo and Borromean nuclei: the 6 He case Radioactive: 6 He β 6 Li (t 1/2 87 ms) Weakly bound: ǫ b =.973 MeV g.s He + n + n Neutron halo Borromean system: n-n and α-n unbound 6 He β g.s. 6 Li 3 body system: α almost inert Programa de doctorado en Fsica Nuclear Reacciones Nucleares 16 /?? 9

10 Normal versus halo nuclei How does the halo structure affect the elastic scatterig? 4 28 He+ E=22 MeV 6 He+ 28 E=22 MeV σ/σ R σ/σ R V =5.89 MeV r =1.33 fm, a=1.15 fm W =9.84 MeV r i =1.33 fm a i =1.7 fm.2 V =96.4 MeV r =1.376 fm a =.63 fm W =32 MeV r i =1.216 fm a i =.42 fm θ c.m. (deg) θ c.m. (deg) 4 He+ 28 Pb shows typical Fresnel pattern strong absorption 6 He+ 28 Pb shows a prominent reduction in the elastic cross section due to the flux going to other channels (mainly break-up) 6 He+ 28 Pb requires a large imaginary diffuseness long-range absorption Programa de doctorado en Fsica Nuclear Reacciones Nucleares 17 /?? Optical model calculations for 6 He+ 28 Pb RADIUS OF SENSITIVITY OF V(R) AND W(R) 16MeV 18MeV 22MeV 27MeV 2.5 -V(r) (MeV) W(r) (MeV) r (fm) r (fm) r (fm) r (fm) Imaginary part long range compared to strong absorption radius Programa de doctorado en Fsica Nuclear Reacciones Nucleares 18 /?? 1

11 Normal versus halo nuclei: Fraunhofer No continuum 1 σ/σ R Including continuum Be + 49 MeV/A θ (degrees) c.m. In Fraunhofer scattering the presence of the continuum produces a reduction of the elastic cross section Programa de doctorado en Fsica Nuclear Reacciones Nucleares 19 /?? Fresco, Xfresco and Sfresco What is FRESCO? Program developed by Ian Thompson since 1983, to perform coupled-reaction channels calculations in nuclear physics. Some general features: Multi-platform (Windows, Linux, Unix, VAX) Treats many direct reaction models: elastic scattering (optical model), transfer, inelastic excitation to bound and unbound states, etc Can be run in text mode and graphical mode (XFRESCO interface) FRESCO and XFRESCO can be freely downloaded at SFRESCO: Extension of Fresco, to provide χ 2 searches of potential and coupling parameters. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 2 /?? 11

12 Optical model calculations with Fresco Essential ingredients of an OM calculation: Physical: Identify projectile and target (mass, spin, etc) Incident energy Parametrization of the optical potential Numerical: Radial step for numerical integration (HCM in fresco) Maximum radius R for integration (RMATCH) Maximum angular momentum L. (JTMAX) RMATCH and JTMAX are linked by: kr g (1 2η/kR g) L g + 1/2 (L g=grazing angular momentum) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 21 /?? Elastic scattering: optical model Effective potential: U(R) = U nuc(r) + U coul (R) Coulomb potential: charge sphere distribution U c(r) = ( Z1 Z 2 e 2 2R c Z 1 Z 2 e 2 R 3 R2 R 2 c if R R c if R R c Nuclear potential (complex): Woods-Saxon parametrization Typically: R = r (A 1/3 p + A 1/3 t ) V U nuc(r) = V (r) + iw(r) = 1 + exp r =reduced radius (r fm) A p,a t: projectile, target masses (amu) i R R a W 1 + exp R Ri a i Programa de doctorado en Fsica Nuclear Reacciones Nucleares 22 /?? 12

13 Elastic scattering: effective potential Effective potential: U(R) = U nuc(r) + U coul (R) Optical potential (MeV) W(r) Z 1 Z 2 e 2 /R V(r) Nuclear + Coulomb Nuclear (imaginary) Nuclear (Real) E=25 MeV E=1 MeV V=191.5 MeV, W=23.5 MeV, r =1.37 fm, a=.56 fm R (fm) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 23 /?? OM example: 4 He+ 58 Ni Input example: 4he58ni_e1.in 4he58ni_e1.in: 4He + 58Ni elastic scattering Ecm=1. MeV NAMELIST &FRESCO hcm=.1 rmatch=25. jtmax=3 thmin=1. thmax=18. thinc=2. smats=2 xstabl=1 elab=1.7 / &PARTITION namep= ALPHA massp=4 zp=2 namet= 58Ni masst=58 zt=28 nex=1 / &STATES jp=. bandp=1 ep=. cpot=1 jt=. bandt=1 et=. / &partition / &POT kp=1 at=58 rc=1.4 / &POT kp=1 type=1 p1=191.5 p2=1.37 p3=.56 p4=23.5 p5=1.37 p6=.56 / &pot / &overlap / &coupling / Programa de doctorado en Fsica Nuclear Reacciones Nucleares 24 /?? 13

14 Elastic scattering example General variables &FRESCO hcm=.1 rmatch=25. jtmax=3 thmin=1. thmax=18. thinc=2. smats=2 xstabl=1 elab=1.7 / Mass partitions & states &PARTITION namep= ALPHA massp=4 zp=2 namet= 58Ni masst=58 zt=28 nex=1 / &STATES jp=. bandp=1 ep=. cpot=1 jt=. bandt=1 et=. / &partition / Potentials &POT kp=1 itt=f at=58 rc=1.4 / &POT kp=1 type=1 p1=191.5 p2=1.37 p3=.56 p4=23.5 p5=1.37 p6=.56 / &pot / Programa de doctorado en Fsica Nuclear Reacciones Nucleares 25 /?? Elastic scattering example Essential input variables: FRESCO namelist &FRESCO hcm=.1 rmatch=25. jtmax=3 thmin=1. thmax=18. thinc=2. smats=2 xstabl=1 elab=1.7 / hcm: step for integration of radial equations. rmatch: matching radius (for R > RMATCH asymptotic behaviour is assumed) elab: laboratory energy jtmax: maximum total angular momentum (projectile+target+relative) smats: trace variable smats=2 print elastic S-matrix xstbl: trace variable xstbl=1 print cross sections Programa de doctorado en Fsica Nuclear Reacciones Nucleares 26 /?? 14

15 Elastic scattering with Fresco Essential input variables: partitions and states &PARTITION namep= ALPHA massp=4 zp=2 namet= 58Ni masst=58 zt=28 nex=1 / namep / namet: projectile / target name massp / masst: projectile / target mass (amu) zp / zt: projectile / target charge nex: number of (pairs) of states in this partition &STATES jp=. bandp=1 ep=. cpot=1 jt=. bandt=1 et=. / jp / jt: projectile / target spins bandp / bandt: projectile / target parities (± 1) cpot: index of potential for this pair of states. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 27 /?? Elastic scattering with fresco &POT kp=1 type= ap= at=58 rc=1.4 / &POT kp=1 type=1 shape= p1=191.5 p2=1.37 p3=.56 p4=23.5 p5=1.37 p6=.56 / &pot / kp: index to identify this potential ap, at: projectile and target mass, for conversion from reduced to physical radii: R = r(ap 1/3 + at 1/3 ) type, shape: potential cathegory and shape: type=: Coulomb potential shape=: uniform charge sphere type=1: volume nuclear potential shape=: Woods-Saxon shape rc: reduced radius for charge distribution p1,p2,p3: V, r, a (real part) p4,p5,p6: W, r i, a i (imaginary part) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 28 /?? 15

16 Xfresco interface General variables: Programa de doctorado en Fsica Nuclear Reacciones Nucleares 29 /?? Optical model with XFRESCO Partitions & states: Programa de doctorado en Fsica Nuclear Reacciones Nucleares 3 /?? 16

17 Optical model with XFRESCO Potentials: Programa de doctorado en Fsica Nuclear Reacciones Nucleares 31 /?? Useful output information in OM calculations Useful output files: Main output file (stdout) fort.21 : Elastic scattering angular distribution thmax > : relative to Rutherford. thmax < : absolute units (mb/sr). fort.7: Elastic S-matrix (real part, imaginary part, angular momentum) fort.56: Fusion (absorption), reaction and inelastic cross section for each angular momentum Programa de doctorado en Fsica Nuclear Reacciones Nucleares 32 /?? 17

18 Elastic scattering: optical model Dynamical effects: 4 He+ 58 Ni at E=5, 1.7, 25 and 5 MeV E lab η k λ = 1/k 2a (MeV) (fm 1 ) (fm) (fm) η 1: Rutherford scattering: σ(θ) 1/ sin 4 (θ/2) η 1: Fresnel scattering (rainbow) η 1: Fraunhofer scattering (oscillatory behaviour): Programa de doctorado en Fsica Nuclear Reacciones Nucleares 33 /?? Elastic scattering: energy dependence 4 He+ 58 E=5 MeV 4 He+ 58 E=1.7 MeV 4 He+ 58 E=25 MeV 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) 1 4 He+ 58 E=5 MeV 1 4 He+ 58 E=1.7 MeV He+ 58 E=25 MeV σ/σ R.5 σ/σ R.5 σ/σ R θ c.m. (deg) θ c.m. (deg) θ c.m. (deg) Rutherford scattering Fresnel Fraunhöfer Programa de doctorado en Fsica Nuclear Reacciones Nucleares 34 /?? 18

19 Elastic scattering: S-matrix elements Elastic (nuclear) S-matrix (fort.7): f L el(r) = I L(r) S L elo L(r) S L el 1 E=5 MeV α+ 58 Ni elastic scattering E=1 MeV E=25 MeV L Reaction cross section (mb) α+ 58 Ni elastic scattering E=1 MeV E=25 MeV E=5 MeV L kr g (1 2η/kR g) L g + 1/2 the number of partial waves required for convergence grows approximately as E Programa de doctorado en Fsica Nuclear Reacciones Nucleares 35 /?? 19

20 Inelastic scattering: DWBA and Coupled-Channels method 36 /?? Inelastic scattering to bound states 11 Be 1 Be Pb n Programa de doctorado en Fsica Nuclear Reacciones Nucleares 37 /?? 2

21 Coupled-channels method The Hamiltonian: H = T R + h(ξ) + (R, ξ) Internal states: h(ξ)φ n(ξ) = ǫ nφ n(ξ) Model wavefunction: Ψ(R, ξ) = φ (ξ)χ (R) + P n> φn(ξ)χn(r) +... Coupled equations: [H E]Ψ(R, ξ) [E ǫ n T R V n,n(r)] χ n(r) = X n nv n,n (R)χ n (R) Coupling potentials: Z V n,n (R) = dξφ n (ξ) (R, ξ)φ n(ξ) φ n(ξ) will depend on the structure model (collective, single-particle,etc). Programa de doctorado en Fsica Nuclear Reacciones Nucleares 38 /?? Boundary conditions and scattering amplitude Boundary conditions: χ (+) (R) e ik R + A,(θ) eik R R χ (+) (elastic) n (R) A n,(θ) eiknr, n (non elastic) R If Coulomb is present, then Cross sections: e ikr R dσ n(θ) dω 1 R ei(kr η2kr) = Kn K A n,(θ) 2 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 39 /?? 21

22 DWBA approximation DWBA approximation: [E ǫ n T n V n:n(r)] eχ n(k,r) = [E ǫ n T n V n :n (R)] eχ n (K,R) = a a DWBA Scattering amplitude: A(K,K) n,n = 2µ Z 4π 2 dreχ ( ) n (K,R)V n :n(r)eχ (+) n (K, R) The DWBA approximation amounts at solving the CC equations to first order (Born approximation) In practice, phenomenological optical potentials that fit the elastic cross section in the respective channels are used instead of V n,n and V n,n V n,n(r) (n (R, ξ) n) U n(r) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 4 /?? Partial wave decomposition: the channel basis I ξ R L J=L+I The channel basis: Φ JM J nli ( ˆR, ξ) = X M I M L i L Y LML ( ˆR) nim I LM LIM I JM J Partial wave expansion of the total WF: Ψ(R, ξ) = X nlijm C JM J fj nli(r) Φ JM nli( R ˆR, ξ) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 41 /?? 22

23 Coupled equations The coupled equations: (Comp. Phys. Comm. 4 (1986) ) 2 «d 2 2µ dr + 2 L(L + 1) + ǫ 2 2µR 2 n E fβ J (R) + X V J β,β (R)fJ β (R) = β β {n, L, I} Coupling potentials: V J β,β (R) = Z d ˆR dξφ JM J β ( ˆR, ξ) (R, ξ)φ JM J β ( ˆR, ξ) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 42 /?? Boundary conditions Solution of the coupled equations: 1. Integrate the differential equation for R [, R m] with the condition: lim R fj β;β i (R) = 2. Match the solution at R m with the asymptotic form S-matrix: f J β;β i (R) δ β,βi I β (R) S J β,β i O β (R) I β (R) = (K nr)h L(K nr)/ v n O β (R) = (K nr)h L(K nr)/ v n Programa de doctorado en Fsica Nuclear Reacciones Nucleares 43 /?? 23

24 Scattering wavefunction Wavefunction that corresponds to the experimental condition: Ψ Ki,n i I i M i (R, ξ) = e ik i R n ii im i + χ (+) K i,n i I i M i (R, ξ) The outgoing wave: vi X χ (+) 2πi K i,n i I i M i (R, ξ) = k JM J L i N ir LiNiIiMi JMJ Y L i N i ( ˆK i) i X δ n,ni δ I,Ii δ n,ni S J nil:ni Ii Li Φ JM J nli ( ˆR, ξ)o nil(r) nil The scattering amplitude: 2πi X A(K i,k) ni I i M i ;nim = L in ii im i JM J Y L i N i ( ˆK i) KKi JM J L i N i X LNIM JM J Y LN( ˆK) δ n,ni δ I,Ii δ L,Li S J nil;ni Ii Li niln Programa de doctorado en Fsica Nuclear Reacciones Nucleares 44 /?? Scattering amplitude and cross sections Elastic and inelastic cross sections: dσ 1 X = A(K i,k) ni I dω i n 2I i + 1 i M i ;nim 2 MM i Coupled channels calculations give elastic and inelastic cross sections, if the states are properly described, if the interactions are known, and if all relevant channels are included Programa de doctorado en Fsica Nuclear Reacciones Nucleares 45 /?? 24

25 Inelastic scattering: collective models Projectile-target Coulomb interaction: V (R, ξ) = Ze2 X 1 4πǫ R r ; ξ {ri} i i r i R R r i Target Multipolar expansion: 1 R r i = X λµ ri λ 4π R λ+1 2λ + 1 Y λµ(ˆr i)yλµ( ˆR) (R > r i) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 46 /?? Inelastic scattering: collective models Electric multipole operator: M(Eλ, µ) = e P i rλ i Y λµ (ˆr i) Monopole and transition operator: V (R, ξ) = V (R) + (R, ξ) = Zze2 4πǫ R + Ze ǫ X λ,µ M(Eλ, µ) 2λ + 1 Y λµ( ˆR) R λ+1 Transition potentials: nm(r) = Ze ǫ X λ,µ ni nm n M(Eλ, µ) mi mm m Yλµ( ˆR) 2λ + 1 R λ+1 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 47 /?? 25

26 Inelastic scattering: collective models Central potential: Typically U nuc(r) = V (R R ), R = R 1 + R 2. Eg: WS parametrization V U(R) = 1 + exp R R a r i W 1 + exp R Ri a i R R 1 R 2 R 1 R R 2 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 48 /?? Inelastic scattering: collective models Non-spherical nucleus deformed surface r(θ, φ) = R + P λµ ˆδ λµ Y λµ(θ, φ) (ˆδ λµ =deformation length operators) Deformed potential: V (R, ξ) = V (R r(θ, φ)) Multipole expansion of the potential: Transition potential: V (R, ξ) = U (R R ) du(r R) nm(r) n V m = dr du(r R) dr X ˆδ λµ Yλµ(θ, φ) λµ X ni nm n ˆδ λµ mi mm m Yλµ( ˆR) λ The nuclear transition potentials are proportional to the matrix element of the deformation length operator. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 49 /?? 26

27 Physical ingredients for collective excitations Coulomb excitation electric reduced matrix elements nm(r) = Ze ǫ X λ,µ ni nm n M(Eλ, µ) mi mm m Yλµ( ˆR) 2λ + 1 R λ+1 ni n M(Eλ) mi m = p (2I n + 1)B(Eλ;I n I m) Nuclear excitation (collective model) deformation lengths within the rotational model: dv(r R) nm(r) = dr X ni nm n δ b λµ mi mm m Yλµ( ˆR) λ ni n b δ λ mi m = δ λ 2In + 1 I nkλ I mk δ λ = β λ R Programa de doctorado en Fsica Nuclear Reacciones Nucleares 5 /?? What do we learn by measuring inelastic scattering? Structure: Properties of h(ξ) and its eigenstates: Spin and parity of excited states, M(Eλ) matrix elements, deformation parameters. Interaction: Properties of V ( r, ξ): Response of the nucleus to the nuclear and coulomb field (transition potentials). Coulomb- nuclear interference effects. Reaction mechanism: Dynamics of the collision: States that should be included in the calculation, Dynamic polarization effects due to the coupling (coupled channels effects). Programa de doctorado en Fsica Nuclear Reacciones Nucleares 51 /?? 27

28 16 O + 28 Pb inelastic scattering Physical example: 16 O + 28 Pb 16 O+ 28 Pb(3,2 + ) i= i= i=1 g.s. 28 Pb + Nucl. Phys. A517 (199) 193 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 52 /?? Collective excitations: example Programa de doctorado en Fsica Nuclear Reacciones Nucleares 53 /?? 28

29 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering Coulomb and Nuclear excitations can produce constructive or destructive interference: 28 Pb( 16 O, 16 O) 28 Pb*(3 - ) 1 DWBA (Coulomb + nuclear) DWBA: Nuclear DWBA: Coulomb dσ inel /dω (mb/sr) 1.1 E lab =78 MeV θ c.m. (deg) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 54 /?? 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering Angular distribution of the ejectile in c.m. frame Absolute cross section Ratio to elastic cross section dσ inel /dω (mb/sr) (E=2.61 MeV) 2 + (E x =4.7 MeV) 28 Pb( 16 O, 16 O) 28 Pb* dσ inel /dσ el Pb( 16 O, 16 O) 28 Pb(3 - ) θ c.m. (deg) θ c.m. (deg) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 55 /?? 29

30 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering CC versus DWBA: Full coupled-channels: iblock = 3, iter = DWBA: iblock =, iter = 1 2 Full CC DWBA 28 Pb( 16 O, 16 O) 28 Pb(3 - ) dσ inel /dω (mb/sr) θ c.m. (deg) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 56 /?? Inelastic scattering Effect of the excitation energy: &STATES jp=. copyp=1 ep=. cpot=1 jt=3. bandt=-1 et=2.61 / 5 4 E x =1 MeV E x =2.61 MeV E x =4 MeV 28 Pb( 16 O, 16 O) 28 Pb(3 - ) dσ inel /dω (mb/sr) θ c.m. (deg) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 57 /?? 3

31 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering Effect of the incident energy: In FRESCO main output: CUMULATIVE REACTION cross section = <L> = 47.7 <L**2> = CUMULATIVE outgoing cross sections in partition 1 : σ (mb) Pb( 16 O, 16 O) 28 Pb(3 -,2 + ) E lab (MeV) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 58 /?? Inelastic scattering: cluster model Some nuclei permit 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,r) = U 1(R 1) + U 2(R 2) Ej:Deuteron case: R 1 = R r; R2 = R 1 2 r Internal states: [T r + V pn(r) ǫ n]φ n(r) = 2 H n r R p T Transition potentials: Z V n,n (R) = drφ n(r)v (r,r)φ n (r) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 59 /?? 31

32 Inelastic scattering: cluster model Example 1: 7 Li(α+t) + 28 Pb at 8 MeV (Phys. Lett. 139B (1984) 15): Z V n,n (R) = drφ n(r) [V α(r α) + V t(r t)] φ n (r); n =, 1 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 6 /?? Failure of the calculations without continuum Example: Three-body calculation (p+n+ 58 Ni) with Watanabe potential: Z V dt (R) = drφ gs(r) (V pt + V nt)φ gs(r) 1 1 d + 58 Ni at E=8 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. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 61 /?? 32

33 Example for inelastic scattering Proposed exercise: For the reaction p+ 12 C at E lab =185 MeV calculate: 1. The elastic scattering angular distribution, using Kooning-Delaroche optical potential (you can get the parameters from RIPL-2 database: 2. The angular distribution for the inelastic scattering populating the first excited state located at E x = 4.44 MeV and with J π = 2 +. Assume that this is a collective quadrupole excitation, and that the DWBA approximation is valid. Compare with the experimental data from Nucl. Phys. 69 (1965) Data: Deformation parameter: β 2 =.6. For the radius of the 12 C assume R = /3. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 62 /?? Input example for inelastic scattering with single-particle form factors 11 Be+ 12 C 11 Be(1/2 +,1/2 ) + 12 C at 49.3 MeV/A Programa de doctorado en Fsica Nuclear Reacciones Nucleares 63 /?? 33

34 Inelastic scattering example: 11 Be+ 12 C Example: 11 Be+ 12 C 11 Be(1/2 +,1/2 ) + 12 C at 49.3 MeV/A Phys. Rev. C 67, 3761 (23) Input file: be11c12_inel.in 1p 1/2 2s 1/2 2s 1/2 1p 3/2 1p 1/2 1p 3/2 1s 1/2 1s 1/2 n p Programa de doctorado en Fsica Nuclear Reacciones Nucleares 64 /?? 11 Be+ 12 C inelastic scattering General variables: &FRESCO hcm=.5 rmatch=6. jtmin=. jtmax=15. thmin=. thmax=45. thinc=.5 iblock=2 nnu=24 chans=1 smats=2 xstabl=1 elab=542.3 / iblock=2: number of channels coupled exacly. Partitions & states: &PARTITION namep= 11Be massp=11. zp=4 namet= 12C masst=12. zt=6 nex=2 / &STATES jp=.5 bandp=1 cpot=1 jt=. bandt=1 / &STATES jp=.5 bandp=-1 ep=.32 cpot=1 jt=. copyt=1 / nex=2: This partition will contain two pairs of states. copy=1: The target of the second pair of states is just the same (a copy) of the first target stat. &PARTITION namep= 1Be massp=1. zp=4 namet= 12C+n masst=13. zt=6 nex=1 / &STATES jp=. bandp=1 cpot=2 jt=. bandt=1 / Programa de doctorado en Fsica Nuclear Reacciones Nucleares 65 /?? 34

35 11 Be+ 12 C inelastic scattering Projectile-target Coulomb potential (monopole): &POT kp=1 ap=11. at=12. rc=1.111 / Neutron-target & core-target potentials: &POT kp=3 ap=. at=12. rc=1.111 / &POT kp=3 type=1 p1=37.4 p2=1.2 p3=.75 p4=1. p5=1.3 p6=.6 / &POT kp=2 ap=1. at=12. rc=1.111 / &POT kp=2 type=1 p1=123. p2=.75 p3=.8 p4=65. p5=.78 p6=.8 / Neutron binding potential: &POT kp=4 ap= at=1. rc=1. / &POT kp=4 type=1 p1=87. p2=1. p3=.53 / Programa de doctorado en Fsica Nuclear Reacciones Nucleares 66 /?? 11 Be+ 12 C inelastic scattering Bound state wave functions: &OVERLAP kn1=1 ic1=1 ic2=2 in=1 nn=2 sn=.5 l= j=.5 kbpot=4 be=.5 isc=1 / &OVERLAP kn1=2 ic1=1 ic2=2 in=1 nn=1 l=1 sn=.5 j=.5 kbpot=4 be=.18 isc=1 ipc=2 / kn1=1,2: Index for this WF ic1/ic2: Index of partition containing core ( 1 Be) / composite ( 11 Be) in=1/2: WF for projectile/target nn,sn,l,j: Quantum numbers for bound state be: separation energy. kbpot=3: Index KP of binding potential. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 67 /?? 35

36 11 Be+ 12 C inelastic scattering Couplings: &COUPLING icto=1 icfrom=2 kind=3 ip1=4 ip2=1 p1=3. p2=2. / kind=3: Single-particle excitations of projectile icto=1: Partition containing nucleus being excited ( 11 Be) icfrom=2: Partition containing core ( 1 Be) ip1=4: Maximum multipole for coupling potentials p1/p2: KP index for fragment-target / core-target potentials Spectroscopic amplitudes: &CFP in=1 ib=1 ia=1 kn=1 a=1. / &CFP in=1 ib=2 ia=1 kn=2 a=1. / in=1/2: Projectile/target ib/ia: Index for composite/core state a=1.: Spectroscopic amplitude Programa de doctorado en Fsica Nuclear Reacciones Nucleares 68 /?? Input example for inelastic scattering with collective form factors Example: 28 Pb( 16 O, 16 O) 28 Pb(3,2 + ) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 69 /?? 36

37 Inelastic scattering Input example 2: 28 Pb( 16 O, 16 O) 28 Pb(3,2 + ) (o16pb_cc1a.in ) o16pb_cc1a.in: 16O+28Pb 8 MeV NAMELIST &FRESCO hcm=.5 rmatch=1. jtmin=. jtmax=3. thmin=5. thmax=-18. thinc=2.5 iblock=3 smats=2 xstabl=1 elab= 8. / &PARTITION namep= 16-O massp= zp=8 namet= PB-28 masst= zt=82 nex=3 / &STATES jp=. bandp=1 ep=. cpot=1 jt=. bandt=1 et=. / &STATES jp=. copyp=1 ep=. cpot=1 jt=3. bandt=-1 et=2.61 fexch=f / &STATES jp=. copyp=1 bandp=1 ep=. cpot=1 jt=2. bandt=1 et=4.7 / &partition / &POT kp=1 itt=f ap=28. at=16. rc=1.2 / &POT kp=1 type=13 shape=1 itt=f p2=54.45 p3=815. / &STEP ib=1 ia=2 k=3 str=815. / &STEP ib=2 ia=1 k=3 str=815. / &STEP ib=1 ia=3 k=2 str=54.45 / &STEP ib=3 ia=1 k=2 str=54.45 / &step / &POT kp=1 type=1 shape=1 p4=1. p5=1. p6=.4 / &POT kp=1 type=-1 p1=6.5 p2=1.179 p3=.658 / &POT kp=1 type=13 shape=11 p2=.4 p3=.8 / &STEP ib=1 ia=2 k=3 str=.8 / &STEP ib=2 ia=1 k=3 str=.8 / &STEP ib=1 ia=3 k=2 str=.4 / &STEP ib=3 ia=1 k=2 str=.4 / &step / &pot / &overlap / &coupling / Programa de doctorado en Fsica Nuclear Reacciones Nucleares 7 /?? 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering General variables: &FRESCO hcm=.5 rmatch=1. jtmin=. jtmax=3. thmin=5. thmax=-18. thinc=2.5 iblock=3 smats=2 xstabl=1 elab= 8. / iblock: Number of states (including gs) that will be coupled to all orders. iblock=1: only elastic scattering iblock=2: elastic scattering + 1st inelastic channel ( 28 Pb(3 )) iblock=3: elastic scattering + 28 Pb(3 ) + 28 Pb(2 + ) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 71 /?? 37

38 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering Partitions and states: &PARTITION namep= 16-O massp= zp=8 namet= PB-28 masst= zt=82 nex=3 / &STATES jp=. bandp=1 ep=. cpot=1 jt=. bandt=+1 et=. / &STATES copyp=1 cpot=1 jt=3. bandt=-1 et=2.61 / &STATES copyp=1 cpot=1 jt=2. bandt=+1 et=4.7 / &partition / nex: number of states within the partition ep, et: excitation energy for projectile / target copyp=1 tells FRESCO that the 2nd and 3rd projectile states are just a copy of the ground state. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 72 /?? 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering Coulomb excitation: &POT kp=1 ap=28. at=16. rc=1.2 / &POT kp=1 type=13 shape=1 p2=54.45 p3=815. p4= p5= p6= / type=13: couple target states by deforming previous potential p1,...,p6: consider couplings for multipolarities k with pk shape=1: usual deformed charge sphere: nm(r) M(Ek)/R k+1 &STEP ib=1 ia=2 k=3 str=815. / &STEP ib=2 ia=1 k=3 str=815. / &STEP ib=1 ia=3 k=2 str=54.45 / &STEP ib=3 ia=1 k=2 str=54.45 / &step / i=3 i= i=1 g.s. 28 Pb + ia, ib: couple from state number ia to state ib k: multipolarity str= ib M(Ek ia = p (2I a + 1)B(Eλ; ia ib) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 73 /?? 38

39 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering Nuclear excitation: &POT kp=1 type=1 shape=1 p4=1. p5=1. p6=.4 / &POT kp=1 type=-1 shape= p1=6.5 p2=1.179 p3=.658 / &POT kp=1 type=13 shape=1 itt=f p2=.4 p3=.8 / type=13: couple target states by deforming preceding potential shape=1: usual deformed nuclear potential: nm(r) δ k du(r)/dr &STEP ib=1 ia=2 k=3 str=.8 / &STEP ib=2 ia=1 k=3 str=.8 / &STEP ib=1 ia=3 k=2 str=.4 / &STEP ib=3 ia=1 k=2 str=.4 / str= ib δ k ia (reduced deformation length) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 74 /?? 28 Pb( 16 O, 16 O) 28 Pb inelastic scattering Useful output files: Main output file: CUMULATIVE REACTION cross section = <L> = 47.7 CUMULATIVE outgoing cross sections in partition 1 : Cumulative ABSORBTION by Imaginary Potentials = <L> = 6.99 Angular distributions: - fort.21 : Elastic scattering angular distribution - fort.22 : 1st state angular distribution - fort.23 : 2nd excited state angular distribution fort.56: 3 columns: Fusion (absorption), reaction and inelastic cross section for each total angular momentum J. σ reac = σ inel + σ abs Programa de doctorado en Fsica Nuclear Reacciones Nucleares 75 /?? 39

40 Transfer reactions: the DWBA method 76 /?? 8 B 7 Be n p 2 H Transfer reactions 8 B 7 Be n p 2 H Programa de doctorado en Fsica Nuclear Reacciones Nucleares 77 /?? 4

41 Formalism for transfer reactions Transfer process:. (a + v) +b a + (b + v) {z } {z } A B A a ξ v r R α R β ξ v b a r ξ ξ b B Projectile target interaction: Prior form: V prior = V vb + U ab = U α + (V vb + U ab U α) {z } prior Post form: V post = V av + U ab = U β + (V av + U ab U β ) {z } post U α, U β : average proyectile-target interaction in entrance/exit channel Programa de doctorado en Fsica Nuclear Reacciones Nucleares 78 /?? 41

42 Coupled Reaction Channels Model wavefunction: Ψ = φ A(ξ,r)φ b (ξ )χ α(r α) + φ a(ξ)φ B(ξ,r )χ β (R β ) Coupled-reaction channels (CRC) equations: [H E] Ψ = Z [E ǫ α T R U α(r α)] χ α(r α) = Z [E ǫ β T R U β (R β )] χ β (R β ) = dr β K α,β (R α,r β )χ β (R β ) dr αk α,β (R α,r β )χ α(r α) Non-local kernels: Z K α,β (R β,r α) = dξdξ dr φ a(ξ)φ B(ξ,r )(H E)φ A(ξ,r)φ b (ξ ) CRC equations have to be solved iteratively due to NL kernels. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 79 /?? DWBA approximation Distorted wave Born approximation: [E ǫ α T R U α(r α)] eχ α(r α) = [E ǫ β T R U β (R β )] eχ β (R β ) = DWBA amplitude (prior): Z Z T prior = β (R β)(φ aφ B prior φ Aφ b )eχ (+) α (R α)dr αdr eχ ( ) Structure form-factor: Z (φ aφ B prior φ Aφ b ) dξdξ φ a(ξ)φ B(ξ,r ) priorφ A(ξ,r)φ b (ξ ) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 8 /?? 42

43 Spectroscopic factors Parentage amplitudes: Projectile: φ JM A (ξ,r) = 1 na PIlj A IJ;lsj ˆφI a (ξ) ϕ lsj (r) JM Target: φ J M B (ξ,r ) = 1 nb PIlj A IJ ;lsj ˆφI b (ξ ) ϕ lsj (r ) J M A IJ;lsj = spectroscopic amplitudes S IJ;lsj = A IJ;lsj 2 = spectroscopic factors DWBA amplitude (prior) T prior = A IJ;lsj A I J ;l sj Z Z β (R β)ϕ l sj (r ) priorϕ lsj (r)eχ (+) α (R α)dr αdr eχ ( ) In DWBA, the transfer cross section is proportional to the spectroscopic factors S IJ;lsj S I J ;l sj Programa de doctorado en Fsica Nuclear Reacciones Nucleares 81 /?? Extracting structure information from transfer reactions Example: d+ 28 Pb p + 29 Pb β 2 =. β 3 =. β 4 =. 3d3/2(184) 1/2(172) 2g7/2(18) V WS =-4.58 MeV -2 3d5/2(17) a WS =.7 fm r WS =1.347 fm 1j15/2(164) 1i11/2(148) V SO =-4.58 MeV a SO =.7 fm -4 2g9/2(136) r SO =1.31 fm κ= ε F E cut-off =15. hω Jeans et al, NPA 128 (1969) 224 3p1/2(126) -8 2f5/2(124) 3p3/2(118) 1i13/2(114) z (fm) Neutron Single-Particle Levels of 28Pb Programa de doctorado en Fsica Nuclear Reacciones Nucleares 82 /?? 43

44 Extracting structure information from transfer reactions Angular distributions of transfer cross sections are very sensitive to the single-particle configuration of the transferred nucleon/s. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 83 /?? 44

45 Beyond DWBA: CCBA formalism When there are strongly coupled excited states in the initial or final partition, the CC and DWBA formalisms can be combined CCBA Ej: 172 Yb(p,d) Ascuitto et al, Nucl Phys. A226 (1974) 454 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 84 /?? 45

46 Brief summary on transfer reactions Inclusion of transfer couplings in the Schrodinger equation gives rise to a set of coupled equations with non-local kernels (Coupled Reactions Channels) If transfer couplings are weak, the CRC equations can be solved in Born approximation DWBA approximation The DWBA amplitude is proportional to the product of the projectile and target spectroscopic factors. The analysis of transfer reactions provide information on: Spectroscopic factors. Quantum number for single-particle configurations (n, l, j). Binding interactions. Reactions mechanisms. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 85 /?? Transfer example Physical example: 56 Fe(d,p) 57 Fe at E d = 12 MeV p n p 56 d + Fe Q 56 Fe n p + 57 Fe Programa de doctorado en Fsica Nuclear Reacciones Nucleares 86 /?? 46

47 Transfer example: 56 Fe(d,p) 57 Fe DWBA transfer amplitude: T DWBA = A ia f χ ( ) p 57 Fe φ57 Fe V prior/post χ (+) d 56 Fe φ d χ d 56 Fe, χ p 57 Fe: initial and final distorted waves φ d : projectile bound wavefunction( p n) φ57 Fe: final (residual) wavefunction (n+ 56 Fe) A i, A f : initial / final spectroscopic amplitudes. V prior/post : transition potential in PRIOR or POST form PRIOR POST p r R n 56 Fe n r 56 Fe R p V prior = V n 56Fe + U p 56Fe U d 56Fe {z } remnant V post = V p n + U p 56Fe U p 57Fe {z } remnant Programa de doctorado en Fsica Nuclear Reacciones Nucleares 87 /?? Transfer example: 56 Fe(d,p) 57 Fe Essential physical ingredients in a DWBA calculation: Potentials (5): Distorted potential for entrance channel (complex): d+ 56 Fe Distorted potential for exit channel (complex): p+ 57 Fe Core-core interaction (complex): p+ 56 Fe Binding potential for projectile (real): p+n Binding potential for target (real): n+ 56 Fe Spectroscopic amplitudes: A i,a f Programa de doctorado en Fsica Nuclear Reacciones Nucleares 88 /?? 47

48 Transfer example: 56 Fe(d,p) 57 Fe Physical ingredients: Optical and binding potentials (NPA(1971) 529) System V r a W d r i a i r C (MeV) (fm) (fm) (MeV) (fm) (fm) (fm) d+ 56 Fe p+ 56,57 Fe p + n a n+ 56 Fe B.E U(R) = V f WS(R) + 4 i a W d df WS(R) dr f WS(R) = exp ` R R a a Gaussian geometry: V (r) = V exp[ (r/a ) 2 ]. Optical potential (MeV) d+ 56 Fe optical potential V nuc (R)+V coul (R) W d (R) V =9 MeV, r =1.15 fm, a =.81 fm W d =21 MeV, r i =1.34 fm, a i =.68 fm R (fm) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 89 /?? Transfer example: 56 Fe(d,p) 57 Fe Spectroscopic factors: φ JM B (ξ,r) = X h i A IJ lsj φ I b(ξ) ϕ lsj (r) JM Ilj So, for example: 57 Fe;1/2 = α ˆ 56 Fe;gs ν2p 1/2 + β ˆ 56 Fe;2 + ν2p 1/2 3/ /2 α, β,... : spectroscopic amplitudes α 2, β 2,... : spectroscopic factors Programa de doctorado en Fsica Nuclear Reacciones Nucleares 9 /?? 48

49 Transfer example: 56 Fe(d,p) 57 Fe Post and prior equivalence: 2 56 Fe(d,p) 57 Fegs dσ/dω (mb/sr) POST: RNL=4 fm PRIOR: RNL=2 fm θ c.m. (deg) Post and prior give identical results, provide that the parameters are adequate for convergence. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 91 /?? Transfer example: 56 Fe(d,p) 57 Fe Dependence with binding energy: &OVERLAP kn1=2 (...) be=7.646 / &OVERLAP kn1=2 (...) be=5.646 / &OVERLAP kn1=2 (...) be=5.646 / &OVERLAP kn1=2 (...) be=3.646 / &OVERLAP kn1=2 (...) be=1.646 / &OVERLAP kn1=2 (...) be=.1 / Fe(d,p) 57 Fegs dσ/dω (mb/sr) S n =7.646 MeV S n =5.646 MeV S n =3.646 MeV S n =1.646 MeV S n =.5 MeV θ c.m. (deg) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 92 /?? 49

50 Transfer example: 56 Fe(d,p) 57 Fe Dependence with beam energy: 2 56 Fe(d,p) 57 Fe 15 σ inel (mb) Incident energy (MeV) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 93 /?? Transfer example: 56 Fe(d,p) 57 Fe Dependence with beam energy E V b : diffractive structure, forward peaked. E V b : smooth dependence with θ, backward peaked. dσ/dω (mb/sr) Fe(d,p) 57 Fegs E=2 MeV E=3 E=4 E=5 E=6 E= θ c.m. (deg) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 94 /?? 5

51 Transfer example: 56 Fe(d,p) 57 Fe Selectivity of l: dσ/dω (mb/sr) l= l=1 l=2 l=3 56 Fe(d,p) 57 Fe θ c.m. (deg) H.M. Sen Gupta et al, Nucl. Phys. A16, 529 (1971) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 95 /?? Beyond DWBA: CCBA formalism When there are strongly coupled excited states in the initial or final partition, the CC and DWBA formalisms can be combined CCBA CC DWBA d + 56 Fe p + 57 Fe Programa de doctorado en Fsica Nuclear Reacciones Nucleares 96 /?? 51

52 Transfer example: 56 Fe(d,p) 57 Fe Appendix: 56 Fe(d,p) 57 Fe with FRESCO: fe56dp_dwba.in Ed=12 MeV; AMELIST FRESCO hcm=.1 rmatch=2. rintp=.2 hnl=.1 rnl=4 centre=-.45 jtmax=15 thmin=1. thmax=18. thinc=1. it=1 iter=1 chans=1 smats=2 xstabl=1 elab= 12 / PARTITION namep= d massp=2.14 zp=1 namet= 56Fe masst= zt=26 nex=1 / STATES jp=1. bandp=1 ep=. cpot=1 jt=. bandt=1 et=. / PARTITION namep= p massp=1.78 zp=1 namet= 57Fe masst= zt=26 qval=5.421 pwf=f nex=1 / STATES jp=.5 bandp=1 ep=. cpot=2 jt=.5 bandt=-1 et=. / partition / POT kp=1 itt=f at=56 rc=1.15 / POT kp=1 type=1 itt=f p1=9 p2=1.15 p3=.81 / POT kp=1 type=2 itt=f p4=21 p5=1.34 p6=.68 / POT kp=2 itt=f at=57 rc=1.15 / POT kp=2 type=1 itt=f p1=47.9 p2=1.25 p3=.65 / POT kp=2 type=2 itt=f p4=11.5 p5=1.25 p6=.47 / POT kp=3 itt=f at=56 rc=1. / POT kp=3 type=1 itt=f p1=65. p2=1.25 p3=.65 / POT kp=4 itt=f ap=1. at=. rc=1. / POT kp=4 type=1 shape=2 itt=f p1=72.15 p2=. p3=1.484 / POT kp=5 itt=f at=56 rc=1.15 / POT kp=5 type=1 itt=f p1=47.9 p2=1.25 p3=.65 / POT kp=5 type=2 itt=f p4=11.5 p5=1.25 p6=.47 / pot / OVERLAP kn1=1 ic1=1 ic2=2 in=1 nn=1 sn=.5 j=.5 kbpot=4 be=2.225 isc=1 / OVERLAP kn1=2 ic1=1 ic2=2 in=2 nn=2 l=1 sn=.5 j=.5 kbpot=3 be=7.646 isc=1 / overlap / COUPLING icto=2 icfrom=1 kind=7 ip2=-1 ip3=5 / CFP in=1 ib=1 ia=1 kn=1 a=1. / CFP in=2 ib=1 ia=1 kn=2 a=1. / cfp / coupling / Programa de doctorado en Fsica Nuclear Reacciones Nucleares 97 /?? Transfer example: 56 Fe(d,p) 57 Fe General variables: Ed=12 MeV; NAMELIST &FRESCO hcm=.1 rmatch=2. rintp=.2 hnl=.1 rnl=4 centre=-.45 jtmax=15 thmin=1. thmax=18. thinc=1. it=1 iter=1 chans=1 smats=2 xstabl=1 elab= 12 / rnl: range of non-locality centre,rintp,hnl: parameters for numerical integration (see fresco manual) iter: Number of iterations so, for DWBA, iter=1 Programa de doctorado en Fsica Nuclear Reacciones Nucleares 98 /?? 52

53 Transfer example: 56 Fe(d,p) 57 Fe Partitions and states: Incoming (initial) partition: d+ 56 Fe &PARTITION namep= d massp=2.14 zp=1 namet= 56Fe masst= zt=26 nex=1 / &STATES jp=1. bandp=1 ep=. cpot=1 jt=. bandt=1 et=. / Outgoing (final) partition: p+ 57 Fe &PARTITION namep= p massp=1.78 zp=1 namet= 57Fe masst= zt=26 qval=5.421 nex=1 / &STATES jp=.5 bandp=1 ep=. cpot=2 jt=.5 bandt=-1 et=. / qval: Q-value for gs-gs transfer Programa de doctorado en Fsica Nuclear Reacciones Nucleares 99 /?? Transfer example: 56 Fe(d,p) 57 Fe Interactions: Entrance channel distorted potential: d+ 56 Fe &POT kp=1 itt=f at=56 rc=1.15 / &POT kp=1 type=1 itt=f p1=9 p2=1.15 p3=.81 / &POT kp=1 type=2 itt=f p4=21 p5=1.34 p6=.68 / Exit channel distorted potential: p+ 57 Fe &POT kp=2 itt=f at=57 rc=1.15 / &POT kp=2 type=1 itt=f p1=47.9 p2=1.25 p3=.65 / &POT kp=2 type=2 itt=f p4=11.5 p5=1.25 p6=.47 / Core-core potential: p+ 56 Fe &POT kp=5 itt=f at=56 rc=1.15 / &POT kp=5 type=1 itt=f p1=47.9 p2=1.25 p3=.65 / &POT kp=5 type=2 itt=f p4=11.5 p5=1.25 p6=.47 / Programa de doctorado en Fsica Nuclear Reacciones Nucleares 1 /?? 53

54 Transfer example: 56 Fe(d,p) 57 Fe Interactions (continued...) Binding potentials: n+ 56 Fe: Woods-Saxon &POT kp=3 at=56 rc=1. / &POT kp=3 type=1 p1=65. p2=1.25 p3=.65 / n+p: Gaussian &POT kp=4 ap=1. at=. / &POT kp=4 type=1 shape=2 p1=72.15 p2=. p3=1.484 / Programa de doctorado en Fsica Nuclear Reacciones Nucleares 11 /?? Transfer example: 56 Fe(d,p) 57 Fe Bound wavefunctions (overlaps): d=p+n: simple 1S model &OVERLAP kn1=1 ic1=1 ic2=2 in=1 nn=1 l= sn=.5 j=.5 kbpot=4 be=2.225 isc=1 / 57 Fe= 56 Fe+n: assume 2p 1/2 configuration &OVERLAP kn1=2 ic1=1 ic2=2 in=2 nn=2 l=1 sn=.5 j=.5 kbpot=3 be=7.646 isc=1 / in=1: projectile in=2: target nn,l,sn,j: quantum numbers: l + sn = j be: binding (separation) energy kbpot: potential index Programa de doctorado en Fsica Nuclear Reacciones Nucleares 12 /?? 54

55 Transfer example: 56 Fe(d,p) 57 Fe Transfer coupling between the two partitions: &COUPLING icfrom=1 icto=2 kind=7 ip1= ip2=-1 ip3=5 / icfrom: index for partition of initial state icto: index for partition of final state kind: kind of coupling. kind=7 means finite-range transfer. ip1=: post representation ip1=1: prior ip2=-1: include full remnant ip3: index for core-core potential (p+ 56 Fe) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 13 /?? Transfer example: 56 Fe(d,p) 57 Fe Spectroscopic amplitudes: &CFP in=1 ib=1 ia=1 kn=1 a=1. / &CFP in=2 ib=1 ia=1 kn=2 a=1. / in=1: projectile state in=2: target state ib: index for state of composite ia: index for state of core a: spectroscopic amplitude Programa de doctorado en Fsica Nuclear Reacciones Nucleares 14 /?? 55

56 Inelastic scattering to continuum states 15 /?? Inelastic scattering to the continuum 11 Be 1 Be Pb n Programa de doctorado en Fsica Nuclear Reacciones Nucleares 16 /?? 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 87 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. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 17 /?? 56

57 Exotic nuclei, halo nuclei, and Borromean systems Proton number He He H D T n Be F Ne O 14 O 15 O 16 O 17O 12N 13 N 14 N 15 N 16 N 17 N 18 N 19 N 2 N 21 N 9 C 1 C 11C 12 C 13C 8B 6 Li 14 C 15 C 1 C 6 17 C 18 C 19 C 2C C B 11 B 12 B 13 B B B B 19 B Be 1 12 Be 11 Be Be Be Li 8 Li 9 Stable Li Li He 1111 He Unstable Neutron halo Borromean Neutron number Programa de doctorado en Fsica Nuclear Reacciones Nucleares 18 /?? Why study reactions with exotic nuclei? Many properties of nuclear structure (level spacing, magic numbers,etc) could be different from normal nuclei. Many reactions of astrophysical interest are known to involve nuclei far from the stability valley. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 19 /?? 57

58 Some difficulties inherent to the study of reactions with exotic beams Experimentally: Exotic nuclei are short-lived and difficult to produce. Beam intensities are typically small. Theoretically: Exotic nuclei are easily broken up in nuclear collisions coupling to the continuum plays an important role. Effective NN interactions, level schemes, etc are different from stable nuclei. Many exotic nuclei exhibit complicated cluster (few-body) structure. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 11 /?? Indirect measurements for nuclear astrophysics Many reactions of astrophysical interest at energies too low to be measured e at present experimental facilities. Coulomb breakup and transfer reactions provide indirect information for these processes. Example: The rate for the 7 Be+ p 8 B + γ reaction depends mainly on the overlap ( 8 B 7 Be), which can be investigated by means of other direct reactions: Transfer reactions: E.g.: d( 7 Be, 8 B)n Coulomb breakup reactions: E.g.: 8 B+ 28 Pb 7 Be+ p + 28 Pb 8 B 8 7 B Be 7 Be n p γ p 2 H 28 Pb Programa de doctorado en Fsica Nuclear Reacciones Nucleares 111 /?? 58

59 Some difficulties inherent to the study of reactions with exotic beams First evidences of the existence of halo nuclei came from reaction cross sections measurements. Tanihata et al, PRL55, 2676 (1985) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 112 /?? Inelastic scattering of weakly bound nuclei Single-particle (or cluster) excitations become dominant. Excitation to continuum states important. Exotic nuclei are weakly bound coupling to continuum states becomes an important reaction channel Programa de doctorado en Fsica Nuclear Reacciones Nucleares 113 /?? 59

60 Inelastic scattering example: 11 Be+ 12 C Fukuda et al, Phys. Rev. C7 (24) 5466) 11 Be+ 12 C ( 1 Be+n) + 12 C Programa de doctorado en Fsica Nuclear Reacciones Nucleares 114 /?? Failure of the calculations without continuum Three-body calculation (p+n+ 58 Ni) with Watanabe potential: Z V (R) = drφ gs(r) (V pt + V nt) φ gs(r) 1 1 d + 58 Ni at E=8 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. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 115 /?? 6

61 The role of the continuum in the scattering of weakly bound nuclei The origins of the CDCC method: Pioneering work of Johnson & Soper for deuteron scattering: PRC1,976(197) p-n continuum represented by a single s-wave state. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 116 /?? The role of the continuum in the scattering of weakly bound nuclei More realistic formulation by G.H. Rawitscher [ PRC9, 221 (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) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 117 /?? 61

62 Inclusion of the continuum in CC calculations: continuum discretization Quantum Hamiltonian Bound states - Discrete - Finite - Normalizable Unbound states - Continuous - Infinite - Non-normalizable Continuum discretization: represent the continuum by a finite set of square-integrable states True continuum Discretized continuum Non normalizable Normalizable Continuous Discrete Programa de doctorado en Fsica Nuclear Reacciones Nucleares 118 /?? Continuum discretization SOME POPULAR METHODS OF CONTINUUM DISCRETIZATION: Box method: Eigenstates of the H in a large box. Sturmian basis Gamow states: complex-energy eigenstates of the Schrödinger equation. Pseudostate method: Expand continuum states in a complete basis of square-integrable states (eg. HO) Bin method: Square-integrable states constructed from scattering states. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 119 /?? 62

63 Bound versus scattering states φ ε (r) ε φ gs (r) 8 B V(r) [MeV] -5-1 V(r) Unbound states are not suitable for CC calculations: Continuous (infinite) distribution in energy. Non-normalizable: φ k (r) φ k (r) δ(k k ) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 12 /?? CDCC formalism Example: discretization of the deuteron continuum in terms of energy bins. φ l,2 φ l,2 φ l,1 n=3 n=2 n=1 l= l=1 l=2 s waves ground state p waves Breakup threshold d waves ε max ε= ε = 2.22 MeV Programa de doctorado en Fsica Nuclear Reacciones Nucleares 121 /?? 63

64 CDCC formalism Bin wavefunction: u lsj,n (r) = r 2 πn Z k2 k 1 w(k)u lsj,k (r)dk k: linear momentum u lsj,k : scattering states (radial part) w(k): weight function 1.5 u(r) -.5 Continuum WF at ε x =2 MeV Bin WF: ε x =2 MeV, Γ=1 MeV r (fm) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 122 /?? CDCC formalism for deuteron scattering Hamiltonian: H = T R + h r + V pt(r pt) + V nt(r nt) Model wavefunction: NX Ψ(R,r) = φ gs(r)χ (R) + φ n(r)χ n(r) n> 2 H n r R p T Coupled equations: [H E]Ψ(R, r)» 2 2µ «d 2 L(L + 1) + ǫ dr2 R 2 n E f αj(r) + X i L L V J α:α (R)f α J(R) = α α= {L, l, s, j, n} Transition potentials: V J n;n (R) = Z drφ n(r) h V pt(r + r 2 ) + Vnt(R r 2 ) i φ n (r) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 123 /?? 64

65 Application of the CDCC formalism: d+ 58 Ni 1 1 (dσ/dω)/(dσ R /dω) Exper. No continuum CDCC: s+d waves θ c.m. Inclusion of the continuum is important to describe the data Programa de doctorado en Fsica Nuclear Reacciones Nucleares 124 /?? 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 6 He=α + n + n 1 6 Li+ 4 Elab =156 MeV 6 He MeV σ/σ R Experiment No continuum CDCD: n s =15, n d = θ c.m. (deg) σ/σ Ruth LLN data no continuum.2 3-body CDCC 4-body CDCC θ c.m. (deg) Programa de doctorado en Fsica Nuclear Reacciones Nucleares 125 /?? 65

66 Normal versus halo nuclei: Fraunhofer No continuum 1 σ/σ R Including continuum Be + 49 MeV/A θ (degrees) c.m. In Fraunhofer scattering the presence of the continuum produces a reduction of the elastic cross section Programa de doctorado en Fsica Nuclear Reacciones Nucleares 126 /?? Breakup observables: resonant and non-resonant continuum What is a resonance? Definition 1: (Experimentalist) It is a maximum in the cross section as a function of the energy dn/de x (counts per 1 kev) (θ c.m. = o ) REX-ISOLDE (IS367) E x (MeV) (above 9 Li+n threshold) Ej: 9 Li + d 1 Li + p (H. Jeppesen et al). Searching for resonances in 1 Li. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 127 /?? 66

67 ...but Not all the maxima in the cross section can be associated to resonances. Coupling to a non-resonant continuum produces a maximum at some energy, which is related to the size of the system, and the properties of the interaction. For weakly bound systems, resonances can be very broad (1 MeV), and occur al relatively low excitation energies. It is not clear, a priori, whether a bump in the cross section is a signature of a genuine resonance, or not. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 128 /?? What is a resonance? Definition 2: (Theoretician) It is a pole on the S-matrix (of the n+ 9 Li system)...but The fact that there is a pole in the S-matrix in the complex E plane does not allow, by itself, to calculate the cross section to the resonance. The wave function corresponding to a pole in the complex E-plane is not square-normalizable. Finding poles in a complex energy-plane for multi-channel or three-body systems is difficult. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 129 /?? 67

68 What is a resonance? Definition 3: 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. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 13 /?? The resonance occurs in the range of energies for which the phase shift is close to π/2. In this range of energies, the continuum wavefunctions have a relatively large probability of being in the radial range of the potential. The continuum wavefunctions are not square normalizable. However, a normalized bin of wavefunctions can be constructed to represent the resonance. Programa de doctorado en Fsica Nuclear Reacciones Nucleares 131 /?? 68