Status of deuteron stripping reaction theories Pang Danyang School - PowerPoint PPT Presentation

Status of deuteron stripping reaction theories Pang Danyang School of Physics and Nuclear Energy Engineering, Beihang University, Beijing November 9, 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Description of the (d,p) reactions Transition amplitude: ⟨ χ ( − ) A | U pA + V pn − U pF | Ψ (+) pF I F M dp = ⟩ i √ I F A ( r n ) = A + 1 ⟨ Φ A ( ξ ) | Φ F ( ξ, r n ) ⟩ . H Ψ (+) E Ψ (+) ( r , R ) = ( r , R ) , i i H = T R + H np + U nA + U pA H np = T r + V np . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Description of the (d,p) reactions Transition amplitude: ⟨ χ ( − ) A | U pA + V pn − U pF | Ψ (+) pF I F M dp = ⟩ i √ I F A ( r n ) = A + 1 ⟨ Φ A ( ξ ) | Φ F ( ξ, r n ) ⟩ . H Ψ (+) E Ψ (+) ( r , R ) = ( r , R ) , i i H = T R + H np + U nA + U pA H np = T r + V np expand Ψ (+) with eigenfunctions of H np : i ∫ Ψ (+) ( r , R ) = ϕ 0 ( r ) χ (+) dkϕ k ( ε k , r ) χ (+) ( R )+ ( ε k , R ) . i 0 k DWBA, ADWA, CDCC: different approx. to Ψ (+) i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Distorted wave Born approximation: DWBA ❳❳❳❳❳❳❳❳❳❳❳❳ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘ ∫ Ψ (+) ( r , R ) = ϕ 0 ( r ) χ (+) dkϕ k ( ε k , r ) χ (+) ( R ) + ( ε k , R ) . i 0 k ❳ DWBA takes the first term of Ψ (+) : i Ψ (+) ϕ 0 ( r ) χ (+) ( r , R ) ≃ ( R ) i 0 ⟨ ⟩ χ ( − ) pF ψ nA | ∆ V | ϕ 0 ( r ) χ (+) M DWBA = ( R ) . dp 0 with DWBA: use optical model potential for U dA Assume breakup effect taken into account in U dA Omit all except elastic component in the 3-body wave function Tobocman, Phys.Rev. 94, 1655 (1954); Austern, Direct nuclear reaction theories , 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Improvement: the adiabatic model: ADWA The 3-body wave function: [ ] E + ε d − ˆ T cm − U nA − U pA ϕ d χ 0 ( R ) ∫ [ ] E − ε k − ˆ + dk T cm − U nA − U pA ϕ k ( ε k ) χ k ( ε k , R ) = 0 . the adiabatic approx.: replacing − ε k with ε d : [ ] χ ad (+) E + ε d − ˆ T cm − ( U nA + U pA ) ˜ ( R ) = 0 d With the adiabatic approximation: ⟨ ⟩ χ ( − ) χ ad(+) M ADWA = pF ψ nA | U pA + V pn − U pF | ϕ 0 ( r )˜ dp d effective d − A interaction (zero-range): U dA = U nA + U pA Johnson, and Soper, Phys. Rev. C 1 , 976 (1970). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Further Improvement: CDCC In the CDCC method C ontinuum states are D iscretised into bin states ∫ Ψ (+) ( r , R ) = ϕ 0 ( r ) χ (+) dkϕ k ( ε k , r ) χ (+) ( R )+ ( ε k , R ) i 0 k ⇒ Ψ (+) CDCC ( r , R ) = ϕ 0 ( r ) χ (+) j ( r ) χ (+) ∑ ϕ bin ( R )+ ( R ) . i 0 j j =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Further Improvement: CDCC In the CDCC method C ontinuum states are D iscretised into bin states ∫ Ψ (+) ( r , R ) = ϕ 0 ( r ) χ (+) dkϕ k ( ε k , r ) χ (+) ( R )+ ( ε k , R ) i 0 k ⇒ Ψ (+) CDCC ( r , R ) = ϕ 0 ( r ) χ (+) j ( r ) χ (+) ∑ ϕ bin ( R )+ ( R ) . i 0 j j =1 3-body equation turned into C oupled- C hannel equations: ( T R + ϵ i − E + U ii ) χ (+) U ij χ (+) ∑ ( R ) = − ( R ) i j 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Weinberg expansion method Expend the 3-body wave function with Weinberg states: Ψ i ( r , R ) (+) = ∑ ϕ W i ( r ) χ W i ( R ) i [ − ε d − T r − α i V np ] ϕ W = 0 , i = 1 , 2 , . . . i 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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Comparisons between DWBA, ADWA, and CDCC 14 C 58 Ni 116 Sn CDCC CDCC 10 1 CDCC 10 1 ADWA ADWA ADWA DWBA DWBA DWBA d σ /d Ω (mb/sr) 10 1 d σ /d Ω (mb/sr) d σ /d Ω (mb/sr) 10 0 10 0 10 0 10 −1 58 Ni, 10 MeV 23.4 MeV 12.2 MeV 10 −1 0 5 10 15 20 25 30 35 0 30 60 90 120 150 0 20 40 60 80 100 120 θ c.m. (deg) θ c.m. (deg) θ c.m. (deg) 10 0 10 1 CDCC CDCC DWBA ADWA ADWA ADWA DWBA DWBA d σ /d Ω (mb/sr) d σ /d Ω (mb/sr) d σ /d Ω (mb/sr) 10 −1 10 0 10 0 10 −2 10 −1 10 −3 10 −1 60 MeV 10 −2 56 MeV 79.2 MeV 10 −4 0 5 10 15 20 25 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 θ c.m. (deg) θ c.m. (deg) θ c.m. (deg) Pang and Mukhamedzhanov, Phys.Rev.C 90, 044611 (2014); Mukhamedzhanov, Pang, Bertulani, and Kadyrov, Phys.Rev.C 90 , 034604 (2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Problem 1: nonlocality of optical model potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Problems: nonlocality of optical potentials ⟨ ⟩ χ ( − ) χ ad(+) M ADWA = pF ψ nA | U pA + V pn − U pF | ϕ 0 ( r )˜ dp d ⟨ ⟩ χ ( − ) M CDCC ∑ ϕ n ( r ) χ bin(+) = pF ψ nA | U pA + V pn − U pF | dp n n { U ADWA,ZR ( R ) = U nA + U pA dA ( pn ) - A interaction U CDCC ( R ) = ⟨ ϕ i ( r ) | U nA + U pA | ϕ j ( r ) ⟩ ij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Problems: nonlocality of optical potentials ⟨ ⟩ χ ( − ) χ ad(+) M ADWA = pF ψ nA | U pA + V pn − U pF | ϕ 0 ( r )˜ dp d ⟨ ⟩ χ ( − ) M CDCC ∑ ϕ n ( r ) χ bin(+) = pF ψ nA | U pA + V pn − U pF | dp n n { U ADWA,ZR ( R ) = U nA + U pA dA ( pn ) - A interaction U CDCC ( R ) = ⟨ ϕ i ( r ) | U nA + U pA | ϕ j ( r ) ⟩ ij Optical model potentials: U nA and U pA energy dependent ⇐ nonlocality of the potential N.K. Timofeyuk and R.C. Johnson, PRL 110, 112501 (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Problems: nonlocality of optical potentials ⟨ ⟩ χ ( − ) χ ad(+) M ADWA = pF ψ nA | U pA + V pn − U pF | ϕ 0 ( r )˜ dp d ⟨ ⟩ χ ( − ) M CDCC ∑ ϕ n ( r ) χ bin(+) = pF ψ nA | U pA + V pn − U pF | dp n n { U ADWA,ZR ( R ) = U nA + U pA dA ( pn ) - A interaction U CDCC ( R ) = ⟨ ϕ i ( r ) | U nA + U pA | ϕ j ( r ) ⟩ ij 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) N.K. Timofeyuk and R.C. Johnson, PRL 110, 112501 (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Problems: nonlocality of optical potentials ⟨ ⟩ χ ( − ) χ ad(+) M ADWA = pF ψ nA | U pA + V pn − U pF | ϕ 0 ( r )˜ dp d ⟨ ⟩ χ ( − ) M CDCC ∑ ϕ n ( r ) χ bin(+) = pF ψ nA | U pA + V pn − U pF | dp n n { U ADWA,ZR ( R ) = U nA + U pA dA ( pn ) - A interaction U CDCC ( R ) = ⟨ ϕ i ( r ) | U nA + U pA | ϕ j ( r ) ⟩ ij 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 N.K. Timofeyuk and R.C. Johnson, PRL 110, 112501 (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015

Effect of nonlocality to spectroscopic factors change of spectroscopic factors by 5-27% due to nonlocality effect N.K. Timofeyuk and R.C. Johnson, PRC 87, 064610 (2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Y. Pang JCNP2015