Nucleon scattering on actinides using a dispersive optical model with extended couplings

25 Pages·2016·0.6 MB·English

PHYSICALREVIEWC94,064605(2016)
Nucleonscatteringonactinidesusingadispersiveopticalmodelwith extendedcouplings
E.Sh.Soukhovitski˜ı,1 R.Capote,2,*J.M.Quesada,3 S.Chiba,4andD.S.Martyanov1
1JointInstituteforEnergyandNuclearResearch,Minsk-Sosny220109,Belarus
2NAPC-NuclearDataSection,InternationalAtomicEnergyAgency,ViennaA-1400,Austria
3DepartamentodeF´ısicaAto´mica,MolecularyNuclear,UniversidaddeSevilla,Apartado1065,SevillaE-41080,Spain
4ResearchLaboratoryforNuclearReactors,TokyoInstituteofTechnology2-12-1-N1-9Ookayama,Meguro-ku,Tokyo152-8550,Japan
(Received30November2015;revisedmanuscriptreceived2October2016;published9December2016)
TheTamuracouplingmodel[Rev.Mod.Phys.37,679(1965)]hasbeenextendedtoconsiderthecouplingof
additionallow-lyingrotationalbandstotheground-stateband.Rotationalbandsarebuiltonvibrationalbandheads
(even-eventargets)orsingle-particlebandheads(odd-Atargets)includingbothaxialandnonaxialdeformations.
These additional excitations are introduced as a perturbation to the underlying axially symmetric rigid-rotor
structure of the ground-state rotational band. Coupling matrix elements of the generalized optical model are
derived for extended multiband transitions in even-even and odd-A nuclei. Isospin symmetric formulation of
the optical model is employed. A coupled-channels optical-model potential (OMP) containing a dispersive
contributionisusedtofitsimultaneouslyallavailableopticalexperimentaldatabasesincludingneutronstrength
functions for nucleon scattering on 232Th, 233,235,238U, and 239Pu nuclei. Quasielastic (p,n) scattering data on
232Th and 238U to the isobaric analog states of the target nucleus are also used to constrain the isovector
part of the optical potential. Lane consistent OMP is derived for all actinides if corresponding multiband
coupling schemes are defined. For even-even (odd-A) actinides almost all low-lying collective levels below
1MeV(0.5MeV)ofexcitationenergyarecoupled.OMPparametersshowasmoothenergydependenceand
energy-independentgeometry.Aphenomenologicaloptical-modelpotentialthatcouplesmultiplebandsinodd-A
actinidesispublishedforafirsttime.CalculationsusingthederivedOMPpotentialreproducemeasuredtotal
cross-section differences between several actinide pairs within experimental uncertainty for incident neutron
energiesfrom50keVupto150MeV.Theimportanceofextendedcouplingisstudied.Multibandcouplingis
stronger in even-even targets owing to the collective nature of the coupling; the impact of extended coupling
onpredictedcompound-nucleusformationcrosssectionreaches5%below3MeVofincidentneutronenergy.
Excitation of multiple bands in odd-A targets is weaker owing to the single-particle nature of the coupling.
Coupling of ground-state rotational band levels in odd-A nuclei is sufficient for a good description of the
compound-nucleusformationcrosssectionsaslongasthecouplingissaturated(aminimumofsevencoupled
levelsaretypicallyneeded).
DOI:10.1103/PhysRevC.94.064605
I. INTRODUCTION scattering cross sections and neutron emission spectra and
reductionoftheiruncertaintiesforneutron-inducedreactions
The cross sections for the interaction of neutrons with
on actinides is an important issue that should initiate new
actinide nuclei are crucially important for the design of
theoreticalstudies.
various fission reactor systems. Tight target uncertainties on
Many theoretical models are being developed to describe
the capture and inelastic scattering data for major actinides
nucleon scattering on actinide nuclei. The optical model is
were derived from advanced reactor sensitivity studies by
one of the fundamental theoretical tools which provides the
Nuclear Energy Agency WPEC Subgroup-26 [1]. However,
basis of nucleon-scattering data evaluations [3]. An accurate
the present status of evaluated nuclear data files for inelastic
calculation of the nucleon scattering from a well-deformed
scattering on major actinides is not satisfactory, as discussed
actinide nucleus using the optical model must include the
at the IAEA Technical Meeting on “Inelastic Scattering and
couplingtothelow-lyingcollectivestates.Averysuccessful
Capture Cross-section Data of Major Actinides in the Fast
computational method to account for the importance of the
Neutron Region” [2] held in 2011. Significant differences in
multistep processes is the coupled-channels (CC) method
evaluated inelastic cross sections were observed in evaluated
[4] using Tamura’s formalism [5], which permits an exact
nucleardatafilesfrom200keVuptoafewMeVofincident
solution of the CC equations. Many deformed optical poten-
neutronenergyformajoractinides[2].Suchdifferenceslead
tials suggested for actinides are based on Lagrange work at
to large uncertainties in the inelastic scattering cross section
Commissariata`l’e´nergieatomiqueetauxe´nergiesalternatives
(n,n(cid:2)),aswellasincalculatedmultipleneutronemissioncross
(CEA), France [6], but coupled-channels equations were a
sections (n,2n) and (n,3n). The improvement of evaluated
formidablechallengeforthecomputingcapabilitiesof1970s
and 1980s. Many approximations were suggested and used;
it was customary to couple only a few levels (usually three)
of the ground-state rotational band in those calculations. In
*[email protected] 2004 two of the authors studied the convergence of neutron
2469-9985/2016/94(6)/064605(25) 064605-1 ©2016AmericanPhysicalSociety
E.SH.SOUKHOVITSKI˜Ietal. PHYSICALREVIEWC94,064605(2016)
crosssectionson238Uasafunctionofthenumberofcoupled quantity for statistical reaction modeling. It has been shown
targetstatesintheground-staterotationalband.Itwasshown thatσCN(E)stronglydependsonemployedcoupled-channels
thatthecommonpracticeofcalculatingneutroncrosssections couplings [8]. The proposed extension should account for
with three coupled levels in K =0+ ground-state bands is multiple-band couplings, while keeping the achieved quality
inadequate [7]. This result was confirmed and generalized in of description of the whole set of scattering data (e.g., total
a new comprehensive study by Dietrich et al. [8], but none cross-sectiondataabove5MeVweredescribedbytheRIPL
of these papers included additional vibrational bands in the 2408potentialwithinthequotedexperimentaluncertaintyof
coupling scheme of even-even nuclei or additional rotational about1.5%).Thecouplingofvibrationalbandsisexpectedto
bandsinodd-Anuclei. improve the description of neutron scattering on even-even
Needed higher accuracy of data for fast reactors requires actinides in the energy region from 500 keV up to about
improvingthedescriptionofscatteringdataatincidentneutron 6MeV,whichiscriticalforfastneutronfissionreactors.The
energies from a few keV up to 5–6 MeV to cover the couplingofrotationalbandsbuiltonsingle-particlebandheads
region with the maximum yield of fission neutrons. While forodd-Anucleiisexpectedtoreduceobserveddiscrepancies
the energies of excited states of the ground-state band of from 100 KeV up to 1 MeV incident neutron energy in
even-even actinides below 500 keV are well described by a inelastic scattering on major fissile actinides 233U, 235U, and
rigid-rotor model, above 500 keV several vibrational bands 239Pu [2]. The authors are unaware of previously published
are observed that need to be considered. The situation for optical-model studies for odd-A actinide targets involving
odd actinides is even more complex, as no pairing gap couplingofmultiplebands.
exists; therefore, low-lying excited states are dominated by The paper is structured as follows. Sections II and III
rotationalbandsbuiltonsingle-particle(1QP)bandheads(e.g., provide descriptions of the nuclear shape parametrization,
particle-holeconfigurationswithK =1/2+,7/2−,and5/2+, nuclearHamiltonian,andnuclearwavefunctionsneededinthe
and corresponding rotational bands, dominate the low-lying solutionof theSchro¨dinger equation describing thenucleon-
excitation spectra for 235U and 239Pu nuclei). Therefore, scattering problem. Section IV contains the new coupling
low-lying rotational bands, built on vibrational bandheads matrixelementsderivedwithinTamura’sformalismallowing
for even-even targets, and on single-particle bandheads for for multiple-band coupling; this is one of the key results of
odd-A targets, need to be taken into account to describe thispaper.Forinterestedreadersadetailedderivationisgiven
neutron inelastic scattering on actinides. This fact has long inAppendixesAandB.SectionVdescribestheformulation
beenrecognizedforeven-evennuclei;avibrational-rotational of dispersive Lane-consistent potential for the simultaneous
description within the coupled-channels approach has been description of neutron- and proton-induced reactions on
used to describe scattering data on even-even actinides by actinides. In Sec. VI, we discuss derived optical potential
UniversityofLowellgroup[9–11]andlaterusedbyKawano parameterstodescribenucleonscatteringonactinidesusinga
etal.[12],Minskgroup[13,14],andBruye`res-le-Chaˆtelgroup propermultibandlevelscheme;wealsocomparethederived
(e.g.,Refs.[15,16]).Authorspresentedpreliminaryfindingsof potential with selected experimental data, including highly
this work for axial-symmetric even-even nuclei in Ref. [17]. accurate averaged total cross-section differences measured
However, authors are not aware of similar published work for incident neutron energies from 50 keV up to 150 MeV.
undertaken for odd actinides,1 for which typically only the Comprehensivecouplingschemesandmultibandcouplingpa-
ground-state rotational band had been considered in optical- rametersderivedfromaleast-squaresfitfor232Th,233,235,238U,
modelstudies.2 Additionally,thoseworksusednondispersive and239PutargetsaretabulatedinAppendixC.Finally,Sec.VII
potentials to describe scattering data, except the dispersive containsourconclusions.
potential used to describe neutron scattering on 238U target
[15]andotheractinides[16].
The main purpose of this comprehensive contribution is
to improve the description of neutron scattering on actinides
II. NUCLEARSHAPEPARAMETRIZATION
at incident neutron energies below 6 MeV by extending
FOREXTENDEDCOUPLINGSCHEME
a previously derived dispersive optical-model potential for
nucleon-inducedreactionsonactinidesbasedonarigid-rotor Actinides are well-deformed nuclei, where low-lying col-
description [19–21]. The previously published rigid-rotor lectivelevelsarestronglyexcitedinnucleoninelasticscatter-
potential of Refs. [19–21] will be referred in the rest of this ing. The employed nuclear shape parametrization combines
workasthe“RIPL2408”potentialbyusingthekeywordfrom a rigid-rotor description of the ground-state rotational band,
theRIPLoptical-modeldatabase[22]. with small quadrupolar and octupolar vibrations around the
Special focus will be on predicted compound-nucleus deformed equilibrium shape. Vibrations are described in the
formation cross section σ (E), which is a critical input spiritofthesoft-rotatormodel[23,24],whichconsidersnuclei
CN
to be deformable with both axial and nonaxial vibrations;
deformations with λ(cid:2)4 are considered axial, assuming that
1P.Romainhasusedextendedcouplingschemesforcalculationsof such deformations are usually small. Rotational bands are
then+235Ureaction,butnoresultshavebeenpublished. built on top of vibrational (single-particle) bandheads for
2Preliminary findings for odd nuclei were presented at ND2013 even-even (odd) nuclei, respectively. The deformed nuclear
[18];however,currentresultssupersedeolderpublicationsasfurther opticalpotentialarisesfromdeformedinstantnuclearshapes
understandingbothinderivationandresultswasachieved. ofactinidenucleus.Theinstantnuclearshapeinabody-fixed
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systemcanbedescribedas
(cid:2) (cid:4) (cid:5)
(cid:3)
1
Ri(θ(cid:2),ϕ(cid:2))=R0i 1+ βλ0Yλ0(θ(cid:2))+β2 cosγY20(θ(cid:2))+ √ sinγ[Y22(θ(cid:2),ϕ(cid:2))+Y2−2(θ(cid:2),ϕ(cid:2))]
2
λ=4,6,8
(cid:4) (cid:5)(cid:6)
1
+β3 cosηY30(θ(cid:2))+ √ sinη[Y32(θ(cid:2),ϕ(cid:2))+Y3−2(θ(cid:2),ϕ(cid:2))] , (1)
2
where Y (θ(cid:2),ϕ(cid:2)) means spherical harmonics, θ(cid:2) and ϕ(cid:2) are follows:
λμ
the angular coordinates in the body-fixed (intrinsic) system, (cid:7) (cid:8)
(cid:3)
βλ0 are the static axial deformations, β2 and β3 are the R (θ(cid:2),ϕ(cid:2))=R 1+ β Y (θ(cid:2))
quadrupolar and octupolar deformations, and γ and η are i 0i λ0 λ0
λ=2,4,6,8
the quadrupolar (γ) and octupolar (η) nonaxial deformation (cid:9) (cid:10)
parameters,respectively.WeintroducedthenotationYλ0(θ(cid:2))≡ +R β δβ2 cosγ +cosγ −1 Y (θ(cid:2))
Y (θ(cid:2),ϕ(cid:2)).WefollowedBohr[25]andassumedthatthebody- 0i 20 β 20
λ0 20
fixed frame is chosen to be the principal nuclear symmetry sinγ
axes, such that β21 =β2−1 =0 and β22 =β2−2; therefore, +R0i(β20+δβ2) √2 [Y22(θ(cid:2),ϕ(cid:2))+Y2−2(θ(cid:2),ϕ(cid:2))]
quadrupolar deformations are described by two parameters
β2 and γ. We also followed Lipas and Davidson [26] and +R0iβ3cosηY30(θ(cid:2))
assumed that octupolar vibrations with even projection (λ= sinη
3,μ=0,±2) best describe the octupole contribution to the +R0iβ3 √ [Y32(θ(cid:2),ϕ(cid:2))+Y3−2(θ(cid:2),ϕ(cid:2))]. (2)
2
nuclear vibrations for the low-lying negative-parity states.
Therefore, the nonaxial deformations β3±1 =β3±3 =0 and The first term of the sum in curly brackets in Eq. (2)
octupolar deformations are described by two parameters β correspondstotheaxiallysymmetricequilibriumshape(rigid
3
and η (such assumption could be easily removed if needed rotor). There are two additional quadrupole and octupole
by introducing one additional parameter). In what follows axially symmetric terms [proportional to Y20(θ(cid:2)) and Y30(θ(cid:2)),
we will use the fact that for nonaxial contributions (μ(cid:5)=0) respectively],andnonaxialquadrupoleandoctupoleterms.If
in Eq. (1) only even values of the projection μ=0,±2 are wefurtherassumethattheoctupoledeformationparameterβ3
allowed for both quadrupolar λ=2 and octupolar λ=3 is small, we can expand the nuclear potential V(r,R(θ(cid:2),ϕ(cid:2)))
vibrations. around the equilibrium rigid-rotor shape (instead of the
Nuclear radiiR =r A1/3 (i =HF,v,s,C,so)aredefined equilibrium spherical shape used by Tamura; see Eq. (5) of
0i i
for five potential geometries described in the next section, Ref.[5]).IfweinsertthenuclearshapegivenbyEq.(2)into
A being the target mass number. The soft-rotator model of the optical potential and make a first-order Taylor expansion
nuclear structure has been successfully applied in coupled- around the equilibrium (axial) shape, we can obtain an
channels optical-model analyses for many nuclei [13,27,28]. expressionforthedeformedopticalpotentialexpansion,
However,untilrecentlywewereunabletoderiveadispersive
coupled-channels optical-model potential for actinides based V(r,R(θ(cid:2),ϕ(cid:2)))=[V(cid:9)(r,R(θ(cid:2),ϕ(cid:2)))](δβ2=0,γ=0(cid:10),β3=0)
onsoft-rotatorcouplings.Suchworkisstillinprogress.Dis- ∂
+ R V(r,R(θ(cid:2),ϕ(cid:2)))
persiveoptical-modelfeaturesareducednumberofparameters 0
∂R
compared to traditional optical-model analyses; therefore, (cid:11) (cid:9) (δβ2=0,γ(cid:10)=0,β3=0)
parameter compensation of model defects becomes more × β δβ2 cosγ +cosγ −1 Y (θ(cid:2))
difficult for dispersive potentials. Mathematically, the failure 20 β 20
20
of the traditional soft-rotator model applied to actinides is sinγ
attributable to the very slow convergence of the multipolar +(β20+δβ2) √ [Y22(θ(cid:2),ϕ(cid:2))+Y2−2(θ(cid:2),ϕ(cid:2))]
2
expansion of the soft-rotor potential around the spherical (cid:4)
shape for large-equilibrium deformations typically encoun- +β cosηY (θ(cid:2))+ s√inη[Y (θ(cid:2),ϕ(cid:2))
3 30 32
tered in actinides. To solve this problem it was assumed 2
(cid:5)(cid:6)
that the quadrupole variable β of the soft-rotator model
2
can be considered as a large equilibrium axial component +Y3−2(θ(cid:2),ϕ(cid:2))]
β plus a small contribution δβ (i.e., β =β +δβ ).
20 2 2 20 2
This assumption allows a much better convergence of the + (potentialhigherderivatives’terms), (3)
potential expansion, as we use a truly small parameter δβ ,
2
instead of the relatively large parameter β originally used where
2
by Tamura [5]. The departures from the axially symmetric
rigid-rotor shape are considered to all orders in quadrupole (i) [V(r,R(θ(cid:2),ϕ(cid:2)))](δβ2=0,γ=0,β3=0) ≡Vrot(r,Raxial(θ(cid:2))),
nonaxiality parameter γ and octupole nonaxiality parameter (ii) [R0∂∂RV(r,R(θ(cid:2),ϕ(cid:2)))](δβ2=0,γ=0,β3=0) ≡
η. Under these assumptions we can rewrite Eq. (1) as [R0∂∂RV(r,R(θ(cid:2),ϕ(cid:2)))]R=Raxial(θ(cid:2)),
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E.SH.SOUKHOVITSKI˜Ietal. PHYSICALREVIEWC94,064605(2016)
aretheaxiallysymmetriccomponentsofthedeformedoptical- A. Even-evennuclei
modelpotential,andthenonaxialcomponentsareincludedin
Followingourassumptionsforactinides,wedonotconsider
Eq.(3)withinlargeparentheses.Ageneralexpressionforthe
unpairednucleonsforeven-evennuclei(norsingle-particleex-
multipolarexpansionofthedeformedoptical-modelpotential citations);therefore,thenuclearHamiltonianH ≡H +H
rot vib
isderivedinAppendixA;seeEq.(A6).
describesthecollectivemotiononly,neglectingtheinteraction
of vibrational and rotational states. Adiabatic approximation
for the collective motion means that the even-even nuclear
III. NUCLEARHAMILTONIANANDTARGET-NUCLEUS wave function can be factorized into a rotational and a
WAVEFUNCTIONS vibrational part (cid:12) =(cid:13) ((cid:14))|n(λ )(cid:7), where the rotational
rot ph
wave function (cid:13) ((cid:14)) depends on the Euler angles (cid:14),
Excitedstatesinactinidesareusuallyclassifiedintosingle- rot
and the vibrational wave function |n(λ )(cid:7) depends on the
particleandcollectiveexcitations,thelatterbeingdividedinto ph
numbern ofexcitedvibrationalphononsofmultipolarityλ
vibrationalandrotationalstates.Suchdivisionworkswellfor ph ph
and,implicitly,onthecorrespondingvibrationaldeformation
even-even nuclei owing to the pairing gap; only collective
statesarelocatedinsidethegap.However,forodd-Aactinides variablesβ2,β3,γ,andη.Thephononparityisπph =(−1)λph.
Inthisworkweconsideronlyone-phononvibrationalstates.
theenergyofsingle-particleexcitationscouldbeofthesame
Thephonondescriptionofvibrationalstatesalsoimpliesthat
order of the rotational energies, and so the independence of
interbandtransitionswhenthenumberofphononschangesby
thesingle-particleexcitationsfromthecollectivemovementis
morethanone((cid:15)n >1)areforbidden(thisisawell-known
hardly justifiable. These cases have been extensively studied ph
selection rule of the vibrational phonon model). Therefore,
by Davydov and co-workers [29–32] and Davidson [33]
strongest interband transitions occur between the bands built
(for additional information see references therein). Davydov
onone-phononstatesandtheground-stateband.
showed that in even-even nuclei whose equilibrium shape is
Assuming that the nucleus behaves as a triaxial rotor and
closetotheaxiallysymmetricshape(e.g.,actinides),rotation
denotingK asthetotalangularmomentumprojectiononthe
cannotberegardedindependentlyofγ vibrations[30].Forodd
symmetryaxisandI (I (cid:2)K)asthetotalangularmomentum,
nuclei,theinteractionofthenuclearrotationwiththeunpaired
then the nuclear wave function (cid:12) is mixed in K and can be
nucleon may change both the structure of the rotational
writtenintheform[25,34]
spectra and the energies of the single-particle excited states
(cid:9) (cid:10)
[32]. Therefore, it is expected that low-lying excited states (cid:3)I 2I +1 1/2(cid:12)
of odd-A nuclei have a complex structure which does not (cid:12)(IMτ(cid:14))= AIτ DI (θ(cid:2),ϕ(cid:2))
K 16π2(1+δ ) MK
permit the separation of the single-particle, rotational, and K=0 K0
(cid:13)
vinibaracttiionnidaelsdeagrereeexspoefctferdeetdoomha.vHeodweefivneirt,eavllaleuxecsitoedf psatartietys +(−1)I+λphDMI −K(θ(cid:2),ϕ(cid:2)) |n(λph)(cid:7). (5)
π =(−1)l,l beingtheorbitalangularmomentum,andofthe ThecoefficientsAIτ aretheK-mixingcoefficientsthatdepend
K
totalangularmomentumI.Theanalysisaboveclearlyshowed on all deformation parameters and satisfy orthonormality
that even-even and odd-A actinides have a very different conditions
(cid:3) (cid:3)
nuclearstructure.
Letusdefinethenuclearstructuremodeltobeusedinthis AIKτAIKτ(cid:2) =δττ(cid:2) and AIKτAIKτ(cid:2) =δKK(cid:2), (6)
work.Weassumethatanucleusiscomposedoftheeven-even K(cid:2)0 τ
core where only paired nucleons are present. Additionally, where τ denotes the additional quantum numbers. The
forodd-Anuclei,weconsiderasingleunpairednucleonthat expression (5) can be considered a generalization of
moves in the nuclear mean field created by the even-even the wave function given in Eq. (91) of Ref. [25] and
core.Wefurtherassumethatthenucleargroundstatemaybe Eqs. (9–59) and (9–60) of Ref. [34] (all restricted to
staticallydeformed.Dynamical deformations areassumedto quadrupolardeformations)foranyphononmultipolarity.The
besmall,andErot (cid:6)Evib;i.e.,theadiabaticassumptionholds phonon wave function |n(λph)(cid:7) has been ignored in those
fortheseparationoftherotationalandvibrationalmotionfor references, because for even-even nuclei the lowest-lying
even-even and odd-A targets. Under those assumptions, the states are given by rotational and vibrational modes with
nuclearHamiltoniancanbewrittenas intrinsic spin 0 [34]. Note that if K =0, then the condition
I +λ =evenholdsandthephasefactorbecomesunity.The
ph
H =H +H +H +H , (4) phononparityoperatorπph =(−1)λph inEq.(5)ismissingin
rot vib p int Eq. (3) of Ref. [35] as only quadrupolar vibrations featuring
positive parity were considered. In this work, the phonon
where H is the rotational energy operator, H is the parityhasbeenconsideredasneededfortheproperinclusion
rot vib
vibrational energy operator, H is the energy operator of the ofbothquadrupolarλ =2andoctupolarλ =3vibrations.
p ph ph
unpaired nucleon (single-particle operator), and H is ThecoefficientsAIτ,whichdetermineK-mixinginagiven
int K
the interaction energy operator of the unpaired nucleon rotationalstateowingtothenonaxiality,aretobedetermined
with the nuclear even-even core field. The corresponding fromthesolutionoftherotationalproblemandcanbeobtained
Schro¨dingerequationcanbewrittenasH(cid:12) =E(cid:12),wherethe asdescribedinRef.[36]bysolvingtheSchro¨dingerequation
nucleareigenfunction(cid:12)describessingle-particle,vibrational, corresponding to the nonaxial soft nuclear Hamiltonian.
androtationalmotions. The code SHEMMAN adjusts the soft-rotator Hamiltonian
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NUCLEONSCATTERINGONACTINIDESUSINGA ... PHYSICALREVIEWC94,064605(2016)
parametersbycomparingtheenergiesoftheexperimentaland wave function given by Eq. (5). Such analogy allows using
calculatedcollectivelevels.Therefore,thecoefficientsAIτ are the same matrix element derived for the even-even nuclear
K
determinedindependentlyfromcoupled-channelsequations. wave function to obtain the odd-A matrix element, as will
For axially symmetric nuclei the projection K becomes a be shown in Appendix B. There are only two differences
goodquantumnumberandonlyoneAIτ coefficientisdifferent regarding the odd and even-even nuclear wave functions:
K
from zero. Therefore, the sum over K and the K mixing (1)thephaseinfrontofthesecondWignerfunctionchanges
disappearsinEq.(5),andweobtain to (−1)I−1/2π (odd-A wave function) from the even-even
χ
(cid:9) 2I +1 (cid:10)1/2(cid:12) case phase (−1)I+λph; and (2) there is a single-particle wave
(cid:12)(IMKλph(cid:14))= 16π2(1+δ ) DMI K(θ(cid:2),ϕ(cid:2)) function χν (χ−ν) that multiplies the first (second) Wigner
K0 (cid:13) functionsintheodd-AcaseinEq.(8).
+(−1)I+λphDMI −K(θ(cid:2),ϕ(cid:2)) |n(λph)(cid:7). (7) For axially symmetric nuclei the sum over K disappears,
K becomes a good quantum number that labels the single-
The optical-model calculations and fitting undertaken in particle state ν (i.e., χ ≡χ ), which automatically leads to
ν K
thisworkassumedthateven-evennucleiwereaxiallysymmet-
Eq.(9-139)ofRef.[34]:
ricrotorsintheirgroundstate.However,nonaxialformulation (cid:9) (cid:10)
of the wave function given by Eq. (5) was retained for the 2I +1 1/2(cid:12)
(cid:12)(IMK(cid:14))= DI (θ(cid:2),ϕ(cid:2))χ
calculationofmostgeneralcouplingmatrixelements. 16π2 MK K
(cid:13)
B. Odd-Anuclei +(−1)I−1/2DMI −K(θ(cid:2),ϕ(cid:2))πχχ−K . (9)
Following our assumptions for odd actinides, we neglect
Note that axial symmetry is an acceptable approximation
vibrational excitations of the core. The total angular mo-
−→ for odd-A nuclei, as demonstrated by an extensive use of
mentum of the nucleus I−→, a constant of the motion, is the rigid-rotor assumption in odd-A actinide optical-model
now the sum of two parts, C, the even-even core’s angular potentials.
−→
momentum, and j , the nucleon angular momentum, where
−→ −→ −→
I = C + j . It is customary to take the projections of IV. COUPLED-CHANNELSMATRIXELEMENTS
−→ −→
I and j along the intrinsic z(cid:2) axis to be K and (cid:17), FOREXTENDEDCOUPLINGSCHEME
−→
respectively, while the z component of I is M. Because Starting from the derived potential multipolar expansion
the potential in which the unpaired nucleon moves is, in oftheoptical-modelpotential[seeEq.(A6)inAppendixA],
general,neithersphericallynoraxiallysymmetric,j2,K,and andusingnuclearwavefunctionsgivenabove,wecanderive
(cid:17) are not constants of the motion. Following Refs. [34,35], the general expressions (B12) and (B16) for even-even and
we assume the extreme extracore single-particle model for odd targets, correspondingly (see Appendix B for a detailed
the Hamiltonian H describing the unpaired nucleon. The derivation) for coupling matrix elements (cid:9)i|V(r,θ,ϕ)|f(cid:7) be-
p
nucleonsingle-particlewavefunctionχ istheeigenfunction tween nuclear states |i(cid:7) and |f(cid:7). In those expressions, the
ν
ofthesingle-particleHamiltonianH thatdescribesanucleon reduced matrix elements have to be defined, and the radial
p
moving in the deformed nuclear mean field (e.g., Nilsson functions are given by Eq. (A4), which for i =1 are being
or Woods-Saxon potential), i.e., H χ =E χ , E being equal to the radial functions used in matrix elements of the
p ν ν ν ν
thesingle-particleenergycorrespondingtothesingle-particle conventionalrigid-rotorpotential.
stateν. ReducedmatrixelementsenteringEqs.(B12)and(B16)are
Following Eq. (3) of Ref. [35], the nuclear wave function different for couplings in even-even and odd-A nuclei as the
intheextremesingle-particlemodelisgivenby corresponding nuclear wave functions given by Eqs. (5)–(9)
(cid:9) (cid:10)
2I +1 1/2(cid:3)(cid:2) (cid:12) aredifferent.Ready-to-useexpressionsforcalculationsofthe
(cid:12)(IMτ(cid:14))= CIτ DI (θ(cid:2),ϕ(cid:2))χ reducedmatrixelementsaregivenbyEq.(B14)foreven-even
16π2 K MK ν
K>0 targetsandbyEq.(B19)forodd-Atargets.Thereducedmatrix
(cid:13)
+ (−1)I−1/2DMI −K(θ(cid:2),ϕ(cid:2))πχχ−ν , (8) ewlehmicehnatrseadlsioscduesfisnedetihneAseplpeecntidoinxrBu.lesdeterminingcouplings,
where π is the parity of the intrinsic wave function, and Thegeneralexpressionsmentionedaboveconstituteanim-
χ
CIτ aretheK-mixingcoefficientsthatdependondeformation portantresultofthisworkallowingtheextensionofTamura’s
K
parameters and are determined from the solution of the coupled-channels formalism to describe the coupling of
rotationalproblem(e.g.,seeRefs.[29,31]).Intheasymmetric additionalexcitedbandsbothforeven-evenandodd-Atargets.
case ellipsoidal symmetry imposes additional restrictions on Anexampleofthecouplingschemeemployedinthecurrent
quantum numbers K and (cid:17) (that characterizes the single- work is shown in Fig. 1 for a neutron-induced reaction on a
particle wave function χ ; see Appendix B), namely that 238U target. The corresponding effective deformations (e.g.,
ν
(K −(cid:17)) must be an even integer [35]. This restriction on [β ] )thatdefinetheinterbandcouplingstrengthareshown
2 eff
K value(cid:14)s is indicated by the apostrophe on the summation nearthetransitions.Thehighest-energystateconsideredinthe
symbol (cid:2) inEq.(8). ground-state rotational band of 238U was the 10+ located at
K>0
It is remarkable that the odd-A nucleus wave function 775.9keV;therigid-rotorassumptionbreaksdownabovethat
given by Eq. (8) has exactly the same collective angular excitationenergy[levelenergiesdonotfollowtherigid-rotor
operatorstructure(i.e.,Wignerfunctions)astheeven-nucleus predictionE(I)∼I(I +1)].
064605-5
E.SH.SOUKHOVITSKI˜Ietal. PHYSICALREVIEWC94,064605(2016)
2+ (1037.3) 4+ (1056.4)
0+ (993.0) 2+ (966.3)
0+ (927.0)
(775.9) 10+ (775.9) 10+
β-band (Kπ=0+)
-band (Kπ=0+)
(517.4) 8+ (517.4) 8+
(307.4) 6+ (307.4) 6+
(148.4) 4+ (148.4) 4+
(44.9) 2+ (44.9) 2+
0+ 0+
GS band (Kπ=0+) GS band (Kπ=0+)
3+ (1105.7)
9- (1150.7)
2+ (1060.3)
7- (966.4)
5- (826.7)
(775.9) 10+ (775.9) 10+
3- (731.3) Anomalous band
1- (679.8) (Kπ=2+)
(517.4) 8+ (517.4) 8+
Nega(cid:4)ve parity
band (Kπ=0-)
(307.4) 6+ (307.4) 6+
(148.4) 4+ (148.4) 4+
(44.9) 2+ (44.9) 2+
0+ 0+
GS band (Kπ=0+) GS band (Kπ=0+)
FIG.1. Multibandcouplingscheme(nottoscale)isshownforageneralizedoptical-modeldescriptionofn+238Ureaction.Transitions
betweenexcitedbandsandtheground-statebandareindicated;thosebetweenexcitedvibrationalbandsareforbiddenwithinthevibrational
phononmodel.NumbersinparenthesesnexttothelevelspinarecorrespondinglevelexcitationenergiesinkeV.Thetrivialin-bandquadrupolar
transitions with (cid:15)K =0 are omitted. The selection rules are discussed in Appendix B. Eighteen levels are coupled for neutron-induced
reactionsincludingtheground-staterotationalband(K =0+)andfourexcitedvibrationalbands(K =0+β band,K =0+γ band,K =0−
octupoleband,andanomalousK =2+nonaxialband).Forproton-inducedreactionsthecouplingfromtheground-statebandtoisobar-analog
states(IASs)shouldbeadded.
V. DISPERSIVEOPTICAL-MODELFORMALISM phenomenological OMP parameters from the experimental
data and effectively also reduces the number of phenomeno-
Pioneering dispersive optical-model analyses for nucleon
logicalparameters.
scattering have been carried out by Lipperheide [37,38],
However, a few studies have been devoted to derive dis-
Passatore[39],andLipperheideandSchmidt[40].Bothbound
persiveopticalmodelparametrizationsforstronglydeformed
and scattering states were calculated using the same nuclear
nuclei,wherecoupled-channelsformalismshouldincludepo-
meanfieldconstrainedbydispersiverelations[41–53].Recent
tentialcorrectionsarisingfromdispersioneffects[55–59].Re-
workstoimprovethedescriptionoftheboundstatesusingthe
cently,wederivedanisospin-dependentoptical-modelpoten-
dispersive optical model and describe reactions on unstable
tialsfor actinides based on rigid-rotorcouplings [20,21].We
targets off the β-stability valley have been published by a
willsummarizebelowtheformalismemployedinthiswork.
groupatWashingtonUniversity,St.Louis,MO(seeRef.[54]
Assumingthatthegeometryoftheimaginarytermsofthe
and references therein). Additional constraint imposed by
OMPisenergy-independent,thenthedeformedoptical-model
dispersionrelationshelpstoreducetheambiguitiesinderiving
potentialforincidentnucleonsmaybewrittenas
V(r,R(θ,ϕ),E)=−V (E)f (r,R (θ,ϕ))−[(cid:15)V (E)+iW (E)]f (r,R (θ,ϕ))−[(cid:15)V (E)+iW (E)]g (r,R (θ,ϕ))
HF WS HF v v WS v s s WS s
(cid:11) (cid:6)
(cid:2) 2 1 d
+ [V (E)+(cid:15)V (E)+iW (E)] f (r,R )(lˆ·σˆ)+V (r,R (θ,ϕ)), (10)
m c so so so r dr WS so Coul c
π
064605-6
NUCLEONSCATTERINGONACTINIDESUSINGA ... PHYSICALREVIEWC94,064605(2016)
where the first term is the real smooth volume potential dispersionrelation.Underfavorableconditionsofanalyticity
V (E). Successive complex-valued terms are the volume, inthecomplexEplanetherealpart(cid:15)V(E)canbeconstructed
HF
surface, and spin-orbit potentials, all containing the cor- from the knowledge of the imaginary part W(E) on the real
responding dispersive contributions (cid:15)V (E),(cid:15)V (E), and axisthroughthedispersionrelation
v s (cid:15)
t(cid:15)erVmso(VE) diisscnueseseddedinforSiencc.idVenAt.prTohtoensC.oWuleomfobllopwoetednotiuarl (cid:15)V(E)= P ∞ W(E(cid:2))dE(cid:2), (12)
Coul π −∞E(cid:2)−E
previous studies [19–21] and connected the imaginary spin-
orbit potential W (E) to the real spin-orbit potential V (E) whereP meansthattheprincipalvalueoftheintegralshould
so so
byadispersionrelationasdiscussedbyWalter[60]. be taken. Assuming that (cid:15)V(E =E )=0, where E is the
F F
ThegeometricalWoods-Saxonformfactorsaregivenas Fermienergy,Eq.(12)canalsobebewritteninthesubtracted
form
fWS(r,Ri(θ,ϕ))={1+exp[(r −Ri(θ,ϕ))/ai]}−1, P (cid:15) ∞ (cid:11) 1 1 (cid:6)
i =HF,v, (cid:15)V(E)= π −∞W(E(cid:2)) E(cid:2)−E − E(cid:2)−EF dE(cid:2). (13)
d
g (r,R (θ,ϕ))=−4a f(r,R (θ,ϕ)), HereE denotestheFermienergy,determinedasE (Z,A)=
WS s sdr s −1[S (FZ,A)+S (Z,A+1)] for neutrons and EF(Z,A)=
fWS(r,R0,so)={1+exp[(r −R0,so)/ai]}−1, (11) −212[Snp(Z,A)+Snp(Z+1,A+1)]forprotons,wherFeSi(Z,A)
denotes the separation energy of nucleon i from a nucleus
where deformed radii R (θ,ϕ) are described by Eq. (2).
i labeledbyZandA.ThesymmetryconditionW(2E −E)=
A spherical spin-orbit potential was used with constant F
W(E)(forsymmetricimaginarypotentials)isusedtoextend
radius equal to R . The deformed Coulomb potential
0,so
the imaginary part of the OMP for energies below the Fermi
V (r,R (θ,ϕ)) was calculated using a multipole expansion
Coul c
energy,allowingforthecalculationofthedispersiveintegral.
of charged ellipsoid with a uniform charge density within
ItisknownthattheenergydependenceofthedepthV (E)
the Coulomb radius R and zero outside as suggested by HF
0,C
isattributabletothereplacementofamicroscopicnonlocalHF
Basseletal.[61].ThesphericaltermoftheCoulombpotential
potentialbyalocalequivalent[64].ForaGaussiannonlocality
was calculated by taking account of the diffuseness of the
charge density distribution of the form f =[1+exp(r − VHF(E)isalinearfunctionofEforlargenegativeEandisan
c
R )/a ]−1[62]. exponentialforlargepositiveE.FollowingMahauxandSartor
0,C c
TheincidentnucleonenergyE =E isequaltotheincident [51],theenergydependenceofthesmooth“Hartree-Fock”part
n
ofthenuclearmeanfieldistakenasthatfoundbyLipperheide
energy for neutrons; for incident protons we have to take
[38],
into account the modification of the nuclear potential by
Coulombrepulsion.Weassumethatthe“effective”interacting V (E)=A exp[−λ (E−E )], (14)
energy of the proton is E =E −C Z(cid:2) , A, Z(cid:2) being HF HF HF F
p CoulA1/3 wheretheparametersA andλ areundeterminedconstants.
the target mass and atomic numbers. The term C Z(cid:2) is HF HF
CoulA1/3 Equation(14)canbeusedtodescribetheHFpotentialinthe
an estimate of the kinetic-energy loss of the incident proton
scatteringregime[51]forE >0.
in the interaction region owing to Coulomb repulsion. The
It is useful to represent the variation of surface W (E)
s
use of the “effective” interacting energy of the proton is a
and volume absorption potential W (E) depth with energy
v
generalization of the previously used Coulomb corrections,
in functional forms suitable for the dispersive optical-model
which considers such corrections in all orders. The constant
analysis, which are integrable analytically [65]. An energy
C is an adjustable constant meant to account for the
Coul dependencefortheimaginaryvolumetermhasbeensuggested
“effective”radiusofinteractionofprotoninnucleus.Avoiding
instudiesofnuclearmattertheorybyBrownandRho[66],
theuseofCoulombcorrectionsisapreconditiontotheLane
(E−E )2
consistency;otherwise,theresultingpotentialisnotsymmetric W (E)=A F , (15)
withrespecttothenucleoncharge. v v(E−E )2+(B )2
F v
The present optical potential includes relativistic correc-
where A and B are undetermined constants. An energy
v v
tions as discussed by Elton [63]. First, the nucleon wave
dependencefortheimaginary-surfacetermhasbeensuggested
numberkwascalculatedintherelativisticform((cid:2)k)2 =[E2−
byDelarocheetal.[49]tobe
(M c2)2]/c2, where E denotes the total energy of projectile,
p (E−E )2
Mp theprojectilerestmass,andc thelightvelocity.Second, W (E)=A F exp(−C |E−E |), (16)
projectile and target masses were replaced by corresponding s s(E−EF)2+(Bs)2 s F
relativistic energies in reduced mass formulas. However, the
whereA ,B ,andC areundeterminedconstants.
s s s
change of the potential depth related to the transformation
The isospin dependence of the potential (the Lane term
fromtheDiracequationwasnotconsideredinthiswork.
[67,68])wasconsideredinrealV (E)andimaginarysurface
HF
W (E)potentialsas
s (cid:9) (cid:10)
A. Dispersiverelations C N −Z
A =V 1+(−1)Z(cid:2)+1 viso , (17)
Inadispersionrelationtreatment,therealpotentialstrength HF 0
V A
0
consists of a term which varies slowly with energy, the (cid:9) (cid:10)
so-called Hartree-Fock (HF) term, VHF(E), plus a dynamic A =W 1+(−1)Z(cid:2)+1CwisoN −Z , (18)
(polarization) term, (cid:11)V(E), which is calculated using a s 0 W A
0
064605-7
E.SH.SOUKHOVITSKI˜Ietal. PHYSICALREVIEWC94,064605(2016)
where V ,C ,W , and C are undetermined constants. contribution to describe σ (E) data for energies above
0 viso 0 wiso T
Manyauthorsfoundthattheimaginaryvolumepotentialdoes 100MeVusinganonsymmetricversionofthevolumeabsorp-
notdependontheisospin. tivepotentialforlargepositiveandlargenegativeenergiesas
For the energy dependence of the spherical spin-orbit proposed by Mahaux and Sartor [51]. Similar behavior was
potential we used the functional form suggested by Koning confirmed in 232Th dispersive coupled-channels analysis of
andDelaroche[69],whichisconvenientforthecalculationof nucleon-induced reactions [19] and in combined dispersive
thedispersivecontribution[65],namely, coupled-channels analysis of nucleon-induced reactions on
232Th and 238U [20,21]. We use the same formalism in this
Vso(E)=Vspoexp[−λso(E−EF)], (19) work,whichisbrieflydescribedbelow.
Following Mahaux and Sartor [51], the assumption that
Wso(E)=Wspo(E−(EEF−)2E+F)(2Bso)2, (20) t[haceciomrdaignignatroy peqouteantitoianl WWv((2EE)Fis−syEm)m=eWtric(Ea)b]ouist Em(cid:2)o=difiEeFd
whereVspo,λso,Wspo,andBsoareundeterminedconstants. above some fixed energy Ea, which is expected to be close
to60MeV.However,thisvalueisfairlyarbitrary[51]andwe
B. High-energybehaviorofthevolumeabsorption
willuseitasafittingparameter.Letusassumetheimaginary
The dispersive-optical-model analysis of neutron scatter- potential to be used in the dispersive integral is denoted by
ing on 27Al [70] showed the importance of the dispersive W(cid:16) (E);thenwecanwrite[52]
v
W(cid:16) (E)=W (E)−W (E) (EF −E−Ea)2 , forE <E −E (boundregime), (21)
v v v (E −E−E )2+E2 F a
F a a
and
(cid:9) (cid:10)
W(cid:16) (E)=W (E)+α √E+(EF+Ea)3/2−3(cid:17)(E +E ) , forE >E +E (scatteringregime). (22)
v v v F a F a
2E 2
Thesefunctionalformsarechoseninsuchawaythatthefunc- intheform
tionanditsfirstderivativearecontinuousatE(cid:2) =|EF −Ea|. (cid:15)V(cid:16)(E)=(cid:15)V (E)+(cid:15)V (E)+α (cid:15)V (E), (23)
At large positive energies nucleons sense the “hard-core” v v < v >
(cid:16)
repulsiveregion√ofthenucleon-nucleoninteractionandWv(E) where (cid:15)Vv(E) is the dispersive correction owing to the
divergeslikeα E.UsingamodelofadiluteFermigashard symmetricimaginarypotentialofEq.(15),whichiscalculated
v
sphere,thecoefficientα canbeestimatedtobeequalto1.65 followingRef.[65],andtheterms(cid:15)V (E)and(cid:15)V (E)are
v < >
MeV1/2 [43], but the actual value is a model parameter. On the dispersive corrections owing to the asymmetric terms of
thecontrary,atlargenegativeenergiesthevolumeabsorption Eqs. (21) and (22), respectively, calculated as described in
decreases and goes asymptotically to zero. The asymmetric Ref. [71]. Unfortunately, the equation giving the dispersive
formofthevolumeimaginarypotentialofEqs.(21)and(22) correction(cid:15)V (E)inRef.[71]containsseveraltypos,sowe
>
resultsinadispersionrelationthatmustbecalculateddirectly givethecorrectexpressionbelow.
fromEqs.(12)and(13)andseparatesintothreeadditiveterms Using the notation E =E +E , the (cid:15)V (E) contribu-
L F a >
[71,72].Therefore,wewritethedispersivevolumecorrection tionforvolumenonlocalitycorrectionisexactly
(cid:18) (cid:17) (cid:19)
1 (cid:17) 2 E |E | E3/2 E
(cid:15)V (E)= × |E |arctan L F + L ln a
> π F E −|E | 2E E
L F F L
⎧√ √ √ (cid:17)
1 ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨√32(cid:17)EElnLl√√nEE2+−4/E√√3aEEELLL, + 32(cid:17)ELlnE−EEaL + E2L3E/2 lnE−ELEL, ffoorrEE =>EELL,,
+π ×⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩−32(cid:17)√EE|lEnL|l√naErEEEc+LLtaa−n+√E2EL12√+(cid:17)EEL32|LE,|E+Ll3n(cid:17)EELE−aEln+ELE2−L3EE/2 l+nEELEL3/−L2Eln, EL , fffooorrrEEEL=<>00E,. >0, (24)
EL−|E| 2 L Ea 2E EL−E
Theresultingdispersivecorrectionfortheasymmetriccase a significant contribution to the real part of the OMP at
startstoincreasealreadyforenergiesabove50MeV,making highenergies.Itshouldbenotedthatnonlocalitycorrections
064605-8
NUCLEONSCATTERINGONACTINIDESUSINGA ... PHYSICALREVIEWC94,064605(2016)
TABLEI. Dispersivecoupled-channelsoptical-modelpotentialparametersfornucleon-inducedreactiononactinides.
Volume Surface Spin-orbit Coulomb
Real V =50.47+0.0292(A−238)MeV V =6.1MeV C =1.36MeV
0 spo Coul
potential λ =0.00977MeV−1 – λ =0.005MeV−1
HF so
parameters C =17.3MeV
viso
A =11.81MeV W =17.43MeV W =−3.1MeV
v 0 spo
Imaginary B =81.81MeV B =10.57MeV B =160MeV
v s so
potential E =55MeV C =0.01331MeV−1
a s
parameters α =0.355MeV1/2 C =28.9MeV
v wiso
r =1.2468−0.00183(A−238)fm r =1.1717+0.0041(A−238)fm r =1.1214fm r =1.2894fm
HF s so c
Potential a =0.638+0.002134(A−238)fm a =0.618fm a =0.59fm a =0.547fm
HF s so c
geometry r =1.2657fm
v
a =0.6960−0.00021(A−238)fm
v
[Eqs. (21) and (22)] can be used either for the volume or bandsasshowninAppendixCforstudiedactinides.Optical-
surfaceimaginarypotential;however,MahauxandSartor[51] model code OPTMAN [76,77] was used for OMP parameter
have shown that nonlocality consideration for the surface fitting. We were using symmetric surface and nonsymmetric
imaginary potential has a very small effect on the calculated volume imaginary absorptive potentials, as described in the
crosssections.Therefore,inthisworkwefollowedRef.[52] previous section. The dispersion integrals were calculated
and only considered the effects of nonlocality in the volume usinganalyticalsolutions[65,71]ofdispersionrelationswith
absorption. Eq.(24)forthe(cid:15)V (E)contribution.
>
InourformulationoftheOMPinEq.(10)thegeometrical
parameters of the “Hartree-Fock” potential r and a are,
HF HF
C. Dispersivecoupled-channelsoptical-modelanalysis in general, different from geometrical parameters r ,a ,r ,a
v v s s
of the volume and surface absorptive potentials; however,
A survey of the experimental data for nucleon interaction
the real and imaginary spin-orbit terms share the same
on actinide nuclei spanning from 0.001 to 150 MeV used in
geometric r and a parameters. Also, the surface (cid:15)V and
thecurrentworkcoincidewiththedatausedbySoukhovitskii so so s
volume(cid:15)V dispersivecorrectionssharethesamegeometryof
andco-workersabout10yearsago[7].Thetotalcross-section v
correspondingimaginarypotentialsW andW ,respectively.
data considered cover all the critical energy points that are s v
Theemployedactinideopticalpotentialenergydependenceis
necessary to reveal the structure owing to the Ramsauer
verysimpleandweuseenergy-independentgeometryandthe
effectandextendsupto150MeV.Theexperimentaldatabase
available for actinide nuclei other than 232Th and 238U is sameOMPparametersforbothneutronandprotonprojectiles.
Aratherweakdependenceofthedispersivepotentialgeometry
very scarce, especially in the high-energy range; but some
onmassnumberAwaspreviouslyobserved[20,21].Thefitted
angular distributions of scattered neutron and total cross-
dispersive potential parameters are listed in Table I. These
section measurements are available for fissile actinides and
parameters have to be complemented by a corresponding
were considered in the fitting. Quasielastic scattering (p,n)
data on 232Th and 238U were also considered in the fit to fix coupling scheme, ground-state deformations, and coupling-
strengthparameters(listedinAppendixC,TablesIII–VIII)to
theisovectortermsofthepotentialasdescribedinRef.[73].
becomeacompleteOMPparametersetforstudiedactinides.
The starting values of static deformation parameters β ,
2
β , and β were taken from Finite Range Droplet Model
4 6
(FRDM) deformations theoretically derived by Mo¨ller et al.
VI. RESULTSANDDISCUSSION
[74].Evaluatedneutronstrengthfunctionsforactinidenuclei,
S and S , and potential scattering radius R(cid:2) [22,75] were The derived dispersive potential parameters shown in
0 1
used for fine-tuning deformation parameters together with Table I for actinide nuclei combined with deformation pa-
measured (p,p(cid:2)) angular distributions on excited states close rametersandcouplingstrengthstabulatedinAppendixCwere
totheCoulombbarrier(E >20MeV)foractinidenucleiof testedagainst selected experimental data.Average resonance
p
interest.Theminimizationprocedureusedthequantityχ2 as parameters are consistently described by our calculations,
describedinpreviousreferences[19–21]. as can be seen in Table II. A similar level of agreement
Bothdirectandstatisticalprocessescontributetonucleon- in calculated strength functions is seen for rigid-rotor OMP
nucleuselasticscatteringattheseenergies.However,accord- results (RIPL 2408) within quoted evaluated uncertainties.
ingtoourestimation,thestatisticalprocessesarenotimportant Current results for S are closer to evaluated ones for 235U
0
above 3 MeV on actinides, so we neglect them in the OMP and238U,butmarginallyworsefor239Pu.ForS ,calculations
1
derivation. The direct processes, increasingly dominant at usingthederivedpotentialareclosertoevaluationsthanthose
higherenergies,canbedescribedbytheopticalmodel. usingtheRIPL2408potential,butS evaluateduncertainties
1
The customary coupled-channels calculations were per- arelarge.
formed by coupling the ground-state Kπ rotational band Theaimofthisworkistoimprovethedescriptionofneutron
with the selected members of vibrational (single-particle) inelastic scattering on actinides at incident neutron energies
064605-9
E.SH.SOUKHOVITSKI˜Ietal. PHYSICALREVIEWC94,064605(2016)
TABLEII. Comparisonofcalculatedandexperimentalaverageresonanceparameters.calculationsusingderivedOMParecomparedwith
resultsusingtheRIPL2408potential[20,21].Theabsoluteuncertaintiesofevaluatedstrengthfunctionsandradiiaregiveninparentheses.
Quantity Reference 232Th 233U 235U 238U 239Pu
Thiswork 0.86 0.97 0.95 1.02 1.15
S [(eV)−1/210−4] RIPL2408OMP[20,21] 0.86 0.93 0.93 0.92 1.18
0
RIPL-3evaluation[22] 0.84(0.07) 0.90(0.05) 0.98(0.07) 1.03(0.08) 1.20(0.10)
Porodzinskietal.[78] 0.80(0.08) 1.07(0.14) 1.01(0.10) 1.17(0.10) 1.25(0.13)
Mughabghab2006[79] 0.71(0.04) 0.98(0.09) 0.98(0.07) 1.29(0.13) 1.3(0.10)
Thiswork 1.66 1.41 1.60 1.68 2.05
S [(eV)−1/210−4] RIPL2408OMP[20,21] 1.72 2.20 1.93 1.72 2.19
1
RIPL-3evaluation[22] 1.50(0.30) – – 1.60(0.20) –
Mughabghab2006[79] 1.35(0.04) – 1.8(0.30) 2.17(0.19) 2.30(0.40)
CSEWG1991[75] 1.60(0.60) – 1.8(0.30) 1.70(0.30) 2.30(0.40)
Thiswork 9.68 9.60 9.58 9.52 9.50
R(cid:2)(fm) RIPL2408OMP[20,21] 9.68 9.56 9.51 9.64 9.49
Mughabghab2006[79] 9.65(0.08) 9.75(0.15) 9.63(0.05) 9.60(0.10) 9.48(0.10)
CSEWG1991[75] 9.65(0.30) 9.75(0.20) 9.65(0.10) – 9.60(0.10)
from keVs to a few MeVs. Unfortunately, many neutron- A lowest-energy vibrational mode in actinides is the
scattering physical observables (e.g., angular distributions octupolar vibration (e.g., the octupolar bandhead in 238U is
of elastic and inelastic scattering, inelastic scattering cross locatedat680keV,asshowninFig.1).Thisvibrationalmode
sections, etc.) measured on actinides for incident neutron
energiesbelow3MeVareinfluencedbyacompoundnuclear
decaythatcannotbedescribedbyacoupled-channelsoptical
modelalone,whichisadirectreactionmodel.Thetotalcross
sectionisonemeasuredobservablenotaffectedbycompound
nuclearprocesses.
Total cross sections for 233,235,238U and 232Th nuclei were
calculatedwiththederivedpotentialparameters.Anexcellent
agreementwithdatainthewholeenergyrangeisachievedas
seeninFig.2.SimilaragreementisalsoshownfortheRIPL
2408optical-modelpotentialbasedonarigid-rotorstructure.
Calculated total cross sections do not show an improvement
for a new potential as both calculated results are within
quoted experimental uncertainty. However, it is remarkable
that the newly derived potential with extended couplings
preserves the achieved agreement with total cross section
and other scattering observables shown in Refs. [20,21],
whileitallowspredictingdirect-interactioncrosssectionsfor
additionalcoupled-levelsinthewholeenergyrangeofinterest.
From the physical point of view, a newly derived OMP
utilizes a better description of the underlying nuclear struc-
ture compared to commonly used rigid-rotor potentials. The
inadequacyoftherigid-rotordescriptionisclearlyseeninthe
deviation from the I(I +1) rule of the energies of excited
states with large I in the ground-state rotational band of
even-even actinides. For example, for 238U and 232Th targets
increasingly larger differences from the I(I +1) rule can
be seen for levels with I >8 in the ground-state rotational
band, corresponding to excitation energies above (cid:13)500 keV.
Such differences are mainly attributable to the change in
momentofinertia(stretchingofthesoftnucleus).Additionally,
K mixing increases for higher-spin states as, e.g., shown
FIG.2. Calculated total cross sections using the dispersive
in a soft-rotator structure description [23,24]. Therefore,
potentialwithmultiple-bandcouplingforneutron-inducedreactions
the accuracy of the rigid-rotor description of the nuclear on233,235,238Uand232Thtargetsarecomparedwithcalculationsusing
structure in even-even actinides rapidly deteriorates above
theRIPL2408potential[20,21].Experimentaldataaretakenfrom
500keV. Refs.[80–87].
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