In e play Be ween Valence and Co e Exci a ion Mechanisms in he B eakup o Halo Nuclei
A. M. Mo o*and J. A. Lay
†
Depa amen o de FAMN, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain
(Recei ed 27 July 2012; e ised manusc ip ecei ed 20 Oc obe 2012; published 5 Decembe 2012)
The phenomenon o co e exci a ion in he b eakup o a wo-body halo nucleus is in es iga ed. We show
ha his e ec plays a signi ican ole in he eac ion dynamics and, u he mo e, i s in e e ence wi h he
alence exci a ion mechanism has sizable and measu able e ec s on he b eakup angula dis ibu ions.
These e ec s ha e been s udied in he esonan b eakup o 11Be on a ca bon a ge , popula ing he
esonances a 1.78 MeV (5=2þ) and 3.41 MeV (3=2þ). The calcula ions ha e been pe o med using a
ecen ex ension o he dis o ed-wa e Bo n app oxima ion me hod, which akes in o accoun he e ec o
co e exci a ion in bo h he s uc u e o he halo nucleus and in he eac ion mechanism. The calcula ed
angula dis ibu ions ha e been compa ed wi h he a ailable da a [Fukuda e al., Phys. Re . C 70, 054606
(2004).]. Al hough each o hese esonances is domina ed by one o he wo conside ed mechanisms, he
angula pa e ns o hese esonances depend in a e y delica e way on he in e e ence be ween hem. This
is he i s clea e idence o his e ec bu he phenomenon is likely o occu in o he simila eac ions.
DOI: 10.1103/PhysRe Le .109.232502 PACS numbe s: 24.50.+g, 25.40.Ep, 25.60.Gc, 27.20.+n
In oduc ion.—The s udy o exo ic nuclei has played a
key ole in nuclea physics o e he pas 25 yea s. Ou
cu en knowledge o hei peculia s uc u al p ope ies
comes mainly om measu emen s o emo al c oss sec-
ions, ans e , and b eakup eac ions. B eakup eac ions
ha e p o ided use ul in o ma ion on he g ound s a e p op-
e ies, such as binding ene gies, spec oscopic ac o s, and
angula momen um (e.g., Re s. [1,2]). Mo eo e , when
exclusi e measu emen s a e possible, i.e., all ou going
agmen s a e de ec ed a e b eakup, hese expe imen s
can be used o in e spec oscopic p ope ies o he con-
inuum, such as he loca ion and spin o assignmen o
esonan s a es [3–5] and dipole s eng hs [3,6,7].
In he case o halo nuclei, loosely bound exo ic nuclei
composed o a igh ly bound co e su ounded by one o wo
loosely bound nucleons, hese p ocesses ha e been con en-
ien ly modeled using a h ee-body model, comp ising he
wo-body weakly bound p ojec ile and he a ge . This has
mo i a ed he de elopmen and e i al o ew-body heo-
ies, such as he con inuum-disc e ized coupled-channels
(CDCC) me hod [8], he adiaba ic ( ozen-halo) app oxi-
ma ion [9–11], a a ie y o semiclassical app oaches
[12–15] and, mo e ecen ly, he Faddee equa ions [16–19].
In hei s anda d o mula ions, hese me hods assume a
single-pa icle desc ip ion o he alence pa icle ela i e o
he co e. Possible exci a ions o he co e a e neglec ed o , a
mos , aken in o accoun e ec i ely h ough he co e- a ge
op ical po en ial. Al hough his app oach has been used
wi h ela i e success in he analysis o many eac ions, i
has been ecen ly shown [20,21] ha his simpli ied pic u e
is no always accu a e, due o he e ec s o co e exci a ion.
Co e exci a ion can a ec he eac ion p ocess in wo ways.
Fi s , he p esence o co e admix u es in he s a es o he
p ojec ile means ha hese s a es canno be simply ea ed
as single-pa icle s a es calcula ed in some mean ield
po en ial. Second, he in e ac ion o he co e wi h he a ge
may gi e ise o ansi ions be ween hese co e s a es, lead-
ing also o he b eakup o he p ojec ile. Al hough hese wo
e ec s ha e been commonly igno ed in he analysis o
eac ions wi h exo ic beams, some p og ess has been
made in ecen yea s owa d hei inco po a ion in exis ing
eac ion o malisms. Fo example, Summe s e al. [22] ha e
p oposed an ex ended e sion o he CDCC me hod which
ea s he s uc u e o he ew-body p ojec ile wi hin he
pa icle- o o model. Mo e ecen ly, a simple ex ension o
he s anda d dis o ed-wa e Bo n app oxima ion (DWBA)
ampli ude which akes in o accoun hese e ec s app oxi-
ma ely has been p oposed [20,21]. Using his me hod, i was
ound ha dynamic co e exci a ions a e indeed e y impo -
an o desc ibe he b eakup o 11Be on p o ons a
70 MeV=nucleon. Some e ec s ha e been also ound in
he elas ic sca e ing o 8Bon a ca bon a ge [23], using an
ex ension o he adiaba ic model o Re . [10].
Al hough he calcula ions p esen ed in Re s. [20,21]
p o ide clea e idence o he impo ance o co e exci a ion
in nuclea b eakup, he es ic ed ene gy and angula eso-
lu ion o he analysed da a p e en ed a de ailed assessmen
o he ela i e impo ance and he in e play be ween he
alence and co e exci a ion mechanisms. In his Le e , we
p esen new esul s showing ha he p esence o co e
admix u es in he halo nucleus and he subsequen dynamic
co e ansi ions gi e ise o e y dis inc i e e ec s on he
shape and magni ude o he b eakup c oss sec ions’ angula
dis ibu ions. Mo eo e , i is ound ha hese e ec s
depend e y c i ically on he amoun o co e exci a ion
o he halo nucleus. This sensi i i y opens new possibili ies
o ex ac ing spec oscopic in o ma ion o halo and o he
weakly bound nuclei, by compa ing he measu ed angula
dis ibu ions wi h a eliable eac ion model. To illus a e
hese e ec s, we p esen he e calcula ions o he b eakup
PRL 109, 232502 (2012) PHYSICAL REVIEW LETTERS week ending
7 DECEMBER 2012
0031-9007=12=109(23)=232502(5) 232502-1 Ó2012 Ame ican Physical Socie y
o he one-neu on halo nucleus 11Be on a ca bon a ge a a
bomba ding ene gy o 70 MeV/nucleon [4], in pa icula
o he exci a ion o he esonances a Ex¼1:78 MeV and
3.41 MeV. The calcula ions a e pe o med using he
ex ended e sion o he DWBA me hod o Re s. [20,21].
Reac ion model.—We now desc ibe b ie ly he co e-
exci a ion model used he e. We ou line he e he main
o mulas, and e e he eade o Re s. [20,21] o u he
de ails. We conside he inelas ic exci a ion o a p ojec ile
nucleus, ini ially in i s g ound s a e, i
JM, o a s a e 
J0M0
(bound o unbound). Wi hin a wo-body (co e þ alence)
desc ip ion o he p ojec ile, hese wa e unc ions a e
expanded as
JMð~
; ~
Þ¼X

½’ð~
ÞIð~
ÞJM;(1)
whe e he unc ions ’ð~
Þdesc ibe he ela i e mo ion
be ween he alence pa icle and he co e, and IMcð~
Þ
a e he co e eigens a es wi h angula momen um Iand
p ojec ion Mc. The index deno es he se o quan um
numbe s ‘; s; j; Ig, wi h ‘,s, and jbeing he o bi al
angula momen um, he in insic spin o he alence pa i-
cle, and hei sum ( ~
j¼~
‘þ~
s), espec i ely. The unc ions
’ð~
Þand IMcð~
Þdepend on he assumed s uc u e
model, o be speci ied la e .
Consis en ly wi h he assumed wo-body desc ip ion o
he p ojec ile, he ansi ion po en ial o his b eakup
p ocess is he sum o he alence- a ge and co e- a ge
in e ac ions, i.e., VT¼V ð~
R ÞþVc ð~
Rc ;~
Þ. While he
alence- a ge in e ac ion is aken o be cen al and o
depend exclusi ely on he alence- a ge sepa a ion, he
co e- a ge in e ac ion is assumed o depend on he co e
in e nal deg ees o eedom, and can he e o e induce
ansi ions be ween di e en co e s a es. By expanding
his in e ac ion in mul ipoles () and sepa a ing he cen al
(¼0) pa , he DWBA ampli ude, desc ibing he exci a-
ion o he halo nucleus du ing he collision, spli s in o wo
e ms:
TJM;J0M0ð~
K0;~
KÞ¼TJM;J0M0
al þTJM;J0M0
co ex ;(2)
wi h
TJM;J0M0
al ð~
K0;~
KÞ¼hðÞ
~
K0ð~
RÞ
J0M0ð~
; ~
ÞjV ðR Þ
þVð0Þ
c ðRc ÞjðþÞ
~
Kð~
RÞi
JMð~
; ~
Þi;(3)
whe e ~
K(~
K0) is he ini ial ( inal) linea momen um and
ðþÞ
ið~
RÞððþÞ
ð~
RÞÞ he ini ial ( inal) dis o ed wa e desc ib-
ing he p ojec ile- a ge ela i e mo ion. The ansi ion
ampli ude gi en by Eq. (3)( alence ampli ude he ea e )
desc ibes exci a ions be ween di e en alence con igu a-
ions, bu wi hou al e ing he s a e o he co e. This e m is
e alua ed ollowing he s anda d echniques used in
coupled-channels codes. The second e m in (2), deno ed
co e exci a ion ampli ude, con ains he noncen al pa
(>0) o he co e- a ge in e ac ion and can he e o e
p oduce co e ansi ions. Consequen ly, his e m accoun s
o he dynamic exci a ion o he co e du ing he collision.
In Re s. [20,21], i was shown ha his e m acqui es a e y
simple o m when e alua ed in he no- ecoil app oxima-
ion ( ~
Rc ~
R),
TJM;J0M0
co ex ¼X
>0;
hJ0M0jJMi
X
;0
hRJ0
0jRJ
iG
J;0J0~
TðÞ
c ðI!I0Þ;(4)
whe e GðÞ
J;0J0is a geome ic ac o [20,21], RJ
a e he
adial pa s o he ’ð~
Þ unc ions, and ~
TðÞ
c is ela ed
o he co e- a ge wo-body ansi ion ampli ude o a
co e ansi ion IMc!I0M0
co mul ipola i y as
TIMc;I0M0
c
c ¼hIMcjI0M0
ci~
TðÞ
c .
Resul s.—The model has been applied o he esonan
b eakup o 11Be on a 12C a ge a 70 MeV/nucleon,
measu ed a he RIKEN acili y by Fukuda e al. [4]. The
ela i e ene gy spec um o he 10Be þncen e o mass
shows peaks a Ex¼1:78 MeV and Ex¼3:41 MeV.
Thei angula dis ibu ions we e compa ed wi h DWBA
calcula ions, based on he ib a ional collec i e model,
sugges ing a ¼2 ansi ion o bo h s a es. These s a es
we e iden i ied wi h he 5=2þ
1and 3=2þ
1 esonances p e-
dic ed by shell-model calcula ions.
In he p esen wo k, we eanalyze hese da a using he
a o emen ioned co e-exci a ion DWBA model. The s uc u e
o he 11Be nucleus is desc ibed wi hin he pa icle- o o
model (PRM) o Boh and Mo elson, wi h he pa ame e s
gi en by he model Be11b o Re . [24]. This model assumes a
pe manen quad upole de o ma ion o he 10Be co e wi h
2¼0:67.
The 11Be wa e unc ions a e ob ained by diagonalizing
he 11Be Hamil onian in a pa icle þco e basis o he o m
jIð~
Þ’THO
nð‘sÞjð~
ÞiJM, whe e ’THO
nð‘sÞjð~
Þ(wi h n¼1;...;N)
a e a unca ed se o ans o med ha monic oscilla o
(THO) unc ions [25], which a e used as a basis o he
alence- a ge ela i e mo ion. This basis is ob ained by
applying an analy ic local scale ans o ma ion (LST) o
he con en ional ha monic oscilla o basis. The model
space is es ic ed o I¼0þ,2þand ‘2. Fo he
g ound s a e, a basis o N¼15 oscilla o unc ions was
used. Resonan s a es a e iden i ied wi h s abilized ene gies
wi h espec o he basis size (N). The pa ame e s o he
LST a e he same as hose used in Re . [25].
The componen s in ol ed in ou calcula ions allowed by
he p esen model space, along wi h hei espec i e
weigh s (spec oscopic ac o s) a e lis ed in Table I. Also
lis ed in his able a e he shell-model spec oscopic ac o s
ob ained wi h he code OXBASH, using he e ec i e NN
in e ac ion WBT p oposed by Wa bu on and B own [26].
Bo h models p edic e y simila spec oscopic ac o s o
he g ound s a e and he 5=2þ esonance, and only some
PRL 109, 232502 (2012) PHYSICAL REVIEW LETTERS week ending
7 DECEMBER 2012
232502-2
small di e ences a e ound in he 3=2þs a e. The g ound
s a e co esponds p edominan ly o a j10Beð0þÞs1=2i
con igu a ion, wi h some admix u e o he j10Beð2þÞ
d5=2icon igu a ion. The 5=2þs a e is mainly based on
he 10Be g ound s a e. On he o he hand, he 3=2þ eso-
nance is mainly buil on op o he exci ed co e. Acco ding
o his esul , i is expec ed ha he popula ion o he 5=2þ
s a e is mainly due o he alence exci a ion mechanism,
whe eas he exci a ion o he 3=2þs a e will be mos ly due
o a co e-exci a ion mechanism.
To illus a e he sensi i i y o he calcula ion wi h he
s uc u e model, we ha e conside ed wo addi ional models
assuming pu e single-pa icle con igu a ions o he 11Be
g.s. and he 5=2þand 3=2þ esonances. Fo he 11Beðg:s:Þ
we conside a pu e j0þ2s1=2icon igu a ion. Fo he
5=2þ esonance we conside wo single-pa icle models:
(i) j0þ1d5=2i(deno ed SP1) and (ii) j2þ2s1=2i(SP2).
In he o me , he esonance is popula ed by means o a
alence exci a ion mechanism, whe eas in he second
model he exci a ion is due o a pu e co e exci a ion e ec .
Simila ly, o he 3=2þwe conside also wo ex eme
models: (i) j0þ1d3=2i(SP1) and (ii) j2þ2s1=2i
(SP2). The equi ed adial wa e unc ions a e aken om
he PRM calcula ion, con enien ly no malized o one.
The nþ12Cpo en ial was aken om Re . [27]. The
cen al and ansi ion componen s o he 10Be þ12Cpo-
en ial we e gene a ed by a double olding p ocedu e,
con olu ing an e ec i e nucleon-nucleon (NN) in e ac ion
wi h he 10Be and 12Cma e densi ies. The la e we e
aken, espec i ely, om he an isyme ized molecula dy-
namics (AMD) calcula ion o Re . [28] and om he
pa ame iza ion o Re . [29]. Fo he e ec i e NN in e ac-
ion we adop he spin-isospin independen pa o he M3Y
in e ac ion [30] based on he Reid so -co e NN po en ial.
Fo he imagina y pa o he 10Be þ12Cpo en ial we
assume he same geome y as o he eal pa . A eno -
maliza ion ac o was included o ep oduce he elas ic
sca e ing da a o 10Be þ12Ca 59.4 MeV/nucleon om
Re . [31]. Fu he de ails o hese calcula ions will be
p o ided elsewhe e.
In Fig. 1we compa e he calcula ed angula dis ibu ions
wi h he expe imen al da a o Re . [4]. The uppe and
bo om panels co espond o he 5=2þ(Ex¼1:78 MeV)
and 3=2þ(Ex¼3:41 MeV) esonances. I is eadily seen
ha he pu e single-pa icle models SP1 and SP2 do no
ep oduce he shape o he esonances. In he model SP1
(pu e alence exci a ion) he maxima and minima a e
shi ed o smalle angles wi h espec o he da a and he
angula dis ibu ion decays oo as . On he o he hand, in
he model SP2 (pu e co e exci a ion mechanism) he max-
ima and minima a e shi ed o la ge angles. Finally, he
ull PRM model, which includes bo h alence and co e
exci a ion mechanisms and hei in e e ence, he posi ion
o he maxima and minima is e y well ep oduced. I is
also seen ha he absolu e magni ude o he da a is o e -
es ima ed. Excep o his disc epancy in he no maliza-
ion, i is clea ha he shape is app eciably imp o ed wi h
espec o he pu e single-pa icle desc ip ion and ha he
024681012
10 0
10 1
10 2
10 3
10 4
dσ/dΩ (mb/s )
PRM (Be11b)
SP1 [0+ x 1d3/2]
SP2 [2+ x 2s1/2]
10 1
10 2
10 3
10 4
dσ/dΩ (mb/s )
RIKEN da a
PRM (Be11b)
SP1 [0+ x 1d5/2]
SP2 [2+ x 2s1/2]
1.78 MeV (5/2+)
3.41 MeV (3/2+)
FIG. 1 (colo online). Angula dis ibu ion o he Ex¼
1:78 MeV and 3.41 MeV s a es in 11Be. The ci cles a e he
da a om Re . [4]. The cu es co espond o he ex ended
DWBA calcula ions, including co e exci a ion e ec s, using
di e en s uc u e models o he 11Be nucleus. Fo he single-
pa icle models (SP1 and SP2) he esonance con igu a ion is
indica ed in he labels.
TABLE I. Spec oscopic ac o s o he g ound s a e and esonan wa e unc ions o 11Be, acco ding o he pa icle- o o model
(PRM) and he shell-model calcula ions (WBT) p esen ed in his wo k.
S a e Model j0þð‘sÞjij2þs1=2ij2þd3=2ij2þd5=2i
1=2þ(g.s.) PRM 0.857 ... 0.021 0.121
WBT 0.762 ... 0.002 0.184
5=2þ(Ex¼1:78 MeV) PRM 0.702 0.177 0.009 0.112
WBT 0.682 0.177 0.009 0.095
3=2þ(Ex¼3:41 MeV) PRM 0.165 0.737 0.017 0.081
WBT 0.068 0.534 0.008 0.167
PRL 109, 232502 (2012) PHYSICAL REVIEW LETTERS week ending
7 DECEMBER 2012
232502-3
in e e ence be ween he alence and co e exci a ion
mechanisms is c ucial o accoun o he co ec shape o
he oscilla ions.
I is enligh ening o conside sepa a ely he con ibu ion o
he alence and co e exci a ion ampli udes, Eqs. (3)and(4).
These a e depic ed in Fig. 2 o he PRM. In his plo , he
calcula ions ha e been con olu ed wi h he expe imen al
angula esolu ion [4] o a mo e meaning ul compa ison
wi h he da a. As an icipa ed, he 5=2þ esonance is mainly
popula ed by he alence exci a ion mechanism, due o i s
dominan 10Beð0þÞcon igu a ion, whe eas o he 3=2þs a e
he dynamic co e exci a ion mechanism is he dominan one.
I is also seen ha bo h con ibu ions a e ou o phase, and he
in e e ence be ween hem is e y impo an . In ac , none o
hem sepa a ely is able o ep oduce by i sel he posi ion o
he maxima and minima o he da a, whe eas hei cohe en
sum (solid line) ep oduces e y well his pa e n. This esul
illus a es e y nicely he delica e in e play be ween he
alence and co e exci a ion mechanisms in he b eakup o a
de o med halo nucleus, like 11Be.No e ha heweakcon-
ibu ion o he alence mechanism in he 3=2þcase is a
consequence o he small spec oscopic ac o associa ed
o he j0þd3=2icon igu a ion (see Table I). This ac
explains also ha his esonance is e y weakly popula ed
in ans e eac ions, such as 10Beðd; pÞ11Be [32], making he
ex ac ion o spec oscopic in o ma ion di icul om hese
expe imen s. In hese cases, he app oach p esen ed in his
wo k, based on he analysis o b eakup eac ions, p o ides a
powe ul al e na i e o access his in o ma ion.
Conclusions.—In conclusion, we ha e s udied he in e -
play be ween he alence and co e exci a ion mechanisms
in he b eakup o halo nuclei using and using a ecen ly
p oposed ex ension o he DWBA me hod. We ha e shown
ha he p esence o co e admix u es in he ini ial and inal
s a es has a sizable impac in he in e e ence pa e n o he
b eakup c oss sec ion and hence a high sensi i i y on he
unde lying s uc u e model o he halo nucleus. This e ec
has been e idenced o he i s ime in he sca e ing o
11Be on 12Ca 70 MeV/nucleon, whe e we ha e shown ha
he inclusion o hese co e exci a ion e ec s imp o es
signi ican ly he ag eemen wi h he da a [4] and p o ides
e y aluable spec oscopic in o ma ion, which would be
e y di icul o ex ac om o he me hods. Finally, we
emphasize ha , al hough he calcula ions ha e been p e-
sen ed o he 11Be nucleus, we do expec hese e ec s o
be impo an in o he ele an cases, such as in he b eakup
o he odd ca bon iso opes 15;17;19C.
We a e g a e ul o D . Y. Kanada En’yo o p o iding us
he 10Be mic oscopic densi ies and o T. Nakamu a o his
help ega ding he 11Be þ12Cda a and he con olu ion
wi h he expe imen al esolu ion. This wo k has been
pa ially suppo ed by he Spanish Minis e io de Ciencia
e Inno acio
´n unde P ojec No. FPA2009-07653, and by
he Spanish Consolide -Ingenio 2010 P og amme CPAN
(CSD2007-00042). J. A. L. acknowledges a esea ch g an
by he Minis e io de Ciencia e Inno acio
´n.
*[email p o ec ed]
†
[email p o ec ed]
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101
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o al
alence
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10 12
θc.m. (deg)
10 0
10 1
10 2
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1.78 MeV (5/2+)
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FIG. 2 (colo online). Valence (dashed line) and co e (do -
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70 MeV/nucleon, using a pa icle-co e desc ip ion o he 11Be
nucleus. The solid line is he cohe en sum o bo h con ibu ions.
PRL 109, 232502 (2012) PHYSICAL REVIEW LETTERS week ending
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