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