In es iga ing he nucleus-nucleus po en ial a e y sho dis ances
C. H. Dasso1and G. Polla olo2
1Depa men o A omic, Molecula and Nuclea Physics, Uni e si y o Se illa, Apa ado 1065, 41080 Se illa, Spain
2Dipa imen o di Fisica Teo ica, Uni e si á di To ino, and Is i u o Nazionale di Fisica Nuclea e, Sezione di To ino,
Via Pie o Giu ia 1, 10125 To ino, I aly
(Recei ed 27 Ma ch 2003; published 17 No embe 2003)
Recen p og ess in expe imen al echniques ha e made possible accu a e measu emen s o usion c oss
sec ions a below he Coulomb ba ie , e ealing an unexpec ed beha io o hese quan i ies as a unc ion o
he bomba ding ene gy. Besides p o iding a plausible cause o he obse ed ene gy dependence we p o i om
he na u e o his explana ion and he high sensi i i y o he expe imen al da a o show how one can use hese
measu emen s o in es iga e he adial dependence o he nuclea ion-ion po en ial a ex emely close dis ances.
DOI: 10.1103/PhysRe C.68.054604 PACS numbe (s): 25.70.Hi, 21.30.Fe, 24.10.⫺i, 25.70.Jj
In hea y-ion pe iphe al collisions close o abo e he
nominal Coulomb ba ie , he esul s o mic oscopic calcula-
ions o eac ion c oss sec ions o quasielas ic p ocesses a e
mos ly sensi i e o he alue o he nucleus-nucleus po en ial
and inelas ic/ ans e o m ac o s a dis ances well ou side
he ouching adius be ween he p ojec ile and a ge . While
he alue o he Coulomb ba ie is sa ely es ima ed [1](al-
hough no be e han o a le el o a ew pe cen ) he e is
conside able ambigui y wi h ega ds o he ac ual adial de-
pendence o he nuclea pa o he ion-ion po en ials ha can
be made consis en wi h he expe imen al indings. I is
wo h men ioning he e ha heo e ical de elopmen s a e
able, wi hin he ame o hei assump ions, o p edic he
shape o he po en ial o dis ances well inside o he con ac
adius [2–4].
In his con ex , measu emen s o sub-ba ie usion c oss
sec ions—a ield ha con inues o a ac much expe imen al
and heo e ical a en ion [5–7]—became one o he mos e -
ec i e ools o shed ligh on he cha ac e is ics o he nuclea
po en ial a sho e dis ances. In ac , he s anda d app oach
o in es iga e his class o phenomena exploi s he mecha-
nism o ba ie pene a ion in he p esence o couplings o
in insic deg ees o eedom [8–12]. C ucial o he success o
hese analyses is a p ope adjus men o he heigh and hick-
ness o he po en ial ba ie o small pa ial wa es 共ᐉ⬇0兲.
While he o me con ols displacemen s in he ene gy scale,
he ba ie wid h is ela ed ( o usion domina ed by nega i e
Q- alue channels, a leas ) o he exponen ial slope 2
␲
/ប
␻
B
o he unc ion
␴
共E兲a he lowes ene gies.
Sub-ba ie usion c oss sec ions a e essen ially de e -
mined by only hese wo pa ame e s in a small ange o
ene gies lowe han bu close o he ba ie . To be p ecise, in
he ange o alidi y o he pa abolic app oxima ion when he
e ec i e po en ial is exp essed in he neighbo hood o i s
maximum, namely,
V共 兲⬇VB−
␮
␻
B
2
2共 − B兲2,
whe e Bindica es he loca ion o he ba ie and
␮
he
educed mass o he sys em. 共Ac ually, he pa ame e s VB,
B, and ប
␻
Bshould be all labeled by he pa ial wa e
numbe ᐉ.兲The se e e exponen ial d op associa ed wi h
he cha ac e is ic alues o ប
␻
Bhas limi ed in p ac ice he
ange o measu ed c oss sec ion o a span o ene gies o
only a ew MeV, a small pe cen change o he bomba d-
ing ene gies wi h espec o he ba ie alue VB. In such
case he pa abolic app oxima ion emains alid.
A pionee ing expe imen in A gonne has gone well be-
yond his limi p omp ing he expe imen al g oup o epo
hei esul s as “unexpec ed” in a ecen pape [13]. How-
e e , a la ge numbe o ion-ion po en ials p edic a pocke in
he inne egion, jus as is illus a ed in Fig. 1. He e we
see— o he pa icula eac ion co e ed in Re . [13]— he
p o ile o a equen ly used ion-ion po en ial ( he Akyüz-
Win he po en ial o Re . [4]). Shown in he igu e is also he
absolu e limi o usion ha comes om he equi ed exo-
he mic cha ac e o he p ocess ha was men ioned by he
au ho s o Re . [13]. This ene gy is a below he ela i e one
quenching he unneling p ocess, VP, and he e o e i is clea
ha he la e akes p ecedence as a limi ing mechanism
causing ln
␴
共E兲→−⬁. Fu he mo e, since somewhe e hal
he way be ween he maximum and he minimum he e
should be an in lec ion poin , one can sa ely conclude ha
de ia ions om he pa abolic app oxima ion should be ex-
pec ed o show up e en much ea lie as he bomba ding en-
e gy is g adually educed below he ba ie . A simple, isual
es ima ion o he alue o ene gy whe e his would happen is
shown by he a ow, less han 5 MeV below he ba ie . This
numbe is e y much consis en wi h he indings o Re .
[13].
I can be app ecia ed in Fig. 1 ha he la ges depa u e
om he pa abolic beha io is associa ed wi h he slow adial
dependence o he monopole Coulomb in e ac ion e m o
dis ances la ge han B. Howe e , his componen o he po-
en ial has a well-de ined shape, does no in oduce any am-
bigui y in he ansmission coe icien s, and he e o e i is
qui e unde con ol.
Ou con en ion in his pape is ha , on he o he hand, i is
possible o u n his si ua ion o ou ad an age and use he
ou s anding quali y o he da a o he expe imen pe o med
in A gonne (o simila ) o lea n abou he shape o he inne
side o he po en ial ba ie . O , equi alen ly, o gain empi i-
cal knowledge o he nucleus-nucleus po en ial a much
PHYSICAL REVIEW C 68, 054604 (2003)
0556-2813/2003/68(5)/054604(3)/$20.00 ©2003 The Ame ican Physical Socie y68 054604-1
sho e dis ances han hi he o achie ed.
To his end we shall show ha he e is enough sensi i i y
in he de ia ions om he pu e exponen ial decay o he
c oss sec ions o e y low ene gies (whe e hey occu , by
how much, e c.) o disce n be ween di e en po en ial p o-
iles in ha inne egion. We will limi ou sel es o show he
way in which hese conside a ions in luence he usion c oss
sec ions in he absence o in e ac ions wi h in e nal deg ees
o eedom. How he couplings o di e en eac ion channels
a ec he so-called “ e e ence” cu e has been ex ensi ely
co e ed in he li e a u e [8–12]and we know ha he inal
esul s a he lowes bomba ding ene gies always inhe i (o
build upon)wha e e cha ac e is ics a e al eady p esen in
he simples ba ie -pene a ion o mula ion o he p oblem.
Clea ly, we canno ely any longe in he analy ic o m o
he ansmission coe icien s o a pa abolic ba ie and we
shall use, in wha ollows, hei Wen zel-K ame s-B illouin
exp ession [4]. The implemen a ion o his p esc ip ion e-
qui es a nume ical in eg a ion o each alue o he bomba d-
ing ene gy and each e ec i e po en ial, as de ined by he
pa ial wa e numbe ᐉ. The p ocedu e, howe e , canno be
used o all alues o E⬍VBand needs o be complemen ed
by he analy ic exp essions which a e alid nea he op o
he ba ie . Fo una ely, he e is a wide ange o ene gies
whe e he esul s o ei he al e na i e coincide and hus a
smoo h ma ching o he ansmission unc ions poses no
echnical di icul y.
Ou s a egy consis s in aking a amily o hypo he ical
po en ials ha a e iden ical o la ge alues o and which
sha e he same alues o VB, B, and ប
␻
B. The only di e ence
be ween hese po en ials occu s o dis ances somewha in-
side o he ba ie posi ion whe e hey a e ma ched in alue
FIG. 2. (Colo online)( op)Th ee possible ion-ion po en ials o
he eac ion 60Ni+89Y ha ing an iden ical adial dependence o
⬎11.01x m. This is a ma ching dis ance chosen somewha inside
he ba ie adius and, consequen ly, he alues o VB, B, and ប
␻
B
a e sha ed by all h ee po en ial unc ions. The ma ching is done
wi h he pu pose o gene a ing di e en al e na i es o he inne
side o he po en ial ba ie . The cu es shown a e o he pa ial
wa e ᐉ=0 bu a simila p esc ip ion can also be implemen ed o
he e ec i e po en ials co esponding o all he o he low alues o
ᐉwhich a e ele an o a calcula ion o sub-ba ie usion c oss
sec ions. Also shown in he ame is he pa abolic app oxima ion
ha is common o he h ee po en ials. (Bo om)The usion c oss
sec ions as a unc ion o bomba ding ene gy in he cen e o mass
o he ou po en ial p o iles shown abo e.
␴
pa 共E兲is he c oss
sec ion wi hin he pa abolic app oxima ion while
␴
1,2,3共E兲co e-
spond o each o he po en ial unc ions as iden i ied in he uppe
ame.
FIG. 1. (Colo online)The nucleus-nucleus po en ial o a
head-on collision 共ᐉ=0兲in he eac ion 60Ni+89Y. The nuclea pa
o he in e ac ion is gi en by he Akyüz-Win he po en ial. Shown
in he igu e is also he p o ile o he pa abolic app oxima ion ha
esul s om app oxima ing he po en ial nea he op o he ba ie ,
VB, by a quad a ic dependence in . The inne pocke VPis in his
case abou 15 MeV below he ba ie , a an ene gy conside ably
la ge han he alue o ⬇90 MeV ha se s he absolu e lowe limi
o which he usion p ocess can ene ge ically occu .
C. H. DASSO AND G. POLLAROLO PHYSICAL REVIEW C 68, 054604 (2003)
054604-2
and i s de i a i e o pa ame ized unc ions ha a e hus
able o ec ea e a ious adial dependences o he le -hand
side o he po en ial ba ie .
An example o h ee such po en ials is shown in he op
ame o Fig. 2. The eac ion is again 60Ni+89Y and he
ma ching has been done a = B−0.2 m, wha makes all
h ee ion-ion in e ac ions iden ical o ⬎11 m. Fo he sake
o compa ison we also show in he same ame he pa abolic
po en ial app oxima ion ha is, by de ini ion, common o all
h ee unc ions. Wi hin his app oxima ion all he po en ials
lead o he same usion eac ion c oss sec ion as a unc ion o
he bomba ding ene gy,
␴
pa 共E兲, shown wi h a dashed line in
he bo om ame o Fig. 2. This unc ion e lec s he beha -
io ha is cha ac e is ic o his app oxima ion; an exponen-
ial allo owa ds he low ene gies wi h a slope en i ely
con olled by he cu a u e o he ba ie a i s op.
C oss sec ions a e displayed in Fig. 2 in he in e al
120 MeV⬍Ec.m.⬍140 MeV, which co esponds o he
shaded ange o ene gies indica ed in he uppe ame, whe e
c.m. s ands o cen e o mass. In his ene gy in e al one
should be able o obse e he o al quenching o he usion
c oss sec ions co esponding o he po en ials 2 and 3, since
he alues o VP o hese po en ials all wi hin he selec ed
ene gy ange. This is indeed he case as i can be seen in he
lowe ame, whe e he cu es ha e been comple ed wi h a
do ed e ical line o guide he eye o he alues o he
abscissa whe e ln
␴
2,3→−⬁. The d as ic cu o occu s, in
hese examples, o c oss sec ions in he ange o mic oba ns
o nanoba ns. Small as hey may seem, we ecall ha he
o de s o magni ude o hese quan i ies do no di e much
om he c oss-sec ion alues epo ed in he A gonne expe i-
men .
I is impo an o no e ha he quan i y ⌬ᐉ=VB
ᐉ−VP
ᐉac-
qui es i s maximum alue o ᐉ=0. In ac , as he cen i ugal
con ibu ion o he e ec i e po en ial inc eases, he ela i e
dep h o he pocke educes (un il i comple ely disappea s a
a c i ical alue ᐉc ). This explains why, in he accumula ion
o pa ial-wa e con ibu ions, which builds up he usion
c oss sec ion, he e ec s we a e discussing end up being
pe haps mo e p onounced han one could ha e expec ed. To
s a , phase-space conside a ions a o he con ibu ion o
he la ge pa ial wa es. Bu , in addi ion, de ia ions om he
pa abolic beha io se in a an ea lie s age, i.e., a a smalle
numbe o MeV’s below he co esponding ba ie heigh .
The na u e o his explana ion is qui e gene al and does
no ely on he cha ac e is ics o any in insic channels ha
may be ac i e in a pa icula eac ion. Thus, he elemen s we
ha e discussed should be p esen , in a mo e o less p omi-
nen way, in many o he eac ions besides he speci ic one we
ha e chosen o illus a ion. This is in ag eemen wi h he
analysis o Re . [13], whe e plen y o e idence is p o ided o
his e ec .
I is possible, wi h his app oach, o lea n abou he inne
shape o he nucleus-nucleus in e ac ion because o he
sha p, exponen ial dependence o he unneling p obabili ies
wi h espec o de ails o he po en ial in he zone whe e
classical mo ion is o bidden. This ex ao dina y sensi i i y
is wha causes he ange o li e imes o
␣
decay o span
o y o de s o magni ude (and wha , inciden ally, p o ided
one o he ea lies p ocedu es o he es ima ion o nuclea
adii). In a sense, he di icul ies posed by he p ecise de e -
mina ion o li e imes exceeding millions o yea s a e a coun-
e pa o hose o measu ing c oss sec ions o he o de o
nanoba ns in he con ex o ou p esen p oblem. The easi-
bili y o achie ing such goal was, howe e , p o ed by he
wo k o Re . [13]. I is impo an o encou age he con inu-
a ion o hese e o s and he accumula ion o u he da a in
o de o e eal, as unambiguously as possible, he cha ac e -
is ics o he ion-ion in e ac ion a ex emely sho dis ances.
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