Semi-Classical perturbation theory Coulomb only First-order most used

Transcription

1 direct reactions

2 Models for breakup Semi-Classical perturbation theory Coulomb only First-order most used TDSE (Time Dependent Schrodinger Equation) Coulomb + Nuclear Semi-classical orbit needed DWBA (Distorted Wave Born Approximation) Coulomb + Nuclear First-order CDCC (Continuum Discretized Coupled Channels) Coulomb + Nuclear Higher order Faddeev

3 Semiclassical Coulomb Dissociation Photo absorption cross section 2λ 1 dσ E E γ de ( γ ) 3 λ ( 2π ) ( λ + 1) = λ( ( 2λ + 1)!!) 2 c h db de ( Eλ ) γ Projectile structure information B ( Eλ ) = ψ f Eλ ψ i i=ground state γ f=continuum state Note that, if b small, nuclear has to be taken into account!

4 Breakup of 7 Be: three body reaction 4 He core r 3 He valence R 208 Pb target 7 Be treated as 2-body projectile 3-body Hamiltonian for reaction H h vc = T R = T r + V + V ct vc + V vt + h vc fix V ct and V vt from elastic scattering φ is 7 Be (α+ 3 He) cluster wavefunction defined by potential V vc h h vc vc φ ( r) = ε n φ ( k, r) = ε lj φ ( r) n n k φ ( k, r) l = core-valence relative angular momentum j = projectile total angular momentum lj ε n ε < k 0 > 0 fixed by binding energy for bound states resonances and scattering phase shifts for continuum

5 Breakup of 7 Be: continuum discretization Discretize continuum into bins average wavefuntion over a bin with weight w i (k) k 2 i φi, lj ( r) = wi ( k ) φlj πn i k i 1 w i (k) chosen so that the bin wavefunctions are real and normalized correctly using ( k, r) dk 20 E vc label the quantum numbers for each bin by α i,lj now have 72 coupled channels N i = k k i i 1 w i ( k ) 2 dk bound states p 1/2 p 3/ s 1/2 p 1/2 p 3/2 d 3/2 d 5/2 f 5/2 f 7/2

6 Breakup of 7 Be: CDCC equations We have 72 coupled channels, each labeled by the set of quantum numbers Solve set of radial coupled equations [ L J ] J J J T + V R) E u ( R) = V ( R) u ( R) ( 1) R α ( i l j L) αα ( α α αα ' α ' α ' α Where the coupling potential from state α to state α is [ φ ( r ) Y ( Rˆ )] V ( r, R) [ φ ( r ) Y ( Rˆ )] J αα ( = α ' L' JM α V ' R) r r V (, R) = V ct r r ( R ct ) + V vt r ( R and the cluster target potentials include both Coulomb and Nuclear parts vt ) r r r L JM Coulomb breakup neglect Nuclear in coupling potentials on rhs of Eq. (1) Nuclear breakup neglect Coulomb in coupling potentials on rhs of Eq. (1)

7 Breakup with CDCC: one case in detail Capture rate 3 He(α,γ) 7 Be Capture rates at astrophysical energies are hard to measure and thus rely on extrapolations to low energies 8 B 7 Be 8 Be 6 Li 7 Li 3 He 4 He Motivates alternative methods using inverse reaction 7 Be + γ α + 3 He (i) Coulomb Dissociation method (ii) Asymptotic Normalization Coefficient (ANC) method 1 H 2 H pp II (α,γ) (e -,γ) (p,α) 3 He 7 Be 7 Li 4 He pp III (p,γ) β + decay 8 B 8 Be 2 4 He NSCL2004

8 Coulomb Dissociation method Cross sections larger for inverse breakup reaction Can relate breakup cross section to capture rate using semi-classical reaction theory complications Nuclear breakup E2 contributions Final state interactions (continuum-continuum couplings) 3 He low relative energy 4 He 7 Be γ 208 Pb 7 Be + γ α + 3 He Motivated experiment at NSCL : 7 Be MeV/nucleon NSCL2004

9 ANC method (nuclear breakup) dσ dω exp = C b 2 lj 2 nlj dσ dω nlj theory Outside range of nuclear force, in asymptotic region I overlap integral c = core 4 He v = valence 3 He P = projectile 7 Be W l 1 1 / 2 P η, + 2 ( r ) Snlj φnlj ( r) Ccvlj r P cvlj ) ANC Spectroscopic factor (2kr) Whittaker function g.s. single particle wavefunction Motivated experiment at Texas A&M : 7 Be MeV/nucleon C P cvlj = S 1/ 2 nlj b nlj NSCL2004

10 Breakup of 7 Be: inputs 7 Be modeled as 2-body projectile 3 He spin 1/2 l=1 relative motion 4 He spin 0 Buck et al. potential fitted to : ground and excited state energies 7/2- and 5/2- resonances scattering phase shifts charge radius, quadrupole and octupole moments and B(E2) for 7 Li wavefunction has node due to Pauli blocking parity dependent potential cluster-target potentials from elastic scattering 3 He MeV 4 He MeV 3 He MeV 4 He MeV NSCL Texas A & M

11 Breakup of 7 Be: convergence 7 Be Pb 3 He + 4 He Pb E lab = 100 MeV/nucleon Convergence checked for many variables, such as: Potential multipoles (Q 2) Max radius for coupled equations (R max =1000fm) Maximum relative energy (E max =20 MeV) Number of partial waves in coupled equations (L max =10000,2000 using interpolation) Bin radius and step length for coupling integrals (r bin =50fm, h=0.04fm) Number of partial waves for breakup states (l 3) Iterative solution of coupled equations versus solving set of equations exactly [PRC 70 (2004) ]

12 Breakup of 7 Be: comparison with other models Semiclassical A&W Coulomb-1step Coulomb-CDCC 7 Be Pb 3 He + 4 He Pb E lab = 100 MeV/nucleon [PRC 70 (2004) ]

13 Breakup of 7 Be: higher order couplings 7 Be + 12 C 3 He + 4 He + 12 C E lab = 25 MeV/nucleon 7 Be Pb 3 He + 4 He Pb E lab = 100 MeV/nucleon [PRC 70 (2004) ]

14 Breakup of 7 Be: peripherality ANC method requires that reaction is peripheral and therefore only probes the tail of the wavefunction 7 Be + 12 C 3 He + 4 He + 12 C E lab = 25 MeV/nucleon approx 28% of breakup cross section comes from impact parameters less than sum of radii impact parameter from semi-classical relation J=Kb [PRC 70 (2004) ]

15 Breakup of 7 Be: Coulomb free? Note that Coulomb is not negligible 7 Be + 12 C 3 He + 4 He + 12 C E lab = 25 MeV/nucleon [PRC 70 (2004) ]

16 Breakup of 7 Be: nuclear free? 7 Be Pb 3 He + 4 He Pb E lab = 100 MeV/nucleon impact parameter from semi-classical relation J=Kb Coulomb dissociation method requires that the reaction is Coulomb dominated, outside the range of the nuclear force 16fm maximum range of nuclear force assuming Rutherford trajectories this impact parameter relates to a scattering angle of 2.5 [PRC 70 (2004) ]

17 Breakup of 7 Be: nuclear component 7 Be Pb 3 He + 4 He Pb E lab = 100 MeV/nucleon Cut-off angle from Rutherford orbit = 2.5 cross section dominated by nuclear breakup even for small scattering angles below the cut-off angle [PRC 70 (2004) ]

18 Breakup of 7 Be: small relative energies 7 Be Pb 3 He + 4 He Pb E lab = 100 MeV/nucleon restricted maximum relative energy between 3 He and 4 He fragments in final state E vc All bins included in coupled channels solution 20 Angular cross section summed for only first 2 bins in each j π set This dramatically reduces the amount of nuclear breakup in the forward angles p 1/2 p 3/ s 1/2 p 1/2 p 3/2 d 3/2 d 5/2 f 5/2 f 7/2 [PRC 70 (2004) ]

19 Breakup of 7 Be: E1 versus E2 7 Be Pb 3 He + 4 He Pb E lab = 100 MeV/nucleon restricted maximum relative energy between 3 He and 4 He fragments in final state [PRC 70 (2004) ]

20 Breakup of 7 Be: cont-cont couplings restricted maximum relative energy between 3 He and 4 He fragments in final state DWBA first order No CC couplings includes couplings to all orders to/from ground state only [PRC 70 (2004) ]

21 Breakup of 8 B: DWBA 8 B + 58 Ni 8 B * + 58 Ni 7 Be+p + 58 Ni (Notre Dame E lab =25.8 MeV) bound state and continuum structure for the projectile fragment-target interactions Note that 8 B is a loosely bound proton rich nucleus (0.137 MeV) DWBA analysis: finite range effects are large [Nunes and Thompson, PRC 57 (1998) R2818]

22 Breakup 8 B: multistep effects CDCC 8 B + 58 Ni 7 Be+p + 58 Ni (E beam =26 MeV) 8 B* angular distributions show multi-step effects are crucial [PRC 59 (1999) 2652] continuum-continuum couplings are critical!! strong nuclear destructive interference

23 Breakup of 8 B: theory versus data 8 B breakup on 58 Ni (E beam =26 MeV) ESB+BG KIM+BG ESB+VG 3-body observables sensitivity to 8 B structure: overall normalisation sensitivity to p-target optical potential at larger angles [Tostevin et al., PRC (2001) ]

24 Breakup of 8 B : theory versus data 8 B breakup on 58 Ni (E beam =26 MeV) Results of CDCC calculations assuming a single particle structure for 8 B= 7 Be+p Disagreement for large angles due to transfer. [ PRC (2001) ]

25 ANC method (checks) breakup reactions are very peripheral cross sections depend essentially on the ANC of the bound state for low relative energies, small residual dependence on continuum properties σ B U α C 2 [Capel and Nunes, Rev. C 73, (2006)]

26 one test case: 14 C(n,γ) 15 C 14 C(n,γ) 15 C has impact on: neutron-induced CNO cycle heaviest element in non-homogenous bigbang abundancies from the r-process in supernovae

27 Coulomb dissociation for (n,γ) 208 Pb( 15 C, 14 C+n) 208 MeV/u CDCC + set of single particle parameters extract ANC from χ 2 minimum error from ε=χ min2 +1 Yao, JPG33 (2006) 1 ANC = 1.32 ± 0.04 fm -1/2 Summers and Nunes, PRC78(2009) Reifarth 14 C(n,γ) 15 C Nakamura et al, NPA722(2003)301c Reifarth et al, PRC77, (2008)

28 Breakup on light targets breakup p( 11 Be, 11 Be)p

29 Breakup on light targets elastic breakup [Summers and Nunes, PRC 76, ] data: Shrivastava et al, PLB 596 (2004) 5

30 Breakup on light targets CDCC Formalism Faddeev Formalism each Faddeev component includes correct asymptotics for the corresponding transfer channel breakup is split in all Faddeev components limitation of Faddeev calculations: convergence with Coulomb is hard (regularization techniques needed to improve) number of channels increases rapidly with partial waves

31 Breakup on light targets [Deltuva et al, PRC 76, ]

32 Breakup on light targets 11 Be breakup on 39 MeV/u [Deltuva et al, PRC 76, ]

33 summary of some recent advances Continuum discretized coupled channels method (CDCC) many applications to weakly bound nuclei: good description of data extensions to core excitation (also 4-body CDCC) Coulomb dissociation can be used to extract peripheral (n,γ) new methodology based on xs scaling with the ANC 14 C(n,γ) 15 C from Coulomb dissociation consistent with direct capture data Transfer reactions and combined method one benchmark with (n,γ) but many applications with future experiments finite range effects can be very important at intermediate energies Testing CDCC against Faddeev disagreement needs to be better understood new formalism Transfer reactions compared to knockout uncertainties in reaction theory have been quantified results move toward agreement Microscopic overlap functions 2n overlap functions show increased spectroscopic strength (compared to 3body) significant progress needs to be made in reaction theory before structure models can be tested

34 reaction theory and other fields Reaction theory can benefit from cross fertilization!

35 efimov states in nuclear systems? bound systems, INT 2010 in nuclei how can we vary the scattering length!? D Incao@Weakly bound systems, INT 2010

36 Great challenges for the future

37 Great challenges for the future Creating bridges between formalisms Quantifying errors Learning from other fields Having useful microscopic input understanding optical potentials from microscopic structure overlaps and transition densities with correct asymptotic behaviour Effective way of dealing with antisymmetrization effects understanding non-localities induced through antisymmetrization magnitude of effects for loosely bound particles Handling multiconfiguration structure in reaction models

38 microscopic 2n overlap functions 6 He 2n overlap functions 6 He(p,t) 4 25 MeV [Brida, PhD thesis, MSU 2009] [Brida and Nunes, NPA]

39 transfer versus knockout [Jenny Lee et al, PRL 2009] [Gade et al, PRL 93, ]

40 error from reaction theory [FN, Deltuva, Hong, 2010]

41 transfer versus knockout [Jenny Lee et al, PRL 2009] [Gade et al, Phys. Rev. Lett. 93, ] [FN, Deltuva, Hong, 2010]