Revisiting the problem of inclusive breakup within the IAV model › event › 11559 › contribution › 4 › ... · 2016-07-25 · Revisiting the problem of inclusive breakup within

Revisiting the problem of inclusive breakup

within the IAV model

Antonio M. Moro

University of Seville (Spain)

Workshop on “Deuteron-induced reactions and beyond: inclusive breakup fragment crosssections, MSU, July 2016”

Outline

1 Motivation

2 Formulation

3 Numerical implementation

4 Applications

5 Possible extensions for incomplete fusion

6 Perspectives

2/37

How and why did we get involved?

In 2012:

Understanding (and quantification) of large inclusive breakup

cross sections observed in reactions with neutron-halo nuclei

(6He, 11Li, 11Be)

In 2016:

Understanding of larger inclusive yields for other (non-halo)

weakly-bound nuclei (6,7Li, 7Be, . . .).

Possible extension to incomplete fusion.

Surrogate reactions.

Ph.D thesis of Jin Lei:

“Study of inclusive breakup reactions induced by weakly boundnuclei” (Univ. of Seville, July 2016)

3/37

How and why did we get involved?

In 2012:

Understanding (and quantification) of large inclusive breakup

cross sections observed in reactions with neutron-halo nuclei

(6He, 11Li, 11Be)

In 2016:

Understanding of larger inclusive yields for other (non-halo)

weakly-bound nuclei (6,7Li, 7Be, . . .).

Possible extension to incomplete fusion.

Surrogate reactions.

Ph.D thesis of Jin Lei:

“Study of inclusive breakup reactions induced by weakly boundnuclei” (Univ. of Seville, July 2016)

3/37

How and why did we get involved?

In 2012:

Understanding (and quantification) of large inclusive breakup

cross sections observed in reactions with neutron-halo nuclei

(6He, 11Li, 11Be)

In 2016:

Understanding of larger inclusive yields for other (non-halo)

weakly-bound nuclei (6,7Li, 7Be, . . .).

Possible extension to incomplete fusion.

Surrogate reactions.

Ph.D thesis of Jin Lei:

“Study of inclusive breakup reactions induced by weakly boundnuclei” (Univ. of Seville, July 2016)

3/37

Evidence of NEB contributions in inclusive 209Bi(6Li,α)X

6Li+209Bi @ 32 MeV

Santra et al, PRC85,014612(2008)

α yields much larger than d yields

⇒ expect other breakup modes

rather than EBU

4/37

Evidence of NEB contributions in inclusive 209Bi(6Li,α)X

Elastic scattering

0.9

1

1.1

dσ/

dσ R

OMPCDCC

0.9

1

1.1

0.6

0.8

1

1.2

dσ/

dσ R

0

0.5

1

0

0.4

0.8

1.2

dσ/

dσ R

0

0.5

1

0

0.5

1

dσ/

σ R

0

0.5

1

0 50 100 150θ

cm (deg.)

0

0.5

1

dσ/

dσ R

0 50 100 150θ

cm (deg.)

0

0.5

1

24 MeV 26 MeV

28MeV 30MeV

32MeV 34MeV

36MeV 38MeV

40MeV 50MeV

Inclusive α’s

0

5

10

dσ/

dΩ

(mb

/sr) EBU (CDCC)

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

5/37

Evidence of NEB contributions in inclusive 208Pb(6He,α)X

Elastic scattering

0 50 100 150θ

c.m. (deg)

0

0.5

1

σ/σ R

PH189 dataPH215 data1 channel (no continuum)

4-body CDCC

6He+

208Pb @ 22 MeV

Inclusive α’s

0 50 100 150θ

lab (deg)

0

100

200

300

dσ/

dΩ

(m

b/s

r)

Exp. PH215Exp. PH1894b-CDCC

6/37

Breakup modes: 11Be+A → 10Be+n+ A example

Ag.s.

Ag.s.

10

Be (g.s.)

10

Be

10

Be

11

Be

11

(A+ )*

10

Be*

("diffraction")

ELASTIC BREAKUP (EBU)n

A

n

n

COMPLETE FUSION

EVAPORATION

(A+n)*

pαBe

INELASTIC BREAKUP

INCOMPLETE FUSION

TRANSFER

+

&

A*

n NO

N−E

LA

ST

IC B

RE

AKU

P (N

EB

)

7/37

Breakup modes: 11Be+A → 10Be+n+ A example

Ag.s.

Ag.s.

10

Be (g.s.)

10

Be

10

Be

11

Be

11

(A+ )*

10

Be*

("diffraction")

ELASTIC BREAKUP (EBU)n

A

n

n

COMPLETE FUSION

EVAPORATION

(A+n)*

pαBe

INELASTIC BREAKUP

INCOMPLETE FUSION

TRANSFER

+

&

A*

n NO

N−E

LA

ST

IC B

RE

AKU

P (N

EB

)

FEW−BODY MODELS

(CDCC, FADDEEV,...)

7/37

Explicit evaluation of inclusive breakup

Inclusive breakup:

(b + x)︸ ︷︷ ︸

a

+A → b + (x + A)∗︸ ︷︷ ︸

c

Inclusive differential cross section: σBUb = σEBU

b + σNEBb

Post-form expression for inclusive breakup:

d2σ

dΩbEb

=2π

~vaρ(Eb)

∑

c

|〈χ(−)b Ψ

c,(−)xA |Vbx |Ψ

(+)〉|2δ(E − Eb − Ec)

Ψc,(−)xA wavefunctions for c ≡ x + A states

Ψ(+) exact scattering wavefunction

Inclusion of all relevant c = x +A channels is not feasible in

general ⇒ use closed-form models

8/37

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

bA

rx

ra

rb

rbx

x

rbA

Treat b particle as spectator ⇒ χ(−)b (~kb,~rb)

Model Hamiltonian (still many-body for x + A):

H = Kb + Vbx + UbA(rbA) + Kx + HA(ξ) + VxA(ξ, rx )︸ ︷︷ ︸

HB

,

x − A wave function for a given final b state:

Z(b)x (ξ, rx ) ≡ (χ

(−)b (~kb)|Ψ〉 =

[E

+ − Eb − HB

]−1

(χ(−)b |Vpost|Ψ〉 Vpost ≈ Vbx (rx)

9/37

Reduction to effective 3-body problem (Austern & Vincent, PRC23, 1847 (1981))

Define projector onto target g.s. (Ag.s.): P = |φ(0)A 〉〈φ

(0)A |

3-body reduction: PZ(b)x (ξ, rx) = ϕ

(0)x (rx )φ

(0)A (ξ)

[Kx + UxA − Ex ]ϕ(0)x (rx ) = (χ

(−)b |Vbx |Ψ

(+)3b 〉

Ψ(+)3b =three-body scattering wave function (DWBA, CDCC,

Faddeev...)

UxA=optical model potential operator.

The asymptotics of ϕ(0)x (rx ) gives ONLY the EBU part.

σNEBb is the flux loss (“absorption”) in the x + Ags channel:

dσNEB

dΩbdEb

= −2

~vaρb(Eb)〈ϕx |WxA|ϕx 〉 (optical theorem)

10/37

Numerical implemetation

Ignore internal spins ⇒~la +~lbx =~lb +~lx

DWBA: Ψ(+)3b ≈ χ

(+)d (R)φd (rbx )

χ(−)b (~kb,~rb) distorted waves averaged in energy bins.

Details in:

J. Lei and A.M.M., PRC 92, 044616 (2015); PRC 92, 061602 (2015)

11/37

Regularization of the post-form amplitude

Two regularization procedures tested and compared:

1 Damping factor method (Huby and Mines):

ρ(~kb,~rx ) → limα→0

e−αrx ρ(~kb,~rx)

2 Binning method (I.J. Thompson): For b − B distorted waves:

Rℓb(rb, k

ib) = N

∫ k ib+∆kb/2

k ib−∆kb/2

dkb Rℓb(rb, kb),

12/37

Application of IAV model to deuteron inclusive breakup

EBU → CDCC (Fresco).

NBU → post-form IAV model.

0 30 60 90 120 150 180θ

c.m.(deg.)

0.01

0.1

1

10

100

dσ/

dΩ

pd

Ep(m

b/s

r/M

eV

)

Pampus et al.

EBU (CDCC)

NEB (IAV )

TBU

93Nb(d,pX) @ E

d=25.5 MeV

(Ep=14 MeV)

Data: Pampus et al, NPA311 (1978)141

13/37

CDCC vs post-DWBA for EBU

0 30 60 90 120 150 180θ

c.m. (deg.)

0.01

0.1

1

10

100d

σ/d

ΩpdE

p(m

b/s

r/M

eV

)

EBU (CDCC)EBU (FR-DWBA)

93Nb(d,pX) @ E

d=25.5 MeV

(Ep=14 MeV)

Good agreement in this case, but needs to be tested for other cases

14/37

Importance of finite-range and remnant term

Zero-range accurate for deuterons, but not for other projectiles like 6Li

Remnant term also important for 6Li

15/37

Comparison of regularization procedures

0 30 60 90 120 150 180θ

c.m.(deg.)

0

10

20

30

40

dσ/

dΩ

pdE

p(m

b/s

r/M

eV

)

∆k=0.06 fm-1

∆k=0.04 fm-1

∆k=0.02 fm-1

∆k=0.015 fm-1

0 30 60 90 120 150 180θ

c.m.(deg.)

0

10

20

30

40

α=0.1 fm-1

α=0.01 fm-1

α=0.001 fm-1

α=0.0001-1

93Nb(d,pX) @ E

d=25.5 MeV (E

p=14 MeV)

Damping factorBinning method

16/37

Comparison of regularization procedures

0 5 10 15 20

Ep

c.m.(MeV)

0

20

40

60

dσ/

dE

p (m

b/M

eV

)

(a)α=0.001 fm-1

∆k=0.02 fm-1

5 10 15 20

Ep

lab (MeV)

0

10

20

30

40

d2σ/

dE

pd

Ωp (

mb

/sr

Me

V) (b)

θp=20

o

Kleinfeller et al.EBU (CDCC)

NEB (FR-DWBA)

TBU

62Ni(d,pX) @ E

d=25.5 MeV

16/37

Application to 209Bi (6Li,α)X

Elastic scattering

0.9

1

1.1

dσ/

dσ R

OMPCDCC

0.9

1

1.1

0.6

0.8

1

1.2

dσ/

dσ R

0

0.5

1

0

0.4

0.8

1.2

dσ/

dσ R

0

0.5

1

0

0.5

1

dσ/

σ R

0

0.5

1

0 50 100 150θ

cm (deg.)

0

0.5

1

dσ/

dσ R

0 50 100 150θ

cm (deg.)

0

0.5

1

24 MeV 26 MeV

28MeV 30MeV

32MeV 34MeV

36MeV 38MeV

40MeV 50MeV

Inclusive α’s

0

5

10

dσ/

dΩ

(mb/s

r) 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/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/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

Good agreement but... WdA has to be readjusted to give good elastic scattering.

17/37

6Li+209Bi: incident energy dependence of cross sections

25 30 35 40 45 50E

lab (MeV)

100

101

102

103

104

σ (m

b)

σR (CDCC)

6Li+

209Bi

Vb

18/37

6Li+209Bi: incident energy dependence of cross sections

25 30 35 40 45 50E

lab (MeV)

100

101

102

103

104

σ (m

b)

σR (CDCC)

α + d EBU (CDCC)

6Li+

209Bi

Vb

18/37

6Li+209Bi: incident energy dependence of cross sections

25 30 35 40 45 50E

lab (MeV)

100

101

102

103

104

σ (m

b)

σR (CDCC)

α + d EBU (CDCC)

NEB-α (d "absorbed")

6Li+

209Bi

Vb

18/37

6Li+209Bi: incident energy dependence of cross sections

25 30 35 40 45 50E

lab (MeV)

100

101

102

103

104

σ (m

b)

σR (CDCC)

α + d EBU (CDCC)

NEB-α (d "absorbed")

NEB-d (α "absorbed")

6Li+

209Bi

Vb

18/37

6Li+209Bi: incident energy dependence of cross sections

25 30 35 40 45 50E

lab (MeV)

100

101

102

103

104

σ (m

b)

σR (CDCC)

α + d EBU (CDCC)

NEB-αNEB-dCF (exp.): Dasgupta et al.

6Li+

209Bi

Vb

18/37

6Li+209Bi: incident energy dependence of cross sections

25 30 35 40 45 50E

lab (MeV)

100

101

102

103

104

σ (m

b)

σR (CDCC)

α + d EBU (CDCC)

NEB-αNEB-dCF (exp.): Dasgupta et al.SUM(CF+NEB-α+NEB-d+EBU)

6Li+

209Bi

Vb

σreac ≈ σα+d (EBU) + σα(NBU) + σd (NBU) + σ(CF )

18/37

Post-prior equivalence

d2σ

dEbdΩb

∣∣∣∣NEB

= −2

~vaρb(Eb)〈ψx |Wx |ψx 〉,

(E+x −Kx −Ux)ψ

postx (rx ) = (χ

(−)b |Vpost|χ

(+)a φa〉; Vpost ≡ Vbx +UbA −UbB

(E+x −Kx −Ux )ψ

priorx (rx ) = (χ

(−)b |Vprior|χ

(+)a φa〉; Vprior ≡ UxA+UbA−UaA

ψpostx = ψ

priorx + ψNO

x ψNOx (~rx ) ≡ 〈χ

(−)b |χ

(+)a φa〉

d2σ

dEbdΩb

∣∣∣∣

IAV

NEB

=d2σ

dEbdΩb

∣∣∣∣

UT

NEB

+d2σ

dEbdΩb

∣∣∣∣

NO

NEB

+d2σ

dEbdΩb

∣∣∣∣

IN

NEB

19/37

Numerical assessment of post-prior equivalence

0 5 10 15 20

Ec.m.

p (MeV)

0

20

40

60

dσ/

dE

p (

mb

/Me

V)

IAVUT

NO

IN

UT+NO+IN

5 10 15 20

Ep

lab (MeV)

0

10

20

30

40

d2/d

Epd

Ωp (

mb

/sr

Me

V)

Kleinfeller et al.EBU (CDCC)

EBU+NEB (IAV)

EBU+NEB (UT)

62Ni(d,pX) @ E

d=25.5 MeV

(a)

θp=20

o

(b)

J. Lei,A.M.M.,PRC92, 061602(R)(2015)

20/37

Numerical assessment of post-prior equivalence

15 20 25 30

Ec.m.

α (MeV)

0

30

60

90

120

dσ/

dE

α (m

b/M

eV

)

IAVUTNOINUT+NO+IN

0 40 80 120 160θ

lab (deg)

0

30

60

90

Santra et al.EBUEBU+NEB(IAV)

EBU+NEB(UT)

209Bi(

6Li,αX) @ E=36 MeV

J. Lei,A.M.M.,PRC92, 061602(R)(2015)

20/37

Application to the 7Be case

Data: Mazzoco et al: σα ≈ 5σ3He

0 50 100 150θ

lab (deg)

0

15

30

45

dσ/

dΩ

(mb

/sr) Mazzocco

EBUNEBTBU

0 50 100 150θ

lab (deg)

0

5

10

15

(a) 58

Ni(7Be,αX)@21.5 MeV (b)

58Ni(

7Be,

3HeX)@21.5 MeV

21/37

Effect of binding energy and incident energy: 209Bi(6Li+,α)X

25 30 35E

lab (MeV)

100

101

102

103

104

σ bu (

mb)

EBUNEBTBU

25 30 35E

lab (MeV)

25 30 35E

lab (MeV)

Sαd=0.47 MeV Sαd

=1.47 MeV Sαd=2.47 MeV

22/37

The role of deformation: core excitations

Core excitations will affect:

1 the structure of the projectile core-excited admixtures

ΨJM(~r , ξ) =∑

ℓ,j,I

[ϕJℓ,j,I(~r )⊗ ΦI(ξ)

]

JM10Be(I) 11

l

n

Be(J)

2 the dynamics collective excitations of the 10Be during the

collision compete with halo (single-particle) excitations.

11Be

10Be*

n

Pb

Both effects have been recently implemented in an extended version of of the CDCC for-malism (CDCC): Summers et al, PRC74 (2006) 014606, R. de Diego et al, PRC 89, 064609(2014)

23/37

Application to halo nuclei: 11Be+64Zn → 10Be + X @ E=28 MeV

Standard CDCC + NEB Extended CDCC + NEB

0 10 20 30 40 50 60 70θ

lab (deg)

0

500

1000

dσ/

dΩ

(m

b/s

r)

Di Pietro et al.EBU (CDCC)

NEB-FREBU (CDCC) + NEB (DWBA)

0 10 20 30 40 50 60 70θ

lab (deg)

0

500

1000

Di Pietro et al.EBU-XCDCCNEB-FREBU(XCDCC) + NEB (DWBA)

(a) (b)

Core deformation/excitations very important for EBU. What about for NEB?

24/37

Extraction of incomplete fusion

The IAV model provides the total NEB cross section, but for some

applications σICF is needed:

σNEB = σ

ICF + σDR

Some ideas to evaluate σICF:

1 WxA = W DRxA + W CN

xA , with W CNxA constrained by CC calculations or x + A

fusion.

dσICF

dΩbdEb

= −2

~vaρb(Eb)〈ϕx |W

CNxA |ϕx 〉

2 Extended CDCC with target excitation (should provide EBU + INBU

consistently).

3 Extended IAV model.

25/37

Application to (d , p) surrogate reactions: the 238U(d,pf) case at Ed=15 MeV

0 2 4 6 8 10 12 14

E*(239

U) (MeV)

0

2

4

6

dσ/

dE

pd

Ωp (

mb/M

eV

sr) EBU

NEBICFTBU

0 2 4 6 8 10 12 14

E*(239

U) (MeV)

0

2

4

6

dσ/

dE

pd

Ωp (

mb/M

eV

sr)(a)

θp=140

o θp=150

o

(b)

26/37

Application to (d , p) surrogate reactions: the 238U(d,pf) case at Ed=15 MeV

Pcorrf (En) = Pmeas

f (En)σTBU(En)

σICF(En),

Q. Ducasse et al,

arXiv:1512.06334

26/37

Extraction of σDR from CDCC

Effective model 3-body Hamiltonian with target excitation:

H = Hproj(r) + Htar(ξt ) + TR + UcA(rcA, ξt ) + UvA(rvA, ξt ),

Benchmark with Faddeev formalism:

p

nr

A

R

M. Gomez-Ramos and A.M.M., in preparation

27/37

Possible extensions and improvements

Inclusion of intrinsic spins of clusters.

Explicit inclusion of target excitation, either prior to breakup (in a + A

interaction) and/or in the x − A channel WF.

Calculations for “stripping” part in nucleon removal cross sections

(knockout) at intermediate energies may provide a benchmark for

standard semiclassical approaches.

Extension to 3-body projectiles (9Be, 6He, 6Li, etc).

Proper application to more weakly bound (e.g. halo) nuclei might require

going beyond (DWBA) in the calculation of ϕx (eg. CDCC):

[Kx + UxA − Ex ]ϕx (rx) = (χ(−)b |Vbx |Ψ

(+)3b 〉

28/37