PHYSICAL REVIEW C 95, 044605 (2017)
Comp ehensi e analysis o la ge αyields obse ed in 6Li-induced eac ions
Jin Lei*and An onio M. Mo o†
Depa amen o de FAMN, Uni e sidad de Se illa, Apa ado 1065, 41080 Se illa, Spain
(Recei ed 10 Janua y 2017; published 6 Ap il 2017)
Backg ound: La ge αyields ha e been epo ed o e he yea s in eac ions wi h 6Li and 7Li p ojec iles. P e ious
heo e ical analyses ha e shown ha he elas ic b eakup (EBU) mechanism (i.e., p ojec ile b eakup lea ing he
a ge in i s g ound s a e) is able o accoun only o a small ac ion o he o al α-inclusi e b eakup c oss
sec ions, poin ing owa d he dominance o nonelas ic b eakup (NEB) mechanisms.
Pu pose: We aim o p o ide a sys ema ic s udy o he α-inclusi e c oss sec ions obse ed in nuclea eac ions
induced by 6Li p ojec iles. In addi ion o es ima ing he o al αsingles’ c oss sec ions, i is ou goal o e alua e
angula and ene gy dis ibu ions o hese αpa icles and compa e hem wi h expe imen al da a, when a ailable.
Me hod: We compu e sepa a ely he EBU and NEB componen s o he inclusi e b eakup c oss sec ions. Fo
he o me , we use he con inuum-disc e ized coupled-channels (CDCC) me hod, which ea s his mechanism
o all o de s. Fo he NEB pa , we employ he model p oposed by Ichimu a e al. [Phys. Re . C 32,431 (1985)],
wi hin he dis o ed-wa e Bo n app oxima ion (DWBA).
Resul s: O e all, he sum o he compu ed EBU and NEB c oss sec ions is ound o ep oduce e y well he
measu ed singles’ c oss sec ions. In all cases analyzed, we ind ha he inclusi e b eakup c oss sec ion is la gely
domina ed by he NEB componen .
Conclusions: The p esen ed me hod p o ides a global and sys ema ic desc ip ion o inclusi e b eakup eac ions
induced by 6Li p ojec iles. I p o ides also a na u al explana ion o he p e iously obse ed unde es ima ion o
he measu ed αyields by CDCC calcula ions. The me hod used he e can be ex ended o o he weakly bound
p ojec iles, including halo nuclei.
DOI: 10.1103/PhysRe C.95.044605
I. INTRODUCTION
Reac ions induced by he 6Li nucleus ha e been ex ensi ely
s udied, gi ing ise o a la ge body o expe imen al da a.
Gi en i s ma ked α+ds uc u e, wi h a sepa a ion ene gy o
1.474 MeV ( o be compa ed wi h he single-nucleon sepa a ion
ene gy o 5.39 MeV), one may an icipa e ha he b eakup o
his nucleus in o αand dis a majo eac ion channel. In ac ,
expe imen al da a show ema kably la ge yields o αpa icles
bu , con a y o expec a ions, hese yields a e ypically much
la ge han he co esponding dyields. This sugges s ha he
b eakup o he 6Li is no a simple di ec b eakup mechanism.
F om he heo e ical poin o iew, a p ope in e p e a ion
o hese αyields is s ill lacking. Con inuum-disc e ized
coupled-channels (CDCC) calcula ions, which ea he 6Li
b eakup as an inelas ic exci a ion o he con inuum, ep oduce
success ully he coincidence α+dmeasu emen s [1]bu hey
la gely unde es ima e he inclusi e αc oss sec ions. I is
wo hwhile ecalling ha he CDCC me hod p o ides only
he so-called elas ic b eakup (EBU) componen o he o al
b eakup c oss sec ion. Fo he eac ion o a 6Li p ojec ile
impinging on a a ge A, his co esponds o he p ocesses o
he o m 6Li +A→α+d+Ag.s.in which he wo-p ojec ile
clus e s su i e a e he collision and he a ge emains in
*P esen add ess: Ins i u e o Nuclea and Pa icle Physics, and
Depa men o Physics and As onomy, Ohio Uni e si y, A hens,
Ohio 45701, USA; [email p o ec ed]
†[email p o ec ed]
he g ound s a e.1Thus, he unde es ima ion o he inclusi e
αyields by he CDCC calcula ions means ha he e o he
mechanisms con ibu ing o he inclusi e b eakup c oss sec ion
o he han he EBU. These include he exchange o nucleons
be ween dand A, he p ojec ile dissocia ion accompanied by
a ge exci a ion, and he usion o dby A, among o he s, ha
we will globally deno e as nonelas ic b eakup (NEB) channels.
An explici accoun o hese p ocesses is e y challenging due
o he huge numbe o accessible inal s a es and he a ie y o
compe ing di e en mechanisms.
When one is only in e es ed in he e alua ion o he singles’
c oss sec ion ( o example, he ene gy o angula dis ibu ion
o αpa icles), a he han on he sepa a e con ibu ing
mechanisms, one may eso o he inclusi e b eakup models
p oposed in he 1980s and ecen ly eexamined by se e al
g oups [2–6]. In hese models, he sum o e all he possible
inal s a es h ough which he unobse ed agmen dmay
in e ac wi h he a ge is done in a o mal way, making use o
he Feshbach p ojec ion o malism [7] and closu e.
In his wo k, we will show ha inclusi e αsingles c oss
sec ions om 6Li-induced eac ions can be ema kably well
ep oduced using he inclusi e b eakup model p oposed by
Ichimu a, Aus e n, Vincen model (IAV) [8]. To ou knowl-
edge, his is he i s s udy o his kind p o iding a sys ema ic
explana ion o hese da a.
1I a h ee-body desc ip ion o he 6Li is used, α+p+n, he h ee-
body b eakup mode 6Li +A→α+p+n+Ag.s.would be also pa
o he elas ic b eakup channel. Since we eso he e o a wo-body
model o 6Li we include his channel in he NEB pa .
2469-9985/2017/95(4)/044605(11) 044605-1 ©2017 Ame ican Physical Socie y
JIN LEI AND ANTONIO M. MORO PHYSICAL REVIEW C 95, 044605 (2017)
Al hough he IAV model p o ides a common o malism
o he calcula ion o he elas ic and nonelas ic b eakup
componen s o he inclusi e b eakup c oss sec ion, in ou
analysis we will employ his model only o he NEB pa ,
whe eas o he EBU pa we will use he con inuum-
disc e ized coupled-channels (CDCC) me hod, which ea s
b eakup o all o de s.
The pape is o ganized as ollows. In Sec. II we gi e a
sho o e iew o he IAV heo y, highligh ing only i s main
o mulas. In Sec. III he ex ension o he o malism o nega i e
deu e on ene gies (bound s a es) is discussed. In Sec. IV, he
o malism is applied o desc ibe he αc oss sec ions in se e al
6Li-induced eac ions, compa ing wi h he a ailable da a. In
Sec. V he ole o he ans e channels on he NEB c oss
sec ion is discussed. In Sec. VI we in es iga e he sys ema ic
beha io o he inclusi e c oss sec ion wi h espec o he
inciden ene gy and o all analyzed a ge s. Finally, in Sec. VII
we summa ize he main esul s o his wo k.
II. THE ICHIMURA, AUSTERN, AND
VINCENT (IAV) MODEL
In his sec ion, we b ie ly summa ize he model o Ichimu a,
Aus e n, and Vincen (IAV), whose o iginal de i a ion can be
ound in Re s. [8,9] and has been also ecen ly e isi ed by
se e al au ho s [2,3,5,6]. We ou line he e he main esul s o
his model and e e he eade o hese e e ences o u he
de ails on hei de i a ions.
We w i e he p ocess unde s udy in he o m
a(=b+x)+A→b+B∗,(1)
whe e he p ojec ile a, composed o band x, collides wi h a
a ge A, emi ing b agmen s and any o he agmen s. Thus,
B∗deno es any inal s a e o he x+Asys em.
This p ocess will be desc ibed wi h he e ec i e Hamil o-
nian
H=K+Vbx +UbA( b)+HA(ξ)+VxA(ξ, x),(2)
whe e Kis he o al kine ic ene gy ope a o , Vbx is he
in e ac ion binding he wo clus e s band xin he ini ial
composi e nucleus a,HA(ξ) is he Hamil onian o he a ge
nucleus (wi h ξdeno ing i s in e nal coo dina es), and VxA
and UbA a e he agmen – a ge in e ac ions. The ele an
coo dina es a e depic ed in Fig. 1.
FIG. 1. Coo dina es used in he nonelas ic b eakup calcula ions.
In w i ing he Hamil onian o he sys em in he o m (2)we
make a clea dis inc ion be ween he wo clus e cons i uen s;
he in e ac ion o he agmen b, he one ha is assumed o
be obse ed in he expe imen , is desc ibed wi h a (complex)
op ical po en ial. Nonelas ic p ocesses a ising om his in-
e ac ion (e.g., a ge exci a ion) a e included only e ec i ely
h ough UbA. The pa icle bis said o ac as spec a o .On
he o he hand, he in e ac ion o he pa icle xwi h he a ge
e ains he dependence o he a ge deg ees o eedom (ξ).
S a ing om Hamil onian (2) IAV de i ed he ollowing
exp ession o he double di e en ial c oss sec ion o he NEB
wi h espec o he angle and ene gy o he b agmen s:
d2σ
dEbdbNEB
=− 2
¯h a
ρb(Eb)ψ(0)
x(
kb, x)Wxψ(0)
x(
kb, x),
(3)
whe e ais he p ojec ile- a ge ela i e eloci y, ρb(Eb)=
kbμb/((2π)3¯h2) is he densi y o s a es o he pa icle b,Wx
is he imagina y pa o he op ical po en ial desc ibing x+A
elas ic sca e ing, and ψ(0)
x(
kb, x)is heso-calledx-channel
wa e unc ion, which go e ns he e olu ion o xa e he
p ojec ile dissocia ion, when bsca e s wi h momen um 
kband
he a ge emains in he g ound s a e. This unc ion sa is ies
he ollowing inhomogeneous di e en ial equa ion
(Ex−Kx−UxA)ψ(0)
x(
kb, x)=(χ(−)
b(
kb, b)|Vpos |3b,(4)
whe e Ex=E−Eb,χ(−)
bis he dis o ed-wa e desc ibing
he sca e ing o bin he inal channel wi h espec o he
x+Asubsys em, and Vpos ≡Vbx +UbA −Ub(wi h Ub he
op ical po en ial in he inal channel) is he pos o m ansi ion
ope a o . The no a ion (|| indica es in eg a ion o e he  b
coo dina e only. This equa ion is o be sol ed wi h ou going
bounda y condi ions.
Aus e n e al. [9] sugges app oxima ing he h ee-body
wa e unc ion appea ing in he sou ce e m o Eq. (4), 3b,
by he CDCC one. Since he CDCC wa e unc ion is also
a complica ed objec by i sel , a simple choice is o use
he dis o ed-wa e Bo n app oxima ion (DWBA), i.e., ψ3b
x≈
χ(+)
a( a)φa( bx), whe e χ(+)
ais a dis o ed wa e desc ibing
a+Aelas ic sca e ing and φais he p ojec ile g ound-s a e
wa e unc ion.
The IAV model has been ecen ly e isi ed by se e al
g oups [2,5,6]. All he calcula ions pe o med so a by
hese g oups make use o he DWBA app oxima ion o
he incoming wa e unc ion. In Re s. [5,6], he heo y was
applied o deu e on-induced eac ions o he o m A(d,pX),
and in Re . [2] he model was ex ended o 6Li p ojec iles,
p esen ing a i s applica ion o he 209Bi(6Li,αX) eac ion. In
gene al, he ag eemen wi h he da a has been ound o be e y
encou aging, al hough u he compa isons wi h expe imen al
da a a e ad isable o be e assess he alidi y and unde s and
he limi a ions o he model.
III. EXTENSION OF IAV MODEL TO Ex<0
The so o b eakup c oss sec ion conside ed by IAV can be
ega ded as ans e o con inuum p ocess popula ing x+A
s a es wi h posi i e ela i e ene gy (Ex>0). In gene al,
he inclusi e c oss sec ion will con ain also con ibu ions
044605-2
COMPREHENSIVE ANALYSIS OF LARGE αYIELDS . . . PHYSICAL REVIEW C 95, 044605 (2017)
coming om he popula ion o s a es below he b eakup x+A
h eshold (Ex<0). Fo example, in a (6Li, αX) eac ion, he
α’s emi ed a he highe ene gies will ac ually co espond
o deu e on ans e o bound s a es o he a ge nucleus.
One would like o ha e a common amewo k o desc ibe
ans e o con inuum s a es as well as o bound s a es.
The explici inclusion o all possible inal bound s a es is
imp ac ical because o hei la ge numbe and he unce ain ies
in hei spin-pa i y assignmen s and spec oscopic ac o s.
An al e na i e p ocedu e was p oposed by Udagawa and
cowo ke s [10]. The key idea is o ex end he complex po en ial
o nega i e ene gies. Then, he bound s a es o he sys em a e
simula ed by he eigens a es in his complex po en ial. The
imagina y pa will be associa ed wi h he sp eading wid h o
he single-pa icle s a es, which accoun s o he agmen a ion
o hese s a es in o mo e complica ed con igu a ions due
o he esidual in e ac ions. The me hod has been ecen ly
eexamined by Po el e al. [11], who ha e p o ided an e icien
implemen a ion o his idea. He e, we closely ollow hei
o mula ion. Fo ha , we i s ew i e Eq. (4) in in eg al o m
ψ(0)
x(
kb, x)=∞
0
Gx( x,

x)ρ(
kb,

x)d3 
x,(5)
whe e ρ(
kb,

x)=(χ(−)
b(
kb, b)|Vpos |3bis he sou ce e m
o he inhomogeneous Eq. (4) and Gx( x,

x) is he G een’s
unc ion
Gx( x,

x)=1
x 
x
lxmx
glx( x, 
x)Ymx∗
lx(ˆ

x)Ymx
lx(ˆ
x),(6)
whe e glx( x, 
x) sa is ies he equa ion
(Ex−Kx−UxA)glx( x, 
x)=δ( x− 
x).(7)
As usual, he solu ion o his equa ion is ob ained om
he egula [ lx( x)] and i egula [h(+)
lx( x)] solu ions o
he co esponding homogeneous equa ion. F om hese wo
solu ions, glx( x, 
x) can be exp essed as
glx( x, 
x)=Nlx lx( <)h(+)
lx( >),(8)
whe e <is he lesse alue o xand 
xand >is he la ge one.
The no maliza ion cons an Nlxcan be ound by in eg a ing
Eq. (7) o e an in ini esimal in e al a ound 
x
2μx
¯h2= 
x+δ

x−δ
d x
d2
d 2
x
glx( x, 
x)
=d
d x
glx( x, 
x)

x+δ

x−δ
=Nlx lx( 
x)d
d x
h(+)
lx( 
x+δ)
−h(+)
lx( 
x)d
d x
lx( 
x−δ)
δ→0
−−→ NlxW lx( 
x),h(+)
lx( 
x),(9)
whe e Wdeno es a W onskian, which is independen o he
alue o 
x.
I is wo h no ing ha he in eg al o m o he x-channel
wa e unc ion (5) can be also be used o posi i e x−A
ene gies. P oceeding in his way, he applica ion o he
IAV o malism o posi i e and nega i e ene gies is o mally
analogous. Despi e his o mal simili ude, he in e p e a ion
o he channel unc ion and o he unde lying imagina y pa
o he po en ial is somewha di e en in bo h egions. Fo
Ex>0 he channel unc ion ψ(0)
xdesc ibes x−Aelas ic
sca e ing and he imagina y pa is he e o e associa ed wi h
he lux lea ing his channel in a o o nonelas ic channels.
Fo Ex<0, he channel wa e unc ion desc ibes he mo ion
o he xpa icle in a bound single-pa icle con igu a ion
s a e o he esidual nucleus, and he imagina y pa is
connec ed wi h he sp eading wid h o his con igu a ion,
which accoun s o he agmen a ion o hese s a es in o
mo e complica ed con igu a ions. The connec ion be ween
bo h egimes becomes mo e anspa en wi hin a dispe si e
o mula ion o he op ical po en ial, as sugges ed long ago by
Mahaux and Sa o [12,13] and ecen ly eexamined by se e al
g oups (see, e.g., Re . [14]).
IV. COMPARISON WITH EXPERIMENTAL DATA
In his sec ion, we compa e he o malism wi h exis ing 6Li
inclusi e b eakup da a on di e en a ge s. The 6Li nucleus
is ea ed in a wo-clus e model (α+d), wi h αand d
playing he oles o spec a o and pa icipan in he IAV model,
espec i ely.
The elas ic b eakup (EBU) con ibu ion o he inclu-
si e b eakup c oss sec ion is e alua ed wi h he CDCC
me hod [9], using he coupled-channels code FRESCO [15].
In his me hod, he b eakup is ea ed as an inelas ic exci a ion
o he con inuum s a es o he p ojec ile. Al hough ou -body
CDCC calcula ions o 6Li sca e ing ha e become ecen ly
a ailable [16], we ely he e on he mo e con en ional α+d
di-clus e model. Thus, diagonal and o -diagonal coupling
po en ials a e gene a ed om he d+ a ge and α+ a ge
in e ac ions, e alua ed a 1/3 and 2/3 o he p ojec ile inciden
ene gy, espec i ely. In o de o ep oduce co ec ly he elas ic
sca e ing da a, CDCC calcula ions based on his wo-body
model ypically equi e some eno maliza ion o he agmen -
a ge po en ials [16,17]. This has been ecen ly ound o be a
consequence o he sho comings o he wo-body desc ip ion
o he 6Li nucleus, which esul s in an e ec i e supp ession o
he deu e on- a ge abso p ion [16]. In ou p e ious wo k [2],
we ound ha his e ec could be well simula ed by emo ing
he su ace pa o he deu e on- a ge op ical po en ial. In
he calcula ions p esen ed in his wo k, we also allow o such
kind o modi ica ion, in o de o ep oduce co ec ly he elas ic
sca e ing da a.
Fo he α+dpo en ial, we use he po en ial model om
Re . [18], which con ains bo h cen al and spin-o bi e ms,
wi h he la e equi ed o place co ec ly he =2 esonances.
Fo he nonelas ic b eakup calcula ions, we ely also on a
α+dmodel, bu he spin o he deu e on is igno ed, since ou
cu en implemen a ion o he IAV model igno es he in insic
spin o he agmen s. This app oxima ion was also used in
ou p e ious wo ks [2–4].
044605-3
JIN LEI AND ANTONIO M. MORO PHYSICAL REVIEW C 95, 044605 (2017)
0 50 100 150
0
20
40
60
dσ/dΩ (mb/s )
29 MeV
C.Signo ini
G.R.Kelly
NEB
EBU
TBU
0 50 100 150
0
20
40
60 33 MeV
0 50 100 150
θlab(deg)
0
20
40
60
80
100
dσ/dΩ (mb/s )
0 50 100 150
θlab(deg)
0
50
100
150
35 MeV 39 MeV
FIG. 2. Angula dis ibu ion o αpa icles p oduced in he eac-
ion 6Li +208Pb a he inciden ene gies indica ed by he labels. The
do ed, dashed, and solid lines co espond o he NEB (IAV model),
EBU (CDCC), and hei sum (TBU), espec i ely. Expe imen al da a
a e om Re s. [19,20]. See ex o de ails.
A. 208Pb (6Li, αX)
Fi s , he esul s o he eac ion 208Pb(6Li,αX), a se e al
ene gies be ween 29 and 39 MeV a e p esen ed, compa ed wi h
he da a om Re s. [19,20]. The nominal Coulomb ba ie o
his sys em is a ound 29.5 MeV [19]. The CDCC calcula ions
use he same s uc u e model and bin disc e iza ion as in
ou p e ious calcula ions o 6Li +209Bi [2]. The d−208Pb
and α−208Pb op ical po en ials a e aken om Re s. [21,22],
espec i ely. To imp o e he ep oduc ion o he elas ic da a,
he su ace e m o he imagina y pa o he d+208Pb
po en ial was emo ed. Fo he NEB calcula ions, he op ical
po en ial o 6Li +208Pb is aken om Re . [23].
Figu e 2shows he compa ison o he calcula ed and
expe imen al angula dis ibu ions o αpa icles p oduced in
his eac ion a he measu ed inciden ene gies. The squa es
and ci cles a e he expe imen al da a om Re s. [19,20],
espec i ely. I is e iden ha he e is an app eciable di e ence
be ween he wo se s o da a. The dashed and do ed lines a e
he EBU (CDCC) and NEB (IAV model) esul s. As in he
6Li +209Bi case [2], he NEB is ound o accoun o mos
o he inclusi e b eakup c oss sec ion. The sum EBU+NEB
(TBU) ep oduces easonably well he magni ude and shape
o he da a o Re . [19], excep o some o e es ima ion o he
lowes ene gies. Thus, ou calcula ions clea ly a o he da a
p esen ed in Re . [19] o e hose p esen ed in Re . [20].
F om he esul s shown he e and in Re . [2], i can be
concluded ha he nonelas ic b eakup p ocess is he dominan
α-emi ing channel in 6Li induced eac ions on hea y a ge s.
To in es iga e whe he his conclusion is a gene al ea u e o
6Li eac ions o i holds only o hea y a ge s, we ex end ou
analysis o ligh e a ge s.
050
100 150
0.3
0.6
0.9
1.2
dσ/dσR
Figuei a e al.
OM
CDCC
050
100 150
θ
c.m.
(deg)
0
0.3
0.6
0.9
1.2
dσ/dσR
22.1 MeV
35.1 MeV
FIG. 3. Elas ic sca e ing o 6Li +144Sm a 22.1 ( op) and
35.1 MeV (bo om). The solid and dashed lines a e, espec i ely, he
CDCC calcula ion and he op ical model calcula ion wi h he op ical
po en ial om Re . [23]. Expe imen al da a a e om Re . [25].
B. 159Tb (6Li, αX)
This eac ion has been measu ed by P adhan e al. [24]a
se e al ene gies be ween 23 and 35 MeV.
In Re . [24], he ollowing p ocesses we e in oked o
explain he obse ed αyields: (i) b eakup o 6Li in o αand
dwhe e bo h agmen s escape wi hou being cap u ed by
he a ge , e e ed o in some wo ks as noncap u e b eakup;
(ii) αpa icles esul ing om dcap u e by he a ge (deu e on
incomple e usion), ollowing he b eakup o 6Li in o αand d
o a deu e on ans e o he a ge ; (iii) single-p o on s ipping
om 6Li o p oduce he unbound 5He nucleus ha decays
in o an αpa icle and a neu on; (i ) single-neu on s ipping
om 6Li o p oduce 5Li, which will subsequen ly decay in o
an α+p; and ( ) single-neu on pickup om 6Li o p oduce
7Li, which b eaks in o an αpa icle and a i on i 7Li is
exci ed abo e i s b eakup h eshold o 2.468 MeV. In Re . [24],
hese p ocesses we e ea ed sepa a ely using se e al eac ion
o malisms and hei sum easonably ep oduced he o al
α-pa icle c oss sec ions, bu no hei angula dis ibu ions.
Wi hin he inclusi e b eakup model adop ed he e, he
p ocesses discussed by P adhan e al. [24] can be ede ined
as ollows: P ocess (i) can be di ided in o wo pa s. Fi s , he
noncap u e b eakup wi h he a ge emaining in i s g ound
s a e, i.e., EBU. Second, he noncap u e b eakup accompanied
by a ge exci a ion, which we call inelas ic b eakup and is pa
o ou nonelas ic b eakup c oss sec ion; p ocesses (ii)–(i ) may
be also embedded in he NEB pa , in which he deu e on is
abso bed by he a ge o i b eaks up in o p+n ollowing
he b eakup o 6Li in o αand d; i can also happen ha
a e he b eakup o 6Li, he deu e on picks a neu on o
become a i ium, con ibu ing o he p ocess ( ). P ocesses
044605-4
COMPREHENSIVE ANALYSIS OF LARGE αYIELDS . . . PHYSICAL REVIEW C 95, 044605 (2017)
(ii)–( ) as well as he inelas ic b eakup can be conside ed as
nonelas ic b eakup and should be he e o e accoun ed by he
IAV o malism.
Elas ic da a o his eac ion a e no a ailable. Thus, he
CDCC calcula ion is es ed agains he da a o he nea by
sys em 6Li +144Sm [25]. The α+144Sm and d+144Sm
op ical po en ials we e aken om Re s. [26] and [21],
espec i ely. The esul s a e shown in Fig. 3. The op ical
model calcula ion using he po en ial o Cook [23] (dashed
lines) is also shown. I can be seen ha he CDCC esul
is simila o he op ical model calcula ion, pa icula ly a
E=35.1 MeV. A his ene gy, he calcula ions ep oduce e y
well he elas ic da a. Fo he lowe ene gy (E=22.1MeV),
bo h calcula ions unde es ima e he da a a backwa d angles.
No e ha , in con as o he 6Li +208Pb case, no appa en
modi ica ion o he deu e on po en ial was equi ed in his
case.
Now he inclusi e b eakup c oss sec ions 159Tb(6Li,αX)
a e discussed. The EBU con ibu ion was ob ained om he
CDCC calcula ions discussed in he p e ious pa ag aph. Fo
he NEB calcula ion, he same op ical po en ials α/d +159Tb
we e used. The Cook po en ial [23] was used o calcula e he
dis o ed wa e o he incoming channel.
In Fig. 4 he calcula ed and expe imen al angula dis ibu-
ions o αpa icles a e compa ed o se e al inciden ene gies
o 6Li. The dashed and do ed lines a e he EBU (CDCC)
and NEB (IAV model) esul s. The summed EBU +NEB
c oss sec ions (solid lines) ep oduce ai ly well he shape
0 50 100 150
0
10
20
30
dσ/dΩ (mb/s )
EBU (CDCC)
NEB (FR-DWBA)
TBU
0 50 100 150
0
10
20
30
40
0 50 100 150
0
20
40
60
dσ/dΩ (mb/s )
0 50 100 150
θlab (deg)
0
50
100
0 50 100 150
θlab (deg)
0
50
100
150
dσ/dΩ (mb/s )
23 MeV 25 MeV
27 MeV 30 MeV
35 MeV
159Tb(6Li,αX)
FIG. 4. Angula dis ibu ion o αpa icle p oduc ion o he
eac ion 6Li +159Tb a he inciden ene gies indica ed by he labels.
The dashed, do ed, and solid lines a e EBU calcula ed wi h CDCC,
NEB calcula ed wi h ini e- ange DWBA, and hei sum (TBU),
espec i ely. The expe imen al da a a e aken om Re . [24].
0 50 100 150
0.6
0.8
1
dσ/dσR
CDCC
OM
0 50 100 150
0.4
0.6
0.8
1
0 50 100 150
0
0.5
1
dσ/dσR
0 50 100 150
0
0.5
1
0 50 100 150
θlab (deg)
0
0.5
1
dσ/dσR
0 50 100 150
θlab (deg)
0
0.5
1
18 MeV 19 MeV
20 MeV 21 MeV
22.5 MeV 24 MeV
FIG. 5. Elas ic sca e ing o 6Li +118Sn a di e en inciden
ene gies. The solid and dashed lines a e, espec i ely, he CDCC
calcula ion and he op ical model calcula ion wi h he op ical po en ial
om Re . [27]. Expe imen al da a a e om Re . [27].
and magni ude o he da a, excep o a sligh o e es ima ion
a some ene gies. Simila ly o he hea y- a ge sys ems, i.e.,
6Li +209Bi [2] and 6Li +208Pb (Sec. IV A), he NEB is ound
o accoun o mos o he inclusi e b eakup c oss sec ion.
C. 118Sn (6Li, αX)
Inclusi e b eakup da a o he 118Sn(6Li, αX) eac ion a e
a ailable in Re . [27] a ene gies be ween 18 and 24 MeV. The
op ical model pa ame iza ions o Re s. [26] and [21] a e used
o he α-118Sn and d-118Sn sys ems. Fo he NEB calcula ions,
he op ical po en ial o 6Li +118Sn is aken om Re . [27].
In Fig. 5we compa e he elas ic da a wi h he CDCC (solid
lines) and op ical model (dashed lines) calcula ions. O e all,
bo h ypes o calcula ions ep oduce he da a well, wi h small
disc epancies obse ed a some o he ene gies.
Figu e 6shows he compa ison o he calcula ed and
expe imen al angula dis ibu ions o αpa icles p oduced in
his eac ion, o se e al inciden ene gies. Again, he NEB pa
(do ed lines) accoun s o mos o he inclusi e b eakup c oss
sec ion and he EBU (dashed lines) becomes he dominan
b eakup mode o angles smalle han ∼50 deg. The summed
EBU +NEB esul (solid line) ep oduces ema kably well
he shape and magni ude o he da a.
044605-5

JIN LEI AND ANTONIO M. MORO PHYSICAL REVIEW C 95, 044605 (2017)
0 50 100 150
0
4
8
12
16
dσ/dΩ (mb/s )
EBU
NEB
TBU
0 50 100 150
0
10
20
0 50 100 150
0
10
20
30
dσ/dΩ (mb/s )
0 50 100 150
0
10
20
30
0 50 100 150
θ
lab
(deg)
0
15
30
45
dσ/dΩ (mb/s )
0 50 100 150
θ
lab
(deg)
0
20
40
60
18 MeV 19 MeV
20 MeV 21 MeV
22.5 MeV 24 MeV
FIG. 6. Angula dis ibu ion o αpa icles p oduced in he eac-
ion 6Li +118Sn a he inciden ene gies indica ed by he labels. The
do ed, dashed, and solid lines co espond o he NEB (IAV model),
EBU (CDCC), and hei sum (TBU), espec i ely. Expe imen al da a
a e om Re . [27].
D. 59Co (6Li, αX)
Expe imen al da a o he α-p oduc ion channel o he
eac ion 6Li +59Co ha e been epo ed by Souza e al. [29]
a Elab =21.5 MeV, which is abo e he Coulomb ba ie
(VB=12 MeV).
Elas ic da a a e a ailable a he somewha smalle ene gy
Elab =18 MeV [28] so we i s compa e hese da a wi h he
op ical model and CDCC calcula ions. Fo he o me , we
employed he global op ical po en ial o Cook [23]. Fo he
CDCC calcula ions, he op ical po en ials o α+59Co and
d+59Co we e aken om Re s. [26] and [21], espec i ely.
The esul s a e shown in Fig. 7. I can be seen ha bo h
he CDCC and op ical model calcula ions ep oduce he da a
ai ly well. We no ice ha no eno maliza ion o he deu e on
po en ial was equi ed in his case.
The expe imen al and calcula ed angula dis ibu ions o
inclusi e αpa icles a e shown in Fig. 8. The NEB is seen
o domina e he inclusi e αp oduc ion. I should be no iced
ha , in his case, he NEB pa includes also he ans e
popula ing bound s a es o he a ge , which was ob ained
using he o malism discussed in Sec. III. A mo e de ailed
discussion o his con ibu ion is le o Sec. V. The o al
c oss sec ion, TBU =EBU +NEB, ep oduces well he
0 306090
120 150
θc.m. (deg)
0.5
1
dσ/dσR
Souza e al.
CDCC
OM
6Li+59Co @ 18 MeV
FIG. 7. Elas ic sca e ing o 6Li +59Co a an inciden ene gy o
18 MeV. The solid and dashed lines a e, espec i ely, he CDCC
calcula ion and he op ical model calcula ion wi h he op ical po en ial
omRe .[23]. Expe imen al da a a e om Re . [28].
shape o he expe imen al da a, al hough he magni ude is
unde es ima ed by ∼30% a he maximum. This migh indica e
he p esence o o he ele an mechanisms leading o he
p oduc ion o αpa icles in his eac ion, such as he o ma ion
o a compound nucleus ollowed by αe apo a ion. In ac ,
s a is ical model calcula ions pe o med in Re . [29] p edic ed
a signi ican amoun o αpa icles coming om his channel.
The e alua ion o his con ibu ion is, howe e , beyond he
scope o he p esen wo k.
The ene gy spec a o selec ed αsca e ing angles a e
also a ailable o his eac ion. These a e compa ed wi h ou
calcula ions in Fig. 9, wi h each panel co esponding o a gi en
αsca e ing angle, as indica ed by he labels. Excep a θlab =
15◦, he sum o EBU and NEB ep oduces he peak o he
αene gy dis ibu ion. Howe e , he low-ene gy ail is clea ly
unde es ima ed. A hese ene gies, he main con ibu ion o
he inclusi e αp oduc ion may a ise om compound nucleus
0 306090
120 150
θlab (deg)
0
50
100
150
dσ/dΩ (mb/s )
Souza e al.
EBU
NEB
TBU
59Co(6Li,αX) @ 21.5 MeV
FIG. 8. Angula dis ibu ion o αpa icles p oduced in he
eac ion 6Li +59Co a an inciden ene gy o 21.5 MeV. The dashed,
do ed, and solid lines a e, espec i ely, he EBU (CDCC), NEB(IAV
model), and hei sum. Expe imen al da a a e aken om Re . [29].
044605-6
COMPREHENSIVE ANALYSIS OF LARGE αYIELDS . . . PHYSICAL REVIEW C 95, 044605 (2017)
0 102030
0
10
20
d2
σ/dEdΩ (mb/(s MeV))
EBU
NEB
TBU
0 102030
0
10
20
0 102030
0
15
30
d2
σ/dEdΩ (mb/(s MeV))
0 102030
0
15
30
0 102030
0
10
20
d2
σ/dEdΩ (mb/(s MeV))
0 102030
Eα
lab (MeV)
0
5
10
0 102030
Eα
lab (MeV)
0
2
4
6
d2
σ/dEdΩ (mb/(s /MeV))
θα=15o
59Co(6Li,αX)@21.5MeV
θα=35oθα=45o
θα=55oθα=65o
θα=75o
θα=25o
FIG. 9. Expe imen al and calcula ed inclusi e αene gy spec a
o Elab =21.5 MeV, a selec ed sca e ing angles. The dashed,
do ed, and solid lines a e espec i ely he EBU (CDCC), NEB (IAV
model), and hei sum. Expe imen al da a a e aken om Re . [29].
ollowed by e apo a ion and p e-equilib ium, which a e no
conside ed in he p esen calcula ions. We no e ha high-
ene gy αpa icles s em om a deu e on ans e mechanism
o he a ge and a e well ep oduced by ou calcula ions.
E. 58Ni (6Li, αX)
The αp oduc ion o he 6Li +58Ni eac ion a se e al
inciden ene gies be ween 12 and 20 MeV was measu ed by
P ei e e al. [27]. Elas ic sca e ing da a, which we e also
measu ed, a e compa ed wi h CDCC and OM calcula ions in
Fig. 10 (no e ha he angles and c oss sec ions a e e e ed
o he labo a o y ame, as in he o iginal e e ence). Fo he
o me , we use he same op ical po en ials as in he nea by
6Li +59Co case. Fo he OM calcula ions we use he global
OM po en ial by Cook [23]. Bo h calcula ions ep oduce
a he well he da a, al hough he CDCC calcula ions sligh ly
unde es ima es he da a a la ge angles.
We p esen now he inclusi e αc oss sec ions. Fo he
NEB calcula ion, he 6Li op ical po en ial om Re . [23]was
used. Figu e 11 shows he compa ison o he calcula ed and
expe imen al angula dis ibu ions o αpa icles p oduced in
his eac ion, o se e al inciden ene gies. No ice ha he NEB
(do ed lines) includes also he con ibu ion coming om he
0 50 100 150
0.5
1
dσ/dσR
P ei e e al.
CDCC
OM
0 50 100 150
0
0.5
1
0 50 100 150
0
0.5
1
dσ/dσR
0 50 100 150
θlab (deg)
0
0.5
1
0 50 100 150
θlab (deg)
0
0.5
1
dσ/dσR
12 MeV 14 MeV
16 MeV 18 MeV
20 MeV
6Li+58Ni
FIG. 10. Elas ic sca e ing o 6Li +58Ni a se e al ene gies
indica ed by he labels. The solid and dashed lines a e, espec i ely,
he CDCC calcula ion and he op ical model calcula ion wi h he
op ical po en ial om Re . [27]. Expe imen al da a a e om Re . [27].
ans e o a ge bound s a es. Again, he NEB pa domina es
he inclusi e αp oduc ion. In gene al, he summed EBU +
NEB c oss sec ion (solid lines) ep oduces well he shape
and magni ude o he da a. A 16, 18, and 20 MeV, some
unde es ima ion is obse ed, which migh be associa ed wi h
o he α-p oduc ion channels, as poin ed ou in he 6Li +59Co
case.
F om he esul s p esen ed in he p e ious sec ions, we
may conclude ha he s ong α-p oduc ion channel obse ed
in 6Li expe imen s o igina es mos ly om nonelas ic b eakup
mechanisms. In all cases analyzed, he EBU mode u ns ou
o accoun o a ela i ely small ac ion o he o al inclusi e
αc oss sec ion and i s con ibu ion is only impo an o he
αpa icles emi ed a small angles. Fo he ligh e a ge s,
we ound also a indi ec e idence o o he αp oduc ion
mechanisms, such as usion.
V. TRANSFER CONTENT OF THE NEB CROSS SECTION
The ela i e impo ance o he ans e o bound s a es
wi hin he NEB c oss sec ion will depend on se e al pa ame-
e s, such as he p ojec ile inciden ene gy and he cha ge o
he a ge nucleus. Fo hea y a ge s, he ans e channel is
044605-7
JIN LEI AND ANTONIO M. MORO PHYSICAL REVIEW C 95, 044605 (2017)
0 50 100 150
0
5
10
15
dσ/dΩ (mb/s )
EBU
NEB
TBU
0 50 100 150
0
10
20
30
0 50 100 150
0
20
40
60
dσ/dΩ (mb/s )
0 50 100 150
θlab (deg)
0
20
40
60
80
0 50 100 150
θlab (deg)
0
50
100
150
dσ/dΩ (mb/s )
12 MeV 14 MeV
16 MeV 18 MeV
20 MeV
58Ni(6Li,αX)
FIG. 11. Angula dis ibu ion o αpa icles p oduced in he
eac ion 6Li +58Ni a he inciden ene gies indica ed by he labels.
The dashed, do ed, and solid lines a e, espec i ely, he EBU, NEB,
and hei sum (TBU). Expe imen al da a a e om Re . [27].
supp essed due o he s ong Coulomb in e ac ion be ween he
deu e on and he a ge , whe eas o ligh a ge s his channel
is expec ed o play a mo e impo an ole.
This is illus a ed in Fig. 12 o wo such cases; he
uppe panel displays he calcula ed 208Pb(6Li, αX) NEB c oss
sec ions as a unc ion o d-208Pb ela i e ene gy a h ee
di e en inciden ene gies, 29, 35, and 39 MeV. The e ical
do ed line indica es he nominal Coulomb ba ie o he
d-208Pb sys em. The black solid cu e is he eac ion c oss
sec ion o he d-208Pb sys em, a bi a ily no malized o i
wi hin he same scale. The bo om panel shows simila cu es
o he 6Li +58Ni eac ion a 12, 16, and 20 MeV. In bo h cases,
i can be seen ha he NEB is a T ojan ho se ype p ocess [30],
which means ha he 6Li p ojec ile b ings he deu e on inside
he Coulomb ba ie and le s i in e ac wi h he a ge nucleus,
gi ing sizable c oss sec ions o deu e on ene gies o which
he eac ion c oss sec ion has al eady become negligibly small.
Fo he 208Pb a ge , due o he s ong Coulomb epulsion, he
NEB c oss sec ion becomes negligible a nega i e d-208Pb
ela i e ene gies and his beha io is independen o he
incoming 6Li ene gy. In con as , o he 58Ni a ge , he e
FIG. 12. Top: NEB c oss sec ion as a unc ion o he d-208Pb
ela i e ene gy in he c.m. ame o he eac ion 6Li +208Pb. The
e ical do ed line indica es he ene gy o he Coulomb ba ie o
he d+208Pb eac ion. The solid line is he eac ion c oss sec ion o
d+208Pb, a bi a ily no malized. Bo om: same as in op panel bu
o he 6Li +58Ni sys em.
is a low-ene gy ail ex ending o nega i e deu e on ene gies
( ans e ).
We expec also some co ela ion be ween he α-pa icle
angula and ene gy dis ibu ion. This is shown in Fig. 13 in he
o m o con ou plo s o double di e en ial c oss sec ions and
FIG. 13. Con ou plo s o he double di e en ial c oss sec ion
(uppe panels) and he angle-in eg a ed ene gy di e en ial c oss
sec ion as a unc ion o he ou going αene gy in he c.m. ame
(lowe panels) o he eac ions (a) 6Li +208Pb, (b) 6Li +159Tb,
(c) 6Li +118Sn, and (d) 6Li +59Co. The e ical lines indica e he
b eakup h eshold o he d+ a ge sys em (Ex=0).
044605-8
COMPREHENSIVE ANALYSIS OF LARGE αYIELDS . . . PHYSICAL REVIEW C 95, 044605 (2017)
0.8 1 1.2 1.4 1.6
Ec.m./VB
10
100
1000
σα
TBU
6Li+58Ni
6Li+59Co
6Li+118Sn
6Li+159Tb
6Li+208Pb
6Li+209Bi
FIG. 14. Inclusi e b eakup αc oss sec ions in ol ing 6Li p ojec-
ile wi h se e al a ge s as a unc ion o Ec.m./VB.
angle-in eg a ed c oss sec ion as a unc ion o he ou going
αene gy in he c.m. ame o he eac ions (a) 6Li +208Pb,
(b) 6Li +159Tb, (c) 6Li +118Sn, and (d) 6Li +59Co. I can seen
ha he mos ene ge ic αpa icles a e p e e ably emi ed a
o wa d angles, whe eas hose wi h lowe ene gies con ibu e
o bo h o wa d and backwa d angles. Mo eo e , when he
cha ge o he a ge is small (59Co), he ans e channel
becomes mo e ele an .
VI. SYSTEMATICS OF INCLUSIVE αPRODUCTION
Sys ema ic s udies o αp oduc ion yields in 6Li eac ions
show an in e es ing uni e sal beha io when plo ed as a
unc ion o he inciden ene gy scaled by he Coulomb ba ie
ene gy as epo ed o ins ance by Pakou e al. [31]. In his
sec ion, we will in es iga e whe he ou calcula ions exhibi
also his uni e sal beha io . Fo his s udy, we ha e conside ed
he a ge sys ems 59Co, 118Sn, 159Tb, and 208Pb, which ha e
been analyzed in he p eceding sec ions, and 209Bi, analyzed
in Re . [2]. The esul s a e shown in Fig. 14, whe e we plo
he calcula ed σTBU
αc oss sec ions as a unc ion o he educed
ene gy (Ec.m./VB), wi h VB he ene gy o he Coulomb ba ie ,
es ima ed as VB=ZpZ e2/[ B(A1/3
p+A1/3
)], whe e Zp(Z )
and Ap(A ) a e he a omic numbe and a omic mass o he p o-
jec ile ( a ge ), espec i ely, and B=1.44 m. As expec ed,
he b eakup c oss sec ion d ops quickly as he inciden ene gy
dec eases below he ba ie . This e ec is enhanced o he
209Bi nucleus, possibly due o he la ge Coulomb epulsion.
Abo e he ba ie , he inclusi e b eakup c oss sec ions show a
simila end o he medium-hea y and hea y a ge s, bu
no o he medium mass a ge s 58Ni and 59Co a la ge
ene gies. We ecall, howe e , ha o hese ligh e sys ems,
he e migh be addi ional con ibu ions om o he channels,
such compound nucleus ollowed by e apo a ion, which a e
no accoun ed o by he IAV o malism.
We ha e also s udied he ela i e impo ance o EBU e sus
NEB as a unc ion o he inciden ene gy. Fo ha , we display
in Fig. 15 he a io o EBU o e TBU (=EBU +NEB) o he
0.8 1 1.2 1.4 1.6
Ec.m./VB
0
0.1
0.2
0.3
0.4
σEBU / σTBU
6Li+58Ni
6Li+59Co
6Li+118Sn
6Li+159Tb
6Li+208Pb
6Li+209Bi
FIG. 15. Ra ios o calcula ed EBU o e TBU (=EBU +NEB)
o di e en sys ems. See ex o de ails.
analyzed sys ems. I is seen ha , o inciden ene gies below
he Coulomb ba ie , he elas ic b eakup c oss sec ion becomes
compa a i ely mo e impo an as he ene gy dec eases. This
can be a ibu ed o he ac ha , below he ba ie , he
b eakup akes place a la ge p ojec ile- a ge sepa a ions, and
he deu e on abso p ion ( esponsible o he NEB pa ) will
be less impo an [4]. By con as , o ene gies abo e he
Coulomb ba ie , he a io shows an almos cons an beha io .
I can also be seen ha while o he hea y mass a ge s elas ic
b eakup plays an impo an ole in he inclusi e αp oduc ion,
especially below he Coulomb ba ie , o he medium mass
a ge s elas ic b eakup is less impo an and he nonelas ic
b eakup is dominan .
Ano he ele an ques ion ega ds he ac ion o he
eac ion c oss sec ion ha is exhaus ed by he αc oss sec ion.
To add ess his ques ion, we plo in Fig. 16 he a io o he
calcula ed TBU and eac ion c oss sec ions as a unc ion
0.8 1 1.2 1.4 1.6
Ec.m./VB
0
0.2
0.4
0.6
0.8
1
σα
TBU/σR
6Li+58Ni
6Li+59Co
6Li+118Sn
6Li+159Tb
6Li+208Pb
6Li+209Bi
FIG. 16. Ra ios o calcula ed TBU (=EBU +NEB) αc oss
sec ions o e he eac ion c oss sec ion o he sys ems and ene gies
analyzed in his wo k. See he ex o de ails.
044605-9