Dirac Phenomenological Analyses of Unpolarized Proton Inelastic Scattering from 22 Ne
aa r X i v : . [ nu c l - t h ] J a n Dirac Phenomenological Analyses of Unpolarized Proton InelasticScattering from Ne Moon-Won Kim and Sugie Shim ∗ Department of Physics, Kongju National University, Gongju 314-701 (Received 2014)
Abstract
Unpolarized 800 MeV proton inelastic scatterings from an s-d shell nucleus Ne are analyzedusing phenomenological optical potentials in the Dirac coupled channel formalism. The first-order rotational collective model is used to obtain the transition optical potentials for the low lyingexcited collective states that belong to the ground state rotational band of the nucleus. The opticalpotential parameters of Woods-Saxon shape and the deformation parameters of the excited statesare varied phenomenologically using the sequential iteration method to reproduce the experimentaldifferential cross section data. The effective central and spin-orbit optical potentials are obtainedby reducing the Dirac equations to the Schr¨odinger-like second-order differential equations and thesurface-peaked phenomena are observed at the real effective central potentials when the scatteringfrom Ne is considered. The obtained deformation parameters of the excited states are comparedwith those of the nonrelativistic calculations and another s-d shell nucleus Ne. The deformationparameters for the 2 + and the 4 + states of the ground state rotational band at the nucleus Neare found to be smaller than those of Ne, indicating that the couplings of those states to theground state are weaker at the nucleus Ne compared to those at the nucleus Ne. The multistepchannel coupling effect is confirmed to be important for the 4 + state excitation of the ground staterotational band at the proton inelastic scattering from the s-d shell nucleus Ne.
PACS numbers: 25.40.Ep, 24.10.Jv, 24.10.Ht, 24.10.Eq, 21.60.EvKeywords: Dirac phenomenology, Coupled channel calculation, Optical potential model, Collective model,Proton inelastic scattering ∗ Electronic address: [email protected]; Fax: +82-41-850-8489 . INTRODUCTION Relativistic analyses based on the Dirac equation have shown that they can achievebetter agreement with experimental intermediate energy proton scattering data than thenonrelativistic analyses based on the Schr¨odinger equation[1–3]. Because the Dirac analyseshave proven to be very successful for the intermediate energy proton elastic scatterings fromthe spherically symmetric nuclei and a few deformed nuclei[3–6], the relativistic approacheshave been expanded to the inelastic scatterings and have shown considerable improvementscompared to the conventional nonrelativistic analyses[7–10]. One of the merits of the Diracapproach instead of using the nonrelativistic approach is that the spin-orbit potential appearsnaturally in the Dirac approach when the Dirac equation is reduced to a Schr¨odinger-likesecond-order differential equation, while the spin-orbit potential should be inserted by handin the nonrelativistic Schr¨odinger approach.In this work we performed a relativistic Dirac analysis for the inelastic proton scatteringsfrom an s-d shell nucleus Ne by using the optical potential model[1] and the first-ordercollective model. Dirac phenomenological optical potentials in Woods-Saxon(W-S) shapeare used, employing the scalar-vector (S-V) model where only Lorentz-covariant scalar andtime-like vector optical potentials are included in the calculation. The first-order collectiverotational model is employed in order to describe the collective motion of the excited statesof the ground state rotational band(GSRB) in the nucleus. A computer program called ECISis used to solve the complicated Dirac coupled channel equations, where the Dirac opticalpotential and deformation parameters are determined phenomenologically using a sequentialiteration method[11]. The Dirac equations are reduced to the Schr¨odinger-like second-orderdifferential equations to obtain the effective central and spin-orbit optical potentials andthe results are analyzed and compared with those of another s-d shell nucleus Ne. Thecalculated results for the deformation parameters for the excited states of the ground staterotational band in the nucleus are analyzed and compared with those of Ne and thenonrelativistic approaches. 1
I. THEORY AND RESULTS
Dirac Analyses are performed phenomenologically for the 800 MeV unpolarized protoninelastic scatterings from Ne by using optical potential model and the collective model.Because Ne is one of the spin-0 nuclei, only scalar, time-like vector and tensor opticalpotentials survive[4, 12, 13], as in spherically symmetric nuclei[14]; hence, the relevant Diracequation for the elastic scattering from the nucleus is given as[ α · p + β ( m + U S ) − ( E − U V − V c ) + iα · ˆ rβU T ]Ψ( r ) = 0 . (1)Here, U S is a scalar potential, U V is a time-like vector potential, U T is a tensor potential,and V c is the Coulomb potential. The scalar and time-like vector potentials are used asdirect potentials in the calculation, neglecting the tensor potentials since they have beenfound to be always very small compared to scalar or vector potentials[8, 10, 15] even thoughthey are always present due to the interaction of the anomalous magnetic moment of theprojectile with the charge distribution of the target. The scalar and vector optical potentialsare complex and given as U S = V S f s ( r ) + iW S g s ( r ) (2) U V = V V f v ( r ) + iW V g v ( r ) , (3)where V S and W S are the strengths of the real and the imaginary scalar potentials, V V and W V are the strengths of real and the imaginary time-like vector potentials, respectively.We assume that these potentials have Fermi distribution as they are assumed to follow thedistribution of nuclear density. Fermi model form factors of Woods-Saxon shape for theDirac optical potentials are given as f i ( r ) , g i ( r ) = 11 + exp[ ( r − R , i ) Z i ] , (4)where R ,i and Z i are potential radius and diffusiveness, respectively and the subscript i stands for the real and imaginary scalar, and the real and imaginary vector potentials. Inthe first-order rotational model of ECIS, the deformation of the radius of the optical potentialis given using the Legendre polynomial expansion method; R ( θ ) = R (1 + β Y + β Y + · · · ) , (5)2 ABLE I: Calculated phenomenological optical potential parameters of a Woods-Saxon shape for800 MeV proton elastic scatterings from Ne.Potential Strength (MeV) Radius (fm) Diffusiveness (fm)Scalar -177.6 2.793 0.8065realScalar 160.0 2.213 0.7792imaginaryVector 75.95 3.098 0.7176realVector -101.6 2.649 0.6332imaginary with R the radius at equilibrium, β λ is a deformation parameter and λ is the multipolarity.We assume that the shape of the deformed potentials follows the shape of the deformednuclear densities and that the transition potentials can be obtained by assuming that theyare proportional to the first-order derivatives of the diagonal potentials. However, dependingon the model assumed, pseudo-scalar and axial-vector potentials may also be present in theequation when we consider inelastic scattering. In the collective model approach used inthis work, we assume that we can obtain appropriate transition potentials by deforming thedirect potentials that describe the elastic channel reasonably well[14]. In order to comparethe calculated results with those of the previous nonrelativistic calculations, we reduce theDirac equation to a Schr¨odinger-like second-order differential equation by considering theupper component of the Dirac wave function to obtain the effective central and spin-orbitoptical potentials[3]. The experimental data for the differential cross sections are obtainedfrom Ref. 16 for the 800 MeV unpolarized proton inelastic scatterings from Ne.As a first step, the 12 parameters of the diagonal scalar and vector potentials in Woods-Saxon shapes are determined by fitting the experimental elastic scattering data. The calcu-lated results are shown as dash-dot lines in Fig. 1, and it is found that the observable elasticexperimental differential cross section data are reproduced quite well. The Dirac equationsare phenomenologically solved to obtain the best fitting parameters to the experimental databy using the minimum chi-sq( χ ) method. 3 -3 -2 -1 el. only 0 + , 2 + cpd. 0 + , 4 + cpd. 0 + , 2 + , 4 + cpd. d / d m b / s r c.m. deg Ne(p,p’) 800MeV 0 + FIG. 1: Differential cross section of the ground state for 800 MeV p + Ne scattering. Thedash-dot, dashed, dash-dot-dot and solid lines represent the results of Dirac phenomenologicalcalculation where elastic scattering is considered, where the ground and the 2 + states are coupled,where the ground and the 4 + states are coupled, and where the ground, the 2 + and the 4 + statesare coupled, respectively. Ne Ne r(fm) SOISOR s p i n - o r b i t po t e n t i a l ( M e V ) Ne Ne CRCI ce n t r a l po t e n t i a l ( M e V ) r(fm) FIG. 2: Comparison of the effective central and spin-orbit potentials of Ne and Ne. CR andCI represent central real and imaginary potentials, and SOR and SOI represent spin-orbit real andimaginary optical potentials, respectively. Ne are shown in Table I. χ /N , where N is the number ofexperimental data, was about 4.3. We observe that the real parts of the scalar potentialsand the imaginary parts of the vector potentials turn out to be large and negative, andthat the imaginary parts of the scalar potentials and the real parts of the vector potentialsturn out to be large and positive, showing the same pattern as in the spherically symmetricnuclei[3]. In Fig. 2 we compared the effective central and spin-orbit potentials of Newith those of Ne. It should be noted that one of the merits of the relativistic approachbased on the Dirac equation instead of using the nonrelativistic approach based on theSchr¨odinger equation is that the spin-orbit potential appears naturally in the Dirac approachwhen the Dirac equation is reduced to a Schr¨odinger-like second-order differential equation,while the spin-orbit potential should be inserted by hand using Woods-Saxon shape in thenonrelativistic Schr¨odinger approach. Surface-peaked phenomena are clearly observed forthe real parts of the effective central potentials (CR) at Ne. Ne and other s-d shellnuclei such as Mg[8, 15] also showed the same surface-peaked phenomena, even thoughthey are less clearly shown. The potential strength for the real central potential was about-50MeV at the center of the nucleus, showing large value compared to that of nonrelativisticcalculations that was about -3.9 MeV[16]. Somehow, the potential strength for the imaginarycentral potential turned out to be negative, about -20MeV, at the center of the nucleus Ne, while those of Ne and other s-d shell nuclei have positive values[8, 15] and that ofnonrelativistic calculations was about 49.1 MeV. However, the imaginary central potentialstrength at the surface area turned out to be positive as shown in the figure. The surface-peaked phenomena are clearly shown at the effective spin-orbit potentials, and the effectivespin-orbit potential strengths of Ne turned out to be about the same order with those ofnonrelativistic calculations[16]. We should note that the surface-peaked phenomena neverappear at the the nonrelativistic approaches since they use the Woods-Saxon shapes for boththe central and spin-orbit potentials.Next, a six-parameter search is performed including one excited state, the 2 + or the 4 + state, in addition to the ground state, starting from the 12 parameters obtained for the directoptical potentials in the elastic scattering calculation. Here, the six parameters are the twodeformation parameters, β S and β V , of the excited state and the four potential strengths; thescalar real and imaginary potential strengths and the vector real and imaginary potential5 -2 -1 + , 2 + cpd. 0 + , 2 + , 4 + cpd. d / d m b / s r c.m. deg Ne(p,p’) 800MeV 2 + (1.27MeV) FIG. 3: Differential cross section of the 2 + state for 800 MeV p + Ne inelastic scattering. Dashedand solid lines represent the results of Dirac coupled channel calculation where the ground and the2 + states are coupled and where the ground, the 2 + and the 4 + states are coupled, respectively. strengths, keeping the potential geometries unchanged. The optical potential strengthsobtained by fitting the elastic scattering data in the elastic scattering calculation are variedbecause the channel coupling of the excited states to the ground state should be includedin the inelastic scattering calculation. Finally, an eight-parameter search is performed byconsidering all three states, the ground, the 2 + and the 4 + states, together in the calculationin order to investigate the effect of the channel coupling between the excited states and theresults are compared with those of the calculation where only the ground and one excitedstates are coupled. Figure 1 shows the results of the coupled channel calculations for theground state and it is seen that the coupling effects with the excited states appears at thelarge angles, making the lines to go down[3, 10]. In the figures, ‘cpd’ means ‘coupled’. InFig. 3 and 4, the calculated results for the the 2 + and the 4 + states are shown. For the2 + state, the agreement with the experimental data didn’t change much by including thecoupling with 4 + state. χ /N for the two cases turned out to be about the same. However,the agreement with the experimental data for the 4 + state improved drastically by includingthe coupling with 2 + state, indicating two-step excitation via 2 + state is essential for the4 + state excitation at the ground state rotational band. It means that it is essential toinclude the multistep transition process because the low lying excited states of the GSRB6 -3 -2 -1 + , 4 + cpd. 0 + , 2 + , 4 + cpd. d / d m b / s r c.m. deg Ne(p,p’) 800MeV 4 + (3.36MeV) FIG. 4: Differential cross section of the 4 + state for 800 MeV p + Ne inelastic scattering. Dashedand solid lines represent the results of Dirac coupled channel calculation where the ground and the4 + states are coupled and where the ground, the 2 + and the 4 + states are coupled, respectively. are strongly coupled each other, as shown in the inelastic scatterings from other axially-symmetric deformed nuclei[8–10, 15]. This was not the case for the spherically symmetricnuclei where the excited states could be well described by considering the coupling via single-step transitions[3]. The potential strengths are changed to -311.8, 321.6, 125.0, and -155.0MeV for scalar real and imaginary and vector real and imaginary potentials, respectively, inthe 2 + state coupled case, -154.3, 302.3, 64.88, and -151.8 MeV in the 4 + state coupled case,and -293.1, 305.5, 118.4, and -150.3 MeV in the 2 + and 4 + states coupled case. These resultsconfirm that the changes in the potential strengths depend on the coupling strength to theground state; that is, the smallest change is seen for the 4 + state coupled case, and aboutthe same change is seen for the 2 + state coupled case and all three states coupled case. Theresults of our relativistic coupled channel calculation showed slightly better agreement withexperimental data than those of nonrelativistic calculations[16].In Table II, we show the deformation parameters for the 2 + and the 4 + states of Ne and Ne. It is shown that the deformation parameters for the 2 + and the 4 + states of Ne aresmaller than those of Ne, even though the excitation energies for the states are smaller atthe Ne. We can say that the couplings of the 2 + and the 4 + states to the ground state areweaker at the nucleus Ne compared to those at the nucleus Ne. The values obtained using7
ABLE II: Comparison of the deformation parameters for the 2 + and the 4 + states for 800 MeVproton inelastic scatterings from Ne with those of Ne and nonrelativistic calculations.Target Energynuclei (MeV) β S β V β NR + state N e . N e . , . + state N e . N e . , . the Dirac coupled channel calculations are also compared with those obtained by using thenonrelativistic coupled channel calculations[16–18]. The obtained deformation parametersfrom Dirac phenomenological calculation for the 2 + and 4 + state excitations of Ne arefound to have slightly smaller values compared to those of the nonrelativistic calculations.
III. CONCLUSIONS
A relativistic Dirac coupled channel calculation using an optical potential model coulddescribe the low-lying excited states of the ground state rotational band for 800 MeV unpo-larized proton inelastic scatterings from an s-d shell nucleus Ne reasonably well. The Diracequations are reduced to second-order differential equations to obtain Schr¨odinger-equivalentcentral and spin-orbit potentials, and surface-peaked phenomena were observed at the realeffective central potentials for the scattering from Ne, as shown in the cases of Ne and Mg. The first-order rotational collective models are used to describe the low-lying excitedstates of the ground state rotational band in the nucleus, and the obtained deformation pa-rameters are compared with those of Ne. It is observed that the deformation parametersfor the 2 + and the 4 + states of Ne are smaller than those of Ne, even though the excita-tion energies for the states are smaller at the nucleus Ne. We can say that the couplingsof the 2 + and the 4 + states to the ground state are weaker at the nucleus Ne compared tothose at the nucleus Ne. The deformation parameters for the 2 + and the 4 + excited statesare also compared with those of nonrelativistic calculations, and the obtained deformationparameters from the Dirac phenomenological calculation for the 2 + and 4 + state of Ne are8ound to have slightly smaller values compared to those of the nonrelativistic calculations.The multistep coupling effect is confirmed to be important for the 4 + state excitation of theground state rotational band at the inelastic scattering from an s-d shell deformed nucleus Ne.
Acknowledgments
This work was supported by a research grant from Kongju National University in 2014. [1] L. G. Arnold, B. C. Clark, R. L. Mercer, and P. Swandt, Phys. Rev. C , 1949 (1981).[2] J. A. McNeil, J. Shepard, and S. J. Wallace, Phys. Rev. Lett , 1439 (1983); , 1443 (1983).[3] S. Shim, Ph.D. dissertation, The Ohio State University 1989; L. Kurth, B. C. Clark, E. D.Cooper, S. Hama, S. Shim, R. L. Mercer, L. Ray, and G. W. Hoffmann, Phys. Rev. C ,2086 (1994).[4] S. Shim, B. C. Clark, E. D. Cooper, S. Hama, R. L. Mercer, L. Ray, J. Raynal, and H. S.Sherif, Phys. Rev. C , 1592 (1990).[5] R. de Swiniarski, D. L. Pham, and J. Raynal, Z. Phys. A - Hadrons and Nuclei , 179(1992).[6] D. L. Pham and R. de Swiniarski, Nuovo Cimento A , 1405 (1994).[7] J. J. Kelly, Phys. Rev. C , 064610 (2005).[8] S. Shim, M. W. Kim, B. C. Clark, and L. Kurth Kerr, Phys. Rev. C , 317 (1999).[9] S. Shim, Shin-Ho Ryu and Min-Soo Kim, J. Korean. Phys. Soc. , 271 (2007); S. Shim,Shin-Ho Ryu and Min-Soo Kim, J. Korean. Phys. Soc. , 1146 (2008).[10] S. Shim and M. W. Kim, Int. Jou. of Mod. Phys. E , 1250098 (2012).[11] J. Raynal, Computing as a Language of Physics , ICTP International Seminar Course,281(IAEA, Italy, 1972); J. Raynal,
Notes on ECIS94 , Note CEA-N-2772, 1994.[12] C. J. Horowitz and B. D. Serot, Nucl. Phys. A , 503 (1981).[13] R. J. Furnstahl, C. E. Price, and G. E. Walker, Phys. Rev. C , 2590 (1987).[14] L. Ray and G. W. Hoffmann, Phys. Rev. C , 538 (1986).[15] S. Shim and M. W. Kim, J. Korean. Phys. Soc. , 483 (2014).
16] G. S. Blanpied, B. G. Ritchie, M. L. Barlett, R. W. Fergerson, G. W. Hoffmann, J. A. McGill,B. H. Wildenthal, Phys. Rev. C , 2180 (1988).[17] G. S. Blanpied, G. A. Balchin, G. E. Langston, B. G. Ritchie, M. L. Barlett, G. W. Hoffmann,J. A. McGill, M. A. Franey, M. Gazzaly, B. H. Wildenthal, Phys. Rev. C , 1233 (1984).[18] G. S. Blanpied, B. G. Ritchie, M. L. Barlett, R. W. Fergerson, G. W. Hoffmann, J. A. McGill,B. H. Wildenthal, Phys. Rev. C , 1987 (1988).[19] R. de Swiniarski, A. D. Bacher, F. G. Resmini, G. R. Plattner, D. L. Hendrie and J. Raynal,Phys. Rev. Lett. , 1139 (1972)., 1139 (1972).