Comprehensive analysis of large $\alpha$ yields observed in $^{6}$Li induced reactions

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by  Jin Lei

Comprehensive analysis of large α yields observed in 6Li induced reactions
Jin Lei1,∗ and Antonio M. Moro1,†
1Departamento de FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain.
(Dated: January 10, 2017)
Background: Largeαyieldshavebeenreportedovertheyearsinreactionswith6Liand7Liprojectiles. Previous
theoretical analyses have shown that the elastic breakup (EBU) mechanism (i.e., projectile breakup leaving the
target in its ground state) is able to account only for a small fraction of the total α inclusive breakup cross
sections, pointing toward thedominance of non-elastic breakup (NEB) mechanisms.
Purpose: We aim to provide a systematic study of the α inclusive cross sections observed in nuclear reactions
induced by 6Li projectiles. In addition to estimating the total α singles cross sections, it is our goal to evaluate
angular and energy distributions of these α particles and compare with experimental data, when available.
7 Method: WecomputeseparatelytheEBUandNEBcomponentsoftheinclusivebreakupcrosssections. Forthe
1 former, we use the continuum-discretized coupled-channels (CDCC) method, which treats this mechanism to all
0 orders. For the NEB part, we employ the the model proposed in the eighties by Ichimura, Austern and Vincent
2 [Phys. Rev. C32, 432 (1982)], within theDWBA approximation.

n Results: Overall, the sum of the computed EBU and NEB cross sections is found to reproduce very well the
a measured singles cross sections. In all cases analyzed, we find that the inclusive breakup cross section is largely
J dominated by theNEB component.

8 Conclusions: Thepresentedmethodprovidesaglobalandsystematicdescriptionofinclusivebreakupreactions
induced by 6Li projectiles. It provides also a natural explanation of the previously observed underestimation of
]
the measured α yields by CDCC calculations. The method used here can be extended to other weakly-bound
h
t projectiles, includinghalo nuclei.
-
l
c
u I. INTRODUCTION thegroundstate.1Thus,theunderestimationoftheinclu-
n siveαyieldsbytheCDCCcalculationsmeansthatthere
[
Reactionsinducedbythe6Linucleushavebeenexten- other mechanisms contributing to the inclusive breakup

2 sivelystudied givingriseto alargebody ofexperimental cross section other than the EBU. These include the ex-
v data at present. Given its marked α+d structure, with changeofnucleonsbetweendandA,theprojectiledisso-
7 a separation energy of 1.474 MeV (to be compared with ciationaccompaniedby targetexcitation, and the fusion
4
the single nucleon separation energy of 5.39 MeV), one of d by A, among others, that we will globally denote as
5
0 may anticipate that the breakup of this nucleus into α non-elastic breakup(NEB)channels. Anexplicitaccount
0 and d is a major reaction channel. In fact, experimen- oftheseprocessisverychallengingduetothe hugenum-
. tal data show remarkably large yields of α particles but, berofaccessiblefinalstatesandthevarietyofcompeting
1
contrary to what naively expected, these yields are typi- different mechanisms.
0
7 cally much larger than the corresponding d yields. This When one is only interested in the evaluation of the
1 suggeststhatthebreakupofthe6Liisnotasimpledirect singles cross section (for example, the energy or angular
: breakup mechanism. distribution of α particles), rather than on the separate
v
From the theoretical point of view, a proper inter- contributingmechanisms,onemayresorttotheinclusive
i
X pretation of these α yields is still lacking. Continuum- breakupmodelsproposedinthe1980sandrecentlyreex-
r discretizedcoupled-channels(CDCC)calculations,which aminedbyseveralgroups[2–6]. Inthesemodels,thesum
a treat the 6Li breakup as an inelastic excitation to the overallthepossiblefinalstatesthroughwhichthe unob-
continuum, reproduce successfully the coincidence α+d served fragment d may interact with the target is done
measurements [1] but they largely underestimate the in- in a formal way, making use of the Feshbach projection
clusive α cross sections. It is worthwhile recalling that formalism [7] and closure.
the CDCC method provides only the so-called elastic Inthiswork,wewillshowthatinclusiveαsinglescross
breakup(EBU)componentofthetotalbreakupcrosssec- sections from 6Li-induced reactions can be remarkably
tion. For the reaction of a 6Li projectile impinging on a well reproduced using the inclusive breakup model pro-
target A, this corresponds to the processes of the form posed by Ichimura, Austern and Vincent (IAV) [8]. To
6Li+A→α+d+A in which the two-projectileclus- our knowledge,this is the first study of this kind provid-
g.s.
ters survive after the collision and the target remains in ing a systematic explanation of these data.
∗ [email protected]; Present address: Institute of Nuclear and Particle 1 Ifathree-bodydescriptionofthe6Liisused,α+p+n,thethree-
Physics, and Department of Physics and Astronomy, OhioUni- bodybreakupmode6Li+A→α+p+n+Ag.s. wouldbealso
versity,Athens,Ohio45701,USA part of the elastic breakup channel. Since we resort here to a
† [email protected] two-bodymodelof6LiweincludethischannelintheNEBpart.
2
Although the IAV model provides a common formal-
ism for the calculation of the elastic and non-elastic x
breakup components of the inclusive breakup cross sec-
tion, in our analysis we will employ this model only
for the NEB part, whereas for the EBU part we will rbx rx
usethe continuum-discretizedcoupled-channels(CDCC)
r
method, which treats breakup to all orders. a
The paper is organized as follows. In Sec. II we give r
b
a short overview of the IAV theory, highlighting only its
b
mainformulas. InSec.IIItheextensionoftheformalism r A
bA
tonegativedeuteronenergies(boundstates)isdiscussed.
In Sec. IV, the formalism is applied to describe the α
crosssectionsinseveral6Li-inducedreactionscomparing
withthe availabledata. InSec.Vtheroleofthetransfer
FIG. 1. (Color online) Coordinates used in the non-elastic
channelsontheNEBcrosssectionisdiscussed. InSec.VI
breakupcalculations.
we investigate the systematic behaviour of the inclusive
cross section with respect to the incident energy and for
all analyzed targets. Finally, in Sec. VII we summarize
the main results of this work. the NEB with respect to the angle and energy of the b
fragments:
II. THE ICHIMURA, AUSTERN, VINCENT d2σ =− 2 ρ (E )hψ(0)(~k ,~r )|W |ψ(0)(~k ,~r )i,
(IAV) MODEL dE dΩ (cid:12) ~v b b x b x x x b x
b b(cid:12)NEB a
(cid:12) (3)
(cid:12)
In this section we briefly summarize the model of where va is the projectile-target relative velocity,
Ichimura, Austern and Vincent (IAV), whose original ρb(Eb) = kbµb/((2π)3~2) is the density of states for the
derivation can be found in [8, 9], and has been also re- particleb,Wx isthe imaginarypartofthe opticalpoten-
centlyrevisitedbyseveralauthors[2,3,5,6]. Weoutline tial describing x+A elastic scattering and ψ(0)(~k ,~r )
x b x
here the main results of this model, and refer the reader is the so-called x-channel wave function, which governs
tothesereferencesforfurtherdetailsontheirderivations. the evolution of x after the projectile dissociation, when
We write the process under study in the form, b scatters with momentum ~k and the target remains in
b
the ground state. This function satisfies the following
a(=b+x)+A→b+B∗. (1)
inhomogeneous differential equation
wheretheprojectilea,composedofbandx,collideswith
atargetA,emittingbfragmentsandanyotherfragments. (Ex−Kx−UxA)ψx(0)(~kb,~rx)=(χb(−)(~kb,~rb)|Vpost|Ψ3bi,
Thus, B∗ denotes any final state of the x+A system. (4)
ThisprocesswillbedescribedwiththeeffectiveHamil- whereE =E−E ,χ(−) isthedistorted-wavedescribing
x b b
tonian thescatteringofbinthefinalchannelwithrespecttothe
x+A sub-system, and V ≡ V +U −U (with U
H =K+V +U (~r )+H (ξ)+V (ξ,~r ), (2) post bx bA b b
bx b b A xA x theopticalpotentialinthefinalchannel)isthepost-form
where K is the total kinetic energy operator, V is the transition operator. The notation (||i indicates integra-
bx
interactionbinding the twoclustersb andxinthe initial tion over the ~rb coordinte only. This equation is to be
compositenucleusa,H (ξ)istheHamiltonianofthetar- solved with outgoing boundary conditions.
A
getnucleus(withξ denotingitsinternalcoordinates)and Austern et al. [9] suggest approximating the three-
V and U are the fragment–target interactions. The body wave function appearing in the source term of
xA b
relevant coordinates are depicted in Fig. 1. Eq. (4), Ψ3b, by the CDCC one. Since the CDCC wave
In writing the Hamiltonian of the system in the form function is also a complicated object by itself, a sim-
(2) we make a clear distinction between the two cluster pler choice is to use the DWBA approximation, i.e.,
constituents; the interaction of the fragment b, the one ψ3b ≈ χ(+)(~r )φ (~r ), where χ(+) is a distorted wave
x a a a bx a
that is assumed to be observed in the experiment, is de- describing a+A elastic scattering and φ is the projec-
a
scribed with a (complex) optical potential. Non-elastic tile ground state wave function.
processes arising from this interaction (e.g. target exci- The IAV model has been recently revisited by sev-
tation), are included only effectively through U . The eral groups [2, 5, 6]. All the calculations performed so
b
particle b is said to act as spectator. On the other hand, far by these groups make use of the DWBA approxima-
the interaction of the particle x with the target retains tion for the incoming wave function. In Refs. [5, 6], the
the dependence of the target degrees of freedom (ξ). theory was applied to deuteron induced reactions of the
Starting fromHamiltonian(2)IAV derivedthe follow- form A(d,pX), and in Ref. [2] the model was extended
ingexpressionforthe doubledifferentialcrosssectionfor to 6Li projectiles, presenting a first application to the
3
209Bi(6Li,αX) reaction. In general, the agreement with foundbyintegratingEq.(7)overaninfinitesimalinterval
thedatahasbeenfoundtobeveryencouraging,although around r′
x
further comparisons with experimental data are advis-
able to better assess the validity and limitations of the 2µ rx′+δ d2 d rx′+δ
model. ~2x =Zrx′−δ drxdrx2glx(rx,rx′)= drxglx(rx,rx′)(cid:12)(cid:12)(cid:12)(cid:12)rx′−δ
d (cid:12)
III. EXTENSION OF IAV MODEL TO Ex <0 =Nlxhflx(rx′)drxh(lx+)(rx′ +δ)
d
The sort of breakup cross section considered by −h(+)(r′) f (r′ −δ)
Ichimura,AusternandVincentcanberegardedastrans- lx x drx lx x i
fer to continuum process populating x+A states with −δ−→−→0 N W[f (r′),h(+)(r′)]
lx lx x lx x
positive relative energy (Ex > 0). In general, the inclu- (9)
sive cross section will contain also contributions coming
from the population of states below the breakup x+A Where W denotes a Wronskian, which is independent of
threshold (Ex < 0). For example, in a (6Li, αX) reac- the value of rx′.
tion, the α’s emitted at the higher energies will actually It is worth noting that the integral form of the
correspond to deuteron transfer to bound states of the x−channel wave function (5) can be also be used for
targetnucleus. One wouldliketohaveacommonframe- positive x−A energies. Proceeding in this way, the ap-
worktodescribetransfertocontinuumstatesaswellasto plication of the IAV formalism to positive and negative
bound states. The explicit inclusion of all possible final energiesisformallyanalogous. Despite this formalsimil-
bound states is unpractical because of their large num- itude, the interpretation of the channel function and of
berandtheuncertaintiesintheirspin/parityassignments the underlying imaginary part of the potential is some-
and spectroscopic factors. An alternative procedure was what different in both regions. For E > 0 the channel
x
proposedby Udagawaand co-workers[10]. The key idea function ψ(0) describes x−A elastic scattering and the
x
is to extend the complex potential to negative energies.
imaginarypartis thereforeassociatedwiththe flux leav-
Then, the bound states of the system are simulated by
ing this channel in favor of non-elastic channels. For
the eigenstatesinthiscomplexpotential. Theimaginary
E < 0, the channel wave-function describes the mo-
x
part will be associated with the spreading width of the
tion of the x particle in a bound single-particle config-
single-particle states, which accounts for the fragmenta-
uration state of the residual nucleus, and the imaginary
tionofthese statesinto morecomplicatedconfigurations
part is connected with the spreading width of this con-
due to the residual interactions. The method has been
figuration,whichaccountsforthe fragmentationofthese
recently reexamined by Potel et al. [11], who have pro-
states into more complicated configurations. The con-
vided an efficient implementation of this idea. Here, we
nectionbetweenbothregimesbecomesmoretransparent
closelyfollowtheirformulation. Forthat,wefirstrewrite
within a dispersive formulation of the optical potential,
Eq. (4) in integral form
assuggestedlongagobyMahouxandSartor[12,13]and
∞ recently reexamined by severalgroups (see e.g. [14]).
ψ(0)(~k ,~r )= G (~r ,r~′ )ρ(~k ,r~′ )d3r , (5)
x b x Z x x x b x x
0
where ρ(~k ,r~′ ) = (χ(−)(~k ,~r )|V |Ψ3bi is the source IV. COMPARISON WITH EXPERIMENTAL
termoftheb inhxomogenbeousbEqb.(4)poasntdG (~r ,r~′ )isthe DATA
x x x
Green’s function
Inthissection,wecomparetheformalismwithexisting
1
G (~r ,r~′ )= g (r ,r′)Ymx∗(rˆ′ )Ymx(rˆ ), 6Li inclusive breakup data on different targets. The 6Li
x x x r r′ lx x x lx x lx x
x x lXxmx nucleus is treated in a two-cluster model (α+d), with α
(6) and d playing the roles of spectator and participant in
where g (r ,r′) satisfies the equation the IAV model, respectively.
lx x x
The elastic breakup (EBU) contribution of the inclu-
(E −K −U )g (r ,r′)=δ(r −r′). (7)
x x xA lx x x x x sive breakup cross section is evaluated with the CDCC
method[9],usingthecoupled-channelscodeFRESCO[15].
As usual, the solution of this equation is obtained from
the regular (f (r )) and irregular (h(+)(r )) solutions In this method, the breakup is treated as an inelastic
lx x lx x excitation to the continuum states of the projectile. Al-
ofthe correspondinghomogeneousequation. Fromthese
though four-body CDCC calculations for 6Li scattering
two solutions, g (r ,r′) can be expressed as
lx x x have become recently available [16], we rely here on the
g (r ,r′)=N f (r )h(+)(r ), (8) moreconventionalα+ddi-clustermodel. Thus,diagonal
lx x x lx lx < lx > and off-diagonal coupling potentials are generated from
where r is the lesser value of r and r′ and r is thed+targetandα+targetinteractions,evaluatedat2/3
< x x >
the larger one. The normalization constant N can be and1/3ofthe projectileincidentenergy,respectively. In
lx
4
60 tials are taken from Refs. [21] and [22], respectively. To
C.Signorini 60 33 MeV
mb/sr)40 GNEBE.RUB.Kelly 29 MeV 40 itmerpmroovfetthheeimreapgriondaurcytipoanrtofofththeeelda+st2i0c8Pdabtap,otthenetsiaulrfwaaces
Ω ( TBU removed. FortheNEBcalculations,theopticalpotential
σ/d20 20 of 6Li+208Pb is taken from Ref. [23].
d Figure 2 shows the comparison of the calculated and
0 0 experimental angular distributions of α particles pro-
0 50 100 150 0 50 100 150
duced in this reactionatthe measuredincident energies.
100 The squares and circles are the experimental data from
150
mb/sr) 6800 35 MeV 100 39 MeV Riseafns.a[1p9p]reacniadb[l2e0d],iffreersepneccetivbeeltyw.eIetnitsheevtiwdeonstetthsaotftdhaetrae.
σ/Ωd ( 40 50 TNhEeBd(aIsAhVedmaonddeld)ortetseudltlsin.eAssarine tthhee E6LBi+U2(0C9BDiCcCas)ea[n2]d,
d 20 the NEB is found to account for most of the inclusive
breakup cross section. The sum EBU+NEB (TBU) re-
0 0
0 50 100 150 0 50 100 150 producesreasonablywellthemagnitudeandshapeofthe
θ (deg) θ (deg)
lab lab data of Ref. [19], except for some overestimation for the
lowestenergies. Thus,ourcalculationsclearlyfavourthe
FIG. 2. (Color online) Angular distribution of α particles
produced in the reaction 6Li+208Pb at the incident ener- data presented in Ref. [19] over those presented in [20].
gies indicated by the labels. The dotted, dashed and solid From the results shown here and in Ref. [2], it can
linescorrespondtotheNEB(IAVmodel),EBU(CDCC)and be concluded that the nonelastic breakup process is the
their sum (TBU), respectively. Experimental data are from dominant α−emitting channel in the 6Li induced reac-
Refs. [19, 20]. See text for details. tions on heavy targets. To investigate whether this con-
clusion is a general feature of 6Li induced reactions or it
holds only for heavy targets we extend our analysis to
lighter targets.
order to reproduce correctly the elastic scattering data,
CDCC calculations based on this two-body model typi-
callyrequiresomerenormalizationofthefragment-target
B. 159Tb (6Li, αX)
potentials [16, 17]. This has been recently found to be
a consequence of the shortcomings of the two-body de-
scription of the 6Li nucleus, which results in an effective ThisreactionhasbeenmeasuredbyPradhanetal. [24]
at several energies between 23 MeV and 35 MeV.
suppressionofthedeuteron-targetabsorption[16]. Inour
In Ref. [24], the following processes were invoked to
previouswork[2], we found that this effect couldbe well
explain the observed α yields: (i) breakup of 6Li into α
simulated by removing the surface part of the deuteron-
and d fragments where both fragments escape without
target optical potential. In the calculations presented in
being captured by the target, referred to in some works
this work,we also allow for such kind of modification, in
as non-capture breakup; (ii) α particles resulting from d
order to reproduce correctly the elastic scattering data.
capture by the target (deuteron incomplete fusion), fol-
For the α+d potential, we use the potential model
lowing the breakup of 6Li into α and d or a deuteron
fromRef.[18],whichcontainsbothcentralandspin-orbit
transfer to the target; (iii) single-proton stripping from
terms, with the latter requiredto place correctlythe ℓ=
6Litoproducetheunbound5Henucleusthatdecaysinto
2 resonances.
anαparticleandaneutron;(iv)single-neutronstripping
For the non-elastic breakup calculations, we rely also
from 6Li to produce 5Li, which will subsequently decay
onaα+dmodel,butthespinofthedeuteronisignored,
into an α+p; and (v) single-neutron pickup from 6Li to
since our current implementation of the IAV model ig-
produce 7Li, which breaks into an α particle and a tri-
nores the intrinsic spin of the fragments. This approxi-
ton if 7Li is excited above its breakup threshold of 2.468
mation was also used in our previous works [2–4].
MeV.InRef.[24]theseprocessesweretreatedseparately,
using several reaction formalisms and their sum reason-
ably reproduced the total α−particle cross sections, but
A. 208Pb (6Li, αX) not their angular distributions.
Within the inclusive breakupmodel adopted here, the
First, the results for the reaction 208Pb(6Li,αX), at processes discussed by Pradhan et al. [24] can be re-
several energies between 29 and 39 MeV are presented, defined as follows: process (i) can be divided into two
comparingwiththedatafromRefs.[19,20]. Thenominal parts. First, the non-capture breakup with the target
Coulombbarrierforthissystemisaround29.5MeV[19]. remaining in its ground state, i.e., EBU. Second, the
The CDCC calculations use the same structure model non-capture breakup accompanied by target excitation,
and bin discretization as in our previous calculations for which we call inelastic breakup and is part of our non-
6Li+209Bi[2]. Thed−208Pbandα−208Pbopticalpoten- elastic breakup cross section; processes (ii)-(iv) may be
5
1.2 30
40
23 MeV 25 MeV
b/sr) 20 30
0.9 m
R Ω ( 20
σσ/d 0.6 σd/d 10 10
d
0 0
0 50 100 150 0 50 100 150
22.1 MeV
0.3 60
27 MeV 30 MeV
0 50 100 150 sr)
b/ 40 100
m
1.2 Ω (
Figueira et al. σ/d 20 50
0.9 d
OM
R CDCC 00 50 100 150 00 50 100 150
σ
d 0.6 θ (deg)
σ/ lab
d sr) 150 35 MeV EBU (CDCC)
0.3 b/ NEB (FR-DWBA)
m
35.1 MeV Ω ( 100 TBU
00 50 100 150 σ/dd 50 159Tb(6Li,αX)
θ (deg)
c.m. 0
0 50 100 150
θ (deg)
FIG.3. (Coloronline)Elasticscatteringof6Li+144Smat22.1 lab
MeV (top) and 35.1 MeV (bottom). The solid and dashed
lines are, respectively, theCDCC calculation and the optical FIG.4. (Coloronline)Angulardistributionofαparticlepro-
model calculation with the optical potential from [23]. Ex- duction of the reaction 6Li+159Tb at the incident energies
perimental data are from Ref. [25] . indicated by the labels. The dashed, dotted and solid lines
areEBU calculated with CDCC, NEBcalculated with finite-
rangeDWBAandtheirsum(TBU),respectively. Theexper-
imental data are taken from Ref. [24]
alsoembeddedintheNEBpart,inwhichthedeuteronis
absorbed by the target or it breaks up into p+n follow-
ing the breakup of 6Li into α and d; it can also happen in the previous paragraph. For the NEB calculation,
that after the breakup of 6Li, the deuteron picks a neu- the same optical potentials α/d+159Tb were used. The
trontobecomeatritium,contributingtotheprocess(v). Cook potential [23] was used to calculate the distorted
Processes (ii)-(v) as well as the inelastic breakup can be wave of the incoming channel.
consideredasnonelasticbreakupandshouldbetherefore In Fig. 4 the calculated and experimental angular dis-
accounted by the IAV formalism. tributionsofαparticlesarecomparedforseveralincident
Elastic data for this reaction are not available. Thus, energiesof6Li. ThedashedanddottedlinesaretheEBU
the CDCC calculation is tested against the data for (CDCC) and NEB (IAV model) results. The summed
the nearby system 6Li+144Sm [25]. The α+144Sm and EBU + NEB cross sections (solid lines) reproduce fairly
d+144Sm optical potentials were taken from Refs. [26] well the shape and magnitude of the data, except for a
and [21], respectively. The optical model calculation us- slight overestimation at some energies. Similarly to the
ingthepotentialofCook[23](dashedlines)isalsoshown. heavy-target systems, i.e., 6Li+209Bi [2] and 6Li+208Pb
Itcanbe seenthatthe CDCC resultissimilarto the op- (Sec.IVA), the NEB is found to accountformost ofthe
tical model calculation, particularly at E = 35.1 MeV. inclusive breakup cross section.
At this energy, the calculations reproduce very well the
elastic data. For the lowerenergy(E =22.1MeV), both
calculations underestimate the data at backwardangles. C. 118Sn (6Li, αX)
Note that, in contrast to the 6Li+208Pb case, no appar-
ent modification of the deuteron potential was required Inclusivebreakupdataforthe118Sn(6Li,αX)reaction
in this case. are available in Ref. [27] at energies between 18 and 24
Now the inclusive breakup cross sections MeV. The optical model parameterizations of Refs. [26]
159Tb(6Li,αX) are discussed. The EBU contribu- and [21] are used for the α−118Sn and d−118Sn sys-
tionwasobtainedfromthe CDCCcalculationsdiscussed tems. For the NEB calculations, the optical potential
6
16
1 1 EBU 18 MeV 20 19 MeV
sr) 12 NEB
σR 0.8 mb/ TBU
σ/d 0.8 Ω ( 8 10
d d
0.6 18 MeV CDCC 0.6 19 MeV σ/d 4
OM
0 0
0.4 0 50 100 150 0 50 100 150
0 50 100 150 0 50 100 150
30
20 MeV 30 21 MeV
1 1 sr)
b/ 20
R m 20
σd Ω (
σ/ 0.5 0.5 d 10
d σ/ 10
d
20 MeV 21 MeV
0 0
0 0 0 50 100 150 0 50 100 150
0 50 100 150 0 50 100 150
45 22.5 MeV 60 24 MeV
sr)
1 1 b/
m 30 40
σR Ω (
σ/dd 0.5 0.5 σ/dd 15 20
22.5 MeV 24 MeV
0 0
0 50 100 150 0 50 100 150
0 0 θ (deg) θ (deg)
0 50 100 150 0 50 100 150 lab lab
θ (deg) θ (deg)
lab lab
FIG.6. (Coloronline)Angulardistributionofαparticlespro-
FIG. 5. (Color online) Elastic scattering of 6Li+118Sn at dif- ducedinthereaction6Li+118Snattheincidentenergiesindi-
ferent incident energies. The solid and dashed lines are, re- catedbythelabels. Thedotted,dashedandsolidlinescorre-
spectively, the CDCC calculation and the optical model cal- spondtotheNEB(IAVmodel),EBU(CDCC)andtheirsum
culation with the optical potential from [27]. Experimental (TBU), respectively. Experimental data are from Ref. [27].
data are from Ref. [27].
Elastic data are available at the somewhatsmaller en-
of 6Li+118Sn is taken from Ref. [27]. ergy E = 18 MeV [28] so we first compare these data
lab
In Fig. 5 we compare the elastic data with the CDCC with the optical model and CDCC calculations. For the
(solid lines) and optical model (dashed lines) calcula- former,weemployedtheglobalopticalpotentialofCook
tions. Overall, both types of calculations reproduce well [23]. For the CDCC calculations, the optical potentials
the data, with some discrepancy observed at 18 and 21 for α+59Co and d+59Co were taken from Refs. [26] and
MeV. [21],respectively. The resultsareshowninFig.7. It can
Figure 6 shows the comparison of the calculated and be seen that both the CDCC and optical model calcula-
experimental angular distributions of α particles pro- tions reproduce fairly well the data. We notice that no
duced in this reaction, for several incident energies. renormalizationofthedeuteronpotentialwasrequiredin
Again, the NEB part (dotted lines) accounts for most of this case.
theinclusivebreakupcrosssectionandtheEBU(dashed Theexperimentalandcalculatedangulardistributions
lines) becomes the dominant breakup mode for angles of inclusive α particles are shown in Fig. 8. The NEB is
smaller than ∼50 degrees. The summed EBU + NEB seen to dominate the inclusive α production. It should
result (solid line) reproduces remarkably well the shape be noticed that, in this case, the NEB part includes also
and magnitude of the data. thetransferpopulatingboundstatesofthetarget,which
was obtained using the formalism discussed in Sec. III.
A more detailed discussionof this contributionis left for
D. 59Co (6Li, αX) Sec. V. The total cross section, TBU= EBU + NEB,
reproduces well the shape of the experimental data, al-
Experimental data for the α-production channel for thoughthemagnitudeisunderestimatedby∼30%atthe
the reaction 6Li+59Co have been reported by Souza et maximum. Thismightindicatethepresenceofotherrele-
al. [29]atE =21.5MeV,whichis abovethe Coulomb vantmechanismsleadingtotheproductionofαparticles
lab
barrier (V =12 MeV). in this reaction, such as the formation of a compound
B
7
V)20 20
e
1 6Li+59Co @ 18 MeV b/sr/M θα=15o θα=25o
m10 10
(
Ω
d
Souza et. al. E
d
σdR CODMCC 2σd/ 00 10 20 30 00 10 20 30
σd/ 0.5 eV)30 30
M θ =35o θ =45o
b/sr/ α α
m15 15
(
Ω
d
E
d
0 30 60 90 120 150 2σd/ 00 10 20 30 00 10 20 30
θc.m. (deg) eV)20 10
FIG. 7. (Color online) Elastic scattering of 6Li+59Co at an mb/sr/M10 θα=55o 5 θα=65o
incident energy of 18 MeV. The solid and dashed lines are, Ω (
d
respectively,theCDCCcalculationandtheopticalmodelcal- dE
culation with the optical potential from [23]. Experimental 2σd/ 00 10 20 30 00 10 20 30
data are from Ref. [28]. V) Elab (MeV)
Me 6 θ =75o α
ΩEd (mb/sr/ 42 αENTBBEUUB 59Co(6Li,αX)@21.5MeV
150 59Co(6Li,αX) @ 21.5 MeV 2σd/d 00 10 20 30
mb/sr)100 SEoBuUza et al. Eαlab (MeV)
Ω ( NEB FIG.9. (Color online) Experimentalandcalculated inclusive
d TBU α energy spectra for E =21.5 MeV, at selected scattering
σ/ lab
d angles. The dashed, dotted, and solid lines are respectively
50
theEBU (CDCC), NEB(IAV model) and theirsum. Exper-
imental data are taken from Ref. [29].
0
0 30 60 90 120 150
θ (deg)
lab
target and are well reproduced by our calculations.
FIG. 8. (Color online) Angular distribution of α particles
produced in the reaction 6Li+59Co at an incident energy of
21.5 MeV. The dashed, dotted, and solid lines are, respec-
E. 58Ni (6Li, αX)
tively, the EBU (CDCC), NEB(IAV model) and their sum.
Experimental data are taken from Ref. [29].
The α production of the 6Li + 58Ni reaction at sev-
eral incident energies between 12 MeV and 20 MeV was
measured by Pfeiffer et al. [27]. Elastic scattering data,
nucleus followed by α evaporation. In fact, statistical whichwerealsomeasured,arecomparedwithCDCCand
modelcalculationsperformedinRef.[29]predictedasig- OM calculations in Fig. 10 (note that the angles and
nificant amount of α particles coming from this channel. cross sections are referred to the laboratory frame, as in
The evaluation of this contribution is beyond the scope the original reference). For the former, we use the same
of the present work. optical potentials as in the nearby 6Li+59Co case. For
Theenergyspectraforselectedαscatteringanglesare the OM calculations we use the global OM potential by
alsoavailableforthis reaction. Thesearecomparedwith Cook [23]. Both calculations reproduce rather well the
ourcalculationsin Fig.9, witheachpanelcorresponding data, although the CDCC calculations slightly underes-
to a given α scattering angle, as indicated by the labels. timates the data at large angles.
Except at θ = 15◦, the sum of EBU and NEB repro- We present now the inclusive alpha cross sections.
lab
ducesthepeakoftheαenergydistribution. However,the For the NEB calculation, the 6Li optical potential from
low-energy tail is clearly underestimated. At these ener- Ref. [23] was used. Figure 11 shows the comparison of
gies,the main contributionof the inclusive α production the calculated and experimental angular distributions of
may arise from compound nucleus followed by evapora- αparticlesproducedinthis reaction,forseveralincident
tionandpre-equilibrium,whicharenotconsideredinthe energies. Notice that the NEB (dotted lines) includes
present calculations. We note that high energy α par- also the contribution coming from the transfer to target
ticles stem from a deuteron transfer mechanism to the bound states. Again, the NEB part dominates the in-
8
15 30
12 MeV 14 MeV
1 1 12 MeV 14 MeV
σdR mb/sr) 10 20
σd/ 0.5 Ω (
0.5 σ/d 5 10
d
0
0 50 100 150 0 50 100 150 0 0
0 50 100 150 0 50 100 150
1 16 MeV 1 18 MeV 60 16 MeV 18 MeV
80
σR b/sr) 60
d m 40
σd/ 0.5 0.5 Ω ( 40
d
σ/ 20
d 20
0 0
0 50 100 150 0 50 100 150 0 0
θ (deg) 0 50 100 150 0 50 100 150
lab θ (deg)
lab
20 MeV
1 150 20 MeV
σR PCfDeCiffCer et al. b/sr) ENBEUB
σd/d 0.5 OM Ω (m 100 TBU
d
6Li+58Ni σd/ 50 58Ni(6Li,αX)
0
0 50 100 150 0
θ (deg) 0 50 100 150
lab θ (deg)
lab
FIG.10. (Coloronline)Elasticscatteringof6Li+58Niatsev-
FIG. 11. (Color online) Angular distribution of α particles
eral energies indicated by the labels. The solid and dashed
produced in the reaction 6Li + 58Ni at the incident energies
lines are, respectively, theCDCC calculation and the optical
indicated by the labels. The dashed, dotted and solid lines
model calculation with the optical potential from [27]. Ex-
are, respectively, the EBU, NEB and their sum (TBU). Ex-
perimental data are from Ref. [27].
perimental data are from Ref. [27].
V. TRANSFER CONTENT OF THE NEB
CROSS SECTION
clusive α production. In general, the summed EBU + Therelativeimportanceofthetransfertoboundstates
NEBcrosssection(solidlines)reproduceswellthe shape within the NEB cross section will depend on several pa-
and magnitude of the data. At 16, 18 and 20 MeV some rameters,such as the projectile incident energy a nd the
underestimation is observed, which might be associated chargeofthetargetnucleus. Forheavytargets,thetrans-
with other α-production channels, as pointed out in the fer channel is suppressed due to the strong Coulomb in-
6Li+59Co case. teraction between the deuteron and the target, whereas
for light targets this channel is expected to play a more
important role.
From the results presented in the previous sections, This is illustrated in Fig. 12 for two such cases; the
we may conclude that the strong α-production channel upperpaneldisplaysthe calculated208Pb(6Li, αX) NEB
observedin 6Li experiments originates mostly from non- cross sections as a function of d-208Pb relative energy at
elastic breakup mechanisms. In all cases analyzed, the three different incident energies, 29 MeV, 35 MeV and
EBU mode turns out to account for a relatively small 39 MeV. The vertical dotted line indicates the nominal
fraction of the total inclusive alpha cross section and Coulombbarrierfor the d-208Pbsystem. The blacksolid
its contribution is only important for the alpha parti- curve is the reaction cross section for the d-208Pb sys-
cles emitted at small angles. For the lighter targets, we tem, arbitrarily normalized to fit within the same scale.
found also a indirect evidence of other alpha production The bottom panel shows similar curves for the 6Li+58Ni
mechanisms, such as fusion. reaction at 12, 16 and 20 MeV. In both cases, it can be
9
80 208Pb(6Li,αX)
)
V
e
M60
29 MeV
b/ 35 MeV
m
(40 39 MeV
E
d
σ/
d20
0
00 55 1100 1155 2200
208
E (d- Pb) (MeV)
c.m.
80 58Ni(6Li,αX)
V) FIG. 13. (Color online) Contour plots for the double differ-
Me60 12 MeV ential cross section (upper panels) and the angle-integrated
b/ 16 MeV enery differential cross section as a function of the outgo-
m 20 MeV ing α energy in the c.m. frame (lower panels) for the reac-
E (40 tions: (a) 6Li+208Pb, (b) 6Li+159Tb, (c) 6Li+118Sn and (d)
d 6Li+59Co. The vertical lines indicate the breakup threshold
σd/20 for the d+target system (Ex =0).
0
--55 00 55 1100 1000
58
E (d- Ni) (MeV)
c.m.
FIG.12. (Coloronline)Top: NEBcrosssectionasafunction
of thed–208Pb relative energy in thec.m.frame for thereac-
tion6Li+208Pb. Theverticaldottedlineindicatestheenergy
of the Coulomb barrier for the d+208Pb reaction. The solid U
lineisthereaction cross section ford+208Pb,arbitrarily nor- TBσα 100 6Li+58Ni
malized. Bottom: same as in top panel but for the 6Li+58Ni 6Li+59Co
system. 6Li+118Sn
6Li+159Tb
6Li+208Pb
6Li+209Bi
seen that the NEB is a Trojan Horse type process [30],
10
which means that the 6Li projectile brings the deuteron 0.8 1 1.2 1.4 1.6
E /V
inside the Coulomb barrier and let it interact with the c.m. B
targetnucleus,givingasizablecrosssectionsfordeuteron
FIG. 14. (Color online) Inclusive breakup α cross sections
energies for which the reaction cross section has already
involving 6Li projectile with several targets as a function of
become negligibly small. For the 208Pb target, due to
thestrongCoulombrepulsion,theNEBcrosssectionbe- Ec.m./VB.
comesnegligibleatnegatived-208Pbrelativeenergiesand
this behavior is independent of the incoming 6Li energy.
Bycontrary,forthe58Nitarget,thereisalowenergytail relevant.
extending to negative deuteron energies (transfer).
Weexpectalsosomecorrelationbetweentheα-particle
angularandenergydistribution. ThisisshowninFig.13 VI. SYSTEMATICS OF INCLUSIVE α
in the form of contour plots of double differential cross PRODUCTION
sections and angle-integrated cross section as a function
of the outgoing α energy in the c.m. frame for the reac- Systematic studies of α production yields in 6Li re-
tions (a) 6Li+208Pb, (b) 6Li+159Tb, (c) 6Li+118Sn and actions show an interesting universal behaviour when
(d)6Li+59Co. Itcanseenthatthe mostenergeticαpar- plotted as a function of the incident energy scaled by
ticles are preferably emitted at forward angles, whereas the Coulomb barrier energy as reported for instance by
thosewithlowerenergiescontributetobothforwardand Pakou et al. [31]. In this section, we will investigate
backward angles. Moreover, when the charge of the tar- whether our calculations exhibit also this universal be-
get is small (59Co), the transfer channel becomes more haviour. For this study, we have considered the target
10
1
6 58
Li+ Ni
0.4 6Li+59Co 6Li+58Ni
6Li+118Sn 0.8 6Li+59Co
0.3 6Li+159Tb 6Li+118Sn
U 6 208 6Li+159Tb
B Li+ Pb
σ / UT0.2 6Li+209Bi σR0.6 66LLii++220089PBbi
σEB TBU/α
σ
0.4
0.1
0.2
0
0.8 1 1.2 1.4 1.6
E /V
c.m. B
0
0.8 1 1.2 1.4 1.6
FIG. 15. (Color online) Ratios of calculated EBU over TBU E /V
c.m. B
(= EBU + NEB) for different systems. Seetext for details.
FIG. 16. (Color online) Ratios of calculated TBU (= EBU
+ NEB) α cross sections over the reaction cross section for
the systems and energies analyzed in this work. See text for
systems 59Co, 118Sn, 159Tb, 208Pb, which have been an- details.
alyzed in the preceding sections, and 209Bi, analyzed in
Ref. [2]. The results are shown in Fig. 14, where we plot
the calculated σTBU cross sections as a function of the
α section. To address this question, we plot in Fig. 16 the
reduced energy (E /V ), with V the energy of the
c.m. B B ratioofthecalculatedTBUandreactioncrosssectionsas
Coulomb barrier, estimated as VB =ZpZte2/(rB(A1p/3+ a function of the reduced energy Ec.m./VB, for the sys-
A1/3)),whereZ (Z )andA (A )aretheatomicnumber tems studied in this work. Several interesting features
t p t p t
and atomic mass of the projectile (target), respectively, emerge from this plot: (i) first, for all systems analyzed
and r = 1.44 fm. As expected, the breakup cross sec- the ratio decreases smoothly as the incident energy in-
B
tiondropsquicklyastheincidentenergydecreasesbelow creases; (ii) second, the heavier the target nucleus, the
thebarrier. Thiseffectisenhancedforthe209Binucleus, larger the ratio. For example, for the 208Pb and 209Bi
possiblyduetothelargerCoulombrepulsion. Abovethe target nuclei the ratio exceeds 80% at sub-Coulomb en-
barrier, the medium-heavy and heavy targets the inclu- ergies. Result (i) may be understood as a consequence
sive breakupcrosssections show a similartrend, but not of the competition with other channels which will open
for the medium mass targets 58Ni and 59Co at largeren- and increase their importance as the incident energy in-
ergies. We recall howeverthat, for these lighter systems, creases,suchasotherbreakupmodesnotassociatedwith
theremightbeadditionalcontributionsfromotherchan- the production of α particles (e.g. 3H+3He), target ex-
nels, such as compound nucleus following evaporation, citationnot accompaniedby projectile breakup, neutron
which are not accounted for by the IAV formalism. pickup from the target, etc.
We have also studied the relative importance of EBU
versus NEB as a function of the incident energy. For
that, we display in Fig. 15 the ratio of EBU over TBU VII. SUMMARY AND CONCLUSIONS
(= EBU + NEB) for the analyzed systems. It is seen
that, for incident energies below the Coulomb barrier, To summarize, we have performed a comprehensive
the elastic breakup cross section becomes comparatively analysisofinclusivebreakupcrosssectionsin6Li-induced
more important as the energy decreases. This can be reactions with the aim of understanding the experimen-
attributedtothefactthat,belowthebarrier,thebreakup tally observed α yields. For that, we have calculated
takesplaceatlargeprojectile-targetseparations,andthe separately the EBU and NEB contributions using the
deuteron absorption (responsible for the NEB part) will CDCC method (for the EBU part) and the closed-form
be less important [4]. By contrast, for energies above model proposed by Ichimura, Austern and Vincent IAV
the Coulombbarrier,the ratioshowsanalmostconstant theory(fortheNEBpart). Forthelattermodel,weused
behavior. It can also be seen that, while for the heavy the DWBA approximation, including finite-range effects
mass targets elastic breakup plays an important role in and the remnant term of the transition operator.
theinclusiveαproduction,especiallybelowtheCoulomb Overall, the calculations show a very good agreement
barrier, for the medium mass targets elastic breakup is with the available data, providing a consistent and neat
less important and the nonelastic breakup is dominant. explanation of the large α yields reported over the years
Another relevant question regards the fraction of the for 6Li reactions, without the need of evaluating the in-
reaction cross section that is exhausted by the α cross dividual channels contributing to the inclusive cross sec-