Deuteron–induced reactions, spin distributions and thesurrogate method
Grégory Potel Aguilar (NSCL, LLNL)Filomena Nunes (NSCL)Ian Thompson (LLNL)
Varenna, June 17th 2015
Varenna, June 17th 2015 slide 1/25
Introduction
We present a formalism for inclusive deuteron–induced reactions. We thuswant to describe within the same framework:
elastic breakup
direct transfer
compound nucleus
Direct neutron transfer: should becompatible with existing theories.
Elastic deuteron breakup: “transfer”to continuum states.
Non elastic breakup (direct transfer,inelastic excitation and compoundnucleus formation): absorption aboveand below neutron emissionthreshold.
Important application in surrogatereactions: obtain spin–paritydistributions, get rid ofWeisskopf–Ewing approximation.
Varenna, June 17th 2015 slide 2/25
Historical background
breakup-fusionQreactions
protonsQandQαQyieldsbombardingQ209BiQwithQ12CQandQ16OQ
BrittQandQQuinton,QPhys.QRev.Q124Q819617Q877
Kerman and McVoy, Ann. Phys.122 (1979)197
Austern and Vincent, Phys.Rev. C23 (1981) 1847
Udagawa and Tamura, Phys.Rev. C24(1981) 1348
Last paper: Mastroleo,Udagawa, Mustafa Phys. Rev.C42 (1990) 683
Controversy between Udagawaand Austern formalism leftsomehow unresolved.
Varenna, June 17th 2015 slide 3/25
Inclusive (d , p) reaction
let’s concentrate in the reaction A+d→ B(=A+n)+p
ϕdχd
ϕA
χpχn
ϕAχp
ϕ1,ϕ2,...,ϕn
ϕn+1,ϕn+2,...
χp
BB
B B B
elastic breakup
direct transfer
neutron captureinelastic excitationcoumpound nucleus
non
elas
tic b
reak
up
we are interested in the inclusive cross section, i.e., we will sum over allfinal states φcB .
Varenna, June 17th 2015 slide 4/25
Neutron states in nuclei
E-E
Fx9M
eV8
scatteringxstates
weaklyxboundxstates
deeplyxboundxstates
Wx9MeV8
EF
Imaginaryxpartxofxopticalxpotential neutronxstates
narrowxsingle-particle
scatteringxandxresonances
broadxsingle-particle
Mahaux,xBortignon,xBrogliaxandxDassoxPhys.xRep.x120x919858x1x
Varenna, June 17th 2015 slide 5/25
Derivation of the differential cross section
the double differential cross section with respect to the proton energy andangle for the population of a specific final φcB
d2σ
dΩpdEp=
2π
~vdρ(Ep)
∣∣∣〈χpφcB |V |Ψ(+)〉∣∣∣2 .Sum over all channels, with the approximation Ψ(+) ≈ χdφdφA
d2σ
dΩpdEp= − 2π
~vdρ(Ep)
×∑c
〈χdφdφA|V |χpφcB〉 δ(E − Ep − E cB) 〈φcBχp|V |φAχdφd〉
χd → deuteron incoming wave, φd → deuteron wavefunction,χp → proton outgoing wave φA → target core ground state.
Varenna, June 17th 2015 slide 6/25
Sum over final states
the imaginary part of the Green’s function G is an operator representationof the δ–function,
πδ(E − Ep − E cB) = lim�→0=∑c
|φcB〉 〈φcB |E − Ep − HB + i�
= =G
d2σ
dΩpdEp= − 2
~vdρ(Ep)= 〈χdφdφA|V |χp〉G 〈χp|V |φAχdφd〉
We got rid of the (infinite) sum over final states,
but G is an extremely complex object!
We still need to deal with that.
Varenna, June 17th 2015 slide 7/25
Optical reduction of G
If the interaction V do not act on φA
〈χdφdφA|V |χp〉G 〈χp|V |φAχdφd〉= 〈χdφd |V |χp〉 〈φA|G |φA〉 〈χp|V |χdφd〉= 〈χdφd |V |χp〉Gopt 〈χp|V |χdφd〉 ,
where Gopt is the optical reduction of G
Gopt = lim�→0
1
E − Ep − Tn − UAn(rAn) + i�,
now UAn(rAn) = VAn(rAn) + iWAn(rAn) and thus Gopt are single–particle,tractable operators.
The effective neutron–target interaction UAn(rAn), a.k.a. opticalpotential, a.k.a. self–energy can be provided by structurecalculations
Varenna, June 17th 2015 slide 8/25
Capture and elastic breakup cross sections
the imaginary part of Gopt splits in two terms
=Gopt =
elastic breakup︷ ︸︸ ︷−π∑kn
|χn〉δ(E − Ep −
k2n2mn
)〈χn|+
neutron capture︷ ︸︸ ︷Gopt
†WAn Gopt ,
we define the neutron wavefunction |ψn〉 = Gopt 〈χp|V |χdφd〉
cross sections for neutron capture and elastic breakup
d2σ
dΩpdEp
]capture= − 2
~vdρ(Ep) 〈ψn|WAn |ψn〉 ,
d2σ
dΩpdEp
]breakup= − 2
~vdρ(Ep)ρ(En) |〈χnχp|V |χdφd〉|2 ,
Varenna, June 17th 2015 slide 9/25
2–step process (post representation)
p
n
d
A A
non elastic breakupelastic breakup
p
n
A
G
p
B*
to detector
step1
step2
breakup
propagation of n in the field of A
n
Varenna, June 17th 2015 slide 10/25
Austern (post)–Udagawa (prior) controversy
The interaction V can be taken either in the prior or the postrepresentation,
Austern (post)→ V ≡ Vpost ∼ Vpn(rpn) (recently revived by Moroand Lei, University of Sevilla)
Udagawa (prior) → V ≡ Vprior ∼ VAn(rAn, ξAn)in the prior representation, V can act on φA → the optical reduction givesrise to new terms:
d2σ
dΩpdEp
]post=− 2
~vdρ(Ep)
[=〈ψpriorn |WAn |ψpriorn
〉+ 2<
〈ψNONn |WAn|ψpriorn
〉+〈ψNONn |WAn |ψNONn
〉],
where ψNONn = 〈χp| χdφd〉.
The nature of the 2–step process depends on the representation
Varenna, June 17th 2015 slide 11/25
neutron wavefunctions
the neutron wavefunctions
|ψn〉 = Gopt 〈χp|V |χdφd〉
can be computed for any neutron energy
0 5 10 15 20 25 30-2
-1
0
1
2
En=2.5 MeVEn=-7.5 MeV
rBn
bound state
scattering state
transfer to resonant and non-resonant continuumwell described
these wavefunctions are not eigenfunctions of the HamiltonianHAn = Tn +
Breakup above neutron–emission threshold
proton angular differential cross section
0 20 40 60 80 100 120 140 160 1800.001
0.01
0.1
1
10
100
L=1L=2L=0L=3L=4L=5total
dσ/dΩ
(mb/
sr,,M
eV)
Ep=9,MeV
93Nb,(d,p),,Ed=15,MeV
θ
En=3.8,MeV
Varenna, June 17th 2015 slide 13/25
neutron transfer limit (isolated–resonance, first–orderapproximation)
Let’s consider the limit WAn → 0 (single–particle width Γ→ 0). For anenergy E such that |E − En| � D, (isolated resonance)
Gopt ≈ limWAn→0
|φn〉〈φn|E − Ep − En − i〈φn|WAn|φn〉
;
with |φn〉 eigenstate of HAn = Tn +
Validity of first order approximation
For WAn small, we can apply first order perturbation theory,
d2σ
dΩpdEp(E ,Ω)
]capture≈ 1π
〈φn|WAn|φn〉(En − E )2 + 〈φn|WAn|φn〉2
dσndΩ
(Ω)
]transfer
-5 -4 -3 -2 -1 0 1 2 3 40
20
40
60
80
100
120
En
σ(m
b/M
eV)
-5 -4 -3 -2 -1 0 1 2 3 40
5
10
15
20
En-5 -4 -3 -2 -1 0 1 2 3 4
0
2
4
6
8
En
σ(m
b/M
eV)
σ(m
b/M
eV)
completed calculationfirstd order
WAn=0.5dMeVWAn=3dMeV WAn=10dMeV
we compare the complete calculation with the isolated–resonance,first–order approximation for WAn = 0.5 MeV, WAn = 3 MeV andWAn = 10 MeV
Varenna, June 17th 2015 slide 15/25
Application to surrogate reactions
B*
B*
Desired reaction: neutron induced fission, gamma emission and neutron emission.
The surrogate method consists in producing the same compound nucleus B* by bombarding a deuteron target with a radio active beam of the nuclear species A.
A theoretical reaction formalism that describes the production of all open channels B* is needed.
n
A
A d
fission
gammaemission
neutronemission
fission
gammaemission
neutronemission
Surrogate for neutron capture
Varenna, June 17th 2015 slide 16/25
Disentangling elastic and non elastic breakup
We show some results for the 93Nb(d , p) reaction with a 15 MeV deuteronbeam (Mastroleo et al., Phys. Rev. C 42 (1990) 683)
4 6 8 10 12 14 16 18 20 220
10
20
30
40
50
60
70
))))shift)real)partof)optical)potential
0 2 4 6 8 10 12 14 16 180
10
20
30
40
50
60
70
dσ/d
E)(m
b/M
eV)
Ep
93Nb(d,p)Ed=15)MeV
total)proton)singleselastic)breakupnon)elastic)breakup
dσ/d
E)(m
b/M
eV)
Ep
we have used the Koning–Delaroche (Koning and Delaroche, Nucl.Phys. A 713 (2003) 231) optical potential
the real part of the optical potential has been shifted to reproducethe position of the L = 3 resonance
contributions from elastic and non elastic breakup disentangled.
Varenna, June 17th 2015 slide 17/25
Obtaining spin distributions
4 6 8 10 12 14 16 180
5
10
15
20
25
30
dσ/d
E (m
b/M
eV)
Ep
93Nb(d,p)Ed=15 MeV
L=0
L=1
L=2
L=3
spin distribution of compound nucleus
dσldEp
=2π
~vdρ(Ep)
∑lp ,m
∫ ∣∣ϕlmlp(rBn; kp)∣∣2W (rAn) drBn.Varenna, June 17th 2015 slide 18/25
Getting rid of Weisskopf–Ewing approximation
Younes and Britt, PRC68(2003)034610
Weisskopf–Ewingapproximation:P(d , nx) = σ(E )G (E , x)
inaccurate for x = γ and forx = f in the low–energy regime
can be replaced by P(d , nx) =∑J,π σ(E , J, π)G (E , J, π, x) if
σ(E , J, π) can be predicted.
0 1 2 3 4 5 6 70
5
10
15
20
Ln
σ (m
b/M
eV) Ep=10 MeV
Varenna, June 17th 2015 slide 19/25
Summary, conclusions and some prospectives
We have presented a reaction formalism for inclusivedeuteron–induced reactions.
Valid for final neutron states from Fermi energy → to scatteringstates
Disentangles elastic and non elastic breakup contributions to theproton singles.
Probe of nuclear structure in the continuum.
Provides spin–parity distributions.
Useful for surrogate reactions.
Need for optical potentials.
Extend for (p, d) reactions (hole states)?
Varenna, June 17th 2015 slide 20/25
The 3–body model
rAp
rBn
rpn
An
p
rdθpd
ξA
Bd
rAn
From H to H3B
H = Tp +Tn +HA(ξA) +Vpn(rpn) +VAn(rAn, ξA) + VAp(rAp, ξA)
H3B = Tp + Tn + HA(ξA) +Vpn(rpn) + UAn(rAn) + UAp(rAp)
Varenna, June 17th 2015 slide 21/25
Observables: angular differential cross sections (neutronbound states)
0 20 40 60 80 100 120 140 160 1800.01
0.1
1
10
100
transferNfrescocapture
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
θ
dσ/dΩ
(mb/
sr)
L=0
L=1
93NbN(d,p)N@N15NMeV
WAn=0.5NMeV
capture at resonantenergies compared with
direct transfer (fresco)calculations,
capture cross sectionsrescaled by a factor〈φn|WAn|φn〉π.
double proton differential cross section
d2σ
dΩpdEp=
2π
~vdρ(Ep)
∑l ,m,lp
∫ ∣∣∣ϕlmlp(rBn; kp)Y lp−m(θp)∣∣∣2W (rAn) drBn.Varenna, June 17th 2015 slide 22/25
Observables: elastic breakup and capture cross sections
4 6 8 10 12 14 16 18 20 220
10
20
30
40
50
60
70
captureelasticNbreakuptotal
Ep
σN(m
b/M
eV)
93NbN(d,p)N@N15NMeV
elastic breakup and capture cross sections as a function of the protonenergy. The Koning–Delaroche global optical potential has been used asthe UAn interaction (Koning and Delaroche, Nucl. Phys. A 713 (2003)231).
Varenna, June 17th 2015 slide 23/25
Sub–threshold capture
-7 -6 -5 -4 -3 -2 -1 00
10
20
30
40
50
60
En
WAn=0.5 MeV
WAn=3 MeV
-7 -6 -5 -4 -3 -2 -1 00
2
4
6
8
10
12
L=2
L=0
L=4 L=5dσ
/dE
(mb/
MeV
)
Varenna, June 17th 2015 slide 24/25
Non–orthogonality term
-4 -2 0 2 4 6 8 10 12 14 16 18 200
3
6
9
12
15
18
21
24
En
θp=10o
93Nbld,pyEd=25.5wMeV
dσ/d
Ewlm
b/M
eVy
0 10 20 30 40 50 60 70 80 90 1000.1
1
10
100
θCM
d2σ/
dEdΩ
NEB without nonworthogonalityNEB with non orthogonality
En=-3 MeV
En=5 MeV
lmb/
MeV
wsry
Varenna, June 17th 2015 slide 25/25
IntroductionHistorical background
