PHYSICAL REVIEW C 75, 054611 (2007)
Semiclassical calcula ion o hea y-ion sca e ing in he chao ic egime
C. H. Dasso,1M. I. Galla do,1and M. Sa aceno2,3
1Depa amen o de F´
ısica A ´
omica, Molecula y Nuclea , Uni e sidad de Se illa, Apdo. 1065, E-41080 Se illa
2Depa amen o de F´
ısica, Comisi´
on Nacional de Ene g´
ıa A ´
omica, A . del Libe ado 8250, Buenos Ai es, A gen ina
3Escuela de Ciencia y Tecnologia, Uni e sidad Nacional de San Ma in, Alem 3901(B1653HIM), Villa Balles e , A gen ina
(Recei ed 20 Sep embe 2006; published 18 May 2007)
The semiclassical app oach has p o en o be a mos aluable ool o he cons uc ion o he sca e ing ma ix
and accu a e e alua ion o c oss sec ions in a la ge a ie y o hea y-ion collision p oblems. In i s amilia
implemen a ion, howe e , i s use is es ic ed o wha is now known as he “ egula egime”, as i makes use o
classical eac ion unc ions ha mus be con inuous and in e polable. In his pape we iden i y wha e sion o
he semiclassical o malisms may be especially sui able o ex ension in o he chao ic egime ha de elops a
ene gies close o he Coulomb ba ie . We also show he c ucial ole o he abso p i e pa o he ion-ion po en ial
o e ain he use ulness o he semiclassical me hods unde condi ions o i egula i y.
DOI: 10.1103/PhysRe C.75.054611 PACS numbe (s): 25.70.−z, 21.10.Re, 24.10.−i, 24.60.Lz
I. INTRODUCTION
The way o e ec i ely compu e sca e ing quan i ies wi hin
he semiclassical app oxima ion in he egula egime has
been known o many yea s. The semiclassical S-ma ix
cons uc ion o Mille [1] achie es he goal o exp essing
he S-ma ix elemen s pu ely in e ms o classical sca e -
ing ajec o ies wi h gi en bounda y condi ions. The main
quan um mechanical e ec inco po a ed in his kind o
calcula ions is he in e e ence be ween di e en ajec o ies
and his simple ea u e allows accu a e calcula ions o many
eac ions in he a omic, molecula [1] and nuclea [2] ealms.
In he egula egime he only issue is he alidi y o he
semiclassical app oxima ion i sel and he accu a e nume ical
calcula ion o classical ajec o ies. When he sca e ing is
chao ic—an almos ine i able occu ence i a eac ion ba ie
is p esen —many new ea u es a ise. F om he nume ical poin
o iew, howe e , he main di icul y lies in he ac ha a
e y la ge numbe o sca e ing ajec o ies can be ound o
ul ill he bounda y condi ions speci ying a gi en eac ion
channel and i is no a all s aigh o wa d o expec ha
hei in e e ence will esul in he eliable compu a ion o
obse ables.
In his pape we apply Mille ’s me hod o he semiclassical
calcula ion o he sca e ing o hea y ions when he adial
mo ion is coupled o a ib a ional shape deg ee o eedom. We
ha e p e iously explo ed [3,4] se e al aspec s o his p oblem,
bo h classical and quan um mechanical, in he con ex o
a simple model ha se s co ec ly he nuclea sizes and
in e ac ion scales.
This pape is o ganized as ollows. In Sec. II we ecall he
model ha was used in Re s. [3,4] and in Sec. III we p o ide
an o e iew o i s classical cha ac e is ics. The semiclassical
p ocedu es a e p esen ed in Sec. IV, whe e we u ilize he
ini ial alue ep esen a ion (IVR) [5] o compu e he S-ma ix
s essing he ole o he abso p ion as a egula izing ac o
in he chao ic egime. A summa y and conclusions close he
pape in Sec. V.
II. THE MODEL
The ela i e mo ion o wo hea y ions is desc ibed in
i s app oxima ion by a po en ial model whose mo e ele an
ea u e is he ba ie c ea ed by he balance be ween Coulomb
and cen i ugal epulsion and he nuclea a ac ion. When
shape deg ees o eedom o he nuclea su ace a e conside ed,
he ela i e mo ion in e ac s wi h hei dynamics and he
sca e ing is hen desc ibed by a se o coupled equa ions.
The de o ma ions o he nuclea shape a e desc ibed by he
mac oscopic a iables αλµ which cha ac e ize de ia ions om
he equilib ium su ace acco ding o
R(θ,ϕ)=R◦
1+
λµ
αλµY∗
λµ(θ,ϕ)
.(2.1)
The in insic deg ees o eedom associa ed wi h hese a i-
ables can be signi ican ly exci ed in pe iphe al collisions [6].
Consequen ly, hey play an impo an ole in he modula ion
o he ion-ion in e ac ion ha a ec s ba ie p ocesses, such
as hose leading o usion [7].
As a simple model we conside he coupling o he ela i e
mo ion o wo ions o an in insic ha monic mode, exp essed
by an e ec i e Hamil onian o he o m
H()( , p, α, )=H()
el ( , p)+Hin (α, )+Vcoup(α, ).
(2.2)
He e is he dis ance be ween he cen e s o mass o he
colliding sys ems and αis a dimensionless a iable ha
measu es he (small) ampli ude o he ib a ional mo ion. The
a iables p,  a e, espec i ely, hei conjuga e momen a. We
ake
H()
el ( , p)=p2
2m+(+1)¯h2
2m 2+U( ),(2.3)
Hin (α, )=Cα2
2+2
2D,(2.4)
0556-2813/2007/75(5)/054611(6) 054611-1 ©2007 The Ame ican Physical Socie y
C. H. DASSO, M. I. GALLARDO, AND M. SARACENO PHYSICAL REVIEW C 75, 054611 (2007)
whe e mis he educed mass, is he angula momen um
and Cand D he es o ing o ce and mass pa ame e s o he
collec i e ib a ion. These las wo quan i ies a e ela ed o he
ene gy ¯hω and de o ma ion pa ame e βo he mode by
C=¯hω
2β2,D=¯h
2ωβ2.(2.5)
The eal po en ial U( ) ep esen s he combined e ec s o
he nuclea and Coulomb in e ac ions. Fo conc e eness he
nuclea po en ial is aken as a Woods-Saxon e sion o he
Ch is ensen-Win he empi ical po en ial o [8]. The Coulomb
po en ial is sc eened a la ge dis ances. The e m ha couples
he in insic and ela i e mo ion a iables, Vcoup, a ises om
he Coulomb and su ace-su ace nuclea in e ac ions be ween
p ojec ile and a ge . In leading o de bo h hese con ibu ions
a e p opo ional o he de o ma ion ampli udes. I a low
mul ipola i y λ o he mode is speci ied, i is possible wi hin
he p esen scheme o in oduce an e ec i e Coulomb o m
ac o ha ep oduces a mo e app op ia e adial dependence,
Vcoup( , α)≈−R◦
∂VN
∂ +3Z1Z2e2
(2λ+1)
Rλ
◦
λ+1α. (2.6)
No e ha he di e en µ-componen s o a mode o mul i-
pola i y λha e been combined in an e ec i e “monopole”
ampli ude. The ac ual mul ipola i y o he mode is he e o e
aken in o accoun h ough he adial dependence o he
Coulomb componen o he o m ac o . This p ocedu e,
jus i ied in a coupled-channels app oach because o hei
degene acy in ene gy, is also sui able o he head-on case
we ea below. Howe e he main jus i ica ion o hese
conside a ions is o make possible a ealis ic classical analysis
ha is no encumbe ed by he need o display many deg ees o
eedom.
When he nuclea densi ies o e lap signi ican ly a la ge
numbe o channels open up ha emo e lux om he deg ees
o eedom ea ed explici ly. This is ou inely ea ed in
quan al calcula ions by adding an imagina y pa o he ion-ion
po en ial. We use he e an abso p i e po en ial o Woods-Saxon
ype
W( )=− W0
1+exp −RW
aW
.(2.7)
The pa ame e s RW=4 m and aW=1 m ha e been chosen
so ha he abso p ion ac s mos ly inside he po en ial pocke s
bu hei alues a e o he wise no c i ical. Wi h his ixed
geome y he s eng h o he abso p ion will be adjus ed only
by he selec ed alues o W0.
III. OVERVIEW OF CLASSICAL FEATURES
The classical sca e ing ajec o ies a e be e desc ibed in
he ac ion-angle a iables o he ha monic mode. They a e
ela ed o α,  by
α=(2n+1) ¯hωC−1cos φ,
(3.1)
=(2n+1) ¯hωD sin φ.
A his poin nis a con inuous a iable, which will e en ually
become disc e ized a in ege alues in he quan um ea men .
Incoming ajec o ies a e de e mined a i=∞by he
ini ial ela i e momen um pi=−
√2m(E−(ni+1
2)¯hω) and
by he oscilla o a iables φi,n
i. Thus a cons an E he
sca e ing is cha ac e ized by he classical map ni,φ
i→
n ,φ
.1
Using his model we ha e s udied [3] he chao ic laye ha
de elops close o ba ie ene gies and which b ings d as ic
consequences in he o ganiza ion o he classical mo ion. We
summa ize hose esul s in Fig. 1, whe e we plo he eac ion
unc ion n (E o ,φ
i) o a ixedni=0 a ene gies below and
abo e he Coulomb ba ie . The wo pic u es gi e an idea
o he way in which he ba ie egion is dis up ed by he
coupling. In he uncoupled case he pic u e would be uni o mly
whi e (n =ni=0) and he e would be a sha p s aigh line
co esponding o he singula ajec o y a he op o he
ba ie (E≈56 MeV). This singula i y e lec s he ac ha
he uns able poin a he op o he ba ie is a bound mo ion and
can only be eached asymp o ically om he sca e ing egion.
The impo an ac is ha his singula ajec o y sha ply
1To emo e he ee o a ion o he angle φwi h equency ωa
di e en a iable φ→φ−ω is used so ha φi,φ
a e cons an in
he asymp o ic egion.
FIG. 1. O e all s uc u e o he
sca e ing map. The shades o g ay gi e
he alue o n (whi e o n =0, black
o n ≈8) o ni=0. The le plo
co esponds o β=0.01 and he igh
one o β=0.04.
054611-2
SEMICLASSICAL CALCULATION OF HEAVY-ION ... PHYSICAL REVIEW C 75, 054611 (2007)
sepa a es he classical mo ion ha occu s abo e and below he
ba ie . Small alues o he coupling (β=0.01) b ing abou he
possibili y o exci a ion (n = ni) along wi h a mos ly smoo h
dependence on φi. The singula egion is sligh ly dis o ed bu
s ill p o ides a clea sepa a ion o ajec o ies ha sca e
below o abo e he ba ie . La ge alues o he coupling
(β=0.04) s a o display he ull complexi y o he ba ie
egion. The “ba ie ” now ex ends o e se e al MeV and is
cha ac e ized by egions o smoo h dependence sepa a ed by
singula egions wi h s uc u e on mul iple scales. Well abo e
and well below his chao ic laye he eac ion unc ions a e
smoo h and he ajec o ies ha e bu one u ning poin ei he
ou side o inside he po en ial pocke . Wi hin he i egula laye
one inds smoo h islands cha ac e ized by ajec o ies wi h
mo e u ning poin s and o inc easing complexi y which a e
esponsible o he esul ing ac al s uc u es in he eac ion
unc ions.
IV. SEMICLASSICAL APPROXIMATION
The classical sca e ing map has as an exac quan iza ion
he uni a y S-ma ix n |S(E)|ni ha yields he ampli ude
o popula ing he oscilla o s a e n , when he incoming
p ojec ile is p epa ed in he s a e ni. Bo h s a es a e now
labeled by in ege s ni,n
. The calcula ion o he quan al
S-ma ix is done nume ically by means o coupled channel
calcula ions ha we ha e pe o med in [4]. These calcula ions
allow us o compa e wi h he semiclassical esul s and
o judge he special di icul ies encoun e ed in he chao ic
egion.
An exp ession o he semiclassical S-ma ix was gi en by
Mille [1]
n |S(E)|nisc =
γ2πi∂n
∂φi1
2
exp(iγ(n ,n
i)),(4.1)
whe e
γ(n ,n
i)=−νγ
π
2−γ dp
¯h+φdn(4.2)
and νγis he addi ional Maslo in ege ha akes in o accoun
he poin s whe e he p e ac o ampli ude di e ges. The sum
in ol es all ajec o ies γ ha sa is y Hamil on’s equa ions
wi h he bounda y condi ions app op ia e o he eac ion
channels: o al ene gy E, ini ial and inal alues ni,n
o
he ac ion a iable o he oscilla o . All quan i ies can be
calcula ed o each sca e ing ajec o y en i ely in e ms o
classical mechanics. These quan i ies, al hough no ou inely
calcula ed in classical eac ion codes, a e easily inco po a ed
along wi h he in eg a ion o ajec o ies.
This app oach has wo impo an d awbacks when one
wan s o apply i o i egula sca e ing. One is he sea ch o
ajec o ies ha sa is y bounda y—and no ini ial—condi ions,
and he second is ha i di e ges a he classical ainbows,
and canno p o ide eliable alues in he classically o bidden
egions. In he egula egime he e a e only a ew (mos ly wo)
independen ajec o ies con ibu ing, he nume ical sea ch is
qui e easible and he me hod has yielded p ecise es ima es o
exci a ion p obabili ies o o a ional bands [9–11].
I is now clea wha is he essen ial addi ional di icul y ha
he p esence o chao ic beha io ep esen s o his ype o
calcula ions. The e y e a ic na u e o he eac ion unc ion
means ha he numbe o classical ajec o ies ha sa is y
he bounda y condi ions n ,n
iis no well de ined (i will
inc ease as we inc ease he numbe o ajec o ies pe in e al
o φi). The sea ch o hese ajec o ies is inc easingly delica e
nume ically and he p esence o many ainbows will make his
calcula ion inaccu a e a many poin s. Mo e c ucially he sum
in Eq. (4.1) will ha e o ake in o accoun he in e e ence
o many ajec o ies wi h widely di e en ac ions. Ques ions
abou he con e gence and eliabili y o his p ocedu e a e
e y se e e. F om he p ac ical side i is ce ainly no e y
economical o ha e o compu e and add an exponen ially
inc easing numbe o sca e ing ajec o ies o come up wi h
a numbe ha is a bes an app oxima ion o he quan um
esul . This s a e o a ai s is qui e simila o wha happens
o he compu a ion o ene gy le els om pe iodic o bi s
[12] whe e an exponen ially inc easing numbe o pe iodic
ajec o ies ha e o be added o ob ain a easonable desc ip ion
o he spec um o bound sys ems in e ms o classical
mechanics.
Ano he semiclassical app oach ha mi iga es some o
hese p oblems is he ini ial alue ep esen a ion o he
S-ma ix [1,5]:
n |S(E)|ni= 1
2π2π
0
dφi∂φ
∂φi1
2
exp(iF(φi,n
i)),(4.3)
whe e F(φi,n
i) is a new phase unc ion
F(φi,n
i)=−φ˙
n+ ˙
p
¯hd +φ (φi,n
i)
×[n (φi,n
i)−n ]−µπ
2
and µano he opological in ege index associa ed wi h he
ajec o y. The phase F(φi,n
i) can be eadily calcula ed om
a classical in eg a ion o Hamil on’s equa ions.
Thein eg alinEq.(4.3) can be pe o med analy ically by
s a iona y phase p o ided he saddle poin s a e su icien ly
sepa a ed, in which case each poin con ibu es one e m
in he sum in Eq. (4.1). Close o he ainbows, wo saddle
poin s coalesce and he in eg al yields an Ai y co ec ion o
he di e gen p e ac o , hus p o iding eliable ampli udes
also in he o bidden egion. I , on he o he hand, he
in eg al is calcula ed nume ically by a uni o m sampling o
he in e al φi=0, ... ,2πwi h many ajec o ies he Ai y
pa e n eme ges au oma ically. The undamen al ad an age o
he IVR is hen ha he sea ch p oblem has been a oided,
he di e gence a he ainbows has been egula ized in
exchange o a uni o m sampling o he ini ial unobse able
angle φi. Howe e , he essen ial di icul y a ising om he
addi ion o a la ge numbe o in e e ing ajec o ies subsis s
unaba ed.
We show in Fig. 2 he di e ences be ween he wo semiclas-
sical me hods, as hey apply o his model in he egula egion,
and how hey compa e wi h he exac coupled-channel esul s.
The IVR was ob ained sampling φiwi h ≈2000 ajec o ies. As
expec ed, he only signi ican di e ences occu nea he edges
054611-3
C. H. DASSO, M. I. GALLARDO, AND M. SARACENO PHYSICAL REVIEW C 75, 054611 (2007)
FIG. 2. (Colo online) Compa ison o he wo semiclassical
app oxima ions wi h he exac esul s in he egula egion. The
shaded a ea is he semiclassical esul using he IVR. The wo Mille ’s
S-ma ix compu ed as a con inuous unc ion o n o show mo e
clea ly he esul ing in e e ence pa e n. A in ege alues i can be
compa ed o he quan um esul s (hea y do s).
o he allowed egion and he IVR is almos indis inguishable
om he exac esul , e en in he o bidden egion.
The IVR me hod has yielded excellen esul s in calcu-
la ions in a omic and molecula physics and in e es in i
has e i ed in ecen yea s in connec ion o i s applica-
ion o complex molecula s uc u es. Fo a ecen e iew
o he me hod and i s applica ions see [5]. This is hen
he me hod we p opose o calcula ions in he chao ic
egime. To sample he ex eme a ie y o ajec o ies we
es ima e he in eg al in Eq. (4.3) by disc e izing φiand
p opaga ing up o 20000 ajec o ies. Figu e 3illus a es he
ad an ages and he di icul ies o his semiclassical p ocedu e.
I shows esul s in he egula and he chao ic egimes a E=
84 MeV and a E=74.5 MeV wi h β=0.04. The op
pa shows how he ini ial se o condi ions ni=4,φ
i=
0,...,2πis mapped by he sca e ing p ocess in o he cu e
n ,φ
. (No ice ha , jus as i was also done in Re s. [3,4],
un ealis ic bounda y condi ions ha e been chosen o imp o e
he pedagogical alue o he illus a ion.) The bo om pa
gi es he IVR esul o he exci a ion p obabili ies oge he
wi h he quan um esul s om a coupled-channel calcula-
ion. The pu ely classical esul ob ained by simply adding
p obabili ies is also shown. In his pa o he calcula ion
he e is no abso p ion (W0=0) and he e o e he S-ma ix
is uni a y wi hin he popula ed channels and in he classi-
cal and semiclassical calcula ions all ajec o ies con ibu e
uni o mly.
In he egula case he sca e ing occu s well abo e he
ba ie and he ajec o ies ha e a single u ning poin . The
esul s can be well unde s ood in e ms o he in e e ence o
jus wo ajec o ies which p oduce he in e e ence pa e n
supe imposed on he classical esul . The Ai y pa e n a ising
om he in e e ences nea he ainbows is clea ly isible. The
ag eemen wi h he esul s o a coupled channel calcula ion is
sys ema ically excellen .
In he chao ic case a E=74.5 MeV he si ua ion is
adically di e en . The IVR esul is essen ially andom and no
ag eemen can be disce ned wi h he exac esul s. The eason
appea s as soon as one looks a he classical o ganiza ion o
he sca e ing ajec o ies: each sca e ing channel is ed by
a la ge amoun o ajec o ies all wi h di e en phases and
ampli udes. Because o he ini e sampling o ini ial ajec o ies
i is impossible o desc ibe well he complica ed in e e ence
pa e n.
To ci cum en hese di icul ies we ake ad an age o he
ac ha no ealis ic ion-ion po en ial is pu ely eal. The
FIG. 3. (Colo online) Semiclassi-
cal calcula ion o he exci a ion p oba-
bili ies o he di e en eac ion chan-
nels in he absence o abso p ion, W=
0 MeV. The eac ion conside ed is
40Ca +40Ca and he ini ial s a e o
he ha monic mode wi h ¯hω =2MeV
and β=0.04 is speci ied by ni=4.
The wo ames o he le co espond
o a o al ene gy o E=82.0MeV
( egula egime) while he wo ames
o he igh a e o E=74.5MeV
(chao ic egime). In each column he
op ame shows he ini ial alues
φi[0,2π], ni=4 mapped o he inal
ones φ ,n
. The lowe ames show
he classical dis ibu ion o p obabili y
(unshaded his og am) and he IVR
esul (shaded his og am). The do s
gi e he quan al esul s om a ull
coupled-channel calcula ion.
054611-4
SEMICLASSICAL CALCULATION OF HEAVY-ION ... PHYSICAL REVIEW C 75, 054611 (2007)
FIG. 4. (Colo online) Semiclassical
calcula ion o he exci a ion p obabili ies
o he di e en eac ion channels in he
p esence o abso p ion. This igu e is
analogous o Fig. 3, excep o he alue o
W0=6 MeV. The e ec o he abso p ion
is ep esen ed by d awing each poin
wi h an a ea p opo ional o i s su i al
p obabili y ργ.
complex op ical po en ial is eadily inco po a ed in o s anda d
hea y-ion coupled channel eac ion codes and esul s in non
uni a y sca e ing ma ices. Classically—and in leading o de
in W0—i is cus oma y o inco po a e he e ec s o abso p ion
by assigning o each classical ajec o y a su i al p obabili y
ργ=exp 2
¯h∞
−∞
W( γ( ))d ,(4.4)
whe e W( ) is he imagina y po en ial in Eq. (2.7). Changing
he alue o W0 esul s in a selec i e emo al o he mo e
complica ed ajec o ies ha ha e long dwell imes in he
in e ac ion egion. In ac , he ole o W0is qui e analogous
o ha o a small imagina y pa in he ene gy o bound
s a e calcula ions in he chao ic egime. I egula izes he
pe iodic o bi sums a he expense o in oducing an ene gy
smoo hing. He e i appea s na u ally as a esul o a physical
mechanism and i s s eng h in a gi en si ua ion is a measu e
o he obse abili y o chao ic e ec s.
In Fig. 4we show he same esul s as in Fig. 3bu now wi h
an added abso p i e po en ial wi h s eng h W0=6MeV.We
ha e ansla ed i s e ec by d awing he poin co esponding
o each ajec o y wi h an a ea p opo ional o he su i al
p obabili y ργ. In he egula case all ajec o ies a e well
abo e ba ie and hey all sha e a simila po ion in he adial
egion whe e he abso p ion is ac i e. Thus hey a e all damped
essen ially by he same ac o . The e o e he whole in e e ence
pa e n is simply scaled by his ac o . This is clea ly seen
in he pa e n o exci a ion p obabili ies, which is almos
iden ical o ha in Fig. 3, jus educed by a cons an ac o . The
ag eemen wi h he quan um calcula ion, which now includes
he imagina y pa o he op ical po en ial, emains excellen .
Abso p ion ac s in a e y di e en way in he chao ic case.
The ajec o ies ha eed one gi en channel a e now o widely
di e en na u e, he simple jus explo ing he in e ac ion
egion once, while he mo e complica ed can ha e many
u ning poin s and spend long imes in i . They a e he e o e
damped wi h e y di e en ac o s, hus changing subs an ially
he esul ing ampli ude. We clea ly obse e his ac in he op
igh pa o Fig. 4whe e he complica ed ajec o ies a e now
absen . As he s eng h o W( ) inc eases, ou o he many
ajec o ies eeding one gi en channel and in e e ing wi h
almos andom pa e ns only he simples su i e he damping.
The in e e ence is mainly p oduced by wo simple b anches
bu some channels a e s ill ed by he emaining i egula
ajec o ies. The esul ing pa e ns a e now in good ag eemen
wi h he coupled-channel calcula ion.
V. CONCLUSIONS
The lesson o be lea ned om his exe cise is wo old. F om
he poin o iew o he semiclassical compu abili y o chao ic
sca e ing we conclude ha he s aigh o wa d supe posi ion
o sca e ing ajec o ies, e en wi h he imp o emen s in o-
duced by he IVR, canno yield e en semiquan i a i e esul s
in he chao ic egime. Whe he mo e sophis ica ed echniques
in ol ing esumma ions [13] can yield be e esul s in his
case emains an open challenge, bu p obably o e y limi ed
p ac ical use o ealis ic nuclea eac ions. On he o he hand
we ha e shown ha abso p ion—s ongly p esen in nuclea
hea y ion eac ions – p o ides a na u al way o selec i ely
damping he mo e i egula ajec o ies, hus egula izing
he sca e ing ajec o y sum and es o ing he easibili y
o semiclassical calcula ions. One could a gue ha s ic ly
speaking he mos se e e aspec s o chaos ha e been elimina ed
by in oducing abso p ion, and ha wha emains is a ini e
collec ion o sca e ing ajec o ies which will be mo e o
less complica ed acco ding o he s eng h o he imagina y
po en ial. We hink ha his is indeed he igh (and p ac ical)
054611-5

C. H. DASSO, M. I. GALLARDO, AND M. SARACENO PHYSICAL REVIEW C 75, 054611 (2007)
way o hink abou his issue in he con ex o nuclea eac ions,
and ha he p esence o special “ anspa ency windows” [14]
in some selec ed eac ions p o ide oppo uni ies o obse e
i [15–17].
ACKNOWLEDGMENTS
We acknowledge suppo om he Minis y o Educa ion
and Science unde p ojec nos. FIS2005-01105, FPA2005-
04460, om ANPCyT-PICT25373 and Conice -PIP6137-04.
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