Status of deuteron stripping reaction theories

Transcription

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

2 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

3 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

4 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 DY Pang

5 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

6 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

7 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

8 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) DY Pang

9 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

10 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, (2013) Johnson, J Phys G: Nucl Part Phys 41, (2014) DY Pang

11 Comparisons between DWBA, ADWA, and CDCC 14 C 58 Ni 116 Sn dσ/dω (mb/sr) dσ/dω (mb/sr) MeV θ cm (deg) 60 MeV CDCC ADWA DWBA CDCC ADWA DWBA θ cm (deg) dσ/dω (mb/sr) dσ/dω (mb/sr) Ni, 10 MeV θ cm (deg) 56 MeV CDCC ADWA DWBA CDCC ADWA DWBA θ cm (deg) dσ/dω (mb/sr) dσ/dω (mb/sr) MeV CDCC ADWA DWBA θ cm (deg) DWBA ADWA MeV θ cm (deg) Pang and Mukhamedzhanov, PhysRevC 90, (2014); Mukhamedzhanov, Pang, Bertulani, and Kadyrov, PhysRevC 90, (2014) DY Pang

12 Problem 1: nonlocality of optical model potentials DY Pang

13 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) DY Pang

14 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, (2013) DY Pang

15 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, (2013) DY Pang

16 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, (2013) DY Pang

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

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

19 Problem 2: inconsistency in neutron-nucleus potentials DY Pang

20 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 DY Pang

21 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, (2014) DY Pang dispersive optical model potentials?

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

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

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

25 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

26 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

27 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 ) DY Pang

28 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

29 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 Assumption: such proportionality extends to all r na : I F A(l na j na ) (r na) = C l na j na b nrl na j na ψ na (r na ) SF nrl na j na = C l na j na b nrl na j na DY Pang

30 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

31 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) CDCC ADWA DWBA Ni, 10 MeV θ cm (deg) DY Pang

32 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 exp [ (r r 0 A 1/3 )/a 0 ] DY Pang

33 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 exp [ (r r 0 A 1/3 )/a 0 ] φ(r na ) Ni, 2p3/2 r 0 =10 fm r 0 =11 fm r 0 =12 fm 10 2 r 0 =13 fm r na (fm) normalized SF b 2 1 3/2 (fm 1/2 ) DWBA ADWA CDCC 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

34 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

35 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

36 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

37 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

38 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

39 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

40 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

41 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

42 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

43 Peripherality of a transfer reaction φ(r) R x dσ/dω (mb/sr) dσ/dω (mb/sr) θ cm (deg) 58 Ni, 10 MeV r na (fm) θ cm (deg) 10 MeV 56 MeV r na (fm) DY Pang 58 Ni, 56 MeV

44 peripherality shown by ANC: the 58 Ni case normalized ANC Ni 10 MeV 56 MeV 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

45 Application of the Combined method: ideally For the 58 Ni(d,p) 59 Ni reaction: C 2 (fm 1 ) SF MeV ( 80) 10 MeV r 0 (fm) DY Pang

46 Application of the Combined method: in reality For the 58 Ni(d,p) 59 Ni reaction: C 2 (fm 1 ) SF MeV 10 MeV r 0 (fm) Pang, Mukhamedzhanov, PRC 90, (2014) DY Pang

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

48 few-body method DY Pang

49 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

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

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

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

53 proposals for Coulomb problems DY Pang

54 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

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