reactiontheoryandadvancedcdcc · a.m. moro, inpc, july 29th, 2019 p. 11. continuum spectroscopy of...

..

INPC, Glasgow,29 July - 2 August, 2019

.

......Reaction Theory and Advanced CDCC

Antonio M. Moro

Universidad de Sevilla, Spain

...1 CDCC approach to few-body problem

...2 Core excitation effects

...3 Exploring continuum structures of 2b and 3b systems

...4 Non-elastic breakup and incomplete fusion

...5 Summary

A.M. Moro, International Nuclear Physics Conference 2019

..

11Be10Be

n





...5 Summary

A.M. Moro, INPC, July 29th, 2019 p. 1

.. Few-body reduction of many-body reactions

Some selection of relevant degrees of freedom is inherent to approximatemethodsE.g.: Breakup of a one-neutron halo nucleus.

Original many-body problem

11Be

10Be*

n

é Start from (effective) NN interaction.é Complicated many-body scattering problem

[See next talk by P. Navratil]

Approximate few-body problem

é Inert target approximationé Projectile described with few-body modelé Phenomenological NA interactions

11Be10Be

n


.. The Continuum-Discretized Coupled-Channels method (CDCC)

Three-body reaction model: two-body projectile with inert constituents on inert target.

Breakup treated as inelastic excitations to projectile unbound (continuum) states.

Continuum states are represented by a finite set of square-integrable functions (discretization).

p

nr

A

R

maxε

ε = −2.22 MeV

φl,1

φl,2

φl,2

s−

waves

p−

waves

d−

waves

Breakup threshold

l=0 l=1 l=2

n=1

n=2

n=3

ε=0

ground state

Three-body wavefunction (after discretization): [H − E]ΨCDCC = 0.

ΨCDCC

(r, R) = ϕ0(k0, r)︸︷︷︸χ0(K0, R) +∑

n,j,πϕ

jπn (kn, r)︸︷︷︸χn,j,π(Kn, R)

Bound state(s) Unbound states

In principle, CDCC provides only elastics and elastic breakup observables.

CDCC does NOT provide (by itself) transfer, and other nonelastic breakup observables.


.. Testing CDCC against Faddeev for d+ 12C → p+n+12C

é Exact solution available for this case (Faddeev equations).

Data: N. Matsuoka et al., Nucl. Phys. A 391, 357 (1986).

θn

θp

C12

θ <0p

θ > 0pd

p

n

-60 -40 -20 0 20 40 60θ

p (deg)

100

101

102

103

d4/d

Ωnd

Ωp (

mb

/sr2

)

Matsuoka (1982)

FaddeevCDCC

12C(d,pn)

12C @ E

d=56 MeV

θn=15

o

A.Deltuva, A.M.M., E.Cravo, F.M.Nunes, A.C.Fonseca, PRC 76, 064602 (2007)


.. CDCC beyond the strict few-body picture

...1 Inclusion of target excitations, eg. d + A → p + n + A∗

+ Kyushu: Yahiro et al, Prog. Theor. Phys. Suppl. 89, 32 (1986)+ Seville: M. Gómez-Ramos and AMM, PRC95, 034609 (2017)

...2 Inclusion of core excitations in the projectile (eg. 11Be=10Be∗ +n).+ Michigan/Surrey (bins) : Summers et al, PRC74, 014606 (2006)+ Seville (pseudo-states): R. de Diego et al, PRC 89, 064609 (2014)

...3 Use of microscopic projectile WFs (microscopic CDCC):+ Kyushu: Sakuragi et al, Prog. Theor. Phys. Suppl. 89, 136 (1986).+ Brussels: Descouvemont and Hussein, PRL 111, 082701 (2013)


..

10Be*11Be

n





...5 Summary


.. Beyond the inert-cluster model

Deviations from the inert-cluster model are expected to show up when cluster d.o.f. arestrongly excited during the reaction.

Microscopic models

11Be

10Be*

n

4 Fragments described microscopically4 Realistic NN interactions (Pauli properly accounted for)6 Numerically demanding / not simple interpretation.

Few-bodyM

any-body

Inert cluster models

6 Ignores cluster excitations (only few-body d.o.f).6 Phenomenological inter-cluster interactions (aprox. Pauli).4 Exactly solvable (in some cases).4 Achieved for A ≤ 5 (eg. coupled-channels, Faddeev).

11Be10Be

n

Non-inert-core few-body models

10Be*11Be

n

4 Few-body + some relevant collective d.o.f.4 Implemented in extensions of Faddeev and CDCC (XCDCC)


.. Effect of core excitations on breakup

11Be+ p @ 64 MeV/u

0

20

40

60d

σ/d

Ωc.m

. (m

b/s

r) XCDCC: fullXCDCC: no deformation in V

ct

10 20 30 40

θc.m.

(deg)

0

10

20

dσ/

dΩ

c.m

. (m

b/s

r)

b) Erel

=2.5-5 MeV

a) Erel

=0.0-2.5 MeV

+ De Diego et al, PRC85, 054613 (2014).+ Data from Shrivastava et al, PLB596 (2004) 54.

11Be+ 12C @ 520 MeV/u

0 1 2 3 4 5 6 7E

x (MeV)

0

5

10

15

dσ

/dE

(m

b/M

eV)

GSI data: Palit et al., PRC 68, 034318 (2003)XCDCC: with core excitationXCDCC: w/o core excitation

11Be →

10Be(g.s.)+n

Prelim

inary

19C+ p @ 69 MeV/u

0 15 30 45 60 75θ

c.m. (deg)

10-2

10-1

100

101

dσ

/dΩ

(m

b/s

r)

5/2+

1 XCDCC

5/2+

2 XCDCC

s1/2

→d5/2

s.p.

Satou et al., PLB660, 320 (’08)

19C+p ->

18C+n +p @ 70 MeV/u

valence exc.

+ JA Lay et al, PRC94,021602(R)(2016).

.

......é Core excitations may enhance dramatically the breakup cross sections in reactions ofdeformed halo nuclei with light targets


..

9

2

Li10

nLi

H

p





...5 Summary


.. Continuum spectroscopy of two-body systems: the 10Li case

9Li +d → 10Li* + p → 9Li + n + p é 3-body final state é CDCC

9

2

Li10

nLi

H

p

The CDCC method can be combined with standard transition amplitudesfor the (d, p) process:

....Tif = ⟨ΨCDCC

n-10Li|Vn-9Li + Up-9Li − Ud-9Li|ϕd(r)χ(+)d−9Li(R)⟩,

dσdΩdEx

∝ |Tif|2 sensitive to structures in 10Li* continuum é spectroscopy



E=2.4 MeV/u (ISOLDE) Jeppesen et al, PLB 642 (2006) 449⇒ large s-wave near-threshold strength (s-wave virtual state)

E=11.2 MeV/u (TRIUMF) Cavallaro et al, PRL118 (2017) 012701⇒ no apparent evicence of near-threshold virtual state

10Li model

s-wave doublet:s1/2 ⊗9 Li(3/2−) ⇒ 1−, 2−

a(2−) = −37.9 fm (v.s.)p-wave resonance doublet:p1/2 ⊗9 Li(3/2−) ⇒ 1+, 2+

Er = 0.37 (1+), 0.61 (2+)

0 0.5 1 1.5E

x (MeV)

0

20

dN

/dE

(co

unts

/MeV

)

0 0.5 1 1.50

5

10

15

20

25

30

35

ISOLDE data

0 1 2 3 4E

x (MeV)

0

1

2

3

4

5

dσ/

dE

x (

mb

/MeV

)

TRIUMF data

0 30 60 90 120 150θ

c.m. (deg)

0.1

1

10

100

dσ/

dΩ

(m

b/s

r)

0 30 60 90 120 150θ

c.m. (deg)

0.001

0.01

0.1

1

10

100

dσ/

dΩ

(m

b/s

r)

(θc.m.

=98o-134

o)

(Ex <1 MeV)

(θc.m.

=5.5o-16.5

o)

.

......

é Same 10Li model gives to different features in both reactions: p-wave dominates at small angles but at largeangles competes with s-wave virtual state.

é Underestimation of TRIUMF data for Ex > 2 MeV suggests contribution from ℓ > 1 waves.

AMM,J. Casal,M. Gomez-Ramos,PLB793,13(2019)



E=2.4 MeV/u (ISOLDE) Jeppesen et al, PLB 642 (2006) 449⇒ large s-wave near-threshold strength (s-wave virtual state)

E=11.2 MeV/u (TRIUMF) Cavallaro et al, PRL118 (2017) 012701⇒ no apparent evicence of near-threshold virtual state

10Li model

s-wave doublet:s1/2 ⊗9 Li(3/2−) ⇒ 1−, 2−

a(2−) = −37.9 fm (v.s.)p-wave resonance doublet:p1/2 ⊗9 Li(3/2−) ⇒ 1+, 2+

Er = 0.37 (1+), 0.61 (2+)

0 0.5 1 1.5E

x (MeV)

0

20

dN

/dE

(co

un

ts/M

eV)

0 0.5 1 1.50

5

10

15

20

25

30

35

ISOLDE data

s1/2

1-

s1/2

2-

p1/2

1+

p1/2

2+

Total

0 1 2 3 4E

x (MeV)

0

1

2

3

4

5

dσ/

dE

x (

mb/M

eV)

TRIUMF data

s1/2

1-

s1/2

2-

p1/2

1+

p1/2

2+

Total

0 30 60 90 120 150θ

c.m. (deg)

0.1

1

10

100

dσ/

dΩ

(m

b/s

r)

0 30 60 90 120 150θ

c.m. (deg)

0.001

0.01

0.1

1

10

100

dσ/

dΩ

(m

b/s

r)

(θc.m.

=98o-134

o)

(Ex <1 MeV)

(θc.m.

=5.5o-16.5

o)

.

......

é Same 10Li model gives to different features in both reactions: p-wave dominates at small angles but at largeangles competes with s-wave virtual state.

é Underestimation of TRIUMF data for Ex > 2 MeV suggests contribution from ℓ > 1 waves.

AMM,J. Casal,M. Gomez-Ramos,PLB793,13(2019)A.M. Moro, INPC, July 29th, 2019 p. 11

.. Two-body correlations in Borromean nuclei from (p,pn) reactions

(C + N1 + N2)︸︷︷︸A

+p → (C + N2)︸︷︷︸B

+N1 + p

3-body structure explicitly includedParticipant (N1) / spectator (B) as-sumption

N2

C

N1

A

~x

~y

p

~R

N2C

~xB

p

N1~r′

~R′

ã Prior-form transition matrix:

....Tif =

⟨φ2b

B (q, x)Ψ(−)f (r ′, R′)

∣∣∣VpN1 + UpB − UpA

∣∣∣Φ3bA (x, y)χ(+)

pA (R)⟩,

é Ψf ≡ final p + (N2+B) relative wave function ⇒ CDCC

Observables

dσdΩBdϵN2B

,dσ

dpBdϵN2B∝ |Tif|2

[M. Gómez-Ramos, J. Casal, A.M.M., PLB 772 (2017) 115]See talk by Jesús Casal


.. Application to 11Li(p,pn)10Li

ALADIN-LAND setup at GSI at 280 MeV/u [Aksyutina et al., PLB 666 (2008) 430]

Phenomenological R-matrix analysisã spectroscopic information extracted

through fitting with assumed shapes(eg. Breit-Wigner)

s-wave: a = −22.4(4.8) fmp-wave: Er = 0.566(14) MeV

ã reaction dynamics not considered

Reaction calculation with 3-body 11Li wf:10Li continuum

s-wave (1−, 2−): aeff = −29.3 fm

p-wave (1+, 2+): Er = 0.52 MeV

11Li g.s.: 67% s, 31% p

J. Casal et al, PLB 772 (2017) 115–120


.. 13Be spectroscopy through 14Be(p,pn)13Be* @ 250 MeV/uReaction calculations based on 14Be three body model including 12Bedeformation/excitations:

0 1 2 3 4E

n-12

Be (MeV)

0

10

20

30

40

50

dσ/

dE

n-1

2B

e (m

b/M

eV)

1/2+ [s1/2 ⊗ 0

+]

1/2- [p1/2 ⊗ 0

+]

5/2+ [d5/2 ⊗ 0

+]

5/2+ [s1/2 ⊗ 2

+]

Full calculation

s-waved-wave

14Be(p,pn)

13Be @ 250 MeV/u

p-wave

-200 0 2000.0

0.2

0.4

0.6

0.8

1.0

1.2

dN

/dp

x (

arb. unit

s)

s-wavep-wave

d-wave

0 - 0.2 MeV

-200 0 200p

x (MeV/c)

0.4 - 0.5 MeV

-200 0 200

1.8 - 2.2 MeV

χ2/N = 3.6 χ2

/N = 3.2 χ2/N = 7.47

A. Corsi et al, submitted[See talk by Jesús Casal]

Supports ℓ = 1 assignment of Ex ∼0.5 MeV peaks-wave virtual state relevant forEX ∼ 0.Bump at Ex ∼ 2 MeV due to d-wave resonance

Consistent with previous RIKEN data at69 MeV/u: [Kondo et al., PLB 690 (2010) 245]


..

11Be10Be

n





...5 Summary


.. “Breakup” modes of few-body projectiles

d

α

6Li

Ag.s.

Ag.s.

A*

d

α

ELASTIC BREAKUP (EBU)

("diffraction")

INELASTIC BREAKUP

α

A (A+d)*

A

d

(A+α)*

(6Li+A)*α α

d

INCOMPLETE FUSION

&

TRANSFER

COMPLETE FUSION

+

EVAPORATION

é For a reaction of the form (b + x)︸︷︷︸a

+A → b + anything

σb = σEBU + σNEB + σCN


.. “Breakup” modes of few-body projectiles

d

α

6Li

Ag.s.

Ag.s.

A*

d

α

ELASTIC BREAKUP (EBU)

("diffraction")

INELASTIC BREAKUP

α

A (A+d)*

A

d

(A+α)*

(6Li+A)*α α

d

INCOMPLETE FUSION

&

TRANSFER

COMPLETE FUSION

+

EVAPORATION

STATISTICAL

MODELS

FEW−BODY MODELS

(CDCC, FADDEEV,...)

?

é For a reaction of the form (b + x)︸︷︷︸a

+A → b + anything

σb = σEBU + σNEB + σCN


.. Ichimura, Austern, Vincent model for NEB (IAV, PRC32, 431 (1985))

For a reaction of the form (b + x)︸︷︷︸a

+A → b + (x + A)∗︸︷︷︸not observed

Inclusive differential cross section: σincb = σEBU

b + σNEBUb

For NEB, assume b behaves as spectator, and described by a distorted waveχ(−)b (kb, rb)

σNEBb can be interpreted as the absorption occurring in the x + A channel:

dσNEB

dΩbdEb= − 2

ℏvaρb(Eb)⟨φ(kb)

x |WxA|φ(kb)x ⟩

where φ(kb)(rx) describes x − A relative motion when b scatters with kb:

[Kx + UxA − Ex]φ(kb)x (rx) = (χ

(−)b (kb)|Vbx|Ψ(+)

3b ⟩

Ψ(+)3b =three-body wavefunction of (b+ x)+A (DWBA, CDCC, Faddeev...)


.. Application to 209Bi (6Li+,α X)

Large α yield (σα ≫ σd) ⇒ evidence of NEB channels

Assume: σα ≃ σEBU + σNEB:EBU é CDCCNEB é IAV with DWBA approximation: Ψ(+)

3b ≈ χa(R)ϕa(r)

0

5

10

dσ/

dΩ

(mb

/sr) EBU (CDCC)

NEB (IAV model)

EBU+NEB

0

5

10

0

10

20

dσ/

dΩ

(mb/s

r)

0

20

40

0

20

40

dσ/

dΩ

(mb

/sr)

0

20

40

60

80

0

30

60

90

dσ/

dΩ

(mb

/sr)

0

40

80

120

0 50 100 150θ

lab (deg.)

0

60

120

180

dσ/

dΩ

(mb/s

r)

0 50 100 150θ

lab (deg.)

0

120

240

360

24 MeV

26 MeV

28 MeV 30 MeV

32 MeV34 MeV

36 MeV 38 MeV

40 MeV 50 MeV

J. Lei and AMM, PRC92, 044616 (2015)

Inclusive data well accounted for byEBU+NEBInclusive α’s dominated by NEBEBU (6Li → α + d) only relevantfor small scattering angles.

Santra et al, PRC85, 014612 (2012)




3b ≈ χa(R)ϕa(r)

0

5

10

dσ/

dΩ

(mb/s

r) EBU (CDCC)

NEB (IAV model)

EBU+NEB

0

5

10

0

10

20

dσ/

dΩ

(mb

/sr)

0

20

40

0

20

40

dσ/

dΩ

(mb/s

r)

0

20

40

60

80

0

30

60

90

dσ/

dΩ

(mb/s

r)

0

40

80

120

0 50 100 150θ

lab (deg.)

0

60

120

180

dσ/

dΩ

(mb

/sr)

0 50 100 150θ

lab (deg.)

0

120

240

360

24 MeV

26 MeV

28 MeV 30 MeV

32 MeV34 MeV

36 MeV 38 MeV

40 MeV 50 MeV

J. Lei and AMM, PRC92, 044616 (2015)


Santra et al, PRC85, 014612 (2012)




3b ≈ χa(R)ϕa(r)

0

5

10

dσ/

dΩ

(mb

/sr) EBU (CDCC)

NEB (IAV model)

EBU+NEB

0

5

10

0

10

20

dσ/

dΩ

(mb

/sr)

0

20

40

0

20

40

dσ/

dΩ

(mb

/sr)

0

20

40

60

80

0

30

60

90

dσ/

dΩ

(mb

/sr)

0

40

80

120

0 50 100 150θ

lab (deg.)

0

60

120

180

dσ/

dΩ

(mb

/sr)

0 50 100 150θ

lab (deg.)

0

120

240

360

24 MeV

26 MeV

28 MeV 30 MeV

32 MeV34 MeV

36 MeV 38 MeV

40 MeV 50 MeV

J. Lei and AMM, PRC92, 044616 (2015)


Santra et al, PRC85, 014612 (2012)


.. Complete fusion supression at above-barrier energies

CF of weakly bound nuclei suppressed at energies above the Coulomb barrier-

Observed for weakly-bound projec-tiles (6,7,8Li,9Be)CF reduced by ∼30% with respectto BPM or CC calculations.

M. Dasgupta et al., PRC 70, 024606 (2004)

Common interpretation:é CF is mostly reduced by breakup and incomplete fusion (ICF)é ICF is modeled as two-step process: breakup followed by capture of one

charged fragment (breakup-fusion, BF).

…but, calculations based on this two-step picture give only a modest reductionof CF (eg. Cook et al. , PRC 93, 064604 (2016)).


.. Complete fusion supression at above-barrier energies

CF of weakly bound nuclei suppressed at energies above the Coulomb barrier-

Observed for weakly-bound projec-tiles (6,7,8Li,9Be)CF reduced by ∼30% with respectto BPM or CC calculations.

M. Dasgupta et al., PRC 70, 024606 (2004)

Common interpretation:é CF is mostly reduced by breakup and incomplete fusion (ICF)é ICF is modeled as two-step process: breakup followed by capture of one

charged fragment (breakup-fusion, BF).…but, calculations based on this two-step picture give only a modest reductionof CF (eg. Cook et al. , PRC 93, 064604 (2016)).


.. Application of the IAV model to the CF / ICF problem

ICF is part of the NEB computed by IAVCF and NEB are constrained by the reaction cross section. For a two-bodyprojectile (a = b + x) one may expect:

σR ≈ σCF + σinel + σEBU + σ(b)NEB + σ

(x)NEB

CF can be deduced from

σCF ≈ σR︸︷︷︸CDCC/OM

− σinel︸︷︷︸CC/CDCC

−σEBU︸︷︷︸CDCC

− (σ(b)NEB + σ

(x)NEB)︸︷︷︸

IAV model


.. Results for 6,7Li+209Bi

EBU: CDCCCalculations: NEB from IAV-DWBA model

0

500

1000

1500

2000

σ (m

b)

6Li+

209Bi

7Li+

209Bi

σR (CDCC)

EBU (CDCC)

NEB (6Li,αX)

NEB(6Li,dX)

20 25 30 35 40 45 50

Elab

(MeV)

0

500

1000

1500

2000

σ (m

b)

σR (CDCC)

EBU (CDCC)

NEB (7Li,αX)

NEB(7Li,tX)

EBU (α+d) mechanism gives a mi-nor contribution to σreac.Dominant non-elastic channel is209Bi(6,7Li,α X)

J. Lei, AMM, PRL122, 042503 (2019)


.. Results for 6,7Li+209Bi

EBU: CDCCCalculations: NEB from IAV-DWBA model

0

500

1000

1500

2000

σ fus (

mb

)

6Li+

209Bi

7Li+

209Bi

CF(exp.): Dasgupta et al.

BPMCF (theor.): σ

R-σ

EBU-σ

NEB

20 25 30 35 40 45 50

Elab

(MeV)

0

500

1000

1500

2000

σ fus (

mb

)

CF(exp.): Dasgupta et al.

BPMCF (theor.): σ

R-σ

EBU-σ

NEB-σ

inel

CF data:Dasgupta et al., PRC 70, 024606 (2004)

CF well reproducedSuppression mostly due to NEB(eg. ICF) competitionDominant channel reducing CF is209Bi(6,7Li,α X)The EBU mechanism plays a minorrole

J. Lei, AMM, PRL122, 042503 (2019)


.. Is ICF a direct or a two-step process?

[Kx + UxA − Ex]φ(kb)x (rx) = ρ(rxA) ≡ (χ

(−)b (kb)|Vbx|Ψ(+)

3b ⟩

Since Ψ3b = Ψ3bbound +Ψ3b

cont, one can separate the contribution from theprojectile continuum states from those of the bound state

ρ(rxA) = ρ(rxA)bound + ρ(rxA)cont

rr

EE

(a) (b)

a+A

b+x+A

b+(x+A)*

a+A

b+x+A

b+(x+A)*

rr

EE

(a) (b)

a+A

b+x+A

b+(x+A)*

a+A

b+x+A

b+(x+A)*

XXXX


.. Is ICF a direct or a two-step process?

Test case: 6Li + 209Bi → α + X at E = 36 MeV

0 50 100 150c.m. angle (deg)

0

20

40

60

80

100

dσ/

dΩ

(m

b/s

r)

IAV-CDCC (full)

IAV-CDCC (w/o continuum)

IAV-DWBA

209Bi(

6Li, α X) Inclusion of breakup-fusion mech-

anism (IAV-CDCC) gives a mod-est increase of the α production(∼10%)DWBA approximation of IAV workswell for this system

.

......

ICF of weakly-bound projectiles does not take place in a two-step process(as commonly assumed) but as a direct capture from the projectile groundstate ⇒ Trojan Horse effect.Supported by recent experimental work (Coot et al, PRL122, 102501 (2019))


..

11Be10Be

n





...5 Summary


.. Summary

The CDCC method provides an accurate approximation of the scatteringWF of reactions induced by few-body nuclei.The standard CDCC WF considers inert fragments and target and henceprovide only elastic and elastic breakup observables.Recent developments have permitted the extension to other reactions andobservables:

“Core excitations”: deformation/excitation of the projectile fragmentsTransfer leading to unbound states (eg. 9Li(d,p)10Li)Proton-induced knockout 3-body projectiles (eg. 11Li(p,pn)10Li)

Non-elastic breakup and incomplete fusionThe inclusive breakup model of IAV has been revisited and succesfully appliedby several groupsIt provides a quantiative description of CF suppression of weakly-bound pro-jectilesCombination of IAV with CDCC is providing further insight on the elusivenature of ICF.


reactiontheoryandadvancedcdcc · a.m. moro, inpc, july 29th, 2019 p. 11. continuum spectroscopy of...

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