Inclusive deuteron induced reactions

Seattle, March 28th 217 slide 1/9 Inclusive deuteron induced reactions Grégory Potel Aguilar (FRIB/NSCL) Seattle, March 28th 217

(d, p) reactions: a probe for neutron nucleus interactions Seattle, March 28th 217 slide 2/9 proton detected deuteron beam elastic scattering inelastic excitation n A B=A+n direct transfer compound nucleus...

General framework for inclusive experiments Seattle, March 28th 217 slide 3/9 this is detected A B summed over states of this dσ(e) de A n = n φ A φ n B V Ψ 2 Ψ V φ A φ n B δ(e E A E n B ) φn B φ A V Ψ = lim ɛ Im Ψ V φ A n φ n B φn B E E A H B + iɛ φ A V Ψ Main challenges describe propagator G B = lim ɛ n describe wave function Ψ φ n B φn B E E A H B +iɛ

Some applications January 31, 217 slide 3/3 p n (d,p) p n F A=p F B=F+1 p (p,d) p n F A=d F B=F-1 (p,pn)/inelastic p p n F A=p F B=F*

Implemented so far: Inclusive (d, p) reaction January 31, 217 slide 4/3 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.

Derivation of the differential cross section East Lansing, July 19th 216 slide 3/27 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.

Sum over final states East Lansing, July 19th 216 slide 4/27 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.

Optical reduction of G ast Lansing, July 19th 216 slide 5/27 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

Capture and elastic breakup cross sections East Lansing, July 19th 216 slide 6/27 the imaginary part of G opt splits in two terms elastic breakup {}}{ IG opt = π non elastic breakup ( ) {}}{ 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 non elastic breakup (NEB) and elastic breakup (EB) d 2 σ dω p de p ] NEB = 2 v d ρ(e p ) ψ n W An ψ n, d 2 σ dω p de p ] EB = 2 v d ρ(e p )ρ(e n ) χ n χ p V χ d φ d 2,

neutron transfer limit (isolated resonance, first order approximation) East Lansing, July 19th 216 slide 12/27 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 )

Validity of first order approximation East Lansing, July 19th 216 slide 13/27 For W An small, we can apply first order perturbation theory, d 2 ] NEB 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) 12 1 complete)calculation first)order 2 15 8 6 σ)(mb/mev) 8 6 4 W An =.5)MeV σ)(mb/mev) 1 5 W An =3)MeV σ)(mb/mev) 4 2 W An =1)MeV) 2-5 -4-3 -2-1 1 2 3 4 E -5-4 -3-2 -1 1 2 3 4 E -5-4 -3-2 -1 1 2 3 4 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 E

Spectral function and absorption cross section 6 5 4 L=2 W An =.5 MeV L= 3 dσ/de (mb/mev) 2 1-7 -6-5 -4-3 -2-1 12 1 8 L=4 L=5 W An =3 MeV 6 4 2-7 -6-5 -4-3 -2-1 E n East Lansing, July 19th 216 slide 14/27

Choosing the potential/propagator Seattle, March 28th 217 slide 4/9 Konig Delaroche (KD) Phenomenological fit of elastic scattering. Local. Available for a wide range of stable nuclei and energies. Not defined for negative energies. Good reproduction of scattering observables. Dispersive optical model (DOM) Elastic scattering fit dispersion negative energies. Local and non local versions. Limited availability. Good reproduction of scattering and low energy observables. Satisfy sum rules (dispersive). Coupled cluster (CC) Ab initio calculation predictive power. Non local. Accuracy of reproduction of observables?.

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 East Lansing, July 19th 216 slide 7/27

Application to surrogate reactions East Lansing, July 19th 216 slide 26/27 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.

Disentangling elastic and non elastic breakup ast Lansing, July 19th 216 slide 1/27 93 Nb(d, p) (Mastroleo et al., Phys. Rev. C 42 (199) 683) dσ/deg(mb/mev) 6 4 2 compoundgnucleus formation protongsingles 93 Nb(d,p) E d =15gMeV compoundgnucleusgspingdistribution 3 totalgprotongsingles elasticgbreakup 2 4 6 8 1 12 14 16 18 E p g(mev) dσ/deg(mb/mev) 25 2 15 1 5 J=1 J=3 J=2 4 8 12 16 E p g(mev) We obtain spin parity distributions for the compound nucleus. Contributions from elastic and non elastic breakup disentangled. J=

Using the DOM: Calcium isotopes Seattle, March 28th 217 slide 5/9 dσ/de (mb/mev) 3 25 2 15 4 1 2 5-8 -6-4 -2 2 4 6 8 1 12 14-6 -4-2 2 4 6 8 1 12 14 5 3 4 3 4 Ca(d,p) E d =2 MeV 4 Ca(d,p) E d =4 MeV 1 8 6 25 2 15 48 Ca(d,p) E d =2 MeV 48 Ca(d,p) E d =4 MeV 4 3 2 1 5 4 3 6 Ca(d,p) total EB NEB 2 4 6 8 1 12 14 6 Ca(d,p) E d =2 MeV E d =4 MeV 2 1 2 1 5 1-8 -4 4 8 12 16 2 24 28 32-8 -4 4 8 12 16 2 24 28 32 E n (MeV) 4 8 12 16 2 24 28 32

Using the DOM: angular momenta Seattle, March 28th 217 slide 6/1 7 7 4.5 dσ/de (mb/mev) 6 5 4 3 2 4 Ca(d,p) l= 6 48 Ca(d,p) E d=4 MeV l=1 l=2 5 E d=4 MeV l=3 l=4 4 3 2 4 3.5 3 2.5 2 1.5 6 Ca(d,p) E d=4 MeV 1 1 1.5-8 -4 4 8 12 16 2 24 28 32-4 4 8 12 16 2 24 28 32 E n (MeV) 4 8 12 16 2 24 28? 32 Uozimi et al., NPA 576 (1994) 123

Using the DOM: comparing with data Seattle, March 28th 217 slide 7/9 3 25 4 Ca(d,p) 3 25 48 Ca(d,p) 5 4 6 Ca(d,p) 2 15 1 2 15 1 3 2 5 5 1-8 -6-4 -2 2 4 6 8 1 12 14-8 -4 4 8 12 16 2 24 28 32 4 8 12 16 2 24 28 32 5 dσ/dω (mb/sr) 4 3 2 f7/2 (gs) data 4 Ca(d,p), E d=1 MeV dσ/dω (mb/sr) 1 48 Ca(d,p), E d=56 MeV g 9/2 resonance dσ/dω (mb/sr) 1 1.1 6 Ca(d,p), E d=4 MeV g 9/2 ground state resonance 1 2 4 6 8 1 12 14 16 θ CM.1 1 2 3 4 5 θ CM.1 2 4 6 8 1 12 14 16 18 θ CM

CC: preliminary results (with J. Rotureau, MSU and G. Hagen, ORNL) Seattle, March 28th 217 slide 8/9 first results using a propagator and self energy generated within the coupled cluster formalism (J. Rotureau, MSU and G. Hagen, ORNL) 4 Ca(d,p), E d =1 MeV dσ/de (mb/mev) 1 8 6 4 2 integrated σ=12.3 mb N max =12 dσ/dedω (mb/mev sr) 12 1 8 6 4 2 E=-7.6 MeV -8-7.9-7.8-7.7-7.6-7.5-7.4-7.3 E (MeV) 3 6 9 12 15 18 θ CM

Open questions (personal selection) Seattle, March 28th 217 slide 9/9 R S = σ exp / σ th 1..8.6.4.2 Separate different channels in the NEB (possibly with M. Dupuis and G. Blanchon?). Better description of deuteron channel (adiabatic approximation? CDCC? Coupled cluster?) Implement (p, pn). Could shed light on the problem of spectroscopic factors as a function of asymmetry. Two particle Green s function? Final neutron as a spectator? Gade et al. PRC 77, 4436 (28) Flavigny et al. PRL 11, 12253 (213) 15 C 24 Si 28 S R S R S R S 9 C 8 B 46 Ar 16 O 31 P 16 O 4 Ca 7 Li 12 C 48 Ca 9 Zr 51 V 12 C 28 Pb (e,e'p): ΔS=Sp-Sn p-knockout: ΔS=Sp-Sn n-knockout: ΔS=Sn-Sp -2-1 1 2 ΔS (MeV) 3 Si 12 C 16 O 34 Ar 28 S 32 Ar 24 Si R s = σ exp (θ) / σ th (θ) 1,8,6,4,2-2 -1 1 2 ΔS = ε (S p - S n ) (MeV)

Inclusive cross section Seattle, March 28th 217 slide 1/9 Single particle Green s function of B system: G B = lim η (E T U B + iη) 1 Strength at given E, r n, r n, J, π ImG B (E, r n, r n, J, π) inclusive cross section folding of strength with breakup probability density ρ bu (r n, k i, k f ) 2 : ρ bu (r n, k i, k f ) = ( χ p (r p, k f ) V χ d (r d, k i )φ d (r pn ) dσ dedω ρ bu (r n, k i, k f )ImG B (E, r n, r n, J, Π)ρ bu (r n, k i, k f )dr n dr n ImG B = elastic breakup spectrum + G B ImU BG B How to write U B = U compound + U rest?

East Lansing, July 19th 216 slide 1/27 Inclusive deuteron induced reactions Grégory Potel Aguilar (NSCL/FRIB) Filomena Nunes (NSCL) Ian Thompson (LLNL) East Lansing, July 19th 216

Inclusive (d, p) reaction East Lansing, July 19th 216 slide 2/27 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.

neutron wavefunctions ast Lansing, July 19th 216 slide 8/27 the neutron wavefunctions ψ n = G opt χ p V χ d φ d can be computed for ANY neutron energy, positive or negative 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 -2 5 1 15 2 25 3 r Bn ψ n are the solutions of an inhomogeneous Schrödinger equation (H An E An ) ψ n = χ p V χ d φ d

Breakup above neutron emission threshold East Lansing, July 19th 216 slide 9/27 1 proton angular differential cross section dσ/dω (mb/sr,,mev) 1 1.1 93 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.1 2 4 6 8 1 12 14 16 18 θ

Dropping a proton dσ/dealmb/mev) 1 9 8 7 6 5 4 3 2 compounda 168 Tmaformationacrossasection regionaunderalp, 167 Er)aCoulombabarrier 167 Erl3He,d)@2aMeV 167 Erld,n)@15aMeV 1-2 2 4 6 8 1 12 E p We can also transfer charged clusters East Lansing, July 19th 216 slide 11/27

Austern (post) Udagawa (prior) formalisms East Lansing, July 19th 216 slide 15/27 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, from Sevilla and Carlson from São Paulo) Udagawa (prior) V V prior V An (r An, ξ An ) (used in calculations showed here) 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

Summary, conclusions and some prospectives East Lansing, July 19th 216 slide 16/27 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. Need to address non locality. Can be generalized to other three body problems. Can be extended for (p, d) reactions (hole states).

The 3 body model ast Lansing, July 19th 216 slide 17/27 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

Observables: angular differential cross sections (neutron bound states) ast Lansing, July 19th 216 slide 18/27 1 dσ/dω (mb/sr) 1 1.1 7 6 5 4 3 2 1 L= 93 NbN(d,p)N@N15NMeV 1 2 3 4 5 6 7 W An =.5NMeV L=1 transfernfresco capture.1 2 4 6 8 1 12 14 16 18 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.

Observables: elastic breakup and capture cross sections East Lansing, July 19th 216 slide 19/27 7 6 93 NbN(d,p)N@N15NMeV σn(mb/mev) 5 4 3 capture elasticnbreakup total 2 1 4 6 8 1 12 14 16 18 2 22 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).

Non orthogonality term dσ/dewlmb/mevy 24 21 18 15 12 9 6 93 Nbld,py E d =25.5wMeV θ p =1 o 3-4 -2 2 4 6 8 1 12 14 16 18 2 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.1 1 2 3 4 5 θ CM 6 7 8 9 1 East Lansing, July 19th 216 slide 2/27

Obtaining spin distributions East Lansing, July 19th 216 slide 21/27 3 25 93 Nb(d,p) E d =15 MeV L=3 dσ/de (mb/mev) 2 15 1 5 L=1 L=2 L= 4 6 8 1 12 14 16 18 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

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)3461 1 2 3 4 5 6 7 L n ast Lansing, July 19th 216 slide 22/27

Introduction ast Lansing, July 19th 216 slide 23/27 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.

Historical background ast Lansing, July 19th 216 slide 24/27 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.

2 step process (post representation) East Lansing, July 19th 216 slide 25/27 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

Weisskopf Ewing approximation East Lansing, July 19th 216 slide 27/27 W-ECapproximation Weisskopf-EwingCapproximation:CprobabilityCofCγCdecayCindependentCofCJ,π EscherCandCDietrich,CPRCC81C24612C(21) DifferentCJ,π DifferentCcrossCsectionCforCγCemission Weisskopf Ewing is inaccurate for (n, γ)

January 31, 217 slide 1/3 Inclusive three body cross sections January 31, 217

Introduction January 31, 217 slide 2/3 General description of experiments in which we measure the energy of final product A and do not measure the energy of final product B: dσ de A (E) n = n φ A φ n B V Ψ 2 Ψ V φ A φ n B δ(e E A E n B ) φn B φ A V Ψ = I lim ɛ Ψ V φ A n φ n B φn B E E A H B φ A V Ψ Main challenge: get the B system propagator G B = n φ n B φn B E E A H B

ICNT Workshop Deuteron induced reactions and beyond: Inclusive breakup fragment cross sections (July 216) January 31, 217 slide 5/3 structure W. Dickhoff (St. Louis) J. Rotureau (MSU) J. Escher (LLNL) experiment G. Perdikakis (CMU, NSCL) A. Macchiavelli (LBNL) S. Pain (ORNL) reactions B. Carlson (Sao Paulo) A. Moro (Sevilla) F. Nunes (MSU) M. Husein (Sao Paulo) P. Capel (Bruxelles) G. Potel (MSU)

Recent results (I): Ca isotopes January 31, 217 slide 6/3 dσ/de (mb/mev) 3 25 2 15 4 1 2 5-8 -6-4 -2 2 4 6 8 1 12 14-6 -4-2 2 4 6 8 1 12 14 5 3 4 3 4 Ca(d,p) E d =2 MeV 4 Ca(d,p) E d =4 MeV 1 8 6 25 2 15 48 Ca(d,p) E d =2 MeV 48 Ca(d,p) E d =4 MeV 4 3 2 1 5 4 3 6 Ca(d,p) total EB NEB 2 4 6 8 1 12 14 6 Ca(d,p) E d =2 MeV E d =4 MeV 2 1 2 1 5 1-8 -4 4 8 12 16 2 24 28 32-8 -4 4 8 12 16 2 24 28 32 E n (MeV) 4 8 12 16 2 24 28 32

Recent results (II): Ca isotopes January 31, 217 slide 7/3 3 3 25 48 Ca(d,p) 5 25 4 Ca(d,p) 2 4 6 Ca(d,p) 2 15 3 15 1 1 5 2 5-8 -4 4 8 12 16 2 24 28 32 1-8 -6-4 -2 2 4 6 8 1 12 14 4 8 12 16 2 24 28 32 5 dσ/dω (mb/sr) 4 3 2 f7/2 (gs) data 4 Ca(d,p), E d=1 MeV dσ/dω (mb/sr) 1 48 Ca(d,p), E d=56 MeV g 9/2 resonance dσ/dω (mb/sr) 1 1.1 6 Ca(d,p), E d=4 MeV g 9/2 ground state resonance 1 2 4 6 8 1 12 14 16 θ CM.1 1 2 3 4 5 θ CM.1 2 4 6 8 1 12 14 16 18 θ CM

Outlook January 31, 217 slide 8/3 Implement (p, d) (with DOM optical potential from Wim Dickhoff and Mack Atkinson). Include non locality in the propagator (with Weichuan). Use Coupled Clusters propagator (with Jimmy).

Derivation of the differential cross section January 31, 217 slide 9/3 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.

Sum over final states January 31, 217 slide 1/3 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.

Optical reduction of G anuary 31, 217 slide 11/3 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

Capture and elastic breakup cross sections January 31, 217 slide 12/3 the imaginary part of G opt splits in two terms elastic breakup {}}{ IG opt = π non elastic breakup ( ) {}}{ 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 non elastic breakup (NEB) and elastic breakup (EB) d 2 σ dω p de p ] NEB = 2 v d ρ(e p ) ψ n W An ψ n, d 2 σ dω p de p ] EB = 2 v d ρ(e p )ρ(e n ) χ n χ p V χ d φ d 2,

2 step process (post representation) January 31, 217 slide 13/3 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

Austern (post) Udagawa (prior) controversy January 31, 217 slide 14/3 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

neutron wavefunctions anuary 31, 217 slide 15/3 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 -2 5 1 15 2 25 3 r Bn these wavefunctions are not eigenfunctions of the Hamiltonian H An = T n + R(U An )

Breakup above neutron emission threshold January 31, 217 slide 16/3 1 proton angular differential cross section dσ/dω (mb/sr,,mev) 1 1.1 93 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.1 2 4 6 8 1 12 14 16 18 θ

neutron transfer limit (isolated resonance, first order approximation) January 31, 217 slide 17/3 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 )

Validity of first order approximation January 31, 217 slide 18/3 For W An small, we can apply first order perturbation theory, d 2 ] NEB 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) 12 1 complete)calculation first)order 2 15 8 6 σ)(mb/mev) 8 6 4 W An =.5)MeV σ)(mb/mev) 1 5 W An =3)MeV σ)(mb/mev) 4 2 W An =1)MeV) 2-5 -4-3 -2-1 1 2 3 4 E -5-4 -3-2 -1 1 2 3 4 E -5-4 -3-2 -1 1 2 3 4 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 E

Application to surrogate reactions January 31, 217 slide 19/3 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.

Weisskopf Ewing approximation January 31, 217 slide 2/3 W-ECapproximation Weisskopf-EwingCapproximation:CprobabilityCofCγCdecayCindependentCofCJ,π EscherCandCDietrich,CPRCC81C24612C(21) DifferentCJ,π DifferentCcrossCsectionCforCγCemission Weisskopf Ewing is inaccurate for (n, γ)

Disentangling elastic and non elastic breakup anuary 31, 217 slide 21/3 93 Nb(d, p) (Mastroleo et al., Phys. Rev. C 42 (199) 683) dσ/deg(mb/mev) 6 4 2 compoundgnucleus formation protongsingles 93 Nb(d,p) E d =15gMeV compoundgnucleusgspingdistribution 3 totalgprotongsingles elasticgbreakup 2 4 6 8 1 12 14 16 18 E p g(mev) dσ/deg(mb/mev) 25 2 15 1 5 J=1 J=3 J=2 4 8 12 16 E p g(mev) We obtain spin parity distributions for the compound nucleus. Contributions from elastic and non elastic breakup disentangled. J=

Extending the formalism dσ/dealmb/mev) 1 9 8 7 6 5 4 3 2 compounda 168 Tmaformationacrossasection regionaunderalp, 167 Er)aCoulombabarrier 167 Erl3He,d)@2aMeV 167 Erld,n)@15aMeV 1-2 2 4 6 8 1 12 E p We can also transfer charged clusters January 31, 217 slide 22/3

Summary, conclusions and some prospectives January 31, 217 slide 23/3 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. Can easily be generalized to other three body problems. Can be extended for (p, d) reactions (hole states).

The 3 body model anuary 31, 217 slide 24/3 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

Observables: angular differential cross sections (neutron bound states) anuary 31, 217 slide 25/3 1 dσ/dω (mb/sr) 1 1.1 7 6 5 4 3 2 1 L= 93 NbN(d,p)N@N15NMeV 1 2 3 4 5 6 7 W An =.5NMeV L=1 transfernfresco capture.1 2 4 6 8 1 12 14 16 18 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.

Observables: elastic breakup and capture cross sections January 31, 217 slide 26/3 7 6 93 NbN(d,p)N@N15NMeV σn(mb/mev) 5 4 3 capture elasticnbreakup total 2 1 4 6 8 1 12 14 16 18 2 22 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).

Sub threshold capture 6 L=2 5 L= 4 W An =.5 MeV 3 dσ/de (mb/mev) 2 1-7 -6-5 -4-3 -2-1 12 1 8 L=4 L=5 W An =3 MeV 6 4 2-7 -6-5 -4-3 -2-1 E n January 31, 217 slide 27/3

Obtaining spin distributions January 31, 217 slide 29/3 3 25 93 Nb(d,p) E d =15 MeV L=3 dσ/de (mb/mev) 2 15 1 5 L=1 L=2 L= 4 6 8 1 12 14 16 18 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