Deuteron induced reactions, spin distributions and the surrogate method

Transcription

1 Varenna, June 17th 215 slide 1/25 Deuteron induced reactions, spin distributions and the surrogate method Grégory Potel Aguilar (NSCL, LLNL) Filomena Nunes (NSCL) Ian Thompson (LLNL) Varenna, June 17th 215

2 Introduction arenna, June 17th 215 slide 2/25 We present a formalism for inclusive deuteron induced reactions. We thus want to describe within the same framework: elastic breakup compound nucleus direct transfer Direct neutron transfer: should be compatible with existing theories. Elastic deuteron breakup: transfer to continuum states. Non elastic breakup (direct transfer, inelastic excitation and compound nucleus formation): absorption above and below neutron emission threshold. Important application in surrogate reactions: obtain spin parity distributions, get rid of Weisskopf Ewing approximation.

3 Historical background arenna, June 17th 215 slide 3/25 breakup-fusionqreactions BrittQandQQuinton,QPhys.QRev.Q124Q819617Q877 protonsqandqαqyields bombardingq 29 BiQwithQ 12 CQandQ 16 OQ Kerman and McVoy, Ann. Phys. 122 (1979)197 Austern and Vincent, Phys. Rev. C23 (1981) 1847 Udagawa and Tamura, Phys. Rev. C24(1981) 1348 Last paper: Mastroleo, Udagawa, Mustafa Phys. Rev. C42 (199) 683 Controversy between Udagawa and Austern formalism left somehow unresolved.

4 Inclusive (d, p) reaction Varenna, June 17th 215 slide 4/25 let s concentrate in the reaction A+d B(=A+n)+p χ p χ n elastic breakup ϕ A ϕ d χ d ϕ A χ p χ p ϕ 1,ϕ 2,...,ϕ n B B B ϕ n+1,ϕ n+2,... B B direct transfer neutron capture inelastic excitation coumpound nucleus non elastic breakup we are interested in the inclusive cross section, i.e., we will sum over all final states φ c B.

5 Neutron states in nuclei Wx9MeV8 scatteringxandxresonances broadxsingle-particle scatteringxstates E-E F x9mev8 E F weaklyxboundxstates narrowxsingle-particle deeplyxboundxstates Imaginaryxpartxofxopticalxpotential neutronxstates Mahaux,xBortignon,xBrogliaxandxDassoxPhys.xRep.x12x919858x1x Varenna, June 17th 215 slide 5/25

6 Derivation of the differential cross section Varenna, June 17th 215 slide 6/25 the double differential cross section with respect to the proton energy and angle for the population of a specific final φ c B d 2 σ = 2π ρ(e p ) χ p φ c B dω p de p v V Ψ(+) 2. d Sum over all channels, with the approximation Ψ (+) χ d φ d φ A d 2 σ dω p de p = 2π v d ρ(e p ) c χ d φ d φ A V χ p φ c B δ(e E p E c B ) φc B χ p V φ A χ d φ d χ d deuteron incoming wave, φ d deuteron wavefunction, χ p proton outgoing wave φ A target core ground state.

7 Sum over final states Varenna, June 17th 215 slide 7/25 the imaginary part of the Green s function G is an operator representation of the δ function, πδ(e E p E c B ) = lim ɛ I c φ c B φc B E E p H B + iɛ = IG d 2 σ = 2 ρ(e p )I χ d φ d φ A V χ p G χ p V φ A χ d φ d dω p de p v d We got rid of the (infinite) sum over final states, but G is an extremely complex object! We still need to deal with that.

8 Optical reduction of G arenna, June 17th 215 slide 8/25 If the interaction V do not act on φ A χ d φ d φ A V χ p G χ p V φ A χ d φ d where G opt is the optical reduction of G = χ d φ d V χ p φ A G φ A χ p V χ d φ d = χ d φ d V χ p G opt χ p V χ d φ d, G opt = lim ɛ 1 E E p T n U An (r An ) + iɛ, now U An (r An ) = V An (r An ) + iw An (r An ) and thus G opt are single particle, tractable operators. The effective neutron target interaction U An (r An ), a.k.a. optical potential, a.k.a. self energy can be provided by structure calculations

9 Capture and elastic breakup cross sections Varenna, June 17th 215 slide 9/25 the imaginary part of G opt splits in two terms elastic breakup {}}{ IG opt = π neutron capture ( ) {}}{ χ n δ E E p k2 n χ n + G opt W An G opt, 2m n k n we define the neutron wavefunction ψ n = G opt χ p V χ d φ d cross sections for neutron capture and elastic breakup d 2 σ dω p de p ] capture = 2 v d ρ(e p ) ψ n W An ψ n, d 2 σ dω p de p ] breakup = 2 v d ρ(e p )ρ(e n ) χ n χ p V χ d φ d 2,

10 2 step process (post representation) Varenna, June 17th 215 slide 1/25 step1 breakup d p A n A step2 propagation of n in the field of A to detector p G n non elastic breakup B* p elastic breakup A n

11 Austern (post) Udagawa (prior) controversy Varenna, June 17th 215 slide 11/25 The interaction V can be taken either in the prior or the post representation, Austern (post) V V post V pn (r pn ) (recently revived by Moro and Lei, University of Sevilla) Udagawa (prior) V V prior V An (r An, ξ An ) in the prior representation, V can act on φ A the optical reduction gives rise to new terms: d 2 ] post σ = 2 ρ(e p ) [ I ψn prior W An ψ prior n dω p de p v d ] + 2R ψn NON W An ψn prior + ψn NON W An ψn NON, where ψ NON n = χ p χ d φ d. The nature of the 2 step process depends on the representation

12 neutron wavefunctions arenna, June 17th 215 slide 12/25 the neutron wavefunctions ψ n = G opt χ p V χ d φ d can be computed for any neutron energy 2 1 bound state E n =2.5 MeV E n =-7.5 MeV -1 scattering state transfer to resonant and non-resonant continuum well described r Bn these wavefunctions are not eigenfunctions of the Hamiltonian H An = T n + R(U An )

13 Breakup above neutron emission threshold Varenna, June 17th 215 slide 13/25 1 proton angular differential cross section dσ/dω (mb/sr,,mev) Nb,(d,p),,E d =15,MeV L=1 L=2 L= L=3 L=4 L=5 total.1 E p =9,MeV E n =3.8,MeV θ

14 neutron transfer limit (isolated resonance, first order approximation) Varenna, June 17th 215 slide 14/25 Let s consider the limit W An (single particle width Γ ). For an energy E such that E E n D, (isolated resonance) G opt φ n φ n lim W An E E p E n i φ n W An φ n ; with φ n eigenstate of H An = T n + R(U An ) d 2 σ lim dω p de χ dφ d V χ p p W An φ n φ n W An φ n φ n (E E p E n ) 2 + φ n W An φ n 2 χ p V χ d φ d, we get the direct transfer cross section: d 2 σ dω p de p χ p φ n V χ d φ d 2 δ(e E p E n )

15 Validity of first order approximation Varenna, June 17th 215 slide 15/25 For W An small, we can apply first order perturbation theory, d 2 ] capture σ (E, Ω) 1 ] φ n W An φ n dσ transfer n dω p de p π (E n E) 2 + φ n W An φ n 2 dω (Ω) σ (mb/mev) completed calculation firstd order W An =.5dMeV σ (mb/mev) W An =3dMeV σ (mb/mev) W An =1dMeV E n E n E n we compare the complete calculation with the isolated resonance, first order approximation for W An =.5 MeV, W An = 3 MeV and W An = 1 MeV

16 Application to surrogate reactions Varenna, June 17th 215 slide 16/25 Surrogate for neutron capture Desired reaction: neutron induced fission, gamma emission and neutron emission. fission n A B* gamma emission neutron emission The surrogate method consists in producing the same compound nucleus B* by bombarding a deuteron target with a radio active beam of the nuclear species A. fission A d B* gamma emission neutron emission A theoretical reaction formalism that describes the production of all open channels B* is needed.

17 Disentangling elastic and non elastic breakup Varenna, June 17th 215 slide 17/25 We show some results for the 93 Nb(d, p) reaction with a 15 MeV deuteron beam (Mastroleo et al., Phys. Rev. C 42 (199) 683) 7 7 dσ/de)(mb/mev) non)elastic)breakup elastic)breakup total)proton)singles dσ/de)(mb/mev) ))))shift)real)part of)optical)potential Nb(d,p) E d =15)MeV E p E p we have used the Koning Delaroche (Koning and Delaroche, Nucl. Phys. A 713 (23) 231) optical potential the real part of the optical potential has been shifted to reproduce the position of the L = 3 resonance contributions from elastic and non elastic breakup disentangled.

18 Obtaining spin distributions Varenna, June 17th 215 slide 18/ Nb(d,p) E d =15 MeV L=3 dσ/de (mb/mev) L=1 L=2 L= E p spin distribution of compound nucleus dσ l = 2π ρ(e p ) ϕlmlp de p v (r Bn; k p ) 2 W (r An) dr Bn. d l p,m

19 Getting rid of Weisskopf Ewing approximation Weisskopf Ewing approximation: P(d, nx) = σ(e)g(e, x) inaccurate for x = γ and for x = f in the low energy regime can be replaced by P(d, nx) = J,π σ(e, J, π)g(e, J, π, x) if σ(e, J, π) can be predicted. 2 σ (mb/mev) 15 1 E p =1 MeV 5 Younes and Britt, PRC 68(23) L n arenna, June 17th 215 slide 19/25

20 Summary, conclusions and some prospectives Varenna, June 17th 215 slide 2/25 We have presented a reaction formalism for inclusive deuteron induced reactions. Valid for final neutron states from Fermi energy to scattering states Disentangles elastic and non elastic breakup contributions to the proton singles. Probe of nuclear structure in the continuum. Provides spin parity distributions. Useful for surrogate reactions. Need for optical potentials. Extend for (p, d) reactions (hole states)?

21 The 3 body model arenna, June 17th 215 slide 21/25 p r pn d r Ap θ pd B r d r Bn ξa From H to H 3B H = T p + T n + H A (ξ A ) + V pn (r pn ) + V An (r An, ξ A ) + V Ap (r Ap, ξ A ) H 3B = T p + T n + H A (ξ A ) + V pn (r pn ) + U An (r An ) + U Ap (r Ap ) n r An A

22 Observables: angular differential cross sections (neutron bound states) arenna, June 17th 215 slide 22/25 1 dσ/dω (mb/sr) L= 93 NbN(d,p)N@N15NMeV W An =.5NMeV L=1 transfernfresco capture θ double proton differential cross section capture at resonant energies compared with direct transfer (fresco) calculations, capture cross sections rescaled by a factor φ n W An φ n π. d 2 σ = 2π ρ(e p ) dω p de p v d l,m,l p ϕlmlp (r Bn; k p )Y lp m (θ p) 2 W (r An ) dr Bn.

23 Observables: elastic breakup and capture cross sections Varenna, June 17th 215 slide 23/ NbN(d,p)N@N15NMeV σn(mb/mev) capture elasticnbreakup total E p elastic breakup and capture cross sections as a function of the proton energy. The Koning Delaroche global optical potential has been used as the U An interaction (Koning and Delaroche, Nucl. Phys. A 713 (23) 231).

24 Sub threshold capture 6 L=2 5 L= 4 W An =.5 MeV 3 dσ/de (mb/mev) L=4 L=5 W An =3 MeV E n Varenna, June 17th 215 slide 24/25

25 Non orthogonality term dσ/dewlmb/mevy Nbld,py E d =25.5wMeV θ p =1 o E n 1 lmb/mevwsry 1 E n =5 MeV d 2 σ/dedω 1 E n =-3 MeV NEB with non orthogonality NEB without nonworthogonality θ CM Varenna, June 17th 215 slide 25/25