Coupling o b eakup channels using a ans o med ha monic oscilla o basis
A. M. Mo o,1,3 J. M. A ias,1J. Go
´mez-Camacho,1I. Ma el,2F. Pe
´ ez-Be nal,2,*R. C espo,3and F. Nunes4
1Depa amen o de FAMN, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain
2Depa amen o de FAIE, Uni e sidad de Huel a, E-21071 Huel a, Spain
3Depa amen o de Fı
´sica, Ins i u o Supe io Te
´cnico, A . Ro isco Pais, 1049-001 Lisboa, Po ugal
4Uni e sidade Fe nando Pessoa, P ac¸a 9 de Ab il, 4200 Po o, Po ugal
共Recei ed 4 Sep embe 2001; published 12 Decembe 2001兲
The applica ion o a ecen ly p oposed p ocedu e o disc e izing he con inuum o collision p ocesses
in ol ing weakly bound nuclei is s udied. In pa icula , he coupling o b eakup s a es in he collision o d
⫹208 Pb a 50 MeV is discussed. Fo illus a i e pu poses, only he s-wa e componen o he bound s a e o he
deu e on is conside ed, and he s udy is es ic ed o he case o nuclea s-wa e b eakup. The con inuum
disc e iza ion p ocedu e p o ides a basis o ans o med ha monic oscilla o wa e unc ions o accomplish he
necessa y calcula ions. App op ia e con e gence o he elas ic and b eakup c oss sec ions wi h inc easing
dimension o he basis is epo ed. In addi ion, i is shown ha he esul s ob ained con e ge o hose o a
s anda d con inuum disc e ized coupled channels calcula ion, wi h he ad an age ha he con e gence o he
me hod is de e mined by only one pa ame e , namely he dimension o he basis.
DOI: 10.1103/PhysRe C.65.011602 PACS numbe 共s兲: 24.10.Eq, 25.10.⫹s, 25.45.De, 25.60.Gc
The de elopmen o adioac i e beam acili ies has shed
ligh on a a ie y o new nuclea s uc u e p oblems 关1兴
which include neu on and p o on ich nuclei close o he
d ip lines, and in pa icula halo nuclei. Gi en ha hese a e
weakly bound sys ems, hei modeling should necessa ily in-
clude in some way he con inuum pa o he spec um. This
becomes essen ial when s udying nuclea collisions, whe e
he inclusion o he coupling o b eakup channels is no only
necessa y o he desc ip ion o p ocesses leading o ag-
men a ion, bu also when dealing wi h elas ic sca e ing o
ans e eac ions.
Reac ion calcula ions whe e b eakup couplings a e in-
cluded ha e he addi ional complica ion ha b eakup s a es
a e no squa e-no malizable. This p oblem is gene ally su -
moun ed eplacing he s a es in he con inuum by a ini e se
o no malized s a es, and p o ing ha he calcula ion o sca -
e ing obse ables in he ini e basis con e ges when inc eas-
ing he numbe o s a es conside ed, o when modi ying he
pa ame e s which de ine he no malized s a es.
Se e al p ocedu es ha e been de eloped o ob ain a ini e
basis o no malized s a es which desc ibes he con inuum.
One o he mos widely used o hese app oaches is he
me hod o con inuum disc e iza ion coupled channels
共CDCC兲关2兴. I consis s in disc e izing he con inuum by
means o aking ixed in e als, o bins, o k alues. Each bin
is cha ac e ized by a single adial wa e unc ion, which is
ob ained as an a e age o he con inuum wa e unc ions o e
he bin. In his a e aged adial wa e unc ion, he oscilla-
ions o he di e en componen s end o cancel beyond a
ce ain dis ance, and so he bin adial wa e unc ion becomes
no malizable. Fo mally, he CDCC me hod equi es he so-
lu ion o he Sch o
¨dinge equa ion o all 共o , a leas , many兲
ene gies in he con inuum as a p e ious equi emen o cal-
cula e he bin wa e unc ions. Thus a p ac ical CDCC calcu-
la ion equi es o ix he maximum k alue conside ed and
he in e al ⌬ko he bins, so ha he numbe o bins is
gi en by M⫽kmax /⌬k. To demons a e con e gence in a
CDCC calcula ion one has o show ha he sca e ing mag-
ni udes a e no modi ied wi h inc easing maximum ene gy
(kmax) o dec easing he in e al ⌬k. Howe e , when ⌬kis
made smalle , he adius a which he bin wa e unc ions
anish becomes la ge , inc easing he ange o he coupling
po en ial. As a consequence, CDCC calcula ions a e ypically
e y ime consuming. In addi ion, as bo h ⌬kand kmax pa-
ame ize he basis, demons a ing con e gence is by no
means i ial 共see 关3兴 o a ecen s udy o he con e gence
wi hin CDCC兲. Despi e hese di icul ies, he me hod has
been success ully applied o a la ge numbe o nuclea eac-
ions and is one o he mos eliable app oaches o eac ions
in ol ing bina y composi e sys ems.
We ha e ecen ly p oposed he use o a ans o med ha -
monic oscilla o 共THO兲basis as an al e na i e o desc ibe he
e ec o he con inuum 关4,5兴. In Re . 关4兴some applica ions
o simple one-dimensional p oblems a e p esen ed and i is
shown ha , as he numbe o s a es in he THO basis in-
c eases, he eigens a es appea densely packed close o he
b eakup h eshold, al hough some o hem lie a highe ene -
gies. Besides, i is shown ha global s uc u e magni udes
ela ed o he coupling o he con inuum, such as sum ules,
a e e y accu a ely desc ibed using ela i ely small THO ba-
sis s a es. In Re . 关5兴 his s udy was ex ended o he mo e
ealis ic case o he deu e on, showing simila esul s o he
one-dimensional case. In he p esen wo k, we show ha he
same ideas can be use ul in he s udy o nuclea collisions
in ol ing weakly bound nuclei.
In he case o a h ee-dimensional p oblem unde cen al
o ces, he ele an pa o he wa e unc ion is he adial
one, ha can be w i en as R( )⫽
␾
( )/ . In he ollowing
when e e ing o he wa e unc ion we always mean he
adial unc ion
␾
( ). The basic idea o he me hod p esen ed
in Re . 关4兴is o de ine a local scale ans o ma ion 关6兴which
con e s he g ound s a e wa e unc ion o a bound nucleus
*P esen add ess: Cen e o Theo e ical Physics, Yale Uni e si y,
New Ha en, CT 06520-8120.
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␾
b( ) in o a ha monic oscilla o wa e unc ion
␾
0
HO(s). The
unc ion s( ), which de ines he local scale ans o ma ion, is
gi en by
冕
0
兩
␾
b共 ⬘兲
兩
2d ⬘⫽e 共s兲⫺2sexp共⫺s2兲
冑
␲
.共1兲
Nex , one gene a es a se o o hogonal wa e unc ions mul-
iplying he g ound s a e wa e unc ion by he app op ia e
polynomials Pn„s( )…,
␾
n
THO共 兲⫽Pn„s共 兲…
␾
b共 兲,共2兲
such ha he s a e wi h n⫽0 coincides wi h he g ound s a e,
and he s a es wi h n⬎0 desc ibe he con inuum, o o he
bound s a es i hey exis . F om he h ee-dimensional HO
wa e unc ions 关7兴
Pn共s兲⫽
␲
1/4
2兺
k⫽0
N
␣
nks2k,共3兲
␣
nk⫽共⫺1兲k
k!
冑
2共n!兲⌫共n⫹3/2兲
共n⫺k兲!⌫共k⫹3/2兲.共4兲
Finally, one diagonalizes he Hamil onian ma ix using a i-
ni e basis o THO s a es: he esul ing posi i e ene gy eigen-
s a es a e aken as con inuum wa e unc ions. He ea e N
deno es he numbe o con inuum s a es ob ained a e diago-
naliza ion, hus he THO basis dimension is N⫹1.
The aim o his wo k is o in es iga e he adequacy o he
THO basis o desc ibing he e ec o coupling o b eakup
s a es in nuclea eac ions. Fo his pu pose, we ha e consid-
e ed a es case whe e he coupling o he con inuum is el-
e an : he elas ic sca e ing and b eakup o d⫹208 Pb a 50
MeV. In o de o s udy his p ocess wo ing edien s a e
needed: he p o on-neu on in e ac ion and he in e ac ion o
he p ojec ile cons i uen s wi h he a ge . A his s age, we
do no in end o do a ull calcula ion o he e ec s o deu-
e on b eakup no compa e wi h he expe imen al da a bu o
show ha he THO disc e iza ion me hod is an adequa e he-
o e ical ool o he modeling o eac ions in ol ing weakly-
bound nuclei. As an illus a ion, we conside he e a simple
heo e ical scena io: he deu e on is aken o be a pu e ss a e,
he e ec o Coulomb b eakup is neglec ed, and he nuclea
in e ac ion o he p o on and neu on wi h he a ge induce
he coupling o s-wa es b eakup s a es only. Wi hin hese
es ic ions, we pe o m a con e gence s udy o he THO
me hod and compa e wi h he CDCC me hod. The ealis ic
case, whe e he dcomponen is inco po a ed in he g ound
s a e o he deu e on, and bo h Coulomb and nuclea b eakup
o l⫽0 s a es a e coupled in, will be p esen ed in a mo e
ex ensi e publica ion.
Conce ning he s uc u e o he p ojec ile, we desc ibe he
p o on-neu on in e ac ion by means o a Poeschl-Telle po-
en ial,
V共 兲⫽⫺D1
cosh2共
␣
兲,共5兲
wi h dep h 共D兲and ange (
␣
) adjus ed o ep oduce he deu-
e on binding ene gy and mean squa e adius: D
⫽102.85 MeV and
␣
⫽0.9407 m⫺1. A ep esen a ion o
his po en ial is shown in he inse o Fig. 1. The use o his
po en ial p o ides an analy ic g ound s a e wa e unc ion o
he deu e on, which is con enien , al hough no essen ial, o
he calcula ions in he THO basis. The in e ac ion o p o on
and neu on wi h he a ge is aken om he Becche i-
G eenless pa ame iza ion 关8兴, e alua ed a hal o he deu-
e on inciden ene gy.As men ioned abo e, only he coupling
o s-wa e b eakup s a es is conside ed.
Once he g ound s a e adial wa e unc ion is ob ained by
sol ing he Sch o
¨dinge equa ion wi h he app op ia e po en-
ial, Eq. 共5兲in ou case, he unc ion de ining he local scale
ans o ma ion is ob ained by di ec in eg a ion o Eq. 共1兲.
Wi h hese wo ing edien s he THO wa e unc ions can be
gene a ed by using Eq. 共2兲. In Fig. 1 we p esen he THO
basis unc ions, om n⫽0 on⫽5. The wa e unc ion wi h
n⫽0 is he exac wa e unc ion co esponding o he only
bound s a e o he deu e on. I is obse ed ha , as he num-
be o nodes 共n兲inc eases, he wa e unc ions ex end o
la ge dis ances, bu hey all show a smoo h beha io and
decay exponen ially. In p inciple he THO basis is in ini e
bu one can unca e i o a ini e dimension and s udy he
con e gence o di e en obse ables as a unc ion o his
dimension. In he p esen case, he in e nal deu e on Hamil-
onian is diagonalized in a unca ed basis and p o ides a
ini e se o no malized eigens a es which desc ibe app oxi-
ma ely he con inuum. As in he CDCC, he p ocedu e is
use ul only i he numbe o con inuum s a es necessa y o
p o ide a good app oxima ion o he con e ged esul is
small enough. I is wo h o s ess ha uniquely he n⫽0
basis wa e unc ion is an eigens a e o he Hamil onian,
whe eas he n⫽0 basis s a es a e no . Consequen ly he s a e
n⫽0 is decoupled om he o he s a es included in he basis.
In Fig. 2 he eigens a es o he Hamil onian o he case o
a unca ed basis o six s a es (N⫽5) a e plo ed. I can be
FIG. 1. Radial THO wa e unc ions,
␾
n
THO( ), wi h n⫽0 on
⫽5, ob ained by means o he local scale ans o ma ion 共1兲. The
deu e on g ound s a e wa e unc ion (n⫽0) is he solu ion o he
Poeschl-Telle po en ial ske ched in he inse , wi h pa ame e s ha
ep oduce he deu e on binding ene gy 共ma ked wi h dashed line兲
and i s oo -mean-squa e adius.
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no ed ha he ange o hese
␾
k( ) wa e unc ions is de e -
mined by he ange o he las s a e included in he THO
basis. All eigen unc ions o he Hamil onian p esen a
smoo h beha io and a e no malized.
In Fig. 3 we p esen he ene gies o he deu e on Hamil-
onian diagonalized in a THO basis as a unc ion o he basis
dimension. One can e i y ha he ene gy densi y o hese
s a es is highe close o he b eakup h eshold.
An a ac i e ea u e o his me hod is ha he only e-
qui emen o cons uc he THO basis is sol ing he Sch o
¨-
dinge equa ion o he g ound s a e, ei he analy ically o
nume ically. The THO basis is hen ob ained by means o
Eqs. 共1兲and 共2兲. As he con inuum wa e unc ions a e ob-
ained h ough he Hamil onian diagonaliza ion, hei calcu-
la ion does no equi e he in eg a ion o he Sch o
¨dinge
equa ion. Fu he mo e, he sca e ing calcula ion is equi a-
len o a s anda d coupled channels calcula ion wi h bound
s a es, whose in e nal ene gies and wa e unc ions a e gi en
by he diagonaliza ion o he deu e on Hamil onian in his
THO basis. The only disc e e pa ame e o be changed in
o de o in es iga e con e gence is he numbe o s a es o be
included in he basis. Using his inpu , we sol e he coupled
channels calcula ions using he compu e code FRESCO 关9兴.
In Table I we p esen he esul s o a s anda d CDCC
calcula ion o he o al b eakup c oss sec ion, as well as he
a e age inal exci a ion ene gy and he a e age in e se exci-
a ion ene gy, using in bo h cases he b eakup c oss sec ions
as weigh s. We ha e kep ixed he maximum exci a ion en-
e gy a 30 MeV and ha e done calcula ions a ying he num-
be o con inuum bins and, acco dingly, he bin size ⌬k.
In Table II we p esen he same obse ables as in Table I,
bu now using he THO me hod. I is ound ha he con e -
gence wi h espec o he numbe o s a es is e y sa is ac-
o y. Taking he esul s o N⫽10 as e e ence, we ind ha
he THO calcula ions wi h ou , six, and eigh con inuum
s a es gi e simila esul s. E en he THO calcula ion wi h
jus wo con inuum s a es gi es as easonable an es ima e as
he co esponding CDCC.
An unusual ea u e o he THO me hod is ha he diago-
naliza ion o he in e nal Hamil onian in he THO basis may
gi e ise o closed channels, i.e., s a es wi h exci a ion ene -
gies la ge han he sca e ing ene gy. Al hough hese s a es
do no con ibu e di ec ly o he ou going lux, hey migh be
signi ican ly exci ed in he p oximi y o he a ge , hus a -
FIG. 2. Radial eigen unc ions o he Po
¨schl-Telle Hamil onian,
␾
k( ), wi h k⫽0 ok⫽5, ob ained by diagonalizing he Poeschl-
Telle Hamil onian in a six-dimensional THO basis.
FIG. 3. G ound s a e and con inuum ene gies ob ained upon
diagonalizing he deu e on Hamil onian as a unc ion o he dimen-
sion o he THO basis (N⫹1). No e ha o N⬎5 he las s a e is
ou o scale.
TABLE I. Con e gence o he b eakup c oss sec ion (
␴
bu), a -
e age exci a ion ene gy (
具
⑀
⫺
⑀
0
典
), and a e age in e se exci a ion
ene gy 关
具
(
⑀
⫺
⑀
0)⫺1
典
兴in he CDCC calcula ion, wi h espec o he
numbe o con inuum bins (M). All calcula ions we e pe o med
keeping kmax⫽0.85 m⫺1共co esponding o a maximum exci a ion
ene gy o
⑀
max⫽30 MeV) and using a di e en numbe o con-
inuum bins.
M
␴
bu
具
⑀
⫺
⑀
0
典具
(
⑀
⫺
⑀
0)⫺1
典
共mb兲共MeV兲(MeV⫺1)
2 86.8 6.20 0.166
4 86.1 5.81 0.219
6 87.0 5.85 0.222
8 84.8 5.94 0.224
10 82.5 6.04 0.223
12 81.8 6.07 0.224
TABLE II. Calcula ed o al b eakup c oss sec ion (
␴
bu), a e -
age exci a ion ene gy (
具
⑀
⫺
⑀
0
典
), and a e age in e se exci a ion en-
e gy 关
具
(
⑀
⫺
⑀
0)⫺1
典
兴, using he THO basis, o di e en choices o
he numbe o con inuum s a es (N). The column Ncoup indica es
he numbe o con inuum s a es which a e coupled oge he in he
coupled channels calcula ion.
NN
coup
␴
bu
具
⑀
⫺
⑀
0
典具
(
⑀
⫺
⑀
0)⫺1
典
共mb兲共MeV兲共MeV⫺1)
2 2 89.2 3.98 0.273
4 3 83.8 5.48 0.231
6 5 84.0 5.46 0.226
8 7 83.5 5.66 0.224
10 8 83.5 5.85 0.220
10 10 82.8 5.87 0.221
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ec ing he elas ic and b eakup c oss sec ions. The di ec
me hod o sol ing di e en ial equa ions used in FRESCO is
nume ically uns able o hese s a es wi h high exci a ion en-
e gies. The e o e, in o de o in es iga e he impo ance o
hese s a es, R-ma ix coupled channel calcula ions we e pe -
o med. We ound ha s a es wi h exci a ion ene gies abo e
30 MeV play a mino ole and, consequen ly, closed chan-
nels do no a ec he sca e ing obse ables o his eac ion.
Wi h he excep ion o he las ow, he calcula ions p esen ed
in Table II a e pe o med excluding he coupling o closed
channels. The numbe o s a es which a e coupled in each
case is indica ed by he ow labeled Ncoup .
In Fig. 4 we compa e he elas ic di e en ial c oss sec ions
ob ained wi hin he CDCC app oach (
⑀
max⫽30 MeV and
M⫽10 bins o uni o m wid h in k) and he co esponding
THO esul s o N⫽10 共dashed line兲. I is obse ed ha he
ag eemen be ween bo h calcula ions is excellen . We also
p esen he esul o a minimal THO calcula ion, including
only wo s a es 共do ed-dashed line兲. We see ha his e y
schema ic calcula ion does indeed desc ibe he e ec o cou-
pling o he con inuum easonably well. As an illus a ion o
he ele ance o b eakup s a es in he elas ic sca e ing, we
ha e also included he calcula ion wi h he olded po en ial,
i.e., igno ing he coupling o he b eakup channels.
In Fig. 5 he b eakup c oss sec ion as a unc ion o he
exci a ion ene gy o he deu e on is depic ed o he CDCC
and THO app oaches. In he THO me hod, we p esen esul s
wi h wo di e en numbe s o basis s a es, N⫽8共open
ci cles兲and N⫽10 共 illed ci cles兲, in o de o show he con-
e gence o he esul s. In he CDCC case, i is ob ained
di iding he c oss sec ion o each bin by he wid h o he
bin. In he THO case, i is ob ained di iding he c oss sec ion
o each con inuum s a e, co esponding o an ene gy
⑀
iby an
ene gy wid h which is gi en by (
⑀
i⫹1⫺
⑀
i⫺1)/2. The consis-
ency be ween he THO calcula ions wi h di e en numbe s
o s a es and he CDCC calcula ion indica es he alidi y o
he THO me hod o calcula e b eakup exci a ion unc ions.
A ea u e o he THO me hod is ha i p o ides a de ailed
desc ip ion o he con inuum in he low ene gy egion whe e
b eakup is mo e ele an . We ha e checked ha , changing
he deu e on Hamil onian o dec ease 共inc ease兲 he binding
ene gy, hen he b eakup s eng h mo es o lowe 共highe 兲
ene gies, and so do he eigen alues o he Hamil onian in he
THO basis. To achie e his in a CDCC calcula ion, one
needs o adjus he bin size by hand.
In Fig. 6 we show he angula dis ibu ion o he b eakup
c oss sec ion. The THO calcula ion wi h N⫽10 s a es is in
good ag eemen wi h he CDCC calcula ion, especially a
o wa d angles o which he c oss sec ions a e mos impo -
an . The THO calcula ion wi h N⫽2 s a es p o ides a ea-
sonable app oxima ion o he con e ged esul .
FIG. 4. Elas ic di e en ial c oss sec ion angula dis ibu ion as a
a io o Ru he o d c oss sec ion o he eac ion d⫹Pb a 50 MeV.
All calcula ions we e pe o med wi h he Becche i-G eenless p e-
sc ip ion o he p o on- a ge and neu on- a ge op ical po en ials.
FIG. 5. B eakup c oss sec ion as a unc ion o he p o on-
neu on ela i e ene gy in he ou going channel, o he eac ion d
⫹Pb a 50 MeV. The CDCC calcula ion wi h en con inuum bins o
uni o m size in k, up o a maximum exci a ion ene gy o
⑀
max
⫽30 MeV is compa ed wi h wo calcula ions using he THO basis
wi h N⫽8 and N⫽10 s a es.
FIG. 6. B eakup c oss sec ion as a unc ion o he deu e on
cen e -o -mass sca e ing angle, o he eac ion d⫹Pb a 50 MeV.
The CDCC calcula ion wi h en con inuum bins o uni o m size in
k, up o a maximum exci a ion ene gy o
⑀
max⫽30 MeV is com-
pa ed wi h wo calcula ions using he THO basis wi h N⫽2 and
N⫽10 s a es.
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In conclusion, we ha e shown in his Rapid Communica-
ion ha he con inuum disc e iza ion me hod p oposed in
Re . 关4兴can be applied o p oblems o cu en in e es in
nuclea physics, in pa icula o he case o collisions in ol -
ing weakly bound nuclei. The only p e ious equi emen o
he applica ion o he me hod is he knowledge o he g ound
s a e wa e unc ion o he nucleus, ei he in an analy ical o
nume ical o m. Then, Eqs. 共1兲and 共2兲p o ide he THO
basis which can be used o diagonalize he nuclea Hamil-
onian. We ha e pe o med a se ies o schema ic calcula ions
o he case o he d⫹208 Pb elas ic sca e ing and b eakup
a 50 MeV. The compa ison o THO and CDCC esul s has
demons a ed he alidi y o he i s me hod and has shown
ha he THO disc e iza ion me hod wo ks p ope ly e en
hough he basis is unca ed o e y ew s a es. One o he
g ea es ad an ages o he THO me hod is ha i au oma i-
cally dis ibu es he basis s a es in he ele an ene gy e-
gions o he con inuum, o e ing an easy, compu a ionally
e icien , al e na i e o he s anda d CDCC me hod.
This wo k was suppo ed in pa by he Spanish DGICYT
unde P ojec s Nos. PB98-1111 and FPA2000-1592-C03-02
and by he Po uguese Science and Technology Founda ion
unde G an No. SAPIENS/36282/99. F.P.B. wishes o hank
he inancial suppo p o ided by he Spanish Di eccio
´n Gen-
e al de Uni e sidades 共Minis e io de Educacio
´n, Cul u a y
Depo e兲 h ough a g an . We acknowledge use ul discus-
sions wi h R. C. Johnson and I. J. Thompson.
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