S abiliza ion me hod in wo-body sys ems wi h
co e exci a ions
J.A. Lay∗, J.M. A ias∗, J. Gómez-Camacho∗,† and A.M. Mo o∗
∗Depa amen o de FAMN, Facul ad de Física, Uni e sidad de Se illa, Apdo. 1065, E-41080
Se illa, Spain
†Cen o Nacional de Acele ado es, A da. Thomas A. Edison, E-41092, Se illa, Spain
Abs ac . The alidi y o he s abiliza ion me hod in co e+ alence sys ems including he possibili y
o exci ing he co e is s udied. A pseudos a e me hod, based on he ans o med ha monic oscilla o
basis, is ex ended o include he co e deg ees o eedom. The me hod is applied o he case o
11Be s uc u e conside ing he 0+g ound s a e and he 2+fi s exci ed s a e o he 10Be co e. The
s abiliza ion me hod is defined in e ms o one pa ame e ha can be chosen ei he disc e e o
con inuous. In he applica ion o 11Be, bo h cases a e analyzed.
Keywo ds: Exo ic nuclei, con inuum disc e iza ion, esonances
PACS: 24.10.Eq, 25.10.+s, 25.45.De
INTRODUCTION
Recen ly, he s udy o eac ions in ol ing loosely bound exo ic nuclei ha e been one o
he mos ac i e fields in Nuclea Physics. This kind o eac ions is known o be s ongly
influenced by he coupling o he unbound s a es o he weakly bound nucleus. The e o e
he e has been a g ea in e es in finding a ealis ic desc ip ion o he con inuum o hese
nuclei.
In eac ion calcula ions, one-neu on halo nuclei a e usually desc ibed using a wo-
body model, comp ising an ine co e and one alence pa icle. This Hamil onian can be
sol ed using pseudos a es (PS). The PS me hod consis s in diagonalizing he Hamil o-
nian in a squa e-in eg able basis. The eigens a es ob ained a e a fini e app oxima ion o
he bound and unbound s a es o he nucleus. Mo eo e , he posi i e ene gy eigens a es
can help o unde s and he s uc u e o he con inuum and i s influence in eac ions in-
ol ing hese nuclei. Se e al bases ha e been es ed in he PS me hod. He e we use a
ans o med ha monic oscilla o (THO) basis, which is o med by applying he ollow-
ing ans o ma ion
s( )=⎡
⎢
⎣
1
1
4+1
γ
√ 4
⎤
⎥
⎦
1
4
,(1)
o he solu ions o he iso opic ha monic oscilla o . This ans o ma ion was fi s in o-
duced in [1]. This basis is easy o calcula e and, in mos o he cases, is able o educe
he minimum numbe o unc ions needed o con e gence o he sca e ing obse ables
wi h espec o o he bases [2].
Beau y in Physics: Theo y and Expe imen
AIP Con . P oc. 1488, 436-440 (2012); doi: 10.1063/1.4759428
© 2012 Ame ican Ins i u e o Physics 978-0-7354-1100-5/$30.00
436
As men ioned abo e, in he desc ip ion o halo nuclei usually an ine co e is consid-
e ed. Howe e , ecen expe imen al and heo e ical de elopmen s sugges ha possible
exci a ions o he co e may a ec no only o he s uc u e bu also o he eac ion dy-
namics. In a ecen wo k [3] we ha e gene alized he THO basis in o de o include he
exci a ions o he co e in a simple pa icle- o o model.
He e we s udy in mo e de ail he so-called s abiliza ion me hod ha allows o iden i y
esonances in he con inuum using PS basis [4]. In his me hod, we assume he e is
a pa ame e ela ed o he ex ension o he basis. The esul ing eigen alues depend
on his pa ame e , so ha hey ypically dec ease in ene gy when he basis ex ension
is inc eased. When an eigen alue c osses he ene gy o a esonance, his beha iou
changes. The ene gy o he eigen alue is s abilized o a ce ain ange o alues o
he pa ame e gi ing ise o a pla eau a ound he ene gy o he esonance. Mo eo e ,
he pseudos a e co esponding o his s abilized ene gy is ound o ep oduce well he
p ope ies o he esonance.
THE THO APPROACH TO PARTICLE-ROTOR MODEL
In he weakly coupling limi , when we conside a co e+ alence sys em including di e -
en exci a ions o he co e, we ha e a Hamil onian o he o m:
H=T +V c( ,
ξ
)+hco e(
ξ
),(2)
whe e hco e(
ξ
) ep esen s he in e nal Hamil onian o he co e, wi h eigens a es
φ
I(
ξ
).
V c( ,
ξ
)is he in e ac ion be ween he pa icle and he co e and i is esponsible o
he coupling be ween he alence pa icle mo ion and he exci a ions o he co e. I
depends on he model assumed o he exci a ions o he co e. In a pa icle- o o model,
we suppose he co e o be pe manen ly de o med. In his case he in e ac ion wi h he
alence pa icle would depend on he s a e o o a ion o he de o med co e and he
ela i e o ien a ion be ween pa icle and o o .
I he coupling is weak we can conside he o al wa e unc ion as a combina ion
o co e s a es and alence configu a ions coupled o he co esponding o al angula
momen um J:
Ψ
ε
;JM( ,
ξ
)=∑
α
RJ
ε
,
α
( )Ysj(ˆ )⊗
φ
I(
ξ
)JM .(3)
Each e m o his sum is called channel and i is cha ac e ized by a gi en se o quan um
numbe s
α
={,s,j,I}.
I we now cons uc a THO unc ion o each channel in he o m:
Φ
α
n,JM( ,
ξ
)=RTHO
n,
α
( )Ysj(ˆ )⊗
φ
I(
ξ
)JM ,(4)
we can eadily diagonalize he Hamil onian in a unca ed basis o N THO unc ions so
ha he final PS will be:
437
1.6 2 2.4 2.8
γ ( m1/2)
0
1
2
3
4
5
6
ε (MeV)
0246810
N
J=5/2+ spec um
FIGURE 1. (Colo online) Eigen alues ob ained om he diagonaliza ion o he 11Be Hamil onian in
a THO basis, as a unc ion o he LST con inuum pa ame e (
γ
) in he le panel, and as a unc ion o
he numbe o oscilla o s a es included in he basis (N) in he igh panel. The dashed line indica es he
ene gy o he 5/2+ esonance and he do ed line, he ene gy o he 10Be(2+)+n h eshold.
Ψ(N)
i,JM( ,
ξ
)=
N
∑
n=1∑
α
ci
n,
α
,JΦ
α
n,JM( ,
ξ
).(5)
APPLICATION TO 11BE
Pa ame e s
γ
and Ncon ol he adial ex ension o he THO basis. The e o e, he
s abiliza ion me hod can be applied wi h bo h pa ame e s. In p e ious wo ks we ha e
used N o his pu pose, whe eas
γ
was al eady fixed in o de o ailo he densi y o
s a es o he ene gy egion o in e es .
The main d awback is ha Nis a disc e e a iable. The e is no gua an ee ha he
poin whe e he ene gy is s abilized coincides wi h a alue o N. Tha is he main eason
o use also
γ
, a con inuous a iable. Howe e , he fi s de i a i e o he e olu ion o
he eigen alue is ela ed o he wid h o he esonance. This implies ha , o na ow
esonances, he ene gy does no a y in a egion nea he s abilized poin and he nea es
alue o Nis a good app oxima ion o he s abilized ene gy.
In o de o analyze he alidi y o
γ
and Nas s abiliza ion pa ame e s in wo-body
sys ems wi h exci a ions o he co e, we apply he PS THO basis o 11Be conside ing a
pa icle- o o model. He e we conside ha he co e 10Be has a quad upole de o ma ion
wi h
β
=0.67. We include only he fi s exci ed s a e o he co e 10Be, 2+, in addi ion
o i s 0+g ound s a e. Fu he de ails o he po en ial can be ound in [5]. The fi s
esonance in 11Be has spin and pa i y 5/2+. In figu e 1, we show he spec um ob ained
a ying bo h
γ
and Npa ame e s. We see ha he s abiliza ion appea s a he ene gy
o he esonance (1.2 MeV abo e he 10Be(0+)+n h eshold), so ha he s abiliza ion
me hod emains alid in his exci ed co e amewo k.
438
0102030
( m)
-0.2
0
0.2
0.4
R( )
N=10 γ=1.84 m1/2
N=10 γ=2.75 m1/2
N=5 γ=1.84 m1/2
|0+ x d5/2 >
|2+ x s1/2 >
|2+ x d3/2 >
|2+ x d5/2 >
FIGURE 2. (Colo online) Radial wa e unc ions o he 5/2+ esonance in 11Be o h ee di e en
s abilized se o pa ame e s Nand
γ
.
Howe e i is necessa y o check how ep esen a i e a e he co esponding eigens a es
when we a y Nand i is no possible o ob ain he exac s abilized poin . In o de o
do so, we show in figu e 2, wo di e en wa e unc ions o wo di e en alues o N
close o his poin . We also compa e wi h a wa e unc ion whe e we a y
γ
o a simila
Nwhe e we can ge he exac pa ame e o he s abiliza ion. We see ha he main
di e ence be ween he h ee wa e unc ions is i s ex ension, so ha he in e io is always
he same, hus e aining he p ope ies o he esonance. The e o e, he selec ion o he
exac pa ame e o s abiliza ion is no c ucial o ob ain a pseudos a e ep esen a i e o
he esonance.
This fi s esonance has mainly a d5/2configu a ion o he halo neu on wi h he
10Be co e in i s 0+g ound s a e. The e o e i is a case e y simila o he wo-body case
wi h an ine co e. A mo e complica ed case a e esonances whose main configu a ion
is bound. This is he case o he fi s 3/2+ esonance in 11Be. The main con ibu ion
o his esonance is an s1/2wa e wi h he 10Be co e in i s 2+exci ed s a e. This
esonance is below he 10Be(2+)+n h eshold so ha his configu a ion is e ec i ely
bound. Ne e heless, we find in figu e 3 ha he eigen alues a e also s abilized o he
ene gy o his esonance (3.0 MeV abo e he 10Be(0+)+n h eshold).
CONCLUSIONS
We ha e seen how we can cons uc a THO basis o wo-body sys ems wi h exci ed
co e in o de o apply he PS me hod. The s abiliza ion me hod is used o iden i y he
esonan ene gies o a wo-body halo nucleus, including he e ec o co e exci a ions.
As an example, he me hod has been applied o he 5/2+ esonance in 11Be, using a
pa icle- o o model o desc ibe his nucleus. The s abiliza ion in he e olu ion o he
eigenene gies p o ides he alue o he esonance, as seen in wo-body sys ems wi h an
439
1.6 2 2.4 2.8
γ ( m1/2)
0
1
2
3
4
5
6
ε (MeV)
0246810
N
J=3/2+ spec um
FIGURE 3. (Colo online) Eigen alues ob ained om he diagonaliza ion o he 11Be Hamil onian in a
THO basis, as a unc ion o he LST con inuum pa ame e (
γ
) in he le panel, and as a unc ion o he
numbe o oscilla o s a es included in he basis in he igh panel. The dashed line indica es he ene gy o
he 3/2+ esonance and he do ed line, he ene gy o he 10Be(2+)+n h eshold.
ine co e. We also ound ha he co esponding pseudos a es o di e en s abilized se s
o pa ame e s a e ep esen a i e o his esonance. E en using he disc e e pa ame e N
whe e we canno selec he he exac alue o s abiliza ion, we ob ain good esul s.
Fo he 3/2+ esonance in 11Be we ound he same s abiliza ion pa e n. The e o e
he s abiliza ion me hod applies also o esonances whose main configu a ion is bound
wi h espec o he co esponding co e+ alence h eshold. Mo eo e , we no e ha he
3/2− esonance p edic ed by he pa icle- o o model used he e gi es simila esul s,
al hough we ha e no include i in he discussion [3].
ACKNOWLEDGMENTS
This wo k is p esen ed on he occasion o F anco Iachello’s 70 h bi hday. This wo k
has been pa ially suppo ed by he Spanish Minis e io de Ciencia e Inno ación and
FEDER unds unde P ojec s No. FIS2011-28738-c02-01, No. FPA2009-07653, and
No. FPA2009-08848, by he Spanish Consolide -Ingenio 2010 P og amme CPAN
(CSD2007-00042), and by Jun a de Andalucía (FQM160, P07-FQM-02894). J.A.L.
acknowledges a esea ch g an by he Minis e io de Ciencia e Inno ación.
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