application/xmlDynamic polarization potential effects on vector analyzing powers of 6Li–28Si elastic scattering from non-monotonic potentialsA.K. BasakP.K. RoyS. HossainM.N.A. AbdullahA.S.B. TariqM.A. UddinI. ReichsteinF.B. Malik6Li elastic scattering on 28SiAnalysis of cross section and vector analyzing power dataCoupled-channels and coupled discretized continuum channels calculationsProperties of 6LiPhysics Letters B 692 (2010) 47-50. doi:10.1016/j.physletb.2010.07.011journalPhysics Letters BCopyright © 2010 Elsevier B.V. All rights reserved.Elsevier B.V.0370-2693692116 August 20102010-08-1647-50475010.1016/j.physletb.2010.07.011http://dx.doi.org/10.1016/j.physletb.2010.07.011doi:10.1016/j.physletb.2010.07.011http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.2PLB26904S0370-2693(10)00811-710.1016/j.physletb.2010.07.011Elsevier B.V.TheoryFig. 1Experimental σ/σR and iT11 for the 6Li–28Si elastic scattering at 22.8 MeV are compared with the predictions from the two-channel CC calculations using the NM 66Li potential with no static SO potential. Solid and dashed lines are, respectively, the predictions with and without reorientions. Dotted lines are calculations with reorientations and the opposite signs of RSC and QDL.Fig. 2Same as Fig. 1 for the four-channel CC calculations. Dotted lines here represent the one-channel OM calculations of [16] with EDF-generated NM real part of the 6Li potential, and empirical imaginary and static SO parts. Dash-dotted lines are the results of calculations using the CF 6Li potential with reorientations and without any SO potential.Fig. 3Same as Fig. 1 for the CDCC calculations including reorientations for the twelve-channel (solid lines), nine-channel (dashed lines) and twenty one-channel (dotted lines) using the CF potential without any static SO potential.Table 1NM potential parameters for 6Li–28Si [15,16] and α-28Si [17], and WS parameters for d-28Si [18] with adjusted imaginary parameters. V0, V1, W0, WS and WD are in MeV while R0, a0, R1, RW, DS, RS, RD and aD are in fm (see text following Eq. (2) for definition of the parameters).Real parametersImaginary parametersParameter6LiαdParameter6LiαdV060.826.057.8WS/WD2.73–10.0R04.905.353.43DS/RD5.66–4.89a00.8520.3400.535RS/aD0.685–0.564V160.042.0–W016.341.0–R13.932.80–RW4.314.0–RC8.609.353.80––––Dynamic polarization potential effects on vector analyzing powers of 6Li–28Si elastic scattering from non-monotonic potentialsA.K.Basaka*akbasak@gmail.comP.K.RoybS.HossaincM.N.A.AbdullahdA.S.B.TariqaM.A.UddinaI.ReichsteineF.B.MalikfgaDepartment of Physics, University of Rajshahi, Rajshahi, BangladeshbDepartment of Physics, B.L. College, Daulatpur, Khulna, BangladeshcDepartment of Physics, Shahjalal University of Science & Technology, Sylhet, BangladeshdDepartment of Physics, Rajshahi University of Engineering & Technology, Rajshahi, BangladesheSchool of Computer Science, Carleton University, Ottawa, ON K1S 5B6, CanadafDepartment of Physics, Southern Illinois University, Carbondale, IL 62901, USAgDepartment of Physics, Washington University, St. Louis, MO 63130, USA*Corresponding author.Editor: W. HaxtonAbstractExperimental cross section (CS) and vector analyzing power (VAP) data of the 6Li–28Si elastic scattering at 22.8 MeV are analyzed in the coupled-channels (CC) and coupled discretized continuum channels (CDCC) methods. Non-monotonic (NM) 6Li and α potentials of microscopic origin are employed, respectively, in the CC calculations and to generate folding potentials for the CDCC calculations. The study demonstrates that the use of central NM potentials can generate an appropriate dynamic polarization potential (DPP) required to describe both the CS and VAP data without the necessity of renormalization. This also produces an effective spin–orbit (SO) potential to account for the iT11 data without the requirement of an additional static SO potential at the incident energy considered.Keywords6Li elastic scattering on 28SiAnalysis of cross section and vector analyzing power dataCoupled-channels and coupled discretized continuum channels calculationsProperties of 6LiThe large vector analyzing powers (VAP) observed in elastically scattered 6Li using the polarized beam at the Heidelberg EN-Tandem [1] has been a subject of considerable interest, since one expects small VAP in terms of the static spin–orbit (SO) potential of a light-heavy projectile like 6Li. A complete understanding of the 6Li potential should include consideration of spin-dependent potentials along with the central one. 6Li, being a spin-1 nucleus, has both first- and second-order SO terms in its potential. The works of Refs. [2,3] suggest that the second-rank tensor analyzing powers T2q of 6,7Li elastic scattering on 58Ni at Ecm(Li)=12.7 and 18.1 MeV can be accounted well for by static tensor potentials. However, both studies failed to reproduce the VAP data of the 6,7Li–58Ni elastic scattering with static SO potentials. As noted in Ref. [3], the SO potentials derived by the double folding (DF) model have the same sign for 6,7Li–58Ni systems, while the experimental VAP data for the two systems are of opposite signs [4]. However, the iT11 data of the two systems could be reproduced using the coupled-channels (CC) calculations [2,3] in which a dynamic polarization potential (DPP) is generated through projectile excitation. In both works, the analyses have been accomplished using the projectile-target interactions based on the cluster-folding (CF) model. Ward et al. [5] could reproduce the iT11 data of elastically scattered 6Li on 26Mg at 60 MeV by generating DPP from the Woods–Saxon (WS) potential in CC by adjusting the parameters of the potential including its real part.Sakuragi [6] and his group [7] investigated the effect of 6Li excitation on its elastic scattering in the frame work of the coupled discretized continuum channels [8] (also referred to as continuum discretized coupled channels [6]) (CDCC) method. In the CDCC calculations, DF potentials from the M3Y internucleon potential [9] were employed for the 6Li-target interaction. Sakuragi demonstrated elegantly that the problem of renormalization with the DF potential can be done away through the generation of a repulsive DPP in the CDCC calculations. However, Hirabayashi and Sakuragi [8] have reported that the real part of CF potentials needs renormalization of about NR=0.5–0.6 at energies below 10 MeV/nucleon even when resonant and non-resonant breakup channels are explicitly included in the CDCC calculations. They [8] further noted that the CF potentials in the CC calculations involving the resonant states of 6Li with the inclusion of a static SO potential can account well for the cross section (CS) and VAP data of 6Li elastic scattering on 26Mg and 120Sn at 44 MeV only when a renormalization factor NR=0.4–0.5 is considered with the real part. However, they noted that the CC calculations using DF potentials for the real part, without renormalizations, and static SO potentials can describe features of the CS and VAP data reasonably well. On the other hand, the conclusion of Van Verst et al. [10], based on their CC analysis of the CS and VAP data for the 6Li elastic scattering on 12C and 16O using the DF potentials, is that VAP arises from a complicated interference between the CC effects and the SO interaction.Analyses on the CS and VAP data of the 6Li–28Si elastic scattering are scarce. In particular, microscopic calculations in the frame work of CC and CDCC are hardly found in literature. Both Weiss et al. [11] and Petrovich et al. [12] used DF-generated SO potentials in the optical model (OM) calculations but had to adjust the 6Li central potential of [13] to describe the iT11 data. On the other hand, Windham et al. [14] enjoyed a reasonable success in describing the CS and VAP data in terms of the CC calculations, involving the resonant states of 6Li with the same form-factor for all the states and the CF potential, derived from the WS type of α and d potentials including a SO term for the latter.The present investigation aims at exploring the origin of the dynamic SO potential, using the real central non-monotonic (NM) potential from the energy-density functional (EDF) theory, which can account for both CS and VAP data simultaneously. In doing so, no static SO potential has been included anywhere to examine the exclusive DPP effects on the VAP data in the light of the premise of [8] that the static SO potential contributes a little at lower energies. Moreover, the parameters of the NM potential have been left unadjusted in the present study. This Letter reports for the first time the use of central non-monotonic (NM) potentials in generating appropriate DPP effects, which can describe simultaneously the CS and VAP data in an elastic scattering. The analyses comprise: (i) CC calculations, involving the 3+ (2.18 MeV), 2+ (4.31 MeV) and 12+ (5.70 MeV) resonant states of 6Li, based on the EDF-generated 6Li–28Si NM potential [15,16] which has been found, without any renormalization, to account well for the CS data of the 6Li–28Si elastic scattering at twelve energy points up to 99.0 MeV and to describe satisfactorily the VAP data at 22.8 MeV in conjunction with a static SO potential, and (ii) CDCC calculations, involving the above-mentioned resonant and non-resonant continuum states, with all the diagonal and coupling potentials, CF-generated from the NM α-28Si [17] and WS d-28Si [18] potentials.The CC and CDCC calculations have been performed using the coupled-channels code FRESCO 2.5 [19]. We denote the NM potentials for 6Li and α by(1)U(r)=VC(r)−V0f0(r)+V1f1(r)−iW0fW(r)−iWSfS(r). Here VC(r) is the Coulomb potential of a homogeneously charged sphere of radius RC. The form-factors in (1) are f0(r)=[1+exp((r−R0)/a0)]−1; fi(r)=exp[−(r/Ri)]2 with i=1 and W; and fS(r)=exp[−((r−DS)/RS)]2. On the other hand, the d-28Si potential from [18] is used in the form:(2)Ud(r)=VC(r)−V0f0(r)−i4E(r)WD(1+E(r))−2 with E(r)=exp[−(r−RD)/aD]. The potential parameters of the 6Li–28Si at ELi=22.8 MeV used in the CC calculations, and α-28Si at Eα=15.0 MeV and d-28Li at Ed=7.5 MeV employed in the CDCC calculations are given in Table 1. The latter two energies are selected to satisfy the Eα=23ELi and Ed=13ELi conditions in deriving the real nuclear part of the CF potential for 6Li at ELi=22.8 MeV. The Coulomb part of the CF potential is again the usual potential due to a homogeneously charged sphere of radius RC.In the CC calculations, rotation-model quadrupole form-factors have been used for the transitions between the coupled states, treating the 3+, 2+ and 12+ resonant states of momentum bins (with respect to the momentum ℏk of the α+d relative motion) corresponding to their widths of 0.1, 2.0 and 3.0 MeV, respectively. In deriving the CF potentials for the CDCC calculations, the D-state of the 6Li ground state (GS) has been omitted in view of the observation in [8] that such an admixture does not affect tangibly the elastic scattering. Hence only the S-state for the GS, resonant and non-resonant continuum states with the spin-parity Iπ=1+ and also only the D-state for the resonant and non-resonant continuum states with Iπ=2+ and 3+ are considered. Although the E1 breakup is not allowed for the even-parity states, following [20] only the E1 breakup is taken into consideration for the non-resonant continuum states with Iπ=0−, 1− and 2− to investigate the breakup effects of the odd-parity states of 6Li. The non-resonant continuum is truncated to about 10.0 MeV corresponding to the relative momentum, k=0.78 fm−1, of the α and d clusters. The momentum bins are discretized with Δk=0.25 fm−1. Following Keeley and Rusek [21], the binning scheme in the presence of the resonance states has been used to avoid double counting.As observed in Ref. [22], the reduced transition probabilities B(E2) of 6Li for the transitions between its GS (1+) and the first two 3+ and 2+ resonant states are reported to be large compared to the GS electric quadrupole moment (Q0). Another problem is that the quadrupole deformation length (QDL) for the [1+↔3+] transition from various sources is found to be spreaded over the range of values QDL=1.89–3.69 fm as compiled in [10]. For convenience of presentation, we label the 1+ GS and the 3+, 2+ and 12+ resonant states, respectively, by 1 through 4. We also denote the reduced strength (RSC) for the Coulomb coupling between the i and j states by ξij and QDL for the nuclear coupling of the two states by δij. For the CC calculations, we derived the ξ11, ξ12 and ξ22 values from B(E2)[1+→3+]=25.6e2 fm4[23]; ξ13 and ξ33 from B(E2)[1+→2+]=7.9e2 fm4[23]; and ξ14 and ξ44 from B(E2)[1+→12+]=4.6e2 fm4, as estimated in Ref. [8]. For nuclear couplings, we employed δ12=−3.41 fm[24] for the best fits to the data; δ13=−1.83 fm[25]; and δ14=−1.00 fm as used in Ward et al. [5]. The negative signs are guided by that for Q0. For reorientation terms, following the argument of the small Q0[5,25], we used δ11=δ22=−1.70 and δ33=−0.92 fm as well as δ44=−1.0. However, some of the nuclear coupling and reorientation δij's are opposite in sign to those used in [5].In both CC and CDCC analyses, the parameters of the real part of the relevant potentials are left unaltered and only those of the imaginary part have been adjusted to improve simultaneously the fits to the CS and VAP data. Moreover, no static SO potential is used anywhere.Fig. 1 shows the predictions from the two-channel CC calculations involving GS and the 2.18 MeV resonant state. The solid and dashed lines denote, respectively, the calculations with and without reorientations. The parameters of the imaginary part of the potential for the former calculations are noted in Table 1. The parameters WS and W0 of the imaginary potential corresponding to the dashed and dotted lines are, respectively, WS=3.61 and 3.82 MeV, and W0=17.72 and 25.69 MeV. The dotted lines represent the predictions with the reorientations and couplings of the opposite signs in RSC and QDL. Obviously, the VAP data is more sensitive to the sign of the coupling strengths than the CS data and the iT11 data favour negative values, which are used in the subsequent CC calculations.Fig. 2 displays the results from the four-channel CC calculations involving GS coupled to all the three resonant states of 6Li. The solid and dashed lines are the predictions using the NM potential with and without the reorientations, respectively. The imaginary parameters for the solid lines are WS=1.872, W0=44.34 MeV, DS=5.43, RS=0.40 and RW=3.40 fm; and those for the dashed lines are WS=2.179, W0=25.67 MeV, DS=5.56, RS=0.568 and RW=3.80 fm. Both the two- and four-channel CC calculations with the reorientations generate better fits to the VAP data. In the overall picture, the CC calculations reproduce the CS and VAP data better than the one-channel OM calculations of Ref. [16] with the NM 6Li potential and an effective SO potential. Inclusion of couplings amongst the resonant states of 6Li did not improve the fits to the data. One may note that the dominant effect stems from the coupling of the lowest 3+ resonant state with GS. To compare the effects of using NM and CF potentials of 6Li, four-channel CC calculations have also been performed using the latter potential. The predictions are given as dash-dotted lines in Fig. 3 with the adjusted imaginary depth WD=12.5 MeV and the RD and aD parameters given in Table 1. Although the CS data are best described by the CF calculations, the position of the peak in the VAP data near 68° is not correctly reproduced. The substantial difference in the predictions with the NM and CF potentials may be ascribed to the underlying methods in deriving the potentials and details in channel couplings. Nevertheless, the features of the oscillations in the VAP data are reproduced in both the cases.To investigate the effects of different number of bin-states and the inclusion of the non-resonant states with odd-parity, where the E1 breakup is allowed [20], three CDCC calculations have been carried out. In the twenty one-channel CDCC calculations, sixteen bin-states have been considered, in addition to the GS and resonant states of 6Li. The mean energies EB along with intercluster spin-parities Iπ=(−1)L of these bin-states each with width Δk=0.25 fm−1 are: EB=0.54 MeV (Iπ=0−, 1−, 2−, 1+), 0.93 MeV (1+), 1.37 MeV (2+), 2.65 MeV (0−, 1−, 2−, 1+), 6.89 MeV (0−, 1−, 2−, 1+, 2+, 3+). The negative parity bins are not included in the nine- and twelve-channel calculations. The difference in the latter calculations is that, while EB=0.54 (1+), 2.65 (1+) and 6.89 MeV (1+) are included in the twelve-channel case, these are omitted in the nine-channel one. The predictions are compared with the data in Fig. 3. It is evident that the iT11 predictions are more sensitive to the binning scheme, as noted in Ref. [8]. The inclusion of the negative-parity bins increases greatly disagreement between the calculated and experimental iT11 values. Both the nine- and twelve-channel CDCC calculations reproduce the features of the CS and iT11 data without the inclusion of a static SO potential anywhere and without the need of adjustment in the potential parameters of [18] in Table 1 excepting WD of the d potential. While best fits to both the CS and iT11 data need WD=11.0 and 10.0 MeV, respectively, for the nine- and twelve-channel calculations, the twenty one-channel CDCC calculation prefers WD=7.0 MeV, although producing an unsatisfactory description of the iT11 data. The large difference in the predicted iT11 results between the twenty one-channel and nine/twelve-channel CDCC calculations conforms to the observation of Hirabayashi [26] that the non-resonant breakup states of 6Li make significant contributions to VAP.In the present investigation, both the CC calculations using the NM 6Li potential and CDCC analyses using the CF potential, derived from the component d and NM α potentials, account well for the CS and VAP data without any need for renormalization and any static SO potential. This is remarkably different from the observation, noted in [8], that reports the essentiality of a renormalization for the CF potential used in CC calculations involving the resonant breakup states of 6Li. The renormalization problem with the CF potential in the work of [8] may arise from the monotonic nature of the component α potential used therein.As noted in Ref. [15], the NM characteristic of a potential stems from the Pauli principle appropriately incorporated in a simple way in the EDF theory of Brueckner, Coon and Dabrowski (BCD) [27] to calculate the binding energies per nucleon, ϵ, curves for homogeneous nuclear matter. The works of Hossain et al. [15,16] and Billah et al. [28], and references therein for α NM potentials, relating the EDF-generated NM potentials, respectively, for 6Li and α, employ the parameters of the nucleonic mean field corresponding to a unique nuclear incompressibility of K=184.7 MeV[29], which fits the ϵ curves of BCD. Since a successful analysis of the nucleus-nucleus interaction data can lead to a reliable extraction of the K-value [30]; the profound achievement of the NM potentials through the CC and CDCC calculations in describing both the CS and iT11 data indicates the relevance of the NM nature of nucleus-nucleus potential in studying the equation of state for nuclear matter.This Letter reports for the first time the DPP effects arising from the use of central NM potentials in the CC and CDCC calculations. The NM potential without the renormalization not only reproduces the CS data but also generate an effective SO potential sufficient to account for the iT11 data at 22.8 MeV without the need of an additional static one. The present work clearly suggests that the generated DPP in the CC and CDCC calculations and appropriate adjustment of the imaginary potential are the essential ingredients to account for the CS data and to understand the origin of dynamic SO potential to reproduce the VAP data. The effect of DPP on iT11 is a reminiscence of a classical picture illustrating how absorptive distortion can lead to polarization in direct reactions [31]. A good potential should be qualified to reproduce both the CS and VAP data. The NM characteristics of the 6Li- and α-nucleus potentials seem to play a major role along with the scattering dynamics in determining the CS and VAP of 6Li elastic scattering. The conjecture merits further investigation.AcknowledgementsWe would like to thank Professor K. Rusek of the University of Warsaw for his valuable comments and Professor I.J. Thompson now at Lawrence Livermore Lab for providing the recent version of the codes FRESCO and SFRESCO. 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