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Benchmarks of breakup modelsProg. Part. Nucl. Phys. 101 (2018) 154, Phys. Scr. T152 (2013) 014019
Jin Lei and AB, in preparation
Angela Bonaccorso
INFNSezione di Pisa
https://reactionseminar.github.io/30th April 2020
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Plan of the Presentation1 Breakup mechanisms
A very versatile reactionExamplesMotivation
2 TC mechanism and FormalismKnockoutQM TCSemiclassical TCEikonal
3 Case study14O(9Be,X)13OAsymmetric spectra
4 PeripheralityL or b-dependenceAngular distribution
5 Implementationn-target optical potential, AB, F. Carstoiu, PRC61.034605Incident energy dependenceResonancesCore-Target S-matrix
6 Kinematics and its effectsKinematicsPhase space effect
7 TC vs eikonalA.B. G.F.Bertsch PRC63.044604
8 More Examples13Be
9 CONCLUSIONS and OUTLOOK
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Breakup mechanisms A very versatile reaction
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Breakup mechanisms Examples
Examples of reactions
TC: n+target interaction
• T(d,p)T+n→surrogate,TrojanHorse
• 9Li(d,p)10Li→ Mario Gomez
• 9Be(18O,17O)10Be
• 9Be(18O,16O)11Be
• 9Be(14O,13O)X(9Be+n )
Fragmentation: n+core interaction
• 11Li(12C,X)9Li+n→ A.Corsi ,M.Gomez
• 14Be(12C,X)12Be+n → A.C .
•••
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Breakup mechanisms Motivation
Structure study motivation for exotic nuclei at the drip lineand beyond (unbound).
• Check the limits of validity ofstructure models such as theSHELL MODEL or ”ab initio”models, understanding of theresidual nuclear force.
• Challenges in breakup reactiontheories.
For normal nuclei: study of low lying resonance properties and/or dampingof high L single particle states in the continuum.
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TC mechanism and Formalism Knockout
bc
11Beà10Be
10Be
n.n.11Be
T: 9Be, 12C, 208Pb,…p
bnz
R(t)
• Peripheral reaction
• How important is theinternal part of the initialand final n wave function
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TC mechanism and Formalism QM TC
Theoretical models for inclusive (nonelastic) breakup 20
• Requires inclusion of all possible processes through which the breakup fragment can interact with the target. Impractical in most cases.
Goals
• Find a suitable model for inclusive breakup
• Explore relations between these models
Challenges
• Numerically difficult
• No numerical implementation in 1980s-2000s even for Finite Range DWBA
In 1980s
• Ichimura, Austern, and Vincent developed a spectator-participant model (post-form)
• Udagawa and Tamura suggested a breakup-fusion model (prior-form)
Phys. Rev. C 23, 1847 (1981) Phys. Rev. C 32, 431 (1985)
Phys. Rev. C 24, 1348 (1981) Phys. Lett. B 135, 333(1984)
• Hussein and McVoy adopted a spectator model with the Feshbach projection method
Nucl. Phys. A 445, 124 (1985)
• Three different approaches with different predictions
Semiclassical methods proposed: W. Baur et al., D.M. Brink and A.B.
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TC mechanism and Formalism QM TC
The Ichimura, Austern, Vincent (IAV) model• Inclusive breakup :
• a + A ⟶ b + anything
b+x
(x+A)*
• Project all degrees of freedom into three body model space
d2σdEbdΩb NEB
= − 2ℏva
ρb(Eb)⟨φx( k b) |Wx |φx( k b)⟩
Any possible states between x and A (including all nucleons
degree of freedom)
Elastic Breakup
Nonelastic Breakup
Inclusive Breakup
Imaginary part of x-A effective interaction
Jin Lei and A. M. Moro, PRC.92.044616, PRL 123, 232501 (2019); cf. Eq. (2.21) of IAV
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TC mechanism and Formalism QM TC
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TC mechanism and Formalism Semiclassical TC
A consistent formalism for all breakup reaction mechanismsThe core-target movement is treated in a semiclassical way, butneutron-target and/or neutron-core with a full QM method.AB and DM Brink, PRC38, 1776 (1988), PRC43, 299 (1991), PRC44, 1559 (1991).
Early eikonal model: I. Tanihata, Prog. Part. Nucl. Phys. 35, 505 (1995), halo-core decoupling.
dσ
dξ= C 2S
∫ ∞0
dbcdP−n(bc )
dξPct(bc ),
⊗ξ → εf , kz ,P// also ANC =
√C 2SC 2
i
Use of the simple parametrizationPct(bc ) = |Sct |2 = e(− ln 2exp[(Rs−bc )/a]),
Rs ≈ rs(A1/3p + A
1/3t ) rs ≈ 1.4fm
’strong absorption radius’
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TC mechanism and Formalism Semiclassical TC
Transfer to the continuum: from resonances to knockoutreactionsFirst order time dependent perturbation theory amplitude: ∗∗
Afi =1
i~
∫ ∞−∞
dt < φf (r)|V (r)|φi (r − R(t)) > e−i(ωt−mvz/~) (1)
ω = εi − εf + 12mv2 R(t) = bc + vt
dP−n(bc )
dεf=
1
8π3
m
~2kf
1
2li + 1Σmi |Afi |2
≈ 4π
2k2f
Σjf (2jf + 1)(|1− Sjf |2 + 1− |Sjf |
2)F ,
φf see (∗)
F = (1 + Flf ,li ,jf ,ji )Blf ,li Blf ,li =1
4π
[kf
mv2
]|Ci |2
e−2ηbc
2ηbcMlf li
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TC mechanism and Formalism Semiclassical TC
Neutron wave functions
Final continuum state:
φlf (r) = Cf ki
2(h
(+)lf
(kr)− Slf h(−)lf
(kr))Ylf ,mf(Ωf ),
Slf (εf ) is an optical model (n-core in fragmentation reactions, n-target inknockout reactions) S-matrix.
Initial state:
φli (r) = −Ci ilγh
(1)li
(iγr)Yli ,mi(Ωi ).
Surface approximation: G. Baur & Co., NPA311 (1978) 141,PRC.28, 946, PR111(1984)333; A. Winter & Co.,
L. Lo Monaco and D.M. Brink JPG11, 935, 1985; A. Mukhamedzhanov PRC 84, 044616, 2011; I. Thomposon talk at
DREB2012 (Pisa).
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TC mechanism and Formalism Eikonal
Eikonal limit
Small neutron scattering angles
Mlf li ≈ Pli (Xi )Plf (Xf ); Plf (Xf )→ I0(2ηbv)
large n-t angular momenta
4π
2k2f
Σjf (2jf + 1)→∫ ∞
0dbv
both conditions might not be well satisfied for stripping of deeply bound nucleons unless
the core-target scattering is very peripheral. Verify core angular distributions.
P−n(bc) =∫∞
0 dbv(|1− S(bv )|2 + 1− |S(bv )|2)|φi (|bv − bc|, k1)|2Notice k1 → −∞ not strictly necessary.
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Case study 14O(9Be,X)13O
3200 3400 3600 3800 4000 4200P//(MeV/c)
0
0.02
0.04
0.06
0.08
0.1
dσ/d
P //(mb/
MeV
/c)
datatotalelasticabsorptionJin absorptionJin absorption ren.
14O(9Be,X)13O
totabs=9.4mb, totelast=2.3mb
4040
Einc=53A.MeV, Sn=23.2MeV
C2S=3.15
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Case study Asymmetric spectra
Example ”deformation” effects due to n-target interaction and kinematical cut-off.
F. Flavigny, A. Obertelli, AB et al. PRL 108, 252501 (2012).
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Case study Asymmetric spectra
Asymmetries at high incident energy
!"#
$%!
!
Data courtesy of A. Gade Calcula3ons G. Salvioni MSc Thesis in prepara3on. 28S (-‐1n) 80.7 d5/2+d3/2 28S (-‐1p) s1/2+d3/2 A.MeV 24Si(-‐1p) d5/2 85.3 A.MeV 34Ar (-‐1n) s1/2 gs 70 A.MeV 34Ar (-‐1n) d3/2+d5/2
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Case study Asymmetric spectra
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Peripherality L or b-dependence
M. Hussein, Mc Voy. NPA445(1985)124
40 60 80L
0
0.5
1
σ(L)
(mb)
A.B.J.L. (14O -9Be)J.L. (13O -10Be)
L=b k, k=8.7 fm-1
b=5.75fm
b=5.2fm
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Peripherality Angular distribution
100 150 200 250 300E(14O -9Be)(MeV)
0
0.005
0.01
0.015
0.02
dσ/d
E(m
b/M
eV)
l=40, b=4.6fml=45l=50, b=5.75fmL=53, b=Rs=6fml=55l=60l=65, b=7.5fm
14O+9Be @ 742 MeV
l: angular momentum between 14O-9Be
k (14O-9Be) = 8.73028 fm^-1
Rs=6fm, l_g=53
200 250E(14O -9Be)(MeV)
0.005
0.01
0.015
dσ/d
E(m
b/M
eV)
l=40, b=4.6fml=45l=50, b=5.75fmL=53, b=Rs=6fml=55l=60l=65, b=7.5fm
Origin of the tail
14O+9Be @ 742 MeV
l: angular momentum between 14O-9Be
k (14O-9Be) = 8.73028 fm^-1
Rs=6fm, l_g=53
600 620 640 660 680 700 720E (13O -10Be*) (MeV)
0.001
0.01
0.1
dσ2 /d
EdΩ
(mb/
MeV
)
integratedθ=1θ=2θ=3θ=4θ=5θ=6θ=7
0 5 10θ(deg)
0
1000
2000
3000
4000
dσ/dΩ(mb/srad)
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Peripherality Angular distribution
Initial wave function
100 150 200 250 300E(13O -10Be*) (MeV)
0
0.1
0.2
0.3
0.4
dσ/d
E (m
b/M
eV)
r=1.4fm ANC=17.74 fm-1/2σ=9.63mb
r=1.0 ANC=10.14 4.59mbr=1.6 ANC=24.14 13.84mb
150 200 250 300E (13O -10Be*) (MeV)
0
0.1
0.2
dσ/d
E (m
b/M
eV)
r=1.4 ANC=17.74r=1.0 ANC=10.14r=1.6 ANC=24.14
0 5 10 15 20r (fm)
0
0.2
0.4
0.6
0.8
Φ(f
m-3
/2)
r=1.4r=1.0 fmr=1.6 fm
40 60 80L
0
0.5
1
σ(L)(mb)
r=1.6r=1.4r=1.0
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Implementation n-target optical potential, AB, F. Carstoiu, PRC61.034605
n-9Be optical potential: A.B & R.J. Charity, PRC89, 024619 (2014)
0.1 1 10 100Elab(MeV)
0.1
1
σ(ba
rn)
data σtot
data σreact p-9Bedata σreactdata σelAB_elDOM_elDOM_reactAB_reactDOM_totAB_tot
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Implementation Incident energy dependence
0 20 40 60 80E_lab(MeV)
0
1
2
3
σ(b
arn)
data p reactiondata n reactiondata totaldata elasticreact eikelastic eiktot eik
4 6 8 10
0.2
0.4
0.6
0.8
1.0
35 40 45 50
0.2
0.4
0.6
0.8
1.0
75 80 85 90
0.2
0.4
0.6
0.8
1.0
Resonances, trojan horse knockout
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Implementation Resonances
Transfer to 10Be, 11Be resonances: missing mass experiment.Phys. Rev. C90, 064621 (2014), Phys. Rev. C100, 024617 (2019).
DianaCarbone,AB,MariangelaBondì,F.Cappuzzello,MCavallaroetal.MAGNEXCollaboration:1nand2ntransferexperimentalcampaign
16Ohasadegenerategs(1d5/2,2s1/2)17Ohas5/2+gsand1/2+firstexcitedstateatE*=0.87MeV
Positionandwidthsofp1/2andd5/2resonancesin10Beperfectlyreproducedd5/2resonancein11BeperfectlyreproducedEvidencefora10Be(2+)+n(d5/2)atEx=5.8MeV
-1 0 1 2 3 4 5 6 7 8 9 10 11 12Ex(MeV)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
dσ/d
E x(mb/
MeV
)
0.06 σ(n+10Be)(b)TC, P1n
phen=0.01TC * Exp Fold
g.s.
(1/2
+ ) + 0
.320
(1/2
- )
1.78
3 (5
/2+ )
3.40
0 (3
/2,+ 3/
2- )
2.65
4 (3
/2- )
3.88
9 +
3.99
5 (3
/2- )
5.25
5 (5
/2- )
5.84
9 +
5.98
0 +
6.05
0 (9
/2+ )
6.70
5
8.81
3
S n = 0
.512
MeV
S 2n =
7.3
14 M
eV
6.70
5
S n = 0
.5 M
eV
1.78
3(5/
2+ )
6.70
5
A.B,D.Carbone,F.Cappuzzello,MCavallaro,G.Hupin,P.Navrátil,andS.Quaglioni
n-10BeSmatrixfromA.Calcietal.,PRL117.242501
Based on A. Calci et al., PRL .
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Implementation Core-Target S-matrix
Single folding vs double folding
A target used very often is 9Be à single folding of a n-9Be phenomenological potential with a microscopic projectile density
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Implementation Core-Target S-matrix
8Li and 8B data from 9C data from Fukuda, Nishimura, private communication
10 20 30 40 50Mass Number
1
1.1
1.2
1.3
1.4
1.5[fm]
sr s.f.d.f.
HFr
0 10 20 30 40 50 60Mass Number
800
1000
1200
1400
1600
1800
2000
[mb]
Rσ
s.f.d.f.
Imane Moumene,AB in preparation
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Kinematics and its effects Kinematics
KinematicsFrom Eq.1 ∗∗ by the change of variables dtdxdydz → dxdydzdz ′
e−i(ωt−mvz/~) → e−ik1z ′e ik2z neutron energies to neutron parallel momentawith respect to core
k1 =εf − εi − 1
2mv2
~v;
to target
k2 =εf − εi + 1
2mv2
~v;
to core parallel momentum
P// =√
E 2r −M2
r =√
(Tr + Mr )2 −M2r
=√
(Tp + εi − εf )2 + 2Mr (Tp + εi − εf ), (2)
breakup threshold at εf = 0++∗∗Exact 4-vec conservation, see https://arxiv.org/ftp/arxiv/papers/1011/1011.1943.pdf
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Kinematics and its effects Phase space effect
Origin of kinematical cut-off (phase space) and deformation effects
PRC60(1999) 054604,PRC44(1991) 1559,AB and GF Bertsch, PRC63(2001) 044604, F. Flavigny, A. Obertelli, AB et al., PRL
108, 252501 (2012). (+)
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TC vs eikonal A.B. G.F.Bertsch PRC63.044604
12Be(9Be,X)[email protected]
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TC vs eikonal A.B. G.F.Bertsch PRC63.044604
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TC vs eikonal A.B. G.F.Bertsch PRC63.044604
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More Examples 13Be
13Be puzzle or of the ”elusive 1/2+ state in Be isotopes
Ourlevelsequence2s1/2a_s=-0.8fm1p1/21d5/2
G.Blanchonetal.PRC82,034313NPAA784(2007)49
A.Corsietal.,PLB.797.1344843
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CONCLUSIONS and OUTLOOK
• QM TC numerically challenging at high energy (large n of partial waves) andfor small separation energies (DWBA source term), all observables.Semiclassical TC has a large range of validity, numerically easy, accurate.NO core angular distributions. Eikonal valid from ≈ 80A.MeV, onlymomentum distributions and total cross sections for knockout.
• Inclusive breakup reactions are dominated by final state interaction with thetarget at small incident energy: used as surrogate reaction
• At intermediate incident energy: strong interplay between projectile andtarget characteristics: ”deformed’ momentum distributions and cutoffeffects.
• From the valence particle projectile momentum distribution at high incidentenergy: information on angular momentum of the initial state and possibledynamical core-target excitations.
• Coincidence experiments of breakup particle experiments (using invariantmass method ) are more INdependent on incident energy (i.e.13Be) case.
• Elastic scattering experiments and or total reaction cross sectionmeasurements: they can tell us about the typical interaction distances andhelp fixing the optical potentials.
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CONCLUSIONS and OUTLOOK
Some of my co-authors and collaborators in historicalorder.
D. M. BrinkN. Vinh MauG. BlanchonF. CarstoiuG. F. BertschRavinder KumarF. Flavigny, A. ObertelliR. J. CharityMAGNEX collaboration at INFN-LNS: F. Cappuzzello, D. Carbone, M.Cavallaro,G. Hupin, P. Navratil, S. QuaglioniG. Salvioni... see his talk at DREB2014 in Darmstadt and Master ThesisJin Lei.
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