First microscopic coupled-channel calculation of α inelastic cross sections on 16 O
KKUNS-2758, NITEP 17
First microscopic coupled-channel calculation of α inelastic cross sections on O Yoshiko Kanada-En’yo
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Kazuyuki Ogata
Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, JapanDepartment of Physics, Osaka City University, Osaka 558-8585, Japan andNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City University, Osaka 558-8585, Japan
The α inelastic scattering on O is investigated with the coupled-channel calculation using the α -nucleus coupled-channel potentials, which are microscopically derived by folding the the Melbourne g -matrix NN interaction with the O and α densities. The matter and transition densities of Oare calculated by a microscopic structure model of the variation after the spin-parity projectionscombined with the generator coordinate method of C+ α in the framework of the antisymmetrizedmolecular dynamics. The calculation reproduces the observed elastic and inelastic cross sections atincident energies of E α = 104 MeV, 130 MeV, 146 MeV, and 386 MeV. The coupled-channel effecton the cross sections is also discussed. I. INTRODUCTION
The α scattering has been used for study of isoscalar(IS) monopole and dipole excitations in nuclei. Theinelastic cross sections have been analyzed by reactionmodel calculations to determine the strength functionsin a wide range of excitation energy covering the giantresonances [1]. The α scattering is also a good tool toprobe cluster states because these have generally strongIS monopole and dipole transition strengths and can bepopulated by the α scattering reaction [2–4]. Indeed, the( α, α (cid:48) ) reaction experiments have been intensively per-formed to investigate cluster structures of excited statesin light nuclei such as C and O recently.For the study of cluster structures in C, the C( α, α (cid:48) ) reaction has been investigated with reactionmodels [5–12], but many of the reaction calculations en-countered the overshooting problem of the 0 +2 cross sec-tions. Also for sd -shell nuclei such as O, α scatteringexperiments have reported the similar overshooting prob-lem of the 0 + cross sections in the reaction model analysis[12]. Recently, Minomo and one of the authors (K. O.)have carried out microscopic coupled-channel calculationand succeeded in reproducing the 0 +2 cross sections of the C( α, α (cid:48) ) reaction with no adjustable parameter [13].In the study, α -nucleus CC potentials are constructedby folding the Melbourne g -matrix effective N N inter-action [14] by a phenomenological matter density of α and the matter and transition densities of C obtainedwith the resonating group method [15]. In our previouspaper [16], we have applied the g -matrix folding modelto the same reaction and reproduced the cross sectionsof the 0 +2 , , 1 − , 2 +1 , , and 3 − states of C with the transi-tion density obtained by the antisymmetrized moleculardynamics (AMD) [17–21], which is a microscopic struc-ture model beyond the cluster models. These works in-dicate that, if reliable transition densities are availablefrom structure model calculations, the approach of the g -matrix folding model can be a useful tool to investi- gate cluster states by the ( α, α (cid:48) ) reaction.In the structure studies of O, a variety of clusterstructures such as the 4 α -tetrahedral, C+ α , and a 4 α -cluster gas state have been suggested by the cluster mod-els [3, 22–38]. Recently, the experimental studies of Ohave been performed by the O( α, α (cid:48) ) reaction [12, 39].In Ref. [39], the 0 +4 state at 13.6 MeV has been discussedin relation with the 4 α -gas state with the reaction modelanalysis using phenomenological CC potentials. In thestudy, the α -scattering cross sections are naively assumedto scale the IS monopole strengths.However, no microscopic CC calculation of the O( α, α (cid:48) ) reaction was performed so far, mainly be-cause of the theoretical difficulty of microscopic struc-ture models in description of O. For instance, a well-known problem is that microscopic cluster models largelyovershoot the excitation energy of the K π = 0 +2 band.Recently, one of the authors (Y. K-E.) investigated thecluster structures of O [40–42] with the AMD. Shehas performed the variation after spin-parity projections(VAP) combined with the generator coordinate method(GCM) of the C+ α cluster in the AMD framework,which we called the VAP+GCM. The VAP+GCM calcu-lation qualitatively described the energy spectra of Oand obtained the 0 +2 , 2 +1 , 4 +1 , 1 − , and 3 − states in the C+ α bands, and also the 3 − and 4 +2 states in the 4 α -tetrahedral ground band. Moreover, it predicts the 4 α -gas state as the 0 +5 state near the 4 α threshold energy.In this paper, we apply the g -matrix folding modelto the O( α, α (cid:48) ) reaction using the matter and transi-tion densities calculated with the VAP+GCM in a similarway to in our previous work on the C( α, α (cid:48) ) reaction[16]. The present work is the first microscopic CC cal-culation of the O( α, α (cid:48) ) reaction that is based on themicroscopic α -nucleus CC potentials derived with the g -matrix folding model. The calculated cross sections arecompared with the observed data at incident energies of E α = 104 MeV, 130 MeV, 146 MeV, and 386 (400) MeV[12, 39, 43–45]. The IS monopole and dipole transitionsto the 0 +2 , , , and 1 − states are focused. We try to an- a r X i v : . [ nu c l - t h ] A p r swer the following questions. Can the microscopic reac-tion calculation describe the α scattering cross sections?Does the overshooting problem of the monopole strengthexist? Is the scaling law of the α scattering cross sectionsand the IS monopole transition strength satisfied?The paper is organized as follows. Section II describesthe structure calculation of O with the VAP+GCM,and Sec. III discusses the O( α, α (cid:48) ) scattering investi-gated with the microscopic CC calculation. Finally, asummary is given in Sec. IV. II. STRUCTURE CALCULATION OF O WITHVAP+GCMA. Wave functions of O The wave functions of O are those obtained by thevariation after spin-parity projections (VAP) [19] com-bined with the C+ α GCM in the AMD framework,which we called the VAP+GCM [41]. As shown inRef. [42], the VAP+GCM calculation reasonably repro-duces the energy spectra and transition strengths of O,and obtains various cluster structures such as the 4 α and C+ α cluster structures. For the details of the formula-tion of the structure calculation and the resulting struc-tures and band assignments in O, the reader is referredto Refs. [40–42]. Using the VAP+GCM wave functions,the transition strengths, matter and transition densities,and form factors are calculated. The definitions of thesequantities are given in Refs. [16, 42].
B. Excitation energies and radii
The excitation energies and radii of O from theVAP+GCM calculation and the experimental data arelisted in Table I. We assign the fourth 0 + (0 +IV ), thethird 0 + (0 +III ), the third 2 + (2 +III ), and the second 2 + (2 +II )states in the theoretical spectra to the experimental lev-els of the 0 +3 (12.05 MeV), 0 +4 (13.6 MeV), 2 +2 (9.85 MeV),and 2 +3 (11.52 MeV) states, respectively, because theVAP+GCM calculation gives incorrect ordering of the K π = 0 +3 and 0 +4 bands; for the detailed discussion, seeRef. [40]. The nuclear sizes of the 3 − and 1 − states arecomparable to the ground state and relatively smallerthan those of other excited states because the 3 − is the4 α -tetrahedral state in the ground band and the 1 − is thevibration mode on the tetrahedral ground band. The 4 +2 state is also regarded as the 4 α -tetrahedral band but itssize is slightly larger than those of the other two in theground band because of the mixing with the 4 +1 state inthe C+ α cluster band. Other states are developed clus-ter states and have relatively larger radii than those ofthe ground-band states. The density distribution of the0 +1 , , , , , 1 − , , 2 +1 , , , 3 − , , and 4 +1 , states obtained by theVAP+GCM is shown in Fig. 1. The 0 +2 , , , , 1 − , 2 +1 , , , (a) ρ (r) (f m - ) r (fm) +1 +2 +3 +4 +5 (b) ρ (r) (f m - ) r (fm) +1 +3 +2 (c) ρ (r) (f m - ) r (fm) +1 +2 (d) ρ (r) (f m - ) r (fm) -1 -2 (e) ρ (r) (f m - ) r (fm) -1 -2 FIG. 1: Proton density ρ p ( r ) = ρ ( r ) / +1 , , , , ,1 − , , 2 +1 , , , 3 − , , and 4 +1 , states of O calculated with theVAP+GCM. − , and 4 +1 states tend to have the slightly enhanced sur-face density in the range of r = 4–5 fm because of thedeveloped cluster structures. C. Transition strengths, transition densities, andcharge form factors of O In Table II, the transition strengths, B ( Eλ ), of Ocalculated with the VAP+GCM are shown comparedwith the experimental data. For the IS dipole transi-tion strengths of the 1 − → + transitions, the valuesof B (IS1) / E E < B ( E < e fm and B ( E > e fm ,respectively. Rather strong E λ = 2transitions give significant CC effects to the α scatteringcross sections as discussed later. The K π = 0 +2 bandof the C+ α cluster is composed of the 0 +2 , 2 +1 , and4 +1 states, and its parity-partner K π = 0 − band is con-structed by the 1 − and 3 − states. The 4 α -gas state isobtained as the 0 +5 state.To reduce ambiguity from the structure model calcula-tion in application of the theoretical transition density tothe reaction calculation, we scale the calculated result as TABLE I: Excitation energies E x (MeV) and rms matter radii R (fm) of O calculated with the VAP+GCM. The experi-mental values of the excitation energies from Ref. [46] are alsoshown. The experimental data R = 2 .
55 fm of the rms radiusof the ground state is deduced from the experimental chargeradius measured by the electron scattering [47]. We assignthe fourth 0 + , third 0 + , third 2 + , and the second 2 + statesobtained by the VAP+GCM to the experimental 0 +3 , 0 +4 , 2 +2 ,and 2 +3 states, which we label as 0 +3 , IV , 0 +4 , III , 2 +2 , III , and 2 +3 , II ,respectively. exp VAP+GCM E x (MeV) E x (MeV) R (fm)0 +1 +2 +3 , IV +4 , III +5 +1 +2 , III +3 , II +4 +5 +6 +7 +8 +1 +2 +3 − − − − ρ (tr) ( r ) → f tr ρ (tr) ( r ) to fit the observed B exp ( Eλ ). Thescaling factor f tr = (cid:112) B exp ( Eλ ) /B cal ( Eλ ) introducedhere is defined by the square root of the ratio of theexperimental B ( Eλ ) value to the theoretical one. Theadopted value of f tr for each transition is shown in Ta-ble II. For the transitions for which experimental data of B ( Eλ ) do not exist, we take f tr = 1 and use the originaltransition density. For the 1 − → +1 transition, B (IS1)is unknown, but the charge form factors are availablefrom the ( e, e (cid:48) ) reaction data. For this transition, we use f tr = 1 in the default CC calculation, and also test amodified value f tr = 1 . +1 → − transition den-sity, which consistently reproduces the charge form fac-tors and α scattering cross sections. Figure 3 shows thescaled transition density f tr ρ (tr) ( r ) for transitions fromthe ground state.We show in Fig. 4 the charge form factors of O cal-culated with the VAP+GCM and the experimental datameasured by the electron scattering [48]. The calculatedform factors are scaled by multiplying f consistently TABLE II: The transition strengths B ( Eλ ) of O calcu-lated with the VAP+GCM and the experimental data fromRefs. [46, 48]. For the IS dipole transition strengths of the1 − → + transitions, the values of B (IS1) / f tr = (cid:112) B exp ( Eλ ) /B cal ( Eλ ) determined by theratio of the experimental value B exp ( Eλ ) to the calculatedvalue B cal ( Eλ ) for each transition is also shown. For the tran-sitions with no experimental data of B ( Eλ ), f tr = 1 is used.The units are e fm λ for B ( Eλ ) ( λ (cid:54) = 0), e fm for B ( E for B (IS1). a The calculated B ( E +2 → +2 ) is toosmall, and therefore, is not adjusted to the experimental valuebut f tr = 1 is used. b For the scaling factor of the 1 − → +1 transition, we use f tr = 1 in the default CC calculation, andalso use the modified value f tr = 1 .
3, which are phenomeno-logically adjusted so as to reproduce the charge form factorsand α scattering cross sections.exp VAP+GCM B ( Eλ ) [46] ( e, e (cid:48) ) [48] B ( Eλ ) f tr E +1 → +1 .
24) 7.79 3.05 1.56 E +1 → +2
65 (7) 140 0.68 E +2 → +1 .
01) 0.05 0.29 0.51 E +2 → +2 .
72) 0.02 1 a E +3 → +1 .
20) 3.40 2.39 1.23 E +3 → +2 .
20) 43.7 0.41 E +1 → +1
156 (14) 1 E +2 → +1 .
72) 1 E − → −
50 (12) 33.7 1.22 E − → − E +2 → +1 E +3 → +1 E +4 → +1 E +5 → +1 E − → +1
205 (11) 207 0.99 E − → +2 E − → +1 E − → +2 E − → +1
378 (133) 420 345 1.10 E − → +2 E − → +1
372 71 2.30 E − → +2 − → +1 b IS1:1 − → +2
74 1IS1:1 − → +1 − → +2
745 1 with the scaled transition density. The experimental dataare reasonably reproduced by the scaled form factors ofthe VAP+GCM. In the E + states(Fig. 4(d)), a clear difference can be seen in the 0 +1 → +2 from the other E +1 → +1 , because thecorresponding transition density of 0 +1 → +2 shows the C+ α (K π =0 + ) C+ α (K π =0 − )4 α gas0 +2 +4,III +3,IV +5 E x ( M e V ) J(J+1)50 e fm , respec-tively. -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0 1 2 3 4 5 6 7 8 (a) f t r ρ t r (r) / ∧ J f (f m - ) r (fm) +2 +3.IV +4,III +5 (d) f t r ρ t r (r) / ∧ J f (f m - ) r (fm) -1 -2 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0 1 2 3 4 5 6 7 8 (e) f t r ρ t r (r) / ∧ J f (f m - ) r (fm) -1 -2 -0.01 0 0.01 0.02 0 1 2 3 4 5 6 7 8 (b) f t r ρ t r (r) / ∧ J f (f m - ) r (fm) +1 +2,III +3,II -0.01 0 0.01 0.02 0 1 2 3 4 5 6 7 8 (c) f t r ρ t r (r) / ∧ J f (f m - ) r (fm) +1 +2 FIG. 3: Transition density for the transitions from the groundstate in O calculated with the VAP+GCM. The scaledtransition density f tr ρ (tr) ( r ) divided by ˆ J f ≡ = (cid:112) J f + 1 isshown. different behavior that it is the compact spatial distri- bution with no nodal structure (see Fig. 3(b)). In the E + states, the 0 +1 → +2 transitionhas the dip at the smallest momentum transfer ( q ) cor-responding to the broadest distribution of the transitiondensity. III. α SCATTERINGA. Coupled-channel calculation
Using the matter and transition densities calculatedwith the VAP+GCM as the input from the structurecalculation, we perform the CC calculation of O( α, α (cid:48) )with the g -matrix folding model in the same way as inour previous work [16]. The α - O CC potentials are con-structed by folding the Melbourne g -matrix N N interac-tion [14] with the densities of α and O in the approx-imation of an extended version of the nucleon-nucleusfolding (NAF) model [49]. For the α density, we adoptthe one-range Gaussian distribution given in Ref. [50].In the default CC calculation of the cross sections ofthe 0 + , 1 − , 2 + , and 3 − states, we adopt the 0 +1 , , , , ,2 +1 , , , , 1 − , , and 3 − , states with the λ ≤ O nucleus. The scaled transition den-sity f tr ρ (tr) ( r ) is used. For the excitation energies of O, we use the experimental values listed in Table I.In the calculation of the 4 + cross sections, we adopt0 +1 , , , , 2 +1 , , , , , , , , and 4 +1 , , states with the λ = 0 , , +5 state, wealso perform the CC calculation using the 0 +1 , , , , and2 +1 , , , , , , , states with the λ = 0 , E +5 and 2 +8 states and compare the result with the defaultCC calculation. B. α scattering cross sections The α scattering cross sections at the incident energiesof E α = 104 MeV, 130 MeV, 146 MeV, and 386 MeV areshown in Figs. 5, 6, and 7, respectively. The cross sec-tions calculated by the DWBA are also shown for com-parison.The elastic cross sections are well reproduced by thepresent calculation except at large scattering angles for E α =104–146 MeV. For the λ = 2 and λ = 3 transitionsto the 2 +1 , 2 +3 , and 3 − states, the calculated cross sec-tions are in good agreement with the experimental data.These states are strongly populated in the direct tran-sitions, and the cross sections are dominantly describedby the DWBA calculation. For these states, the CC ef-fect is minor, in particular, at E α =386 MeV, but notnegligible in the cross sections at the relatively low inci-dent energies of E α =104–146 MeV. For the 2 +2 , the cal-culation reproduces the absolute amplitude of the crosssections at forward angles but does not satisfactorily de-scribe the diffraction pattern of the observed data. Forthe 4 +1 state, the present calculation predicts a very weakpopulation and much underestimates the experimentalcross sections.For the IS dipole excitation to the 1 − state, the exper-imental cross sections are somewhat underestimated bythe default CC calculation (solid lines of Fig. 7) with theoriginal 1 − → +1 transition density, but successfully re-produced by the calculation with the modified 0 +1 → − transition density scaled by the factor of f tr = 1 .
3, whichreproduces the charge form factors of this transition. Incomparison with the DWBA calculation, one can see thesignificant CC effect in the 1 − cross sections. Namely,the absolute amplitude of the cross sections is drasticallyreduced and the dip positions are slightly shifted to for-ward angles. This CC effect, which is mainly through the3 − state, is essential to reproduce the first dip positionof the experimental cross sections at E α = 104 MeV and130 MeV.For the monopole excitations, the calculated 0 +3 and 0 +4 cross sections are in good agreement with the observeddata. The present CC calculation describes well not onlythe diffraction pattern but also the absolute amplitudein the wide range of the incident energies of E α = 104–386 MeV, and there is no overshooting problem of the 0 + cross sections.In comparison with the DWBA calculation, one cansee how the CC effect contributes to the monopole transi-tions in the α scattering. In the 0 +3 and 0 +4 cross sections,the CC effect is not so large but not negligible, in particu-lar, at the low incident energies, E α = 104–146 MeV. Bycontrast, the CC effect gives the drastic change of the 0 +2 cross sections mainly because of the strong in-band E − state in the C+ α -cluster bandbuilt on the 0 +2 state. At E α = 104–146 MeV, the peakamplitude of the 0 +2 cross sections is largely reduced byabout a factor of three from the result of the DWBAcalculation. Even at E α = 386 MeV, the peak ampli-tude of the CC result is smaller by a factor of two thanof DWBA. Also for the 0 +5 cross sections, the CC effectis found to be of importance, because of the strong E +5 and 2 +8 states with the devel-oped cluster structure. Although the CC effect seems tobe not so strong in the default CC calculation withoutthe coupling with the higher 2 + states (the (red) solidlines in Fig. 5), the CC calculation with the 0 +1 , , , , and2 +1 , ,..., states shows the drastic CC effect of the signifi-cant reduction of the 0 +5 cross sections at the low incidentenergies of E α = 104–146 MeV (the (blue) long-dashedlines in Fig. 5). The CC effect in the 0 +5 cross sectionsbecomes weak at the relatively high incident energy of E α = 386 MeV.In the experimental studies of the monopole transi-tions, the α scattering cross sections have been used todeduce the monopole strengths based on the reactionmodel analysis mainly with the DWBA calculation bynaively expecting the scaling law of the α -scattering cross sections and the electric monopole transition strength, B ( E α scatter-ing indicates that the scaling law is not necessarily validfor the cluster states. Firstly, the amplitude of the 0 + cross sections can be significantly affected by the CC ef-fect mainly through the strong λ = 2 transitions betweenthe developed cluster states. Secondly, the scaling law isnot satisfied even in the one-step process of the DWBAcross sections because of the difference in the matter andtransition densities between excited 0 + states. These re-sults indicate that, for study of the monopole transitionsby means of the ( α, α (cid:48) ) reaction, it is necessary to ana-lyze the α scattering cross sections with microscopic re-action models considering such the CC effect and densityprofiles. Nevertheless, we should remark that 0 + clusterstates with significant monopole strengths are stronglypopulated by the ( α, α (cid:48) ) reaction, meaning that it is stilla good probe for the cluster states and can be useful forqualitative discussion even though the scaling law is notquantitatively valid. IV. SUMMARY
The α inelastic scattering cross sections on O wasinvestigated by the folding model with the Melbourne g -matrix N N interaction. This is the first microscopic CCcalculation of the O( α, α (cid:48) ) reaction that is based onthe α -nucleus CC potentials microscopically derived withthe g -matrix N N interactions and the matter and tran-sition densities of the target O nucleus calculated withthe microscopic structure model. As for the structuremodel, we employed the VAP+GCM in the frameworkof the AMD, which is the microscopic approach beyondthe cluster models. In the application to the reactioncalculation, the calculated transition density is scaled tofit the experimental transition strengths to reduce theambiguity of the structure model.The calculation reproduces well the observed cross sec-tions of the 0 +2 , , , 2 +1 , 1 − , and 3 − states as well as theelastic cross sections at incident energies of E α = 104MeV, 130 MeV, 146 MeV, and 386 MeV. In the 0 + crosssections, there is no overshooting problem. In compari-son with the DWBA calculation, the significant CC effectwas found in the 0 +2 , 0 +5 , and 1 − cross sections becauseof the strong λ = 2 coupling between excited states thathave developed cluster structures. We clarified that thescaling law of the α -scattering cross sections and B ( E λ = 2transitions between the developed cluster states. Nev-ertheless, it should be remarked that the ( α, α (cid:48) ) reac-tion can be used for qualitative discussion on the clusterstates because 0 + cluster states with significant monopolestrengths are strongly populated by the ( α, α (cid:48) ) reaction.It is suggested that the microscopic reaction calcula-tion is needed in the quantitative analysis of the α scat-tering cross sections. The present g -matrix folding modelwas proved to be applicable to describe the α scatteringcross sections. This approach is a promising tool to ex-tract information on cluster structures of excited statesin other nuclei by the ( α, α (cid:48) ) reaction. Acknowledgments
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0 1 2 3 4 5 6 7 8 (h) 3 -1,2 f t r | F ( q ) | q (fm -2 ) -1 -2 -9 -8 -7 -6 -5 -4 -3 -2
0 1 2 3 4 5 6 7 8 (b) 0 +2,3 f t r | F ( q ) | q (fm -2 ) +2 +3 × -3 -9 -8 -7 -6 -5 -4 -3 -2
0 1 2 3 4 5 6 7 8 (c) 0 +4,5 f t r | F ( q ) | q (fm -2 ) +4 +5 × -3 -9 -8 -7 -6 -5 -4 -3 -2
0 1 2 3 4 5 6 7 8 (d) 2 +1,2 f t r | F ( q ) | q (fm -2 ) +1 +2 × -2 -9 -8 -7 -6 -5 -4 -3 -2
0 1 2 3 4 5 6 7 8 (e) 2 +3 f t r | F ( q ) | q (fm -2 ) +3 -9 -8 -7 -6 -5 -4 -3 -2
0 1 2 3 4 5 6 7 8 (f) 4 +1,2 f t r | F ( q ) | q (fm -2 ) +1 +2 × -3 -9 -8 -7 -6 -5 -4 -3 -2
0 1 2 3 4 5 6 7 8 (g) 1 -1,2 f t r | F ( q ) | q (fm -2 ) -1 -2 -1 × -8 -7 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8 (a) 0 +1 f t r | F ( q ) | q (fm -2 ) +1 FIG. 4: Squared charge form factors of O. The theoretical values obtained with the VAP+GCM are scaled by f consistentlywith the scaled transition density. For the 1 − → +1 transition, the squared form factors scaled by the modified value f = 1 . of the scaling factor are also shown. The experimental data are from Ref. [48]. -4 -2
0 10 20 30 40 +1
104 MeV( × )130 MeV( × )146 MeV 386 MeV( × -2 ) R u t h e rf o r d r a ti o θ (degree)CCDWBA10 -4 -2
0 10 20 30 40 +4 c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBAexp:400 MeV( × -2 ) 10 -4 -2
0 10 20 30 40 +2 c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBAexp:130 MeV( × ) 10 -4 -2
0 10 20 30 40 +3 c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBA10 -4 -2
0 10 20 30 40 +5 c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBACC:0 +1-5 ,2 +1-8 FIG. 5: α scattering cross sections on O at E α = 104 MeV ( × ), 130 MeV ( × ), 146 MeV, and 386 MeV ( × − ).The differential cross sections of the 0 +1 , , , , obtained by the CC and DWBA calculations are shown by (red) solid and (green)dashed lines, respectively. For the 0 +5 , the cross sections obtained by the CC calculation using the 0 +1 , , , , and 2 +1 , ,..., statesare also shown by (blue) long-dashed lines. The experimental data at E α = 104 MeV [43], 130 MeV [12], 146 MeV [44], and400 MeV [39] are shown by filled circles, open circles, open triangles, and filled squares, respectively. For the 0 +3 cross sections,the data at E α = 104 MeV from Ref. [45] and those at E α = 386 MeV from Ref. [12] are also shown by open squares and opencircles, respectively. -4 -2
0 10 20 30 40 +1
104 MeV( × )130 MeV( × )146 MeV 386 MeV( × -2 ) c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBA 10 -4 -2
0 10 20 30 40 +2 c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBAexp:130 MeV( × ) 10 -4 -2
0 10 20 30 40 +3 c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBA10 -4 -2
0 10 20 30 40 +4 c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBA 10 -4 -2
0 10 20 30 40 +1 c r o ss s ec ti on ( m b / s r) θ (degree)CC:0 +1-4 ,2 +1-8 ,4 +1-3 DWBAexp:130 MeV( × ) 10 -4 -2
0 10 20 30 40 +2 c r o ss s ec ti on ( m b / s r) θ (degree)CC:0 +1-4 ,2 +1-8 ,4 +1-3 DWBA
FIG. 6: α scattering cross sections on O at E α = 104 MeV ( × ), 130 MeV ( × ), 146 MeV, and 386 MeV ( × − ). Thedifferential cross sections of the 2 +1 , , , and 4 +1 , obtained by the CC and DWBA calculations are shown by (red) solid and(green) dashed lines. In the CC calculation for the 4 +1 , cross sections, the 0 +1 , , , , 2 +1 , ,..., , and 4 +1 , , states are used. Theexperimental data at 130 MeV [12], 146 MeV [44], and 386 MeV [12] are shown by open circles, open triangles, and open circles,respectively. -4 -2
0 10 20 30 40 -1
104 MeV( × )130 MeV( × )146 MeV 386 MeV( × -2 ) c r o ss s ec ti on ( m b / s r) θ (degree)CC DWBA 10 -4 -2
0 10 20 30 40 -2 c r o ss s ec ti on ( m b / s r) θ (degree)CCDWBA 10 -4 -2
0 10 20 30 40 -1 c r o ss s ec ti on ( m b / s r) θ (degree) CCDWBACC:1.310 -4 -2
0 10 20 30 40 -2 c r o ss s ec ti on ( m b / s r) θ (degree) CCDWBA FIG. 7: α scattering cross sections on O at E α = 104 MeV ( × ), 130 MeV ( × ), 146 MeV, and 386 MeV ( × − ). Thedifferential cross sections of the 3 − , and 1 − , obtained by the default CC and DWBA calculations are shown by (red) solidand (green) dashed lines. The experimental data at 104 MeV [45], 130 MeV [12], 146 MeV [44], and 386 MeV [12] are shownby open squares, open circles, open triangles, and open circles, respectively. For the 1 − states, the result calculated with themodified value f tr = 1 . +1 → −1