Analysis of a low-energy correction to the eikonal approximation

0.32 MB·

by  Tokuro Fukui

Analysisofa low-energy correction to the eikonalapproximation
Tokuro Fukui,1,∗ Kazuyuki Ogata,1,† and Pierre Capel2,‡
1 Research Centerfor NuclearPhysics, Osaka University, Ibaraki, Osaka 567-0047, Japan
2 Physique Nucle´aire et Physique Quantique C.P. 229,
Universite´ Libre de Bruxelles (ULB), B-1050 Brussels, Belgium
(Dated:September25,2014)
Extensionsoftheeikonalapproximationtolowenergy(20MeV/nucleontypically)arestudied. Therelation
betweenthedynamicaleikonalapproximation(DEA)andthecontinuum-discretizedcoupled-channelsmethod
withtheeikonal approximation (E-CDCC)isdiscussed. WhenCoulomb interactionisartificiallyturned off,
DEA and E-CDCC are shown to give the same breakup cross section, within 3% error, of 15C on 208Pb at
20MeV/nucleon. WhentheCoulombinteractionisincluded, thedifferenceisappreciableandnoneofthese
modelsagreeswithfullCDCCcalculations. Anempiricalcorrectionsignificantlyreducesthisdifference. In
4
addition,E-CDCChasaconvergenceproblem. Byincludingaquantum-mechanicalcorrectiontoE-CDCCfor
1 lowerpartial wavesbetween 15Cand208Pb, thisproblemisresolvedandtheresult perfectlyreproduces full
0
CDCCcalculationsatalowercomputationalcost.
2

p PACSnumbers:24.10.-i,25.60.Gc,21.10.Gv,27.20.+n
e
S
I. INTRODUCTION approximatecompletesetofstates. Withsuchanexpansion,
4
2 the corresponding Schro¨dinger equation translates into a
set of coupled equations [6–8]. This reaction model is very
The development of radioactive-ion beams in the mid-
] generalandhasbeensuccessfullyusedtodescribeseveralreal
h 1980s has enabled the exploration of the nuclear landscape
and virtual breakup reactions at both low and intermediate
t farfromstability. Thistechnicalbreakthroughhasledto the
- energies [9–12]. However it can be very computationally
l discoveryofexoticnuclearstructures,like nuclearhalosand
c challenging,especiallyathighbeamenergy. Thishindersthe
shellinversions.Halonucleiexhibitaverylargematterradius
u extension of CDCC models to reactions beyond the simple
n comparedto their isobars. This unusualfeature is explained
usualtwo-bodydescriptionoftheprojectile. Simplifyingap-
[ by a stronglyclusterized structure: a compactcore that con-
proximations, less computationally demanding, can provide
tains most of the nucleonsto which one or two neutronsare
2 anefficientwaytoavoidthatlimitationofCDCC.
looselybound. Duetoquantumtunneling,thesevalenceneu-
v
Atsufficientlyhighenergy,theeikonalapproximationcan
3 tronsexhibitalargeprobabilityofpresenceatalargedistance
beperformed.Inthatapproximation,theprojectile-targetrel-
9 fromthecore,henceincreasingsignificantlytheradiusofthe
8 nucleus. Examples of one-neutronhalo nuclei are 11Be and ativemotionis assumednotto deviatesignificantlyfromthe
6 15C, while 6He and 11Li exhibit two neutrons in their halo. asymptotic plane wave [14]. By factorizing that plane wave
. out of the three-body wave function, the Schro¨dinger equa-
1 Thoughlessprobable,protonhalosarealsopossible. Theex-
tion can be significantlysimplified. Both the eikonalCDCC
0 otichalostructurehasthusbeenthesubjectofmanytheoreti-
4 calandexperimentalstudiesforthelast30years[1,2]. (E-CDCC)[15,16]andthedynamicaleikonalapproximation
1 (DEA)[17,18]aresucheikonalmodels.Notethatthesemod-
Due to their veryshortlifetime, halo nucleimust be stud-
: elsdifferfromtheusualeikonalapproximationinthattheydo
v ied throughindirecttechniques, such asreactions. The most
notincludethesubsequentadiabaticapproximation,inwhich
i widelyusedreactiontostudyhalonucleiisthebreakupreac-
X theinternaldynamicsoftheprojectileisneglected[19].
tion, in whichthe halodissociatesfromthe corethroughthe
r The E-CDCC model solves the eikonal equation using
a interactionwith a target. The extractionof reliable structure
the same discretization technique as the full CDCC model.
informationfrommeasurementsrequiresa goodunderstand-
Thanks to this, E-CDCC can be easily extended to a hybrid
ing of the reaction process. Various models have been de-
version, in which a quantum-mechanical(QM) correction to
velopedto describe the breakupof two-bodyprojectiles, i.e.
thescatteringamplitudecanbeincludedfortheloworbitalan-
one-nucleonhalonuclei(seeRef.[3]forareview).
gularmomentumLbetweentheprojectileandthetarget.This
The continuum-discretized coupled channel method
helps in obtaining results as accurate as a full CDCC with a
(CDCC)isafullyquantummodelinwhichthewavefunction
minimal task. In addition, E-CDCC can take the dynamical
describing the three-body motion—two-body projectile plus
relativistic effects into account [20, 21], and it has recently
target—is expanded over the projectile eigenstates [4–6].
beenextendedtoinclusivebreakupprocesses[22,23].
For breakup modeling, the core-halo continuum must be
WithintheDEA,theeikonalequationissolvedbyexpand-
included and hence is discretized and truncated to form an
ing the projectile wave function upon a three-dimensional
mesh, i.e., without the CDCC partial-wave expansion [17].
This prescription is expected to efficiently include compo-
nents of the projectile wave function up to high orbital an-
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected] gular momentum between its constituents. Moreover it en-
‡Electronicaddress:[email protected] ables describing both bound and breakupstates on the same
2
footing,withoutresortingtocontinuumdiscretization. Since 14C
DEA treats the three-body dynamics explicitly, all coupled-
R
channelseffectsareautomaticallyincluded. Excellentagree- 14
ment with the experimenthas been obtained at the DEA for
thebreakupandelastic scatteringofone-nucleonhalonuclei r R
onbothheavyandlighttargetsatintermediateenergies(e.g.,
70MeV/nucleon)[18].
n R
In a recent work [24], a comparison between CDCC and n
208Pb
DEA was performed. The breakup of the one-neutron halo
nucleus 15C on 208Pb has been chosen as a test case. At 15C
68 MeV/nucleon, the results of the two models agree very
well with each other. At 20 MeV/nucleon, DEA cannot re-
FIG.1:Schematicillustrationofthe(14C+n)+208Pbthree-body
producetheCDCCresultsbecausetheeikonalapproximation
system.
isnolongervalidatsuchlowenergy.Itappearsthattheprob-
lem is due to the Coulomb deflection, which, at low energy,
significantlydistortstheprojectile-targetrelativemotionfrom
a pure plane wave. Because of the computationaladvantage where µ is the 14C-n reduced mass and U is a phe-
n14 nC
of the eikonal approximation over the CDCC framework, it nomenologicalpotentialdescribingthe14C-ninteraction(see
isimportanttopindownwherethedifferencecomesfromin
Sec.IIIA).Wedenotebyϕ theeigenstatesofHamiltonian
εℓm
moredetailandtrytofindawaytocorrectit. ofEq.(1)atenergyεinpartialwaveℓm,withℓthe14C-nor-
ThegoalofthepresentpaperistoanalyzeindetailtheQM bital angular momentum and m its projection. For negative
correctiontoE-CDCCinitshybridversiontoseeifitcanbe energies, these states are discrete and describe bound states
incorporatedwithintheDEA to correctthe lackofCoulomb of the nucleus. For the presentcomparison, we consider the
deflectionobservedinthatreactionmodel.Todoso,wecom- sole ground state ϕ at energy ε = −1.218 MeV. The
0ℓ0m0 0
pare E-CDCC and DEA with a special emphasis upon their positive-energy eigenstates correspond to continuum states
treatment of the Coulomb interaction. We focus on the 15C thatsimulatethebrokenupprojectile.
breakupon208Pbat20MeV/nucleon. First,wecomparethe Tosimulatetheinteractionbetweenn(14C)and208Pb,we
results of DEA and E-CDCC with the Coulomb interaction adopt the optical potential U (U ) (see Sec. IIIA). Within
n 14
turnedofftoshowthatbothmodelssolveessentiallythesame this framework, the study of 15C-208Pb collision reduces to
Schro¨dingerequation. ThenweincludetheCoulombinterac- solvingthethree-bodySchro¨dingerequation
tiontoconfirmtheCoulombdeflectioneffectobservedbythe
authorsofRef. [24]. We checkthatthe hybridversionof E- HΨ(R,r)=E Ψ(R,r) (2)
tot
CDCC reproducescorrectlythefullCDCC calculations,and
analyzethisQMcorrectiontosuggestanapproximationthat withtheHamiltonian
canbeimplementedwithinDEAtosimulatetheCoulombde-
~2
flection. H =− ∆R+h+U14(R14)+Un(Rn), (3)
In Sec. II we briefly review DEA and E-CDCC, and clar- 2µ
ify the relation between them. We compare in Sec. III the
breakup cross sections of 15C on 208Pb at 20 MeV/nucleon, whereµ is the 15C-208Pb reducedmass. Equation(2) hasto
besolvedwiththeincomingboundarycondition
with and without the Coulomb interaction. A summary is
giveninSec.IV.
lim Ψ(R,r)=eiK0z+···ϕ (r), (4)
z→−∞ 0ℓ0m0
whereK is the wave numberforthe initialprojectile-target
II. FORMALISM 0
motion, whosedirectiondefinesthe z axis. Thatwave num-
ber is related to the total energy E = ~2K2/(2µ)+ ε .
A. Three-bodyreactionsystem tot 0 0
The “···” in Eq. (4) indicates that the projectile-targetrela-
tive motion is distorted by the Coulomb interaction, even at
Wedescribethe15Cbreakupon208Pbusingthecoordinate largedistances.
systemshowninFig.1. Thecoordinateofthecenter-of-mass In the eikonal approximation, the three-body wave func-
(c.m.) of 15C relative to 208Pb is denoted by R, and r is tionΨisassumednottovarysignificantlyfromtheincoming
theneutron-14Crelativecoordinate.RnandR14 are,respec- planewaveofEq.(4). Hencetheusualeikonalfactorization
tively,thecoordinatesofneutronnandthec.m. of14Cfrom
208Pb. We assume both14Cand 208Pb to be inertnuclei. In Ψ(R,r)=eiK0zψ(b,z,r), (5)
thisstudyweneglectthespinofn. TheHamiltoniandescrib-
ingthe15Cstructurethereforereads wherewehaveexplicitlydecomposedRinitslongitudinalz
andtransversebcomponents. Inthefollowingbisexpressed
~2 as b = (b,φ ) with b the impact parameter and φ the az-
h=−2µ ∆r+UnC(r), (1) imuthalangleRofR. R
n14
3
Using factorization Eq. (5) in Eq. (2) and taking into ac- where
countthatψvariessmoothlywithR,weobtainequationssim-
Z Z e2
plertosolvethanthefullthree-bodySchro¨dingerequation(2). V (R)= C T (12)
C
Inthefollowingsubsections,wespecifytheequationssolved R
withintheE-CDCC(Sec.IIB)andtheDEA(Sec.IIC).
is the projectile-target potential that slows down the projec-
tileasitapproachesthetarget. ThecouplingpotentialFcc′ is
definedby
B. Continuum-discretizedcoupled-channelsmethodwiththe
eikonalapproximation(E-CDCC) Fcc′(b,z)=hϕc|U14+Un−VC|ϕc′irei(m−m′)φR, (13)
E-CDCCexpressesthethree-bodywavefunctionΨas[15, and
16]
Ψ(R,r)= ξ¯iℓm(b,z)ϕiℓm(r)eiKizei(m0−m)φRφCi (R), Rii′(b,z)= (K(Ki′RR−−KKi′zz))iiηηii′. (14)
i i
iℓm
X (6)
Within the E-CDCC framework, the boundary condition
where {ϕ } are square-integrable states that describe the
iℓm Eq.(4)translatesinto
eigenstates of Hamiltonian h Eq. (1). The subscript i la-
bels the eigenenergy of 15C. The ground state corresponds lim ξ¯ (b,z)=δ δ δ . (15)
to i = 0 and the correspondingwave functionϕ is the z→−∞ iℓm i0 ℓℓ0 mm0
0ℓ0m0
exact eigentstate of h. The values i > 0 correspond to dis-
cretestatessimulating15Ccontinuum. Theyareobtainedby
C. Thedynamicaleikonalapproximation(DEA)
thebinningtechnique[25],vizbyaveragingexactcontinuum
states over small positive-energy intervals, or bins. The set
{ϕ }satisfying IntheDEA,thethree-bodywavefunctionisfactorizedfol-
iℓm
lowing[17,18]
hϕi′ℓ′m′|h|ϕiℓmir =εiδi′iδℓ′ℓδm′m (7)
Ψ(R,r)=ψ(b,z,r)eiK0zeiχC(b,z)eiε0z/(~v0), (16)
isassumedtoformanapproximatecompletesetforthepro-
jectile internal coordinater. The plane wave eiKiz contains
whereχ istheCoulombphasethataccountsfortheCoulomb
C
thedominantpartoftheprojectile-targetmotionasexplained
projectile-targetscattering
above. The correspondingwave number varies with the en-
ergy of the eigenstate of 15C respecting the conservation of 1 z
emnaetregCyoEutlootm=bi~n2cKidi2e/n2tµwa+veεif.unInctEioqn.g(6iv),enφCibyistheapproxi- χC(b,z)=−~v0 Z−∞VC(R)dz′, (17)
where v = ~K /µ is the initial velocity of the projectile.
φCi (R)=eiηiln[KiR−Kiz] (8) Notetha0tthepha0seexp[ε z/(i~v )]canbeignoredasithas
0 0
where η is the Sommerfeld parameter correspondingto the noeffectonphysicalobservables[18].
i
ithstateof15C FromthefactorizationinEq.(16),weobtaintheDEAequa-
tion[17,18]
Z Z e2µ
η = C T , (9)
i ~2Ki i~v ∂ ψ(b,z,r)=[h+U +U −ε −V ]ψ(b,z,r).
0 14 n 0 C
∂z
withZ e(Z e)thechargeoftheprojectile(target).Thefunc-
C T (18)
tionsξ¯arethecoefficientsoftheexpansionEq.(6)thathave TheinitialconditionofEq.(4)translatesinto
tobeevaluatednumerically.
lefIt,nasenrdtiinngteEgqra.t(i6n)gionvtoerErq,.o(n2e),gmetusl[t1ip5l,ie1d6]byϕi′ℓ′m′ onthe z→lim−∞ψ(b,z,r)=ϕ0ℓ0m0(r). (19)
∂ TheDEAequation(18)issolvedforallbwithrespecttoz
ξ¯c(b,z)= andr expandingthewavefunctionψ onathree-dimensional
∂z
mesh. This allows to include naturally all relevant states
1
i~v (R) Fcc′(b,z)ξ¯c′(b,z)ei(Ki′−Ki)zRii′(b,z), of 15C, i.e., eigenenergies ε up to high values in the n-
i c′ 14C continuum, and large angular momentum ℓ, and its z-
X
(10) componentm. Thisresolutionisperformedassumingacon-
stant projectile-targetrelative velocity v = v . It should be
wheretheindexc denotesi, ℓ, andm together. InE-CDCC, 0
noted that this does not mean the adiabatic approximation,
theprojectile-targetvelocityv dependsonboththeprojectile
i because in Eq. (18) the internal Hamiltonian h is explicitly
excitationenergyanditspositionfollowing
included. The DEA thus treats properly the change in the
1 eigenenergyof 15C during the scattering process. However,
v (R)= ~2K2−2µV (R), (11)
i µ i C it does not change the 15C-208Pb velocity accordingly. This
q
4
givesaviolationoftheconservationofthetotalenergyofthe theexponentEq.(25)becomesthesameasinE-CDCC.Inthe
three-bodysystem. However,evenat20MeV/nucleon,itsef- modelspacetakeninthepresentstudy,Eq.(26)holdswithin
fectisexpectedtobeonlyafewpercentsasdiscussedbelow. 1.5%errorat20MeV/nucleonofincidentenergy.
The calculation of physical observablesrequiresthe wave Third, E-CDCC equation contains Rii′ taking account of
functionΨofEq.(16)atz →∞[17,18]. Thecorresponding the channel dependence of the 15C-208Pb Coulomb wave
Coulombphaseχ reads[26] function, which DEA neglects. Nevertheless, it should be
C
noted that, as shown in Refs. [15, 16], the Coulomb wave
lim χ =2η ln(K b), (20) functionsintheinitialandfinalchannelsinvolvedinthetran-
C 0 0
z→∞
sition matrix(T matrix)of E-CDCCeventuallygivea phase
whereη0 istheSommerfeldparameterfortheentrancechan- 2ηjln(Kjb), with j the energy index in the final channel.
nel[seeEq.(9)]. Thus, if Eq. (26) holds, the role of the Coulomb wave func-
tionintheevaluationoftheT matrixinE-CDCCisexpected
tobethesameasinDEA,sinceDEAexplicitlyincludesthe
D. ComparisonbetweenE-CDCCandDEA Coulombeikonalphase,Eq.(20).
When the Coulomb interaction is absent, we have
ToeasethecomparisonbetweentheDEAandtheE-CDCC, Rii′(b,z) = 1 andno R dependenceof the velocity. There-
fore, it will be interesting to compare the results of DEA
we rewrite the formulas given in Sec. IIC in a coupled-
andE-CDCCwithandwithouttheCoulombinteractionsepa-
channelrepresentation.Weexpandψas
rately.
ψ(b,z,r)= ξ (b,z)ϕ (r)eεiz/(i~v0)ei(m0−m)φR.
iℓm iℓm
Xiℓm (21) III. RESULTSANDDISCUSSION
InsertingEq.(21)intoEq.(18),multipliedbyϕi′ℓ′m′ from
theleft,andintegratingoverr,onegets A. Modelsetting
∂∂zξc(b,z)= i~1v Fcc′(b,z)ξc′(b,z)e(εi′−εi)z/(i~v0), disWtriebuctailocnuldatσe/tdhΩe eonfetrhgeybspreeacktruupmcrdoσs/sdsεecatniodnthoef a1n5Cguloanr
0 Xc′ (22) 208Pb at 20MeV/nucleon,where ε is the relativeenergybe-
tweenn and14Cafterbreakup,andΩ isthescatteringangle
which is nothing but the DEA equation (18) in its coupled-
ofthec.m. ofthen-14Csystem. Weusethepotentialparam-
channelrepresentation.
eters shown in Table I for U (the n-14C interaction), U ,
TheboundaryconditionEq.(19)thusreads nC 14
and U [24]; the depth of U for the d-wave is changedto
n nC
lim ξ (b,z)=δ δ δ . (23) 69.43 MeV to avoid a non-physical d resonance. The spin
z→−∞ iℓm i0 ℓℓ0 mm0 oftheneutronis disregardedasmentionedearlier. We adopt
Woods-Saxonpotentialsfortheinteractions:
ConsideringtheexpansionEq.(21)intheDEAfactorization
Eq.(16),thetotalwavefunctionreads U (R ) = −V f(R ,R ,a )−iW f(R ,R ,a )
x x 0 x 0 0 v x w w
d
Ψ(R,r) = ξc(b,z)ϕc(r)e(εi−ε0)z/(i~v0) +iWsdR f(Rx,Rw,aw) (27)
x
c
X
×ei(m0−m)φReiK0zeiχC(b,z). (24) withf(Rx,α,β)=(1+exp[(Rx−α)/β])−1;Rx =r,R14,
andR forx = nC, 14,andn , respectively. TheCoulomb
n
One may summarize the difference between Eqs. (22) interactionbetween14C and208Pb isdescribedbyassuming
and (10) as follows. First, the DEA uses the constant and auniformlychargedsphereofradiusRC.
channel-independent15C-208Pbrelativevelocityv , whereas
0
E-CDCCusesthevelocitydependingonbothRandthechan- TABLEI: Potential parameters for the pair interactions UnC, U14,
nelithatensuresthetotal-energyconservation. andUn[24].
Second, whereas the right-hand side of Eq. (22) involves
V R a W W R a R
thephaseexp[(εi′ −εi)z/(i~v0)], theE-CDCCEq.(10)in- (Me0V) (fm0) (fm0) (MevV) (MesV) (fmw) (fmw) (fmC)
cludes the phase exp[i(Ki′ −Ki)z]. The former can be UnC 63.02 2.651 0.600 — — — — —
rewrittenas U14 50.00 9.844 0.682 50.00 — 9.544 0.682 10.84
εii′~−vεiz = ~2(K2µi2i−~2KKi2′)µz = Ki2′K+Kii(Ki′ −Ki)z. Un 44.82 6.932 0.750 2.840 21.85 7.466 0.580 —
0 0 0 (25) Unless stated otherwise, the model spaces chosen for our
Ifwecanassumethesemi-adiabaticapproximation calculations give a confidence level better than 3% on the
crosssectionspresentedinSecs.IIIBandIIIC.InE-CDCC,
Ki′ +Ki ≈1, (26) we take the maximum value of r to be 800 fm with the in-
2K crementof 0.2 fm. When the Coulomb interaction is turned
0
5
off, we take the n-14C partial waves up to ℓ = 10. For 102
max
each ℓ the continuumstate is truncatedat k = 1.4 fm−1 w/o Coulomb
max
anddiscretizedinto35stateswiththeequalspacingof∆k = 101 DEA
0.04fm−1;kistherelativewavenumberbetweennand14C. E-CDCC (ℓ =10)
Tmhaexirmesuumltinvagluneusmobfezroafndcobu,pzlmedaxchaanndnbemlsa,xN,rcehs,piesc2t3iv1e1ly.,Tahree b/sr)100 E-CDCC (ℓmmaaxx=6)
bothsetto50fm. WhentheCoulombinteractionisincluded, Ω (
we use ℓ = 6, k = 0.84 fm−1, ∆k = 0.04 fm−1, d 10-1
max max σ/
z = 1000fm,andb = 150fm. We haveN = 589 d
max max ch
inthiscase. 10-2
In the DEA calculations, we use the same numerical pa-
rametersasinRef.[24]. Inthepurelynuclearcase,thewave
10-3
function ψ is expanded over an angular mesh containing up 0 5 10 15 20
to N ×N = 14×27points, a quasi-uniformradialmesh θ (deg)
θ ϕ
that extends up to 200 fm with 200 points, b = 50 fm,
max
andzmax =200fm(seeRef.[13]fordetails). Inthecharged FIG.3:(Coloronline)SameasFig.2butfortheangulardistribution.
case, the angular mesh contains up to N ×N = 12×23
θ ϕ
points,theradialmeshextendsupto800fmwith800points,
b =300fm,andz =800fm.
max max
solve the same equation and give the same result, as ex-
pected from the discussion at the end of Sec. IID. In par-
ticular this comparison shows that Eq. (26) turns out to be
B. ComparisonwithoutCoulombinteraction
satisfied withveryhighaccuracy. Itshouldbenotedthatthe
good agreementbetween the DEA and E-CDCC is obtained
Thegoalofthepresentworkbeingtostudythedifferencein
only when a very large model space is taken. In fact, if we
thetreatmentoftheCoulombbreakupbetweentheE-CDCC
putℓ = 6inE-CDCC,wehave30%smallerdσ/dε(dot-
and DEA, we first check that both models agree when the max
ted line) than the convergedvalue and, more seriously, even
Coulomb interaction is switched off. We show in Fig. 2 the
the shape cannot be reproduced. This result shows the im-
resultsofdσ/dεcalculatedbyDEA(solidline)andE-CDCC
portanceofthehigherpartialwavesofn-14Cforthenuclear
(dashed line); dσ/dε is obtained by integrating the double-
breakupat20MeV/nucleon.
differentialbreakupcrosssection d2σ/(dεdΩ) overΩ in the
whole variableregion. The two results agree very well with
eachother;thedifferencearoundthepeakisbelow3%.
C. ComparisonwithCoulombinteraction
In Fig. 3 the comparison in dσ/dΩ, i.e., d2σ/(dεdΩ) in-
tegrated overε up to 10 MeV, is shown. The agreementbe-
WhentheCoulombinteractionisswitchedon,DEAandE-
tween the two modelsis excellentconfirmingthat, when the
CDCCnolongeragreewitheachother.AsseeninFig.4,the
Coulomb interaction is turned off, the DEA and E-CDCC
DEAenergyspectrum(solidline)ismuchlargerthantheE-
CDCCone(dashedline). Moreovernoneofthemagreeswith
30 thefullCDCC calculation(thinsolid line): DEA istoo high
w/o Coulomb while E-CDCC is too low. The discrepancy of both models
25 DEA with the fully quantal calculation manifests itself even more
E-CDCC (ℓ =10) clearly in the angular distribution. In Fig. 5 we see that not
eV) 20 E-CDCC (ℓmmaaxx=6) onlydotheDEAandE-CDCCcrosssectionsdifferinmagni-
M
tude,but—asalreadyseeninRef.[24]—theiroscillatorypat-
b/ 15
m tern is shifted to forward angle compared to the CDCC cal-
(
ε culation. To understand where the problem comes from we
d 10
σ/ analyze in Fig. 6 the contribution to the total breakup cross
d
section of each projectile-target relative angular momentum
5
L . As expected from Figs. 4 and 5, the DEA calculation is
larger than the E-CDCC one, and this is observed over the
0
0 1 2 3 4 5 whole L range. However, the most striking feature is to see
ε (MeV)
thatboth modelsseem to be shifted to largerL comparedto
thefullCDCCcalculation. Tocorrectthis,wereplaceinour
FIG. 2: (Color online) Energy spectrum of the 15C breakup cross
calculationsthetransversecomponentoftheprojectile-target
sectionon208Pbat 20MeV/nucleon withtheCoulomb interaction
relativecoordinatebbytheempiricalvalue[26–28]
turnedoff. Thesolidanddashedlinesshowtheresultsobtainedby
DEAandE-CDCC,respectively.TheresultofE-CDCCwithℓmax =
6isdenotedbythedottedline. b′ = η0 + η02 +b2. (28)
K0 sK02
6
ThecorrespondingresultsaredisplayedinFigs.4,5and6as 14
DEA
dash-dottedlinesforDEAanddottedlinesforE-CDCC.
12 E-CDCC
ThecorrectionEq.(28)isveryeffective.Itsignificantlyre- fullCDCC
ducestheshiftobservedintheLcontributionstothebreakup 10 DEA
cross section (see Fig. 6). Accordingly, it brings both DEA r) withshift
andE-CDCCenergyspectraclosertothefullCDCCone(see b/s 8 E-CDCC
( withshift
Fig. 4). Note that for this observable the correction seems dΩ 6 hybrid
betterforE-CDCCthanforDEA:evenwiththeshift,thelat- σ/
d
terstillexhibitsanon-negligibleenhancementwithrespectto 4
CDCC at low energy E. More spectacular is the correction
2
of the shift in the angular distribution observed in Ref. [24]
and in Fig. 5. In particular, the shifted DEA cross section 0
0 2 4 6 8 10 12 14
is now very close to the CDCC one, but at forward angles,
θ(deg)
whereDEAoverestimatesCDCC.Onceshifted,E-CDCCstill
underestimatesslightlythefullCDCCcalculation. However,
itsoscillatorypatternisnowinphasewiththatoftheCDCC FIG.5:(Coloronline)SameasFig.4butfortheangulardistribution.
crosssection,whichisabigachievementinitself. Thisshows
thatthelackofCoulombdeflectionobservedinRef.[24]for
eikonal-basedcalculationscanbeefficientlycorrectedbythe
simpleshiftEq.(28)suggestedlongago[26,28]. providesasimple,elegant,andcost-effectivewaytoaccount
forCoulombdeflectionineikonal-basedmodels.
Albeitefficient,thecorrectionEq.(28)isnotperfect. This
isillustratedbytheenhanced(shifted)DEAcrosssectionob- TheunderestimationofthefullCDCCangulardistribution
served in the low-energy peak in Fig. 4 and at forward an- byE-CDCC comesmostlikely froma convergenceproblem
gles in Fig. 5. Both problems can be related to the same withinthatreactionmodel. ThisisillustratedinFig.7,show-
root because the forward-angle part of the angular distribu- ingtheL-contributiontothetotalbreakupcrosssection. The
tion is dominated by low-energy contributions. As shown thinsolidlinecorrespondstothe(converged)CDCCcalcula-
in Ref. [18], that part of the cross section is itself domi- tion,whereastheotherlinescorrespondto(shifted)E-CDCC
nated by large b’s, at which the correction Eq. (28) is not calculationswith binwidthsof∆k = 0.02(solidline), 0.03
fullysufficient. AsshowninFig.6,theshiftedDEAremains (dashed line), and 0.04 fm−1 (dotted line). As can be seen,
slightlylargerthanthefullCDCC.Futureworksmaysuggest belowL ≈ 500~,noconvergencecanbeobtained,although
abetterwaytohandlethisshiftthantheempiricalcorrection CDCC has fully converged. We cannot expect this model
Eq.(28). Nevertheless,theseresultsshowthatthiscorrection to provide accurate breakup cross sections. The results dis-
playedinFigs.4and5arethereforeunexpectedlygood.Note
thatthepresentill-behaviorofE-CDCCoccursonlywhenthe
500 Coulomb interaction involved is strong and the incident en-
DEA ergyislow;nosuchbehaviorwasobservedinpreviousstud-
E-CDCC
400
fullCDCC
V) DEAwithshift 10
e E-CDCCwithshift
M 300
b/ hybrid DEA
m E-CDCC
( 200 full CDCC
ε
d DEA with shift
dσ/ ) E-CDCC with shift
100 mb 1
(
σ L
0
0 1 2 3 4 5
ε(MeV)
FIG. 4: (Color online) Energy spectrum of the 15C breakup cross 0.1
sectionon208Pbat20MeV/nucleonincludingtheCoulombinterac- 0 100 200 300 400 500 600 700 800
tion.Thesolid,dashed,andthinsolidlinesshowtheresultsobtained L(ℏ)
byDEA,E-CDCC,andfull(QM)CDCC,respectively. Theresults
obtained with the correction (28) are displayed with a dash-dotted FIG.6:(Coloronline)Contributiontothetotalbreakupcrosssection
lineforDEAandadottedlineforE-CDCC.Thecalculationusing perprojectile-targetangularmomentumL. NeglectingtheCoulomb
theQMcorrectionofE-CDCC,i.e.,thehybridcalculation,isshown deflection,DEAandE-CDCCareshiftedtolargeLcomparedtothe
bythelight-greenthindashedline(superimposedontothethinsolid full CDCC. The correction Eq. (28) significantly reduces this shift
line). forbothmodels.
7
10
IV. SUMMARY
Δk = 0.02
Δk = 0.03 With the ultimate goal of understandingthe ability of the
Δk = 0.04 hybridversionofthecontinuum-discretizedcoupled-channels
full CDCC
methodwiththeeikonalapproximation(E-CDCC)toaccount
)
mb 1 for the discrepancy between the dynamical eikonal approxi-
σ (L mation(DEA)andthecontinuumdiscretizedcoupledchannel
model (CDCC) observed in Ref. [24], we have clarified the
relationbetweentheDEAandE-CDCC.Byusingacoupled-
channelrepresentationofDEAequations,DEAisshowntobe
formallyequivalenttoE-CDCC,ifthesemi-adiabaticapprox-
0.1 imationofEq.(26)issatisfiedandtheCoulombinteractionis
0 100 200 300 400 500 600 700 800
L(ℏ) absent.
We consider the same test case as in Ref. [24], i.e., the
breakupof15Con208Pbat20MeV/nucleon.Forthisreaction
FIG. 7: (Color online) Convergence problem observed in (shifted)
Eq.(26)holdswithin1.5%error.WhentheCoulombinterac-
E-CDCC calculations: cross sections computed with different bin
widthsdonotconvergetowardstheCDCCcalculation. tion is artificially turned off, DEA and E-CDCC are found
togivethesameresultwithin3%differenceforboththeen-
ergyspectrumandtheangulardistribution. Thissupportsthe
equivalenceofthetwomodelsfordescribingthebreakuppro-
cessduepurelytonuclearinteractions.
NextwemakeacomparisonincludingtheCoulombinter-
ies [6, 15, 16, 20, 21]. Interestingly, DEA does not exhibit action. Inthiscase, DEAandE-CDCCnolongeragreewith
suchaconvergenceissue. Thisisreminiscentoftheworkof each other and they both disagree with the full CDCC cal-
Dassoetal.[29],whereitwasobservedthatreactioncalcula- culation. In particular bothangulardistributionsare focused
tionsconvergefasterbyexpandingthewavefunctionupona at too forward an angle, as reported in Ref. [24]. This lack
meshratherthanbydiscretizationofthecontinuum. of Coulomb deflection of the eikonal approximation can be
solvedusingtheempiricalshiftEq.(28). Usingthisshiftthe
The aforementionedresults indicatethatthe shiftEq. (28) agreementwithCDCCimprovessignificantly.
corrects efficiently for the Coulomb deflection, which is ex- Inaddition,E-CDCCturnsouttohaveaconvergenceprob-
pected to play a significant role at large L. At small L, we lem,whichindicatesthelimitofapplicationoftheeikonalap-
believe the nuclear projectile-target interaction will induce proximationusingadiscretizedcontinuumtoreactionsatsuch
significant couplings between various partial waves, which low energiesinvolvinga strongCoulombinteraction. Fortu-
cannot be accounted for by that simple correction. To in- nately, by including a QM correction to E-CDCC for lower
clude these couplings, a hybrid solution between E-CDCC projectile-target partial waves, this convergence problem is
and the full CDCC has been suggested [15, 16]. At low L completelyresolved.Moreover,theresultofthishybridcalcu-
ausualCDCCcalculationisperformed,whichfullyaccounts lationperfectlyagreeswiththeresultofthefull(QM)CDCC.
forthestrongcouplingexpectedfromthenuclearinteraction The present study confirms the difficulty to properly de-
betweentheprojectileandthetarget. AtlargerL,thesecou- scribe the Coulomb interaction within the eikonal approx-
plingsareexpectedtobecomenegligible,whichimpliesthat imation at low energy. However, it is found that even at
a (shifted) E-CDCC calculation should be reliable. As ex- 20MeV/nucleon,theempiricalshiftEq.(28)helpscorrectly
plained in Refs. [15, 16], the transition angular momentum reproducing the Coulomb deflection that was shown to be
L abovewhichE-CDCC isusedis anadditionalparameter missing in the DEA [24]. Including QM corrections within
C
of the model space that has to be determined in the conver- theE-CDCCleadstoahybridmodelthatexhibitsthesameac-
gence analysis. Dependingon the beam energyand the sys- curacyasthefullCDCC,withaminimalcomputationalcost.
temstudied,usualvaluesofL areintherange400–1000~. This hybrid calculation will be useful for describing nuclear
C
Inthepresentcase,duetotheconvergenceissueobservedin and Coulomb breakupprocesses in a wide range of incident
E-CDCC,thevalueL =500~ischosen. energies. It could be included in other CDCC programs to
C
increasetheircomputationalefficiencywithoutreducingtheir
Thequalityofthishybridsolutionisillustratedbythelight- accuracy.Thiscouldbeanassettoimprovethedescriptionof
greenthindashedlinesinFigs.4and5,whicharebarelyvis- projectileswhilekeepingreasonablecalculationtimes.
ible as theyare superimposedto the fullCDCC results. The
couplingofthehybridsolutiontotheCoulombshiftEq.(28)
enables us to reproduce exactly the CDCC calculations at a Acknowledgment
muchlowercomputationalcostsincethecomputationaltime
foreachbwithE-CDCCisabout1/60ofthatforeachLwith The authors thank D. Baye and M. Yahiro for valuable
fullCDCC.Inaddition,itsolvestheconvergenceproblemof comments on this study. This research was supported in
E-CDCC. part by the Fonds de la Recherche Fondamentale Collective
8
(Grant No. 2.4604.07F), Grant-in-Aid of the Japan Society clei(BRIX),ProgramNo. P7/12,oninteruniversityattraction
forthePromotionofScience (JSPS),andRCNP YoungFor- polesoftheBelgianFederalSciencePolicyOffice.
eign Scientist Promotion Program. This paper presents re-
searchresultsoftheBelgianResearchInitiativeoneXoticnu-
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