PHYSICAL REVIEW C 85, 054613 (2012)
Co e exci a ion e ec s in he b eakup o he one-neu on halo nucleus 11Be on a p o on a ge
A. M. Mo o1,*and R. C espo2,3,†
1Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea , Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain
2Cen o de F´
ısica Nuclea , Uni e sidade de Lisboa, A enida do P o esso Gama Pin o 2, P-1649-003 Lisboa, Po ugal
3Depa amen o de F´
ısica, Ins i u o Supe io T´
ecnico, Uni e sidade T´
ecnica de Lisboa, A enida do P o esso
Ca aco Sil a, Taguspa k, P-2780-990, Oei as, Po ugal
(Recei ed 6 Ma ch 2012; published 9 May 2012)
We in es iga e he phenomenon o co e exci a ion in he inelas ic and b eakup o wo-body halo nuclei,
composed by a alence nucleon and a co e. To e alua e he impo ance o his e ec we p opose a simple eac ion
model based on an ex ension o he s anda d dis o ed wa e Bo n app oxima ion (DWBA) me hod. The model
akes in o accoun co e-exci ed admix u es in he s a es o he composi e p ojec ile, as well as he possibili y
o dynamic co e exci a ion due o he in e ac ion o he co e wi h he a ge . As an applica ion o he model,
we p esen calcula ions o he b eakup o 11Be on a p o on a ge a an inciden ene gy o 63.7 MeV/nucleon,
compa ing i wi h he a ailable da a o his eac ion. We ind ha he da a a e well ep oduced by he model and
ha he e ec o dynamic co e exci a ion is essen ial o explain he obse ed c oss sec ion.
DOI: 10.1103/PhysRe C.85.054613 PACS numbe (s): 24.50.+g, 25.60.Gc, 27.20.+n
I. INTRODUCTION
B eakup and inelas ic sca e ing a e now s anda d ech-
niques o s udy he s uc u e o halo nuclei. In pa icula ,
he p o on a ge is e y appealing o spec oscopic s udies
because he exci a ion occu s mainly due o he nuclea in e -
ac ion, he eby exci ing se e al mul ipola i ies and e ealing
he esonan s uc u e o he p ojec ile con inuum [1,2]. To
ex ac eliable in o ma ion om hese da a i is impo an
o iden i y he ele an exci a ion mechanisms. In he case
o a p ojec ile composed by a alence pa icle weakly bound
o a co e, i is commonly assumed ha he main exci a ion
mechanism is due o he exci a ion o he alence nucleon
ou side he co e. This single-pa icle pic u e is behind many
ew-body eac ion o malisms used in he analysis o eac ions
induced by halo and o he weakly bound nuclei, such as
he con inuum-disc e ized coupled-channels (CDCC) me hod
[3], he adiaba ic app oxima ion [4,5], he Al , G assbe ge ,
Sandhas (AGS) o mula ion o he Faddee equa ions [6,7],
and a a ie y o semiclassical app oaches [8–13].
Despi e he ela i e success o hese me hods, hey ypically
igno e he ac ha he p ojec ile s a es will con ain, in
gene al, signi ican admix u es o se e al co e componen s
and, he e o e, he exci a ion o he p ojec ile may be caused
also by ansi ions be ween hese co e s a es.
These e ec s a e no included in he s anda d o mula ions
o he me hods discussed abo e, al hough some e o s ha e
been made in ecen yea s owa d his di ec ion. A ecen
example is he ex ended e sion o he CDCC me hod (named
XCDCC) ecen ly p oposed by he au ho s o Re . [14].
The e ec o co e exci a ion has also been s udied [15],
using an ex ension o he adiaba ic model o Re . [16]. The
o malism was applied o he elas ic sca e ing o 8Bona
*[email p o ec ed]
†[email p o ec ed]
ca bon a ge , and some con ibu ion was ound a la ge angles
due o he exci a ion o he 7Be co e.
I is he pu pose o his wo k o e alua e he in luence o
hese co e exci a ion e ec s in he inelas ic and b eakup o
halo nuclei using a simple model, bu e aining ne e heless
he key physics. The model p esen ed he e is an ex ension o
he co e exci a ion model p esen ed in a p e ious wo k [17].
Unde he assump ion ha he DWBA app oxima ion is alid,
and ha co e- ecoil e ec s can be neglec ed, his model gi es
a simple ela ion be ween he exci a ion o he p ojec ile and
ha o he co e on he same a ge . The model was applied o
he b eakup o 11Be on a p o on a ge a 63.7 MeV/nucleon
inciden ene gy, and compa ed wi h he da a om Re . [1].
Due o he expe imen al ene gy esolu ion, he expe imen al
angula dis ibu ions o he exclusi e b eakup we e ex ac ed
o wo in e als o neu on-10Be co e ene gy: (i) E el =0–
2.5 MeV and (ii) E el =2.5–5.0 MeV. In he i s in e al one
expec s an impo an con ibu ion coming om he low-lying
na ow 5/2+ esonance a a ela i e ene gy o 1.28 MeV [18].
This esonance has a dominan 10Be(0+)⊗ν1d5/2pa en age
and a small componen 10Be(2+)⊗ν2s1/2. The c oss sec ion
o he second in e al con ains, p esumably, con ibu ions
coming om se e al esonances, namely, Ex=2.64 MeV
(3/2−), 3.40 MeV (3/2−,3/2+), 3.89 MeV (5/2−), and 3.95
(3/2−)[18]. In a p e ious wo k, we analyzed hese da a
using a simple e sion o he co e exci a ion model [17], and
showed ha he main con ibu ion o he second in e al comes
om he esonance a 3.40 MeV, o which we assumed a
3/2+assignmen , ollowing he sugges ion o he au ho s o
Re . [19]. These calcula ions p o ided a easonable ag eemen
wi h he da a o Re . [1] and, mos impo an ly, e idenced he
impo ance o he co e exci a ion mechanism in he b eakup
o halo nuclei.
The calcula ions pe o med by he au ho s o Re . [17]
assumed a simple single-pa icle con igu a ion o he alence
neu on in he 11Be nucleus ela i e o he 10Be co e. I is
he pu pose o his wo k o ex end his model o he mo e
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0556-2813/2012/85(5)/054613(8) ©2012 Ame ican Physical Socie y
A. M. MORO AND R. CRESPO PHYSICAL REVIEW C 85, 054613 (2012)
ealis ic si ua ion in which he p ojec ile s a es consis o
a supe posi ion o di e en alence con igu a ions and co e
s a es.
The pape is s uc u ed as ollows. In Sec. II we in oduce
he Hamil onian used in his wo k. In Sec. III, we desc ibe
he co e exci a ion eac ion model used in he sca e ing
calcula ions. In Sec. IV we de ail he s uc u e model used o
desc ibe he 11Be s a es. In Sec. Vwe p esen he calcula ions
o he 11Be +p eac ion. Finally, Sec. VI is le o summa y
and conclusions.
II. THE FEW-BODY HAMILTONIAN
The h ee-body Hamil onian o he sys em has he o m
H=TR+Hp oj +Vc +V ,(1)
whe e TR ep esen s he kine ic ene gy ope a o o he
p ojec ile- a ge ela i e mo ion, Vc and V a e he co e- a ge
and he alence- a ge in e ac ions, and Hp oj is he in e nal
Hamil onian o he p ojec ile, gi en by
Hp oj =T +V c( , 
ξ)+hco e(
ξ),(2)
whe e  is he ela i e coo dina e be ween he alence and
he co e, T he co e- alence kine ic ene gy ope a o , and
hco e(
ξ) he in insic Hamil onian o he co e, whose ene gies
and eigens a es a e labeled by he angula momen um (I) and
i s p ojec ion (Mc), ha is,
hco e|IMc=εI|IMc.(3)
Addi ional quan um numbe s, equi ed o ully speci y he
co e s a es, a e omi ed o simplici y in he no a ion. The
co e- alence and co e- a ge in e ac ions may depend upon
he in e nal co e deg ees o eedom 
ξ. A gi en p ojec ile s a e
wi h o al angula momen um and p ojec ion JM will consis
on a linea supe posi ion o se e al alence-co e con igu a ions
JM( , 
ξ)=
α=I,,j ψJ
α( )⊗I(
ξ)JM,(4)
whe e ψJ
α( ) a e wa e unc ions desc ibing he mo ion o
he alence neu on ela i e o a gi en co e s a e. The se o
quan um numbe s s,, and j e e o he in insic spin o he
alence pa icle, i s o bi al angula momen um ela i e o he
co e, and hei sum, espec i ely.
The wa e unc ions ψJ
α( ) will depend upon he adop ed
s uc u e model. Fo ou case s udy his will be desc ibed in
Sec. IV.
III. THE REACTION APPROACH: CORE EXCITATION
REACTION MODEL
In his sec ion we epo in de ail he co e-exci ed eac ion
o malism (C-ex) ha shall be used o desc ibe he exci a ion
o a p ojec ile nucleus (p), om i s g ound s a e wi h o al
angula momen um J o a inal s a e Jdue o i s in e ac ion
wi h a a ge ( ). The p ojec ile nucleus is assumed o be well
desc ibed by a alence pa icle ( ) o bi ing a ound a co e
o nucleons (c). The a ge is assumed o be ine . In ou
Rc
R
ξ
c
R
FIG. 1. (Colo online) Rele an coo dina es o he sca e ing o
a wo-body sys em, composed o a co e (c) and a alence pa icle ( ),
sca e ed o a a ge nucleus ( ).
wo king example, his will co espond o he exci a ion o 11Be
(10Be +n) on a p o on a ge o inal s a es in he con inuum,
bu he o malism could be equally applied o he ansi ion
be ween bound s a es. Fo simplici y, he e we igno e he spin
o he p o on a ge .
The di e en ial c oss sec ion o he exci a ion om an
ini ial s a e |i
JM o a inal s a e |
JMo he p ojec ile is
gi en in gene al by
dσp
d =1
ˆ
J2
μp
(2π¯h2)2
K
K
M,M TJM,JM
p 2(5)
wi h ˆ
J=√2J+1, μp is he p ojec ile- a ge educed mass,

K(
K) he ini ial ( inal) linea momen um, and TJM,JM
p is he
ansi ion ampli ude. Wi hin he DWBA app oach, he la e is
gi en by
TJM,JM
p =χ(−)

K(
R)
JM( ,
ξ)VTχ(+)

K(
R)i
JM( , 
ξ),(6)
whe e χ(+)

K(
R) and χ(−)

K(
R) a e dis o ed wa es desc ibing
he p ojec ile- a ge ela i e mo ion in he ini ial and inal
channels, espec i ely. In a co e + alence model o he
p ojec ile, he ansi ion po en ial is
VT=V (
R )+Vc (
Rc ,
ξ),(7)
whe e V (
R ) and Vc (
Rc ,
ξ) a e he alence- a ge and co e-
a ge po en ials. The ele an coo dina es o his p oblem a e
depic ed in Fig. 1.
The alence- a ge in e ac ion is assumed o be cen al, and
hence V (
R )=V (R ). On he o he hand, he co e- a ge
in e ac ion con ains bo h cen al and noncen al pa s, and is
w i en as he mul ipola expansion
Vc (
Rc ,
ξ)=
L,M
V(L)
c (Rc ,ξ)Y∗
LM(ˆ
R)YLM(ˆ
ξ).(8)
We assume ha he adial o m ac o does no depend on he
in e nal coo dina es o he co e [i.e., VL(Rc ,ξ)=VL(Rc )],
which is alid o he o a ional model used he e.
F om Eqs. (6) o (8) he DWBA ampli ude is exp essed as
a sum o wo e ms:
TJM,JM
p (
K,
K)=TJM,JM
al +TJM,JM
co ex .(9)
The i s e m, ha we deno e alence ampli ude o
sho ness, is explici ly gi en by
TJM,JM
al (
K,
K)=χ(−)

K(
R)
JM( ,
ξ)V (R )
+V(0)
c (Rc )χ(+)

K(
R)i
JM( , 
ξ).(10)
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CORE EXCITATION EFFECTS IN THE BREAKUP OF THE ... PHYSICAL REVIEW C 85, 054613 (2012)
This e m con ains only he cen al pa s o he agmen - a ge
po en ials V and Vc and he e o e i canno induce ansi ions
in ol ing exci a ions o he co e. Using he gene al o m (4)
o he ini ial and inal s a es, his ampli ude can be ew i en
as
TJM,JM
al (
K,
K)=
α,αχ(−)

K(
R)ψJ
α( )V (R )
+V(0)
c (Rc )χ(+)

K(
R)ψJ
α( )δI,I.(11)
Fo each s a e Io he co e con ained in bo h he ini ial and
inal s a es o he p ojec ile, his ampli ude is simila o ha
ound in con en ional CDCC calcula ions and he e o e can
be e alua ed using he s anda d codes a ailable o his kind
o calcula ions.
The second e m o he ansi ion ampli ude is due o he
noncen al pa o he co e in e ac ion, ha is,
TJM,JM
co ex (
K,
K)=
L>0,Mχ(−)

K(
R)
JM( ,
ξ)V(L)
c (Rc )
×Y∗
LM(ˆ
Rc )YLM(ˆ
ξ)χ(+)

K(
R)i
JM( , 
ξ).
(12)
This e m accoun s o he dynamic exci a ion o he co e
du ing he collision. I is he pu pose o he p esen wo k
o e alua e his e m app oxima ely, bu e aining he key
physics, o p o ide a simple es ima ion o he co e exci a ion
e ec s.
In addi ion o he co e exci a ion mechanism, he ampli-
ude (12) may also p oduce exci a ions in he co e- alence ela-
i e mo ion. This is because he ansi ion po en ial is e alua ed
a 
Rc . This coo dina e can be exp essed as 
Rc =
R+γ
[γ=m /(m +mc)]. The dependence on he co e- alence
coo dina e ( ) p oduces a co e- ecoil e ec ha can induce
exci a ions o he p ojec ile. Howe e , because in ou es case
γ1, we will assume ha his e ec can be neglec ed in
his ampli ude [al hough i is ully aken in o accoun in he
ampli ude (11)], so we can make he app oxima ion 
R≈
Rc
in Eq. (12), which means ha he dis o ed wa es in Eq. (12)
a e e alua ed a he 
Rc coo dina e. This app oxima ion leads
o wha we shall call he ea e he co e exci a ion (C-exc)
eac ion model. We no e ha his model can be deduced om
he XCDCC o malism [14] by calcula ing he ansi ion in he
Bo n app oxima ion and neglec ing he co e- ecoil e ec s o
L>0.
The sca e ing ampli ude o he C-exc app oach ac o izes
hen in o a sum o p oduc s o a eac ion and a s uc u e e m
TJM,JM
co ex =
L>0,M
TLM(
K,
K)
×
JM( ,
ξ)YLM(ˆ
ξ)i
JM( , 
ξ),(13)
whe e we ha e in oduced he quan i ies
TLM(
K,
K)=χ(−)

K(
Rc )|V(L)(Rc )Y∗
LM(ˆ
Rc )|χ(+)

K(
Rc ),
(14)
which con ain he dependence on he eac ion pa o he
ansi ion ampli ude.
Using he Wigne -Ecka heo em in he o m [20], we can
w i e o he s uc u e componen

JMYL,M (ˆ
ξ)i
JM=JM|JMLM
J||YL(
ξ)||i
J.
(15)
No ing ha he ope a o appea ing in he educed ma ix
elemen depends only on he co e coo dina es and ha he
ini ial and inal s a es a e exp essed in he o m (4), his
ma ix elemen can be w i en as (see, o ins ance, Re . [20],
Appendix VI)

J||YL(
ξ)||i
J=
α,αRJ
αRJ
αG(L)
αJ,αJI||YL(
ξ)||I,
(16)
whe e we in oduced he geome ic ac o
G(L)
αJ,αJ=δj,j(−1)L+j+J+Iˆ
Jˆ
IJJL
II
j.(17)
The educed ma ix elemen I||YL(
ξ)||Iappea ing in
Eq. (16) depends on he s uc u e model assumed o he co e
and will be speci ied la e . Collec ing esul s, he sca e ing
ampli ude o he co e exci a ion eads
TJM,JM
co ex =
L>0,MJMLM|JMTLM(
K,
K)
×
α,αRJ
α|RJ
αG(L)
αJ,αJI||YL(
ξ)||I.(18)
The di e en e ms ha en e in his ansi ion ampli ude
a e ela i ely s aigh o wa d o calcula e. The ampli udes
TLM(
K,
K), de ined by Eq. (14), a e hose appea ing in s an-
da d DWBA calcula ions wi h local o m ac o s, o example,
in inelas ic sca e ing calcula ions. The adial unc ions RJ
α( )
a e he solu ion o a coupled se o di e en ial equa ions and
he echniques o sol e his p oblem a e desc ibed elsewhe e
(see, e.g., he Appendix VI o Re . [21]). The es o he e ms
a e jus kinema ical and geome ical ac o s.
The ampli ude (18) can also be ela ed o he wo-body
inelas ic ampli udes o a co e- a ge sca e ing p oblem. To
make his ela ion explici , le us conside a gi en ansi ion
IMc→IM
c o he inelas ic exci a ion o he co e sca e ed
o he same a ge . In DWBA, he ampli ude o his p ocess
eads, simila ly o Eq. (6),
TIMc,I M
c
c =χ(−)

K(
Rc )
IM
c(
ξ)Vc (
Rc ,
ξ)
×χ(+)

K(
Rc )i
IMc(
ξ).(19)
Using he Wigne -Ecka heo em, his ampli ude can be
also w i en as
TIMc,I M
c
c =IM
c|IMcLM
T(LM)
c (I→I),(20)
wi h he educed ampli udes

T(LM)
c (I→I)=TLM(
K,
K)I||YL(ˆ
ξ)||I,(21)
wi h TLM(
K,
K) gi en by Eq. (14).
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A. M. MORO AND R. CRESPO PHYSICAL REVIEW C 85, 054613 (2012)
Compa ing wi h Eq. (18) and making use again o he no-
ecoil app oxima ion ( 
Rc ≈
R) we ge he ela ion
TJM,JM
co ex =
L>0,MJM|JMLM
×
α,αRJ
αRJ
αG(L)
αJ,αJ
T(LM)
c (I→I).(22)
This ampli ude is equi alen o ha gi en by Eq. (18),bu
emphasizes in a clea way he ela ion be ween he h ee-
body sca e ing ampli ude, Eq. (12), and he co esponding
wo-body ampli udes, unde he app oxima ions assumed by
he model. Equa ion (22) p o ides also a p ac ical way o
nume ically implemen ing he co e-exci ed model since he
wo-body ampli udes can be ob ained om any s anda d
DWBA code.
In he ex eme si ua ion in which he alence exci a ion
is small compa ed wi h he co e exci a ion mechanism,
he ansi ion ampli ude is gi en en i ely by Eq. (18) and
he co esponding c oss sec ion is ob ained inse ing his
ampli ude in o Eq. (5). Fu he mo e, i he e is a con ibu ion
om a single mul ipole ansi ion L, and a single con ig-
u a ion in he ini ial and inal s a es, he p ojec ile- a ge
c oss sec ion becomes p opo ional o he c oss sec ion o
he exci a ion o he co e mul iplied by a geome ic ac o and
he adial o e lap, ha is,
dσp
d co ex =ˆ
J2
ˆ
J2
ˆ
I2
ˆ
I2G(L)
αJ,αJRJ
α|RJ
α2dσc
d (I→I).
(23)
This pa icula case was used in a p elimina y applica ion o
his model o 11Be +p esonan b eakup [17].
No e, howe e , ha in gene al bo h he alence and he co e
exci a ion mechanisms will con ibu e o he b eakup p ocess.
In his case, he co esponding ampli udes need o be added
cohe en ly and hence in e e ence e ec s will a ise. In he
p esen wo k we ake in o accoun simul aneously he alence
and co e exci a ion con ibu ions.
IV. STRUCTURE MODEL
To calcula e he ini ial and inal s a es o he p ojec ile
one needs o speci y he s uc u e model o he co e. In he
calcula ions p esen ed in his wo k o he 11Be +psys em,
he p ojec ile is ea ed wi hin he pa icle- o o model o Boh
and Mo elson [21]. Then we assume a o a ional model o he
10Be co e wi h a pe manen quad upole de o ma ion, which,
o simplici y, is aken o be axially symme ic. The e o e, we
can cha ac e ize he de o ma ion by a single pa ame e β2.In
he body- ixed ame, he su ace adius is hen pa ame ized
as R(ˆ
ξ)=R0[1 +β2Y20(ˆ
ξ)], wi h R0an a e age adius, o
be speci ied la e . S a ing om a cen al po en ial, V(0)
c ( ),
he ull alence-co e in e ac ion is ob ained by de o ming his
cen al in e ac ion
V c( , ˆ
ξ)=V(0)
c [ −δ2Y20(ˆ
ξ)],(24)
wi h δ2=β2R0being he de o ma ion leng h. T ans o ming o
he space- ixed ame o e e ence, and expanding in sphe ical
ha monics, his de o med po en ial eads
V c( , 
ξ)=
L,M
V(L)
c ( )Y∗
LM(ˆ
)YLM(ˆ
ξ).(25)
The in e nal s a es o he p ojec ile a e expanded acco ding
o Eq. (4). I we w i e ψJ
α( )=RJ
α( )Ysjm(ˆ
), wi h Ysjm(ˆ
)=
[Y(ˆ
)⊗χs]jm, he p ojec ile s a es become
JM( , 
ξ)=
α
RJ
α( )[Ysj (ˆ
)⊗I(
ξ)]JM.(26)
The adial unc ions RJ
α( ) a e hen ob ained by sol ing he
Sch ¨
odinge equa ion using he po en ial (25) and wi h he
app op ia e bounda y condi ions.
Fo bound s a es, hese adial unc ions decay exponen ially
o →∞gi ing ise o squa e-in eg able unc ions. Fo
con inuum s a es, he unc ions RJ
α( ) a e also ob ained by
sol ing a se o coupled adial equa ions, bu subjec o he
bounda y condi ion ha inciden wa es occu in a gi en
open channel cha ac e ized by a se o quan um numbe s
α={, s, j, I}.
Al hough he calcula ions p esen ed in his wo k could
be pe o med wi h he sca e ing s a es hemsel es, i is
nume ically ad an ageous o adop a binning p ocedu e,
simila o ha used in CDCC calcula ions. Then, he con inuum
spec um is di ided in o ene gy in e als. Fo each ene gy
in e al, o bin, a ep esen a i e squa e-in eg able s a e is
cons uc ed by a weigh ed supe posi ion o sca e ing s a es
wi hin he bin in e al. The bin wa e unc ion o a gi en
incoming wa e αis gi en by a squa e-in eg able unc ion,
wi h a s uc u e simila o ha o he bound s a es (26)
bin
[k1k2]αJM( , 
ξ)=
α
RJ
[k1k2]αα( )
×[Ysj (ˆ
)⊗I(
ξ)]JM,(27)
whe e [k1k2] deno es he momen um in e al de ining he bin.
V. APPLICATION TO 11Be +pRESONANT BREAKUP
We now apply he co e-exci ed model o he b eakup o 11Be
on a p o on a ge and compa e wi h he da a o [1]. Be o e
p esen ing he calcula ions wi h his model, we conside
he case igno ing he e ec o de o ma ion in bo h he 11Be
s uc u e and in he co e- a ge in e ac ion.
A. Calcula ions wi hou de o ma ion
The calcula ions wi h no-de o ma ion we e done wi h
he s anda d CDCC me hod, using he FRESCO code [22].
These calcula ions a e simila o hose p esen ed in ou
p e ious wo k [23]. The alence-co e in e ac ion con ains
cen al +spin-o bi e ms, o Woods-Saxon shape, wi h
pa ame e s adjus ed o ep oduce he g ound s a e sepa a ion
ene gy, he bound exci ed s a e (1/2−), and he posi ion o
he low-lying 5/2+ esonance. The co e- a ge in e ac ion
co esponds o he Wa son pa ame iza ion o Re . [24]. Fo
he in e ac ion be ween he alence neu on and he p o on
a ge , we conside ini ially a simple Gaussian in e ac ion
054613-4
CORE EXCITATION EFFECTS IN THE BREAKUP OF THE ... PHYSICAL REVIEW C 85, 054613 (2012)
0246
E el (MeV)
0
20
40
dσ/dE el (mb/MeV)
Faddee : 3S1 Gaussian
Faddee : CD Bonn
CDCC: 3S1 Gaussian
CDCC: Modi ied Gaussian
FIG. 2. (Colo online) Di e en ial c oss sec ion as a unc ion
o he n-10Be ela i e ene gy calcula ed wi h he Faddee /AGS and
CDCC me hods, using di e en choices o he alence neu on-
p o on in e ac ion. The solid line is he Faddee calcula ion using
he ealis ic CD Bonn in e ac ion. The do ed and do ed-dashed lines
a e, espec i ely, he Faddee and CDCC calcula ions using a p-n
Gaussian in e ac ion adjus ed o he 3S1s-wa e phase-shi s: V( )=
−72.15 exp[−( /1.484)2]. The dashed line is he CDCC calcula ions
using he modi ied Gaussian in e ac ion wi h he same geome y,
bu wi h he educed s eng h V0=−45 MeV. All hese calcula ions
igno e he 10Be de o ma ion.
V( )=−72.15 exp[−( /1.484)2][3], which ep oduces he
deu e on binding ene gy and low-ene gy s-wa e iple (3
S1)
phase shi s. The 11Be con inuum was disc e ized in ene gy
in e als using he s anda d binning p ocedu e. We included
n-10Be pa ial wa es up o =2.
In Fig. 2we ep esen he calcula ed ene gy spec um
dσ/dE el ha eme ges by in eg a ing he b eakup c oss sec ion
o e he solid angle dc.m.. To assess he eliabili y o he
CDCC me hod in his case, we include also he Faddee
calcula ion (quo ed om Re . [23]), pe o med wi h he same
wo-body in e ac ions. The CDCC calcula ion (do -dashed
line) ep oduces ai ly well he Faddee esul indica ing ha ,
a leas in he limi o no-de o ma ion, he CDCC is a eliable
ool o analyze his eac ion.
I was ound by he au ho s o Re . [23] ha he b eakup
c oss sec ions a e e y sensi i e o he in e ac ion be ween
he alence neu on and he p o on a ge . This is illus a ed
by he solid line in Fig. 2, which ep esen s he Faddee
calcula ion pe o med eplacing he simple Gaussian po en ial
by he ealis ic CD Bonn in e ac ion. This p oduces a sizable
educ ion o he b eakup c oss sec ion. Consequen ly, o ge e-
liable esul s i is manda o y o use a ealis ic nucleon-nucleon
(NN) in e ac ion. Un o una ely, exis ing implemen a ions o
he CDCC me hod do no inco po a e he possibili y o using
hese ealis ic NN in e ac ions, such as Pa is o CD Bonn.
Ne e heless, we ha e ound ha he e ec o using a ealis ic
NN in e ac ion can be well simula ed educing he dep h o
he Gaussian in e ac ion o V0≈−45 MeV. The co esponding
CDCC calcula ion, depic ed by he dashed line in Fig. 2, is seen
o ep oduce ai ly well he Faddee esul wi h he CD Bonn
po en ial.
The co esponding angula dis ibu ions a e shown in Fig. 3.
The uppe and bo om panels co espond o he ela i e
0
20
40
dσ/dΩc.m. (mb/s )
Faddee : CD Bonn
CDCC: modi ied Gaussian
DWBA: modi ied Gaussian
10 20 30 40
θc.m. (deg)
0
10
20
dσ/dΩc.m. (mb/s )
(b) E el=2.5-5 MeV
(a) E el=0.0-2.5 MeV
FIG. 3. (Colo online) Angula dis ibu ion o he b eakup o
11Be on a p o on a ge a 63.7 MeV/nucleon o E el =0–2.5MeV
(uppe panel) and E el =2.5–5.0 MeV (bo om panel) calcula ed
wi hou de o ma ion. The ci cles a e he expe imen al da a o
Re . [1]. The solid line is he Faddee calcula ion wi h he ealis ic
CD Bonn in e ac ion. The dashed line is he CDCC calcula ion
using he modi ied Gaussian in e ac ion (Vg=−45 MeV) and
he do -dashed line is he CDCC calcula ion calcula ed in i s
o de .
ene gy in e als E el =0–2.5 MeV and E el =2.5–5.0MeV,
espec i ely. In each panel, he solid line is he Faddee
calcula ion using he ealis ic CD Bonn in e ac ion, whe eas
he dashed line is he CDCC calcula ion using he modi ied
Gaussian po en ial. I is seen ha bo h calcula ions a e in
easonable ag eemen . The e a e disc epancies below 15◦,
whe e no da a exis ne e heless. Since he calcula ions
including de o ma ion a e pe o med wi hin he DWBA
app oxima ion, we ha e included also in his plo he CDCC
calcula ion pe o med in i s o de , which is equi alen o
a DWBA calcula ion. I is seen ha his calcula ion is e y
close o he ull CDCC esul , jus i ying he use o he Bo n
app oxima ion in his eac ion. This plo shows also e y
clea ly ha he single-pa icle exci a ion mechanism is no
adequa e o desc ibe hese da a, pa icula ly in he highe
exci a ion ene gy in e al.
F om he calcula ions p esen ed in his sec ion, we conclude
ha he CDCC me hod (e en o i s o de ), using an adequa e
e ec i e p-nin e ac ion, p o ides a good app oxima ion o
he mo e sophis ica ed Faddee calcula ion wi h a ealis ic
NN in e ac ion. In he ollowing sec ion, we in oduce he
e ec o co e de o ma ion using he C-ex model desc ibed in
Sec. III.
054613-5

A. M. MORO AND R. CRESPO PHYSICAL REVIEW C 85, 054613 (2012)
B. Calcula ions wi h he co e exci a ion model
To calcula e he ini ial and inal s a es o he p ojec ile
[Eqs. (26) and (27)] we adop he alence-co e po en ial om
Re . [25] (model Be12b) consis ing o a cen al Woods-Saxon
po en ial, wi h adius R0=2.483 m, di useness a=0.65 m,
and pa i y-dependen dep h, wi h V0=−54.239 MeV (V0=
−49.672 MeV) o he e en (odd) wa es. This po en ial is
de o med using a de o ma ion pa ame e β2=0.67, which
co esponds o a de o ma ion leng h o δ2=β2R0=1.64 m.
A spin-o bi e m, using he s anda d Woods-Saxon de i a i e
o m o he adial shape, wi h he same adius and di useness
as he cen al pa and a dep h o Vso =8.5MeV,isalso
included. Fo he 10Be co e we conside only he g ound s a e
(0+) and he i s exci ed s a e (Iπ=2+,Ex=3.368 MeV).
The o bi al angula momen um is unca ed a max =3.
We conside ed con inuum s a es wi h Jπ=1/2+,1/2−,3/2−,
3/2+, and 5/2+.
Wi h he assumed Hamil onian and model space, he
g ound s a e co esponds p edominan ly o a s1/2con igu a ion
coupled o he co e in he g ound s a e (≈85%), bu wi h
a signi ican admix u e o he |2+⊗νd5/2con igu a ion
(≈13%).
This po en ial p oduces also he low-lying na ow eso-
nances 5/2+,3/2−, and 3/2+a ela i e ene gies o 1.2, 2.7,
and 3.2 MeV, espec i ely. These esonances can be iden i ied
wi h he s a es obse ed by Fukuda e al. in he 11Be +
12C eac ion a 70 MeV/nucleon [19]. The 5/2+ esonance
co esponds p edominan ly o a d5/2con igu a ion coupled o
he co e in he g ound s a e, whe eas he 3/2+ esonance has
a dominan 10Be(2+)⊗νs1/2pa en age.
Fo he in e ac ion be ween he alence neu on and
he p o on a ge we use he modi ied Gaussian in e ac ion
ob ained in he p e ious sec ion. Fo he cen al pa o
he 10Be +pin e ac ion we keep he Wa son po en ial
[24]. This in e ac ion is used o calcula e he dis o ed
wa es appea ing in he DWBA ansi ion ampli ude. The
noncen al pa o his in e ac ion, equi ed o allow o he
dynamic exci a ion o he co e, is ob ained de o ming his
cen al po en ial using he same de o ma ion leng h (δ2=
1.64 m).
In Fig. 4we show he calcula ed b eakup c oss sec ion, as
a unc ion o he n-10Be ela i e ene gy, in eg a ed o e he
cen e -o -mass angula ange θc.m.⩽60◦. Figu e 4(a) shows
he con ibu ion o he dominan Jπpa ial wa es o he
calcula ed ene gy spec um, as well as he o al sum o all
included wa es. I is seen ha he b eakup c oss sec ion is
domina ed by he esonan 3/2+and 5/2+con ibu ions and, a
low exci a ion ene gies, by he 3/2−non esonan con inuum.
The 1/2−and 1/2+wa es (no shown in his igu e) gi e
also some con ibu ion a small exci a ion ene gies. I is also
obse ed ha he 3/2− esonance (loca ed a E el =2.7MeV
in his model) has a negligible e ec on he c oss sec ion.
In Fig. 4(b), we show sepa a ely he alence and co e
exci a ion con ibu ions o he angle-in eg a ed b eakup c oss
sec ion. The low-ene gy spec um is mos ly due o he alence
exci a ion, and hence he dynamic co e exci a ion is negligible
a hese exci a ion ene gies. As he exci a ion ene gy inc eases
he e ec o co e exci a ion becomes mo e and mo e impo an .
012345
E el (MeV)
0
20
40
60
dσ/dE el (mb/MeV)
(b)
alence
co e
alence+ co e
0
20
40
60
dσ/dE el (mb/MeV)
(a) 3/2-
5/2+
3/2+
ull
θc.m.=(0o- 60o)
FIG. 4. (Colo online) Di e en ial ene gy c oss sec ion o
p(11Be,p)10Bena 63.7 MeV/nucleon in eg a ed o e he cen e -
o -mass angula ange θc.m.⩽60◦, calcula ed wi h he C-ex model
p oposed in his wo k. The uppe panel shows he indi idual
con ibu ion o he main pa ial wa es and he sum o all included
pa ial wa es. The bo om panel shows he sepa a e con ibu ion
coming om he alence exci a ion and dynamic co e exci a ion
ampli udes. See ex o de ails.
In pa icula , co e dynamic e ec s a e clea ly dominan in he
egion o he 3/2+ esonance and a e also impo an in he
egion o he 5/2+ esonance. We hus expec signi ican
changes in he co esponding angula dis ibu ions.
In Fig. 5we compa e ou esul s o he b eakup c oss
sec ion angula dis ibu ion dσ/dc.m.wi h he expe imen al
da a o Re . [1] con aining con ibu ions wi hin ela i e
neu on–10Be ene gy ange 0–2.5 MeV [Fig. 5(a)] and
2.5–5 MeV [Fig. 5(b)]. Again, we show he sepa a e con-
ibu ions coming om he alence exci a ion (do -dashed
line) and dynamic-co e exci a ion (dashed line), as well
as hei cohe en sum (solid line). In he lowe ene gy
in e al [Fig. 5(a)] he c oss sec ion is domina ed by he
single-pa icle b eakup mechanism. The calcula ed angula
dis ibu ion ep oduces easonably well he shape o he da a,
al hough some o e es ima ion o he absolu e magni ude is
obse ed. On he o he hand, in he highe ene gy in e al
[Fig. 5(b)], bo h he single-pa icle and dynamic co e exci a-
ion mechanisms a e impo an . The sum o bo h con ibu ions
accoun s easonably well o he da a, excep o he i s da a
poin , which is o e es ima ed. The main con ibu ion in his
exci a ion ene gy egion comes om he popula ion o he
3/2+ esonance, as can be expec ed om Fig. 4. These esul s
054613-6
CORE EXCITATION EFFECTS IN THE BREAKUP OF THE ... PHYSICAL REVIEW C 85, 054613 (2012)
0
20
40
60
dσ/dΩc.m. (mb/s )
ull ( alence+ co e)
co e
alence
10 20 30 40
θc.m. (deg)
0
10
20
dσ/dΩc.m. (mb/s )
(b) E el=2.5-5 MeV
(a) E el=0.0-2.5 MeV
FIG. 5. (Colo online) Angula dis ibu ion o he b eakup o
11Be on a p o on a ge a 63.7 MeV/nucleon o E el =0–2.5MeV
(uppe panel) and E el =2.5–5.0 MeV (bo om panel). The ci cles
a e he expe imen al da a o Re . [1]. The dashed-do ed and dashed
cu es ep esen he alence and co e exci a ion con ibu ions,
espec i ely, whe eas he solid line co esponds o hei cohe en
sum.
e idence he impo ance o he co e exci a ion mechanism
and con i m ou ea lie indings using a mo e simpli ied
model [17].
Finally, we no e ha his eac ion has been s udied also by
he au ho s o Re . [26] using an ex ended e sion o he CDCC
me hod, which inco po a es equi alen co e exci a ion e ec s
o hose discussed he e. Howe e , con a y o ou esul s, he
e ec o co e exci a ion was ound o be e y small in ha
wo k. This esul is unexpec ed gi en he la ge de o ma ion o
he 10Be nucleus.
VI. CONCLUSION
In conclusion, we s udied he p oblem o co e exci a ion in
he b eakup sca e ing o halo nuclei. To accoun o his e ec
in a quan i a i e way, we de eloped a co e exci a ion eac ion
model, based on he DWBA app oxima ion, which akes in o
accoun he e ec o co e de o ma ion in he s uc u e o
he halo nucleus, as well as he possibili y o dynamic co e
exci a ion du ing he collision. We showed ha , igno ing
co e- ecoil e ec s, he con ibu ion o he sca e ing ampli ude
a ising om he co e exci a ion can be w i en in e ms o
a supe posi ion o wo-body ampli udes co esponding o he
inelas ic sca e ing o he co e sca e ed by he same a ge .
As an illus a ion o he model, we pe o med calcula ions
o he b eakup o 11Be on p o ons a an inciden ene gy o
63.7 MeV/nucleon. The ini ial and inal s a es a e ea ed
wi hin he pa icle- o o model and hence hey a e conside ed
as a supe posi ion o se e al alence con igu a ions coupled
o he 10Be co e in ei he he g ound s a e (0+)o he
i s exci ed s a e (2+). The noncen al pa o he 10Be +
p o on in e ac ion, which is esponsible o he dynamic co e
exci a ion, is ob ained de o ming he 10Be +p o on po en ial.
We ind ha he co e exci a ion mechanism gi es an
impo an con ibu ion and i s inclusion pe mi s a sui able
desc ip ion o he da a om Re . [1]. We also showed
ha he impo ance o dynamic co e exci a ion becomes
mo e impo an a inc easing exci a ion ene gies and is in
ac essen ial o accoun o he ene gy-in eg a ed angula
dis ibu ion a E el =2.5–5.0MeV.
F om he calcula ions p esen ed in his wo k, we may con-
clude ha hese co e exci a ion e ec s will be also impo an
in o he eac ions induced by weakly bound p ojec iles wi h
de o med cons i uen s. The me hod p oposed he e can p o ide
a use ul and simple es ima e o hese e ec s in hose si ua ions
in which he assump ions o he model (i.e., he alidi y
o he Bo n app oxima ion and he possibili y o neglec ing
co e- ecoil) a e jus i ied.
ACKNOWLEDGMENTS
This wo k has been pa ially suppo ed by he FCT G an
No. PTDC/FIS/103902/2008, by he Spanish Minis e io de
Ciencia e Inno aci´
on unde P ojec No. FPA2009-07653, and
by he Spanish Consolide -Ingenio 2010 P og amme CPAN
(CSD2007-00042).
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