Semi-microscopic folding model for the description of two-body halo nuclei

0.28 MB·

by  J. A. Lay

Semi-microscopic folding model for the description of two-body halo nuclei
J. A. Lay,∗ A. M. Moro,† and J. M. Arias‡
Departamento de FAMN, Facultad de F´ısica, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain
Y. Kanada-En’yo§
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan
(Dated: January 21, 2014)
One-neutron halo nuclei, composed by a weakly-bound particle coupled to a core nucleus, are
studied within a particle-plus-core model. A semi-microscopic method to generate the two-body
Hamiltonian of such a system, including core excitation, is proposed. The method consists in
generatingthespin-independentpartofthevalence-coreinteractionusingasingle-foldingprocedure,
convolutingarealisticnucleon-nucleon(NN)interactionwiththecoretransitiondensities. Thelatter
4 are calculated with the Antisymetrized Molecular Dynamics (AMD) method. The prescription is
1 appliedtothewellknownhalonucleus,11Be,asatestcase. Theresultsshowanimportantpredictive
0 power that opens a door to the understanding of other lesser known halo nuclei. In order to show
2 thepotential usefulness of the method,it is applied to analyze the structureof 19C.

n
a PACSnumbers: 21.10.Jx,21.60.-n24.10.Eq,27.20.+n
J

0
I. INTRODUCTION approaches [6–11]. Furthermore, in many of these appli-
2
cations, the valence-core potential is approximated by

] One of the main topics in Nuclear Physics during the a simple phenomenological potential, with the param-
h
t last years is the study of nuclei far off the stability line. eters adjusted to reproduce the low-lying spectrum of
- For these nuclei the ratio of protons to neutrons is quite the composite nucleus. This simple picture can be im-
l
c different from the usual ratios for stable nuclei. Because provedincluding some excitedstatesofthe core nucleus.
u of that, they are known as exotic nuclei. New physics These configurations are naturally included in micro-
n is expected for these nuclei since, for instance, new close scopic approaches, such as the shell-model and ab-initio
[
(sub-)shells could appear affecting to both the structure approaches. Within an effective two-body Hamiltonian,

1 ofthe nucleusanditsbehaviorwhenparticipatinginnu- these core-excited components are usually included as-
v clear reactions. Among the observed exotic nuclei, spe- sumingacollectivemodelforthe core nucleus(e.g.rotor
0 cial interest has been devoted to halo nuclei. These are orvibrator)givingrisetothe so-calledparticle-rotor[12]
9
weakly-bound systems composed by one or two weakly- or the particle-vibrator [13, 14] models. In these mod-
9
4 bound nucleons orbiting a relatively compact core. In els, in addition to the central potential, the valence-core
. theextremeweak-couplinglimit,itiscommonlyassumed interaction contains some non-central term, which is re-
1 that the properties of this composite system are mainly sponsible for the coupling between different core states
0
4 determinedbythedegreeoffreedomoftheweakly-bound and givesrise to core-excited admixtures in the states of
the composite system. In practice, this is usually done
1 nucleon(s), commonly referred to as halo.
: Inthis workweconcentrateintwo-bodyhalosystems. adding a transition potential with some phenomenologi-
v
calradialshapeandastrengthdependingonsomecollec-
In the simplest approach, the halo particle is assumed
i
X to move in a spherical mean-field potential generated by tiveparameter. Theparametersforthecentralandtran-
sition potentials are usually determined from the known
r the remaining nucleons, leading to a description of the
a properties of the composite system and, consequently,
levels of the composite system in terms of single-particle
require some a priori knowledge of the properties of the
orbitals. This simple picture is atthe basis ofmanyfew-
system,suchasthe energyexcitationandspin-parityas-
bodyreactionformalismsusedintheanalysisofreactions
signment of the low-lying states. This restricts the pre-
induced by halo and other weakly-bound nuclei, such as
dictive power of these models. Moreover,since the rotor
the Continuum-Discretized Coupled-Channels (CDCC)
and vibratormodels are expected to be limiting cases,it
method [1], the adiabatic approximation [2, 3], the Fad-
is not guaranteed their accuracy in specific cases.
deev/AGSequations[4,5],andavarietyofsemi-classical
To overcome these limitations, it would be desirable
to construct a particle-plus-core model starting from
more fundamental principles. Structure models based
∗ [email protected]; Present address: Dipartimento di Fisica e Astrono- on microscopic many-body calculations are also poten-
mia,Universit`adiPadova, I-35131Padova,Italy
tiallyusefulmethodsbuttheirapplicationstohalonuclei
IstitutoNazionalediFisicaNucleare,SezionediPadova,I-35131
witha deformedcore arestilllimited[15]. Alternatively,
Padova,Italy
† [email protected] whensomepropertiesofthecore nucleusareknown(rms
‡ [email protected] radius, excitation energies, etc) one can make use of a
§ [email protected] semi-microscopic picture, and construct the interaction
2
betweenthevalenceparticleandthecore nucleusbyfold- where φ (ξ) denotes the core eigenstates, ~ℓ is the or-
I
ing a suitable nucleon-nucleon effective interaction with bital angular momentum between the valence particle
an appropriate core density, following the same scheme and core, which couples to the spin of the valence par-
used in the calculation of folding potentials for elastic ticle (~s) to give the particle total angular momentum~j.
and inelastic scattering [16]. Thelabelαdenotesthesetofquantumnumbers ℓ,s,j .
It is our goal in this work to apply this idea to calcu- TheradialfunctionsRJ (r)canbedeterminedin{severa}l
ε,α
late the energies and wavefunctions of the states of one- ways. Acommonprocedureistoinserttheexpansion(2)
neutronhalonuclei. The aforementionedfoldingmethod intotheSchr¨odingerequation,givingriseto asetofcou-
is applied to generate the spin-independent part of the pleddifferentialequationsfortheradialfunctionsRJ (r)
ε,α
particle-core interaction. Bothcentralandtransitionpo- (see e.g. [12]).
tentials are calculatedwithin this scheme, making use of In this work,insteadof this coupled-channels method,
theappropriatemonopoleandtransitiondensities. Inthe we use the so-called pseudo-state (PS) method. This
calculations presented in this work, the nucleon-nucleon method consists in diagonalizing the Hamiltonian in a
interactionofJeukenne,LejeuneandMahaux[17]isused finite basis of square integrable functions. This basis is
as effective interaction, whereas the core densities are chosen of the form
calculatedhereusing the antisymmetrizedmoleculardy-
φTHO (~r,ξ)=RTHO(r) (rˆ) φ (ξ) (3)
namics (AMD) method [18]. A phenomenological spin- n,α,J,M n,ℓ Y(ℓs)j ⊗ I JM
orbit part, with standard parameters, is also added to (cid:2) (cid:3)
whereRTHO(r)areasetofsquare-integrableradialfunc-
the model Hamiltonian. It should be noted that the n,ℓ
tions. For the latter, we use the analytic Transformed
method can be equally applied with any other appropri-
Harmonic Oscillator (THO) basis, which is obtained by
ate nucleon-nucleon interaction and/or different method
applyingalocalscaletransformation(LST)tothespher-
to extract the core transition densities.
ical HO basis as
The paper is structured as follows. In Sec. II the
particle-core Hamiltonianisdefinedandthemethodused ds
toobtainthesolutions(eigenfunctionsandeigenenergies) RnT,HℓO(r)=rdrRnH,Oℓ [s(r)], (4)
of this Hamiltonianis explained. In Sec.III, the method
is applied to 11Be and 19C. Finally, Sec. IV is devoted whereRHO(s)(withn=1,2,...)istheradialpartofthe
n,ℓ
to discuss and summarize the main conclusions of this HOfunctionsands(r)definestheLST.Forthelatterwe
work. use the analytical prescription of Karataglidis et al. [19]
1 1 m 1 m −m1
s(r)= + , (5)
II. CORE EXCITATIONS IN THE STRUCTURE √2b(cid:20)(cid:18)r(cid:19) (cid:18)γ√r(cid:19) (cid:21)
OF TWO-BODY HALO NUCLEI
that depends on the parameters m, γ and the oscillator
A. Particle-core model length b. In Ref. [19], it was stated that the LST de-
pends very weakly on m and they suggested the value
m = 4. In this work we adopt this value, so the only
We consider a composite nucleus, described as a two-
active parameters in the LST are b and γ. The ratio
body system, comprising a weakly-bound nucleon cou-
γ/b determines the range of the basis functions and the
pled to a core. In the weak-coupling limit, the Hamilto-
density of eigenstates as a function of the excitation en-
nian of the system can be written as
ergy: as γ decreases, the basis functions explore larger
distances and the corresponding eigenvalues concentrate
=h (ξ)+T(~r)+V (~r,ξ), (1)
H core vc at lower excitation energies. Further details are given in
Refs. [20, 21].
where T(~r) is the kinetic energy operator for the rela-
The eigenstates of the Hamiltonian (1) will be ex-
tive motionbetween the valence andthe core, h (ξ) is
the Hamiltonian of the core and Vvc(~r,ξ) is theceoffreective pressed as an expansion in the THO basis, Φi(,NJ) =
valence-core interaction. Thevariableξ denotetheinter- N Ci φTHO, where N is the number of ra-
n=1 α n,α,J n,α,J
nal coordinates of the core. The dependence of Vvc(~r,ξ) dPial fuPnctions retained in the THO basis, i is an index
on these coordinates account for core-excitation effects. labeling the eigenstates for agivenJ, andCi arethe
n,α,J
TheeigenfunctionsofthisHamiltonian,foragivenen- expansion coefficients in the truncated basis. The nega-
ergy ε, would be characterized by the total angular mo- tiveeigenvaluesoftheHamiltonian(1)areidentifiedwith
mentum J~, resulting from the coupling of the angular the energies of the bound states, whereas the positive
momentum~j of the valence particle to the core angular ones provide a discrete representation of the continuum
momentum I~. These functions can be generically ex- spectrum. For small values of N, some of the positive-
pressed as energy eigenvalues become stable with respect to small
changes of N or of some non-linear parameter of the ba-
Ψ (~r,ξ)= RJ (r) (rˆ) φ (ξ) , (2) sis (e.g. γ). These stabilized energies are identified with
ε;JM ε,α Y(ℓs)j ⊗ I JM the resonances of the system [22, 23].
Xα (cid:2) (cid:3)
3
B. Matrix elements of the Hamiltonian in the PS where v is the effective NN interactionandρ(r~′,ξ) the
nn
basis density operator, defined as usual as
A
ThediagonalizationoftheHamiltonian(1)requiresthe
ρ(~r,ξ)= δ(~r ~r ). (13)
evaluation of the matrix elements of the potential V in − i
vc Xi=1
thePSbasis,denotedinketformas n(ℓs)jI;J . Forthis
| i
purpose, it is convenient to separate the angular part by This is conveniently expanded in multipoles as
performingamultipoleexpansionofthisinteraction,i.e.:
ρ(r~′,ξ)= ρ (r′,ξ)Y∗ (rˆ′). (14)
λµ λµ
Vvc(~r,ξ)= Vλµ(r,ξ)Yλ∗µ(rˆ)= Vλ(r,ξ)·Yλ(rˆ). (6) Xλµ
Xλµ Xλ
Notethat,inthesphericalcase,ρ(r~′)=ρ(r′),andV (r)
vc
Then, following the convention for matrix elements used becomesacentralpotential,soitcontainsonlytheλ=0
in Brink and Satchler [24], we obtain for each λ: term. Inamoregeneralcase,asweconsiderhere,ρ(r~′,ξ)
containsalsonon-centraltermsthatwillgiverisetotran-
n(ℓs)jI;J V (r,ξ) Y (rˆ) n′(ℓ′s)j′I′;J = sition terms with λ>0 in the valence-core potential.
λ λ
h || · || i
( 1)(j′+I+J)(2I+1)1/2 nℓI V (r,ξ) n′ℓ′I′ According to Eq. (7), one requires the reduced ma-
× − h k λ k i trixelementsoftheVvc interactionbetweendifferentcore
(2j+1)1/2 j j′ λ (ℓs)j Y (ℓ′s′)j′ , states. Inthefoldingscheme,thesewillbe relatedtothe
× (cid:26) I′ I J (cid:27)h k λk i matrixelementsofthedensityoperatorbetweendifferent
(7) core states, i.e.
where A
Iν ρ(~r,ξ)I′ν′ = φIν(ξ) δ(~r ~ri)φI′ν′(ξ)
h | | i h | − | i
hnℓIkVλ(r,ξ)kn′ℓ′I′i=ZI dVrλ(rr2,Rξ)nJℓI∗(′r,)RnJ′ℓ′(r) (8) =Xλ,µhI′ν′λXi=µ1|Iνiρλ,I′→I(r)Yλ∗µ(rˆ) (15)
×h k k i
whichcontainsthe dependenceofthe assumedmodelfor where ρλ,I′→I(r) correspond to the reduced matrix ele-
the core. For example, in the rotor model, these matrix ments
elements read (see e.g. [12, 21, 25, 26])
ρλ,I′→I(r) I ρλ I′ . (16)
≡h || || i
I Vλ(r,ξ) I′ =Vλ(r)( 1)I−I′ IKλ0I′K , (9) Our convention for reduced matrix elements is that of
h || || i − h | i
BrinkandSatchler[24]sothatthe reverseddensities are
where K is the projection of the angular momentum on related as √2I′+1 I′ ρ I =√2I+1 I ρ I′ .
λ λ
the core symmetry axis (usually K = 0 for the lowest h || || i h || || i
The density operator can be analogously defined for
energy levels in even-even systems). Each Vλ(r) reads: protons and neutrons (ρ(p) and ρ(n)), in which case the
sum in Eq. (13) runs over protons or neutrons, respec-
V (r)= dΩ V(r R(Ω))Y (θ,0), (10) tively. The corresponding monopole transition densities
λ λ0
Z −
are normalized as
R(Ω)=R + δ Y (θ,0), (11)
0 λ λ0
λX≥2 Z d~r ρ(0p,I)→I(r)Y00(rˆ)=Z, (17)
where usually we use a typical Woods-Saxon form for
(n)
d~r ρ (r)Y (rˆ)=N. (18)
V(r R(Ω))andonlyconsiderλ=0,2,beingδ2 =β2R0 Z 0,I→I 00
−
the deformation length of the core.
For the protoncase, the multipole terms are constrained
by the electric transition probabilities, i.e.:
C. Folding model for the valence-core interaction 2I+1 2
(Eλ,I′ I)= e2 dr rλ+2ρ(p) (r) (.19)
B → 2I′+1 (cid:12)Z λ,I′→I (cid:12)
Inthiswork,weproposeasimplesemi-microscopicpre- (cid:12) (cid:12)
(cid:12) (cid:12)
scription, in which the V (~r,ξ) interaction is calculated Thecore transitiondensitiesc(cid:12)anbeobtainedwithd(cid:12)iffer-
vc
bymeansofafoldingprocedure,convolutinganeffective ent methods. In this work, these densities are obtained
in-medium NN interaction with microscopic transition from antisymmetrized molecular dynamics (AMD) [27,
densities of the core nucleus, i.e. 28] calculations for the core nucleus. This method is
a microscopic structure model based on effective nuclear
interactionsinwhichtheantisymmetrizationbetweennu-
V (~r,ξ)= dr~′ρ(r~′,ξ)v (~r r~′). (12)
vc Z nn − cleons is fully taken into account. AMD wavefunctions
4
are formed from Slater determinants of single-nucleon III. APPLICATION TO HALO NUCLEI
Gaussian wavefunctions. Namely, many-body wavefunc-
tions are treated without assuming existence of any spe- A. Structure of 11Be
cific clusters in the method. Nevertheless, the AMD
model space covers a variety of cluster structures and
As a test example of the formalism presented in the
it can describe those of neutron-rich nuclei. Actually,
preceding section, we first consider the well known one-
the method has been proved to be very useful to un-
neutronhalo nucleus 11Be. Although the low-lyingspec-
derstand the level structure and deformation of Be and
trum of this nucleus is reasonably described in terms
B isotopes [18]. In the application to Be isotopes, it
of single-particle configurations, it is known that these
was shown that the structure of low lying states can be
statescontainsignificantadmixturesofcore-excitedcom-
described as two alpha clusters and remaining neutrons
ponents. To account for these components, within a
around the two alphas as proposed by Von Oertzen [29].
particle-plus-core picture,severalmodelshavebeenused,
Following[16],thecentralpartoftheeffectivenucleon-
such as the particle-vibrator (PVM) model [13, 14] and
nucleon interaction (v ) is decomposed in terms of the
nn the particle-rotor (PRM) model [32, 33]. Pairing effects
total spin (S) and isospin (T) of the colliding pair but,
have also been treated approximately within the quasi-
for simplicity, only the S =0 terms are considered,
particle rotor (QPRM) and quasi-particle vibrator mod-
els (QPVM) [34].
vnn(s)=v00(s)+v01(s)τ~′ ~τ, (20) Here, we compare the semi-microscopic approachpro-
·
posed in this work with the particle-rotor model (PRM)
where vST are the expansion terms and τ is the isospin developed by Nunes et al. [33, 35] (model Be12-b). This
operator. Attending to the isospin dependence, the model accounts well for the energies of the bound states
v00 and v01 terms are called, respectively, isoscalar and and low-lying resonances 3/2+, 5/2+ and 3/2− and has
1 1 1
isovector parts. The radial forms v0T(s) are taken from beenpreviouslyemployedtoillustrate the use ofthe PS-
the work of Jeukenne-Lejeune-Mahaux (JLM) [17] THO basis described above [21].
The required transition densities in 10Be are obtained
v0T(s,ρ,E)=λvVT(ρ,E)(tv√π)−3exp(−s2/t2v), (21) from the antisymmetrized molecular dynamics (AMD)
calculation of Ref. [27]. The AMD model is able to re-
where the strength of the potential, VT, depends on the produceE2transitionprobabilitiesbetweenthedifferent
density ρ, and the nucleon-nucleonrelative energy E. In 10Be energy levels with a substantial improvement with
this case, for simplicity, we choose E = 0. On the other respect to the shell model calculations. In addition, the
hand, normalizations factors, λv, and the effective range central and transition potentials calculated with these
of the Gaussian form factor, tv, are adjustable parame- densities and the JLM potential are able to reproduce
ters with typical values between 0.8 and 1.2 for λv, and the p+10Be inelastic cross sections at intermediate ener-
between 1.2 and 1.4 for tv. This interaction has been gies[31]withintheDWBAframework. Thisparticle-core
found to reproduce satisfactorily the elastic and inelas- folding potential based in AMD transition densities will
ticexperimentalcrosssectionsintheintermediateenergy be referred to as P-AMD.
region for light nuclei [30, 31]. In our calculation, we will only consider valence con-
InordertoevaluateEq.(12)we alsoexpandthe inter- figurations with ℓ 2 and the two lowest lying states in
action in multipoles as we did for the density: 10Be, the 0+ groun≤d state and the 2+ first excited state.
Allpossibletransitiondensitiesbetweenthesecore states
v (~r r~′ ,ρ,E)= v(ℓ)(r,r′)Y (rˆ) Y (rˆ′). (22) are plotted in Fig. 1. Convoluting the JLM interaction
0T | − | 0T ℓ · ℓ
Xℓ (λv = 1.0 and tv = 1.2) with these densities, the poten-
tials shown in Fig. 1 (right panels) are obtained. The
In the test cases considered in this work, the valence ρ density and the corresponding 2+ V 2+ po-
0,2+→2+ 0
h k k i
particleisaneutron,inwhichcasetheresultingpotential tentialareincludedinthe calculationsalthoughtheyare
can be expressed in terms of the corresponding proton not plotted in Fig. 1. In the rotor model this poten-
and transition densities as [30]: tial coincides with the central 0+ V 0+ one. In our
0
h k k i
P-AMP model both potentials are almost identical as
I V (r,ξ~) I′ = dr′ r′2 v(λ)(r,r′) so do the corresponding densities, confirming that the
h || λ || i Z n 00 rotor assumptions are satisfied for the 10Be core. The
ρ(n) (r)+ρ(p) (r) potentials have been calculated with the code MINC by
×h λ,I′→I λ,I′→I i M. Takashina [36].
+v(λ)(r,r′) In addition to the central term, the n-10Be interac-
01
tion will contain spin-dependent parts. For simplicity,
× hρ(λn,I)′→I(r′)−ρ(λp,I)′→I(r′)io.(23) weconsideronly the spin-orbitterm, forwhichwe adopt
the phenomenological parameterization of the potential
Note that, ifthe valenceparticle isa proton,the signsin Be12-b with a standard strength V =6 MeV.
so
the isovector part are changed. The calculated spectrum is shown in Fig. 2 (second
5
0.3 ρ(p) (0+→0+) 0 bined effect of the core deformation, Pauli blocking and
-3fm) 0.2 ρ00(n) (0+→0+) MeV)-20 TISotal pnaoirreidngineffoeucrtstr[e1a4t]m. enPtaiwrihnegreeaffsecPtasualireblcoocmkipnlgetieslyonigly-
ρ (00.1 (a) V (-40 IV(d) ccoannsniodtereexdpeacptparnoxaicmcautrealtye(dseesecrdipistciounssoiofnthbeeleoxwp)ersimoewne-
00 0
(b) (e) tal spectrum. To account in an effective way for these
-3m) -0.02 eV) -5 effects a slightrenormalizationof the folding potentialis
ρ (f2-0.04 ρρ2((pn)) ((00++→→22++)) V (M-10 Non-diag. ainllgowofedt.heInmoenrdtieornteodrsetpatreosd,utcheetrheenoerxmpearliimzaetniotnalfaocrdtoerrs-
2 need to be different for positive (λ ) and negative (λ )
0 + −
(c) Reor. (f) parity states: λ = 1.054 and λ = 0.995. The new
-3m) -0.02 eV)10 spectrum is also+shown in Fig. 2−(third column). The
ρ (f2-0.04 ρρ2((pn)) ((22++→→22++)) V (M 5 preoassiotinoanbolyftwheellreresopnroadnucecsed3./2O−n,ly5/t2h+e a5n/2d−3/r2es+onaarencneowis
-0.06 2 0 not well reproduced by the model. Considering that the
0 2 4 6 8 0 2 4 6 8
r (fm) r (fm) onlyadjustableparametersareλ±,andthattherequired
normalizations are of the order of 5%, we can conclude
FIG. 1. (Color online) Neutron and proton transition densi- that the overall agreement is fairly good. It is worth
tiesforthetwostatesconsideredin10Be,the0+ groundstate mentioning that the PRM model requires also a weaker
and the 2+ first excited state. The central densities, λ = 0, strength for negative parity states to obtain the inver-
are shown in the upper-left panel normalized according to sion. This fact can be related with the Pauli exclusion
(18)andthequadrupoletransitiondensities,inthelower-left principle. The antisymmetrization of the wavefunctions
panels. Thecorresponding transition potentials areshown in
should add an extra repulsion to p configurations, re-
the right panels, with the isoscalar (IS) and isovector (IV) 1/2
pulsion that was added phenomenologically by reducing
contributions indicated separately.
the strength of the negative parity potentials [39].
It is also worth noting that, in addition to the levels
Experimental P-AMD P-AMD PRM
shown in Fig. 2, some other deeply bound eigenvalues
4 (λ+=λ-=1) (λ+=1.05, λ-=0.995)
are obtained in the diagonalization. These are identified
5/2- 3/2+ 3/2+ with Pauli forbidden states and are therefore removed.
3 3/2+ 3/2+ 3/2- These states come from the 1s1/2, 1p3/2 and 1p1/2 or-
bitalsinthesphericalbasis,whicharealreadyoccupiedin
V) 3/2- 3/2- 3/2- 5/2- the 10Benucleus accordingto our simple modelinwhich
2
e 5/2+ exchange and pairing effects are ignored. By construc-
M
E (1 5/2+ 5/2- 55//22-+ 5/2+ tmioonv,edth,ewshtaattesshooubltdaiancecdouhnertefoarrethoertPhaougolinparlintocitphleo.seTrhee-
forbidden states are therefore an admixture of different
0 1/2- 1/2+ 1/2- 1/2- valence+core configurations. Note that, with this proce-
1/2+ 1/2- 1/2+ 1/2+ dure,thepartofthesingle-particlestrengthsofthe1s1/2,
1p and1p orbitalswillappearembeddedamongthe
3/2 1/2
retained valence+core states. Alternatively, Pauli prin-
ciple could be applied by removing these spherical va-
lence configurations completely, as done for example in
FIG. 2. (Color online) Experimental and calculated energy
Refs. [40, 41]. Both methods are indeed approximate.
levelsof11Be. Startingfromtheleft,thesecondcolumnisthe
Since in the cases treated in this work the core is de-
P-AMD calculation without any renormalization. The third
formed, we follow the former approach. This election
column is the P-AMD spectrum obtained with the indicated
slightly affects the energiesand spectroscopicfactorsob-
renormalizationconstantsforpositive(λ+)andnegative(λ−)
parity levels. The last column corresponds to the PRM cal- tained.
culation. Experimental values are from [37, 38]. In Fig. 2, we can see that most of the low lying struc-
tures in 11Be can be understood within the P-AMD
model. ThespectraprovidedbyP-AMDandPRMmod-
column). Resonantenergiesareidentifiedwithstabilized els[21]arecompatible. P-AMDgivesexcitationenergies
eigenvalues with respect to variations in the number of forthepositiveparityresonances5/2+,and3/2+ inbet-
states included in the THO basis [21]. The P-AMD cal- teragreementwithexperiment,althoughamajorpartof
culation succeeds to produce two weakly bound states this effect is related to the spin-orbit term. P-AMD also
(1/2+ and 1/2−), in agreement with experiment, but improves the excitation energy for the negative parity
with the wrong ordering. Several low-lying resonances resonance 3/2−.
(5/2± and 3/2±), are also predicted. The inversion of In addition to the energies of the bound and resonant
the 1/2+ and 1/2− levels has been ascribed to a com- states,wehavecomparedthe weightsofthe variousrele-
6
0
TABLEI.Spectroscopicfactorsforthegroundstateandlow-
(a)
lying positive energy resonances in 11Be, according to the
different models considered.
-20
)
V
e PRM
M P-AMD
V ( Model |0+⊗(ℓs)ji |2+⊗s1/2i |2+⊗d3/2i |2+⊗d5/2i
-40 1/2+ PRM 0.857 – 0.021 0.121
+ +
<0 |V |0 >
0 P-AMD 0.849 – 0.031 0.121
WBT 0.762 – 0.002 0.184
-60
0 2 4 6 8 5/2+ PRM 0.702 0.177 0.009 0.112
20 P-AMD 0.674 0.189 0.014 0.124
(b)
+ + WBT 0.682 0.177 0.009 0.095
<2 |V |2 >
10 2 3/2+ PRM 0.165 0.737 0.017 0.081
) P-AMD 0.316 0.565 0.031 0.089
V
e WBT 0.068 0.534 0.008 0.167
M 0 <2+|V |0+>
( 2
V
-10
where I δˆ I′ is the reducedmatrixelementofthe de-
λ
h k k i
formation length operator for a given transition. Multi-
-20
0 2 4 6 8 plyinginbothsidesbyr2andintegratinginroneobtains
r (fm)
FIG. 3. (Color online) Potentials obtained for the n-10Be hIkδˆλkI′i= λ+1 2R ρλ,ρI′→(rI)(rrλ)+r1λd+r2dr. (25)
0
system with the PRM model and with the P-AMD model.
R
See text for details. Using the microscopic AMD densities employed in this
work, one gets 0 δˆ 2 = 1.90 fm. Interestingly,
2 AMD
h k k i
this value is very close to that obtained by Iwasaki et
vantcomponents(channels)forthedifferentmodelscon-
al from a DWBA analysis of 10Be(p,p’) inelastic data,
sideredhere. Withinourassumedtwo-bodymodel,these
1.80 0.25 fm [44]. On the other hand, in the rotor
weights can be regarded as spectroscopic factors. In Ta- ±
model, these matrix elements are related to the intrinsic
bleIthesespectroscopicfactorsareshownforthepositive
parity states in 11Be calculated for: the P-AMD model, deformation as
the particle-rotor model (PRM), and a shell model cal- I δˆ I′ = IKλ0I′K β R . (26)
λ λ 0
culation with the WBT interactionfrom Warburtonand h k k i h | i
Brown [42]. The latter was performed with the code Inserting the mean radius and deformation parameter
oxbash [43]. The three calculations give slightly dif- employed in our calculations (R0 = 2.483 fm, β2=0.67)
ferent but compatible spectroscopic factors. In particu- one gets 0 δˆ 2 = 1.66 fm, which is somewhat
2 rot
lar,thesemodelsagreeinthedominanceofthe10Be(0+) smaller thahnkthekAiMD value (these values are also listed
component for the ground state and 5/2+ resonance, as in Table II for convenience). Consequently, the effective
well as in the dominant 10Be(2+) contribution in the deformation obtained from the AMD densities is larger
3/2+ resonance. thanthat assumedin the rotormodelusedhere andthis
The agreement between P-AMD and PRM values is explains the largermixing ofthe 10Be(2+) componentin
not unexpected in view of the similarity of the corre- the 11Be wavefunctions.
spondingtransitionpotentials. TheseareshowninFig.3. From the results presented in this subsection, we can
The main difference is that the P-AMD densities yield a concludethatthedevelopedP-AMDmodelgivesanover-
largerdeformationofthecore. InthePRMthisdeforma- all good description of the bound states and low-lying
tion is a parameter that was estimated from the experi- resonances in 11Be with very small adjustments of the
mentalquadrupolemomentofthe core,andcorrectedby parametersinvolvedinthe calculation. Theseresultsen-
thechargetomassdeformationratiogivenbyshellmodel courage us for using this model to make predictions for
calculations [35]. On the other hand, a deformation pa- the structure of poorer known halo nuclei. As an exam-
rameter can be inferred from AMD densities comparing ple, in the next subsection we apply the model to 19C.
the transition density with the derivative of the central
density. The relation between these two magnitudes, as-
suming that the rotor model is a good approximation, B. Structure of 19C
is
dρ Once tested the semi-microscopic model with the well
ρλ,I′→I(r) I δˆλ I′ 0, (24) known nucleus 11Be, we now consider the halo nucleus
≈h k k i dr
7
Experimental WBP P-AMD PRM(1) PRM(2)
TABLE II. Properties of the 10Be and 18C systems, derived 4 5/2+
frdraeodnmisuitstyh(.ermAsM)iDsoabntdairnoetdorfrcoamlcuthlaeticoonrsr.esTpohnedrionogtcmenetarnalsAquMarDe 3 13//22++ 3/2+1/2+ 1/2+ 1/2+
Nucleus rms h0kδˆ2k2iAMD h0kδˆ2k2irot MeV) 2 35//22++ 35//22++ 35//22++
(fm) (fm) (fm)
10Be 2.538 1.90 1.66 a E ( 1 5/2+ 5/2+
18C 2.776 1.20 1.50 b 3/2+ 5/2+ 5/2+ 5/2+
a RotormodelBe12-bofRef.[33]. 0 15//22++ 3/2+51//22++ 31//22++ 331///222+++ 31//22++
b RotormodelofRef.[32],referredinthisworkasPRM(1). -1
FIG.4. (Coloronline)Spectrumobtainedforthe19Cnucleus
within the two PRM calculations (PRM(1) and PRM(2)), a
19C.Thepropertiesofthisnucleusarenotsowellknown
shell model calculation (WBP), and with the single-folding
and the experimental data, needed to adjust the param-
calculation based on microscopic densities of the core (P-
eters involved in phenomenological models (as PRM or AMD) compared with theexperimental one [48, 52].
PVM), are scarceand, sometimes, contradictory. There-
fore,asemi-microscopicmodelasP-AMDcanshedsome
light on the structure of this nucleus. producethe tripletofboundstatesreportedbyElekeset
Although the properties of this nucleus, including the al. [52]aswellasthe5/2+ resonancesuggestedby Satou
low-lying spectrum, are not well known, it has been re- et al. [48] they need to introduce some Pauli hindrance
centlythefocusofseveralworks[45–49]. Itisknownthat of the 1d orbit. This introduces a phenomenological
5/2
the ground state has spin and parity 1/2+ and that the parameter in the model, which accounts for the amount
binding energy with respect to the 18C+n threshold is ofPauliblockingofagivenorbital. Clearly,thesituation
0.58 0.09 MeV [50]. Almost all theoretical calculations calls for further experimental and theoretical works.
predi±ct a 1/2+ (prolate) and a 3/2+ (oblate) almost de- Withtheaimofsheddingsomelightintothisproblem,
generate states [51]. In fact, the ground state spin and we presentour predictionfor the structure of 19C within
parity were not confirmed until recently [48]. The de- the semi-microscopic P-AMD framework. As in the
formation for all lighter carbon isotopes is known to be 11Be case, the neutron+18C folding potential was gener-
prolate. For 19C, prolate and oblate structures seem to ated with the JLM nucleon-nucleon interaction, and the
be almost degenerate (shape coexistence) anticipating a monopoleandtransitiondensities calculatedwithAMD.
kindofshapephasetransition. Itcanbeindicativeofthe These densities, and the correspondingtransitionpoten-
presence of a new magic number, N = 16, for neutron tials, areshownin Fig. 5. The centralfolded potentialis
rich nuclei [51]. supplemented with a phenomenological spin-orbit term,
In addition to the 1/2+ and 3/2+ bound states, the parameterized in terms of the derivative of a Woods-
analysis of the inelastic data of 19C on protons reported Saxon shape, with a standard strength Vso = 6.5 MeV.
in Ref. [52]suggestedthe existence of anotherbound ex- Thegeometryisadjustedtobeconsistentwiththeexten-
cited state. This was assigned a spin-parity 5/2+ with sionofthecentralpartofthefoldingpotential,obtaining
the guidance of shell model calculations. Later on, in Rso =3.0 fm and aso =0.70 fm.
a exclusive breakup experiment of 19C+p, a prominent Diagonalizing the Hamiltonian, the spectrum shown
peakwasobservedinthe relativeenergyspectrumofthe under P-AMD in Fig. 4 is obtained. The experimental
outgoing 18C and n particles. Using microscopic DWBA levels reported in Refs. [48, 52], with the spin-parity as-
calculationsbasedonshellmodeldensities,thisstatewas signment suggested in these works, are indicated in the
associated with a second 5/2+ state predicted by some left part of the figure. The prediction of two PRM cal-
shellmodelcalculations. Thisistheacceptedexperimen- culations,labeledasPRM(1)andPRM(2),areshownat
talknowledgeofthelow-lying19Cspectrumpresentedin the right part of the figure. PRM(1) is from Tarutina
the left part of Fig. 4. However, the experimental data and Hussein [32] that use a Woods-Saxon central poten-
are far from being clearly established. For example, a tial with parameters V = 52.30 MeV, V =6.5 MeV,
0 so
−
recent knockout experiment [53] seems to question the R =3.0fmanda =0.65fmandatransitionpotential,
0 0
existence of a 5/2+ bound state. From the theoretical obtained by deforming the central potential with a de-
point of view, the situation is also unclear. The quasi- formationparameterofβ =0.5 fm. Onthe other hand,
2
particle rotor model (QPRM) of Ref. [34] gives correctly the model PRM(2) is obtained by adjusting the Wood-
the1/2+ groundstate,butpredictsthatthe firstexcited Saxon parameters to reproduce the potential provided
state is 5/2+. In Ref. [49], the 19C was studied within a by the P-AMD model. The aim of this PRM(2) model
multi-channel algebraic method based upon a two-state, is to reproduce P-AMD results with simpler potentials
collective, model for the n+18C system. In order to re- in a more standard and widespread framework, so that
8
0.4 ρ(p) (0+→0+) 0 ofWarburtonandBrown[42]. Asexpected,thePRM(2)
-3ρ (fm)0000...231 ρ00(n) (0+→0+) V (MeV)--4200 TIISVotal ma osTedhpeealrrPae-tpAioroMndDeuncmeersogtdyheeolfpPrS-eAndMi=ctDsasεp1ges/c2t=+ru0gm.r5oq8u2unidtMesetwVaet,leli,.nweitxh-
(a) (d) cellent agreementwith the exp−erimental value. The first
0 5
ρ(p) (0+→2+) (e) excited state is 3/2+, also in agreement with the exper-
-3ρ (fm)2 0.002 ρ22(n) (0+→2+) V (MeV)-1-005 idsmteagetenentsaearlardetaetfoawu,itnahdlthtihnoeutgghhriosiunmnodoudsretlma,toec.doneNltrotahraiysddsttiotaitotehneiaslsabulogmugoenssdt-
-3ρ (fm)2--0000....000002224 ρρ22((pn)) ((22++→→((bc22))++)) V (MeV)-11100555 NRoeno-rd.iag(f.) atosthiupuoegptngaesteaphsorpatsfertodv[5txeh2irinmi]ys.a[c5sTtlt2oiaoh]s.tneeesAtfiiiosmrsstapethccleirtoeeuntdsahdolirnlney5as/ohna2uot+rlwdsmter.aaeoTtskdeoaleynklia,sibnnoaogncuee5inn/cidtsa2on+1osantb,acotwctateoihr,nuuiecnalhdest
-0.04 -5 in P-AMD at ε = 1.704 MeV, but this state does not
0 2 4 6 8 0 2 4 6 8
r (fm) r (fm) have a clear counterpart in the experimental spectrum.
No states are found close to the resonant peak observed
FIG.5. (Coloronline) Neutronandproton transition densi- by Satou et al. [48] which, again, could be attributed to
tiesbetweenthetwoconsideredlowestlyingstatesin18C,the the uncertainties of our model. The PRM(1) model pre-
0+ ground state and the 2+ first excited state. The central dictsalsotwoboundstateswithspinandparity1/2+and
densities,λ=0,areshownintheupper-leftpanelnormalized 3/2+,butwiththelatterbeinglowerinenergy. Asinthe
according to (18) and the quadrupole transition densities, in P-AMD model, the second 5/2+ state appears as a low-
thelower-leftpanels. Thecorrespondingtransitionpotentials
lyingresonanceclosetotheneutronseparationthreshold.
are shown in theright panels.
The transition potentials obtained with the P-
AMD, PRM(1) and PRM(2) models are compared in
0
Fig. 6. The central potentials are similar in the three
(a)
models, whereas the quadrupole transition potentials
( 2+ V 2+ and 2+ V 0+ ) are larger in the PRM(1)
2 2
V)-20 + + mhodekl. kThiis canhbekunkdersitood in terms of the cor-
e <0 |V |0 >
M 0 responding deformation lengths. The values computed
V ( withEqs.(25)and(26)arelistedinTableII.Inthiscase,
P-AMD the deformation predicted by the AMD model is smaller
-40
PRM(1) than the one assumed in the rotor model PRM(1), and
PRM(2) this explains the stronger transition potential in the lat-
ter case.
0 2 4 6 8
Despite slight disagreements in energy, all studied
modelsgivethe samespinandparityforthe fourlowest-
(b)
<2+|V |2+> lying states. In Table III the spectroscopic factors pro-
10 2
vided by the different models for these four levels are
) presented. Good agreement is found for all the states as
V
Me 0 <2+|V |0+> in 10Be. In the case of the two 5/2+ states, the ordering
2
( predicted by the PRM and P-AMD models seems to be
V
inverted with respect to the shell model prediction.
-10 It is worth mentioning that the original value of the
central potential depth in model PRM(1) was V =
0
42.95 MeV [32]. With this choice one obtains, a 1/2+
0 2 4 6 8 −
r (fm) state with the experimental separation energy and a
deeplybound1/2+ state,whichisconsideredtobePauli
FIG. 6. (Color online) Potentials obtained for the n-18C forbidden. In the calculations presented in this work,
system with thePRM and P-AMD models. we assume that there is an additional 1/2+ Pauli for-
bidden state, and hence the depth of the potential has
beenincreasedaccordinglyinorderthethird1/2+ eigen-
these results canbe more easilyreproducedandused for value has the experimental separation energy. The last
different purposes. From this adjustment it is obtained neutrons in 18C partially occupy the 2s and 1d or-
1/2 5/2
V = 51.80MeV,R =3.0fmanda =0.70fmandas- bits,sothatfullremovalofthesesphericalconfigurations
0 0 0
−
sumesadeformationparameterβ =0.4fm. Alsoshown mayproducemisleadingresults. Inordertoestimatethe
2
in Fig. 4, is the spectrum predicted by the shell-model number of Pauli forbidden states we follow the proce-
calculationperformedwiththeWBPeffectiveinteraction dure ofRef. [54], whichmakesuse ofthe strongcoupling
9
low-energy excitation of the system. In the 18C case the
TABLE III. Spectroscopic factors for the ground state and
five lowest Nilsson levels are fully occupied and blocked.
low-lying positive energy resonances in 19C, according to the
The extra 2 neutrons of the core can occupy partially
different models considered in this work.
[2113+], [2111+] and [2025+] but do not block com-
2 2 2
pletely these levels to the extra neutron in 19C. In or-
Model |0+⊗(ℓs)ji |2+⊗s1/2i |2+⊗d3/2i |2+⊗d5/2i der to calculate the Pauli forbidden states for the extra
1/2+1 P-AMD 0.529 – 0.035 0.436 neutron,[00021+],[11021−],[10123−],[10121−]and[22021+]
levels are blocked. Taking into account that we are only
PRM(1) 0.517 – 0.081 0.402
interested in positive parity states in 19C, and that we
PRM(2) 0.505 – 0.033 0.462
are including only the 0+ and the first excited state 2+
WBP 0.580 – 0.085 0.470
states of the core, the negative parity Nilsson orbitals
3/2+1 P-AMD 0.028 0.386 0.121 0.464 arenotimportantsincetheywillproducenegativeparity
PRM(1) 0.043 0.348 0.150 0.459 statesin19C.Consequently,the relevantblockedNilsson
PRM(2) 0.023 0.371 0.106 0.500 levelsare[0001+]and[2201+]. TheNilssonmodelcorre-
2 2
WBP 0.026 0.494 0.001 0.076 spondstoanadiabaticapproximationoftheparticle-core
5/2+ P-AMD 0.276 0.721 0.000 0.003 model in whichthe core states are assumedto be degen-
1
PRM(1) 0.285 0.716 0.000 0.003 erate in energy. Once the energy of the core states is
increased the K quantum number is no longer a good
PRM(2) 0.278 0.719 0.000 0.003
quantum number, but one can monitor how the orbits
WBP 0.383 0.015 0.000 0.751
characterizedbythisquantumnumbersplitsintoseveral
5/2+ P-AMD 0.200 0.142 0.002 0.657
2 non-degenerate states, characterized by the total angu-
PRM(1) 0.217 0.178 0.004 0.602
lar momentum J of the system. In particular, one gets
PRM(2) 0.207 0.100 0.002 0.690 two 1/2+ states (1/2+ 0+ from [0001+] and [2201+]),
WBP 0.035 0.609 0.009 0.291 two 3/2+ states, and t⊗wo 5/2+ states2(1/2+ 2+ 2from
[0001+] and [2201+]) that should be removed⊗. Thus, in
2 2
ourcalculations,the third 1/2+ state is consideredto be
the physical ground state of the 19C system. The fact
20
[2 0 2 3/2+ ] that, with this selection of Pauli forbidden states, the
1d3/2 [2 0 0 1/2+ ] groundstateenergy,itsspectroscopicfactorandthelevel
scheme,amongotherobservables,areingoodagreement
[2 0 2 5/2+ ]
2s [2 1 1 1/2+ ] with experimental data and shell model calculations as
1/2
Fermi level shownin Table III and Fig. 4 supports this procedure of
1d5/2 [2 1 1 3/2+ ] calculating the Pauli forbidden states.
[2 2 0 1/2+ ]
8
s
el
1p [1 0 1 1/2− ] ev IV. SUMMARY AND CONCLUSIONS
1p1/2 [1 0 1 3/2− ] ed l
3/2 2 [1 1 0 1/2− ] ock A semi-microscopic particle-plus-core description of
Bl one-neutron halo nuclei, considering excitations of the
1s [0 0 0 1/2+ ] core, has been presented. In this model, the neutron-
1/2
core interaction is constructed by folding an effective
Deformation nucleon-nucleon interaction with microscopic (monopole
and transition) densities of the core. For the former, the
FIG. 7. Schematic diagram for the lowest Nilsson levels rel-
JLM [17] interactionis chosen, whereas the densities are
evant for the calculation of the Pauli forbidden states in our
calculatedwith the Antisymmetrized MolecularDynam-
P-AMD model for 19C.
ics(AMD)method. Thefoldedpotentialissupplemented
with a phenomenological spin-orbit interaction.
The model has been applied to the well known halo
limit in the simple Nilsson model [12, 54]. Using the
nucleus 11Be (10Be+n) and to the less known halo nu-
asymptoticquantumnumbers[Nn ΛKπ],corresponding
z cleus 19C (18C+n). In the 11Be case, the model is able
to large prolate deformations, the relevant Nilsson levels
to reproduce the experimental spectrum using standard
for 18C (N=12) are: [0001+], [1101−], [1013−], [1011−],
2 2 2 2 parameters for the JLM interaction in the region of the
[2201+], [2113+], [2111+] and [2025+] (see scheme in nucleus. Small normalization factors, different for posi-
2 2 2 2
Fig. 7). In this extreme model the occupancy of each tive and negative parity states, are required in order to
level is two. Levels well below the Fermi level are com- reproducetheseparationenergiesofthetwoboundstates
pletely occupied and blocked, levels around the Fermi in 11Be. However, these normalization factors are close
level are only partially occupied and participate in the to one (λ =1.056and λ =0.995for positive and nega-
+ −
10
tiveparitystates,respectively)suggestingthatthissemi- ever we have a certain knowledge of the core, the model
microscopic description can have predictive power to canbeusedtopredictthespectrumofthecore+neutron
study less known nuclei with an important effect of core (or core+proton) composite system. Furthermore, since
excitations,suchas odd-evennucleiin deformedregions. itusesthesameTHOformalismasthePRMmodelused
The small difference in the renormalization factors can in [55], it will be easily included in reaction calculations
be understood considering the effect of Pauli repulsion including core excitations as we hope to show in future
in the negative parity states due to the nucleons that fill works.
the1p and1p states. Thegoodagreementbetween
3/2 1/2 An important issue for extending the application of
the spectroscopicfactorsfromparticle-rotormodel, shell
this model would be the correct application of the Pauli
model and this semi-microscopic description also sup-
principle. Herewehaveremovedthosefinaldeeplybound
ports the predictive power of this method.
eigenstatesthatweconsideredasoccupiedbycomparing
Based on these results, AMD densities for 18C [28]
with the spherical and Nilsson limits for each case. Dif-
have been used to obtain the spectrum, spectroscopic
ferences between removing these eigenstates, removing
factors and properties of the 19C nucleus. Very few ex-
pure spherical or Nilsson configurations, and more so-
perimentaldataareavailableforthishalonucleus. With-
phisticated treatments of the Pauliprinciple should lead
out any renormalization of the coupling potentials, the
to future developments of the present model.
model reproduces the spin-parity (1/2+) and the sep-
aration energy (Sn = 0.58 MeV) of the ground state. In this work, particular nucleon-nucleon interaction
It also predicts the existence of a bound 3/2+ state, in (JLM) and method to extract core transition densities
agreement with shell model calculations and experimen- (AMD) have been used. However, the formalism pro-
tal evidences. Two low-lying 5/2+ resonances have been posed is general and not linked to them. It can be
found. Thecalculatedspectroscopicfactorsforthesetwo equally applied with any other appropriate NN interac-
resonances suggest a possible inversion in the energy se- tion and/or model able to calculate core transition den-
quenceofthesestateswithrespecttoshellmodel. Apart sities.
from this exception, it should be emphasized that the ACKNOWLEDGMENTS
proposed semi-microscopic model is able to reproduce
an important part of the available experimental data for
11Be and 19C. This work has been partially supported by the Span-
The predictive power of the model makes it particu- ishMinisteriodeCienciaeInnovacio´nandFEDERfunds
larlyusefulforexoticnucleiforwhichscarceinformation under projects FIS2011-28738-c02-01, FPA2009-07653,
is available, such as 19C and other unknown nuclei. Al- FPA2009-08848 and by the Spanish Consolider-Ingenio
though we compare with particle-rotor models, the pro- 2010ProgrammeCPAN (CSD2007-00042)and by Junta
posedmodeldoesnotneedthe core to beneither arotor de Andaluc´ıa (FQM160, P11-FQM-7632). J.A.L. ac-
nor a vibrator. That makes the model far more general knowledgesaresearchgrantbytheMinisteriodeCiencia
and suitable for different regionsof exotic nuclei. When- e Innovacio´n.
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