Table Of ContentExtracting three-body breakup observables from CDCC calculations with core
excitations
R. de Diego∗
Centro de Ciˆencias e Tecnologias Nucleares, Universidade de Lisboa,
Estrada Nacional 10, 2695–066 Bobadela, Portugal
R. Crespo†
Departamento de F´ısica, Instituto Superior T´ecnico,
Universidade de Lisboa, Av. Rovisco Pais 1, 1049–001 Lisboa, Portugal
and Centro de Ciˆencias e Tecnologias Nucleares, Universidade de Lisboa,
Estrada Nacional 10, 2695–066 Bobadela, Portugal
7
A. M. Moro‡
1
0 Departamento de FAMN, Facultad de F´ısica, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain
2 (Dated: January 31, 2017)
n Background Core-excitation effects in the scattering of two-body halo nuclei have been investigated in previous works. In
a particular, these effects have been found to affect in a significant way the breakup cross sections of neutron-halo nuclei
J withadeformedcore. Toaccountfortheseeffects,appropriateextensionsofthecontinuum-discretizedcoupled-channels
0 (CDCC) method havebeen recently proposed.
3
Purpose Weaimtoextendthesestudiestothecaseofbreakupreactionsmeasuredundercompletekinematicsorsemi-inclusive
reactions in which only the angular or energy distribution of one of theoutgoing fragments is measured.
]
h Method We use the standard CDCC method as well as its extended version with core excitations (XCDCC), assuming a
t pseudo-statebasis for describing theprojectile states. Two- and three-bodyobservablesare computedbyprojecting the
-
l discretetwo-bodybreakupamplitudes,obtained withinthesereaction frameworks, ontotwo-bodyscattering stateswith
c
definiterelative momentum of theoutgoing fragments and a definitestate of the core nucleus.
u
n Results Ourworkingexampleistheone-neutronhalo11Be. Breakupreactionsonprotonsand64Zntargetsarestudiedat63.7
[ MeV/nucleon and 28.7 MeV, respectively. These energies, for which experimental data exist, and the targets provide
two different scenarios where the angular and energy distributions of the fragments are computed. The importance of
1
core dynamical effects is also compared for both cases.
v
6 Conclusions The presented method provides a tool to compute double and triple differential cross sections for outgoing
1 fragments following the breakup of a two-body projectile, and might be useful to analyze breakup reactions with other
6 deformedweakly-boundnuclei,forwhichcoreexcitationsareexpectedtoplayarole. Wehavefoundthat,whiledynamical
8 core excitations are important for the proton target at intermediate energies, they are very small for the Zn target at
0 energies around theCoulomb barrier.
.
1
PACSnumbers: 24.10.Eq,25.60.Gc,25.70.De,27.20.+n
0
7
1
I. INTRODUCTION the case of elastic breakup, that we address here, sev-
:
v eralfew-bodyformalismshavebeendevelopedtoextract
i
X thecorrespondingcrosssections: Continuum-Discretized
Recentexperimentalactivitieswithnucleiinthe prox-
Coupled-Channels (CDCC) method [2, 3], the adiabatic
r imity of the drip-lines have increased the interest in this
a approximation [4, 5], the Faddeev/AGS equations [6, 7],
region of the nuclear landscape. In particular, special
andavarietyofsemiclassicalapproximations[8–13]. The
attention has been paid to halo nuclei, spatial extended
CDCCmethodisbaseduponanexplicitdiscretizationof
quantum systems with one or two loosely bound valence
all channels in the continuum and requires the solution
particles, for example 11Be or 11Li. Breakup reactions
of an extensive set of coupling equations. It has been
haveshowntobeausefultoolforextractinginformation
applied at both low and intermediate energies.
fromtheseexoticstructures[1]. Reliableandwellunder-
The standard formulation of these approaches ignore
stoodfew-body reactionframeworkstodescribebreakup
reactions of exotic nuclei, such as haloes, are therefore thepossibleexcitationsoftheconstituentfragments: the
states of the few-body system are usually described by
needed. Particularly, it is timely to estimate what are
pure single-particle configurations, ignoring the admix-
the relevant excitations mechanisms for the reaction. In
tures of different core states in the wave functions of
the projectile. These admixtures are known to be im-
portant, particularly in the case of well-deformed nu-
clei, such as the 11Be halo nucleus [14]. In addition,
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] for such two-body weakly bound system, dynamic core
‡Electronicaddress: [email protected] excitation effects in breakup have been recently stud-
Typeset by REVTEX
2
ied with an extension of the Distorted Wave Born Ap- tions of the 10Be fragments resulting from the breakup,
proximation (DWBA) formalism within a no-recoil ap- for which new measurements have been made [26].
proximation [15–18] and found to be important. This The paper is organized as follows. In Sec. II we
method is based in the Born approximation and ignores briefly discuss the structure approach to describe two-
higher order effects (such as continuum-continuum cou- body loosely-bound systems with core excitation. In
plings). A recent attempt to incorporate core excitation Sec.III,theexpressionofthescatteringwavefunctionsis
effects withinacoupled-channelscalculationwasdonein derived. The three-body breakup amplitudes are shown
Ref.[19],usinganextendedversionoftheCDCCformal- inSec.IVandthe relatedmostexclusiveobservablesare
ism [20], hereafter referred to as XCDCC. The different presented in Sec. V. In Sec. VI, we show some working
calculations have shown that, for light targets, dynamic examples. We study 11Be+p and 11Be+64Zn reactions
core excitations give rise to sizable changes in the mag- at low energies and we calculate the angular and energy
nitude of the breakup cross sections. Additionally, these distributionsofthe10Befragmentsafterthebreakup. In
excitations have been found to be dominant in resonant Sec. VII, the main results and remarks of this work are
breakupof19Conaprotontarget[21]. Theseeffectshave collected.
also been studied within the Faddeev/AGS approach in
theanalysisofbreakupandtransferreactions[15,22,23].
II. THE STRUCTURE FORMALISM
Within the CDCC and XCDCC reaction formalisms,
the breakup is treated as an excitation of the projec-
tile to the continuum so the theoretical cross sections Inthismanuscript,thecompositeprojectileisassumed
are describedin terms of the c.m. scattering angleof the to be well described by a valence nucleon coupled to a
projectileandthe relativeenergyofthe constituents, us- core nucleus and the projectile states are described in
ing two-body kinematics. Because of this, experimental theweak-couplinglimit. Thus,thesestatesareexpanded
data should be transformed to the c.m. frame for com- as a superposition of products of single-particle config-
parison, but this process is ambiguous in the case of in- urations and core states. The energies and wavefunc-
clusive data. This is a common situation, for example, tions of the projectile are calculated using the pseudo-
in the case of reactions involving neutron-halo nuclei in state (PS) method [27], that is, diagonalizing the model
whichveryoftenonlythechargedfragmentsaredetected. Hamiltonianinabasisofsquare-integrablefunctions. For
Furthermore, even in exclusive breakup experiments un- the relative motion between the valence particle and the
der complete kinematics, in which this transformation is core,we use a recently proposedextensionofthe analyt-
feasible, the possibility of comparing the calculated and ical Transformed Harmonic Oscillator (THO) basis [14],
experimental cross sections for different configurations which incorporates the possible excitations of the con-
(angular andenergy)of the outgoing fragments provides stituents of the composite system. This approachdiffers
a much deeper insight of the underlying processes, as from the binning procedure [25], where the continuum
demonstrated by previous analyses performed by exclu- spectrum is represented by a set of wave packets, con-
sive breakup measurements with stable nuclei [24]. The structedasasuperpositionofscatteringstatescalculated
continuous developments at the radioactive beam facili- by direct integration of the multi-channel Schr¨odinger
tiesopenstheexcitingpossibilityofextendingthesestud- equation. ThemainadvantageofthePSmethodrelieson
ies to unstable nuclei. This will require improvements the fact that it provides a suitable representation of the
and extensions of existing formalisms to provide these continuumspectrumwithareducednumberoffunctions
observables. and, as we will see below, it is particularly convenientto
describe narrow resonances.
In the case of the CDCC framework, fivefold fully ex-
We briefly review the features of the PS basis used in
clusive crosssections werealreadyderivedin[25]. Inthe
this work to describe the states of the two-body com-
present work we extend this framework to the XCDCC
posite projectile. The full Hamiltonian, under the weak-
formalism and we provide more insight into the contri-
coupling limit, would be written as:
bution of core admixtures (CA) and dynamic core exci-
tation(DCE)inthecollisionprocess. Aproperinclusion H =Tˆ +V (r,ξ)+H (ξ), (1)
r cv core
of core excitation effects in the description of kinemati-
cally fully exclusive observables in the laboratory frame where Tˆ (kinetic energy) and V (r,ξ) (effective poten-
r cv
is the primary motivation of this manuscript with the tial) describe the relative motion between the core and
aim of pinning down the effect of this degree of freedom the valence while H (ξ) is the intrinsic hamiltonian of
core
inangularandenergydistributions. Additionaly,thisal- thecore,whoseinternaldegreesoffreedomaredescribed
lowstocomputebreakupobservablesforspecificstatesof through the coordinate ξ. The eigenstates of H (ξ)
core
thecorenucleus,whichcouldbeofutilityinexperiments correspondingtoenergiesǫ (definedbytheintrinsicspin
I
usinggammarayscoincidences. Thecalculationsarecar- ofthecore,I)willbedenotedbyφ and,additionalquan-
I
ried out within the combined XCDCC+THO framework tum numbers, required to fully specify the core states,
[19] for breakup reactions of 11Be on protons and 64Zn will be given below.
targets with full three-body kinematics. The formalism The core-valence interaction is assumed to contain a
isappliedtoinvestigatetheangularandenergydistribu- noncentral part, responsible for the CA in the projectile
3
states. In general, this potential can be expanded into a set of N functions RTHO(r) (with n=1,...,N). The
n,α
multipoles: eigenvectors of the Hamiltonian will be of the form:
V (r,ξ)= V (r,ξ)Y (rˆ). (2) nα N RTHO(r)
cv λµ λµ Ψ(N) (r,ξ)= ci n,α Φ (rˆ,ξ ,ξ ),
λµ i,J,M n,α,J r α,J,M v c
X α n=1
XX
(9)
In this work the projectile is treated within the
where i is an index that identifies each eigenstate and
particle-rotor model [28] with a permanent core defor-
ci arethecorrespondingexpansioncoefficientsinthe
mation (assumed to be axially symmetric). Thus, in n,α,J
truncated basis, obtained by diagonalization of the full
thebody-fixedframe,thesurfaceradiusisparameterized
Hamiltonian (1).
in terms of the deformation parameter, β , as R(ξˆ) =
2 For numerical applications, the sum over the index of
R0[1+β2Y20(ξˆ)], with R0 an average radius. The full the THO basis can be actually performed to get
valence-core interaction is obtained by deforming a cen-
tral potential V(0)(r) as follows, gJ (r)
cv Ψ(N) (r,ξ ,ξ )= i,α Φ (rˆ,ξ ,ξ ) (10)
i,J,M v c r α,J,M v c
V (r,ξˆ)=V(0) r δ Y (ξˆ) , (3) Xα
cv cv − 2 20
where the radial function is:
(cid:16) (cid:17)
whereδ =β R istheso-calleddeformationlength. The
2 2 0 N
transformationtothespace-fixedreferenceframeismade gJ (r)= ci RTHO(r). (11)
throughtherotationmatrices λ (α,β,γ)(dependingon i,α n,α,J n,α
the Euler angles α,β,γ ). AfDteµr0expanding in spherical nX=1
{ }
harmonics (see e.g. Ref. [29]), the potential in Eq. (3) The negative eigenvalues of the Hamiltonian (1) are
reads identified with the energies of bound states whereas the
positive ones correspond to a discrete representation of
V (r,ξˆ)=√4π λ(r) λ (α,β,γ)Y (rˆ), (4) the continuum spectrum.
cv Vcv Dµ0 λµ
λµ
X
with the radial form factors (u=cosθ′): III. SCATTERING WAVE FUNCTIONS
λ(r)= √2λ+1 1 V (r δ Y (θ′,0))P (u) du. For the calculation of the three-body scattering ob-
Vcv 2 cv − 2 20 λ servables (see Sec. IV) we need also the exact scattering
Z−1
(5) statesof the valence+coresystemfor agivenasymptotic
In comparison with Eq. (2) we have for a particle-rotor relative wave vector k , and given spins of the core (I)
I
model: and valence particle (s), as well as their respective pro-
jections (µ and σ, respectively), that will be denoted as
Vλrµotor(r,ξˆ)=√4πVcλv(r)Dµλ0(α,β,γ). (6) φ(k+I;)Iµ;sσ(r,ξv,ξc). Thesestatescanbewrittenasalinear
combination of the continuum states with good angular
The eigenstates of the Hamiltonian (1) will be a super-
momentum J,M, that are of the form
position of severalvalence configurationsand core states
α = ℓ,s,j,I , with ~ℓ (valence-core orbital angular mo- fJ (k ,r)
ment{um) and}~s (spin of the valence) both coupled to ~j Ψ(α+,J),M(kI,r,ξv,ξc)= α:α′r I Φα′,J,M(rˆ,ξv,ξc),
α′
(total valence particle angular momentum), for a given X
(12)
totalangularmomentumandparityofthecompositepro- wherethe radialfunctions fJ (k ,r) arethe solutionof
jectile, Jπ , i.e.: α:α′ I
the coupled differential equations,
Ψε,J,M(r,ξ)= nα RεJ,α(r)Φα,J,M(rˆ,ξv,ξc), (7) Eα′ −Trℓ′ −VαJ′:α′ fαJ:α′(kI,r)=
r
Xα (cid:2) VαJ′(cid:3):α′′fαJ:α′′(kI,r), (13)
α′′6=α′
where n is the number of such channel configurations X
α
and the set of functions whereEα′ =Eα ǫI′+ǫI,asaconsequenceoftheenergy
−
conservation in the nucleon-core system when the latter
Φα,J,M(rˆ,ξv,ξc)≡[Yℓsj(rˆ)⊗φI(ξ)]JM (8) is in the state I or I′, Trℓ′ is the relative kinetic energy
operator,andVJ arethe couplingpotentialsgivenby
α′:α′′
is the so-called spin-orbit basis.
ingTthheefHunacmtiioltnosnRiaεJn,αin(ra)baaresihseorfesoqbutaarien-eindtebgyradbialegostnaatleizs-, VαJ′:α′′(r)=hα′JM|Vvc|α′′JMi (14)
such as the THO basis. For each channel α, we consider with α′JM denoting the spin-basis defined in Eq. (8).
| i
4
These radial functions behave asymptotically as a ular Coulomb function and TJ the T-matrix, that is
α,α′
plane wave in a given incoming channel α and outgoing directly related to the S-matrix according to:
waves in all channels, i.e.:
SαJ,α′ =δα,α′ +2iTαJ,α′. (16)
fαJ:α′(kI,r)→eiσℓ Fℓ(kIr)δℓ,ℓ′ +TαJ,α′Hℓ(′+)(kIr) ,
h i(15) In terms of these good-angular momentum states, the
where σ are the Coulomb phase shifts, F (k r) the reg- scattering states result (see Appendix A)
ℓ ℓ I
4π
φ(k+I;)Iµ;sσ(r,ξv,ξc)=kIr ℓ,j,J,MiℓYℓ∗m(kˆI)hℓmsσ|jmjihjmjIµ|JMi α′ fαJ:α′(kI,r)Φα′,J,M(rˆ,ξv,ξc), (17)
X X
where m =M µ, and m=m σ. Next, assuming the validity of the completeness rela-
j j
− −
tion in the truncated basis, we get:
IV. BREAKUP AMPLITUDES TIs;J0 (k ,K) φ(−) Ψ(N)
µσ;M0 I ≃ h kI;Iµ;sσ| i,J′,M′i
i,J′,M′
X
The scattering problem can be described by means Ψ(N) eiK·R U ΨXCD(K )
of the breakup transition amplitude TIs;J0 (k ,K) con- ×h i,J′,M′ | | J0,M0 0 i
nsteactteincgomanpriinseitdiablysttahteet|aJr0gMet0(iaswsiutmheaµdσt;tMhor0ebee-bIsotrduyctfiunrea-l =i,J′,M′hφk(−I;)Iµ;sσ|Ψ(i,NJ)′,M′iTMi,J00,M,J′′(K),
X
less), the valence particle and the core, whose motion is (20)
described in terms of the relative momentum, k , and a
c.m.wavevector,K,thatdiffersfromtheinitialImomen- where the transition matrix elements Ti,J0,J′(K) are to
tumW,eKpr0o,cinee|dJ0tMo 0rei.late TµIσs;;JM00 to the discrete XCDCC bTeheinitnetreproploaltaetdionfrommeththoeddfoisllcorwetsecolonseesMlyT0,tMMih,J0e′0,M,pJ′′r(oθcie,dKuir)e.
two-body inelastic amplitudes Ti,J0,J′(θ ,K ), obtained of Ref. [25], with the difference that in this reference the
M0,M′ i i
after solving the coupled equations in the XCDCC continuum states of the projectile are described through
methodandevaluatedonthe discretevaluesofK,given a set of single-channel bins.
by the K = θ ,K grid. In order to obtain this The overlaps between the final scattering states and
i i i
relations{hip,}we r{eplace}the exact three-body wave func- thepseudo-statesareexplicitlygivenintheAppendixB,
tion by its XCDCC approximation in the exact (prior so Eq. (20) yields the following transition amplitude:
form) breakup transition amplitude. That is, we take
ΨJ0,M0(K0)≃ΨXJ0C,MD0(K0) and therefore we can write: TµIσs;;JM00(kI,K)≃ 4kπ (−i)ℓYℓm(kˆI)hℓmsσ|jmji
I J′ ℓ,m,j
TµIσs;;JM00(kI,K)≃hφk(−I;)Iµ;sσeiK·R|U|ΨXJ0C,MD0(K0)i, (18) ×hjmXjIµ|XJ′M′i Gαi,J′(kI)TMi,J00,M,J′′(K),
with the interaction U between the projectile and the Xi
(21)
targetdescribedbya complexpotentialexpressedasfol-
lows:
where
U =Uct(r,R,ξ)+Uvt(r,R), (19) Gαi,J′(kI)= fαJ:α′(kI,r)giJ,α′(r)dr (22)
where, in addition to the projectile coordinates r and Xα′ Z
ξ, we have the relative coordinate R between the pro-
are the overlaps between the radial parts of the scatter-
jectile center of mass and the target. Furthermore,
ing states and pseudo-stateswavefunctions. Notice that
the core-target interaction (U ) contains a non-central
ct these overlaps are not analytical and they must be cal-
part,responsibleforthedynamic coreexcitationandde-
culated at the energies given by the relative momentum
excitation mechanism, while the valence particle-target
k . In practice,we compute the term involvingthe sum-
I
interaction (U ) is assumed to be central. The scatter-
vt mation over i in the r.h.s. of Eq. (21) on a uniform mo-
ing wave functions φk(−I;)Iµ;sσ are just the time reversalof mentum mesh, and interpolate this sum at the required
those defined in Eq. (17), and whose explicit expression k values when combining them with the scattering am-
I
is given in Appendix A. plitudes. In fact, Eq. (20) is formally equivalent to the
5
relation appearing in Ref. [25] and the main difference Here, the particle masses are given by m (core), m
c v
concerns the calculation of the overlaps. Moreover, the (valence), and m (target) while ~k and ~k are the
t c v
aboveexpressionscanbeusedwithinthestandardCDCC coreandvalenceparticlemomentainthefinalstate. The
method (i.e. withoutcoreexcitations), inwhichcasethe totalmomentumofthesystemcorrespondsto~K and
tot
core internal degrees of freedom (ξ) are omitted. the connection with the momenta in Eq. (21) is made
through:
V. TWO- AND THREE-BODY OBSERVABLES
m m m
K =k +k p K ; k = ck vk (27)
c v tot I v c
− M m − m
The transition amplitudes in Eq. (21), TIs;J0 (k ,K) tot p p
µσ;M0 I
(withtherelativemomentumkI andthec.m.wavevector with mp = mc + mv and Mtot = mc + mv + mt the
K), contain the dynamics of the process for the coordi- totalmassesoftheprojectileandthethree-bodysystem,
natesdescribingtherelativeandcenterofmassmotionof respectively.
thecoreandthevalenceparticle. Fromtheseamplitudes
we can derive the two-body observables for a fixed spin
ofthecore,I,thesolidanglesdescribingtheorientations VI. APPLICATION TO 11BE REACTIONS
of k (Ω ) and K (Ω ), as well as the relative energy
I k K
between the valence and the core, Erel. These observ- As anillustrationof the formalismwe evaluate several
ables factorize into the transition matrix elements and a angular and energy distributions after a proper integra-
kinematical factor: tion of the two- and three-body observablespresented in
d3σ(I) µ k K µ2 theprecedingsection. Inparticular,weconsiderthescat-
cv I pt
dΩ dΩ dE =(2π)5~6K 2J +1 tering of the halo nucleus 11Be on 1H and 64Zn targets,
k K rel 0
comparingwithdatawhenavailable. Theboundandun-
× |TµIσs;;JM00(kI,K)|2, (23) bound states of the 11Be nucleus are known to contain
µ,Xσ,M0 significant admixtures of core-excited components [31–
where µ and µ are the valence-core and projectile- 33], and hence core excitation effects are expected to be
cv pt
target reduced masses. The integration over the angu- important.
lar part of k can be analytically done giving rise to As in previous works [15, 19, 34], the 11Be structure
I
thefollowingexpressionforthetwo-bodyrelativeenergy- is described with the Hamiltonian of Ref. [35] (model
angular cross section distributions: Be12-b), which consists of a Woods-Saxon central part
(R = 2.483 fm, a = 0.65 fm) and a parity-dependent
d2σ(I) = 1 K µ2ptµcv 1 strength(Vc = 54.24MeVforpositive-paritystatesand
dΩKdErel 2π3~6K02J +1kI Vc = 49.67 M−eV for negative-parity ones). The poten-
i,J′(k )Ti,J0,J′(K)2. tial co−ntains also a spin-orbit term, whose radial depen-
× | Gα I M0,M′ | dence is given by 4/r times the derivative of the central
J′,MX′,M0Xℓ,j Xi Woods-Saxonpart,andstrengthV =8.5 MeV.For the
(24) so
10Be core, this model assumes a permanent quadrupole
This expression provides angular and energy distribu-
deformationβ =0.67(i.e.δ =1.664fm). Onlytheground
2 2
tions as a function of the continuous relative energy Erel state (0+) and the first excited state (2+, Ex = 3.368
fromthediscrete(pseudo-state)amplitudes. Asshownin
MeV) are included in the model space.
the next section, this representation is particularly use-
ful to describe narrow resonances in the continuum even
with a small number of pseudo-states. A. 11Be+ p resonant breakup
The three-body observables, assuming the energy of
the core is measured, are given by [25]:
We first perform a proof of principle calculation and
d3σ(I) 2πµ 1 apply the method to the breakup of 11Be on a proton
pt
=
dΩ dΩ dE ~2K 2J +1 target at 63.7 MeV/nucleon. Previous work [16] showed
c v c 0
that the main contributions to the total energy distri-
TIs;J0 (k ,K)2ρ(Ω ,Ω ,E ), (25)
× | µσ;M0 I | c v c bution arise from the single-particle excitation mecha-
µ,Xσ,M0 nism populating the 5/2+1 resonance at Ex=1.78 MeV
where the phase space term ρ(Ω ,Ω ,E ), i.e., the num- [36]andthecontributionfromtheexcitationofthe3/2+
c v c 1
ber of states per unit core energy intervalat solidangles resonance (Ex=3.40 MeV, [36]) due to the collective ex-
Ω and Ω , takes the form [30]: citation of the 10Be core.
c v
We repeat, in here, the calculations of Ref. [16] for
m m ~k ~k
ρ(Ω ,Ω ,E )= c v c v theangulardistributionusingtheXCDCCformalismfor
c v c (2π~)6
thereactiondynamics,andthepseudo-statebasisforthe
mt structureof11Be. ContinuumstatesuptoJ =5/2(both
. (26)
× m +m +m (k K ) k /k2 parities) were found to be enough for convergenceof the
(cid:20) v t v c− tot · v v(cid:21)
6
calculated observables. These states were generated di- 40
agonalizing the 11Be Hamiltonian in a THO basis with
N =12 radialfunctions andvalence-coreorbitalangular a) E =0.0-2.5 MeV
rel
)
momenta ℓ 5. For the interaction between the projec- sr 10 +
tile and the≤target, Eq. (19), we used the approximate b/ Be(I=0 )
m
proton-neutron Gaussian interaction as in Ref. [16]; the (
centralpartofthecore-targetpotentialwascalculatedby m. 20
a folding procedure, using the JLM nucleon-nucleon ef- Ωc.
fectiveinteraction[37]andthe10Beground-statedensity d
σ/
from a Antisymmetrized Molecular Dynamics (AMD) d
calculation [38]. For the range of the JLM interaction,
we used the value prescribedin the originalwork, t=1.2
0
fm, andthe imaginarypartwasrenormalizedbyafactor
N =0.8,obtainedfromthesystematicstudyofRef.[39].
i 10 +
Be(I=0 )
The XCDCC coupled equations were integrated up to
100 fm and for total angular momenta JT ≤65. b/sr) 20 b) Erel=2.5-5 MeV 10B+e(I=2+)+
InFig.1weshowthetwo-bodybreakupangulardistri- m I=0 + I=2
butions, as a function of the 11Be∗ c.m. scatteringangle, (m.
and we compare the total results with the experimen- c.
Ω
taldatawithinthetwoavailablerelativeenergyintervals
d 10
[40],Erel =0 2.5(toppanel)andErel =2.5 5(bottom σ/
− − d
panel). As in previous calculations [15, 19], the agree-
mentwiththedataisfairlyreasonablewiththeexception
of the first data point in the higher energy interval. The
0
peak appearing at small scattering angles for the lower 0 10 20 30 40 50
energyintervalisduetoCoulombbreakupanditwasnot θ (deg)
c.m.
present in our previous calculations due to the smaller
cutoffinthe totalangularmomentum. Weshowalsothe FIG. 1: (Color online) Differential breakup cross sections of
separate contribution for each of the states of the core. 11Be on protons at 63.7 MeV/nucleon, with respect to the
We note that both contributions include the core excita- outgoing 11Be∗ c.m. scattering angle and the neutron-core
tion effect through the admixtures of core-excited com- relativeenergyintervalsErel =0−2.5 MeV(top)andErel=
ponents in the projectile (structure effect) and the core- 2.5−5 MeV (bottom). The contributions corresponding to
the considered outgoing core states are also shown. See text
targetpotential (dynamics). However,the productionof
10Be(2+) is only kinematically allowed when the excita- for details.
tion energy is above the 10Be(2+)+n threshold, which
lies at an excitation energy of 3.87 MeV with respect to
the 11Be(g.s.). Consequently, for the lower energy in- distribution, it becomes apparent that the DCE mech-
terval (top panel) the system will necessarily decay into anism is very important in this reaction. In particular,
10Be(g.s.)+n, irrespective of the importance of the DCE the energy spectrum is dominated by two sharp peaks
mechanism. Notice thatthe emitted 10Be(2+)fragments correspondingtothe5/2+ and3/2+ resonanceswiththe
1 1
would be accompanied by the emission of a γ-ray with latter mostly populated by a DCE mechanism [15]. De-
the energycorrespondingto the excitationenergyofthis spitetherelativelysmallTHObasis,theenergyprofileof
state, thus allowing an unambiguous separation of both these resonances is accurately reproduced and this high-
contributions. lightstheadvantageofthe pseudo-statemethodoverthe
This is better seen in Fig. 2, where the differential en- binning procedure when describing narrow resonances.
ergy cross section is plotted after integration over the Finally, besides the resonant contribution, we also note
angular variables Ω in Eq. (24). The solid line is the that there is a non-resonant background at low relative
K
full XCDCC calculation, considering the core excitation energies and above the 10Be(2+)+n threshold.
effects in both the structure and the dynamics of the re- Regarding the three-body observables [Eq. (25)], we
action, and includes the two possible final states of the present in Fig. 3 the energy distributions of the 10Be
10Be nucleus. The 10Be(2+) contribution (red dashed fragments from the breakup process at four laboratory
line) only appears for E > 3.4 MeV, corresponding to angles, plotting separately the 0+ and 2+ contributions.
rel
the 10Be(2+)+n threshold. As already noted, above this We observe from this plot the increasing relative impor-
energy,the10Befragmentscanbeproducedineitherthe tance of the 10Be(2+) distribution with the angle. This
g.s. or the 2+ excited state. We also show the calcu- is expected since larger scattering angles of the core im-
lation omitting the DCE mechanism (green dot-dashed plies a stronger interaction with the proton target. It
curve) and considering only the CA contributions in the is also apparent that this distribution is shifted to lower
structure of the projectile. By comparing with the total energies with respect to the 10Be(g.s.) due to the higher
7
80
11Be+p @ 63.7 MeV/nucleon 1.2 11Be+p @ 63.7 MeV/nucleon
mb/MeV) 60 XXXCCCDDDCCCCCC::: n1100oBB Dee C((02E+++) 2+) mb/MeV) 0.81 1100BBee ((II==02++))
(el40 E (c0.6
dEr σ/d
σ/ d 0.4
d 20
0.2
0
0 550 600 650 700
0 1 2 3 4 5 E (MeV)
E (MeV) c
rel
FIG. 4: (Color online) Calculated differential energy cross
FIG.2: (Coloronline)Differentialenergydistributionfollow- section, as a function of the 10Be energy in the laboratory
ing the breakup of 11Be on protons at 63.7 MeV/nucleon. frame, for the reaction 11Be+p at 63.7 MeV/nucleon. The
Solid and dashed (red) lines correspond to the full (0++2+)
solid(blue)anddashed(red)linesdescribethosecalculations
and the 2+ contribution of the XCDCC calculation while
when the I =0 or I =2 state of thecore is adopted.
the dot-dashed line (green) represents the result without dy-
namic core excitation. The arrow indicates theenergy of the
10Be(2+)+nthreshold.
in Ref. [26] and have been analyzed within the stan-
300 dardCDCC frameworkin severalworks[41–43]andalso
MeV)) 600 θc=0o θc=1o within the XCDCC framework [19]. The results pre-
mb/(sr 400 1100BBee((02++)) 200 sreefnetreednchee,rbeuftolwloitwhctlwoosemlyatinhodsieffeinrecnlucdese:dfiirnsttlyh,isinlatthtaetr
E (c workthe 10Be scattering angle was approximatedby the
d 100
Ωc200 11Be∗ angle,assumingtwo-bodykinematics,whereasthe
d
3σd/ appropriate kinematical transformation is applied here;
0 0
V)) θ=2o θ=3o secondly, the XCDCC calculations are performed here
Me c c 60 in an augmented model space, including higher values
b/(sr 100 of the relative orbital angular momentum between the
E (mc 40 valence and core particles. In addition, the former anal-
Ωdc 50 20 ysisis extendedbystudying the individualcontributions
σ/d of the 10Be 0+ and 2+ states when computing the two-
3d 0 0 andthree-body observables. For the sakeof comparison,
500 550 600 650 700 550 600 650 700
E (MeV) E (MeV) we also perform standard CDCC calculations similar to
c c
thosepresentedin[42]butusingalargermodelspace,as
FIG. 3: (Color online) Calculated laboratory-frame double detailed below.
differential cross section for the 10Be fragments emitted in
For the CDCC calculations, 11Be continuumstates up
the process 11Be+p at 63.7 MeV/nucleon when four differ-
to J =9/2 (both parities) and J =11/2− were included
ent scattering angles are considered. The blue solid and red
formaximumn-10Berelativeenergiesandorbitalangular
dashedlinesrefertothecontributionsfromthedifferentcore
states. momenta Ermealx =12 MeV and ℓmax =5, respectively. A
THO basis with N =30 was employed and the involved
interactions (i.e., the n-10Be, 10Be-64Zn, and n-64Zn po-
tentials) were the same as in Ref. [42] except for that
excitationenergyrequiredto produce the 10Be(2+) frag- betweentheneutronand10Be. Asin[42],weuseforthis
ments. Theangle-integratedcontributionscanbeseenin
potential that from Ref. [12], but we slightly modify the
Fig.4,wherewenotethedominancefromthe0+ compo-
depth for ℓ = 2 in order to reproduce the energy of the
nent to the overall energy distribution although the 2+ 5/2+ resonanceobtainedwiththedeformedmodelBe12-
1
contributionamountsto13%ofthetotalcrosssectionat
b. As for the XCDCC calculations,the following contin-
this energy.
uum states were considered: for J 5/2 (both parities),
≤
ℓ = 3 and Emax = 12 MeV. For 5/2 < J 11/2, we
umseadx ℓ = 5 arneld Emax = 9 MeV. A THO≤basis with
B. 11Be+ 64Zn breakup N =2m0afxunctions wasruelsed for all Jπ with the same po-
tentials between the projectile constituents and the tar-
We consider the 11Be+64Zn reaction at 28.7 MeV get as those used in Ref. [19]. The coupled equations
for which inclusive breakup data have been reported were solved in this case with the parallelized version of
8
11Be+64Zn @ 28.7 MeV 600 11Be+64Zn @ 28.7 MeV
1000 Di Pietro et al.
sr) XCDCC: 10Be(0+) eV) XCDCC: 10Be (I=0+)
Ωd (mb/lab 500 XCCDDCCCC (S: 1P0 Bmeo(2d+e)l ) (x 25) E (mb/Mrel400 XCCDDCCCC (S: 1P0 Bmeo (dI=e2l)+) (x 25)
σ/ d
d σd/ 200
00 10 20 θlab 3(0deg) 40 50 60 00 1 2Erel (MeV)3 4 5
FIG.5: (Coloronline)Differentialcrosssection,asafunction FIG. 6: (Color online) Calculated 10Be differential energy
ofthelaboratoryangle,forthe10Befragmentsresultingfrom cross section for the reaction 11Be+64Zn at 28.7 MeV. The
the breakup of 11Be on 64Zn at Elab = 28.7 MeV. The solid solid (blue) and dashed (red) lines represent the contribu-
line(blue)isthecalculation whenthecoregroundstate, I = tions with I =0 or I =2 as the core states in the two-body
0, is selected in Eq. (25) after integration over Ωv and Ec. distributions (24) after integration overΩK. The dot-dashed
Thedashedline(red)referstothe2+ contribution. Asingle-
(green) line considers the same calculation without core de-
particle calculation, omitting the core deformation, is also
formation, assumingasingle-particle modelfortheprojectile
shown as a dot-dashed curve(green). Experimental data are
within the standard CDCC framework. The arrow indicates
from Ref. [26]. theenergy of the10Be(2+)+nthreshold.
the Fresco coupled-channels code [44]. The inclusion of
10Be(2+)+n threshold. Consequently, in this reaction
high-lyingexcitedstates producesnumericalinstabilities
the 10Be will be mostly produced in its ground state.
in the solution of the coupled equations. A stabilization
Moreover, it is remarkable the presence of a promi-
procedure similar to that proposed in [45] was used to
nent peak at energies around the low-lying 5/2+ reso-
get stable results.
nance, where the dominant component corresponds to
In Fig. 5 we compare the data from Ref. [26] with the
the10Be(0+)configuration. Asecondbump,correspond-
present calculations. For the XCDCC calculations, the
ing to the population of the 3/2+ resonance, can be
0+ and 2+ contributions are shown separately as solid 1
barely seen. This small contribution reflects the scarce
(blue) or dashed (red) lines, although the latter is found
relevance of the DCE mechanism for this medium-mass
to be negligible in this case. The new CDCC calcula-
targetaspointedoutinRef.[19]but,contrarytothecon-
tion(greendot-dashedcurve)appearstobeclosertothe
clusions therein, the core excitation effects in the struc-
results with core deformation but both of them clearly
ture of the 11Be are not so large and the assumption of
underestimate the experimental breakup cross section,
a single-particle model for the projectile yields a similar
in accordance with previous results [19]. Nevertheless,
breakup cross section (green dot-dashed line). In addi-
these former calculations were carried out within a re-
tion, unlike the case of the proton target, a dominant
duced model space and the proper kinematical transfor-
non-resonantbreakup is found at low relative energies.
mationwasnotapplied. Theremainingdiscrepancywith
the data could be due to the contribution of non-elastic The different role of DCE for elastic breakup in the
breakupevents(neutronabsorptionortargetexcitation) cases of the proton and 64Zn targets can be ascribed to
inthedatasincetheneutronswerenotdetectedintheex- the dominance of the dipole Coulomb couplings in the
perimentofRef. [26]. In this regard,itis worthrecalling latter case, which hinders the effect of the quadrupole
that the CDCC method provides only the so-calledelas- couplingsassociatedwiththeexcitationofthe10Becore.
ticbreakupcomponent,sothetargetisleftintheground We expect that, for heavier targets, this dominance is
state. The inclusion of these non-elastic breakup contri- enhancedandthereforetheeffectoftheDCEmechanism
butions has been recently the subject of several works will be further reduced.
[46–48] but the considerationof this effect is beyond the Finally,asmallcoreexcitationeffectisalsoapparentin
scope of the present work. Fig.7,whereweshowthebreakupenergydistributionsof
In Fig. 6 we show the breakup cross section as a func- the 10Be emitted fragments within the standard CDCC
tion of the 11Be excitation energy, with respect to the (onlyconsideringthe10Be(0+)state)orXCDCC(includ-
10Be(g.s.)+n threshold. The solid (blue) and dashed ing both 0+ or 2+ states in 10Be) reactionformalisms at
(red)linescorrespondtothe10Be(g.s)and10Be(2+)con- four laboratory angles. The relative importance of the
tributions. Most of the cross section is concentrated 2+ state increases with increasing angle but its absolute
at low excitation energies, close to the breakup thresh- magnitudeisnegligibleinallcases. Itisalsoobservedan
old, being negligible for excitation energies above the overallreductionofthemeanenergyofthe2+ corestate,
9
500 200 300
MeV)) 400 θc=13o θc=30o 160 11Be+64Zn @ 28.7 MeV
3σΩd/ddE (mb/(sr cc1230000000 11C00DBBeeC((C02++ ())S P(x m25o)del) 04810020 E (mb/MeV)c200 XXCCCDDDCCCCCC (S:: 11P00 BBmeeo ((dII==e02l)++)) (x 25)
sr MeV)) 18000 θc=40o θc=50o 30 σd/d 100
b/(
E (mc 60 20
Ωdc 40 10 010 15 20 25 30
σ/d 20 E (MeV)
3d c
0 0
15 20 25 3015 20 25 30
E (MeV) E (MeV) FIG. 8: (Color online) Calculated laboratory frame 10Be
c c
differential energy cross section for the breakup reaction
FIG.7: (Coloronline)Calculatedlaboratoryframe10Becross 11Be+64Zn at 28.7 MeV.Thebluesolid andred dashedlines
sectionenergydistributionsfortheprocess11Be+64Znat28.7 account for the components obtained with the different core
MeV with different laboratory angles in Eq. (25). Under the statesintheXCDCCcalculation. Thedistributioncomputed
XCDCC framework we study both the 0+ (blue solid) or 2+ with the standard CDCC method (no deformation) is also
(reddashed)contributionswhilethestandardCDCCmethod shown (green dot-dashed).
provides the distributions without core deformation (green
dot-dashed).
nucleus, thus permitting a more direct connection with
experimental observables.
as a consequenceofthe more negativeQ-value. The cor- Themethodhasbeenappliedtothescatteringof11Be
respondingangle-integratedenergydistributionisshown on protons and 64Zn. The 11Be nucleus is described in
in Fig. 8. As expected, this distribution is completely a simple particle-rotor model, in which the 10Be core
dominated by the 10Be(g.s.) contribution, displaying an is assumed to have a permanent axial deformation [35].
asymmetric shape with a maximum around26 MeV and The core-target interaction is obtained by deforming a
a large low-energy tail extending down to 15 MeV. For central phenomenological potential. Within the devel-
core energies above the peak, the distribution exhibits oped approach the angular and energy distributions of
a pronounced drop as a consequence of the kinematical the 10Be fragments (with an intrinsic spin, I) following
cutofffromtheenergyconservationtogetherwiththein- the breakup of 11Be have been calculated and compared
teractionbetweenthephasespacefactorandthebreakup with experimental data, when available.
amplitude in the semi-inclusive cross section. Actually, In the 11Be+p reaction,we find that a significant part
thesharpfalloffwouldbepresentunlessthebreakupam- of the breakup cross section corresponds to the 10Be ex-
plitude is very small around the maximum energy [49].
citedstate. Moreover,wehaveconfirmedtheimportance
oftheDCEmechanism,arisingfromthenon-centralpart
of the core-target interaction, for the excitation of the
VII. SUMMARY AND CONCLUSIONS low-lying 5/2+ and 3/2+ resonances [16].
We have also studied the 11Be+64Zn reaction at
In this paper we have presented a formalism for the 28.7 MeV, extending the previous analysis performed in
calculation of two- and three-body breakup observables Ref. [19]. Although in that reference a sizable difference
fromXCDCCcalculations. Themethodhasbeenapplied was observedbetween the calculations with and without
tothecaseofthescatteringofatwo-bodyprojectilecon- deformation, the present calculations suggest that this
sistingonacoreandavalenceparticle,andtakingexplic- difference is largely reduced if a sufficiently large model
itly into account core excitations. The method is based spaceisemployedfortheXCDCCcalculation. Inviewof
on a convolution of the discrete XCDCC scattering am- these new results, we may conclude that, unlike the pro-
plitudes with the exact core+valence scattering states. tontargetcase,the effectof coreexcitationis very small
The formalism is a natural extension of that presented in this reaction as far as the breakup cross sections con-
in Ref. [25] for the standard CDCC method, the main cern. Asaconsequence,the10Be(2+)yieldisfoundtobe
difference being the multi-channel character of the pro- negligibly small. We may anticipate that this conclusion
jectile states in the present case. The convoluted transi- will also hold for other medium-mass or heavy systems.
tionamplitudesarethenmultipliedbythecorresponding Thequalitativedifferencewithrespecttotheprotoncase
phase-spacefactorto producethe desired(two-orthree- stems from the larger importance of Coulomb couplings
body)differentialcrosssections. Theformalismprovides in the 64Zn case.
the separate cross section for specific states of the core Although all the calculations presented in this work
10
have been performed for the 11Be nucleus, we believe where, in the r.h.s. of the equation, we have separated
that the results are extrapolable to other weakly-bound for conveniencethe partcontaining the regularCoulomb
nucleiand,consequently,theeffectsdiscussedhereshould function.
betakeninconsiderationforanaccuratedescriptionand
interpretation of the data. Finally, we notice that the To compare this with the asymptotic behaviour we
semi-inclusivedifferentialcrosssectionspresentedinthis need the partial-wave decomposition of the plane wave:
papercanbeusedtoproducetransverseandlongitudinal
momentumdistributions,whichhavebeenusedtoobtain
spectroscopic information using both light and proton 4π
targets. eikI·r = k r iℓ(kIr)jℓ(kIr)Yℓ∗m(kˆI)Yℓm(rˆ). (A3)
I
ℓ,m
X
More generally, in presence of Coulomb, the expansion
Acknowledgments above becomes:
The authors would like to thank Prof. J.A. Tostevin
fsourppusoerftulbcyomthmeeFnutsndaan¸cda˜odispcaursasioanCs.iˆeRn.cDia. aeckanoTweclendogloes- χC(kI,r)= k4πr iℓeiσℓFℓ(kIr)Yℓ∗m(kˆI)Yℓm(rˆ). (A4)
I
ℓ,m
gia (FCT) through Grant No. SFRH/BPD/78606/2011. X
This workhas also been partially supported by the FCT
contract No. PTDC/FIS-NUC/2240/2014as well as the
Spanish Ministerio de Econom´ıa y Competitividad and
FEDER funds under project FIS2014-53448-C2-1-Pand Using this result, the plane wave reads:
by the European Union’s Horizon 2020 research and in-
novation programunder grant agreement No. 654002.
χ (k ,r)ϕ(v)(ξ )ϕ(c)(ξ )= (A5)
C I sσ v Iµ c
Appendix A: Calculation of multi-channel scattering 4π
iℓeiσℓF (k r)Y∗ (kˆ)Y (rˆ)ϕ(v)(ξ )ϕ(c)(ξ ).
states k r ℓ I ℓm I ℓm sσ v Iµ c
I
ℓ,m
X
Herewederivethe coefficientsC ,thatrelatethe
ℓ,j,J,M
scattering states with the solution of the Schr¨odinder
equation for good values of J,M according to the ex- Inoder to expressthis state interms ofthe basis(8), we
pansion use the following expansions:
φ(+) (r,ξ ,ξ )= C Ψ(+) (k ,r,ξ ,ξ ).
kI;Iµ;sσ v c ℓ,j,J,M α,J,M I v c
ℓ,j,J,M
X Y (rˆ)ϕ(v)(ξ )= ℓmsσ jm (rˆ,ξ ) (A6)
(A1) ℓm sσ v h | jiY(ℓs)jmj v
jX,mj
The expansion coefficients are determined by replacing
thefunctionsΨ(+) (k ,r,ξ ,ξ )bytheirasymptoticbe-
α,J,M I v c
haviour, Eq. (15),
F (k r)
φ(k+I;)Iµ;sσ(r,ξv,ξc)→ eiσℓCℓ,j,J,M ℓ rI δℓ,ℓ′
ℓ,jX,J,M Xα′ (cid:20) (rˆ,ξ )ϕ(c)(ξ )= jm IµJM Φ (rˆ,ξ ,ξ ).
+ TαJ,α′Hℓ(′+)r(kIr) Φα′,J,M(rˆ,ξv,ξc) Y(ℓs)jmj v Iµ c XJMh j | i α,J,M (vA7c)
#
(A2) So, collecting results,
4π
χC(kI,r)ϕ(svσ)(ξv)ϕ(Icµ)(ξc)= k r Yℓ∗m(kˆI)iℓeiσℓFℓ(kIr) hℓmsσ|jmjihjmjIµ|JMiΦα,J,M(rˆ,ξv,ξc). (A8)
I
Xℓ,m jX,mjJX,M
The above expression gives the plane-wave part of Eq. (A2), so we get for the C coefficients:
4π
C = iℓY∗ (kˆ) ℓmsσ jm jm IµJM .
ℓ,j,J,M k ℓm I h | jih j | i
I
mX,mj
(A9)
