problems in deuteron stripping reaction theories - … · problems in deuteron stripping reaction...

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Problems in deuteron stripping reaction theories

D.Y. 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

Three-body systems in reactions with rare isotopes ECT∗ Trento – 2016

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1933: possibly the first (d,p) measurement

Ernest O. Lawrence, Phys. Rev. 45, 66 (1933)


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1934


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1934


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

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1947

Heydenburg and Inglis, Phys.Rev. 73, 230 (1948).


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1950

Burrows, Gibson, Rotblat, Bulter, Phys.Rev. 80, 1095 (1950)


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Application of plane wave approximation

R.L.Preston et al., Phys. Rev. 121, 1741 (1961).

Cut off radius: R = 4.37 + 0.042A or 1.7 + 1.22A1/3 fm for lightnuclei at above Coulomb barrier.


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1953

J.Horowitz and A.M.L. Messiah, Phys.Rev. 92, 1326 (1953)


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1961

B.Buck and P.E. Hodgson, Phil.Mag. 6, 1371 (1961)


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1962

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


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1963

G.R. Satchler, Nucl. Phys. 55, 1 (1964)


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1968


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Problems raised to the distorted-wave method in 1968

1 Ambiguities and uncertainties in optical model parameters;

2 elastic parts only of the scattering wave functions may not besufficient;

3 Non-locality of optical model potentials is usually neglected;

4 Core excitation effects to weak transitions;

5 D-state contriubtions of projectile (d, 3He, etc.);

6 Breakup effects of the projectile;

7 Elastic scattering measurements only determine theasymptotic form of the distorted waves, which areextrapolated in the nuclear interior;

R.J. Philpott, W.T. Pinkston and G.R. Satchler, NPA119, 241 (1968)


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Problem 1: uncertainties in optical model potentials

Solution: make use of systematic optical model potentials:

X.D. Liu et al., PRC 69, 064313 (2004).

microscopic optical model potential: cf. Ogata-san’s talk


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Problem 2: sufficiency of elastic part only

Solution: coupled channel calculations: CCBA, CRC

T. Tamura and T. Udagawa, PRC 5, 1127 (1972)


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Problem 3: Non-locality of (optical model) potentials

Solution: cf. N.K. Timofeyuk’s talk.

S.J. Waldecker and N.K. Timofeyuk, PRC 94, 034609(2016)


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Problem 4: effect of core excitation

Solution: cf. A.M. Moro, A. Deltuva, and A. Ogata’s talks.


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Problem 5: D-state contributions of the projectiles

S-state dominats at low energies. Both S- and D-states areimportant at high energies.

G.R. Smith et al., PRC 30, 593 (1984)


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Problem 6: breakup effects to stripping cross sections

Solutions: Adiabatic model (ADWA), CDCC, etc.


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transition amplitude of (d,p) reaction

Transition amplitude:

Mfi = ⟨χ(−)pF I FA |UpA + Vpn − UpF |Ψ

(+)i ⟩

I FA (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

(+)0 (R)+

∫dkϕk(εk , r)χ

(+)k (εk ,R).

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


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Distorted wave Born approximation: DWBA

Ψ(+)i (r ,R) = ϕ0(r)χ

(+)0 (R) +

XXXXXXXXXXXXX

∫dkϕk(εk , r)χ

(+)k (εk ,R) .

DWBA takes the first term of Ψ(+)i :

Ψ(+)i (r ,R) ≃ ϕ0(r)χ

(+)0 (R)

MDWBAfi =

⟨χ(−)pF ψnA|∆V |ϕ0(r)χ(+)

0 (R)⟩.

with DWBA:

UdA: optical model potential (describe d + A elasticscattering)

Assume breakup effect taken into account in UdA

Omit all except elastic component in the three-body wavefunction


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Improvement: the adiabatic model: ADWA

The three-body wave function:[E + εd − Tcm − UnA − UpA

]ϕdχ0(R)

+

∫dk

[E − εk − Tcm − UnA − UpA

]ϕk(εk)χk(εk ,R) = 0.

Adiabatic approx.: replacing −εd with εk :[E + εd − Tcm − (UnA + UpA)

]χad(+)d (R) = 0

With the adiabatic approximation:

MADWAfi =

⟨χ(−)pF ψnA |UpA + Vpn − UpF |ϕ0(r)χ

ad(+)d

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

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


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Further Improvement: CDCC

In the CDCC method Continuum states areDiscretised into bin states

Ψ(+)i (r ,R) = ϕ0(r)χ

(+)0 (R)+

∫dkϕk(εk , r)χ

(+)k (εk ,R)

⇒ Ψ(+)CDCCi (r ,R) = ϕ0(r)χ

(+)0 (R)+

∑j=1

ϕbinj (r)χ(+)j (R).

Three-body equation turned into Coupled-Channel equations:

(TR + Hr + UnA + UpA)∑j=0

ϕj(r)χ(+)j (R) = E

∑j=0

ϕj(r)χ(+)j (R).

⇒ (TR+ϵi−E+Uii )χ(+)i (R) = −

∑j =i

Uijχ(+)j (R)

Uij(R) = ⟨ϕi (r)|UnA + UpA|ϕj(r)⟩.


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Comparisons between DWBA, ADWA, and CDCC

14C

100

101

0 5 10 15 20 25 30 35

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

23.4 MeV

CDCCADWADWBA

10−1

100

0 5 10 15 20 25

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

60 MeV

CDCCADWADWBA

58Ni

10−1

100

101

0 30 60 90 120 150d

σ/d

Ω (

mb/s

r)θc.m. (deg)

58Ni, 10 MeV

CDCCADWADWBA

10−2

10−1

100

101

0 10 20 30 40 50 60 70 80

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

56 MeV

CDCCADWADWBA

116Sn

10−1

100

101

0 20 40 60 80 100 120

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

12.2 MeV

CDCCADWADWBA

10−4

10−3

10−2

10−1

100

0 10 20 30 40 50 60 70

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

79.2 MeV

DWBAADWA

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

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


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The Weinberg expansion method

Expend the three-body wavefunction with Weinberg states:

Ψi (r ,R)(+) =∑i

ϕWi (r)χWi (R)

[−εd−Tr−αiVnp]ϕWi = 0, i = 1, 2, . . .

The first term gives close results as CDCC ⇒ new effectivedeuteron potential UdA

Pang, Timofeyuk, Johnson, and Tostevin, Phys. Rev. C 87, 064613 (2013).

R.C. Johnson, J. Phys. G: Nucl. Part. Phys. 41, 094005 (2014).


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Problem 7: inner part of the wave function

The deuteron stripping amplitude:

M = ⟨χ(−)pF I FA |UpA + Vpn − UpF |Ψ

(+)i ⟩

the overlap function I FA :

I FA (rn) =√A+ 1⟨ΦA(ξ)|ΦF (ξ, rn)⟩.

Model-independent definition of thespectroscopic factor (SF):

SF = ⟨IFA |I FA ⟩.


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Asymptotic behaviors

of the overlap function (ANC):

I FA(ℓnAjnA)(rnA)rnA>RnA−−−−−→ CℓnAjnA iκnAh

(1)ℓnA

(iκnArnA)

of the neutron s.p. w.f. of F = n + A (SPANC):

ψnA(nr ℓnAjnA)(rnA)rnA>RnA−−−−−→ bnr ℓnAjnA iκnAh

(1)ℓnA

(iκnArnA)

Asymptotically: I FA(ℓnAjnA) proportional to ψnA(nr ℓnAjnA):

I FA(ℓnAjnA)(rnA)rnA>RnA=

CℓnAjnAbnr ℓnAjnA

ψnA(nr ℓnAjnA)(rnA)

A big assumption: such proportionality extend to all rnA:

IFA(ℓnAjnA)(rnA) =CℓnAjnA

bnr ℓnAjnAψnA(rnA) ⇒ SFnr ℓnAjnA =

C2ℓnAjnA

b2nr ℓnAjnA


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Extration of SF and ANC from experimental data

spectroscopic factor in transition amplitude:

M = SF1/2nr ℓnAjnA

⟨χ(−)pF ψnA(nr ℓnAjnA)|UpA + Vpn − UpF |Φ

(+)i ⟩.

Experimentally, SFnr ℓnAjnA and CℓnAjnA are obtained by

SFnr ℓnAjnA =dσexp/dΩ

dσth/dΩ⇒ C 2

ℓnAjnA= SFnr ℓnAjnAb

2nr ℓnAjnA

10−1

100

101

0 30 60 90 120 150

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

58Ni, 10 MeV

CDCCADWADWBA


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Single-particle potential for ψnA(nr ℓnAjnA)

ψnA(nr ℓnAjnA) obtained with a Woods-Saxon potential:

V (r , r0, a0) =V0

1 + exp[(r − r0A1/3)/a0

]

10−2

10−1

0 2 4 6 8 10

|φ(r

nA)|

rnA (fm)

59Ni, 2p3/2

r0=1.0 fmr0=1.1 fmr0=1.2 fmr0=1.3 fm

assymptotically:

ψnA(nr ℓnAjnA)(rnA)rnA>RnA−−−−−→ bnr ℓnAjnA iκnAh

(1)ℓnA

(iκnArnA)


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V (r , r0, a0) =V0

1 + exp[(r − r0A1/3)/a0

]

10−2

10−1

0 2 4 6 8 10

|φ(r

nA)|

rnA (fm)

59Ni, 2p3/2

r0=1.0 fmr0=1.1 fmr0=1.2 fmr0=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

no

rma

lize

d S

F

r0 (fm)

−b2 1 3/2 (fm−1/2

)

DWBAADWACDCC


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V (r , r0, a0) =V0

1 + exp[(r − r0A1/3)/a0

]

10−2

10−1

0 2 4 6 8 10

|φ(r

nA)|

rnA (fm)

59Ni, 2p3/2

r0=1.0 fmr0=1.1 fmr0=1.2 fmr0=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

no

rma

lize

d S

F

r0 (fm)

−b2 1 3/2 (fm−1/2

)

DWBAADWACDCC

M = SF1/2nr ℓnAjnA

⟨χ(−)pF ψnA|∆VpF |Φ

(+)i ⟩, C 2 = SF × b2


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Peripherality check

10−1

100

101

0 20 40 60 80

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

58Ni, 10 MeV

10−2

10−1

100

101

0 10 20 30 40

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

58Ni, 56 MeV

−0.4

−0.2

0.0

0.2

0.4

0 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

0 2 4 6 8 10 12

Rx

rnA (fm)

10 MeV56 MeV


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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

norm

aliz

ed A

NC

r0 (fm)

58Ni

10 MeV56 MeV

C 2ℓnAjnA

(r0) =dσexp/dΩ∣∣∣ Mint(r0)

bnr ℓnAjnA(r0)

+ Mext

∣∣∣2Three-body systems in reactions with rare isotopes ECT∗ Trento – 2016

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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

DYP, A.M. Mukhamedzhanov, PRC 90, 044611 (2014)


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Application of the Combined method: actually ...

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 MeV10 MeV

The internal part of the overlap is not well represented by thesingle-particle wave function!


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Messages from Akram

1 When SF and ANC are not compatible, the inner part of theoverlap function is not represented well with the well-depthprescription;

2 To obtain reliable SFs, improvement of the treatment of theinternal region is necessary.


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Inconsistency in neutron potentials VnA and UnA

MADWAfi =

⟨χ(−)pF ψnA |UpA + Vpn − UpF |ϕ0(r)χ

ad(+)d

⟩Distorted waves χ

ad(+)d ⇐ complex UnA ⇐ dσel

dΩ

Single particle wave function ψnA ⇐ real VnA ⇐ Ebinding

A.M. Mukhamedzhanov, DYP, C. Bertulani, A.S. Kadyrov, PRC 90, 034604 (2014)

dispersive optical model potentials: cf. Natasha’s talk


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Necessity of closed channels: at year 1987


N. Austern et al.,

Phys.Rep. 154, 125 (1987)

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Necessity of closed channels

Three-body wave function with Weinberg and CDCC states:

Ψ(r ,R) =∑i

ϕWi (r)χWi (R) = ϕd(r)χ0(R) +

∑i=1

ϕbini (r)χbini (R)

10−4

10−3

10−2

10−1

100

dσ

/dΩ

(m

b/s

r)Ed=100 MeV

(a)

10−1

100

101

0 20 40 60 80 100

dσ

/dΩ

(m

b/s

r)

θc.m. (deg)

Ed=30 MeV

(×103)

(×103)

(b)

DWχ1ADWχ2ADWχ3A

sumCDCC−ZR

PDY, N.K. Timofeyuk, R.C. Johnson, and J.A. Tostevin, PRC87, 064613 (2013)


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Expand Weinberg distorted waves with CDCC distorted waves:

χWi (R) = Ci0χ0(R) +

∑j=1

Cijχbinj (R).

10−3

10−2

10−1

100

101

0 10 20 30 40 50 60 70 80

|Cij|

Ebin (MeV)

i=1i=2i=3i=4i=5



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Expand Weinberg distorted waves with CDCC distorted waves:

χWi (R) = Ci0χ0(R) +

∑j=1

Cijχbinj (R).

0.3

0.6

0.9

1.2

1.5

1.8

0 20 40 60 80

02 4 6 8 10 12 14

|C1j|

Ebin (MeV)

i=1



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Summary

A brief history of the study of deuteron stripping reactions

Some problems in deuteron stripping reactions: solved andunsolved

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. A.I.Sattraov (TAMU–College Station), Profs. Ron Johnson, JeffTostevin, and Dr. Natasha Timofeyuk (Surrey).


problems in deuteron stripping reaction theories - … · problems in deuteron stripping reaction...

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