Pauli-blocking Effects in Nilsson States of Weakly Bound Exotic Nuclei

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“This is an Accep ed Manusc ip o an a icle published in PHYSICAL REVIEW
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Pauli Blocking e ec s in Nilsson s a es o weakly bound exo ic nuclei
P. Pun a,∗J. A. Lay, and A. M. Mo o
Depa amen o de FAMN, Facul ad de F´ısica, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain
G. Col`o
Dipa imen o di Fisica, Uni e si `a degli S udi di Milano, ia Celo ia 16, I-20133 Milano, I aly and
INFN, Sezione di Milano, ia Celo ia 16, I-20133 Milano, I aly
(Da ed: June 18, 2025)
Backg ound: The desc ip ion o weakly bound nuclei using de o med ew-body models has p o en o be c ucial
in he s udy o eac ions in ol ing ce ain exo ic nuclei. Howe e , he co e+ alence nucleons models ace he
challenge o applying he Pauli exclusion p inciple, since he ac o isa ion o he sys em does no allow comple e
an isymme iza ion. The e o e, s a es occupied by co e nucleons should be blocked o he alence nucleons.
Pu pose: We aim o s udy he Ca bon iso opes 17C and 19C which a e good examples o weakly bound exo ic
nuclei wi h signi ican de o ma ion whe e he alence shell is pa ially illed. We will explo e he e ec o di e en
me hods o blocking occupied s a es in de o med wo-body models.
Me hods: The s uc u e o 17C and 19C is desc ibed wi h de o med wo-body models whe e a Nilsson Hamil-
onian is cons uc ed using An isymme ized Molecula Dynamic calcula ions o he co es. Di e en me hods o
blocking occupied Nilsson s a es a e conside ed using he Ba deen–Coope –Sch ie e o malism: wi hou block-
ing, o al blocking and pa ial blocking. The la e also akes in o accoun pai co ela ions o some ex en . These
models a e la e used o s udy 16C(d, p)17C, 17C(p, d)16C and 18C(d, p)19C ans e eac ions wi hin he Adiaba ic
Dis o ed Wa e App oxima ion. In he i s case, he esul s a e compa ed wi h expe imen al da a.
Resul s: A good ep oduc ion o he s uc u e o 17C is ound, signi ican ly imp o ing he ag eemen in he
16C(d, p)17C eac ion including blocking e ec s. The 19C spec um is be e ep oduced conside ing blocking, in
pa icula , he pa ial blocking me hod ha conside s he pai ing in e ac ion p o ides he bes desc ip ion.
Conclusions: P omising esul s a e shown o he s udy o ans e eac ions in ol ing weakly bound exo ic
nuclei, by highligh ing he e ec o blocking occupied Nilsson s a es. We en ision o ex end he models o he
s udy o b eakup eac ions and o newly disco e ed halo nuclei.
Keywo ds: exo ic nuclei, halo nuclei
I. INTRODUCTION
In ecen yea s, he s udy o weakly bound exo ic nu-
clei has been boos ed by he de elopmen o adioac i e
beam acili ies. The exo ic nuclei ha e a a he di e en
a io o p o ons o neu ons om ha o s able nuclei.
Because o ha , hei p ope ies can be e y di e en ,
o example, some o hem exhibi a halo na u e. Halo
nuclei a e weakly bound sys ems composed o a ela i ely
compac co e and one o wo highly delocalized alence
pa icle(s) o ming a halo o ma e a ound he co e.
To s udy he eac ions including weakly bound nu-
clei, hey ha e usually been desc ibed using ew-body
models ha igno e agmen de o ma ions. Howe e , i
is known ha co e de o ma ions can signi ican ly a ec
bo h he s uc u e and he dynamics o hese sys ems [1–
4]. The e o e, o ob ain a mo e eliable desc ip ion o
eac ions in ol ing hese nuclei, de o ma ion needs o be
included in ew-body models.
Nuclei composed o a weakly bound neu on and an
e en-e en co e wi h signi ican quad upole de o ma ion
a e conside ed. Di e en de o med wo-body models
ha e been success ully applied o he desc ip ion o 17C,
∗ppun[email p o ec ed]
19C and 11Be. P omising esul s a e ob ained wi h he
semi-mic oscopic Pa icle+AMD model (PAMD) [5], in
which he Hamil onian is cons uc ed using he ansi-
ion densi ies o he co esponding co es calcula ed in
he An isymme ized Molecula Dynamics (AMD) o -
malism. Howe e , in he case o 17C, be e esul s a e
ob ained wi h he Nilsson model [6]. In he p esen wo k,
bo h models a e combined in o de o ha e a model like
Nilsson’s bu main aining mos o he mic oscopic in o -
ma ion o he PAMD model.
A inhe en di icul y in de o med wo-body models is
he applica ion o he Pauli exclusion p inciple ela ed
o he s a es occupied by he co e nucleons. In many-
body calcula ions, his p oblem is au oma ically sol ed
by an isymme iza ion o he wa e unc ion, while ew-
body models equi e he explici exclusion o o bidden
s a es. The simples way o deal wi h his p oblem is
o disca d he bound s a es ha a e conside ed occu-
pied by compa ison wi h he sphe ical and Nilsson limi s.
Howe e , despi e i s widesp ead applica ion, his me hod
shows limi a ions in i s applica ion o some nuclei. To-
al blocking o single-pa icle s a es eme ges as an al-
e na i e me hod, ini ially p oposed o sphe ical single-
pa icle s a es and ecen ly ex ended o de o med Nilsson
s a es [7].
In his wo k, he possibili y o including mo e sophis-
ica ed Pauli blocking e ec s is explo ed. In pa icu-
2
la , single-pa icle Nilsson s a es a e o ally o pa ially
blocked using he Ba deen–Coope –Sch ie e (BCS) o -
malism, hus including a esidual pai ing in e ac ion. In
his way, pai co ela ions a e also app oxima ely aken
in o accoun in o de o ob ain mo e comple e and de-
sc ip i e models. The implemen a ion o BCS heo y in
a de o med wo-body model has al eady been es ed [8].
Howe e , in his new me hod, ins ead o applying he
BCS calcula ion o he sphe ical single-pa icle le els, we
use he de o med Nilsson single-pa icle le els as a s a -
ing poin .
We p esen he esul s o using di e en Pauli block-
ing me hods applied o he de o med weakly-bound nu-
clei 17C and 19C. The ene gies o he bound s a es and
some low-lying esonances, oge he wi h hei associ-
a ed wa e unc ions, a e ob ained by diagonalizing he
model Hamil onian in a basis o squa e-in eg able unc-
ions. We use he ans o med ha monic oscilla o unc-
ions (THO) [9], which ha e been success ully applied o
he disc e iza ion o he con inuum o weakly bound nu-
clei o i s applica ion o b eakup and ans e eac ions
bo h o wo-body and h ee-body sys ems. [2, 5, 6, 10–
13].
The calcula ed 17C wa e unc ions a e es ed by apply-
ing hem o he ans e eac ion 16C(d, p)17C, compa -
ing wi h he expe imen al da a om GANIL [14]. We
will also p opose simila eac ions no ye expe imen-
ally measu ed, 17C(p, d)16C and 18C(d, p)19C, and anal-
yse how hey can be used o disc imina e be ween he
di e en models p esen ed.
The pape is o ganized as ollows. Sec. II. desc ibes
he s uc u e o malism. I ocuses on he desc ip ion o
he new Nilsson+AMD model and he implemen a ion
o he BCS heo y. Sec ion III shows he esul s o he
applica ion o 17C, including he s udy o ans e eac-
ions 16C(d, p)17C and 17C(p, d)16C. The applica ion o
19C and 18C(d, p)19C a e p esen ed in Sec. IV. Finally, in
Sec. VI we summa ize he main esul s o his wo k.
II. STRUCTURE FORMALISM
We conside sys ems ha can be desc ibed using wo-
body models, whe e a neu on mo es in a de o med po-
en ial gene a ed by he co e. The Hamil onian o he
sys em can be w i en as
H=T( ) + Vℓs( )(
ℓ·s) + V c( , ξ) + hco e(ξ),(1)
whe e T( ) is he kine ic ene gy ope a o o he ela i e
mo ion be ween he alence and he co e and hco e(ξ)
is he Hamil onian o he co e. V c( , ξ) is he e ec i e
alence-co e in e ac ion, which depends on he ela i e
mo ion be ween he alence and he co e, bu also on
he co e deg ees o eedom ξ. A spin-obi e m wi h he
usual adial dependence Vℓs( ) is added o his alence-
co e in e ac ion.
The eigen alues and eigen unc ions o he Hamil o-
nian a e ob ained by diagonaliza ion in he THO ba-
sis. In ou case, he single-pa icle Nilsson Hamil onian
HN=H − hco e is diagonalized in he in insic sys em
which o a es join ly wi h he co e. La e , he wa e unc-
ions a e p ojec ed in he ixed labo a o y ame o inally
add he e m hco e. A mo e comple e desc ip ion o his
me hod can be ound in Sec. II o Re . [6]. The eigen unc-
ion associa ed wi h an eigen alue εJπ
ican be gene ically
exp essed as
ΨJπ
iM ( , ξ) = X
α
RJπ
iα ( )ΦM
αJ (ˆ , ξ).(2)
They a e cha ac e ized by he index i, pa i y π, he o-
al angula momen um Jand i s p ojec ion Mon he
zaxis o he ixed labo a o y ame. The adial unc ions
RJπ
iα ( ) a e a linea combina ion o o hono mal unc ions
o he basis RT HO
nℓ ( ). ΦM
αJ (ˆ , ξ) a e he eigen unc ions
o J2and Jz esul ing om he coupling o he angula
momen um 
jo he alence pa icle o he co e angula
momen um 
I,
ΦM
αJ (ˆ , ξ)≡hYj
ℓs(ˆ )⊗ϕI(ξ)iJM .(3)
He e, ℓis he o bi al angula momen um o he alence
pa icle ela i e o he co e, which couples o he spin o
he alence pa icle s o gi e he pa icle o al angula
momen um j. The label αdeno es he se o quan um
numbe s {ℓ, s, j, I}.Yjm
ℓs (ˆ ) deno es he wa e unc ion e-
sul ing om coupling he spin o he alence pa icle wi h
he co esponding sphe ical ha monic.
The weigh o each αcomponen is gi en by he in-
eg al R∞
0d | RJπ
iα ( )|2. When an isymme iza ion be-
ween he alence neu on and he co e is p ope ly aken
in o accoun , hese weigh s co espond o spec oscopy
ac o s (SF). In ou case, whe e an isymme iza ion be-
ween he alence pa icle and he emaining nucleons is
no conside ed, his co espondence be ween weigh s and
SF is only app oxima e.
A. NAMD model
In he Nilsson+AMD me hod p oposed he e, deno ed
NAMD he ea e , we combine he Nilsson and PAMD
models compa ed in Re . [6]. On he one hand, as in he
Nilsson model, he alence-co e in e ac ion is conside ed
in he in insic ame (
′), which o a es join ly wi h he
co e. Fo his ame, conside ing an axially symme ic
quad upole de o ma ion, we can assume ha he po en-
ial V c only depends on he adial coo dina e = ′and
he angle θ′wi h espec o he symme y axis o he
co e:
V c( , θ′) = V0( )Y00 +V2( )Y20(θ′).(4)
Fu he mo e, as in he Nilsson model, he co e is app ox-
ima ed by a pe ec o o . The e o e, hco e depends on
he angula momen um o he co e 
Iand i s momen o
3
ine ia J,
hco e =ℏ2
2J
I2.(5)
On he o he hand, he coupling po en ials V0( ) and
V2( ) a e calcula ed as in he PAMD model :
Vλ( ) = ⟨0+||Vλ( , ξ)||λ+⟩.(6)
No e ha |λ+⟩co esponds ei he o he g ound s a e o
he co e (0+) o o he i s exci ed (2+), depending on
he selec ed mul ipola i y λ. The calcula ion o hese e-
duced ma ix elemen s ollows he p ocedu e o Re . [5],
whe e he nucleon-nucleon in e ac ion o Jeukenne, Leje-
une, and Mahaux [15] is con olu ed wi h mic oscopic
ansi ion densi ies o he co e nucleus calcula ed wi h
An ysymme ized Molecula Dynamics (AMD) [16]. In
o de o compa e he s eng h o hese couplings wi h
a s anda d pa icle- o o o Nilsson models, a de o -
ma ion leng h δ2≡ ⟨0||ˆ
δ2||2⟩can be ex ac ed om
he same ansi ion densi ies as de ined, o ins ance, in
Re s. [5, 16].
In ela ion o he Pauli exclusion p inciple, i a me hod
wi hou blocking is used (NoB), he Hamil onian in
Eq. (1) is di ec ly he Hamil onian o he sys em. In his
case, he Pauli p inciple is applied a e he diagonaliza-
ion o H: he ob ained bound s a es ha we conside
o bidden by compa ison wi h he sphe ical and Nilsson
limi s a e disca ded. This me hod is called S anda d
PRM in Re . [7], since i is commonly used in pa icle-
o o models (PRM). In ou NoB me hod, we disca d he
s a es ob ained wi h a dominan con ibu ion om he
lowes N/2 Nilsson s a es, whe e N is he numbe o neu-
ons o he co e. Howe e , he Pauli p inciple can ha e
a s ong e ec on he calcula ion o ene gies and wa e
unc ions. The e o e, in some cases, i is con enien o
conside me hods o blocking single-pa icle s a es oc-
cupied by he co e nucleons. In his wo k, we use he
BCS heo y o accoun o he pa ial occupa ion o he
single-pa icle Nilsson s a es.
B. Implemen a ion o BCS o malism
The Ba deen–Coope –Sch ie e (BCS) o malism is
applied as explained in Re . [17]. The eigen alues εν
o he Nilsson Hamil onian HN=H − hco e a e aken
as single-pa icle ene gies in he BCS calcula ion o he
N neu ons o he co e. A cons an pai ing s eng h G
is conside ed o ac be ween a se o le els a ound he
Fe mi ene gy λ. Le els om 0 MeV o 10 MeV below
he neu on sepa a ion h eshold a e included in he BCS
ac i e space. The speci ic le els o each nucleus s udied
can be seen in Fig. 1, be ween he do ed lines. A new
Hamil onian can be de ined in e ms o he c ea ion and
annihila ion ope a o s [17]
HBCS =X
ν
(εν−λ)(a†
νaν+a†
¯νa¯ν)−GX
νν′
a†
νa†
¯νa¯ν′aν′.
(7)
Sphe ical Nilsson
-28
-24
-20
-16
-12
-8
-4
0
4
E (MeV)
Fe mi Le el
2s1/2
1d5/2
1p1/2 [101 1/2]
[220 1/2]
[211 3/2]
[211 1/2]
[202 5/2]
1p3/2
1s1/2
17C
[101 3/2]
[110 1/2]
[000 1/2]
[202 3/2]
[200 1/2]
1d3/2
Sphe ical Nilsson
Femi Le el
2s1/2
1d5/2
1p1/2
1s1/2
1p3/2
[202 5/2]
[211 1/2]
[211 3/2]
[220 1/2]
[101 1/2]
19C
1d3/2
[101 3/2]
[110 1/2]
[200 1/2]
[000 1/2]
[202 3/2]
FIG. 1. Sphe ical single-pa icle le els and Nilsson le els o
17C (le ) and 19C ( igh ) ob ained using he NAMD (PB)
model. The Fe mi le el ob ained in he BCS calcula ion is
also ep esen ed. Posi i e and nega i e pa i y s a es a e ep-
esen ed wi h di e en colou s, and Nilsson le els a e labelled
wi h he asymp o ic quan um numbe s [Nn3Λ Ω] e e ing o
la ge p ola e de o ma ions [18].
The single-pa icle ene gies ενand hei associa ed wa e
unc ions ψν(
′) a e ob ained by diagonalizing he Hamil-
onian HNin he THO basis gi ing ise o he linea
combina ions,
ψν(
′) = X
nj
Cν
njRT HO
nl ( )YjΩ
ℓs (ˆ ′),(8)
whe e Cν
nj a e he componen s o he eigen ec o ma ix.
These Nilsson s a es νdo no ha e well-de ined alues o
ℓand j, bu hey can be cha ac e ised by hei pa i y π
and he p ojec ion Ω o 
jalong he axial symme y axis.
The ene gy ενco esponds o he s a e νwi h p ojec ion
Ω and wa e unc ion ψν(
′), bu also o he ime- e e sed
s a e ¯νwi h p ojec ion −Ω and wa e unc ion
ψ¯ν(
′) = X
j
(−1)j−ΩCν
njRT HO
nl ( )Yj−Ω
ℓs (ˆ ′).(9)
The e o e, unlike sphe ical single-pa icle le els, all he
Nilsson le els can be occupied only by a pai o neu ons
o p o ons. The schemes o sphe ical and Nilsson single-
pa icle le els ob ained o 17C and 19C a e shown in
Fig. 1, wi h he model pa ame e s speci ied below.
The applica ion o he BCS o malism in ol es he
Bogoljubo -Vala in canonical ans o ma ion om pa -
icles o quasipa icles. The e o e, once he BCS cal-
cula ion is pe o med, he pa icle s a e |ν⟩becomes a
one-quasipa icle s a e |BCS, νJM⟩. Addi ionally, he
s a e has been p ojec ed in he ixed labo a o y sys em
by including he o a ional s a e o he nucleus. Then,
4
o a o al angula momen um Jwi h p ojec ion Mon
he Z-axis o he labo a o y sys em, he one-body wa e
unc ion becomes
ΨM
νJ (
′, ω) = √2J+ 1
4πhψν(
′)DJ
MΩ(ω)∗+
(−1)J−Ωψ¯ν(
′)DJ
M−Ω(ω)∗i.(10)
The de ini ion o [19] is used o he o a ion ma ices
DJ
MΩ(ω) and he h ee Eule angles a e deno ed by ω.
The unc ions ΨM
νJ (
′, ω) a e o hono mal and ake in o
accoun he symme y ega ding he Ω and −Ω p ojec-
ions. The e ec o hco e is included by diagonalizing a
new quasipa icle Hamil onian HJπ=HBCS +hco e −e0
using s a es |BCS, νJM⟩as a basis. e0≡ ⟨BCS|HBCS +
hco e|BCS⟩is he ene gy o he BCS g ound s a e, ha is
he quasipa icle acuum. The ma ix elemen s o HJπ
esul :
⟨BCS, ν′JM|HJπ|BCS, νJM⟩=
Eνδν′ν+ℏ2
2J⟨ΨM
ν′J|
I2|ΨM
νJ ⟩Fν′ν.(11)
Eνis he quasipa icle ene gy associa ed wi h ϵνand he
Fν′ν ac o appea s na u ally when including a one-body
ope a o be ween one-quasipa icle s a es [17, 18]. I de-
pends on he occupa ion numbe s uνand νob ained in
he BCS calcula ion as
Fν′ν=uν′uν− ν′ ν.(12)
Fo each angula momen um and pa i y Jπ, a se
o ene gies EJπ
iwi h hei co esponding eigen ec o s
ha ing componen s CiJ
νa e ob ained. The associa ed
wa e unc ions can be w i en in he ixed labo a o y
ame as in Eq. (2), wi h adial unc ions RJπ
iα ( ) =
PnCiJπ
nα RT HO
nℓ ( ). The coe icien s can be calcula ed
om he eigens a es o HJπand HN
CiJπ
nα = 2I+ 1
2J+ 1q1+(−1)IX
ν⟨jΩI0|JΩ⟩CiJ
νCν
nj,
(13)
using exp ession (8) om Re . [6]. We can also de ine
he occupa ion numbe 2
iJπ=Pν(CiJ
ν ν)2 ela ed o
he occupa ion o he s a e by he co e neu ons.
The ob ained quasipa icle s a es mix pa icle and hole
s a es. Howe e , we assume ha he s a es wi h low oc-
cupa ion by co e neu ons ( 2
iJπ<0.5) can be app oxi-
ma ed as pa icle s a es, while hose wi h high occupa-
ion ( 2
iJπ>0.5) a e conside ed hole s a es and hey a e
disca ded. The pa icle s a es ha e an associa ed ene gy
εJπ
i=EJπ
i+λ, whe e λis he Fe mi ene gy ob ained in
he BCS calcula ion. These a e he ene gies ha can be
compa ed wi h he eigen alues o he Hamil onian wi h-
ou blocking (1) and he expe imen al da a.
No e ha he ac o Fν′ν educes he coupling be ween
Nilsson s a es wi h high and low occupancy. The e o e,
his o malism can be ega ded as a pa ial blocking
Expe imen al NoB TB PB
-0.9
-0.6
-0.3
0
E (MeV)
1/2+
5/2+
5/2+
3/2+
1/2+
3/2+
1/2+
5/2+
3/2+
16C(0+)+n
3/2+
1/2+
5/2+
FIG. 2. Ene gies o he bound s a es o 17C. The expe imen-
al alues om [22, 23] a e compa ed wi h he esul s o he
NAMD model using di e en blocking me hods.
me hod (PB). I is also in e es ing o explo e he ex-
eme case o a o al blocking me hod (TB). To do his,
we assume ha he pai ing s eng h is ze o and he low-
es N/2 Nilsson s a es, hose below he Fe mi le el, a e
conside ed ully occupied (uν= 0, ν= 1), while he
es a e comple ely emp y (uν= 1, ν= 0). In his case,
he occupied and emp y Nilsson s a es a e comple ely
decoupled because Fν′ν= 0 be ween hem. This TB
me hod is equi alen in p ac ice o model III (“de o med
PF model”) om Re . [7].
III. APPLICATION TO 17C
The NAMD model has been applied o s udy he 17C
sys em using he h ee di e en blocking me hods. A
sligh eno malisa ion o he cen al po en ial V0( ) was
necessa y o ma ch he ene gy o he g ound s a e wi h
he expe imen al alue. This eno malisa ion is di e en
depending on he blocking model: 1.014 (NoB), 1.000
(TB) and 0.997 (PB). The spin-o bi po en ial is ob-
ained as a unc ion o he de i a i e o V0( ) ollowing
he ela ion om [20], bu using ou semi-mic oscopic
unc ion ins ead o a Wood-Saxon po en ial. Fo all
blocking me hods, he alue ℏ/2J= 0.295 MeV is used
o he co e Hamil onian, compa ible wi h he exci a ion
ene gy o he i s exci ed s a e 2+o 16C (1.766 MeV
[21]). Rega ding o he de o ma ion leng h, δ2= 1.27 m
(p ola e) is ob ained ollowing he p esc ip ion o [5].
In he TB me hod, he i e mos bound Nilsson s a es
( he s a es below he Fe mi le el in Fig. 1) a e conside ed
ully occupied. In he case o PB, only he i s h ee a e
conside ed ully occupied. Fo he nex ou s a es, which
co espond o a sepa a ion ene gy be ween -10 MeV and
0 MeV ( he s a es be ween he do ed lines in Fig. 1), a
BCS calcula ion wi h a pai ing s eng h G= 1.3 MeV
is pe o med. F om he BCS calcula ion, he Fe mi le el
is ound a -4.13 MeV and he pai ing gap is 1.58 MeV,
close o he alue ob ained wi h he h ee-poin o mula
cen e ed on he 16C sys em, 1.76 MeV.
The Hamil onian is diagonalized in he THO basis us-
ing he pa ame e s b= 2.4 m and γ= 3.0 m1/2and
conside ing 0 ≤ℓ≤4 and 1 ≤n≤30. The ene gies

5
-0.2
0
0.2
0.4
u( ) ( m-1/2)
〈17C(1/2+
1)|16C(0+)〉
(a)
0510 15 20 25
( m)
0
0.2
0.4
u( ) ( m-1/2)
(b)
〈17C(5/2+
1)|16C(0+)〉
RGM
NAMD NoB
NAMD TB
NAMD PB
FIG. 3. O e laps o he g ound s a e o 16C wi h he i s
(uppe panel) and second (lowe panel) exci ed s a es o 17C.
The esul s o he RGM calcula ions [24] a e compa ed wi h
he esul s o he NAMD model using di e en ea men s o
Pauli blocking.
o he bound s a es ob ained o he di e en blocking
me hods a e shown in Fig. 2 compa ed o he expe i-
men al alues [22, 23]. Using he TB and PB me hods,
he 5/2+s a e becomes he i s exci ed s a e and he
1/2+ he second, con a y o he expe imen al e idence.
Howe e , i should be no ed ha he h ee bound s a es
a e e y close in ene gy, so ha he di e ence be ween
he calcula ed le els and he expe imen al alues is less
han 0.5 MeV.
To compa e he wa e unc ions ob ained wi h each
blocking me hod, he adial o e lap unc ions be ween
he exci ed bound s a es o 17C and he g ound s a e o
he co e 16C(0+) a e shown in Fig. 3. These a e he el-
e an o e laps o he s udy o he 16C(d,p)17C eac ion
ha will be discussed la e . The o e laps ob ained om
he esona ing g oup me hod (RGM) [24], a e also shown.
The RGM me hod is a mic oscopic clus e model ha is
much mo e complex han he models p esen ed he e. Fo
example, 990 Sla e de e minan s a e used in he RGM
me hod o ob ain hese o e laps. The e o e, i is na u al
o ind di e ences be ween he NAMD and RGM models.
Howe e , in he lowe panel o Fig. 3 i can be seen how,
by applying o al o pa ial blocking, he NAMD model
p oduces o e laps ⟨17C(5/2+
1)|16C(0+)⟩close o mic o-
scopic RGM calcula ion a la ge dis ances. No e ha
ans e eac ions a e pe iphe al, and hence mos ly sen-
si i e o he beha iou o he nucleons a ound he su ace
o he nucleus. On he o he hand, ⟨17C(1/2+
1)|16C(0+)⟩
TABLE I. Spec oscopic ac o s o he o e laps
⟨17C(Jπ)|16C(0+)⊗nℓj⟩. The esul s o he NAMD
and RGM [24] models a e compa ed wi h hose ex ac ed
om he expe imen al analysis [14].
Jπ(ℓ, j) NoB TB PB RGM Exp.
3/2+
1(2, 3/2) 0.00 0.01 0.01 0.01 0.03 +0.05
-0.03
1/2+
1(0, 1/2) 0.68 0.80 0.82 0.94 0.64±0.18
5/2+
1(2, 5/2) 0.33 0.66 0.65 0.56 0.62±0.13
a e simila excep o he PB case, whe e he misma ch in
ene gy o he s a e leads o di e ences in he asymp o ic
beha iou .
A. 16C(d, p)17C
The di e en ial c oss sec ions o he ans e eac ion
16C(d, p)17C popula ing bound exci ed s a es ha e been
calcula ed as in Re . [6], using he ini e- ange adiaba ic
dis o ed wa e app oxima ion (ADWA) [25], and employ-
ing he same op ical model po en ials o consis ency. In
he pos o m, hese calcula ions equi e he o e lap unc-
ions ⟨17C|16C(0+)⟩, which a e aken om he esul s o
he NAMD model. The esul s ob ained using he h ee
di e en blocking me hods a e compa ed wi h he ex-
pe imen al da a om GANIL [14] in Fig. 4, These da a
we e ob ained in in e se kinema ics wi h a 16C beam a
17.2 MeV/nucleon.
Fo he case o he i s exci ed s a e 1/2+, despi e
he di e ence o he PB due o i s ene gy misma ch, we
ind good ag eemen be ween he esul s o all blocking
me hods and he da a (Fig. 4(a)). The esul s wi hou
blocking (NoB) a e simila o hose ob ained in [6], which
unde es ima e he di e en ial c oss sec ion o he sec-
ond exci ed s a e 5/2+(Fig. 4(b)). Howe e , using he
o al and pa ial blocking me hods (TB and PB), he
ag eemen wi h he expe imen al da a o his s a e is
signi ican ly imp o ed. Figu e 4 also shows he ADWA
esul s using mic oscopic RGM o e laps. The ag eemen
o he esul s using his RGM model wi h he da a is
a he good, bu i should be no ed ha , using a simple
model such as NAMD a simila deg ee o ag eemen is
ound, p o ided ha Pauli blocking e ec s a e p ope ly
accoun ed o .
In Table I, he spec oscopic ac o s (SF) o he o e -
laps ⟨17C(Jπ)|16C(0+)⊗nℓj⟩a e shown. The alues ex-
ac ed om he RGM calcula ions [24] and he di e -
en NAMD models a e compa ed wi h he expe imen al
ones om Re . [14]. These expe imen al SF a e ob ained
as he a io o he expe imen al c oss sec ions and pu e
single-pa icle ini e- age ADWA calcula ions using CH89
pa ame e isa ion. In Table I, i can be seen how he dis-
c epancy o he NAMD NoB model in 16C(d, p)17C(5/2+
1)
is ela ed o he di e ence in SF. He e, he blocking o
he Nilsson s a e [220 1/2] (see Fig. 1) plays a key ole. I
6
10-1
100
101
dσ/dΩ (mb/s )
16C(d,p)17C(1/2+)
17.2 MeV/nucleon
16C(d,p)17C(1/21
(a)
(b)
0 10 20 30 40
θCM (deg)
100
101
dσ/dΩ (mb/s )
16C(d,p)17C(5/21
16C(d,p)17C(5/2+)
GANIL da a
RGM
NAMD NoB
NAMD TB
NAMD PB
FIG. 4. Angula dis ibu ion o he 16C(d, p)17C eac ion a
17.2 MeV/nucleon when he 17C bound s a es 1/2+
1(uppe
panel), and 5/2+
1(lowe panel) a e popula ed. The esul s
using he di e en blocking me hods a e compa ed wi h he
expe imen al da a [14] and he esul s using he RGM o e -
laps [24].
his Nilsson con igu a ion is no blocked, he 5/2+
1s a e
includes a signi ican con ibu ion o s a es wi h Ω = 1/2
ha inc ease he weigh o he componen s1/2⊗2+
(0.39 o NoB e sus 0.16 o TB and 0.08 o PB) o e
d5/2⊗0+.
Al hough we ha e ixed he pa ame e s o he NAMD
model in o de o mee se e al cons ain s which min-
imise hei eedom, di e en pa ame e isa ions ha e
been es ed (mainly a ia ions in he spin-o bi po en ial
and he momen o ine ia o he co e). Howe e , wi hou
applying blocking e ec s, i was no possible o inc ease
he weigh o he componen d5/2⊗0+su icien ly in any
case. No signi ican imp o emen s in he spec um we e
achie ed o he models wi h blocking ei he .
Table I also shows he ag eemen in he small con i-
bu ion o he componen d3/2⊗0+in he g ound s a e
o 17C. This makes he 16C(d, p)17C c oss sec ion o his
s a e small and hence di icul o measu e expe imen ally;
o s udy he g ound s a e o 17C one may conside he e-
e se eac ion 17C(p, d)16C, which is s udied in he nex
subsec ion.
TABLE II. Compa ison o spec oscopic ac o s o he o e -
laps ⟨17C(3/2+
1)|16C(2+)⊗nℓj⟩ob ained wi h he NAMD and
RGM [24] models. Fo ℓ= 2, he wo con ibu ions, 1d3/2and
d5/2, a e summed. Fo ℓ= 4 all con ibu ions a e lowe han
0.001.
ℓNoB TB PB RGM
0 0.33 0.04 0.02 0.37
2 0.50 0.88 0.88 1.13
B. 17C(p, d)16C
The ans e eac ion 17C(p, d)16C is s udied a
31 MeV/nucleon so ha he ela i e eloci ies p−17C
and d−16C a e he same as hose o he p e ious sec-
ion. Consequen ly, using he same po en ials and he
o e lap unc ions ⟨17C(3/2+
1)|16C⟩, di e en ial c oss sec-
ions can be calcula ed using he ADWA app oxima ion
in p io o m.
The ans e o he g ound s a e in 17C(p, d)16C is
analogous o ha in 16C(d, p)17C. In pa icula , bo h
depend on he o e lap unc ion ⟨17C(3/2+
1)|16C(0+)⟩.
Again, he small weigh o his componen makes he
ans e c oss sec ion o he 0+g ound s a e o 16C small.
The c oss sec ion o he ans e o he i s exci ed s a e
16C(2+) mus be la ge and is expec ed o p o ide mo e
in o ma ion. The esul s ob ained in his case using he
NAMD and RGM models a e shown in Fig. 5. The di e -
ences be ween he models can be unde s ood by looking
a SF o he o e laps ⟨17C(3/2+
1)|16C(2+)⊗nℓj⟩, which
a e p esen ed in Table II. On he one hand, he c oss
sec ion is la ge using he o e laps om he RGM calcu-
la ions because i s SF exceeds uni y, while he app oxi-
ma e alues ob ained wi h NAMD models a e necessa ily
less han one. On he o he hand, he la ge weigh o
he s1/2⊗2+componen (ℓ= 0) in he RGM and NAMD
NoB models causes a di e ence in he shape o he dis-
ibu ion compa ed o he o he s, which is he esul o
he clea dominance o he ℓ= 2 componen s.
IV. APPLICATION TO 19C
The NAMD model has also been applied o he s udy
he 19C nucleus, using he same THO basis as in he
case o 17C. The spin-o bi po en ial is pa ame ised
in e ms o he de i a i e o he Wood-Saxon unc ion
wi h s anda d alues o adius (Rso=3 m), di useness
(aso=0.7 m) and s eng h (Vso=6.5 MeV). The alue
ℏ/2J= 0.25 MeV is used o he co e Hamil onian,
compa ible wi h he exci a ion ene gy o he i s ex-
ci ed s a e 2+o 18C (1.588 MeV [26]). The de o ma ion
leng h o he NAMD model is ha o he AMD calcula-
ion, δ2= 1.20 m [5] (p ola e). In he TB me hod, six
Nilsson s a es a e conside ed ully occupied ( he s a es
below he Fe mi le el in Fig. 1). In he PB me hod,
h ee deeply bound Nilsson s a es a e conside ed ully
7
0 10 20 30 40
θCM (deg)
100
101
dσ/dΩ (mb/s )
17C(p,d)16C(2+)
31 MeV/nucleon
RGM
NAMD NoB
NAMD TB
NAMD PB
FIG. 5. Di e en ial c oss sec ion o 17C(p, d)16C ans e o
he i s exci ed s a e 2+o 16C a 31 MeV/nucleon. The e-
sul s using he NAMD model wi h di e en blocking me hods
a e compa ed wi h he esul s using he RGM o e laps [24].
Expe imen al NoB TB PB
-1
-0.5
0
0.5
1
1.5
E (MeV)
1/2+
5/2+
5/2+
3/2+
1/2+
3/2+
1/2+
5/2+
3/2+18C(0+)+n
3/2+
1/2+
5/2+
5/2+
5/2+
5/2+
5/2+(b)
Vl=0=0.9V0 Vl=2=1.1V0
Expe imen al NoB TB PB
-1
-0.5
0
0.5
1
1.5
E (MeV)
18C(0+)+n
1/2+
3/2+
5/2+
5/2+5/2+
1/2+1/2+
3/2+
3/2+
5/2+
(a)
Vl=0=Vl=2=V0
1/2+
5/2+
FIG. 6. Spec um ob ained o he 19C nucleus using he
NAMD model conside ing di e en Pauli blocking me hods
in compa ison wi h he expe imen al one [23, 27–29]. The
uppe panel shows he ene gies ob ained wi hou ℓ-dependen
eno maliza ion, while he lowe panel shows he esul a e
including his eno maliza ion (see ex ).
occupied, and he BCS calcula ion is pe o med wi h he
i e bound s a es be ween -10 MeV and he h eshold (see
Fig. 1). Using G= 1.3 MeV, he Fe mi le el is ound a
-3.73 MeV and he pai ing gap is 1.78 MeV, compa i-
ble wi h he alue ob ained wi h he h ee-poin o mula
cen e ed on he 18C sys em, 1.80 MeV.
In p inciple, he NAMD model gi es almos degene -
TABLE III. Spec oscopic ac o s o he o e laps
⟨19C(Jπ)|18C(0+)⊗nℓj⟩ex ac ed using he NAMD
model wi h di e en Pauli blocking me hods.
S a e (ℓ, j) NoB TB PB
1/2+
1(0, 1/2) 0.68 0.72 0.69
3/2+
1(2, 3/2) 0.52 0.52 0.54
a e sphe ical single-pa icle le els 2s1/2and 1d5/2a ound
-2 MeV, and he 1d3/2a ound 3 MeV. Wi h his sd-
shell scheme, 3/2+and 5/2+s a es o 19C a e no ound
up o 1 MeV abo e he 1/2+g ound s a e, as shown in
Fig. 6.(a). In o de o ob ain a be e ag eemen wi h he
expe imen al spec um [23, 27–29], a ℓ-dependen eno -
malisa ion o he cen al po en ial V0( ) is applied: 0.9
o ℓ= 0 and 1.1 o ℓ= 2. Thus, making he 1dle -
els mo e bound wi h espec o he 1sle el, he 3/2+
and 5/2+s a es o 19C a e close o he g ound s a e
1/2+. Fu he mo e, he po en ial V0( ) con inues o be
sligh ly eno malised globally in each model o ma ch he
ene gy o he g ound s a e wi h he expe imen al alue:
0.973 (NoB), 0.972 (TB), 0.998 (PB). We ha e o empha-
size ha , a e se e al es s, his pa icula pa ame e i-
za ion has u ned ou o be he mos sui able o desc ibe
19C sys em wi h ou NAMD model.
In Fig. 6(b), i can be seen ha in his way he spec-
a ob ained wi h he TB and PB models a e close o he
expe imen al one, whe eas signi ican de ia ions a e seen
in he case o he NoB model. No e ha we a e assum-
ing ha he i s 5/2+s a e is a low-ene gy esonance,
as sugges ed in [28, 29], a he han a bound s a e as
p e iously epo ed [23].
A. 18C(d, p)19C
The wa e unc ions ob ained wi h he NAMD models
a e applied o he calcula ion o he ans e eac ion
18C(d, p)19C a 17 MeV/nucleon. A his ene gy, he
d−18C sys em has a ela i e ene gy e y simila o ha
o he d−16C sys em in he p e ious eac ions.
The ans e o he g ound s a e and he i s exci ed
s a e o 19C a e s udied and he di e en ial c oss sec-
ions ob ained a e shown in Fig. 7. The esul s o he
models a e no e y di e en in his case, which can be
explained by compa ing he SF calcula ed o he o e -
laps ⟨19C(Jπ)|18C(0+)⊗nℓj⟩. These SF a e shown in
Table III, and i can be seen ha he alues a e e y
close, app oxima ely 0.7 o he s1/2⊗0+componen in
he g ound s a e and 0.5 o he d3/2⊗0+componen in
he i s exci ed s a e, poin ing o a smalle impo ance
o Pauli blocking e ec s in his case.
8
10-1
100
dσ/dΩ (mb/s )
18C(d,p)19C(1/2+)
1
17 MeV/nucleon
(a)
0 10 20 30 40
θCM (deg)
100
101
dσ/dΩ (mb/s )
18C(d,p)19C(3/2+)
1
(b)
NAMD NoB
NAMD TB
NAMD PB
FIG. 7. Angula dis ibu ion o he 18C(d, p)19C eac ion
when he 19C g ound s a e 1/2+
1(uppe panel), and i s ex-
ci ed s a e 3/2+
1(lowe panel) a e popula ed. Calcula ions
ha e been made o 17 MeV/nucleon using he NAMD model
wi h di e en blocking me hods.
V. SUMMARY AND CONCLUSIONS
A new wo-body model based on he combina ion o
Nilsson and PAMD models om Re . [6], has been ap-
plied o he s udy o weakly bound exo ic nuclei. The
model conside s a neu on mo ing in a de o med po en-
ial gene a ed by he co e. The in e ac ion includes semi-
mic oscopic coupling po en ials calcula ed using AMD
ansi ion densi ies [5], a spin-o bi po en ial, and a col-
lec i e o a ional e m. By inco po a ing he BCS o -
malism in he model, di e en Pauli blocking me hods
a e used, and hei esul s a e compa ed. The e o e, we
ha e co e+nucleon models sui able o applica ion o he
s udy o eac ions, p o iding a e y comple e desc ip-
ion o he nuclei, since hey conside he de o ma ion o
he co e including mic oscopic in o ma ion, he collec i e
o a ion, and he pai ing e ec s in case o PB. The mod-
els ha e been applied o he desc ip ion o 17C, 19C and
ans e eac ions ha in ol e hese nuclei: 16C(d, p)17C,
17C(p, d)16C, 18C(d, p)19C.
Al hough all a ian s o he NAMD model gi e a
easonable desc ip ion o 17C, including Pauli blocking
me hods imp o es he ag eemen wi h he da a o he
s ipping eac ion 16C(d, p)17C(5/2+
1). Ou esul s a e
compa ed wi h hose ob ained using mic oscope RGM
calcula ions [24], which a e in good ag eemen wi h he
expe imen al da a. I should be no ed ha using a much
simple model such as NAMD, i we include blocking e -
ec s, he ag eemen wi h he da a is simila o ha ound
wi h his mic oscopic model. Blocking he Nilsson s a e
[220 1/2] (see Fig. 1) which is domina ed by he s1/2
componen , implies a educ ion o he s1/2⊗2+s eng h
in he second exci ed s a e 5/2+
1, inc easing he weigh
o he componen d5/2⊗0+. This is e lec ed in he SF
shown in Table I, o in he comple e ⟨17C(5/2+
1)|16C(0+)⟩
o e laps shown in Fig. 3. The inc ease in such o e lap
explains he e o e he be e ag eemen wi h he expe i-
men al da a o 16C(d, p)17C(5/2+
1) as compa ed o he
calcula ion in which Pauli blocking is no aken in o
accoun . TB and PB me hods gi e goods esul s in
16C(d, p)17C eac ion, bu hey p edic he spec um o
he bound s a es in a di e en o de han ha obse ed
expe imen ally. No ice, howe e , ha di e ences in en-
e gies a e e y small. The ene gy misma ch is wo se in
he case o PB, and his could be indica i e ha a be -
e implemen a ion o pai ing would be needed. Blocking
me hods also d as ically educe he s1/2s eng h in he
g ound s a e, and i can be seen in he ans e eac ion
17C(p, d)16C(2+). Ob aining expe imen al da a o his
eac ion would help de e mine whe he he g ound s a e
o 17C is s ongly domina ed by ℓ= 2 componen s as
sugges ed by he TB and PB models, o i he e is a sig-
ni ican con ibu ion om ℓ= 0 as sugges ed by he NoB
and he mic oscopic RGM model.
In case o 19C, an ℓ-dependen eno malisa ion is
needed o easonably desc ibe he sys em using he
NAMD model. This limi a ion o ou model can be e-
la ed o he assump ion o a p ola e de o ma ion, when
he possibili y ha p ola e and obla e s uc u es a e al-
mos degene a e (shape co-exis ence) has been p e iously
discussed [30]. Using he ℓ-dependen eno malisa ion,
he NAMD TB model ob ains a spec um up o 1.5 MeV
ha ag ees wi h he expe imen al one, he ag eemen
no being so good in he case o he NoB model. In-
cluding pai ing, he PB model e en imp o es he ag ee-
men . Howe e , he e a e no majo di e ences in he
⟨19C|18C(0+)⟩o e laps ob ained wi h he di e en mod-
els, as e lec ed in he SF and he c oss sec ion o he
eac ion 18C(d, p)19C. I should be no ed ha expe i-
men al measu emen s o his ans e eac ion would help
con i m o ule ou he shape coexis ence in 18C and 19C.
The esul s shown he e p o e he sui abili y o he
NAMD model o desc ibe he 17C and 19C sys ems. Fu -
he mo e, i has been p o en ha blocking o Nilsson
s a es occupied by he co e nucleons has signi ican e ec s
ha imp o e he desc ip ion o he nuclei. Al hough he
pa ial blocking me hod (PB) is mo e comple e, he limi
o o al blocking (TB) p o es o be a good app oxima-
ion in he cases s udied. The applica ion o hese models
o ans e o he con inuum and b eakup eac ions is in