Eu . Phys. J. A (2025) 61:47
h ps://doi.o g/10.1140/epja/s10050-025-01500-0
Re iew
The a o modeling nuclea eac ions wi h weakly bound nuclei:
s a us and pe spec i es
An onio M. Mo oa, Jesús Casal , Ma io Gómez-Ramos
Depa amen o de Física A ómica, Molecula y Nuclea , Facul ad de Física, Uni e sidad de Se illa, Apa ado 1065, 41080 Se illa, Spain
Recei ed: 31 July 2024 / Accep ed: 22 Janua y 2025
© The Au ho (s) 2025
Communica ed by Ma ia Bo ge
Abs ac We gi e an o e iew o he heo e ical desc ip-
ion o nuclea eac ions in ol ing weakly-bound nuclei.
Some o he mo e widesp ead eac ion o malisms employed
in he analysis o hese eac ions a e b ie ly in oduced,
including a ious ecen de elopmen s. We pu special
emphasis on he con inuum-disc e ized coupled-channel
(CDCC) me hod and i s ex ensions o inco po a e co e and
a ge exci a ions as well as i s applica ion o h ee-body p o-
jec iles. The ole o he con inuum o one-nucleon ans-
e eac ions is also discussed. The p oblem o he e alua-
ion o inclusi e b eakup c oss sec ions is add essed wi hin
heIchimu a–Aus e n–Vincen (IAV)model.O he me hods,
such as hose based on a semiclasical desc ip ion o he sca -
e ing p ocess, a e also b ie ly in oduced and some o hei
applica ions a e discussed and a b ie discussion on opics o
cu en in e es , such as nucleon-nucleon co ela ions, unce -
ain y e alua ion and non-locali y is p esen ed.
Con en s
1 In oduc ion ......................
2 The impac o weak-binding on he elas ic c oss sec-
ion: op ical model app oach ..............
2.1 E alua ion o he pola iza ion po en ial in a sim-
ple case: he adiaba ic pola iza ion po en ial ...
2.2 The phenomenological op ical model .......
3 Inclusion o b eakup: he CDCC me hod .......
3.1 B ie esumè o he CDCC me hod ........
3.2 The ole o closed channels ............
3.3 T i ially equi alen pola iza ion po en ial ....
3.4 Inclusion o co e exci a ions ...........
3.5 Inclusion o a ge exci a ions ...........
3.6 Fou -body CDCC .................
3.7 Mic oscopic CDCC ................
ae-mail: [email p o ec ed] (co esponding au ho )
3.8 In e p e a ion o he CDCC wa e unc ion and
c oss sec ions ...................
3.8.1 Smoo hingp ocedu e o wo-bodyobse -
ables ....................
3.8.2 Th ee-body obse ables ..........
3.8.3 Pseudos a es e sus bins ..........
4 T ans e eac ions wi h weakly bound nuclei .....
The Johnson–Sope app oxima ion ..........
The Johnson–Tandy app oxima ion ..........
The CDCC-BA app oxima ion .............
4.1 T ans e eac ions popula ing unbound sys ems .
4.2 T ans e eac ions in ol ing h ee-body p ojec iles
4.3 Simul aneous inclusion o p ojec ile b eakup
and a ge exci a ion in ans e eac ions ....
5 Inclusi e b eakup eac ions ..............
5.1 The Ichimu a–Aus e n–Vincen (IAV) model ..
Applica ion o h ee-body p ojec iles .........
5.2 The Eikonal Hussein–McVoy o mula (EHM) ..
5.3 In e p e a ion o inclusi e b eakup da a .....
6 Fusion in ol ing weakly bound nuclei ........
6.1 Compu a ion o CF and ICF wi h CDCC ....
6.2 E alua ion o CF and ICF c oss sec ions wi h
he IAV model ...................
6.3 Applica ion o su oga e eac ions ........
7 Semiclassical desc ip ion o b eakup and ans e
eac ions ........................
7.1 The semiclassical o malism o Alde and Win he
7.2 Dynamic Coulomb pola iza ion po en ial om
he AW heo y ...................
7.3 Semiclassical ans e - o- he-con inuum model .
7.4 Dynamical Eikonal app oxima ion ........
8 Ex ac ion he elec ic ansi ion p obabili ies om
Coulomb dissocia ion da a ...............
8.1 Semiclassical analysis o Coulomb dissocia ion da a
8.2 Quan um-mechanicale ec s:CDCCanalysiso
Coulomb dissocia ion da a ............
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47 Page 2 o 57 Eu . Phys. J. A (2025) 61:47
9 S udy o nucleon–nucleon co ela ions om nucleon
emo al eac ions ...................
10Unce ain y e alua ion .................
11Inclusion o non-local po en ials ............
12Conclusions and pe spec i es .............
Re e ences .........................
1 In oduc ion
Nuclea eac ions a e key ools o ex ac in o ma ion abou
he s uc u e o a omic nuclei, and he dynamical phenomena
a ising om nucleus-nucleus o ces. When one o he collid-
ing pa ne s is weakly bound, new phenomena and mecha-
nisms a ise, equi ing in some cases speci ic eac ion ame-
wo ks ailo ed o he peculia i ies o hese sys ems and hei
in e ac ions.
Weakly-bound nuclei appea in he p oximi y o he neu-
on and p o on d iplines, a egion whe e new exo ic s uc-
u es and phenomena a e ound. P omimen examples a e
halo nuclei, weakly bound nuclei composed o a com-
pac co e and one o wo loosely bound nucleons wi h an
unusually la ge ma e adius, o Bo omean sys ems, h ee-
body sys ems wi h no bound bina y subsys ems, such as
9Be (α +α+n)o 6He (α +n+n).
Al hough he ield has expe ienced a g ea impulse in he
las decades, he physics o nuclea sca e ing wi h weakly-
bound p ojec iles is no new and many o he p oblems ha
a e being add essed we e al eady ecognized much ea lie as,
o example,in hecon ex o low-ene gydeu e onsca e ing.
The deu e on, while being a s able nucleus, displays many
halo-like ea u es, such as weak binding (Eb=2.22 MeV),
la ge spa ial ex ension ( he p o on-neu on sepa a ion is
abou 3.8 m) and no bound exci ed s a es.
To illus a e he ole o he weak binding on he sca -
e ing obse ables, le us conside he sca e ing o low-
ene gy deu e ons (a ew MeV) by a hea y a ge nucleus, like
208Pb. Ini ially, when he deu e on is a apa om he a ge
nucleus, i is in an in e nal s a e gi en by he g ound-s a e o
he p o on-neu on Hamil onian. As he deu e on app oaches
he a ge , i will eel i s Coulomb epulsion. I he deu e on
we e a poin -like pa icle, he e ec o his in e ac ion would
be o dis o he ajec o y o he deu e on, wi hou al e -
ing i s in e nal s uc u e. Howe e , his is no he case. The
Coulomb in e ac ion ac s on he p o on, whose dis ance om
he cen e -o -mass o he deu e on is ∼2 m. Consequen ly,
in addi ion o he monopole Coulomb po en ial (∝1/R) he
deu e on will eel highe -o de Coulomb mul ipoles a ising
om he expansion, alid o R, gi en by [1]
VC( ,
R)=Z e2
|
R+ /2|=Z e2
R−Z e2
2R2 cos(θ) +...
(1)
Fig. 1 Rele an coo dina es o a deu e on+ a ge sca e ing p oblem
whe e
Ris he coo dina e om he a ge o he deu e on
c.m. he p o on-neu on ela i e coo dina e and θis he
angle be ween hem (see Fig. 1). The main de ia ion om
he poin Coulomb in e ac ion is caused by he dipole e m
which depends on he o ien a ion o he p o on-neu on ela-
i e coo dina e wi h espec o he deu e on- a ge coo di-
na e. Speci ically, when he p o on is close o he a ge
nucleus (cos(θ) < 0), he dipole po en ial will add a posi i e
con ibu ion, whe eas when he p o on is a he om he
c.m. o he deu e on he con ibu ion will be nega i e. Clas-
sically, we may hink o his p oblem as an elec ic dipole
mo ing in a slowly a ying elec ic ield, in which he idal
o ce exe ed on he dipole will a ou he con igu a ion o
Fig. 1. Quan um-mechanically, he pe u ba ion induced by
he dipole o ce will couple he deu e on g ound s a e (posi-
i epa i y)wi h nega i e-pa i ys a es.Since he g ound-s a e
is he only bound s a e, hese s a es necessa ily appea in he
con inuum. The e o e, he pola iza ion e ec will modi y he
s a e o he sys em, p oducing a new one in which he g ound
s a e is mixed wi h con inuum s a es o nega i e pa i y. This
has a wo old e ec on he ou come o he sca e ing p ocess.
Fi s , i will p oduce de ia ions o he elas ic sca e ing wi h
espec o he Ru he o d o mula. Second, he coupling wi h
he posi i e-ene gy s a es will gi e ise o some dissocia ion
p obabili y, ha is, he b eakup o he deu e on in o a p o on
and a neu on. In he nex sec ions, we will discuss se e al
me hods o inco po a e hese wo e ec s in he eac ion o -
malisms.
The discussed example, albei desc ibing a speci ic p ob-
lem, exhibi s some gene al ea u es commonly ound in he
sca e ing o weakly bound nuclei. In pa icula , he cou-
pling o he b eakup channels will play a ole, o a la ge o
smalle ex en , in essen ially all eac ion obse ables. The e-
o e, eac ionmodelsemployed o desc ibe hese obse ables
will ha e o inco po a e his e ec .
We enume a e some inge p in s o he weak binding on
eac ion obse ables:
–La ge in e ac ion c oss sec ions in nuclea collisions
a high ene gies. His o ically, he i s e idence o he
unusual p ope ies o halo nuclei came om he pionee -
ing expe imen s pe o med by Taniha a and co-wo ke s
a Be keley using e y ene ge ic (800 MeV/nucleon) sec-
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Eu . Phys. J. A (2025) 61:47 Page 3 o 57 47
onda y beams o adioac i e species [2,3]. A hese high
ene gies, in e ac ion c oss sec ions a e app oxima ely
p opo ional o he size o he colliding nuclei. I was
ound ha some exo ic iso opes o ligh nuclei (6He, 11Li,
14Be) p esen ed much highe in e ac ion c oss sec ions
han hei neighbou iso opes, which was in e p e ed as
an abno mally la ge adius.
–Na ow momen um dis ibu ions o esidues ollowing
as nucleon emo al. Momen um dis ibu ions o he
esidual nucleus ollowing he emo al o one o mo e
nucleons o a ene ge ic p ojec ile colliding wi h a a -
ge nucleus a e closely ela ed o he momen um dis-
ibu ion o he emo ed nucleon(s) in he o iginal p o-
jec ile. Kobayashi e al. [4] ound ha he momen um
dis ibu ions o 9Li ollowing he agmen a ion p ocess
11Li +12C→9Li +X we e abno mally na ow, which,
acco ding o he Heisenbe g’s unce ain y p inciple, sug-
ges ed a long ail in he densi y dis ibu ion o he 11Li
nucleus. This esul was la e ound in o he weakly
bound nuclei.
–Abno mal elas ic sca e ing c oss sec ions. Elas ic sca -
e ing is a ec ed by he coupling o non-elas ic p o-
cesses. In pa icula , when coupling o b eakup chan-
nels is impo an , elas ic sca e ing c oss sec ions a e
deple ed wi h espec o he case o igh ly bound nuclei.
Some o he key signa u es a e he depa u e o he elas ic
c oss sec ion om he Ru he o d c oss sec ion a sub-
Coulomb ene gies and he disappea ance o he F esnel
peak a nea -ba ie ene gies in eac ions induced by halo
nuclei on hea y a ge s [5–8].
–Enhanced nea - h eshold b eakup c oss sec ion in
Coulomb dissocia ion expe imen s o neu on-halo nuclei.
When a neu on-halo nucleus, composed o a cha ged
co e and one o wo weakly-bound neu ons (11Be, 6He,
11Li,…) collides wi h a high-Z a ge nucleus, he p o-
jec ile s uc u e is hea ily dis o ed due o he idal o ce
o igina ed om he une en ac ion o he Coulomb in e -
ac ion on he cha ged co e and he neu ons. This p o-
duces a s e ching which may e en ually b eak up he
loosely bound p ojec ile. This gi es ise o a la ge popu-
la ion o he con inuum s a es close o he b eakup h esh-
old.
–Comple e usion supp ession
Expe imen s wi h weakly bound ligh s able nuclei (such
as 6,7Li and 9Be) ha e shown a sys ema ic supp ession
o comple e usion (CF) c oss sec ions (de ined as cap-
u e o he comple e cha ge o he p ojec ile) o ∼20-30%
compa ed o he case o igh ly bound nuclei [9–15]. The
e ec has been a ibu ed o he p esence o s ong com-
pe ing channels, such as he b eakup o he weakly bound
p ojec ile p io o eaching he usion ba ie , wi h he
subsequen educ ion o cap u e p obabili y. This in e -
p e a ion is suppo ed by he p esence o la ge αyields
as well as a ge -like esidues which a e consis en wi h
he cap u e o one o he agmen cons i uen s o he p o-
jec ile, a p ocess which is usually e med as incomple e
usion (ICF).
Ap ope andquan i a i eunde s andingo hese ando he
phenomena equi es he use o an app op ia e eac ion he-
o y. When dealing wi h weakly-bound sys ems, one expec s
ace aindecouplingbe ween hedeg eeso eedom desc ib-
ing he ela i e mo ion be ween he weakly-bound nucleon
(o clus e ) om ha o he in e nal exci a ions o he clus e s
hemsel es. Following his a gumen , one may be emp ed o
adop an ex eme model in which only he deg ee o eedom
o he ela i e mo ion o he weakly-bound nucleon(s) o
clus e s a e conside ed, while he o he s a e simply igno ed
o ozen. This app oach, which eminds o he sepa a ion
be ween ac i e nucleons and co e nucleons in he shell-
model, has in ac become e y use ul o unde s and he
main ea u es o he s uc u e o halo, and o he weakly-
bound, nuclei and hei dynamics. In some cases, such as he
deu e on sys em o one-nucleon halos, such as 19Co 11Be,
a wo-body model may p o ide a easonable s a ing poin
o he desc ip ion o hese nuclei. Howe e , in some o he
sys ems, such as he Bo omean nuclei 11Li, 6He o 9Be, i
will be in gene al manda o y o eso o (a leas ) a h ee-
body model o a meaning ul desc ip ion o hei s uc u e
and eac ions.
One o he mos success ul models o desc ibing eac-
ions in ol ing weakly-bound nuclei, which inco po a es in
a na u al way he ew-body s uc u e o hese sys ems as
well as i s b eakup, is he Con inuum Disc e ized Coupled-
Channels (CDCC) [16]. The model and i s ecen ex ensions
will be discussed in Sec . 3. O he models, ailo ed o speci ic
p ocesses, will be discussed along his e iew.
I is he pu pose o his pape o e iew some o he he-
o ies de eloped and applied o desc ibing nuclea eac ions
wi h weakly bound nuclei. As any e iew, he p esen one
will be necessa ily incomple e and possibly biased by he
expe ise and pe sonal as e o he au ho s bu wi h he hope
ha he eade will ob ain, a leas , a la ou o how eac-
ion heo y has helped he in e p e a ion o expe imen s wi h
hese nuclea species.
2 The impac o weak-binding on he elas ic c oss
sec ion: op ical model app oach
Elas ic sca e ing p o ides a aluable sou ce o in o ma ion
on he s uc u e and dynamics o nuclei. As explained in he
In oduc ion, i was long ealised ha he elas ic sca e ing o
deu e ons, admi edly he simples example o weakly bound
nucleus, does no ollow he expec ed beha io o igh ly
bound nuclei.
123
47 Page 4 o 57 Eu . Phys. J. A (2025) 61:47
As in he case o well-bound nuclei, he na u al amewo k
o s udy elas ic sca e ing is he Op ical Model (OM), which
consis s in sol ing a wo-body Sch ödinge equa ion wi h an
e ec i e nucleus-nucleus po en ial, e e ed o as he op i-
cal model po en ial (OMP) [17]. The o m o his OMP can
be o mally de i ed om he mic oscopic nucleus-nucleus
in e ac ion. In his seminal wo k, Feshbach [18] showed ha
he OMP can be o mally w i en as a sum o wo e ms, a
ba e po en ial, which is he expec a ion alue o he nucleus-
nucleus po en ial in he g ound s a e o he p ojec ile+ a ge
sys em, and an addi ional e m, he pola iza ion po en ial,
which accoun s o he e ec o non-elas ic channels (inelas-
ic sca e ing, ans e , b eakup, usion, …) on elas ic sca -
e ing: Explici ly:
V=V00 +V01
E(+)−HV†
0,(2)
whe e he i s and second e ms co espond, espec i ely,
o he ba e and pola iza ion po en ials. He e, Hs ands o
he ull, mic oscopic Hamil onian, V0is an ope a o wi h
non- anishing ma ix elemen s be ween he g ound-s a e and
o he s a es o he sys em, V0=(V01,V02,...). This e ec-
i e po en ial gi es ise o he Sch ödinge equa ion o he
ela i e-mo ion wa e unc ion:
T
R+V−Eχ0(
R)=0(3)
whe e T
Ris he kine ic ene gy ope a o and χ0 he wa e-
unc ion o he ela i e mo ion be ween p ojec ile and a ge
in hei g ound s a es.
The e ec o weak binding o any o he colliding pa -
ne s a ec s bo h he ba e and pola iza ion po en ials. The
g ound-s a e o he p ojec ile will exhibi an ex ended ail
(as compa ed o he case o igh ly-bound nuclei) and his
will in luence he o m o V00. In addi ion, he weak binding
will enhance ce ain couplings desc ibed by he V0ope -
a o , e y p ominen ly he b eakup channels, and his will
also modi y he pola iza ion e m. In gene al, he e alua ion
o he Feshbach ope a o associa ed wi h he pola iza ion
po en ial is e y in ol ed and e y o en i is eplaced by
some phenomenological o m. A ew cases exis howe e in
which an explici e alua ion is easible, a leas o speci ic
nonelas ic p ocesses. One such example is he so-called adi-
aba ic Coulomb pola iza ion po en ial, b ie ly in oduced in
he nex subsec ion.
2.1 E alua ion o he pola iza ion po en ial in a simple
case: he adiaba ic pola iza ion po en ial
In he in oduc o y sec ion, we desc ibed he case o deu e on
sca e ing by a hea y a ge nucleus. I was a gued ha he
idal o ce caused by he ac ion o he Coulomb ield on he
p o on p oduced a modi ica ion o he Coulomb- a ge in e -
ac ion wi h espec o he poin -Coulomb case [c. . Eq. (1)].
Quan um-mechanically, he e ec o his modi ied Coulomb
in e ac ion will be o induce coupling o deu e on b eakup
s a es. This is in ac one o he con ibu ions included in he
Feshbach pola iza ion po en ial gi en by he second e m o
Eq. (3). This can be analy ically e alua ed in he so-called
adiaba ic limi , which assumes ha he exci a ion ene gies
a e high enough so he cha ac e is ic ime o a ansi ion
o a s a e is small compa ed o he cha ac e is ic ime o he
collision[1].Applyingsecond-o de pe u ba ion heo y,one
ge s he ollowing exp ession o his adiaba ic pola iza ion
po en ial [1,19]:
Vpol(R)=−
n=0
|n|Vdip|0|2
En−E0=−1
2α(ZTe)2
R4,(4)
whe e Vdip is he second e m o Eq. (1) and αis he dipole
pola izabili y pa ame e [1], which measu es he elec ic
esponse o he nucleus ( he deu e on in his case) o an ex e -
nal elec ic ield. I is he e o e a s uc u e p ope y o any
nucleus.
I is o be no ed ha he adiabia ic pola iza ion po en-
ial gi en by Eq. (4) is pu ely eal and, as such, does no
desc ibe hee ec o hedeu e onb eakup ha would emo e
lux om he elas ic channel. This poin will be e isi ed in
Sec. 7.2, whe e ano he e sion o he pola iza ion po en ial,
which accoun s o deu e on b eakup, will be p esen ed.
The pola iza ion po en ial (4) adds an a ac i e con ibu-
ion o he Coulomb poin -pa icle po en ial ha , when used
in he Sch ödinde equa ion, will p oduce a small, bu mea-
su able de ia ion o he elas ic c oss sec ion wi h espec o
he well-known Ru he o d o mula. An accu a e measu e-
men o sub-Coulomb elas ic sca e ing da a can he e o e be
used o ex ac he dipole pola izabili y o he deu e on (and
o he pola izable sys ems). This idea was used by Rodning
e al. [20] o in e he deu e on pola izabili y pa ame e om
he analysis o sub-Coulomb deu e on elas ic sca e ing on
lead.
To a oid he de e mina ion o absolu e c oss sec ions, in
hei analysis he au ho s o Re . [20] in oduced he adimen-
sional quan i y
R(E)=σ(E=3MeV,θ
1=60◦)
σ(E=3MeV,θ
2=150◦)
σ(E,θ
2=150◦)
σ(E,θ
1=60◦).
(5)
which is iden ically uni y o pu e Ru he o d sca e ing. The
deu e on pola izabili y will gi e ise o alues o R(E)which
a eless han uni y and he de ia ion inc eases wi h inc easing
sca e ing ene gy. In Fig. 2we show he measu ed alues o
R(E) om [20] along wi h wo calcula ions. One in which
only a nuclea po en ial is included, which plays he ole
o a ba e pa o Eq. (2) and a second calcula ion includ-
123
Eu . Phys. J. A (2025) 61:47 Page 5 o 57 47
Fig. 2 Expe imen al [20] alues o he quan i y R(E),de inedin
Eq. (5) o deu e on sca e ing on 208Pb a sub-Coulomb ene gies. The
dashed line is a single-channel calcula ion including only he nuclea
po en ial. The solid line is he single-channel calcula ion include also
hee ec o he Coulomb dipole pola izabili y by means o he adiaba ic
pola iza ion po en ial o Eq. (4). See ex o de ails
ing he e ec o he deu e on poloa izabili y in he adiaba ic
app oxima ion, Eq. (4), wi h α=0.70±0.05 m3, he alue
ex ac ed in [20] by compa ison wi h he da a.
This cons i u es a nea and beau i ul example in which an
impo an s uc u e quan i y can be in e ed om eac ion
obse ables.
2.2 The phenomenological op ical model
Mos p ac ical applica ions o he OM ely on app oxi-
ma e o ms o he Feshbach po en ial, consis ing ypically
o a cen al po en ial (wi h possibly spin-o bi and enso
e ms) whose adial pa s a e pa ame ized in e ms o simple
analy ical o ms, such as he popula Woods-Saxon po en-
ial. Because he Feshbach po en ial is complex, so is he
phenomenological op ical po en ial. In p inciple, he po en-
ial should also be non-local and angula -momen um and
ene gy dependen . While he ene gy dependence is a com-
mon ea u e o phenomenological po en ials, nonlocalil y
andangula -momen umdependence is mo e a ely akenin o
accoun (see Sec . 11).
In he p esence o s ong abso p ion, elas ic sca e ing
angula dis ibu ions be ween composi e nuclei display some
common ea u es ega dless o he in e ac ing nuclei. A
nea -ba ie ene gies, elas ic dis ibu ions p esen wo dis-
inc egions. A small angles, whe e he Coulomb in e -
ac ion domina es, he c oss sec ion emains close o he
Fig. 3 Expe imen al da a o 4,6He+208Pb compa ed wi h OM calcu-
la ions, wi h Woods-Saxon o ms wi h pa ame e s gi en in Table 1.
Expe imen al da a om [22,23]
Ru he o d o mula p edic ion, bu oscilla es a ound i . The
ampli udes o he oscilla ions inc ease as he sca e ing angle
inc eases bu emain oughly 25% wi hin he Ru he o d p e-
dic ion. This is he so-called “illumina ed” egion. A e his
maximum, he c oss sec ion d ops apidly, becoming much
smalle han he Ru he o d c oss sec ions a la ge angles ( he
“shadow” egion). An example o his beha iou can be seen
in he elas ic sca e ing da a o αpa icles on lead, shown in
he le panel o Fig. 3.
These ea u es ha e been commonly in e p e ed by in ok-
ing classical models. The oscilla o y beha iou in he illumi-
na ed egion can be quali a i ely in e p e ed wi hin a F es-
nel pic u e as an in e e ence be ween dis an pu e Coulomb
ajec o ies wi h close ajec o ies a ec ed by he nuclea
a ac i e po en ial, while he shadow egion can be unde -
s ood as a consequence o he exis ence o a maximum angle
in he classical de lec ion unc ion [17]. A comple e unde -
s anding o he obse ed pa e n equi es also he in oduc-
ion o e ac i e e ec s which a ise in nuclea sca e ing as
a esul o abso p ion [21].
All hese e ec s a e na u ally accommoda ed wi hin he
OM. Fo example, he solid line in he le panel o Fig. 3
ep esen s an OM i o he da a using a s anda d (complex)
Woods-Saxon po en ial. Al hough he OM is no unique, one
may ind po en ials wi h adii close o he sum o he geome -
ical adiio hein e ac ingnucleianddi usenesspa ame e s
close o hose o he nuclea densi ies.
In he case o eac ions in ol ing weakly bound nuclei,
he obse ed elas ic sca e ing angula dis ibu ions dis-
play no able de ia ions wi h espec o he a o emen ioned
beha iou . The e ec becomes mo e e iden in he case o
halo nuclei, such as 6He o 11Li. The oscilla ions in he illu-
mina ed egion a e damped (o e en absen ). The d op o
he c oss sec ion wi h espec o he Ru he o d p edic ion
s a s a smalle angles and hence he c oss sec ion exhibi s
a smoo he angula dependence. An example is gi en by he
6He+208Pb da a aken a Elab =22 MeV shown in he igh
panel o Fig. 3.
123
47 Page 6 o 57 Eu . Phys. J. A (2025) 61:47
Table 1 Woods–Saxon pa ame e s o 4,6He+208Pb op ical models.
Reduced adii ( x) a e con e ed in o absolu e (physical) adii as Rx=
x(A1/3
p+A1/3
)
Sys em V0 0a0W iai
[MeV] [ m] [ m] [MeV] [ m] [ m]
4He+208Pb 96.44 1.085 0.625 32 0.958 0.42
6He+208Pb 124.8 1.085 0.564 6.8 0.958 1.91
This beha iou can also be desc ibed wi hin he OM
bu wi h po en ial pa ame e s e y di e en om hose
used in s anda d pa ame iza ions ex ac ed om well-bound
nuclei. In pa icula , one needs e y di use po en ials wi h
long- ange abso p i e ails. The e ec o his long- ange
abso p ion is wo old: (i) i damps o supp esses he elas-
ic Coulomb ampli ude a small angles (i.e., a e y la ge
impac pa ame e s), and (ii) i educes he nuclea ampli-
ude. The esul is ha he o wa d c oss-sec ion no only
becomes smalle o e all, bu also loses i s Coulomb-nuclea
oscilla ions.
In o de o ep oduce he elas ic sca e ing da a o weakly-
bound nuclei wi hin he OM amewo k using s anda d
Woods-Saxon po en ials, a e y long- ange imagina y pa
is needed. Table 1lis s he OMP pa ame e s o 4He+208Pb
and 6He+208Pb a he same inciden ene gy (22 MeV). The
adii o he eal and imagina y pa s a e kep he same
o bo h sys ems. To ep oduce he da a, he 6He OM
equi es a signi ican ly la ge imagina y di useness pa am-
e e . The esul o he OM calcula ion o 6He is com-
pa ed o he da a o his eac ion in he igh panel o
Fig. 3.
This p ope y is a clea indica ion o he p esence o eac-
ion channels domina ed by long- ange couplings. Na u al
candida es a e he b eakup o he p ojec ile and neu on
ans e (ei he one o wo) om he p ojec ile o he a -
ge nucleus. Bo h p ocesses a e o a pe iphe al na u e, which
explains he long- ange imagina y po en ial. To iden i y he
ac ual channels p oducing his e ec one mus go beyond he
single-channel OM scheme and inco po a e hose channels
in o he eac ion amewo k. In he case o he b eakup o
he p ojec ile, his can be e icien ly accomplished wi h he
Con inuum-Disc e izedCoupled-Channels(CDCC)me hod,
which is desc ibed in he nex sec ions.
3 Inclusion o b eakup: he CDCC me hod
3.1 B ie esumè o he CDCC me hod
The CDCC me hod was o iginally in oduced by G. Raw-
i sche [24] and la e e ined by he Pi sbu gh-Kyushu col-
labo a ion [16,25] o desc ibe he e ec o he b eakup chan-
nels on he elas ic sca e ing o deu e ons. Deno ing he eac-
ion by a+A, wi h a=b+x( e e ed he ea e as he co e
and alence pa icles, espec i ely), he me hod assumes he
e ec i e h ee-body Hamil onian
H=Hp oj +T
R+UbA( bA)+UxA( xA), (6)
wi h Hp oj =T +Vbx he p ojec ile in e nal Hamil onian,
T and T
Ra e kine ic ene gy ope a o s, Vbx he in e -clus e
in e ac ion and UbA and UxA a e he co e- a ge and alence-
a ge op ical po en ials (complex in gene al) desc ibing he
elas ic sca e ing o he co esponding b+Aand x+Asub-
sys ems, a he same ene gy pe nucleon o he inciden p o-
jec ile. In he CDCC me hod he h ee-body wa e unc ion
o he sys em is expanded in e ms o he eigens a es o he
Hamil onian Hp oj including bo h bound and unbound s a es.
Since he la e o m a con inuum, a p ocedu e o disc e iza-
ion is applied, consis ing in ep esen ing his con inuum by a
ini eand disc e ese o squa e-in eg able unc ions.Inac ual
calcula ions, his con inuum mus be unca ed in exci a ion
ene gy and limi ed o a ini e numbe o pa ial wa es asso-
cia ed wi h he ela i e co-o dina e . No malizable s a es
ep esen ing he con inuum should be ob ained o each se
o o bi al and o al angula momen a. Two main me hods a e
used o his pu pose:
–The pseudo-s a e me hod, in which he b+xHamil onian
is diagonalized in a basis o squa e-in eg able unc ions,
such as Gaussians [26] o ans o med ha monic oscilla-
o unc ions[27].Nega i eeigen aluesco espond o he
bounds a eso hesys ems,whe easposi i eeigen alues
a e ega ded as a ini e ep esen a ion o he con inuum.
–The binning me hod, in which no malizable s a es a e
ob ained by cons uc ing a wa e packe (bin) by linea
supe posi iono heac ualcon inuums a eso e ace ain
ene gy in e al [16].
Wedesc ibe hela e me hodinsomemo ede ail.Assum-
ing o simplici y a spinless co e, hese disc e ized unc ions
a e deno ed as
φn( )=unjn( )
[Y(ˆ )⊗χs]jnmjn,(7)
123
Eu . Phys. J. A (2025) 61:47 Page 7 o 57 47
whe e n≡ {[kn,kn+1]n,s,jn,mjn}speci ies he n- h bin,
wi h [kn,kn+1] he wa enumbe in e al o he bin, he
alence-co e o bi al angula momen um, s he alence spin,
and
j=
+s he o al angula momen um. The symbol ⊗
deno es angula momen um coupling. The adial pa o he
bin is ob ained as a linea combina ion (i.e., a wa e packe )
o sca e ing s a es as
unjn( )=2
πNnkn+1
kn
wn(k)uk,j( )dk,(8)
whe e uk,j( )is he sca e ing s a es o a con inuum ene gy
ε=¯
h2k2/2μand angula momen um quan um numbe s
, s,j,wn(k)is a weigh unc ion ( o non- esonan con in-
uum wn(k)is usually aken as eiδ, whe e δa e he phase
shi s [28] o he sca e ing s a es wi hin he bin) and Nn
is a no maliza ion cons an . The e ec o his a e aging is
o damp he oscilla ions a la ge dis ances, making he bin
wa e unc ion no malizable.
Assuming a single bound s a e o simplici y, he CDCC
wa e unc ion can be w i en
ΨCDCC(
R, )=χ(+)
0(
R)φ0( )+
N
n=1
χ(+)
n(
R)φn( ), (9)
whe e he index n=0 deno es he g ound s a e o he b+x
sys em.
This model wa e unc ion mus e i y he Sch ödinge
equa ion: [H−E]ΨCDCC(
R, )=0. This gi es ise o a se
o coupled di e en ial equa ions o he unknowns χ(+)
n(
R)
E−εn−T
R−Unn(
R)χ(+)
n(
R)=
m=n
Unm(
R)χ(+)
m(
R),
(10)
whe e εn=φn|Hp oj|φnand Unm(
R)a e he coupling
po en ials gi en by
Unm(
R)=d φ∗
n( )[UbA +UxA]φm( ). (11)
The inclusion o he b eakup channels in expansion (9)
will a ec he elas ic channel wa e unc ion h ough he cou-
pling po en ials U0n(n>0). The en ance lux, gi en by he
no m o he plane wa e associa ed wi h he en ance channel,
will be dis ibu ed among he elas ic and b eakup channels
and his will na u ally educe he elas ic c oss sec ion wi h
espec o he si ua ion in which hose b eakup channels a e
omi ed. This e ec u ns ou o be essen ial o a co ec
desc ip iono heelas icc osssec ions, as illus a ed in Fig. 4
o he d+58Ni eac ion a 80 MeV. The igu e also depic s he
disc e iza ion scheme employed in hese calcula ions, which
comp ises =0,2 con inuum s a es.
In ac ual calcula ions, such as hose pe o med by he pop-
ula coupled-channels code FRESCO [29], he o al h ee-
Fig. 4 Le : Applica ion o he CDCC me hod o d+58Ni elas ic sca -
e ing a Ed=80 MeV. The solid line is he ull CDCC calcula ion. The
dashed line is he calcula ion omi ing he b eakup channels. The cal-
cula ions we e pe o med igno ing he in e nal spins o he p o on and
neu on and so =j. Righ : Illus a ion o con inuum disc e iza ion
o he same p oblem
Fig. 5 Rele an coo dina es o he desc ip ion o he sca e ing o a
wo-body composi e nucleus by a a ge
body wa e unc ion is expanded in s a es wi h good o al
angula momen um (JT), i.e.,
Ψ(
R, )=
βi,JT,MT
Cβi,JT,MTΨβi,JT,MT(
R, )(12)
wi h
Ψβi,JT,MT(
R, )=
β
χJT
β,βi(R)
RYL(ˆ
R)⊗φn,J( )JT,MT
,
(13)
whe e, o he in e nal s a es, φn,J,M, we ha e included an
addi ional subsc ip J o w i e explici ly he p ojec ile spin
andhence nisnow anindex oenume a e s a eswi h hesame
J. Also, in he o me equa ion
Ris he ela i e coo dina e
be ween he p ojec ile cen e o mass and he a ge (assumed
by now o be s uc u eless), see Fig. 5. The index βga he s
he quan um numbe s compa ible wi h a gi en o al angula
momen um JT,β≡{L,J,n}, whe e L(p ojec ile- a ge
o bi al angula momen um) and Jbo h couple o he o al
spin o he h ee-body sys em JT. The spin o he a ge is
igno ed o simplici y o no a ion. Each o hese se s is called
achannel. In pa icula , βideno es he channels compa ible
wi h he ini ial s a e o he sys em ( ypically, he g ound s a e
o he p ojec ile and a ge nuclei).
123
47 Page 8 o 57 Eu . Phys. J. A (2025) 61:47
The adial coe icien s, χJT
β,βi(R), om which he sca -
e ing obse ables a e ex ac ed, a e calcula ed by inse ing
Eq. (13) in he Sch ödinge equa ion, gi ing ise o a sys em
o coupled di e en ial equa ions:
−¯
h2
2μ
d2
dR2+¯
h2L(L+1)
2μR2+εn−EχJT
β,βi(R)
+
β
UJT
β,β(R)χ JT
β,βi(R)=0(14)
whe e εnis he nominal ene gy o unc ion φnand wi h he
coupling po en ials:
UJT
β,β(R)=β;JT|VbA(
R, )+VxA(
R, )|β;JT,(15)
whe e
ˆ
R, |β;JT=YL(ˆ
R)⊗φn,J( )JT
.(16)
The sys em o equa ions (14) is o be sol ed nume ically
ollowing heme hodsdesc ibedelsewhe e (see,e.g.[30]and
e e ences he ein) and subjec o he asymp o ic bounda y
condi ions:
χJT
β:βi(Kβ,R)→i
2H(−)
L(KβR)δβ,βi−SJT
β,βiH(+)
L(KβR)
(17)
→FL(KβR)δβ,βi+TJT
β,βiH(+)
L(KβR)(18)
whe e Kβ=√2μ(E−n),SJT
β,βia e he S-ma ix elemen s,
TJT
β,βia e he T-ma ix elemen s, H±)
L(KR)a e he ingoing
(−) and ou going (+) Coulomb unc ions and FL(KR)is
he egula Coulomb unc ion [17]. Because o he ela ion
be ween he Coulomb unc ions, he wo equa ions abo e
a e ully equi alen . In ac , he coe icien s SJT
β,βiand TJT
β,βi
a e ela ed by SJT
β,βi=δβ,βi+2iTJT
β,βi. Sca e ing ampli udes
and di e en ial c oss sec ions a e exp essed in e ms o hese
coe icien s (see, e.g., [17,31]).
Thes anda dCDCCme hod isbased onas ic h ee-body
eac ion model (b+x+A), and has p o en a he success ul
in desc ibing elas ic and b eakup c oss sec ions o deu e ons
and o he weakly bound wo-body nuclei, such as 6,7Li and
11Be(seeFig. 6).Howe e ,i haslimi a ions. Theassump ion
o ine bodies is no always jus i ied, since exci a ions o he
p ojec ile cons i uen s (band x) and o he a ge (A)may
ake place along wi h p ojec ile dissocia ion. Fu he mo e,
he wo-body pic u e may be inadequa e o some nuclei as,
o example, in he case o he Bo omean sys ems (e.g. 6He,
11Li). Some ex ensions o he CDCC me hod o deal wi h
hese si ua ions a e ou lined below.
Fig. 6 Applica ion o he CDCC me hod o 6Li+40Ca elas ic sca e ing
a 156 MeV. The solid and dashed lines a e he CDCC calcula ions wi h
and wi hou inclusion o he 6Li (α+d) con inuum. Expe imen al da a
a e om Re . [32]. Adap ed om [27]
3.2 The ole o closed channels
In p inciple, expansion (9) may include in e nal s a es wi h
exci a ion ene gies below (εi<E) o abo e (εi>E) he
o al ene gy, which a e e e ed o as open and closed chan-
nels, espec i ely. Closed channels iola e ene gy conse a-
ion and, as such, canno con ibu e asymp o ically o he
sca e ing wa e unc ion. Howe e , due o he couplings wi h
o he channels, hey can a ec he open channels and hence
he c oss sec ions.
The solu ion o he CDCC equa ions (14) in he p esence
o closed channels is o ally analogous o he case wi h open
channels, excep o he modi ica ion o he bounda y condi-
ion o he adial unc ions associa ed wi h closed channels,
which now eads
χJT
β:βi(Kβ,R)→Cβ,βiW−ηi,L+1/2(−2iKβR), (19)
whe e W−ηi,L+1/2is a Whi ake unc ion wi h η he Som-
me eld pa ame e , de ined as η=ZpZ e2/¯
h . In gene al,
he e ec o closed channels is small a su icien ly high ene -
gies. Howe e , a low ene gies, hey can ha e a s ong in lu-
ence on he sca e ing obse ables. An example is shown in
Fig. 7, co esponding o he eac ion d+12 C→p+n+12 C
a 12 MeV [33]. CDCC calcula ions wi h and wi hou he
inclusion o closed channels a e compa ed wi h Faddee cal-
cula ions pe o med in he momen um-space o mula ion o
Al , G assbe ge and Sandhas (FAGS) [34], which p o ide
an essen ially exac solu ion o he same Hamil onian and
he e o e also accoun o he e ec o closed channels. The
uppe and lowe panels a e o he angula and exci a ion
ene gy dis ibu ions. The impo ance o closed channels is
123
Eu . Phys. J. A (2025) 61:47 Page 9 o 57 47
con e ged
open channels only
w/o odd pa ial wa es
FAGS
T (deg)
dV/d: (mb/s )
(a)
FAGS
con e ged
open channels only
H (MeV)
dV/dH (mb/MeV)
(b)
Fig. 7 (a) Angula dis ibu ion and (b) b eakup ene gy dis ibu ion o
he elas ic b eakup c oss sec ion o 12C(d,pn)12Ca 12MeV.The
solid, dashed, and dash-do ed lines in each panel show he con e ged
CDCC esul , he esul o he CDCC me hod calcula ed wi h only he
open channels, and he esul o he Faddee -AGS (FAGS) heo y aken
om Re . [35], espec i ely. The do ed line in (a) is he same as he
solid line bu omi ing he odd pa ial wa es be ween pand n. Taken
om [33], wi h au ho iza ion om APS
clea ly seen. Only when closed channels a e included, he
CDCC calcula ion app oaches he Faddee solu ion.
3.3 T i ially equi alen pola iza ion po en ial
F om a coupled-channel calcula ion (in pa icula , CDCC)
onemayex ac ane ec i epola iza ionpo en ial,alsocalled
i ial equi alen local pola iza ion po en ial (TELP). Fo
simplici y, we assume ha he e is only one en ance channel
(βi=β0) so ha he label βiin (20) can be omi ed. Then
he adial equa ion o he elas ic channel becomes:
−¯
h2
2μ
d2
dR2+¯
h2L0(L0+1)
2μR2+UJT
β0,β0(R)+ε0−EχJT
β0(R)
=−
β
UJT
β0,β(R)χ JT
β(R)=0(20)
Fig. 8 A e agepola iza ionpo en ialUpol (solid lines), compa ed wi h
he g ound s a e diagonal monopole po en ial U00 (dashed) o he
d+58Ni eac ion a 80 MeV
The igh -hand side o his equa ion is hen used o de ine
he angula -momen um dependen pola iza ion po en ial:
Upol(R)=JTwJT(R)UTELP
JT(R)
JTwJT(R),(21)
whe e UTELP
JT(R)is he “ i ial equi alen po en ial” o he
o al angula momen um JTde ined by
UTELP
JT(R)=1
χJT
β0(R)
β=β0
UJT
β0,β(R)χ JT
β(R), (22)
and wJT,βi(R)a e weigh ac o s chosen as
wJT(R)=(2JT+1)(1−|SJT
β0,β0|2)|χJT
β0(R)|2,(23)
whe e SJT
β0,β0a e he elas ic S-ma ix elemen s o each JT
alue.Asinglechannelcalcula ion(elas icchannel)using he
sum o “UJT
β0,β0(R)+Upol(R)” should app oxima ely ep o-
duce he elas ic sca e ing c oss sec ions.
InFig.9,weshow o illus a ion hepola iza ionpo en ial
de i ed om he CDCC calcula ion o he eac ion d+58Ni
eac ion a 80 MeV (c. . Fig. 4). Fo compa ison, we show he
g ound s a e diagonal monopole coupling po en ial (U00)In
his case, he e ec p oduced by he coupling o he b eakup
channels is mos ly epulsi e and abso p i e.
3.4 Inclusion o co e exci a ions
Exci a ions o he p ojec ile cons i uen s (band xin ou case)
may ake place along wi h p ojec ile b eakup. This mecha-
nism is neglec ed in he s anda d o mula ion o he CDCC
me hod. Fo example, o he sca e ing o halo nuclei, he
co e agmen bis assumed o be ine and, as such, he p o-
jec ile s a es a e desc ibed by pu e single-pa icle o pu e
clus e s a es. Excep o e y speci ic cases (e.g. when bis
an alpha pa icle), he s uc u e o a composi e sys em o he
123
47 Page 16 o 57 Eu . Phys. J. A (2025) 61:47
Fig. 20 8Li+12C( oppanel)and8Li+209Bi(bo ompanel)elas icc oss
sec ion, ela i e oRu he o d,compu edwi hin hemic oscopicCDCC
app oach o h ee-body p ojec iles. Do ed lines ep esen he calcu-
la ions wi hou b eakup channels and he solid lines a e he ull calcu-
la ions. Expe imen al da a om Re s. [72,73]. Figu e om Re . [71],
wi h au ho iza ion om Sp inge
Figu e19 shows an applica ion o he me hod o he eac-
ion 7Li+208Pb a nea -ba ie ene gies. Expe imen al da a
a e compa ed wi h a one-channel calcula ion (only 7Li g.s.),
wo-channel calcula ion (g ound plus i s exci ed s a e) and
se e al CDCC calcula ions including con inuum s a es up o
a ce ain p ojec ile angula momen um. I is seen ha a good
desc ip ion o he da a is achie ed when a su icien ly la ge
numbe o con inuum s a es a e included.
O he applica ions o he mic oscopic CDCC me hod
include i s ex ension o h ee-body p ojec iles including a
single nucleon as one o he h ee pa icles, such as 8Li o
8Bas(4He+ +n)and (4He+3He+p), espec i ely [71].
Figu e20 shows esul s o he elas ic sca e ing o 7Li on
12C and 209Bi a ge s a low ene gies. The dashed line co e-
sponds o he calcula ion including only he p ojec ile g ound
s a e, while he solid line is he ull mic oscopic CDCC cal-
cula ions ha exhibi a be e ag eemen wi h he da a.
3.8 In e p e a ion o he CDCC wa e unc ion and c oss
sec ions
Fo a meaning ul compa ison wi h expe imen al da a, i is
impo an o p ope ly unde s and which in o ma ion and
obse ables a e (and which a e no !) p o ided by he CDCC
wa e unc ion. O cou se, elas ic sca e ing is he mos di ec
ou come o he me hod and he one ha can be compa ed
wi h da a mo e easily. By cons uc ion, he elas ic sca e ing
p oduced by he CDCC calcula ion inco po a es he e ec o
he coupling o he inelas ic and b eakup channels included
in he CDCC modelspace. Acco ding o ou o me no a ion,
hese b eakup channels co espond o he h ee-body inal
s a es o he o m b+x+Ain which he a ge and he
p ojec ile subsys ems band xescape he in e ac ion egion
and emain in hei g ound s a e. Mo o e , in addi ion o
hese channels explici ly included in he modelspace, CDCC
includes e ec i ely he e ec s o o he channels associa ed
wi h possible exci a ions o he a ge and/o o he agmen s
ia he op ical po en ials.
Rega ding b eakup obse ables, in p inciple, he CDCC
wa e unc ion p o ides only he elas ic b eakup ones. The
mos immedia e obse ables a e he b eakup angula dis i-
bu ions o speci ic bin in e als, i.e.,
dσn
dΩc.m. =| 0,n(θ)|2.(35)
When he binning me hod is employed, an app oxima e dou-
ble di e en ial c oss sec ion (wi h espec o he c.m. angle
and ela i e ene gy) can be ob ained by di iding each angula
dis ibu ion by he bin wid h, as ollows:
d2σ
dΩc.m.d≃1
Δn
dσn
dΩc.m. .(36)
Rega ding henonelas ic b eakupcon ibu ions,whicha e
e ec i ely included in he agmen - a ge imagina y po en-
ials, he CDCC me hod by i sel does no p o ide de ailed
b eakup c oss sec ions in ol ing exci a ions. Howe e , one
can de ine and compu e a o al abso p ion c oss sec ion ha
accoun s o hei o e all con ibu ion. Fo ha , one i s
de i es a eac ion c oss sec ion in e ms o he CDCC S-
ma ices as1:
σ eac =π
K2
i
JTπ
Li
2JT+1
(2Ji
p+1)(2Ji
+1)1−|SJT
βi,βi|2,
(37)
and an in eg a ed elas ic b eakup c oss sec ion
σbu =π
K2
i
K
Ki
JTπ
LLi
2JT+1
(2Ji
p+1)(2Ji
+1)|SJT
β,βi|2.(38)
Then, one may de ine an abso p ion c oss sec ion as
σabs =σ eac−σbu. This abso p ion c oss sec ion will accoun
o all nonelas ic p ocesses associa ed wi h he imagina y
pa o he agmen - a ge po en ials, such as a ge exci a-
ion (i.e., inelas ic sca e ing), comple e usion (i.e., cap u e
1See Re . [74] o a di e en con en ion.
123
Eu . Phys. J. A (2025) 61:47 Page 17 o 57 47
Fig. 21 Compa ison o one-channel (i.e., no con inuum) calcula ions
and ull CDCC calcula ions o he elas ic sca e ing o a8B+90Z and
b6He+208Pb a nea -ba ie ene gies. See ex o de ails
o he whole p ojec ile) o incomple e usion (i.e., cap u e o
one o he p ojec ile agmen s). Ve y o en, compa ison wi h
expe imen al da a equi es mo e de ailed c oss sec ions, such
as heangula o ene gydis ibu ions o he agmen ha su -
i es a e he cap u e (abso p ion) o he o he agmen . As
no ed abo e, hese obse ables a e no di ec ly p o ided by
he CDCC me hod, a leas in i s s anda d o mula ion. How-
e e , as will be shown in Sec . 3.8.2, he CDCC wa e unc ion
can be used as an inpu o a sui able o malism capable o
p o iding such obse ables.
The abso p ion c oss sec ion plays an impo an ole in
unde s anding he CDCC esul s. To quan i y he impo ance
o he con inuum on he elas ic c oss sec ions i is common
o compa e ull CDCC esul s wi h no-con inuum calcula-
ions, in which only he g ound-s a e o g ound-s a e cou-
pling po en ial is e ained. An example is shown in Fig. 21,
which depic s he esul s o he 6He+208Pb eac ion a 18
MeV and he 8B+90Z eac ion a 26.5 MeV. Dashed and
solid lines deno e he no-con inuum and ull CDCC calcu-
la ions, espec i ely (in he 6He case, he imp o ed dineu-
on model o Re . [75] was used). Table 2lis s he eac-
ion, abso p ion, and (elas ic) b eakup c oss sec ions om
Table 2 Compa ison o eac ion, abso p ion and elas ic b eakup c oss
sec ions o 6He+208Pb a 18 MeV and 8B+90Z a 26.5 MeV
Reac ion σ eac σabs σbu
(mb) (mb) (mb)
8B+90Z Full 362 136 226
No con 136 124 –
6He+208Pb Full 477 359 117
No con 134 134 –
he CDCC calcula ions. Clea ly, inclusion o he con inuum
has a much la ge e ec on he elas ic c oss sec ion o 6He.
This is in appa en con adic ion wi h he ac ha he elas ic
b eakup c oss sec ion is la ge in he 8B case, as shown in
he Table. The appa en inconsis ency can be explained by
looking a he abso p ion and eac ion c oss sec ions. In he
8B eac ion, he inclusion o he con inuum couplings p o-
duces only a small inc ease o he abso p ion c oss sec ion,
and he eac ion c oss sec ion is essen ially inc eased by he
elas ic b eakup c oss sec ion. By con as , in he 6He eac-
ion, he abso p ion c oss sec ion inc eases by almos a ac o
o 3 when adding he con inuum couplings and his esul s
also in an equi alen inc ease o he eac ion c oss sec ion,
hence he enhanced e ec on he elas ic c oss sec ion. One
may an icipa e ha he inc eased abso p ion c oss sec ion o
6He is associa ed wi h p ocesses in which he alence neu-
ons expe ience some kind o nonelas ic in e ac ion wi h he
a ge , such as ans e o bound s a es. These p ocesses will
be discussed in a subsequen sec ion.
The di e en e ec o he b eakup channels on he elas-
ic channel o neu on e sus p o on halo nuclei has been
add essed in he li e a u e by se e al au ho s [76–78].
3.8.1 Smoo hing p ocedu e o wo-body obse ables
Owing o he disc e iza ion p ocedu e inhe en o he CDCC
me hod, he b eakup c oss sec ions a e gi en only o he dis-
c e e ene gies o he bins o pseudos a es. In eali y, b eakup
c oss sec ions should be con inuous unc ions o he ela i e
ene gy be ween he agmen s. Se e al p ocedu es ha e been
p oposed o con e he disc e e b eakup ene gy dis ibu ions
in o con inuous dis ibu ions o a bi a y alues o he in e -
agmen ela i e ene gy. We discuss he e wo such me hods,
one based on he S-ma ices and he o he on he sca e ing
ampli udes.
TheS-ma icescalcula edinCDCCco espond odisc e e
alueso heene gy.In hebinningme hod,anapp oxima ion
o his con inuous S−ma ix can be ob ained di iding he
disc e e S−ma ix by he squa e oo o he bin wid h, i.e.,
SJT
β,βi(k)≈1
√ΔEnˆ
SJT
β,βi(39)
123
47 Page 18 o 57 Eu . Phys. J. A (2025) 61:47
wi h β={L,J,n},βi={Li,J0,n0}and ΔEnis he wid h
o he n- h bin.
In he PS me hod, one could apply a simila p ocedu e by
assigning a wid h o each pseudos a e. In ac , his app oach
was used, o example, in Re . [79] o calcula e he di e en-
ial b eakup c oss sec ion om he c oss sec ion o indi idual
pseudos a es, assuming ha he wid h o he i h pseudos a e
is app oxima ely gi en by ΔEn=(εn+1−εn−1)/2. A mo e
accu a e p ocedu e, p e iously p oposed in Re . [80], is o
ob ain con inuous S−ma ix elemen s Sβ,βi(k), depending
on he con inuous a iable k, as well as on he ini ial and inal
angula momen a, by an app op ia e supe posi ion o he dis-
c e e S−ma ix elemen s ˆ
Sβ,βi esul ing om he solu ion o
he coupled channel equa ions. Igno ing o simplici y in in-
sic spins o he p ojec ile cons i uen s as [80,81] one ge s
SJT
β,βi(k)≈
N
n=1φ(−)
k, |φ(N)
n, ˆ
SJT
β,βi,(40)
whe e φ(−)
k, ( )and φ(N)
n, ( )a e he adial pa s o he exac
and pseudos a e wa e unc ions, espec i ely. The sum uns
o e hese o pseudos a es included in he coupled–channels
calcula ion.
F om he S-ma ices, he double di e en ial c oss sec ion
o a b−xb eakup s a e wi h o bi al angula momen um
and wa enumbe kis gi en by:
d2σ(k)
dk dΩc.m. =
m=0
π
K2
0
J,L
(2J+1)mL −m|J0
×|YL−m(Ωα)SJT
β,βi(k)2
,(41)
As an applica ion o his me hod, in Fig. 22 we plo he
modulus o he b eakup S−ma ix elemen s o he eac ion
d+58Ni→p+n+58Ni o a o al angula momen um J=17.
In insic spins o he p o on and neu on we e omi ed, so
he ele an channels a e (, L)=(0,17), (2,15), (2,17) and
(2,19). Fo he CDCC-Bin calcula ions, he con inuum was
di ided in o Ns=Nd=35 bins up o a maximum exci a ion
ene gy o 70 MeV. Fo he CDCC-PS calcula ion, a ans-
o med ha monic oscilla o (THO) basis o N=50 s a es
was necessa y o ob ain ull con e gence a high exci a ion
ene gies, al hough N=30 gi es al eady a he good esul s.
A e diagonaliza ion o he in e nal Hamil onian, only he
eigens a es below 70 MeV we e e ained, educing he ac ual
size o he basis o only 24 o each pa ial wa e (along wi h
he g ound s a e). In Fig. 22, he his og am ep esen s he
CDCC-Bin calcula ion, he illed ci cles co espond o he
CDCC-PS calcula ion, in which each disc e e S−ma ix has
been di ided by he squa e oo o he pseudos a e wid h,
and he line is he CDCC-PS calcula ion olded wi h he con-
inuum wa e unc ions, Eq. (40). I is clea ly seen ha bo h
disc e iza ion me hods a e in almos pe ec ag eemen . Fu -
Fig. 22 Modulus o he b eakup Sma ix elemen s o he o al angula
momen um J=17 o he eac ion d+58Ni a 80 MeV as a unc ion o
hep–n ela i emomen umin he inals a e.Thehis og amis heCDCC
calcula ion wi h binning me hod. The illed and open ci cles ep esen
he CDCC calcula ions using he analy ical THO pseudos a e basis o
Re . [27]. The solid lines a e ob ained olding he disc e e S-ma ices
wi h he ue con inuum wa e unc ions. Adap ed om Re . [27]
he de ails abou hese calcula ions can be ound in Re .
[27].
The connec ion be ween he disc e e and con inuous
b eakup c oss sec ions can be also done a he le el o
he ansi ion ampli udes hemsel es. In bo h CDCC and
XCDCC, he solu ion o he coupled-channel equa ions p o-
ides some disc e e b eakup ansi ion ampli udes, deno ed
Ti,J0,J
M0,M(θi,Ki), connec ing an ini ial s a e |J0M0wi h a
h ee-body inal s a e comp ised o he a ge (assumed o
be s uc u eless), he alence pa icle and he co e, a some
disc e e alue o he inal p ojec ile- a ge c.m. momen um
Ki={θi,Ki}. The i s s ep o he o malism is o ela e
hese disc e e ampli udes wi h he ac ual b eakup sca e ing
ampli udes,wi hou con inuumdisc e iza ion, ha wedeno e
as TIs;J0
μσ;M0(
k,
K), whe e
kis he p ojec ile in e nal ela i e
momen um. Fo mally, hese b eakup ansi ion ampli udes
can be w i en in in eg al o m as:
TIs;J0
μσ;M0(
k,
K)=φ(−)
k;Iμ;sσei
K·
R|U|ΨJ0,M0(
K0),(42)
whe e U=UbA( ,
R,ξ)+UxA( ,
R)and φ(−)
k;Iμ;sσa e wo-
body exac sca e ing wa e unc ions o he b+xsys em
o a ela i e inal momen um
kand gi en co e and alence
spins.
In o de o ela e he disc e e and con inuous ampli udes,
one can app oxima e he exac wa e unc ion ΨJ0,M0in he
equa ion abo e by i s (X)CDCC coun e pa and in oduce
he app oxima e comple eness ela ion in he unca ed dis-
c e e basis {φ(N)
n,J,M;i=1,...,N}:
123
Eu . Phys. J. A (2025) 61:47 Page 19 o 57 47
TIs;J0
μσ;M0(
k,
K)≃
n,J,Mφ(−)
k;Iμ;sσ|φ(N)
n,J,Mφ(N)
n,J,Mei
K·
R
×|U|ΨCDCC
J0,M0(
K0)
=
n,J,Mφ(−)
k;Iμ;sσ|φ(N)
n,J,MTn,J0,J
M0,M(
K),
(43)
whe e he ansi ion ma ix elemen s Tn,J0,J
M0,M(
K)a e o be
in e pola ed om he disc e e ones Tn,J0,J
M0,M(θi,Ki). Exp es-
sions o he o e laps be ween he inal sca e ing s a es and
he disc e e s a es, φ(−)
k;Iμ;sσ|φ(N)
n,J,M, a e gi en explici ly in
Re s. [81] and [42] o bin and PS unc ions, espec i ely.
The ansi ion ampli udes o Eq. (43), TIs;J0
μσ;M0(
k,
K), con-
ain he dynamics o he p ocess in he coo dina es desc ib-
ing he ela i e and cen e o mass mo ion o he co e and
he alence pa icle. F om hese ampli udes one can de i e
wo-body obse ables o a ixed spin o he co e, I, he solid
angles desc ibing he o ien a ions o
k(Ωk) and
K(ΩK), as
well as he ela i e ene gy be ween he alence and he co e,
E el. These obse ables ac o ize in o he ansi ion ma ix
elemen s and a kinema ical ac o :
d3σ(I)
dΩkdΩKdE el =μbxkI
(2π)5¯
h6
K
K0
μ2
aA
2J0+1
×
μ,σ,M0|TIs;J0
μσ;M0(
k,
K)|2,(44)
whe e μbx and μaA a e he alence-co e and p ojec ile- a ge
educed masses.
3.8.2 Th ee-body obse ables
Wi hin he CDCCandXCDCC eac ion o malisms,b eakup
is ea ed as an exci a ion o he p ojec ile o he con inuum,
so he heo e ical c oss sec ions a e desc ibed in e ms o
he c.m. sca e ing angle o he p ojec ile and he ela i e
ene gy o he cons i uen s, using wo-body kinema ics. Fo
compa ison wi h expe imen al da a, i is use ul o ha e also
he c oss sec ions in e ms o he angle and ene gy o he
p ojec ile agmen s, since hese quan i ies a e mo e di ec ly
connec ed wi h he ac ual measu emen s.
In he case o he s anda d CDCC amewo k, i e old
ully exclusi e c oss sec ions ha e been de i ed and p e-
sen ed by se e al au ho s [81,82]. The me hod was gene al-
ized in Re . [42] o he case o XCDCC. We b ie ly e iew he
main o mulas o he la e , no ing ha he case wi hou co e
exci a ion is eco e ed when a single co e s a e is conside ed.
Fo simplici y, we igno e he a ge spin.
Using he wo-body ansi ion ampli udes discussed in he
p e ious subsec ion, he h ee-body obse ables, assuming
he ene gy o he co e is measu ed, a e gi en by [81]:
d3σ(I)
dΩbdΩxdEb=2πμaA
¯
h2K0
1
2J0+1
×
μ,σ,M0|TIs;J0
μσ;M0(
k,
K)|2ρ(Ωb,Ω
x,Eb),
(45)
whe e he phase space e m ρ(Ωb,Ω
x,Eb), i.e., he numbe
o s a es pe uni co e ene gy in e al a solid angles Ωband
Ωx, akes he o m [83]:
ρ(Ωb,Ω
x,Eb)=mbmx¯
hkb¯
hkx
(2π¯
h)6
×mA
mx+mA+mx(
kb−
K o )·
kx/k2
x.
(46)
He e, he pa icle masses a e gi en by mb(co e), mx
( alence), and mA( a ge ) while ¯
h
kband ¯
h
kxa e he co e
and alence pa icle momen a in he inal s a e. The o al
momen um o he sys em co esponds o ¯
h
K o and he con-
nec ion wi h he momen a in Eq. (43) is made h ough:
K=
kb+
kx−ma
M o
K o ;
k=mb
ma
kx−mx
ma
kb(47)
wi h ma=mb+mxand M o =mb+mx+mA he o al
masses o he p ojec ile and he h ee-body sys em, espec-
i ely.
An applica ion o his o malism is p esen ed in Fig. 23,
co esponding o he b eakup o 11Be on a p o on a ge a
Ep=63.7 MeV/u. The op panel co esponds o he di e -
en ial c oss sec ion wi h espec o he inal n+10Be ela i e
ene gy and he bo om panel o he b eakup wi h espec o
he inal 10Be ene gy (in eg a ed in he 10Be and neu on
angles). These obse ables we e compu ed by ans o m-
ing he XCDCC b eakup ansi ion ampli udes by means o
Eqs. (43)–(45), and in eg a ing o e he unmeasu ed a i-
ables. The XCDCC calcula ions we e pe o med wi h a 11Be
model including he 10Be g ound (0+) and i s exci ed (2+)
s a es.The o malismallows o sepa a eandquan i y hecon-
ibu ion o hese wo s a es o 10Be. No e ha , due o ene gy
conse a ion, in he ela i e ene gy dis ibu ion, he con ibu-
ion coming om he 10Be(2+) s a e con ibu es only abo e
he exci a ion ene gy o his s a e (Ex=3.367 MeV). The
impo ance o he 10Be exci a ion du ing he eac ion due o
i s in e ac ion wi h he a ge nucleus is illus a ed also in he
op panel by means o a calcula ion in which he de o med
pa o he 10Be+p in e ac ion is omi ed (do -dashed line).
3.8.3 Pseudos a es e sus bins
As discussed in he p eceding sec ions, disc e iza ion p oce-
du es employed in he CDCC me hod a e ei he based on he
123
47 Page 20 o 57 Eu . Phys. J. A (2025) 61:47
Fig. 23 Di e en ial b eakup c oss sec ions, wi h espec o he n-10Be
ela i e ene gy (a) and wi h espec o he inal 10Be ene gy (b), o he
b eakup o 11Be on p o ons a 63.7 MeV/nucleon. The con ibu ions o
he 10Be g ound (0+) and i s exci ed (2+) s a es a e shown in each
panel. In he op panel, he esul o he calcula ion neglec ing he 10Be
exci a ion mechanism (labelled “no DCE”) is also shown. The e ical
a ow deno es he h eshold o 10Be(2+
1)+n. Figu e adap ed om Re .
[42]
pseudos a e (PS) me hod o he binning me hod. Whe eas
eac ion obse ables should be independen o he adop ed
disc e iza ion me hod (p o ided he calcula ions a e ully
con e ged), in p ac ical applica ions one o he wo me h-
ods migh esul mo e con enien han he o he .
The PS disc e iza ion me hod becomes pa icula ly sui -
able when dealing wi h na ow esonances. In he case o
bins,oneneeds ine disc e iza ion ino de oob ain a de ailed
desc ip ion o he esonance egion in he ela i e ene gy
spec um. By con as , using PS’s one can ob ain a de ailed
desc ip ion o he esonance p o ile wi h a ela i ely small
basis, wi h he aid o he con olu ion p ocedu e discussed in
Sec . 3.8.2. This is illus a ed in Fig. 24, whe e we show he
10Be+n ela i e ene gy di e en ial c oss sec ion co espond-
ing o he11Be+p →10Be+n+p eac ion a an inciden ene gy
Fig. 24 Compa ison o pseudos a e (PS) and binning me hods o he
11Be+p →10Be+n+p eac ion a 63.7 MeV/nucleon. Bo h me hods
use he same s uc u e model [84] and include con inuum s a es wi h
con igu a ion s1/2,p1/2,p3/2,d3/2,d5/2. Fo he CDCC-PS calcula ion,
he dominan indi idual con ibu ions a e shown sepa a ely
o 63.7 MeV/nucleon. The 11Be bound and con inuum s a es
we e gene a ed wi h he s uc u e model o Re . [84], which
doesno accoun o 10Beexci a ions.Con inuum wa ess1/2,
p1/2,p3/2,d3/2,d5/2we e included. Fo he CDCC-Bin cal-
cula ion, he con inuum was unca ed a E el =12 MeV
and di ided in o 12 bins o each pa ial wa e, e enly spaced
in momen um. The ela i e-ene gy di e en ial c oss sec ion
was ob ained by di iding he c oss sec ion o each bin by
he wid h o ha bin [see Eq. (36)]. The CDCC-PS calcula-
ions employ a THO basis wi h N=30 (N=25) oscilla o
unc ions o =1,2(=0). A e diagonaliza ion o he
11Be Hamil onian on his basis, only he eigen alues below
12 MeV we e e ained. This esul s in 13 s a es o each pa -
ial wa e (15 o =0). The calcula ed b eakup sca e ing
ampli udes we e hen con olu ed using Eq. (43), om which
he di e en ial c oss sec ions, Eq. (44), we e hen e alua ed.
Finally, he la e we e in eg a ed o e he solid angles Ωk
and ΩK.
As seen in Fig. 24, he binning and PS me hods yield, as
expec ed, almos iden ical esul s bu he PS me hod allows
o a ine desc ip ion o he esonance egion wi h a compa-
able, o e en smalle , numbe o basis unc ions.
Howe e , he e a e o he si ua ions whe e he binning p o-
cedu e u ns ou o be mo e e icien han he PS me hod. One
such si ua ion is he case o eac ions o highly pola izable
weakly-boundnuclei(suchas neu on-halo nuclei) onhigh-Z
a ge s. These eac ions a e cha ac e ized by he p esence o
s ong long- ange Coulomb couplings, which end o empha-
size low-lying exci a ion s a es and p obe la ge sepa a ions.
In his case, he binning me hod has been shown o p o ide
mo e s able esul s wi h espec o he basis size [85].
123
Eu . Phys. J. A (2025) 61:47 Page 21 o 57 47
Fig. 25 Rele an coo dina es o a ans e eac ion o he o m
A(d,p)B
4 T ans e eac ions wi h weakly bound nuclei
T ans e eac ions a e key spec oscopic ools o bo h s a-
ble and uns able nuclei. Angula dis ibu ions o ou going
agmen s p o ide in o ma ion on he angula momen um
con en o he popula ed s a es and he magni ude is closely
ela ed o he single-pa icle con en o hese s a es, quan i-
ied in e ms o spec oscopic ac o s. The s anda d ool o
analysing ans e eac ions, hedis o ed-wa eBo napp ox-
ima ion (DWBA) me hod, assumes ha he ans e occu s in
a single-s ep. Possible exci a ions o he colliding o ou go-
ing nuclei a e igno ed o , a mos , aken in o accoun e ec-
i ely h ough he choice o he e ec i e in e ac ions. These
in e ac ions a e ypically chosen so as o ep oduce he co e-
sponding elas ic sca e ing c oss sec ions in he inciden and
exi channels.
To see how his me hod wo ks in p ac ice, le us conside
asa pa icula caseap ocesso he o m A(d,p)B,schema i-
cally depic ed in Fig. 25. Using he pos - o m ep esen a ion,
he ansi ion ma ix o his p ocess can be w i en as [17]
Tdp =χ(−)
pΦB|Vpn +UpA −UpB|Ψ(+)
d,(48)
whe e Vpn,UpA a e he p o on-neu on and p o on- a ge
in e ac ions, UpB is an auxilia y (and, in p inciple, a bi a y)
po en ial o he p-Bsys em,ΦBis hein e nalwa e unc ion
o he esidual nucleus Band Ψ(+)
dis he o al wa e unc ion
co esponding oaninciden deu e onbeamo kine icene gy
Edand binding ene gy εd. I a ge exci a ion is no consid-
e ed explici ly, his o al wa e unc ion can be app oxima ed
as
Ψ(+)
d( ,
R,ξA)≈Ψ3b( ,
R)dΦ(0)
A(ξA)(49)
whe e Φ(0)
A(ξA)deno es he a ge g ound-s a e wa e unc-
ion.
Igno ingan isymme iza ion o simplici y, hewa e unc-
ion ΦB o a o al angula momen um Jand p ojec ion M
canbeexpanded in As a es using he usual pa en age decom-
posi ion
ΦJM
B( nA,ξ)=
I,, j[ΦI
A(ξ) ⊗φj( nA)]JM,(50)
whe e Iis he spin o A,and j he o bi al and o al (
j=
+s)angula momen umo he alencepa icleandφj( nA)
is a unc ion desc ibing he neu on-co e ela i e mo ion. The
no maliza ion
S, j=|φj( nA)|2d nA (51)
canbe ega dedasaspec oscopic ac o o hecon igu a ion
{, j}.2
I he in e ac ions UpA and UpB a e assumed o be inde-
penden o he a ge deg ees o eedom (ξA), he in eg al in
ξAcan be eadily pe o med, ans o ming Eq. (48)in o
T3b
dp =S, jχ(−)
pφj|Vpn +UpA −UpB|Ψ3b,(+)
d,(52)
In DWBA, he h ee-body wa e unc ion Ψ3b,(+)
dis u -
he app oxima ed by Ψ(+)
d≈χ(+)
d(
R)ϕd( ). When he
ans e eac ion in ol es weakly-bound nuclei, including
he deu e on discussed he e, his choice is no well jus i-
ied. Due o he p esence o he sho - ange Vpn in e ac ion,
he e alua ion o DWBA ma ix elemen is mos ly sensi i e
o small p-nsepa a ions. These con igu a ions do no only
con aincon ibu ionscoming om he deu e on g ounds a e,
bu also om p-nunbound s a es. By con as , he deu e on
op ical po en ial desc ibing deu e on- a ge elas ic sca e ing
is no es ic ed o small p-ndis ances. The e o e, he use o
he deu e on op ical po en ial in he (d,p)o (p,d) ans e
ampli ude is likely o lack impo an deu e on b eakup com-
ponen s. This p oblem was ecognized long ago and se e al
solu ions ha e been p oposed. One o he i s and mos pop-
ula ones is he adiaba ic me hod o Johnson and Sope [86].
The Johnson–Sope app oxima ion
In he Johnson–Sope (JS) app oxima ion, his e ec is
app oxima ely aken in o accoun by means o he choice
Ψ3b,(+)
d(
R, )≃χ(+)
JS (
R)φd( ), (53)
whe e χ(+)
JS (
R)is he solu ion o a wo-body sca e ing p ob-
lem, on he coo dina e
R, in which he in e ac ion is gi en
by a p−nze o- ange app oxima ion:
UJS(R)=UpA(R)+UnA(R). (54)
We see ha , in his limi , he adiaba ic heo y o he ans e
ampli ude adop s a o m akin o ha ound in DWBA, bu his
analogyisonly o malbecause he unc ionχ(+)
JS (
R)includes
con ibu ions om b eakup and he po en ial UJS(R)may
ha e li le o do wi h he op ical po en ial desc ibing he
deu e on elas ic sca e ing, so χ(+)
JS does no p o ide a good
desc ip ion o he elas ic channel. Due o he adiaba ic
app oxima ion, he JS heo y is no expec ed o be accu a e
a low inciden ene gies.
2S ic ly, he unc ions φjand he spec oscopic ac o s Sjdepend in
gene al on he co e s a e Ibu , in he p esen case, his quan um numbe
can be eadily in e ed om he , j alues, so i is omi ed o b e i y.
123
47 Page 22 o 57 Eu . Phys. J. A (2025) 61:47
Fig. 26 Schema ic ep esen a ion o DWBA, ADWA and CDCC-BA
app oaches o a (d,p) ans e eac ion
The adiaba ic app oxima ion is equi alen o neglec ing
he exci a ion ene gy o he p ojec ile s a es, [86], which
amoun s a se ing Hbx →ε0in he h ee-body Hamil o-
nian (6). The adiaba ic wa e unc ion akes in o accoun he
exci a ion ob eakup channels, assuming ha hese s a es a e
degene a e in ene gy wi h he p ojec ile g ound s a e, as illus-
a ed in Fig. 26b. The e o e, he ADWA app oach akes in o
accoun , app oxima ely, he e ec o deu e on b eak-up on
he ans e c osssec ion,wi hin headiaba icapp oxima ion.
So, i should be well sui ed o desc ibe deu e on sca e ing a
highene gies,a ound100MeVpe nucleon.Sys ema ics ud-
ies [87–89] ha e shown ha ADWA is supe io o s anda d
DWBA o (d,p)sca e ing a hese ela i ely high ene gies.
The Johnson–Tandy app oxima ion
Al hough he ze o- ange adiaba ic model o JS p o ides a
sys ema ic imp o emen o e he con en ional DWBA, he e
a esi ua ionsinwhich he o me ails o ep oduce heexpe -
imen alda a[90,91].Modelswhichgobeyond heze o- ange
and adiaba ic app oxima ions a e he e o e needed. One o
such models is he Weinbe g expansion me hod o Johnson
and Tandy (JT) [92]. The idea is o expand Ψ3b,(+)
din e ms
o a se o unc ions which a e comple e wi hin he ange o
Vpn. A con enien choice is he se o Weinbe g s a es (also
called S u mians), gi en by
Ψ3b,(+)
d(
R, )=
N
i=0
φW
i( )χ W
i(
R), (55)
whe e φW
i(ξ) a e he Weinbe g s a es, which a e solu ions o
he eigen alue equa ion
[T +αiVpn]φW
i( )=−εdφW
i( ), (56)
whe e εd=2.225 MeV is he deu e on binding ene gy and
whe e αia e he eigen alues, o be de e mined along wi h
he eigen unc ions. Beyond he ange o he po en ial all
he Weinbe g s a es decay exponen ially, like he deu e on
g ound-s a e wa e unc ion. Fo i=0, α0=1 and so φW
0( )
is jus p opo ional o he deu e on g ound s a e. As iand
αiinc ease, hey oscilla e mo e and mo e apidly a sho
dis ances. The Weinbe g s a es o m a comple e se o unc-
ions o wi hin he ange o he po en ial Vpn.Theya e
Fig. 27 Compa ison o he ADWA and DWBA me hods o he
56Ni(p,d)57Ni (le ) and 48Ca(d,p)49Ca ( igh ) eac ions a inciden
ene gies o 37 MeV and 10 MeV, espec i ely. The nucleon op ical
po en ials a e aken om he global CH89 [50] sys ema ics while he
Daehnick [95] global po en ial is used o he deu e on op ical po en ial
in he DWBA calcula ions. Fo he ADWA calcula ions, he p esc ip-
ionso Johnson–Sope (JS)and Johnson-Tandy(JT) a e shown.Quo ed
om Re . [96]
well sui ed o expand Ψ3b,(+)
din his egion, as is equi ed
by he ampli ude in Eq. (48). They do no sa is y he usual
o hono mali y ela ion bu he less con en ional one
φW
i|Vpn|φW
j=−δij.(57)
I we e aininEq.(55)only heleading e m,Ψ3b,(+)
d(
R, )≈
φW
0χW
0(
R),one inds[92] ha χW
0 e i ies hesingle-channel
equa ion
[T
R+UJT(
R)−Ed]χW
0(
R)=0,(58)
wi h Ed=E−εdand whe e he po en ial UJT is gi en by:
UJT(R)=φW
0( )|Vpn(UnA +UpA)|φW
0( )
φW
0( )|Vpn|φW
0( ).(59)
The b a and ke in his equa ion mean in eg a ion o e ,
wi h ixed
R. In e es ingly, in he ze o- ange limi , UJT(R)
educes o he JS po en ial, Eq. (54). The e o e, he ze o-
o de esul gi enbyEq.(59)canbe ega dedasa ini e- ange
e sion o he adiaba ic (JS) po en ial. These wo models a e
globally e e ed oas Adiaba ic Dis o ed Wa eApp oxima-
ion (ADWA). Howe e , i is wo h no ing ha he ull Wein-
be g expansion makes no e e ence o he inciden ene gy
and, as such, does no in ol e he adiaba ic app oxima ion.
This sugges s ha a s ipping heo y based on his Weinbe g
expansion can be used a low ene gies, whe e he adiaba ic
condi ion is no well sa is ied. The inclusion o highe -o de
e ms (i>0) in he Weinbe g expansion has been in es i-
ga ed in Re s. [93,94].
In Fig. 27 we p esen a compa ison be ween he DWBA
and ADWA me hods o he 56Ni(p,d)57Ni (le ) and 48Ca
(d,p)49Ca ( igh ) eac ions. Fo he ADWA calcula ions, he
p esc ip ions o JS and JT a e shown. I is clea ly seen ha
he ADWA, in bo h i s ze o- ange and ini e- ange o ms,
p o ides an imp o ed desc ip ion o he da a as compa ed o
he DWBA me hod.
An appealing ea u e o he ADWA me hod is ha
i s ing edien s a e comple ely de e mined by expe imen s.
123
Eu . Phys. J. A (2025) 61:47 Page 23 o 57 47
These ing edien s a e he p o on- a ge and neu on- a ge
op ical po en ials, e alua ed a hal he deu e on inciden
ene gy, as well as he well-known p o on-neu on in e ac-
ion. On he nega i e side, he ADWA app oach does no
consis en ly desc ibe elas ic sca e ing and nucleon ans e .
Al hough, physically, elas ic sca e ing, ans e and b eak-
up should be closely ela ed by lux conse a ion, his con-
nec ion is no p esen in ADWA. Fu he mo e, he a gumen s
leading o ADWA a e s ongly dependen on he assump ion
ha he ans e p ocess is go e ned by a sho - ange ope -
a o . Thus, i is no ob ious ha he app oxima ions emain
alid o o he weakly bound sys ems, like 11Be. E en in he
case o (d,p)sca e ing, he ans e ma ix elemen is de e -
mined, in addi ion o he n-pin e ac ion, by he p o on- a ge
and p o on-composi e in e ac ions which de ine he emnan
e m.The oleo hese e ms,whichwouldha econ ibu ions
o h ee-body con igu a ions in which p o on and neu on a e
no so close oge he , is no clea ap io i. The la e p ob-
lem can be a oided by using an al e na i e exp ession o he
sca e ing ampli ude. Following Goldbe g and Wa son [97],
one can choose he auxilia y in e ac ion UpB ha appea s in
he emnan e m o Eq. (52) o cancel exac ly he co e- a ge
in e ac ion, UpA. This esul s in an al e na i e, bu s ill exac ,
sca e ing ampli ude ( he spec oscopic ac o is omi ed o
simplici y), namely,
T3bGW
dp =˜
Φ(−)
pB |Vpn|Ψ3b,(+)
d,(60)
whe e ˜
Φ(−)is a solu ion o he h ee-body equa ion:
[T +T
R+UpA +UnA −E]˜
Φ(−)
pB =0.(61)
In [98] Timo eyuk and Johnson use an adiaba ic app ox-
ima ion o ˜
Φ(−)
pB o p oduce a ac able exp ession ha s ill
includes ecoil exci a ion and b eakup e ec s, while keep-
ing he ma ix elemen cons ained o he ange o he Vpn
in e ac ion:
˜
Φ(−)
pB ≃χ(−)
pA ( pA,kα)φnA( nA)e−iαkα nA.(62)
In his exp ession, kαis he ela i e momen um be ween
Band pand α=mn/mB, and i was ound o p oduce
a good ag eemen be ween his calcula ion and expe imen-
al da a o he 16O(d,p)17O and 10Be(d,p)11Be eac ions.
Ne e heless, gi en ha his exp ession is be e sui ed o
he ans e o weakly bound and halo nuclei, he ADWA
model has enjoyed mo e widesp ead use.
The CDCC-BA app oxima ion
Ano he way o accoun ing o he b eakup channels in ans-
e eac ions is by inse ion o he CDCC wa e unc ion,
Eq. (9), in he ansi ion ampli ude o Eq. (52). The esul-
an ampli ude is o mally analogous o ha ound in he
CCBA me hod [17], so we e e o i as CDCC-BA app ox-
Fig. 28 Compa ison o he DWBA, ADWA and CDCC-BA me hods
o 58Ni(d,p)59Ni eac ion a Ed=10 MeV (uppe panel) and Ed=
56 MeV (lowe panel). Adap ed om Re . [99]
ima ion. This is expec ed o be a good app oxima ion since
he CDCC expansion is accu a e o small p-nsepa a ions,
o which he ma ix ansi ion ampli ude p esen s he la ges
magni ude. The esul an ampli ude, howe e , will be clea ly
much mo e compu a ionally demanding han ha ob ained
wi h he DWBA o ADWA me hods.
In Fig. 28 we compa e he pe o mance o he DWBA,
ADWA and CDCC-BA me hods o he 58Ni(d,p)59Ni eac-
ion a Ed=10 MeV (uppe panel) and Ed=56 MeV
(lowe panel), aken om Re . [99]. I becomes clea ha he
ADWA and CDCC me hods p o ide a mo e eliable desc ip-
ion o he da a compa ed o he adi ional DWBA app oach.
In [100], a sys ema ic compa ison be ween he CDCC-BA
me hod and an adiaba ic app oxima ion (CDCC-AD) was
pe o med o (d,p) eac ions on a ious a ge s, showing
good ag eemen be ween bo h, excep in some cases, such as
when he deu e on beam ene gy is small and simila o he
binding ene gy o he nucleon in he a ge o when he bind-
ing ene gy o he nucleon is e y small (Sn≃0.1MeV).
I should be no ed ha in ha wo k he adiaba ic app oxi-
ma ion is pe o med by se ing he nominal ene gies o he
con inuum bins o ha o he bound s a e, which is o mally
equi alen o he ADWA app oach, al hough he calcula ion
123
47 Page 24 o 57 Eu . Phys. J. A (2025) 61:47
is a he di e en , so ex ensions o hese esul s o s anda d
ADWA calcula ions should be made wi h cau ion.
The CDCC-BA app oxima ion is easonable as long as
hecoupling o he ea angemen ( ans e )channelsisweak,
ypically,when he ans e c osssec ionismuchsmalle han
he eac ion c osssec ion.O he wise,o he me hods ha ea
ans e on equal oo ing o b eakup, such as he Coupled-
Reac ion-Channel me hod (CRC) [17], a e equi ed.
4.1 T ans e eac ions popula ing unbound sys ems
So a , we ha e conside ed ans e eac ions as a ool o
in es iga ing he bound s a es o a gi en nucleus. Howe e ,
in a ea angemen p ocess, he ans e ed pa icle can also
popula e unbound s a es o he inal nucleus. This opens he
possibili y o s udying and cha ac e izing s uc u es in he
con inuum, such as esonances o i ual s a es. In ac , due
o he ma ching condi ions o his ype o p ocesses [101],
eac ionsinducedby weakly-bound nuclei a ou hepopula-
ion o highly exci ed s a es o he esidual nucleus, including
hose abo e he b eakup h eshold.
As in he case o ans e o bound s a es, he simples
o malism o analyze hese p ocesses is he DWBA me hod.
In his case, he bound wa e unc ion φ, j( )appea ing in
he inal s a e in Eq. (52) should be eplaced by a posi i e-
ene gy wa e unc ion desc ibing he s a e o he ans e ed
pa icle (neu on in his case) wi h espec o he co e a -
ge . In p inciple, o his pu pose, one could use he sui able
sca e ing s a e o he +bsys em a he app op ia e ela-
i e ene gy. Howe e , his p ocedu e ends o gi e nume ical
di icul ies in e alua ing he ans e ampli ude due o he
oscilla o y beha iou o bo h he inal dis o ed wa e and he
wa e unc ion φ, j( ). To a oid his p oblem, se e al al e -
na i e me hods ha e been used. We enume a e he e some o
hem:
(i) The bound s a e app oxima ion [102]. In he case o
ans e o a esonan s a e, his me hod eplaces he
sca e ing s a e ϕ, j( )by a weakly bound wa e unc-
ion wi h he same quan um numbe s and j. In p ac-
ice, his can be achie ed by s a ing wi h he po en ial
ha gene a es a esonance a he desi ed ene gy and
inc ease p og essi ely he dep h o he cen al po en-
ial un il he s a e becomes bound.
(ii) Huby and Mines [103] used a sca e ing s a e o
φ, j( )modula ed by a con e gence ac o e−α (wi h
αa posi i e eal numbe ), which is in ended o elimi-
na e i s con ibu ion o he in eg al coming om la ge
alues, and hen ex apola e nume ically o he limi
α→0.
(iii) Vincen and Fo une [104] pu in o ques ion he alid-
i y o he bound s a e app oxima ion a guing ha , in
gene al, he bound s a e and esonan o m ac o s can
Fig. 29 Radial pa o he d3/2single-pa icle esonance wa e unc ion
in 17Oa E =0.95 MeV compa ed wi h a sligh ly bound wa e unc ion
(E=−0.1 MeV) and a bin wa e unc ion, cen e ed a he nominal
ene gy o he esonance and wi h a wid h o 0.5 MeV
be e y di e en and, e en in hose cases in which he
ic i ious o m ac o gi es he co ec shape, hey can
lead o e y di e en absolu e c oss sec ions. They sug-
ges using he ac ual sca e ing s a e, bu choosing an
in eg a ion con ou along he complex plane in such a
way ha he oscilla o y in eg andis ans o medin oan
exponen ial decay, hus pallia ing he slow con e gence
p oblem o he pos - o m ans e ampli ude.
(i ) In a eal ans e expe imen leading o posi i e-ene gy
s a es, one does no ha e access o a de ini e inal
ene gy, bu o a ce ain egion o he con inuum. Tha is
o say, he ex ac ed obse ables, such as ene gy di e -
en ial c oss sec ions, a e in eg a ed o e some ene gy
ange which, a leas , is o he o de o he ene gy es-
olu ion o he expe imen . This sugges s a me hod o
dealing wi h he unbound s a es consis ing o disc e iz-
ing he con inuum s a es in ene gy bins, as in he CDCC
app oxima ion.
In Fig. 29, we show as an example he adial pa o a
3/2+ esonance in 17O, desc ibed in e ms o a d3/2neu on
coupled o a ze o-spin 16O co e. The solid line is a sca e -
ing wa e unc ion e alua ed a he nominal esonance ene gy
(E el =0.95 MeV). No e he oscilla o y beha iou a la ge
dis ances. The do ed line is a bin wa e unc ion, cons uc ed
by a supe posi ion o sca e ing s a es, wi hin he ange o 0.5
MeV a ound he esonance ene gy. I is seen ha , asymp o i-
cally, he oscilla ions a e damped wi h espec o he o iginal
sca e ing s a es. Finally, he do -dashed line is a bound s a e
wa e unc ion, wi h a 1d3/2single-pa icle con igu a ion, and
a sepa a ion ene gy o 0.1 MeV. This wa e unc ion is e y
123
Eu . Phys. J. A (2025) 61:47 Page 25 o 57 47
simila o he sca e ing s a e a sho dis ances bu decays
exponen ially a la ge dis ances.
An ad an age o he me hod (i ) is ha i can be applied o
si ua ions in which one is in e es ed in he desc ip ion o he
popula ion o a ange o con inuum ene gies possibly co -
e ing bo h esonan and non- esonan s a es. Fo ha , i is
con enien o eso o he p io - o m exp ession o he an-
si ion ampli ude. Conside ing again he A(d,p)B eac ion
o simplici y, his ampli ude eads
Tp io
i =Ψ(−)
(
R, )|VnA +UpA −UdA|φd( )χ(+)
dA (
R).
(63)
The unc ion χ(+)
dA is he dis o ed wa e gene a ed by he
op ical po en ial UdA and Ψ(−)
(
R, )is he exac h ee-
body wa e unc ion o he inal p+n+A s a e, wi h and
Rdeno ing he n-Aand p-B ela i e coo dina es, espec-
i ely. The wa e unc ion Ψ(−)
(
R, )is he ime- e e sed
o Ψ(+)
(
R, ), which sa is ies he h ee-body equa ion:
[T
R+T +Vpn +UpA +VnA −E]Ψ(+)
(
R, )=0,
(64)
wi h E he o al ene gy o he sys em. To sol e his equa ion,
he wa e unc ion Ψ(−)
(
R, )can be expanded in n+A
s a es wi h well-de ined ene gy and angula momen um,
as in he con inuum-disc e ized coupled-channels (CDCC)
me hod.
An example is shown in Fig. 30, which co esponds o
he di e en ial c oss sec ion, as a unc ion o he n-9Li ela-
i e ene gy, o he eac ion 2H(9Li, p)10Li∗a 2.36 MeV/u
(uppe ) and 11.1 MeV/u (lowe panel), co esponding o he
measu emen so ISOLDE [105] andTRIUMF[106], espec-
i ely. The lines a e he esul s o ans e - o- he-con inuum
calcula ions popula ing 10Li∗con inuum s a es using he
same 10Li s uc u e in bo h cases. The igu e shows he sepa-
a e con ibu ion o he s-wa e (1−,2
−) and p-wa e (1+,2
+)
con inuum s a es. The s eng h o he measu ed c oss sec ion
close o ze o ene gy is due o he p esence o a i ual s a e
in he n+9Li s-wa e, whe eas he peak a ound 0.4 MeV is
due o a p1/2 esonance. This is an example o how he use
o ans e eac ions can p o ide in o ma ion on he con in-
uum s uc u e o weakly-bound o e en unbound sys ems.
Fo mo e de ails on he calcula ions, see Re . [107].
4.2 T ans e eac ions in ol ing h ee-body p ojec iles
The o malism p esen ed in he p e ious sec ions can be
ex ended and applied o mo e complex sys ems. We discuss
he e he in e es ing case o he s ipping eac ion induced
by a h ee-body Bo omean sys em, in which one o he
p ojec ile agmen s is ans e ed o he a ge , lea ing a
Fig. 30 Illus a ion o he ans e - o- he-con inuum me hod, using a
binning disc e iza ion, o he eac ion 2H(9Li,p)10Li∗. Calcula ions a e
compa edwi h heexpe imen alda a omRe s. [105]and[106]. Figu e
aken om Re . [107]
Fig. 31 Diag am o a (p,d)o (p,pN) eac ion induced by a h ee-
body p ojec ile in in e se kinema ics. Taken om Re . [108]
wo-body esidual unbound sys em. To be mo e speci ic, we
conside he case o a wo-neu on halo p ojec ile, such as
11Li, 6He impinging on a p o on a ge . In hese cases, gi en
he uns able na u e o he p ojec ile, he eac ion expe imen
mus be pe o med in in e se kinema ics. This p ocess can
be schema ically ep esen ed as (see Fig. 31).
(C+N1+N2)
A
+p→(C+N2)
B
+d,(65)
123
47 Page 32 o 57 Eu . Phys. J. A (2025) 61:47
Fig. 40 IAV calcula ions o he eac ion 197Au(9Be,8Be*)198Au pop-
ula ing bound s a es o he esidual nucleus. aNo m o he o e laps
be ween he 9Be g.s. and he 8Be con inuum pseudos a es o di e en
ela i e angula momen a L, as a unc ion o he α-α ela i e ene gy up
o 20 MeV. Dashed lines a e included as a guide. bDi e en ial c oss
sec ion, a Ecm =36 MeV, as a unc ion o he α-α ela i e ene gy in he
inal 8Be∗sys em, o 0+s a es (solid line) and 2+s a es (dashed line).
The symbols indica e he nominal ene gies o he g.s. (0+)and he i s
2+
1 esonance o 8Be. cAngle-in eg a ed o al ans e c oss sec ion as
a unc ion o Ecm. The con ibu ion om he 0+(do -dashed line) and
2+(do ed) s a es o 8Be, oge he wi h hei sum (blue solid), a e com-
pa ed o he expe imen al da a o Re . [129]. The IAV esul escaled by
ab-ini io Va ia ional Mon e Ca lo (VMC) spec oscopic ac o s (dashed
line) is also shown (see ex ). Adap ed om Re . [128]
o z. The exponen in he second ac o is
Δk(z,b)≡−k
2EU(z,b), (87)
which,whenin eg a edalong heen i e ajec o y,gi es
he op ical phase shi
2δ(b)=∞
−∞
Δkz,b dz=2∞
0
Δkz,b dz,
(88)
ha is ela ed o he pa ial-wa e op ical S-ma ix, i.e.,
S(b)=exp[2iδ(b)].
(ii) The Uapo en ial, dis o ing he inciden wa e, is aken
as he sum o he co esponding agmen - a ge po en-
ials:
Ua=UbA +UxA (89)
This pa icula choice has he i ue o aking in o
accoun b eakup e ec s in he en ance channel. How-
e e , since each po en ial in Eq. (89) is e alua ed in he
co esponding agmen - a ge coo dina e, he associ-
a ed ini ial s a e wa e unc ion o he sys em would be
a solu ion o a complica ed h ee-body equa ion.
(iii) The abo e complica ion anishes hanks o he use o
he eikonal app oxima ion: he xand b agmen s mo e
wi h he same a e age eloci y as he p ojec ile and
hence hei momen a a e gi en by
kx=(mx/ma)
ka,
kb=(mb/ma)
ka.(90)
Wi h he pa icula choice o Eq. (89) and he assump ion
in Eq. (90) one ob ains he ollowing esul o he x-channel
wa e unc ion:
ϕEHM
x( x)=d3 bχ(−)∗
b( b)χ(+)
a( a)φa( bx)
=ei
kx· xexp izx
−∞
Δkxz,bx dz!
×d3 beiq· bSbA (bb)φa( bx)
(91)
wi h q=
kb−
k
b, he a e age momen um ans e ed in b−A
elas ic sca e ing.
Inse ing his exp ession in o he gene al exp ession (78)
(see de ails in [118,130]) one ob ains o he double di e -
en ial c oss sec ion:
d2σ
dEbdΩb
EHM
NEB =2
¯
h a
ρb(Eb)Ex
kxd2
bx˜
φa,b(q,
bx)2
×1−|SxA(bx)|2.(92)
123
Eu . Phys. J. A (2025) 61:47 Page 33 o 57 47
whe e ˜
φa,b(q,
bx)is gi en explici ly in Re . [118]
I should be no iced ha he NEB depends only on he
asymp o ic p ope ies, his is, he Sma ices, o he in e ac-
ion o band xwi h he a ge . The e is no sensi i i y on he
wa e unc ions in he in e ac ion egion. This is a esul o
he eikonal app oxima ion, plus he pa icula choice o he
dis o ed in e ac ion, which included he imagina y po en ial
WxA which ul ima ely gene a es he NEB.
In many applica ions, one is in e es ed in he o al yield o
agmen b, which is ob ained upon in eg a ion o he p e i-
ous o mula o e he angula and ene gy a iables, esul ing:
σEHM
NEB =2
a(2π)3Ex
¯
hkxd3 bd3 x|φa( bx)|2
×|SbA(bb)|21−|SxA(bx)|2.(93)
This equa ion has an appealing and in ui i e o m: he
in eg and con ains he p oduc o he p obabili ies o he
co e being elas ically sca e ed by he a ge , |SbA(bb)|2,
imes he p obabili y o he alence pa icle being abso bed,
(1−|SxA(bx)|)2. These p obabili ies a e weigh ed by he
p ojec ile wa e unc ion squa ed and in eg a ed o e all pos-
sible impac pa ame e s. Due o he Glaube app oxima-
ion, Eq. (92) is expec ed o be accu a e a high ene gies
(abo e ∼100 MeV pe nucleon). In ac , his o mula has
been ex ensi ely employed in he analysis o in e media e-
ene gy knockou eac ions (see e.g. [81,131,132] and e e -
ences he ein) mos ly aimed a ob aining spec oscopic in o -
ma ion o nucleon hole s a es.
5.3 In e p e a ion o inclusi e b eakup da a
In he p e ious sec ions, we ha e conside ed elas ic b eakup
and nonelas ic b eakup (wi h he la e possibly including
ans e and incomple e usion) aking place in a b eakup
eac ion. In ac ual inclusi e b eakup expe imen s hese con-
ibu ions will appea en angled in he da a, al hough hey
p oduce some dis inc i e ea u es. Fo example, he ene gy
dis ibu ion o he obse ed agmen a a gi en sca e ing
angle will exhibi a cha ac e is ic shape, as shown in Fig. 41
o a hypo he ical A(d,pX) eac ion. To unde s and his
spec um, i is impo an o ecall ha a gi en p o on ene gy
and angle will uni ocally de e mine he exci a ion ene gy o
he esidual n+Asys em. Acco ding o he p o on ene gy
(o , equi alen ly, he esidual sys em exci a ion ene gy), we
may dis inguish he ollowing egions:
(i) The highes p o on ene gies will be cha ac e ized by
some na ow peaks co esponding o bound s a es o
he n+Asys em. Depending on he expe imen al es-
olu ion, hese peaks will appea sepa a ed o , ins ead,
will me ge wi h neighbou ing peaks. Theo e ically, he
Fig. 41 P o on ene gy spec um om a A(d,pX)inclusi e b eakup
eac ion (blue line). The e ical do ed line ma ks he neu on sepa a-
ion h eshold in he A+nsys em. The low-ene gy peak a ises om
compound nucleus (CN) and p e-equilib ium (PE) p ocesses ollowed
by p o on e apo a ion (magen a line)
c osssec ion o heseisola edbounds a escan be e al-
ua ed wi h he s anda d o malisms o ans e eac-
ions, such as DWBA, CCBA, ADWA o CRC.
(ii) As he exci a ion ene gy o he esidual nucleus
inc eases,sodoes hedensi yo s a esand helow-lying
bound s a es will me ge o o m a quasi-con inuum.
He e, he ea men wi h he a o emen ioned me hods
becomes mo e oublesome because o he impossibil-
i y o disen angling unambiguously he con ibu ion o
each s a e om he da a. When one is in e es ed in
ene gy a e aged c oss sec ions, he combina ion o he
IAV model o Sec . 5.1 wi h dispe si e models p o ides
an appealing al e na i e o he adi ional me hods.
(iii) A a ce ain exci a ion ene gy, he esidual sys em will
each he neu on sepa a ion h eshold (Ex=Sn).
Jus abo e his exci a ion ene gy, he p o on spec um
will exhibi na ow peaks, co esponding o low-lying
esonances, supe imposed o a non esonan con inuum
backg ound. I is impo an o ealize ha he p ope -
ies o he sys em jus abo e he h eshold (Ex>Sn)
and jus below i (Ex<Sn) s a es a e quali a i ely
simila . In pa icula , he disc e e ene gy le els ex end
abo e Sne en i hose s a es a e no s ic ly s a iona y.
The e o e, he wa e unc ions o he na ow esonances
esemble e y much hose o he bound s a es below
h eshold and co espond o si ua ions in which he sys-
em emains bound o a long ime be o e decaying by
ba ie pene a ion. This is impo an om he eac ion
poin o iew because i indica es ha he ans e c oss
sec ion should e ol e smoo hly om nega i e o posi-
i e ene gies, as i ac ually happens in he dis ibu ion
shown in Fig. 41
123
47 Page 34 o 57 Eu . Phys. J. A (2025) 61:47
(i ) As he exci a ion ene gy o he esidual nucleus
inc eases abo e he h eshold, he na ow esonances
disappea , he spec um becomes con inuous and s uc-
u eless, gi ing ise o a bell-shaped bump. The mos
p obableene gyo heeme gingp o on can beob ained
by assuming ha he p o on ge s hal he kine ic ene gy
o he deu e on a he poin o b eakup, plus he
Coulomb ene gy o he deu e on a ha place, and hal
he in e nal ene gy o he deu e on (EB=−2.22 MeV)
(see Re . [133], p.509), i.e.,
Ep≃1
2Ed−Ze2
Rbu +Ze2
Rbu −1
2EB(94)
whe e Edis he kine ic ene gy o he ela i e mo ion o
he deu e on and he a ge nucleus A in he c.m. ame
and Rbu he deu e on- a ge sepa a ion a he ime o
he deu e on b eakup. The bump will con ain con i-
bu ions coming om bo h EBU and NEB componen s
discussed in p e ious sec ions. The EBU can be con e-
nien ly e alua ed wi h he DWBA o CDCC me hods
whe eas, o he NEB pa , he IAV model p o ides a
e y con enien amewo k.
( ) In he o me con ibu ions,weha eimplici lyassumed
ha he obse ed pa icle (p o on in his case) sca e s
elas ically by he a ge nucleus. The e will be si ua-
ions in which his no he case; o example, when
he p ojec ile uses comple ely wi h he a ge nucleus
o ming a compound nucleus ha will e en ually he -
malize by emi ing pa icles and gamma ays. Among
hese pa icles, he e will be p o ons ha will add up
incohe en ly o he p o on spec um. These p o ons a e
ypically emi ed wi h low ene gy in he c.m. ame
and will he e o e con ibu e o he low ene gy pa
o he p o on spec um (see magen a line in Fig. 41).
Somephenomena ela ed o he usiono weakly-bound
nuclei will be discussed in he nex sec ion.
6 Fusion in ol ing weakly bound nuclei
The p e ious sec ions ha e been de o ed o he model-
ing o di ec nuclea eac ions. Fusion eac ions in ol -
ing weakly-bound nuclei display also dis inc i e ea u es
which equi e adequa e eac ion o malisms and conside a-
ions wi h espec o he case o well-bound nuclei. The opic
o usion wi h bo h s able and uns able nuclei has mo i a ed
manywo ksandexcellen e iew pape s ha ebeenpublished
in ecen yea s, so we e e he eade o hese wo ks o a
de ailed accoun o he p esen s a us o he desc ip ion o
usion (e.g. [134–136]). We shall discuss wo phenomena
which a e he ocus o many heo e ical and expe imen al
e o s by se e al g oups. One is he phenomenon o com-
ple e usion supp ession. Comple e usion is con enien ly
de ined as he p ocess in which he whole cha ge o he
p ojec ile and a ge nuclei me ge, gi ing ise o an exci ed
compound nucleus ha will subsequen ly decay by pa i-
cle and/o gamma emission. The o he phenomenon is he
la ge obse ed yields compa ible wi h he pa ial usion o
he p ojec ile (incomple e usion, ICF). Expe imen al esul s
and heo e ical calcula ions indica e ha hese wo phenom-
ena a e ac ually ela ed since hey appea simul aneously and
so a plausible explana ion o he CF supp ession migh be in
ac he leak o lux going o he ICF channels.
A a ie y o models ha e been p oposed o e alua e he
CF c oss sec ion, om he simple single-ba ie pene a ion
model o mo e sophis ica ed coupled-channels me hods, in
which collec i e exci a ions o he p ojec ile and/o a ge
nucleus a e aken in o accoun explici ly. These models a e
e y success ul a p edic ing he c oss sec ion o well bound
nuclei, bu end o o e es ima e hem o weakly bound p o-
jec iles a ene gies abo e he ba ie . Fo example, o he
ligh weakly bound nuclei 6,7,8Li, 9Be he expe imen al CF
c oss sec ions a e ound o be supp essed by ∼20-30% com-
pa ed o he case o igh ly bound nuclei [9–15].
Ea ly analyses o hese expe imen s ied o explain he
phenomenon using coupled-channels calcula ions, including
he coupling o low-lying exci ed s a es o he p ojec ile and
a ge [9,11,77,137,138]. Ye , hese calcula ions sys ema i-
cally ailed o ep oduce he expe imen al supp ession. This
ailu e has been a ibu ed o he omission in hese calcula-
ions o he b eakup o he p ojec ile; a scena io was sug-
ges ed in which he weakly bound p ojec ile b eaks up p io
o eaching he usion ba ie , wi h he subsequen educ-
ion o he comple e usion p obabili y. This in e p e a ion is
suppo ed by he p esence o la ge αyields (in 6,7Li-induced
eac ions) as well as a ge -like esidues which a e consis-
en wi h he cap u e o one o he agmen cons i uen s o
he p ojec ile, ha is, ICF. To accoun o hese obse a ions,
some au ho s ha e p oposed a wo-s ep scena io [11,139]
consis ing on he elas ic dissocia ion o he p ojec ile ol-
lowed by he cap u e o one o he agmen s by he a ge .
Howe e , calcula ions based on a h ee-dimensional classi-
cal dynamical model [139], which inco po a es his wo-s ep
b eakup- usion mechanism, can only explain a small ac-
ion o he obse ed CF supp ession o 9Be [140] and 8Li
[72] eac ions. Mo e encou aging esul s ha e been ob ained
wi h di e en me hods based on he CDCC o malism, as
desc ibed in he ollowing subsec ions.
6.1 Compu a ion o CF and ICF wi h CDCC
Al hough he CDCC me hod was o iginally en isaged as a
p ac ical ool o e alua e he elas ic and b eakup obse ables,
some wo ks ha e been done o use his me hod o ob ain
comple e and incomple e usion c oss sec ions.
123
Eu . Phys. J. A (2025) 61:47 Page 35 o 57 47
Fig. 42 Calcula ed o al usion c oss sec ions o 11Be + 208Pb ( ull
s a s) compa ed wi h he expe imen al da a om [143] o 11Be + 209Bi
( ull squa es) using he CDCC wa e unc ion. Adap ed om Re . [142],
wi h pe mission om APS
Someau ho s[141,142]ha ep oposed oiden i y he o al
usion wi h he amoun o lux ha lea es he coupled chan-
nels se due o a sho - ange imagina y po en ial iWF(R),
while CF is iden i ied wi h he abso p ion due o such po en-
ial, bu es ic ed o bound s a es o he p ojec ile only. Thus,
o al usion is compu ed as
σTF =π
¯
h2K0
JT
(2JT+1)PJT,(95)
whe e K0is he wa enumbe o he inciden channel and PJT
is he comple e usion p obabili y o o al angula momen-
um JT
PJT=− 8μ
¯
h2K0
β,βi
∞
0|χJT
β,βi(R)|2WF(R)dR.(96)
whe eχJT
β,βia e he adialsolu ionsob ained om hecoupled
equa ions (14). In he case o CF, he exp ession o he c oss
sec ion is iden ical bu he sum in βis es ic ed o hose
channels associa ed wi h bound s a es.
The CF ob ained in his way ep esen s a lowe limi o he
physical CF c oss sec ion, since one has assumed no cap u e
o all p ojec ile agmen s om b eakup channels. In eal-
i y, hese e en s should con ibu e o he CF, bu canno be
dis inguished in his model om he cap u e o only one p o-
jec ile agmen . In his model, he incomple e usion σICF
is he e o e de ined as he abso p ion om b eakup channels.
Fig. 43 Schema icillus a iono he ou in eg a ion egionsemployed
in he me hod o Hashimo o e al. [144] o e alua ing comple e and
incomple e usion omCDCCwa e unc ionQuo edwi hau ho iza ion
om Ox o d Uni e si y P ess
An applica ion o his me hod is shown in Fig. 42, whe e
he calcula ions o he o al usion o 11Be + 208Pb ( ull
s a s) a e compa ed wi h he o al usion da a o he nea by
eac ion 11Be + 209Bi om Re . [143]. The ag eemen is
easonable a ound he Coulomb ba ie , bu he calcula ion
unde es ima es he da a by ∼41% o ene gies well abo e
he Coulomb ba ie .
A limi a ion o his me hod is ha i can only be applied
o p ojec iles composed o a hea y cha ged agmen and
a ligh uncha ged one (such as 11Be), since i elies on he
assump ion ha he cen e o mass o he p ojec ile is close
o ha o he hea y agmen and a om he ligh one.
Thus, i canno be used o p ojec iles like 6,7Li ha b eak
up in o wo agmen s o compa able masses. To o e come
his di icul y, Hashimo o e al. [144] p oposed an al e na i e
app oach based also on he CDCC me hod and applied i o
he case o deu e on sca e ing. Thei idea is o ans o m he
CDCC wa e unc ion om i s na u al coo dina es { ,
R} o
he coo dina es { p, n},
|Ψ( ,
R)|2d d
R=|"
Ψ( p, n)|2d pd n,(97)
and hen associa e he CF and ICF c oss sec ions wi h he
abso p ion aking place in di e en egions o he { p, n}
space, as illus a ed in Fig. 43. The dis ances ab
pand ab
n
deno e he abso p ion adii o he p o on and neu on, such
ha o p> ab
p( n> ab
n) he p o on (neu on) abso p ion
becomes negligible. The CF is iden i ied wi h he abso p ion
aking place when bo h he p o on and neu on a e inside
hei espec i e abso p ion adii, i.e.,
σCF =2μ
¯
h2K0 p< ab
p
d p n< ab
n
d n|"
Ψ( p, n)|2
×{Wp( p)+Wn( n)},(98)
123
47 Page 36 o 57 Eu . Phys. J. A (2025) 61:47
Fig. 44 a) Compa ison o σ(p)
ICF (sho -dashed line) and σ(n)
ICF (dash-
do ed line) ob ained in [144] wi h σ(p)
STR (squa es) and σ(n)
STR ( iangles)
gi en by he Glaube model [145]b) Comple e usion c oss sec ions
calcula ed wi h he Glaube model (do s) and wi h he me hod o [144]
(solid line). Quo ed wi h au ho iza ion om Ox o d Uni e si y P ess
whe e K0is he inciden wa e numbe , and Eis he ene gy.
Simila ly, he ICF c oss sec ion is ob ained om he abso p-
ion occu ing in he egion whe e one o he wo agmen s
is inside i s abso p ion adius while he o he is ou side i ,
i.e.,
σ(p)
ICF =2μ
¯
h2K0 p< ab
p
d p n> ab
n
d n|"
Ψ( p, n)|2Wp( p),
(99)
σ(n)
ICF =2μ
¯
h2K0 p> ab
p
d p n< ab
n
d n|"
Ψ( p, n)|2Wn( n).
(100)
The me hod was applied o he eac ion d+7Li [144].
Al hough no compa ison wi h expe imen al da a was
a emp ed, heau ho scompa ed hei esul swi h hosecom-
pu ed wi h he Glaube calcula ions o Re . [145]. This com-
pa ison is shown in Fig. 44 wi h he uppe and lowe panels
co esponding espec i ely o he ICF and CF c oss sec ions
as a unc ion o he deu e on inciden ene gy. The ag eemen
o he ICF is ound o be e y sa is ac o y o ene gies as low
as Ed=10 MeV. Since he Glaube model is a high-ene gy
app oxima ion, he ag eemen a hese ela i ely low ene -
gies is somewha unexpec ed. By con as , o he CF pa
(bo om panel), he Glaube me hod is ound o gi e signi -
ican ly smalle c oss sec ions. A discussion o hese esul s
is p o ided in [145].
Ano he p ocedu e o ex ac he CF and ICF c oss sec ion
om he CDCC me hod was p oposed by Pa ka and co-
wo ke s [146]. The cen al idea o hei me hod is o pe o m
a se ies o CDCC calcula ions wi h di e en choices o he
agmen - a ge po en ials. In pa icula , o ob ain he ICF
c osssec ion o hecap u eo agi en agmen , heype o m
a CDCC calcula ion using a sho - ange imagina y po en ial
o he in e ac ion o ha agmen and he a ge , and a eal
po en ial o he o he agmen . Using his me hod, hese
au ho s ha e been able o ob ain a easonable accoun o
he CF, ICF and TF c oss sec ions o eac ions induced by
6,7Li p ojec iles [146,147], al hough he sho - ange usion
po en ial needs o be adjus ed o each eac ion.
Mo e ecen ly, he au ho s o Re s. [148,149]ha ep o-
posed an al e na i e me hod which equi es a single CDCC
calcula ion o compu e he TF, CF and ICF c oss sec ions.
In his CDCC calcula ion, he agmen - a ge in e ac ions
a e also modeled wi h op ical po en ials wi h a sho - anged
imagina y pa . Fo a wo-body p ojec ile, he o al usion
c oss sec ion is compu ed as:
σTF =2μ
¯
h2K0#Ψ(+)W(1)+W(2)Ψ(+)$,(101)
whe e W(1,2) ep esen he imagina y pa s o he agmen -
a ge in e ac ions.ThisCDCCwa e unc ioncanbespli in o
bound (ΨB) and con inuum (ΨC) componen s:
Ψ(+)(
R, )=ΨB(
R, )+ΨC(
R, ), (102)
whe e ΨBand ΨCa e gi en by he channel expansions
ΨB(
R, )=
β∈B
χβ(
R)φβ( )(103)
ΨC(
R, )=
γ∈C
χγ(
R)φγ( ), (104)
whe e φβand φγa e, espec i ely, he bound and unbound
s a es o he p ojec ile, and χβand χγa e he co espond-
ing wa e unc ion desc ibing he p ojec ile- a ge ela i e
mo ion.
Assuming ha ma ixelemen so heimagina ypo en ials
connec ing bound channels o bins a e negligible, Eq. (101)
can be pu in he o m
σTF =σB
TF +σC
TF,(105)
wi h
σB
TF =2μ
¯
h2K0
β,β∈B#χβW(1)
ββ+W(2)
ββχβ$(106)
σC
TF =2μ
¯
h2K0
γ,γ∈C#χγW(1)
γγ+W(2)
γγχγ$.(107)
whe e W(i)
αα=φαW(i)φα ,wi h α, αs anding o ei he
β,βo γ,γ, a e hema ixelemen so heimagina ypo en-
ials.
123
Eu . Phys. J. A (2025) 61:47 Page 37 o 57 47
Then, by pe o ming an angula momen um expansion o
he wa e unc ions and he imagina y po en ials, Eqs. (106)
and (107) become
σB
TF =π
K2
0
JT
(2JT+1)PTF
B(JT)(108)
σC
TF =π
K2
0
JT
(2JT+1)PTF
C(JT), (109)
wi h
PTF
B(JT)=P(1)
B(JT)+P(2)
B(JT)(110)
PTF
C(JT)=P(1)
C(JT)+P(2)
C(JT). (111)
whe e P(i)
B(JT)and P(i)
C(JT)a e he p obabili ies o abso p-
ion o agmen ciin bound channels and in he con inuum,
espec i ely, esul ing om he con ibu ions o W(i) o he
TF c oss sec ion.
In e ms o hese p obabili ies, he au ho s o Re . [148,
149] in oduce he ICF p obabili ies
PICF1(JT)=P(1)
C(JT)×1−P(2)
C(JT)(112)
PICF2(JT)=P(2)
C(JT)×1−P(1)
C(JT),(113)
and he sequen ial comple e usion p obabili y
PSCF(JT)=P(1)
C(JT)×P(2)
C(JT). (114)
In e ms o he in oduced p obabili ies, he ollowing
usion c oss sec ions a e de ined:
– Di ec comple e usion (CF):
σDCF =σB
TF,(115)
whichdesc ibes hesimul aneouscap u eo he wo ag-
men s.
– Sequen ial comple e usion (SCF):
σSCF =π
K2
0
JT
(2JT+1)PSCF(JT). (116)
– ICF o agmen ci(ICFi)
σICFi =π
K2
0
JT
(2JT+1)PICFi(JT). (117)
In his o malism, he CF, ICF and TF c oss sec ions a e
hen gi en by
σCF =σDCF +σSCF,(118)
σICF =σICF1+σICF2,(119)
σTF =σCF +σICF.(120)
Fig. 45 CF (le ) and ICF ( igh ) c oss sec ions o 7Li+209Bi. Expe -
imen al da a om Re s. [11,12] a e compa ed wi h he calcula ions o
Re . [149]. Wi h pe mission o APS
An applica ion o his model is shown in Fig. 45, whe e
CF (le panel) and ICF ( igh panel) da a o he eac ion
7Li+209Bi [11,12] a e compa ed wi h he p edic ions o he
model.Theuppe andlowe panelsdisplay hesame esul sin
loga i hmic and linea scale, espec i ely. Fo CF, he ag ee-
men wi h he da a is e y sa is ac o y, bo h abo e and below
heba ie (indica edby hea ow).Fo ICF, hesepa a econ-
ibu ions o i on cap u e and αcap u e a e shown, wi h he
o me gi ing a much la ge c oss sec ion. The calcula ions
a e ound o ep oduce well he da a up o Ec.m.≈34 MeV,
bu o e es ima e hem a highe ene gies. A de ailed discus-
sion o hese esul s can be ound in Re . [149].
6.2 E alua ion o CF and ICF c oss sec ions wi h he IAV
model
As discussed in Sec . 5, he IAV model p o ides he o al
inclusi ec oss sec ion co esponding o he de ec ion o he b
agmen in eac ions o he o m A(a,b)X. This esul s om
he ac ha he imagina y pa ha appea s in he expec a ion
alue o Eq. (78) accoun s in p inciple o all p ocesses in
which he pa icipan agmen xin e ac snonelas icallywi h
he a ge nucleus. This will include he ICF c oss sec ion, bu
also o he NEB p ocesses no associa ed wi h he o ma ion
o a compound nucleus o he x+Asys em, such as a ge
exci a ion. The isola ion o he ICF c oss sec ion om he
o al NEB c oss sec ion is indeed no a i ial p oblem. An
in ui i e app oach consis s in iden i ying he ICF wi h he
abso p ion due o a sho - anged imagina y po en ial.
A en a i e applica ion o his idea is shown in he bo -
om panel o Fig. 46, quo ed om [150], co esponding o
he eac ion 7Li+209Bi a ene gies below and abo e he ba -
ie . The symbols co espond o he ICF da a o Dasgup a
e al. [11,12]. Fo he calcula ions we show he indi id-
ual con ibu ions o he ICF c oss sec ion, namely, α-ICF
(i.e., αabso bed) and -ICF ( -abso bed) as well as hei
123
47 Page 38 o 57 Eu . Phys. J. A (2025) 61:47
Fig. 46 CF( op)andICF(bo om)c osssec ions o 7Li+209Bi.Expe -
imen al da a om Dasgup a e al. [11,12] a e compa ed wi h he cal-
cula ions based on he IAV model. In he CF plo , we include also he
eac ion c oss sec ion ( om a CDCC calcula ion) and he usion com-
pu ed wi h he ba ie pene a ion model (BPM). Top panel adap ed
om [120]. Bo om panel quo ed om [150]
sum. To compu e he α-ICF ( -ICF), he imagina y pa o
he α+209Bi ( +209Bi) sys em was eplaced by a sho - ange
imagina y po en ial o Woods-Saxon o m and pa ame e s
W0=−50 MeV, i=1.0 m,a=0.2 m. The esul s
a e e y simila o hose epo ed in Re . [149] and shown
in Fig. 45, in which he au ho s made use o he abso p ion
and su i al p obabili ies ex ac ed om he CDCC calcu-
la ions. Fu he calcula ions a e needed o elucida e he use-
ulness and applicabili y o he IAV model o e alua e ICF
c oss sec ions.
The IAV model has also been used o in e CF c oss sec-
ions o weakly bound nuclei [120,151]. The idea o his
me hod is o decompose he eac ion c oss sec ion as ollows
σR≈σCF +σinel +σEBU +σ(b)
NEB +σ(x)
NEB.(121)
In his exp ession, σinel co esponds o he exci a ion o he
p ojec ile and/o a ge wi hou dissocia ion (i.e., inelas ic
sca e ing). The e ms σEBU and σ(b,x)
NEB co espond o he
elas ic b eakup (EBU) and nonelas ic b eakup (NEB) con i-
bu ions al eady discussed in Sec . 5. In he la e , one dis in-
guishes he cases in which ei he he agmen xo bin e -
ac s nonelas ically wi h he a ge whe eas he o he sca -
e s elas ically. A success ul de e mina ion o he CF c oss
sec ion om he decomposi ion (121) equi es ha all o he
quan i ies in ol ed in his o mula can be e alua ed accu-
a ely. The pu e inelas ic sca e ing c oss sec ions (σinel)a e
s anda dly compu ed by means o coupled-channels calcu-
la ions including low-lying collec i e exci a ions o he p o-
jec ile and a ge . The EBU pa can be accu a ely calcula ed
wi h he con inuum-disc e ized coupled-channels (CDCC)
me hod.Finally, heNEBcon ibu ionscan bee alua edwi h
he IAV model, a leas o p ojec iles wi h a de eloped wo-
boy s uc u e, such as 6,7Li.
An applica ion o he me hod o he 7Li+209Bi eac ion is
shown in he op panel o Fig. 46, adap ed om Re . [120].
Theci clesa e heCFda a omRe .[12], hesolidg eenline
is he eac ion c oss sec ion ob ained om a CDCC calcula-
ion and he solid ed line is he calcula ed CF c oss sec ion
in e ed om Eq. (121) assuming a wo-body model (α+ )
o 7Li. Fo compa ison, a single-channel ba ie pene a ion
model (BPM) calcula ion (dashed lined) is shown. As can
be seen, he da a a e la gely supp essed wi h espec o he
BPM. In con as , he CF ex ac ed om Eq. (121) explains
e y well he da a. As discussed in [151], he educ ion wi h
espec o he BPM is ound o be mainly due o he com-
pe i ion due o he 209Bi(7Li, α)X channel, which includes,
among o he s, he -ICF channel (see he bo om panel).
6.3 Applica ion o su oga e eac ions
Theg adual imp o emen inmodelso ien ed o he compu a-
ion o ICF c oss sec ions has d i en hei applicabili y o he
so-called su oga e me hod (SRM). This me hod p o ides an
indi ec way o e alua ing compound-nucleus c oss sec ions
in eac ions o which he di ec measu emen is di icul o
e en unpossible. An example is he ex ac ion o neu on-
induced c oss sec ions o he o m (n,χ), whe e χis a gi en
decay channel p oduc (γ, ission agmen , e c). Following
he Boh hypo hesis, i is cus oma ily assumed ha in hese
eac ions he o ma ion and decay o a compound nucleus
ake place independen ly o each o he . To ob ain in o ma-
ionon hedecayo hecompoundnucleus (B∗) ha occu sin
he eac ion o in e es (n+A→B∗→χ+C), one uses he
al e na i e(su oga e) eac iond+D→b+B∗→b+χ+C
ha in ol es a p ojec ile- a ge combina ion (d+D) ha is
expe imen ally mo e accessible. Fo example, one can use
he s ipping eac ion d+A→p+B∗→p+χ+C,in
which pand χa e measu ed in coincidence.
Compound nuclea eac ions a e adequa ely desc ibed in
he Hause –Feshbach o malism, which conside s he con-
se a ion o angula momen um Jand pa i y π. The c oss
123
Eu . Phys. J. A (2025) 61:47 Page 39 o 57 47
sec ion o he “desi ed” eac ion A(n,χ)Cis gi en by
σ(n,χ)(En)=
JT,π
σCN(Eex,JT,π) GCN
χ(Eex,JT,π),
(122)
whe e σCN
JT,π (Eex,JT,π) is he c oss sec ion o he CN
o ma ion and GCN
χ(Eex,JT,π) he b anching a io o
he decay o channel χ. The objec i e o he su oga e
me hod is o expe imen ally de e mine he decay p obabili-
ies GCN
χ(Eex,JT,π), which a e o en di icul o calcula e
accu a ely.
In he su oga e eac ion, d+D→b+B∗→b+χ+C,
he same CN nucleus B∗is o med and he decay p oduc
o in e es (χ) is measu ed in coincidence wi h he ou going
pa icle b. The p obabili y o his p ocess can be w i en as:
PS,χ (Eex)=
JT,π
FCN
S(Eex,JT,π)GCN
χ(Eex,JT,π),
(123)
whe e he subsc ip Sdeno es he speci ic su oga e eac ion,
FCN
S(Eex,JT,π) is he p obabili y o o ming B∗in his
su oga e eac ion (wi h speci ic alues o Eex,Jand π)
and whe e GCN
χ(Eex,JT,π) a e he same b anching a ios
appea ing in equa ion (122). The p obabili y PS,χ (Eex)can
be ob ained expe imen ally as he a io be ween he numbe
o coincidences be ween he bpa icle and he decay pa icle
χ,NS,χ , and he o al numbe o su oga e e en s, NS, i.e.:
Pexp
S,χ (Eex)=NS,χ
NSχ
,(124)
whe e χis he e iciency o de ec ing he exi -channel χ o
he eac ions in which bis de ec ed.
Ideally, i a eliable p edic ion o FCN
S(Eex,JT,π) is
possible, wi h an accu a e de e mina ion o Pexp
S,χ (Eex) o
a ange o ene gies and angles o b, i migh be possible o
ex ac heb anching a iosGCN
χ(Eex,JT,π)whichcan hen
be used o calcula e he desi ed c oss sec ion using Eq. (122).
In p ac ice, his app oach is no always easible due o he
lack o some o his equi ed in o ma ion and he app oach
has elied on addi ional app oxima ions. In pa icula , ea ly
applica ions made use o he so-called “Weisskop -Ewing
app oxima ion”, which assumes ha he b anching a ios
GCN
χ(Eex,JT,π) a e independen o he angula momen-
um and spin, gi ing ise o he simpli ied c oss sec ion:
σ(n,χ)(En)=σCN(Eex)GCN
χ(Eex), (125)
whe e σCN(Eex)is he CN c oss sec ion summed o e all
possible JT,π alues. Applying he same app oxima ion o
hesu oga e eac ion,andusingJT,π FCN
S(Eex,JT,π)=
1weha e
PS,χ (Eex)=GCN
χ(Eex), (126)
allowing he de e mina ion o he desi ed c oss sec ion as
σ(n,χ)(En)=σCN(Eex)PS,χ (Eex), (127)
which a oids he need o he p obabili ies FCN
S(Eex,JT,π).
So, unde he alidi y o heWeisskop -Ewing app oxima-
ion, he neu on (n,χ)c oss sec ion can be eadily in e ed
om he measu ed p obabili ies o he su oga e eac ion.
No e also ha he CN c oss sec ion mus be es ima ed in
some way, using, o example, an op ical model calcula ion.
In p ac ice, he Weisskop -Ewing app oxima ion is a ely
jus i ied in mos cases so one needs o eso o he
mo e gene al exp essions (122) and (123). The p obabili ies
FCN
S(Eex,JT,π) ha appea in he la e can be es ima ed
wi h he IAV model discussed in Sec. 5.1. This idea has been
success ully applied o he 95Mo(n,γ) eac ion [152]. The
di ec (n,γ) c oss sec ion o his eac ion a e compa ed in
Fig. 47 wi h he c oss sec ion ex ac ed om he su oga e
eac ion 95Mo(d,p) using he a o emen ioned o malism. As
can be seen, hey a e in excellen ag eemen . As also shown
in his igu e, he esul using he Weisskop -Ewing app ox-
ima ion depa s signi ican ly om he di ec measu emen .
7 Semiclassical desc ip ion o b eakup and ans e
eac ions
When he de B oglie wa eleng h o he p ojec ile is small
compa ed o some cha ac e is ic dis ance o he collision
p ocess one may desc ibe i s mo ion in e ms o classical
ajec o ies. This p o ides a mo e in ui i e and usually ma h-
ema ically simple desc ip ion o he eac ion. This app oxi-
ma ion canno be applied o he in e nal mo ion o he nucle-
ons inside he nucleus because hei ypical wa eleng h is
o he same o de as he size o he nucleus and he e o e
quan um e ec s a e impo an ( o example, o an ene gy
o 30 MeV, ∼c/5 and so =¯
h/p≈1 m). The me h-
ods in which he in e nal exci a ions a e ea ed quan um-
mechanically, while he p ojec ile– a ge ela i e mo ion is
ea ed classically, a e called semiclassical me hods. A la ge
a ie y o such models ha e been p oposed in he li e a u e
[84,155–159]. As examples, we discuss he e he one de el-
oped by Alde and Win he and he semiclassical ans e - o-
he-con inuum model o Bonacco so and B ink.
7.1 The semiclassical o malism o Alde and Win he
In i s simples o m, he heo y o Alde and Win he [19]
assumes ha he p ojec ile mo es along a classical ajec o y,
which is weakly a ec ed by he in e nal exci a ions o he
colliding nuclei. This means ha :
Δ
1 and Δεn
E1.
123
47 Page 40 o 57 Eu . Phys. J. A (2025) 61:47
Fig. 47 Applica ion o he su oga e me hod o he 95Mo(n,γ) eac-
ion. Top (panel (A)): Calcula ions o he cumula i e p obabili y o
o ming he CN h ough 95Mo(d,p)(FCN
95 Mo(d,p)). The shaded
egion, which spans exci a ion ene gies in he ange om Ex=8.55
MeV o Ex=10.65 MeV, indica es he exci a ion ene gies o e which
he su oga e da a a e i . The e ical do ed line ep esen s he neu on
e apo a ion h eshold. Middle (panel (B)): his og am o he o al con-
ibu ion o CN o ma ion o e he shaded ange in (A) as a unc ion o
angula momen um,decomposed in oposi i eandnega i epa i ies,and
no malized o one o e he in eg a ion egion. Bo om: The solid blue
egion is he (n,γ)c oss sec ion ob ained om he SRM, he ed ci cles
and black squa es a e he di ec measu emen s [153,154]. The unce -
ain y esul ing om expe imen al da a and i ing e o is indica ed
by he shaded band. The esul ob ained using he Weisskop -Ewing
app oxima ion is also shown (gold diamonds). Quo ed om Re . [152],
wi h pe mission om APS
The p ojec ile– a ge in e ac ion is spli as: V(
R,ξ) =
V0(
R)+Vcoup(
R,ξ), whe e V0(
R)is independen o he
in e nal coo dina es and de e mines he classical ajec o y
R( ). The ime e olu ion o he o al wa e unc ion o he
sys em e i ies he Sch ödinge equa ion
i¯
hdΨ(ξ,θ, )
d =V(
R(θ, ), ξ) +Hp(ξ)Ψ(ξ,θ, )
(128)
subjec o he ini ial condi ion |Ψ(−∞)=|0.
In hespi i o hecoupled-channelsme hod, he o alwa e
unc ion is expanded in a basis o in e nal s a es o he p o-
jec ile Hamil onian [Hp(ξ) −εn]φn(ξ)=0:
Ψ(ξ,θ, )=
n=0
cn(θ, )e−iεn /¯
hφn(ξ) (129)
which, when inse ed in Eq. (128), gi es ise o a se o cou-
pled equa ions o he expansion coe icien s:
i¯
hdcn(θ, )
d =
m
e−i(εm−εn) /¯
hVnm(θ, )cm(θ, )(130)
wi h he ini ial condi ion cn(θ, −∞)=δn0. The ime-
dependen coupling po en ials Vnm(θ, )a e gi en by:
Vnm(θ, )=dξφ∗
n(ξ)V(
R(θ, ), ξ)φm(ξ). (131)
Once he coe icien s a e ob ained, he exci a ion p oba-
bili y o a 0 →n ansi ion is gi en by:
Pn(θ) =|cn(θ, ∞)|2,
and he di e en ial c oss sec ion by:
dσ
dΩ0→n=dσ
dΩclas
Pn(θ).
whe e (dσ/dΩ)clas is he classical di e en ial elas ic c oss
sec ion which, o a pu e Coulomb case, coincides wi h he
Ru he o d c oss sec ion.
Due o he conse a ion o he o al p obabili y ( lux), one
has
n
Pn( )=
n|cn( )|2=1.
When he couplings a e weak, one may sol e Eq. (130)
pe u ba i ely, assuming ha c0≈1 and cn1 o n>0.
This gi es he i s -o de solu ion
cn(θ) ≡cn(θ, ∞)≃1
i¯
h∞
−∞
e−i(ε0−εn) /¯
hVn0(θ, )d .
(132)
In he impo an case o pu e Coulomb sca e ing, which
was he case s udied in de ail by Alde and Win he [19], one
inds analy ical exp essions o he exci a ion p obabili ies.
Inpa icula , he i s -o de exci a ionp obabili y o a0 →n
ansi ion, due o he elec ic Coulomb ope a o Eλ, esul s
123
Eu . Phys. J. A (2025) 61:47 Page 41 o 57 47
246810
E
J
[MeV]
10
100
1000
n
E1
(E
J
)
28 MeV/nucl.
70 MeV/nucl.
280 MeV/nucl.
Pb a ge
b
min
= 12.3 m
Fig. 48 To al numbe o i ual pho ons o he E1 mul ipola i y, co -
esponding o he sca e ing o a p ojec ile by a 208Pb a ge a di e en
inciden ene gies and in eg a ed o impac pa ame e s b>12.3 m.
Quo ed om Re . [160]
dσ
dΩ0→n=Z e2
¯
h 2B(Eλ,0→n)
e2a2λ−2
0
λ(θ, ξ) (133)
which is alid only o angles smalle han he g azing 3one
(θ<θ
g )andwhe ea0ishal hedis anceo closes app oach
in a classical head-on collision, ξ0→n=(εn−ε0)
¯
h
a0
is he
adiaba ici y pa ame e and λ(θ, ξ) is an analy ic unc ion,
depending on he kinema ical condi ions, bu independen o
he s uc u e o he p ojec ile.
Fo weaklyboundnuclei, he exci a ionwill ypicallypop-
ula e unbound (con inuum) s a es. The p e ious o mula can
be gene alized o:
dσ(Eλ)
dΩdε=Z e2
¯
h 21
e2a2λ−2
0
dB(Eλ)
dε
×d
λ(θ, ξ)
dΩ(θ < θg )(134)
whe e d
λ(θ, ξ)/dΩis also a well-de ined analy ic unc ion
and dB(Eλ)/dεis he elec ic educed p obabili y o he
con inuum s a es.
I is common o exp ess (134) in e ms o he so-called
numbe o equi alen pho ons,
dσ(Eλ)
dΩdε=16π3
9¯
hc
dNEλ(Exθ)
dΩ
dB(Eλ)
dε(135)
whe e dNEλ(Exθ)/dΩis he numbe o equi alen pho ons
pe solid angle. This is ypically e e ed o as Equi alen
Pho on Me hod (EPM).
3The g azing angle e e s o he angle a which he p ojec ile in e ac s
wi h he su ace o he a ge in such a way ha he p ojec ile ba ely
“g azes” he su ace o he a ge , a he han ully pene a ing o col-
liding head-on.
Fig. 49 B eakup o 8B→7Be+p on a 58Ni a ge a 26 MeV. Coupled-
channel semiclassical calcula ions (labeled “p esen wo k”), using
Coulomb+nuclea ajec o ies, a e compa ed wi h CDCC calcula ions
(labeled “Nunes and Thompson”), which include also nuclea and
Coulomb couplings. Quo ed om [130] wi h pe mission om APS
Figu e48,quo ed om[160],shows henumbe o equi a-
len i ual pho ons, in eg a ed o impac pa ame e s beyond
b=12.3 m (no e ha b=a0co (θ/2)), o he collision o
a p ojec ile wi h a Z=82 a ge , o h ee di e en inciden
ene gies. I can be seen ha inc easing he inciden ene gy
dec eases he popula ion o low-lying s a es and enhances
he popula ion o high-ene gy ones.
The assump ion o pu e Coulomb ajec o ies can be
elaxed, a he expense o losing some o he simplici y o he
me hod. A compelling applica ion is shown in Fig. 49 ( aken
om Re . [130]) whe e semiclassical coupled-channels cal-
cula ions, using ajec o ies modi ied by he nuclea in e ac-
ion, a e compa ed wi h CDCC calcula ions o he b eakup
angula dis ibu ion o he 8B+58Ni eac ion a he nea -
ba ie ene gy o 26 MeV, inding a nice ag eemen be ween
bo h.
7.2 Dynamic Coulomb pola iza ion po en ial om he AW
heo y
As discussed in Sec . 2, he Feshbach heo y o nuclea eac-
ions p o ides a o mal exp ession o he nucleus-nucleus
op ical po en ial. This can be exp essed as a sum o a ba e
po en ial ( ha is, he s a ic pa o he in e ac ion due o he
g ound-s a e densi ies o he p ojec ile and a ge ) and a
pola iza ion po en ial. Unde sui able app oxima ions, i is
possible o de i e simple, analy ical exp essions o speci ic
123
47 Page 48 o 57 Eu . Phys. J. A (2025) 61:47
Fig. 59 Rela i e ene gy dis ibu ion o he agmen s ollowing he
b eakupo 11Be on alead a ge a 69MeV/u,in eg a ed in hesca e ing
angle up o a maximum alue o θmax =6◦( op) and 1.3◦(bo om). Full
CDCC calcula ions, including bo h Coulomb and nuclea couplings a e
compa ed wi h CDCC calcula ions including only nuclea couplings
Fig. 60 Rela i e ene gy dis ibu ion o he agmen s ollowing he
b eakup o 11Be on a lead a ge a 520 MeV/u. EPM and XCDCC
calcula ions, based on he same s uc u e model, a e compa ed. The
e ec o he nuclea /Coulomb in e e ence is illus a ed
is seen ha (i) nuclea b eakup is a he signi ican (consis-
en ly wi h he es ima e o [179]) and, mo e impo an ly, (ii)
he incohe en sum o he nuclea and Coulomb con ibu ions
o e es ima es he ull XCDCC calcula ions wi h in e e ence
e ec s. F om hese calcula ions, i becomes appa en ha he
p ocedu eo summingincohe en ly henuclea andCoulomb
Fig. 61 Compa ison o he Coulomb dissocia ion da a o 15Con208Pb
a 68 MeV/u [180] wi h CDCC calcula ions using di e en 14C+n mod-
els o 15C. Figu e om [181]
con ibu ions in oduces some e o in he ex ac ion o he
elec ic ansi ion p obabili ies om Coulomb dissocia ion
expe imen s. The e o will become la ge a lowe inciden
ene gies.
Ideally, one may also analyze di ec ly he expe imen al
c oss sec ions wi h he CDCC me hod o ex ac he elec-
ic ansi ion p obabili ies and o he s uc u e p ope ies.
The p ac ical applicabili y o his idea is hinde ed by he
ac ha he e is no simple connec ion be ween he eac ion
obse ables (c oss sec ions) and he unde lying B(Eλ) dis-
ibu ions. In pa icula , b eakup c oss sec ions a e no longe
p opo ional o he B(Eλ).
One possibili y is o compa e he measu ed c oss sec ions
wi h a se ies o CDCC calcula ions assuming di e en s uc-
u e models, wi h di e en elec ic s eng h dis ibu ions.
Thisisexempli iedinFig.61whe e heda aco espond o he
ela i eene gydis ibu iono 15Con208Pba 68MeV/umea-
su ed a RIKEN [180] and he lines a e CDCC calcula ions
o di e en po en ial models o he 15C nucleus. F om he
compa ison one may selec he bes s uc u e model among
hose conside ed.
Recen ly, an al e na i e o his p ocedu e o ex ac B(Eλ)
dis ibu ions om he compa ison o CDCC calcula ions
wi h b eakup c oss sec ion da a has been p oposed [182].
The s a egy can be summa ized as ollows:
– S a wi h some ial s uc u e model o he p ojec ile.
This will p edic some dis ibu ion B0(E1;ε) depending
on he con inuum ene gy.
– In oduce some small changes (pe u ba ion)in he
model, by mul iplying all Coulomb dipole ma ix ele-
men s by a bi a y ac o s (1+2δ(εi)), whe e εia e he
measu ed ene gies. This will modi y he B(E1)a each
measu ed εi
Bmod(E1,ε
i)≃B0(E1,ε
i)(1+2δ(εi)),(155)
123
Eu . Phys. J. A (2025) 61:47 Page 49 o 57 47
012345
H (MeV)
-0.2
0
0.2
0.4
0.6
0.8
1
dB/dH (e2 m2/MeV)
S a ing 11Be model
Ex ac ed B(E1) om GSI da a
Ex ac ed B(E1) om RIKEN da a (Tc.m.< 1.3o)
Ex ac ed B(E1) om RIKEN da a (Tc.m.< 6o)
NCSMC: Calci e al.
Fig. 62 Un olded B(E1)dis ibu ionsex ac ed om heexpe imen al
b eakup c oss sec ions measu ed in [179] (squa es) and [178](ci cles)
using he p ocedu e based on he XCDCC me hod p oposed in [182].
Fo compa ison, he s a ing B(E1)dis ibu ion is shown by he black
line and a NCSMC ab-ini io calcula ion o Re . [183] ( ed solid line) is
also shown. Figu e adap ed om [182]
– Fo small pe u ba ions, he calcula ed b eakup c oss sec-
ions a e modi ied simply as [182]
σmod
i≃σ0
i+δ(εi)σi.(156)
whe e σ0
iand σia e cons an s which can be de e mined
by pe o ming se e al calcula ions o di e en δ alues.
– By compa ing he model calcula ions wi h he measu ed
b eakup da a, one may in e he op imal alue o δ(εi)
o each exci a ion ene gy and, om hem, he alues o
B(E1;εi) ha bes desc ibe he da a wi hin he CDCC
amewo k.
An applica ion o his me hod is shown in Fig. 62, adap ed
om[182].Thesolidblacklineis he B(E1)dis ibu ionp o-
ided by he s a ing ( ial) B(E1)model used in a XCDCC
calcula ion. Following he ou lined p ocedu e, his B(E1)is
co ec ed o each exci a ion ene gy by compa ing he calcu-
la ed and measu ed c oss sec ions. The p ocedu e is applied
sepa a ely o he da a o Re . [178], o he wo angula cu s
discussed abo e, as well as o he da a o [179]. The co ec ed
B(E1)dis ibu ions a e displayed in his igu e wi h he sym-
bols. I is seen ha he h ee o hem ag ee easonably well
wi hin he es ima ed e o s. This is in con as o he B(E1)
ex ac ed in he o iginal wo ks and shown in Fig. 58.I is
impo an o no e ha he la e a e a ec ed by he expe i-
men al esolu ion,while hoseshowninFig.62dono include
he e ec o he expe imen al ene gy esolu ion. Di e ences
be ween hese expe imen al esolu ions in he wo expe i-
men s a e pa ially esponsible o he di e ences obse ed
in he B(E1)dis ibu ions epo ed in he o iginal wo ks.
9 S udy o nucleon–nucleon co ela ions om nucleon
emo al eac ions
One pa icula poin o in e es o nucleon-Bo omean sys-
ems o he o m N + N + C is he s udy o he co ela-
ion be ween hei wo alence nucleons, since he nucleon-
nucleon in e ac ion is essen ial o hold he Bo omean
sys em bound. As many Bo omean sys ems also p esen
nucleon halos, he s udy o he co ela ion be ween hei
alence nucleons pe mi s he explo a ion o he nucleon-
nucleon in e ac ion in low-densi y, isospin-asymme ic en i-
onmen s. A quan i y ela ed o he nucleon-nucleon co e-
la ion is he a e age opening angle be ween nucleons wi h
espec o he co e θNN. I bo h nucleons ollow indepen-
den unco ela ed o bi s, his angle will on a e age be 90◦
[184,185]. A smalle angle co esponds o a small a e age
dis ance be ween nucleons compa ed o he dis ance o hei
cen e o mass o he co e, in which is known as a “dineu-
on” (o dip o on) con igu a ion [186], which is associa ed
wi h an a ac i e co ela ion be ween nucleons. Meanwhile,
an angle la ge han 90◦indica es a la ge in e nucleon dis-
ance compa ed o hei dis ance o he co e, which co e-
sponds o bo h nucleons being on opposi e sides o he co e,
in he so-called “ciga ” con igu a ion, which co esponds o a
epulsi e nucleon-nucleon co ela ion. No e ha h ee-body
calcula ions show ha bo h “dineu on” and “ciga ” compo-
nen s appea in he wa e unc ion [58], and i is hei ela i e
s eng h which leads o a mo e dominan con igu a ion.
An expe imen al p obe o θNNin he case o neu on-
Bo omean sys ems is he dipole elec ic s eng h B(E1):
assuming poin -like pa icles, i s exp ession educes o:
B(E1)=3
πZc
A2
e2# 2
n+ 2
n+2 n ncos θNN$
≃6
πZc
A2
e2# 2
n$(1+cos θNN),
(157)
whe e ' 2
n(is he mean squa e dis ance be ween neu on and
co e, Zcis hecha geo heco eand A hemasso henucleus
[187], so an enhanced B(E1)indica es a smalle angle and
dineu on co ela ions while a educed B(E1)indica es he
exis ence o a ciga -like con igu a ion. B(E1)s eng hs ha e
been measu ed o a ious Bo omean nuclei, om which
alues o θNNcan be ex ac ed. The alues ex ac ed in
[185] a e p esen ed in Table 3, whe e he angles can be seen
o su e om signi ican unce ain ies, due bo h o expe i-
men al unce ain ies and he model dependence due o ' 2
n(.
Ano he me hod o ex ac ing he opening angle is he
use o nucleon knockou eac ions, whe e one o he alence
nucleons is emo ed ia a sudden eac ion, such as nucleon-
knockou wi h 9Be o 12C a ge s o a (p,pN) eac ion wi h
p o on a ge s, assuming he alidi y o he spec a o app oxi-
ma ion o he emaining(C+N)sys em.Wi hin hisapp ox-
123
47 Page 50 o 57 Eu . Phys. J. A (2025) 61:47
Table 3 B(E1) s eng h unc ion and deduced a e age opening angle
θNN o a ious Bo omean nuclei. Table adap ed om [185]
6He 11Li 14Be 17Ne
B(E1) (e2 m2) 1.20(20) 1.42(18) 1.69 1.56
[188][177][189][190]
θNN(deg.) [185]83
+20
−10 66+22
−18 64+9
−10 110
Fig. 63 Opening angle and
momen a measu ed in
nucleon- emo al eac ions on
Bo omean sys ems. Figu e
adap ed om [191]
ima ion, in he p ojec ile es ame he Bo omean nucleus
s a s a es , so ha when he nucleon is emo ed, he inal
momen um o he (C+N)sys em will be equal and oppo-
si e o ha o he emo ed nucleon wi hin he Bo omean
sys em (ky). Since he (C+N)sys em is unbound, he el-
a i e momen um be ween Cand he emaining nucleon can
also be measu ed (kx), and he angle be ween hem θkde e -
mined. A scheme o he quan i ies o in e es is shown in
Fig. 63.
I should be no ed ha , as θkis an angle be ween he
momen a associa ed wi h he nucleons ins ead o hei posi-
ions, i can be iewed as a Fou ie ans o m o he open-
ing angle θNN. As such, a alue o θkla ge han 90◦is
associa ed wi h a alue o θNN lowe han 90◦, and hus
o a dineu on con igu a ion, while he ciga con igu a ion
co esponds o θk<90◦and θNN >90◦. Va ious mea-
su emen s o θkha e been pe o med o Bo omean nuclei
[192–195], inding o i s a e age alue θk he ollowing
esul s (in deg ees): 90 (6He), 90 (8He), 103.6+0.7
−0.9(11Li),
96.2+1.8
−1.6(14Be) [195]. I can be seen ha , indeed, alues o
θk>90◦co espond o θNN<90◦, as shown in Table 3.
I should also be no ed ha an asymme ic dis ibu ion in
ei he θko θNN ( esul ing in θ= 90◦) equi es he in e -
e ence o wa es wi h opposi e pa i ies [196], which nea ly
explains he symme y o 6,8He, since he alence neu ons
ind hemsel es in he nega i e-pa i y p3/2and p1/2wa es,
so hei angula dis ibu ion mus be symme ic. This also
se s he a e age opening angle as a measu e o he admix u e
o opposi e pa i y componen s [197].
Asseenabo e, o neu on-Bo omeansys ems, hedineu-
onco ela ionseems odomina eo e heciga dis ibu ion.
Recen expe imen s ha e ied o explo e he spa ial loca ion
o his dineu on co ela ion, as some nuclea ma e calcula-
ions poin o he low-densi y su ace o Bo omean sys ems
asa egionwhe e heseco ela ionsshouldbe a ou ed[198].
Recen (p,pn)expe imen s on Bo omean nuclei [191,197]
ha e used he modulus o he momen um o he emo ed
neu on (ky) as a p oxy o he nuclea loca ion (smalle
momen a co espond o he nuclea halo and su ace, while
la ge momen a co espond o he nuclea in e io ) and he
a e age opening angle as a p oxy o dineu on co ela ions.
Th ough he use o an eikonal ze o- ange app oxima ion o
he (p,pn)nucleon-nucleon collision p ocess, he c oss sec-
ion can be exp essed as [199,200]:
σ∝#φCn(kx,x)⊗eiky·y|S(y)φgs(x,y)$2
,(158)
whe e xand ya e de ined as in Sec. 4.2,φgs desc ibes he
bound s a e o he Bo omean nucleus, φCn he inal-s a e
wa e- unc ion be ween he co e and he emaining neu on,
and S(y)is a S−ma ix ha includes he abso p i e po en-
ial be ween a ge p o on and co e C. As such, he c oss
sec ion can be in e p e ed as he squa e o he Fou ie ans-
o m o he bound s a e, dis o ed by he abso p ion o he
a ge p o on and he inal-s a e in e ac ion be ween neu on
and co e. In [197], he maximum θk o 11Li was ound
o small alues o ky∼0.3 m
−1, which was in e p e ed
as he dineu on being loca ed a he nuclea su ace whe e
he nuclea densi y is low, as p edic ed by p e ious mod-
els [198,201]. As he low nuclea densi y is a ea u e o he
su ace o all neu on-Bo omean nuclei, i is expec ed ha
he dineu on also appea s in he su ace o o he such sys-
ems. Indeed, in [191], he maximum o θkwas ound o
oughly he same alue o kyalso o 14Be and 17B, as shown
in Fig. 64, making a an alizing case o he uni e sali y o
dineu on co ela ions in he su ace o neu on-Bo omean
nuclei, as p edic ed in [198], al hough expe imen s on mo e
nuclei, such as 19Bo 22C, a e equi ed in o de o con i m
his ea u e. In he igu e he g een bands co espond o sys-
ema ic e o s in expe imen al da a, while s a is ical e o s
a e indica ed by he e o ba s. The heo e ical calcula ions
a e able o ma ch expe imen al esul s o 11Li and 17Bbu
o e es ima e he maximum a e age angle o 14Be. This has
been associa ed wi h he con ibu ion o exci ed s a es o he
12Be beyond he 2+s a e included in he heo e ical model
[191].
10 Unce ain y e alua ion
Gi en heamoun o ing edien s equi ed o he compu a ion
o eac ion obse ables (bound-s a e wa e uncions, op ical
po en ials, spec oscopic ac o s,...) he e has been a push
o he quan i ica ion o he unce ain ies in he calcula ed
obse ables due o he ambigui ies o hese ing edien s ( he
Igo ambigui y [202] o he op ical po en ials is a classical
exampleo suchambigui ies).While he classical esponse o
his demand was he compa ison o calcula ions using di e -
en ing edien s( o example, op ical po en ials omdi e en
global pa ame iza ions), his e alua ion o he unce ain ies
is a mos quali a i e and highly dependen on he choice o
123
Eu . Phys. J. A (2025) 61:47 Page 51 o 57 47
Fig. 64 A e age opening angle in momen um space θkas a unc ion
o ky. G een bands co espond o he sys ema ic e o s in he expe i-
men al measu emen . In all nuclei he maximum a e age momen um is
ound a simila alues o ky o all conside ed nuclei. Figu e adap ed
om Re . [191]
inpu s. Recen ly, Bayesian me hods ha e been p oposed o
quan i y he unce ain ies ha op ical po en ials su e when
hey a e ob ained om he i ing o elas ic sca e ing da a
(as is usual o op ical po en ials) and o s udy he p opa-
ga ion o hese unce ain ies o obse ables such as (d,p)
ans e angula o cha ge exchange (p,n)di e en ial c oss
sec ions [203–210] o nucleon-knockou momen um dis i-
bu ions [211]. The de e mina ion o Bayesian pos e io dis-
ibu ions and con idence in e als equi es he e alua ion
o he eac ion obse ables o a ange o alues o e e y
conside ed pa ame e , which apidly esul s in a humongous
numbe o calcula ions, which a e easible in acili ies wi h
high compu ing powe o ligh -compu a ion models such
as DWBA and ADWA, bu become p ohibi i e o mod-
els which equi e hea ie compu a ions, such as CDCC o
CRC.Assuch,me hods o accele a e hese calcula ions using
emula o s [212,213] o p edic and in e pola e he esul o
hese hea ie calcula ions o he equi ed pa ame e al-
ues a e being explo ed. Th ough hese me hods, unce ain y
analyses ha e been pe o med using CDCC [214], al hough
admi edly only he unce ain ies due o he alence-co e
in e ac ion we e s udied, u he p oo o he compu a ional
magni ude o he ask. In his ega d, new global po en ial
pa ame iza ions wi h quan i ied unce ain ies ha e ecen ly
been de eloped, wi h unce ain ies o igina ing ei he om
he chi al in e ac ions used o gene a e he po en ials [215]
o om i ing o ex ensi e elas ic sca e ing da a [216],
al hough i should be no ed ha in [216] he i was no pe -
o medwi hinaBayesian amewo k.Thela e po en ialhas
been used o s udy he unce ain y in he educ ion ac o s o
ans e and knockou eac ions [217], which is s ill an open
p oblem [218].
11 Inclusion o non-local po en ials
I has long been known ha nucleon-nucleus and nucleus-
nucleus op ical po en ials a e non-local [219], bo h due o
an isymme iza ion and he e ec o non-elas ic channels
(encoded in he second e m o he Feshbach po en ial in
Eq. (2)). The use o non-local po en ials ans o ms he
Sch ödinge equa ion in o an in eg o-di e en ial equa ion,
which is a mo e cumbe some o sol e, so local po en ials
ha e adi ionally been a ou ed in eac ion calcula ions. In
ac , o a long ime only a ew non-local pa ame iza ions
o op ical po en ials ha e been published [220–223] and
non-locali y is no explici ly conside ed in he majo i y o
po en ial pa ame iza ions [50,224]. In ecen yea s he e has
been a de elopmen o dispe si e op ical po en ials based on
he nuclea sel -ene gy, which is inhe en ly non-local [225–
228], hus eigni ing he in e es in ully non-local po en-
ials.
Fo he s udy o elas ic sca e ing, he neglec o non-
locali y is no p oblema ic, as one can always ind a local
equi alen po en ial which ep oduces he phase-shi s and
obse ables o he non-local one [229,230]. Analogously,
eac ions ha a e only sensi i e o he asymp o ics o he
wa e unc ions a e usually well desc ibed by local po en-
ials. Howe e , non-locali y educes wa e unc ions in he
nuclea in e io , in wha is called he Pe ey e ec [229,231],
so non-locali y should be aken in o conside a ion in eac-
ions ha a e sensi i e o he nuclea in e io , such as ans e
eac ions. This has usually been app oxima ed h ough he
mul iplica ion o wa e unc ions ob ained om a local po en-
ial by he so-called Pe ey ac o [229], which educes he
wa e unc ions in he in e io wi hou al e ing hei asymp-
o ics. Al hough he Pe ey ac o can be ob ained o any
non-locali y shape, he p esc ip ions ypically used assume a
Gaussian dependence on non-locali y [232] (usually e e ed
o as Pe ey-Buck geome y). These app oxima ions (local
equi alen po en ial and Pe ey ac o ) ha e been compa ed
o he compu a ion o he wa e unc ion using he ull non-
local po en ial in [233] o (d,p) ans e eac ions, inding a
di e ence in spec oscopic ac o s o ∼10%, al hough hese
123
47 Page 52 o 57 Eu . Phys. J. A (2025) 61:47
esul s did no conside non-locali y in he ou going deu e on
wa e unc ion.
The s udy o non-locali y in (d,p) ans e eac ions has
ecei ed pa icula a en ion, due o he la ge e ec non-
locali y has on he obse ables, he ac ha he e exis mo e
eliable non-local nucleon-nucleus po en ials han nucleus-
nucleus ones and he ela i e simplici y o hei desc ip ion,
which allows o mo e manageable compu a ions. As such,
non-locali y has been explici ly included in Faddee calcu-
la ions o (d,p) ans e eac ions [234]aswellasin he
widesp ead ADWA calcula ions [235–237], wi h a special
ocus on he local-equi alen adiaba ic deu e on po en ial,
whose main ea u e was he ene gy shi in nucleon-nucleus
po en ials om he nominal alue o hal he deu e on ene gy
due o he in e nal kine ic ene gy o he nucleons in he
deu e on. La e implemen a ions included non-locali y in he
deu e on channel ully [238–240], wi h he in iguing inding
[239,240] ha c oss sec ions o he 26Al(d,p)27Al eac-
ion we e signi ican ly dependen on he deu e on model,
and in pa icula on i s d-wa e componen , a dependence
ha was no obse ed in calcula ions wi h local po en-
ial. This dependence was la e ound o be an a i ac
due o he in e play o he ADWA o malism and he d-
wa e componen o he deu e on, as he inclusion o non-
locali y in CDCC calcula ions ( h ough he use o i s -
o de local equi alen po en ials [241]) and Faddee cal-
cula ions [242] esul ed in a educed dependence on he
deu e on model, showing he need o p ecise eac ion he-
o ies o ex ac eliable in o ma ion om expe imen al da a.
The CDCC o malism has been u he ex ended o include
non-locali y h ough he conside a ion o he Pe ey e ec
[243] and he ex ension o 6Li→α+db eakup eac ions
[244]. The e ec o non-locali y in o he eac ions, such
as (d,p)Xsu oga e eac ions [245] and eikonal desc ip-
ions o knockou eac ions [246], has also been ecen ly
s udied, as co esponds o he moun ing in e es in non-
locali y a ising om he new ab ini io and dispe si e po en-
ials.
12 Conclusions and pe spec i es
Thep esen e iewaimed op o ideaglimpsein o heunique
physics behind eac ions in ol ing weakly-bound nuclei and
he o malisms and me hods commonly used in he modeling
and analysis o hese eac ions. We ha e shown he impo -
ance o b eakup channels o weakly-bound sys ems and he
me hods de eloped o p ope ly conside hem, such as he
CDCC me hod (Sec. 3), se e al semiclassical desc ip ions
(Sec.7), o he ADWA and CDCC-BA o malisms o ans-
e eac ions (Sec. 4), as well as hei ex ension o conside
collec i e exci a ions and/o 3-body p ojec iles, whe e he
pa icula cha ac e is ics o Bo omean sys ems ha e been
exploi ed o explo e nucleon-nucleon co ela ions (Sec. 9).
As well, inclusi e b eakup (Sec. 5) and incomple e and com-
ple e usion (Sec. 6) eac ions wi h weakly-bound sys ems
ha e been explo ed due o he unique phenomenon o com-
ple e usion supp ession ound o weakly-bound species.
Thanks o he inc ease in compu a ional powe , demanding
me hodssuchasCDCC andIAVbecomemo eaccessibleand
use ul in he analysis o expe imen al da a. In addi ion, ques-
ionswhich equi ela ge nume icale o s,suchas heinclu-
sion o non-locali y in hese sophis ica ed me hods (Sec. 11)
and he s udy o heo e ical unce ain ies (Sec. 10), a e s a -
ing o be explo ed.
Admi edly, he selec ion o eac ions and me hods p e-
sen ed in his wo k has been hea ily cu a ed by he expe ise
and p e e ences o he au ho s and many o he me hods o
he s udy o eac ions wi h weakly-bound nuclei exis and/o
a e cu en ly being de eloped:
– Some o he mos p omising ad ances a e hose ela ed
o ew-body and ab-ini io me hods. The applica ion o
he Faddee me hod o nuclea eac ions, while po en-
iallyp o idinganexac , igo oussolu iono agi en ew-
body Hamil onian, has only been possible in ecen yea s
hanks o he inc easing compu a ional capabili ies, he
inclusion o complex op ical po en ials and he de elop-
men o echniques o deal wi h he long- ange Coulomb
couplings. The me hod has been e y use ul o bench-
ma king some o he me hods discussed in his e iew,
such as he CDCC and ADWA me hods [35,247]. I has
also p o en e y use ul o analyzing nucleon knockou
eac ions [36,248,249].
– The e has also been ema kable p og ess in he No-Co e-
Shell-Model (NCSM) and i s ex ension, he NCSM wi h
Con inuum (NCSMC) [250]. While o iginally de eloped
o s udy nuclea s uc u e, he me hod has been applied
wi h g ea success o he calcula ion o sca e ing obse -
ables [183,251,252].
– Ab-ini io me hods p o ide no only al e na i e ways
o e alua ing sca e ing obse ables by hemsel es, bu
hey also u nish e y use ul s uc u e inpu s o adi-
ional eac ion amewo ks. Fo example, he Va ia ional
Mon e Ca lo (VMC) [253] and i s a ian s, such as he
G een’s unc ion Mon e Ca lo and he Clus e Va ia ional
Mon e Ca lo [254], p o ide mic oscopic o e lap unc-
ions which can be di ec ly plugged in o eac ion o -
malisms o ans e [255] and knockou [256] eac ions.
– The expansion o expe imen al s udies, eaching hea -
ie exo ic nuclei, has also shown he need o ex end and
upg ade he exis ing eac ion models o inco po a e he
new ea u es ound in he s uc u e and eac ions o he
newly disco e ed nuclei. Fo example, i has become
clea ha a p ope desc ip ion o newly disco e ed halo
nuclei in he islands o in e sion equi es he simul ane-
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Eu . Phys. J. A (2025) 61:47 Page 53 o 57 47
ous inclusion o de o ma ion, pai ing and Pauli block-
ing. Reac ion o malisms, such as DWBA o CDCC,
mus he e o e be ex ended o accommoda e such e ec s.
Admi edly, he XCDCC me hod discussed in his e iew
has used so a ela i ely simple pa icle-plus- o o mod-
els, and hence i is manda o y o eso o mo e sophis i-
ca ed models.
– The challenge is e en la ge o he case o h ee-body
sys ems which, in addi ion o he p ope ies ci ed abo e,
equi e speci ic o malisms o accoun o hei h ee-
body na u e. Fo example, knockou expe imen s wi h
14Be ha e clea e idenced he need o including he exci-
a ions o he 12Be “co e” [191,257]. E en beyond, mod-
els including ou - and i e-body sys ems may be neces-
sa y, as he ecen disco e y o a ou -neu on co ela ed
sys em in 8He a es s [258].
All o hese a enues o ad ancemen show ha , despi e
i s nume ous and signi ican successes, he s udy o nuclea
eac ions wi h weakly-bound nuclei emains a de eloping
and ac i e ield, pushedby heo e ical and echnical ad ances
and by new expe imen al measu emen s in e e mo e exo ic
nuclei. While we hope ha his wo k has ansmi ed o he
eade a leas a glimpse o he cu en me hods in his as-
cina ing ield, we also hold he pa adoxical hope ha new
de elopmen s p omp ly ende his wo k somewha obsole e
hanks o new and pe haps su p ising ad ancemen s.
Acknowledgemen s Wea eg a e ul oAngelaBonacco soandJinLei
o hei eedbackon hesemiclassical ans e o hecon inuumcalcula-
ions.Thiswo khas been pa ially unded by heMinis e iodeCienciae
Inno ación, MCIN/AEI/10.13039/501100011033 unde I+D+i p ojec
No. PID2020-114687GB-I00. M.G.-R. acknowledges inancial suppo
by MCIN/AEI/10.13039/501100011033 unde g an IJC2020-043878-
I (also unded by “Eu opean Union Nex Gene a ionEU/PRTR”).
Funding Funding o open access publishing: Uni e sidad de Se illa/
CBUA
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