Deuteron induced reactions

INT, March 2015 slide 1/23 Deuteron induced reactions Grégory Potel Aguilar (NSCL, LLNL) Filomena Nunes (NSCL) Ian Thompson (LLNL) INT, March 2015

Introduction NT, March 2015 slide 2/23 We present a formalism for inclusive deuteron induced reactions. We thus want to describe within the same framework: Direct neutron transfer: should be compatible with existing theories. Elastic deuteron breakup: transfer to continuum states. Neutron capture 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 (see J. Escher s talk).

Historical background NT, March 2015 slide 3/23 breakup-fusionqreactions BrittQandQQuinton,QPhys.QRev.Q124Q819617Q877 protonsqandqαqyields bombardingq 209 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 (1990) 683 Controversy between Udagawa and Austern formalism left somehow unresolved.

Inclusive (d, p) reaction INT, March 2015 slide 4/23 let s concentrate in the reaction A+d B(=A+n)+p χ p χ n ϕ A beakup ϕ d χ d ϕ A χ p ϕ 1,ϕ 2,...,ϕ n B B B transfer χ p ϕ n+1,ϕ n+2,... B B capture 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 INT, March 2015 slide 5/23 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 INT, March 2015 slide 6/23 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 0 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 NT, March 2015 slide 7/23 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 G opt χ p V χ d φ d, G opt = lim ɛ 0 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 (previous talks by W. Dickhoff, C. Barbieri, P. Navratil, G. Hagen, J. Rotureau, J. Holt...)

Capture and elastic breakup cross sections INT, March 2015 slide 8/23 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,

2 step process INT, March 2015 slide 9/23 step1 breakup d p A n A step2 propagation of n in the field of A to detector p transfer, capture p elastic breakup G G n B* A

Austern (post) Udagawa (prior) controversy INT, March 2015 slide 10/23 The interaction V can be taken either in the prior or the post representation, Austern (post) V V post V pn (r pn ) 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.

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.x120x919858x1x INT, March 2015 slide 11/23

neutron wavefunctions NT, March 2015 slide 12/23 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 0-1 scattering state transfer to resonant and non-resonant continuum well described -2 0 5 10 15 20 25 30 r Bn these wavefunctions are not eigenfunctions of the Hamiltonian H An = T n + R(U An )

neutron transfer limit (isolated resonance, first order approximation) INT, March 2015 slide 13/23 Let s consider the limit W An 0 (single particle width Γ 0). For an energy E such that E E n D, (isolated resonance) G opt φ n φ n lim W An 0 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 0 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 INT, March 2015 slide 14/23 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) 120 100 80 60 40 completed calculation firstd order W An =0.5dMeV σ (mb/mev) 20 15 10 W An =3dMeV σ (mb/mev) 8 6 4 W An =10dMeV 5 2 20 0-5 -4-3 -2-1 0 1 2 3 4 E n 0 0-5 -5-4 -3-2 -1 0 1 2 3 4 E n -4-3 -2-1 0 1 2 3 4 E n we compare the complete calculation with the isolated resonance, first order approximation for W An = 0.5 MeV, W An = 0.5 MeV and W An = 0.5 MeV

Observables: elastic breakup and capture cross sections INT, March 2015 slide 15/23 70 60 93 NbN(d,p)N@N15NMeV σn(mb/mev) 50 40 30 capture elasticnbreakup total 20 10 0 4 6 8 10 12 14 16 18 20 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 (2003) 231).

Observables: angular differential cross sections (neutron bound states) NT, March 2015 slide 16/23 100 dσ/dω (mb/sr) 10 1 0.1 70 60 50 40 30 20 10 L=0 93 NbN(d,p)N@N15NMeV 0 0 10 20 30 40 50 60 70 W An =0.5NMeV L=1 transfernfresco capture 0.01 0 20 40 60 80 100 120 140 160 180 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: angular differential cross sections (above neutron emission threshold) INT, March 2015 slide 17/23 100 dσ/dω (mb/sr,,mev) 10 1 0.1 0.01 93 Nb,(d,p),@,15,MeV E p =9,MeV L=1 L=2 L=0 L=3 L=4 L=5 total 0.001 0 20 40 60 80 100 120 140 160 180 θ

Observables: compound nucleus spin and parity INT, March 2015 slide 18/23 70 σ (mb/sr) 60 50 40 30 L=0 L=1 L=4 L=2 L=3 L=5 total 6 5 4 3 2 1 0 4 6 8 10 12 14 16 18 20 22 20 10 0 4 6 8 10 12 14 16 18 20 22 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

Application to surrogate reactions INT, March 2015 slide 19/23 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.

Surrogate reactions 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. 20 σ (mb/mev) 15 10 E p =10 MeV 5 Younes and Britt, PRC 68(2003)034610 0 0 1 2 3 4 5 6 7 L n (see J. Escher talk) NT, March 2015 slide 20/23

Preliminary comparison with experiment INT, March 2015 slide 21/23 We show very preliminary results for the 93 Nb(d, p) reaction with a 15 MeV deuteron beam (Mastroleo et al., Phys. Rev. C 42 (1990) 683) 70 70 60 50 60 50 total elastic)breakup capture L=2 L=3 σ)(mb/mev) 40 30 20 σ)(mb/mev) 40 30 ))))shift)real)part 20 of)optical)potential 10 10 0 4 6 8 10 12 14 16 18 20 22 0 4 6 8 10 12 14 16 18 E p E p we have used the Koning Delaroche optical potential the real part of the optical potential has been shifted to reproduce the position of the L = 3 resonance the experimental results seem to be sensitive to the position and strength of a modest number of resonances

Summary, conclusions and some prospectives INT, March 2015 slide 22/23 Reaction formalism for inclusive deuteron induced reaction. final neutron states from Fermi energy to scattering states 2 step reaction mechanism breakup+absorption probe of nuclear structure over a wide energy range need for optical potentials useful for surrogate reactions transfer to individual resonances? extend for (p, d) reactions (hole states)?

The 3 body model NT, March 2015 slide 23/23 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