Problems in deuteron stripping reaction theories

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1 Problems in deuteron stripping reaction theories DY Pang School of Physics and Nuclear Energy Engineering, Beihang University, Beijing October 7, 2016 Topics: Some history of the study of deuteron stripping reactions Some problems in deuteron stripping reactions Compatibility of SF and ANC as a test of the reaction model

2 1933: possibly the first (d,p) measurement Ernest O Lawrence, Phys Rev 45, 66 (1933)

3 1934

4 1934 Cockcroft and Walton, Porc Roy Soc A144, 704 (1934)

5 1947

6 1947 Heydenburg and Inglis, PhysRev 73, 230 (1948)

7 1950

8 1950 Burrows, Gibson, Rotblat, Bulter, PhysRev 80, 1095 (1950)

9 Application of plane wave approximation RLPreston et al, Phys Rev 121, 1741 (1961) Cut off radius: R = A or A 1/3 fm for light nuclei at above Coulomb barrier

10 1953 JHorowitz and AML Messiah, PhysRev 92, 1326 (1953)

11 1961 BBuck and PE Hodgson, PhilMag 6, 1371 (1961)

12 1962 S Hinds and R Middleton, Phys Lett 1, 12 (1962)

13 1963 GR Satchler, Nucl Phys 55, 1 (1964)

14 1968

15 Problems raised to the distorted-wave method in Ambiguities and uncertainties in optical model parameters; 2 elastic parts only of the scattering wave functions may not be sufficient; 3 Non-locality of optical model potentials is usually neglected; 4 Core excitation effects to weak transitions; 5 D-state contriubtions of projectile (d, 3 He, etc); 6 Breakup effects of the projectile; 7 Elastic scattering measurements only determine the asymptotic form of the distorted waves, which are extrapolated in the nuclear interior; RJ Philpott, WT Pinkston and GR Satchler, NPA119, 241 (1968)

16 Problem 1: uncertainties in optical model potentials Solution: make use of systematic optical model potentials: XD Liu et al, PRC 69, (2004) microscopic optical model potential: cf Ogata-san s talk

17 Problem 2: sufficiency of elastic part only Solution: coupled channel calculations: CCBA, CRC T Tamura and T Udagawa, PRC 5, 1127 (1972)

18 Problem 3: Non-locality of (optical model) potentials Solution: cf NK Timofeyuk s talk SJ Waldecker and NK Timofeyuk, PRC 94, (2016)

19 Problem 4: effect of core excitation Solution: cf AM Moro, A Deltuva, and A Ogata s talks

20 Problem 5: D-state contributions of the projectiles S-state dominats at low energies Both S- and D-states are important at high energies GR Smith et al, PRC 30, 593 (1984)

21 Problem 6: breakup effects to stripping cross sections Solutions: Adiabatic model (ADWA), CDCC, etc

22 transition amplitude of (d,p) reaction Transition amplitude: M fi = χ ( ) pf I F 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

23 transition amplitude of (d,p) reaction Transition amplitude: M fi = χ ( ) pf I F 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

24 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) M DWBA fi = with DWBA: χ ( ) pf ψ na V ϕ 0 (r)χ (+) 0 (R) dkϕ k (ε k, r)χ (+) k (ε k, R) U da : optical model potential (describe d + A elastic scattering) Assume breakup effect taken into account in U da Omit all except elastic component in the three-body wave function

Improvement: the adiabatic model: ADWA The three-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 Adiabatic approx: replacing ε d

25 Improvement: the adiabatic model: ADWA The three-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 Adiabatic approx: replacing ε d with ε k : [ E + ε d ˆT ] cm (U na + U pa ) χ ad(+) d (R) = 0 With the adiabatic approximation: M ADWA fi = 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 RC Johnson, and PJR Soper, Phys Rev C 1, 976 (1970)

26 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)

27 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) Three-body equation turned into Coupled-Channel equations: (T R + H r + U na + U pa ) j=0 ϕ j (r)χ (+) j (R) = E j=0 (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) ϕ j (r)χ (+) j (R)

28 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)

gives close results as CDCC new effective deuteron potential U da Pang, Timofeyuk, Johnson,

29 The Weinberg expansion method Expend the three-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) RC Johnson, J Phys G: Nucl Part Phys 41, (2014)

30 Problem 7: inner part of the wave function The deuteron stripping amplitude: M = χ ( ) pf I F 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 = I F A I F A

31 Asymptotic behaviors 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 )

32 Asymptotic behaviors 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 ) of the neutron sp wf of F = n + A (SPANC): ψ na(nr l na j na )(r na ) r na>r na bnr l na j na iκ na h (1) l na (iκ na r na )

33 Asymptotic behaviors 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 ) of the neutron sp wf of F = n + A (SPANC): ψ na(nr l na j na )(r na ) r na>r na bnr l na 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

34 Asymptotic behaviors 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 ) of the neutron sp wf of F = n + A (SPANC): ψ na(nr l na j na )(r na ) r na>r na bnr l na 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 A big assumption: such proportionality extend to all r na : I F A(l na j na ) (r na) = C l na j na b nr l na j na ψ na (r na ) SF nr l na j na = C 2 lna j na b 2 n r l na j na

35 Extration of SF and ANC from experimental data spectroscopic factor in transition amplitude: M = SF 1/2 n r l na j na χ ( ) pf ψ na(n r l na j na ) U pa + V pn U pf Φ (+) i Experimentally, SF nr l na j na and C lna j na are obtained by SF nr l na j na = dσexp /dω dσ th /dω C 2 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 58 Ni, 10 MeV θ cm (deg)

36 Single-particle potential for ψ na(nr l na j na ) ψ na(nr l 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 ] 59 Ni, 2p3/2 φ(r na ) r 0 =10 fm r 0 =11 fm r 0 =12 fm 10 2 r 0 =13 fm r na (fm) assymptotically: ψ na(nr l na j na )(r na ) r na>r na bnr l na j na iκ na h (1) l na (iκ na r na )

37 Single-particle potential for ψ na(nr l na j na ) ψ na(nr l 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 ) 59 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)

38 Single-particle potential for ψ na(nr l na j na ) ψ na(nr l 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 ) 59 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 = SF 1/2 n r l na j na χ ( ) pf ψ na V pf Φ (+) i, C 2 = SF b 2

39 Peripherality check 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) 58 Ni, 56 MeV

40 Peripherality check 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) 58 Ni, 56 MeV

41 Peripherality check 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) 58 Ni, 56 MeV

42 Peripherality check 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) 58 Ni, 56 MeV

43 Peripherality check 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) 58 Ni, 56 MeV

44 Peripherality check 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) 58 Ni, 56 MeV

45 Peripherality check 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) 58 Ni, 56 MeV

46 Peripherality check 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) 58 Ni, 56 MeV

47 Peripherality check 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) 58 Ni, 56 MeV

48 Peripherality check 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) 58 Ni, 56 MeV

49 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 nr lna j na (r 0 ) + M 2 ext

50 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) DYP, AM Mukhamedzhanov, PRC 90, (2014)

51 Application of the Combined method: actually For the 58 Ni(d,p) 59 Ni reaction: C 2 (fm 1 ) SF MeV 10 MeV r 0 (fm)

52 Application of the Combined method: actually For the 58 Ni(d,p) 59 Ni reaction: C 2 (fm 1 ) SF MeV 10 MeV r 0 (fm) The internal part of the overlap is not well represented by the single-particle wave function!

53 Messages from Akram 1 When SF and ANC are not compatible, the inner part of the overlap function is not represented well with the well-depth prescription; 2 To obtain reliable SFs, improvement of the treatment of the internal region is necessary

54 Inconsistency in neutron potentials V na and U na M ADWA fi = 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

55 Inconsistency in neutron potentials V na and U na M ADWA fi = 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 AM Mukhamedzhanov, DYP, C Bertulani, AS Kadyrov, PRC 90, (2014) dispersive optical model potentials: cf Natasha s talk

56 Necessity of closed channels: at year 1987 N Austern et al, PhysRep 154, 125 (1987)

57 Necessity of closed channels Three-body wave function with Weinberg and CDCC states: Ψ(r, R) = i ϕ W i (r)χ W i (R) = ϕ d (r)χ 0 (R) + i=1 ϕ bin i (r)χ bin (R) i dσ/dω (mb/sr) (a) E d =100 MeV dσ/dω (mb/sr) ( 10 3 ) ( 10 3 ) E d =30 MeV DWχ 1 A DWχ 2 A DWχ 3 A sum CDCC ZR (b) θ cm (deg) PDY, NK Timofeyuk, RC Johnson, and JA Tostevin, PRC87, (2013)

58 Necessity of closed channels Expand Weinberg distorted waves with CDCC distorted waves: χ W i (R) = C i0 χ 0 (R) + j=1 C ij χ bin j (R) C ij i=1 i=2 i=3 i=4 i= E bin (MeV) PDY, NK Timofeyuk, RC Johnson, and JA Tostevin, PRC87, (2013)

59 Necessity of closed channels Expand Weinberg distorted waves with CDCC distorted waves: χ W i (R) = C i0 χ 0 (R) + j=1 C ij χ bin j (R) C 1j i= E bin (MeV) PDY, NK Timofeyuk, RC Johnson, and JA Tostevin, PRC87, (2013)

60 Summary A brief history of the study of deuteron stripping reactions Some problems in deuteron stripping reactions: solved and unsolved 1 Uncertainties in optical model parameters; 2 Coupled channel effects (CCBA, CRC); 3 Non-locality of optical model potentials; 4 Core excitation effects; 5 D-state contriubtions of projectile; 6 Breakup effects of the projectile; 7 Internal part of the nucleus: the combined method; 8 Application of dispersive optical model potential; 9 Effect of closed channels

61 Summary A brief history of the study of deuteron stripping reactions Some problems in deuteron stripping reactions: solved and unsolved 1 Uncertainties in optical model parameters; 2 Coupled channel effects (CCBA, CRC); 3 Non-locality of optical model potentials; 4 Core excitation effects; 5 D-state contriubtions of projectile; 6 Breakup effects of the projectile; 7 Internal part of the nucleus: the combined method; 8 Application of dispersive optical model potential; 9 Effect of closed channels Thanks to Prof Akram Mukhamedzhanov and Dr AI Sattraov (TAMU College Station), Profs Ron Johnson, Jeff Tostevin, and Dr Natasha Timofeyuk (Surrey)

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,