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

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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

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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

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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

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Description of the (d,p) reactions

Transition amplitude: Mdp = ⟨χ(−)

pF IF A|UpA + Vpn − UpF |Ψ(+) i

⟩ IF

A(rn)

= √ A + 1⟨ΦA(ξ)|ΦF (ξ, rn)⟩. HΨ(+)

i

(r, R) = EΨ(+)

i

(r, R), H = TR + Hnp + UnA + UpA Hnp = Tr + Vnp

D.Y. Pang JCNP2015

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Description of the (d,p) reactions

Transition amplitude: Mdp = ⟨χ(−)

pF IF A|UpA + Vpn − UpF |Ψ(+) i

⟩ IF

A(rn)

= √ A + 1⟨ΦA(ξ)|ΦF (ξ, rn)⟩. HΨ(+)

i

(r, R) = EΨ(+)

i

(r, R), H = TR + Hnp + UnA + UpA Hnp = Tr + Vnp expand Ψ(+)

i

with eigenfunctions of Hnp: Ψ(+)

i

(r, R) = ϕ0(r)χ(+) (R)+ ∫ dkϕk(εk, r)χ(+)

k

(εk, R). DWBA, ADWA, CDCC: different approx. to Ψ(+)

i

D.Y. Pang JCNP2015

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Distorted wave Born approximation: DWBA

Ψ(+)

i

(r, R) = ϕ0(r)χ(+) (R) +

✘✘✘✘✘✘✘✘✘✘✘✘ ✘ ❳❳❳❳❳❳❳❳❳❳❳❳ ❳

∫ dkϕk(εk, r)χ(+)

k

(εk, R) . DWBA takes the first term of Ψ(+)

i

: Ψ(+)

i

(r, R) ≃ ϕ0(r)χ(+) (R) M DWBA

dp

= ⟨ χ(−)

pF ψnA|∆V |ϕ0(r)χ(+)

(R) ⟩ . with DWBA: use optical model potential for UdA Assume breakup effect taken into account in UdA 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

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Improvement: the adiabatic model: ADWA

The 3-body wave function: [ E + εd − ˆ Tcm − UnA − UpA ] ϕdχ0(R) + ∫ dk [ E − εk − ˆ Tcm − UnA − UpA ] ϕk(εk)χk(εk, R) = 0. the adiabatic approx.: replacing −εk with εd: [ E + εd − ˆ Tcm − (UnA + UpA) ] ˜ χad(+)

d

(R) = 0 With the adiabatic approximation: M ADWA

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF | ϕ0(r)˜

χad(+)

d

⟩ effective d − A interaction (zero-range): UdA = UnA + UpA

Johnson, and Soper, Phys. Rev. C 1, 976 (1970).

D.Y. Pang JCNP2015

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Further Improvement: CDCC

In the CDCC method Continuum states are Discretised into bin states Ψ(+)

i

(r, R) = ϕ0(r)χ(+) (R)+ ∫ dkϕk(εk, r)χ(+)

k

(εk, R) ⇒ Ψ(+)CDCC

i

(r, R) = ϕ0(r)χ(+) (R)+ ∑

j=1

ϕbin

j (r)χ(+) j

(R).

D.Y. Pang JCNP2015

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Further Improvement: CDCC

In the CDCC method Continuum states are Discretised into bin states Ψ(+)

i

(r, R) = ϕ0(r)χ(+) (R)+ ∫ dkϕk(εk, r)χ(+)

k

(εk, R) ⇒ Ψ(+)CDCC

i

(r, R) = ϕ0(r)χ(+) (R)+ ∑

j=1

ϕbin

j (r)χ(+) j

(R). 3-body equation turned into Coupled-Channel equations: (TR+ϵi−E+Uii)χ(+)

i

(R) = − ∑

j̸=i

Uijχ(+)

j

(R) Uij(R) = ⟨ϕi(r)|UnA + UpA|ϕj(r)⟩.

Mitsuji Kawai, Masanobu Yahiro, Yasunori Iseri, Hirofumi Kameyama, Masayasu Kamimura, Prog. Theor. Phys. Suppl. 89, 1986

D.Y. Pang JCNP2015

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Weinberg expansion method

Expend the 3-body wave function with Weinberg states: Ψi(r, R)(+) = ∑

i

ϕW

i (r)χW i (R)

[−εd−Tr−αiVnp]ϕW

i

= 0, i = 1, 2, . . . The first term gives close results as CDCC ⇒ new effective deuteron potential UdA

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

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Comparisons between DWBA, ADWA, and CDCC

14C

100 101 5 10 15 20 25 30 35 dσ/dΩ (mb/sr) θc.m. (deg) 23.4 MeV CDCC ADWA DWBA 10−1 100 5 10 15 20 25 dσ/dΩ (mb/sr) θc.m. (deg) 60 MeV CDCC ADWA DWBA

58Ni

10−1 100 101 30 60 90 120 150 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

CDCC ADWA DWBA 10−2 10−1 100 101 0 10 20 30 40 50 60 70 80 dσ/dΩ (mb/sr) θc.m. (deg) 56 MeV CDCC ADWA DWBA

116Sn

10−1 100 101 20 40 60 80 100 120 dσ/dΩ (mb/sr) θc.m. (deg) 12.2 MeV CDCC ADWA DWBA 10−4 10−3 10−2 10−1 100 10 20 30 40 50 60 70 dσ/dΩ (mb/sr) θc.m. (deg) 79.2 MeV DWBA ADWA

Pang and Mukhamedzhanov, Phys.Rev.C 90, 044611 (2014); Mukhamedzhanov, Pang, Bertulani, and Kadyrov, Phys.Rev.C 90, 034604 (2014)

D.Y. Pang JCNP2015

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Problem 1: nonlocality of optical model potentials

D.Y. Pang JCNP2015

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Problems: nonlocality of optical potentials

MADWA

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF | ϕ0(r)˜

χad(+)

d

⟩ MCDCC

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF |

∑

n

ϕn(r)χbin(+)

n

⟩ (pn)-A interaction { U ADWA,ZR

dA

(R) = UnA + UpA U CDCC

ij

(R) = ⟨ϕi(r)|UnA + UpA|ϕj(r)⟩

D.Y. Pang JCNP2015

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Problems: nonlocality of optical potentials

MADWA

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF | ϕ0(r)˜

χad(+)

d

⟩ MCDCC

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF |

∑

n

ϕn(r)χbin(+)

n

⟩ (pn)-A interaction { U ADWA,ZR

dA

(R) = UnA + UpA U CDCC

ij

(R) = ⟨ϕi(r)|UnA + UpA|ϕj(r)⟩ Optical model potentials: UnA and UpA energy dependent ⇐ nonlocality of the potential

N.K. Timofeyuk and R.C. Johnson, PRL 110, 112501 (2013)

D.Y. Pang JCNP2015

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Problems: nonlocality of optical potentials

MADWA

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF | ϕ0(r)˜

χad(+)

d

⟩ MCDCC

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF |

∑

n

ϕn(r)χbin(+)

n

⟩ (pn)-A interaction { U ADWA,ZR

dA

(R) = UnA + UpA U CDCC

ij

(R) = ⟨ϕi(r)|UnA + UpA|ϕj(r)⟩ Optical model potentials: UnA and UpA energy dependent ⇐ nonlocality of the potential In ADWA and CDCC, En = Ep = Ed/2 : (the Ed/2 rule)

N.K. Timofeyuk and R.C. Johnson, PRL 110, 112501 (2013)

D.Y. Pang JCNP2015

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Problems: nonlocality of optical potentials

MADWA

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF | ϕ0(r)˜

χad(+)

d

⟩ MCDCC

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF |

∑

n

ϕn(r)χbin(+)

n

⟩ (pn)-A interaction { U ADWA,ZR

dA

(R) = UnA + UpA U CDCC

ij

(R) = ⟨ϕi(r)|UnA + UpA|ϕj(r)⟩ Optical model potentials: UnA and UpA energy dependent ⇐ nonlocality of the potential In ADWA and CDCC, En = Ep = Ed/2 : (the Ed/2 rule) Nonlocality effect: En,p shift from Ed

2 by around 40 MeV N.K. Timofeyuk and R.C. Johnson, PRL 110, 112501 (2013)

D.Y. Pang JCNP2015

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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

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Systematic nonlocal nucleon-nucleus potential

Tian Yuan, Pang Danyang, and Ma Zhongyu, IJMPE 24, 1550006 (2015).

D.Y. Pang JCNP2015

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Problem 2: inconsistency in neutron-nucleus potentials

D.Y. Pang JCNP2015

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Inconsistency in neutron potentials VnA and UnA

M ADWA

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF | ϕ0(r)˜

χad(+)

d

⟩ Distorted waves ˜ χad(+)

d

⇐ complex UnA ⇐ dσel

dΩ

Single particle wave function ψnA ⇐ real VnA ⇐ Ebinding

D.Y. Pang JCNP2015

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Inconsistency in neutron potentials VnA and UnA

M ADWA

dp

= ⟨ χ(−)

pF ψnA |UpA + Vpn − UpF | ϕ0(r)˜

χad(+)

d

⟩ Distorted waves ˜ χad(+)

d

⇐ complex UnA ⇐ dσel

dΩ

Single particle wave function ψnA ⇐ real VnA ⇐ Ebinding

Mukhamedzhanov, Pang, Bertulani, Kadyrov, PRC 90, 034604 (2014) dispersive optical model potentials?

D.Y. Pang JCNP2015

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Dispersive optical model potential

Rui Li, Weili Sun, et al., PRC 87, 054611 (2013)

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Dispersive optical model potential

Rui Li, Weili Sun, et al., PRC 87, 054611 (2013)

D.Y. Pang JCNP2015

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

D.Y. Pang JCNP2015

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Transition amplitude of (d,p) reactions

The deuteron stripping amplitude in the post form is: Mdp = ⟨χ(−)

pF IF A|UpA + Vpn − UpF |Ψ(+) i

⟩ the overlap function IF

A:

IF

A(rn) =

√ A + 1⟨ΦA(ξ)|ΦF (ξ, rn)⟩. Model-independent definition of the spectroscopic factor (SF): SF = ∫ IF

A 2(rn)r2 ndrn

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SF, ANC, and single-particle ANC

Asymptotics of the overlap function (ANC): IF

A(ℓnAjnA)(rnA) rnA>RnA

− − − − − − → CℓnAjnAiκnAh(1)

ℓnA(iκnArnA)

D.Y. Pang JCNP2015

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SF, ANC, and single-particle ANC

Asymptotics of the overlap function (ANC): IF

A(ℓnAjnA)(rnA) rnA>RnA

− − − − − − → CℓnAjnAiκnAh(1)

ℓnA(iκnArnA)

Asymptotics of the neutron s.p. w.f. (SPANC): ψnA(nrℓnAjnA)(rnA)

rnA>RnA

− − − − − − → bnrℓnAjnAiκnAh(1)

ℓnA(iκnArnA)

D.Y. Pang JCNP2015

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SF, ANC, and single-particle ANC

Asymptotics of the overlap function (ANC): IF

A(ℓnAjnA)(rnA) rnA>RnA

− − − − − − → CℓnAjnAiκnAh(1)

ℓnA(iκnArnA)

Asymptotics of the neutron s.p. w.f. (SPANC): ψnA(nrℓnAjnA)(rnA)

rnA>RnA

− − − − − − → bnrℓnAjnAiκnAh(1)

ℓnA(iκnArnA)

Asymptotically: IF

A(ℓnAjnA) proportional to ψnA(nrℓnAjnA):

IF

A(ℓnAjnA)(rnA) rnA>RnA

= CℓnAjnA bnrℓnAjnA ψnA(nrℓnAjnA)(rnA)

D.Y. Pang JCNP2015

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SF, ANC, and single-particle ANC

Asymptotics of the overlap function (ANC): IF

A(ℓnAjnA)(rnA) rnA>RnA

− − − − − − → CℓnAjnAiκnAh(1)

ℓnA(iκnArnA)

Asymptotics of the neutron s.p. w.f. (SPANC): ψnA(nrℓnAjnA)(rnA)

rnA>RnA

− − − − − − → bnrℓnAjnAiκnAh(1)

ℓnA(iκnArnA)

Asymptotically: IF

A(ℓnAjnA) proportional to ψnA(nrℓnAjnA):

IF

A(ℓnAjnA)(rnA) rnA>RnA

= CℓnAjnA bnrℓnAjnA ψnA(nrℓnAjnA)(rnA) Assumption: such proportionality extends to all rnA:

IF

A(ℓnAjnA)(rnA) =

CℓnAjnA bnrℓnAjnA ψnA(rnA) ⇒ SFnrℓnAjnA = CℓnAjnA bnrℓnAjnA

D.Y. Pang JCNP2015

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Extration of SF and ANC from experimental data

spectroscopic factor in transition amplitude: Mdp = SF 1/2

nrℓnAjnA⟨χ(−) pF ψnA(nrℓnAjnA)|UpA + Vpn − UpF |Ψ(+) i

⟩.

D.Y. Pang JCNP2015

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Extration of SF and ANC from experimental data

spectroscopic factor in transition amplitude: Mdp = SF 1/2

nrℓnAjnA⟨χ(−) pF ψnA(nrℓnAjnA)|UpA + Vpn − UpF |Ψ(+) i

⟩. Experimentally, SFnrℓnAjnA and CℓnAjnA are obtained by SFnrℓnAjnA = dσexp/dΩ dσth/dΩ ⇒ C2

ℓnAjnA = SFnrℓnAjnAb2 nrℓnAjnA

10−1 100 101 30 60 90 120 150 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

CDCC ADWA DWBA

D.Y. Pang JCNP2015

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Single-particle potential for ψnA(nrℓnAjnA)

ψnA(nrℓnAjnA) obtained with a Woods-Saxon potential: V (r, r0, a0) = V0 1 + exp [ (r − r0A1/3)/a0 ]

D.Y. Pang JCNP2015

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Single-particle potential for ψnA(nrℓnAjnA)

ψnA(nrℓnAjnA) obtained with a Woods-Saxon potential: V (r, r0, a0) = V0 1 + exp [ (r − r0A1/3)/a0 ]

10−2 10−1 2 4 6 8 10 |φ(rnA)| rnA (fm)

59Ni, 2p3/2

r0=1.0 fm r0=1.1 fm r0=1.2 fm r0=1.3 fm 0.00 0.20 0.40 0.60 0.80 1.00 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 11 13 15 18 22 27 32 normalized SF r0 (fm) −b2 1 3/2 (fm−1/2) DWBA ADWA CDCC

Mdp = SF 1/2

nrℓnAjnA⟨χ(−) pF ψnA|∆VpF |Ψ(+) i

⟩, C2 = SF × b2

D.Y. Pang JCNP2015

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Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

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Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

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Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

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Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

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Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

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Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

SLIDE 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

SLIDE 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

SLIDE 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

SLIDE 43

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Peripherality of a transfer reaction

10−1 100 101 20 40 60 80 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 10 MeV

10−2 10−1 100 101 10 20 30 40 dσ/dΩ (mb/sr) θc.m. (deg)

58Ni, 56 MeV

−0.4 −0.2 0.0 0.2 0.4 2 4 6 8 10 12 |φ(r)| rnA (fm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 4 6 8 10 12 Rx rnA (fm) 10 MeV 56 MeV

D.Y. Pang JCNP2015

SLIDE 44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

peripherality shown by ANC: the 58Ni case

0.6 0.8 1.0 1.2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 normalized ANC r0 (fm)

58Ni

10 MeV 56 MeV C2

ℓnAjnA(r0) =

dσexp/dΩ

˜

Mint(r0) bnrℓnAjnA(r0) + ˜

Mext

2

D.Y. Pang JCNP2015

SLIDE 45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Application of the Combined method: ideally ...

For the 58Ni(d,p)59Ni reaction:

0.0 0.2 0.4 0.6 0.8 1.0 1.2 SF 50 100 150 200 250 300 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 C2 (fm−1) r0 (fm) 56 MeV (−80) 10 MeV D.Y. Pang JCNP2015

SLIDE 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Application of the Combined method: in reality ...

For the 58Ni(d,p)59Ni reaction:

0.0 0.2 0.4 0.6 0.8 1.0 1.2 SF 50 100 150 200 250 300 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 C2 (fm−1) r0 (fm) 56 MeV 10 MeV

Pang, Mukhamedzhanov, PRC 90, 044611 (2014)

D.Y. Pang JCNP2015

SLIDE 47

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Problem 4: Coulomb potential in few-body reaction theory

D.Y. Pang JCNP2015

SLIDE 48

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

few-body method

D.Y. Pang JCNP2015

SLIDE 49

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

few-body method

d + A p + B(A + n) →            d + A elastic scattering p + (nA) neutron transfer n + (pA) proton transfer p + n + A breakup reaction

D.Y. Pang JCNP2015

SLIDE 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Faddeev method for the (d,p) reactions

Faddeev: treat all 3-body reaction channels simultaneously

Mukhamedzhanov, Eremenko and Sattraov, PRC 86, 034001 (2012)

D.Y. Pang JCNP2015

SLIDE 51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Faddeev method for the (d,p) reactions

Faddeev: treat all 3-body reaction channels simultaneously

Mukhamedzhanov, Eremenko and Sattraov, PRC 86, 034001 (2012)

D.Y. Pang JCNP2015

SLIDE 52

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Comparison between Faddeev and CDCC

Screening method for Coulomb potential does not converge for Z > 20 nuclei

Upadhyay, Deltuva, and Nunes, Phys.Rev. C 85, 054621 (2012).

D.Y. Pang JCNP2015

SLIDE 53

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

proposals for Coulomb problems

D.Y. Pang JCNP2015

SLIDE 54

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Summary

Current models: DWBA, ADWA, CDCC Faddeev method and Coulomb problem

Thanks to Prof. Akram Mukhamedzhanov and Dr. A.I.

Sattraov (TAMU), Profs. Ron Johnson, Jeff Tostevin, and Dr. Natasha Timofeyuk (Surrey), and Prof. Ma ZhongYu (CIAE)

D.Y. Pang JCNP2015

SLIDE 55

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Difficulty in integrations with Coulomb wave functions

Mukhamedzhanov, Eremenko and Sattraov, PRC 86, 034001 (2012)

D.Y. Pang JCNP2015