Status of deuteron stripping reaction theories

Status of deuteron stripping reaction theories Pang Danyang School of Physics and Nuclear Energy Engineering, Beihang University, Beijing November 9, 2015 DY Pang

Outline 1 Current/Popular models for (d, p) reactions 2 Problems Nonlocality of optical model potentials Inconsistency in neutron-nucleus potentials Inner part of the single-particle wave functions/overlap functions Coulomb problem in Faddeev method for (d,p) reactions DY Pang

Why study (d,p) reactions For nuclear structure: Angular distributions spin and parity of nuclei Amplitudes of cross sections spectroscopic factors, ANCs For nuclear astrophysics: indirect methods for (n,γ) Test case for few/3-body reaction theories It is essential to know the uncertainty of the reaction models DY Pang

Description of the (d,p) reactions Transition amplitude: M dp = χ ( ) pf IF A U pa + V pn U pf Ψ (+) i I F A (r n ) = A + 1 Φ A (ξ) Φ F (ξ, r n ) HΨ (+) i (r, R) = EΨ (+) i (r, R), H = T R + H np + U na + U pa H np = T r + V np expand Ψ (+) i with eigenfunctions of H np : Ψ (+) i (r, R) = ϕ 0 (r)χ (+) 0 (R)+ dkϕ k (ε k, r)χ (+) k (ε k, R) DWBA, ADWA, CDCC: different approx to Ψ (+) i DY Pang

Distorted wave Born approximation: DWBA Ψ (+) i (r, R) = ϕ 0 (r)χ (+) 0 (R) + DWBA takes the first term of Ψ (+) i : Ψ (+) i (r, R) ϕ 0 (r)χ (+) 0 (R) Mdp DWBA = χ ( ) pf ψ na V ϕ 0 (r)χ (+) 0 (R) with DWBA: 1970 use optical model potential for U da dkϕ k (ε k, r)χ (+) k (ε k, R) Assume breakup effect taken into account in U da Omit all except elastic component in the 3-body wave function Tobocman, PhysRev 94, 1655 (1954); Austern, Direct nuclear reaction theories, DY Pang

Improvement: the adiabatic model: ADWA The 3-body wave function: [E + ε d ˆT ] cm U na U pa ϕ d χ 0 (R) + dk [E ε k ˆT ] cm U na U pa ϕ k (ε k )χ k (ε k, R) = 0 the adiabatic approx: replacing ε k with ε d : [ E + ε d ˆT ] cm (U na + U pa ) χ ad(+) d (R) = 0 With the adiabatic approximation: M ADWA dp = χ ( ) pf ψ na U pa + V pn U pf ϕ 0 (r) χ ad(+) d effective d A interaction (zero-range): U da = U na + U pa Johnson, and Soper, Phys Rev C 1, 976 (1970) DY Pang

Further Improvement: CDCC In the CDCC method Continuum states are Discretised into bin states Ψ (+) i (r, R) = ϕ 0 (r)χ (+) 0 (R)+ dkϕ k (ε k, r)χ (+) k (ε k, R) Ψ (+)CDCC i (r, R) = ϕ 0 (r)χ (+) 0 (R)+ ϕ bin j=1 j (r)χ (+) j (R) 3-body equation turned into Coupled-Channel equations: (T R +ϵ i E +U ii )χ (+) i (R) = U ij χ (+) j (R) j i U ij (R) = ϕ i (r) U na + U pa ϕ j (r) Mitsuji Kawai, Masanobu Yahiro, Yasunori Iseri, Hirofumi Kameyama, Masayasu Kamimura, Prog Theor Phys Suppl 89, 1986 DY Pang

Weinberg expansion method Expend the 3-body wave function with Weinberg states: Ψ i (r, R) (+) = i ϕ W i (r)χ W i (R) [ ε d T r α i V np ]ϕ W i = 0, i = 1, 2, The first term gives close results as CDCC new effective deuteron potential U da Pang, Timofeyuk, Johnson, and Tostevin, Phys Rev C 87, 064613 (2013) Johnson, J Phys G: Nucl Part Phys 41, 094005 (2014) DY Pang

Comparisons between DWBA, ADWA, and CDCC 14 C 58 Ni 116 Sn dσ/dω (mb/sr) dσ/dω (mb/sr) 10 1 10 0 10 0 10 1 234 MeV 0 5 10 15 20 25 30 35 θ cm (deg) 60 MeV CDCC ADWA DWBA CDCC ADWA DWBA 0 5 10 15 20 25 θ cm (deg) dσ/dω (mb/sr) dσ/dω (mb/sr) 10 1 10 0 10 1 10 1 10 0 10 1 10 2 58 Ni, 10 MeV 0 30 60 90 120 150 θ cm (deg) 56 MeV CDCC ADWA DWBA CDCC ADWA DWBA 0 10 20 30 40 50 60 70 80 θ cm (deg) dσ/dω (mb/sr) dσ/dω (mb/sr) 10 1 10 0 10 1 10 0 10 1 10 2 122 MeV CDCC ADWA DWBA 0 20 40 60 80 100 120 θ cm (deg) DWBA ADWA 10 3 792 MeV 10 4 0 10 20 30 40 50 60 70 θ cm (deg) Pang and Mukhamedzhanov, PhysRevC 90, 044611 (2014); Mukhamedzhanov, Pang, Bertulani, and Kadyrov, PhysRevC 90, 034604 (2014) DY Pang

Problem 1: nonlocality of optical model potentials DY Pang

Problems: nonlocality of optical potentials M ADWA dp = M CDCC dp = (pn)-a interaction χ ( ) pf ψ na U pa + V pn U pf ϕ 0 (r) χ ad(+) d χ ( ) pf ψ na U pa + V pn U pf n ϕ n (r)χ bin(+) n { U ADWA,ZR da (R) = U na + U pa Uij CDCC (R) = ϕ i (r) U na + U pa ϕ j (r) Optical model potentials: U na and U pa energy dependent nonlocality of the potential NK Timofeyuk and RC Johnson, PRL 110, 112501 (2013) DY Pang

Problems: nonlocality of optical potentials M ADWA dp = M CDCC dp = (pn)-a interaction χ ( ) pf ψ na U pa + V pn U pf ϕ 0 (r) χ ad(+) d χ ( ) pf ψ na U pa + V pn U pf n ϕ n (r)χ bin(+) n { U ADWA,ZR da (R) = U na + U pa Uij CDCC (R) = ϕ i (r) U na + U pa ϕ j (r) Optical model potentials: U na and U pa energy dependent nonlocality of the potential In ADWA and CDCC, E n = E p = E d /2 : (the E d /2 rule) NK Timofeyuk and RC Johnson, PRL 110, 112501 (2013) DY Pang

Problems: nonlocality of optical potentials M ADWA dp = M CDCC dp = (pn)-a interaction χ ( ) pf ψ na U pa + V pn U pf ϕ 0 (r) χ ad(+) d χ ( ) pf ψ na U pa + V pn U pf n ϕ n (r)χ bin(+) n { U ADWA,ZR da (R) = U na + U pa Uij CDCC (R) = ϕ i (r) U na + U pa ϕ j (r) Optical model potentials: U na and U pa energy dependent nonlocality of the potential In ADWA and CDCC, E n = E p = E d /2 : (the E d /2 rule) Nonlocality effect: E n,p shift from E d 2 by around 40 MeV NK Timofeyuk and RC Johnson, PRL 110, 112501 (2013) DY Pang

Effect of nonlocality to spectroscopic factors change of spectroscopic factors by 5-27% due to nonlocality effect NK Timofeyuk and RC Johnson, PRC 87, 064610 (2013) DY Pang

Systematic nonlocal nucleon-nucleus potential Tian Yuan, Pang Danyang, and Ma Zhongyu, IJMPE 24, 1550006 (2015) DY Pang

Problem 2: inconsistency in neutron-nucleus potentials DY Pang

Inconsistency in neutron potentials V na and U na M ADWA dp = χ ( ) pf ψ na U pa + V pn U pf ϕ 0 (r) χ ad(+) d Distorted waves χ ad(+) d complex U na dσ el dω Single particle wave function ψ na real V na E binding Mukhamedzhanov, Pang, Bertulani, Kadyrov, PRC 90, 034604 (2014) DY Pang dispersive optical model potentials?

Dispersive optical model potential Rui Li, Weili Sun, et al, PRC 87, 054611 (2013)

Dispersive optical model potential Rui Li, Weili Sun, et al, PRC 87, 054611 (2013) DY Pang

Problem 3: inner part of the overlap function: SF and ANC DY Pang

Transition amplitude of (d,p) reactions The deuteron stripping amplitude in the post form is: M dp = χ ( ) pf IF A U pa + V pn U pf Ψ (+) i the overlap function I F A : I F A (r n ) = A + 1 Φ A (ξ) Φ F (ξ, r n ) Model-independent definition of the spectroscopic factor (SF): SF = IA F 2 (rn )rndr 2 n DY Pang

SF, ANC, and single-particle ANC Asymptotics of the overlap function (ANC): I F A(l na j na ) (r na) r na>r na ClnA j na iκ na h (1) l na (iκ na r na ) DY Pang

SF, ANC, and single-particle ANC Asymptotics of the overlap function (ANC): IA(l F na j na ) (r na) r na>r na ClnA j na iκ na h (1) l na (iκ na r na ) Asymptotics of the neutron sp wf (SPANC): ψ na(nr l na j na )(r na ) r na>r na bnrlna j na iκ na h (1) l na (iκ na r na ) Asymptotically: I F A(l na j na ) proportional to ψ na(n r l na j na ): IA(l F na j na ) (r na) r na>r = na C lna j na ψ b na(nr l na j na )(r na ) nr l na j na DY Pang

Extration of SF and ANC from experimental data spectroscopic factor in transition amplitude: M dp = SF 1/2 n rl na j na χ ( ) pf ψ na(n r l na j na ) U pa + V pn U pf Ψ (+) i DY Pang

Extration of SF and ANC from experimental data spectroscopic factor in transition amplitude: M dp = SF 1/2 n rl na j na χ ( ) pf ψ na(n r l na j na ) U pa + V pn U pf Ψ (+) i Experimentally, SF nrl na j na and C lna j na are obtained by SF nr l na j na = dσexp /dω dσ th /dω C2 l na j na = SF nr l na j na b 2 n r l na j na dσ/dω (mb/sr) 10 1 10 0 CDCC ADWA DWBA 10 1 58 Ni, 10 MeV 0 30 60 90 120 150 θ cm (deg) DY Pang

Single-particle potential for ψ na(nr l na j na ) ψ na(nrl na j na ) obtained with a Woods-Saxon potential: V (r, r 0, a 0 ) = V 0 1 + exp [ (r r 0 A 1/3 )/a 0 ] DY Pang

Single-particle potential for ψ na(nr l na j na ) ψ na(nrl na j na ) obtained with a Woods-Saxon potential: V (r, r 0, a 0 ) = V 0 1 + exp [ (r r 0 A 1/3 )/a 0 ] φ(r na ) 10 1 59 Ni, 2p3/2 r 0 =10 fm r 0 =11 fm r 0 =12 fm 10 2 r 0 =13 fm 0 2 4 6 8 10 r na (fm) normalized SF 100 080 060 040 020 b 2 1 3/2 (fm 1/2 ) 11 13 15 18 22 27 32 DWBA ADWA CDCC 000 10 11 12 13 14 15 16 17 r 0 (fm) M dp = SF 1/2 n r l na j na χ ( ) pf ψ na V pf Ψ (+) i, C 2 = SF b 2 DY Pang

Peripherality of a transfer reaction φ(r) R x 04 02 00 02 04 35 30 25 20 15 10 05 dσ/dω (mb/sr) dσ/dω (mb/sr) 10 1 10 0 10 1 10 0 10 1 θ cm (deg) 58 Ni, 10 MeV 10 1 0 2 4 6 8 10 12 0 20 40 60 80 r na (fm) θ cm (deg) 10 MeV 56 MeV 00 0 2 4 6 8 10 12 r na (fm) DY Pang 58 Ni, 56 MeV 10 2 0 10 20 30 40

peripherality shown by ANC: the 58 Ni case normalized ANC 12 10 08 06 58 Ni 10 MeV 56 MeV 1 11 12 13 14 15 16 17 r 0 (fm) Cl 2 dσ exp /dω na j na (r 0 ) = M int (r 0 ) b n rl na j na (r 0 ) + M 2 ext DY Pang

Application of the Combined method: ideally For the 58 Ni(d,p) 59 Ni reaction: C 2 (fm 1 ) SF 12 10 08 06 04 02 00 300 250 200 150 100 50 56 MeV ( 80) 10 MeV 10 11 12 13 14 15 16 17 r 0 (fm) DY Pang

Application of the Combined method: in reality For the 58 Ni(d,p) 59 Ni reaction: C 2 (fm 1 ) SF 12 10 08 06 04 02 00 300 250 200 150 100 50 56 MeV 10 MeV 10 11 12 13 14 15 16 17 r 0 (fm) Pang, Mukhamedzhanov, PRC 90, 044611 (2014) DY Pang

Problem 4: Coulomb potential in few-body reaction theory DY Pang

few-body method DY Pang

few-body method d + A p + B(A + n) d + A p + (na) n + (pa) p + n + A elastic scattering neutron transfer proton transfer breakup reaction DY Pang

Faddeev method for the (d,p) reactions Faddeev: treat all 3-body reaction channels simultaneously Mukhamedzhanov, Eremenko and Sattraov, PRC 86, 034001 (2012) DY Pang

Comparison between Faddeev and CDCC Screening method for Coulomb potential does not converge for Z > 20 nuclei Upadhyay, Deltuva, and Nunes, PhysRev C 85, 054621 (2012) DY Pang

proposals for Coulomb problems DY Pang

Summary Current models: DWBA, ADWA, CDCC Faddeev method and Coulomb problem Thanks to Prof Akram Mukhamedzhanov and Dr AI Sattraov (TAMU), Profs Ron Johnson, Jeff Tostevin, and Dr Natasha Timofeyuk (Surrey), and Prof Ma ZhongYu (CIAE) DY Pang

Difficulty in integrations with Coulomb wave functions Mukhamedzhanov, Eremenko and Sattraov, PRC 86, 034001 (2012) DY Pang