Isospin dependent global neutron-nucleus optical model potential
aa r X i v : . [ nu c l - t h ] O c t Isospin dependent global neutron-nucleus optical model potential
Xiao-Hua Li , ∗ , Lie-Wen Chen , †
1. Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China2. School of Nuclear Science and Technology,University of South China, Hengyang, Hunan 421001, China3. Center of Theoretical Nuclear Physics,National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China (Dated: June 20, 2018)
Abstract
In this paper, we construct a new phenomenological isospin dependent global neutron-nucleusoptical model potential. Based on the existing experimental data of elastic scattering angulardistributions for neutron as projectile, we obtain a set of the isospin dependent global neutron-nucleus optical model potential parameters, which can basically reproduce the experimental datafor target nuclei from Mg to
Pu with the energy region up to 200 MeV. ∗ li [email protected] † [email protected] . INTRODUCTION The optical model(OM) is of fundamental importance on many aspects of nuclearphysics [1]. It is the basis and starting point for many nuclear model calculations andalso is one of the most important theoretical approaches in nuclear data evaluations andanalyses. The optical model potential (OMP) parameters are the key to reproduce the ex-perimental data, such as reaction cross sections, elastic scattering angle distributions, andso on.Over the past years, a number of excellent local and global optical potentials for nu-cleons have been proposed [2][3][4]. Koning and Delarche [2] constructed a set of globalphenomenological nucleon-nucleus optical model potential parameters (KD OMP), whichcan perfectly reproduce the experimental data for the region of targets from Mg to
Biwith the incident energy from 1 keV to 200 MeV; Weppner et al [5] obtained a set of isospindependent global nucleon-nucleus optical model potential parameters (WP OMP) with tar-get nuclei region from carbon to nickel and the projectile energy from 30 to 160 MeV; Han et al [6] also obtained a new set of global phenomenological optical model potential param-eters for nucleon-actinide reactions with energies up to 300 MeV. In the nucleon opticalmodel potential, the isospin degree of freedom may play an important role to more accu-rately describe the experimental data [7] [8]. Information on the isospin dependence of thenucleon optical model potential has been shown to be very useful to understand the nuclearsymmetry energy [9–12] which encodes the energy related to the neutron-proton asymme-try in the equation of state of isospin asymmetric nuclear matter and is a key quantity formany issues in nuclear physics and astrophysics (See, e.g., Ref. [13]). On the other hand,to study the systematics of neutron scattering cross sections on various nuclei for neutronenergies up to several hundred MeV is a very interesting and important topic due to theconcept of an accelerator driven subcritical (ADS) system in which neutrons are producedby bombarding a heavy element target with a high energy proton beam of typically above1.0 GeV with a current of 10 mA and the ADS system serves a dual purpose of energymultiplication and waste incineration (See, e.g., Ref. [14]). Therefore, to construct a moreaccurate neutron-nucleus optical model potential is of crucial importance. The motivationof the present paper is to construct a new isospin dependent neutron-nucleus optical modelpotential, which can reproduce the experimental data for a wider range of target nucleus2han the formers.This paper is arranged as follows. In Sec. II, we provide a description of the opticalmodel and the form of the isospin dependent neutron-nucleus optical potential. Section IIIpresents the results, and section IV is devoted to the discussion. Finally, a summary is givenin Sec. V.
II. OPTICAL MODEL AND THE FORM OF THE ISOSPIN DEPENDENTNEUTRON-NUCLEUS OPTICAL POTENTIAL
The phenomenological OMP for neutron-nucleus reaction V ( r, E ) is usually defined asfollows: V ( r, E ) = − V v f r ( r ) − i W v f v ( r ) + i a s W s df s ( r ) dr + λ − π V so + iW so r df so ( r ) dr ~S · ~l, (1)where V v and V so are the depth of real part of central potential and spin-orbit potential,respectively; W v , W s and W so are the depth of imaginary part of volume absorption potential,surface absorption potential and spin-orbit potential, respectively. The f i ( i = v, s, so ) arethe standard Wood-Saxon shape form factors.In this work, according to Lane Model [7], we add the isospin dependent terms in the V v , W v and W s , which can be parameterized as: V v = V + V E + V E + ( V + V L E ) ( N − Z ) /A, (2) W s = W s + W s E + ( W s + W s L E ) ( N − Z ) /A (3) W v = W v + W v E + W v E + ( W v + W v L E ) ( N − Z ) /A (4)The shape form factors f i can be expressed as f i ( r ) = [1 + exp(( r − r i A / ) /a i )] − with i = r, v, s, so (5)where r i = r i + r i A − / with i = r, v, s, so (6) a i = a i + a i A / with i = r, v, s, so (7)3n above equations, A = Z + N with Z and N being the number of protons and neutrons ofthe target nucleus, respectively; E is the incident neutron energy in the laboratory frame; λ − π is the Compton wave length of pion, and usually we use λ − π = 2 . . APMN [15] is a code to automatically search for a set of optical potential parameterswith smallest χ in E ≤
300 MeV energy region by means of the improved steepest descentalgorithm [16], which is suitable for non-fissile medium-heavy nuclei with the light projectiles,such as neutron, proton, deuteron, triton, He, and α . The optical potential in APMN [15]has been modified based on the standard BG form [3], i.e. Woods-Saxon form for the realpart potential V v and the imaginary part potential of volume absorption W v ; derivativeWoods-Saxon form for the imaginary part potential of surface absorption W s ; and Thomasform for the spin-orbital coupling potential V so and W so . It should be noted that all theradius and diffusiveness parameters in the standard BG optical potential form are constant,not varying with the mass of target nuclei. In the present work, they are modified asfunctions of the mass of target nuclei according to our former work [17]. We modify the APMN code according to the the present form of the isospin dependent global neutron-nucleus optical model potential and thus totally 32 adjustable parameters are involved inthe code
APMN [15].In the code
APMN [15], the compound nucleus elastic scattering is calculated with theHauser-Feshbach statistic theory with Lane-Lynn width fluctuation correction [18](WHF),which is designed for medium-heavy target nuclei. For these nuclei, the spaces between levelsare usually small, the concepts of continuous levels and level density can be properly used fordescription of higher levels, say, their excited energies are higher than the combined energy ofthe emitting particle in compound nucleus. In the code
APMN , the Hauser-Feshbach theorysupposed that after the compound nucleus emits one of the six particles–n, p, d, t, α and He, or a γ photon, all discrete levels of the residual nucleus de-excite only through emissionof γ photons, not permitting emission of any particles. For medium-heavy target nuclei,when the incident energy increase to about 5–7 MeV, the cross sections of the compoundnucleus elastic scattering usually will drop to very small values in comparison with the shapeelastic scattering; so there is no need for considering pre-equilibrium particle emission.4 II. RESULTS
Our theoretical calculation is carried out within the non-relativistic frame and the rela-tivistic kinetics corrections have been neglected because they are usually very small whenthe projectile energy E ≤
200 MeV (See, e.g., Ref. [19]). In the present work, we choosethe existing experimental data of neutron elastic scattering angular distributions with theincident energy region from 0 . APMN .Through the calculation of
APMN code, we obtain a new set of isospin dependent globalneutron-nucleus optical model potential parameters which can be expressed as following: V v = 54 . − . E + 0 . E − (18 . − . E )( N − Z ) /A (MeV) (8) W s = 11 . − . E − (16 . − . E )( N − Z ) /A (MeV) (9) W v = − . . E − . E − (0 . − . E )( N − Z ) /A (MeV) (10) a r = 0 . − . A / (fm) , a s = 0 . − . A / (fm) (11) a v = 0 .
912 + 0 . A / (fm) , a so = 0 .
677 + 0 . A / (fm) (12) r r = 1 . − . A − / (fm) , r s = 1 . − . A − / (fm) (13) r v = 1 .
266 + 0 . A − / (fm) , r so = 0 .
828 + 0 . A − / (fm) (14) V so = 8 .
797 (MeV) , W so = 0 .
019 (MeV) (15)5here the unit of the incident neutron energy E is MeV.With above optical model potential parameters, we calculate the angular distributions ofelastic scattering for many nuclei with neutron as projectile. Some of the calculated resultsand experimental data of elastic scattering angular distributions are shown in Fig. 1 to Fig.12 where the corresponding results from KD OMP are also included for comparison. IV. DISCUSSION
The χ represents the deviation of the calculated values from the experimental data, andin this work it is defined as follows: χ = 1 N N X n =1 χ n , (16)with χ n = 1 N n,el N n,el X i =1 N n,i N n,i X j =1 ( σ thel ( i, j ) − σ expel ( i, j )∆ σ expel ( i, j ) ) , (17)where χ n is for a single nucleus, and n is the nucleus sequence number. χ is the averagevalues of the N nuclei with N denoting the numbers of nuclei included in global parameterssearch and its value is 45 in the present work. σ thel ( i, j ) and σ expel ( i, j ) are the theoreticaland experimental differential cross sections at the j -th angle with the i -th incidence energy,respectively. ∆ σ expel ( i, j ) is the corresponding experimental data error. N n,i is the numberof angles for the n -th nucleus and the i -th incidence energy. N n,el is the number of incidentenergy points of elastic scattering angular distribution for the n -th nucleus.Through minimizing the average χ value for the 45 nuclei in Table I with the modifiedcode APMN , we find an optimal set of global neutron potential parameters, which are given inEqs. (8) − (15). With the obtained parameters above, we get the average value of χ = 32 . χ = 30 .
11 for the same 45 nuclei. Therefore, our parameter set has almost thesame good global quality as that of Koning and Delaroche for the global neutron potential.We use the optical model potential parameters of ours and Koning et al to calculate the χ n of a single nucleus for the 45 nuclei in Table I. In addition, in order to see the predictivepower, we also calculate the χ n for other 58 nuclei listed Table II where the incident energyregion and references are also given. The calculated results for all the 103 nuclei in Table I6nd Table II are shown in Table III where our results are denoted by χ n and that of Koning et al are denoted by χ n , respectively.From Table III, we can see that the value of χ n is close to that of χ n for the nuclei inTable I; The value of χ n is much less than that of χ n for the nuclides Os, Pt, Th, U, andPu; The value of χ n is also close to that of χ n for the other nuclei. This means that ournew set of the isospin dependent global neutron-nucleus optical potential parameters can beas equally good as that of Koning et al to reproduce the experimental data for neutron asprojectile with target ranging from Mg to
Bi. However our results are better than thoseof Koning et al for the actinide. We would like to point out that the number of parametersof our optical model potential is significantly less than that of Koning et al .Some of the elastic scattering angular distributions obtained with our global optical po-tential parameters and with those of Koning et al as well as the corresponding experimentaldata are plotted in Figs. 1 to 12. The sold lines are the results calculated with our param-eters, the dashed lines are the results with the parameters of Koning et al , and the pointsrepresent the experimental data. The same symbols are used in all figures. The experimentaldata and the corresponding theoretical calculation results in all figures are in the center ofmass (C.M.) system. From these figures, we can see clearly that our theoretical calculationscan reproduce the experimental data as equally well as those of Koning et al in the targetsrange from Mg to
Bi, except for some energy points of few nuclei.From Fig. 1 and Fig. 2, it is seen that both of our theoretical calculations and thoseof Koning et al can not well reproduce the experimental data for some energy points oftargets Ca and Ca. This is a well-known problem [20][21] for Ca. It may be due tothe fact that both Ca and Ca are double magic nuclei and the shell effect correctionsmay be important. However, both of our work and that of Koning et al aim at constructingglobal spherical optical model potentials. So the shell effects are not included in both of theOMPs. In addition, the effects of giant resonances have been neglected in both theoreticalcalculations and including them could improve the agreement [22].From Figs. 3-8, one can see that there exist some obvious deviations between experimentaldata and theoretical calculations with both our OMP parameters and that of Koning et al for nuclei Ba and W. This may be due to the fact that the Ba and W exist large deformation,and an effective spherical mean field may no longer provide a totally adequate descriptionof the neutron-nucleus many body problem [2]. Both of the OMPs are based on spherical7rame and the effects of deformation are not considered.For the actinide, such as Th, U, and Pu, it is seen from Figs. 9-12 that our theoreticalresults exhibit significantly better agreement with experimental data than those of Koning et al . V. SUMMARY
A new set of isospin dependent global neutron-nucleus optical potential parameters hasbeen obtained based on the existing experimental data of neutron elastic scattering angu-lar distributions by using the modified code
APMN [15]. The calculated elastic scatteringangular distributions with the new optical model potential parameters have been shownto be in good agreement with the corresponding experimental data for many nuclei from Mg to
Pu in the energy region up to 200 MeV. In particular, our new global opticalmodel potential parameters can give a significantly improved description of neutron elasticscattering angular distributions for the actinide, such as Th, U, and Pu, than the existingglobal optical model potential parameters in the literature. Our new global optical modelpotential can be used to calculate the neutron elastic scattering for different target nucleiincluding those for which the experimental data are unavailable so far.In the present work, polarization of the projectile is not considered. The polarized neutronbeams may play a very important role in nuclear reaction and nuclear structure studies aswell as many fundamental issues of particle physics. We plan to investigate the effect ofneutron polarization in a future work.
Acknowledgments
The authors would like to thank Professor Chong-Hai Cai for useful discussions. Thiswork was supported in part by the NNSF of China under Grant Nos. 10975097 and 11047157,Shanghai Rising-Star Program under Grant No. 11QH1401100, and the National Basic8esearch Program of China (973 Program) under Contract No. 2007CB815004.
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C 36(1987) 73.[215] P. T. Guenther, D. G. Havel, A. B. Smith, Nucl. Sci. Eng. 65 (1978) 174.[216] Y. Tomita, S. Tanaka, M. Maruyama, EANDC Report No. EANDC(J)-30, 1973, p. 6. ABLE I: The data base for searching global optical potential parametersnucleus En.(MeV) Refs. nucleus En.(MeV) Refs. Mg 1.5-14.83 [23]-[27] Al 0.3-26 [23][28]-[48] Si 3.4-21.7 [25][27][38][41][49]-[54] P 3.4-20 [25][35][38][40][55][56] S 5.95-21.7 [26][27][38][54][57]-[60] Ca 1.175-225 [61]-[65] Sc 1.6-10 [66] V 1.61-14.37 [67]-[72] Cr 1.5-18.54 [73]-[77] Mn 2.47-14.1 [25][32][34][72][78] Fe 1.3-26 [79]-[86] Fe 1.8-26 [25][81][83]-[85][87]-[89] Co 1-23 [32][45][47][78][90]-[95] Ni 1.42-24 [73][85][86][96]-[100] Ni 1.5-24 [73][85][98]-[103] Cu 5.5-13.92 [83][104] Cu 2.33-13.92 [83][88][104] As 3.2-8.05 [34][105][106] Se 0.34-10 [107]-[111] Sr 11 [112] Y 0.8892-21.6 [44][47][113]-[120] Zr 1.5-24 [121]-[127] Zr 8-24 [127] Zr 1.5-24 [121][122][124][126][127] Zr 1.5-24 [121][127] Nb 1-20 [44][78][90][128]-[140] Mo 0.9-26 [121][124][141]-[145] Mo 0.9-26 [121][124][141][142][144]-[146] Mo 0.9-26 [141][142][145]
Mo 0.9-26 [124][141][142][144][145]
Rh 1.5-9.995 [115][117][147]
Ag 1.5-4 [148]
Sn 0.4-24 [149]-[151]
Sn 0.8-24 [149][151]-[153]
Sn 0.4-16.905 [78][149]-[151][154]
Sn 0.4-24 [149][151]
I 0.8893-16.1 [152][155]-[157]
Pr 0.8788-8 [34][155][158]-[162]
Nd 2.5-7 [163][164]
Nd 2.5-7 [163][164]
Sm 2.47-7 [165]-[167]
Au 0.134-14.7 [44][134][161][168]-[174]
Pb 0.5-21.6 [47][78][161][169][174]-[178]
Pb 1.285-225 [44][65][178]-[189]
Bi 2-24 [47][182][190]-[194] ABLE II: The incident energy points and data references of the other 58 nuclei.nucleus En.(MeV) Refs. nucleus En.(MeV) Refs. Mg 24 [195] S 21.7,25.5 [54] K 14.07 [49] Ca 2.35-7.97 [196] K 2.9 [74] Cr 1.5-3 [73] Cr 1.5-3 [74][198] Ni 1.5-5 [73][98][152] Ni 1.5-7 [73][98] Zn 1.5-3 [73][198] Zn 1.5-3 [73][198] Zn 1.5-3 [73] Se 0.34-10 [107]-[109][111][199] Se 0.34-8 [107]-[109][199] Se 0.34-10 [107]-[109][199] Mo 0.9-8.04 [121][124][141][143][144][146]
Cd 0.4-1.24 [149]
Cd 4 [152]
Cd 0.6-1.24 [149]
In 5.19-8.53 [200]
In 1.8-8.53 [152][200][201]
Sn 0.4-11 [149][151]
Te 0.3-1.97 [202]
Te 0.3-1.97 [202]
Te 0.3-1.97 [202]
Te 0.3-1.97 [202]
Te 0.3-1.97 [202]
Cs 0.8772 [155]
Ba 3-20 [203]
Ba 2-20 [203]
Ba 4-20 [203]
Ba 3-20 [203]
Ba 5-20 [203]
Ba 3-20 [203]
Ba 5-20 [203]
La 0.98-8 [158][162]
Ce 7.5-14.6 [44][204]
La 7.5 [204]
Nd 2.5-7 [163][164]
Nd 2.5-7 [163][164]
Nd 2.5-7 [163][164]
Sm 2.47-7 [165][167]
Sm 2.5-7 [165][167]
Sm 6.25-7 [166][167]
Sm 2.5-7 [165][167]
Ta 0.323-14.8 [44][78][138][205]-[208]
W 1.5-4.87 [209][210]
W 1.5-4.84 [209][210]
W 1.5-4.87 [209][210]
W 1.5-4.84 [209][210]
W 1.5-3.95 [209]
Os 2.5-4 [211]
Os 1.6-3.94 [212]
Pt 2.5-4.55 [213][214]
Pt 2.53-4.64 [211]
Pb 2.53-8 [176]
Pb 0.5-13.7 [169][215][178][216]
Th 0.144-14.1 [90][181][217]-[222]
U 0.7-1.5 [181]
U 0.185-5.5 [181][223]-[226][230]
U 0.055-15 [181][219][227]-[236]
Pu 0.149-14.1 [181][222][223][237][238]
Pu 0.4-1.2 [239]
Pu 0.57-1.5 [240] ABLE III: χ n of a single nucleus. χ n for our global potential parameters, χ n for those of A. J.Koning and J. P. Delarochenucleus χ n χ n nucleus χ n χ n nucleus χ n χ n Mg 48.07 77.88 Mg 73.45 43.48 Al 33.74 50.36 Si 27.24 22.88 P 33.46 43.17 S 13.25 14.51 S 18.78 15.66 K 65.04 45.88 Ca 19.44 16.25 Ca 319.3 301.2 Sc 14.08 7.934 Ti 16.45 12.60 V 29.21 21.82 Cr 3.095 5.057 Cr 11.81 94.89 Cr 3.012 4.558 Mn 17.92 23.83 Fe 31.20 122.7 Fe 29.40 38.53 Co 50.25 51.38 Ni 11.64 20.90 Ni 18.90 27.08 Ni 6.552 9.851 Ni 6.625 5.580 Cu 8.719 6.573 Cu 10.11 5.919 Zn 6.833 7.435 Zn 6.611 4.363 Zn 7.712 3.665 As 12.06 11.64 Se 44.21 63.26 Se 22.08 35.32 Se 29.91 39.48 Se 13.97 14.68 Sr 38.11 32.66 Y 34.55 19.09 Zr 29.93 23.97 Zr 31.67 19.35 Zr 13.97 6.971 Zr 13.06 12.79 Nb 50.10 42.60 Mo 24.39 29.55 Mo 40.40 35.57 Mo 101.8 170.6 Mo 17.66 20.94
Mo 67.80 137.1
Rh 10.33 15.54
Ag 8.043 42.19
Cd 4.622 6.179
Cd 298.4 77.41
In 5.413 5.741
In 51.65 18.74
Sn 8.380 8.194
Sn 11.67 13.38
Sn 28.10 17.12
Sn 9.091 4.693
Sn 10.06 5.025
Te 3.090 8.610
Te 2.177 5.716
Te 3.471 4.347
Te 9.282 3.668
Te 13.68 4.837
I 91.22 50.50
Cs 7.464 7.211
Ba 897.9 726.0
Ba 687.2 292.9
Ba 957.7 300.9
Ba 1221. 378.3
Ba 2055. 457.7
La 44.88 41.60
Ce 44.40 13.34
Ce 199.8 129.7
Pr 125.2 115.1
Nd 27.00 20.94
Nd 12.55 8.944
Nd 14.58 22.04
Nd 23.58 96.56
Nd 137.8 319.3
Sm 18.90 30.05
Sm 18.01 87.68
Sm 29.60 113.2
Sm 34.01 28.10
Ta 33.85 92.52
W 75.12 316.4
W 65.26 252.6
W 65.85 234.4
Os 246.5 636.7
Os 96.69 323.6
Pt 78.25 297.1
Pt 69.51 204.0
Au 53.09 42.39
Pb 48.68 48.38
Pb 48.26 34.32
Pb 41.30 9.177
Pb 39.23 26.93
Bi 50.53 24.29
Th 43.23 293.5
U 77.01 241.0
U 33.50 126.2
U 119.3 551.8
Pu 37.75 143.5
Pu 36.86 175.9
Pu 27.92 91.89 P 7.79MeV S 5.95MeV P 8.0MeV S 6.1MeV P 9.0MeV S 7.6MeV P 9.05MeV S 9.76MeV P 10.0MeV S 14.5MeV P 11.0MeV S 14.6MeV P 12.0MeV S 14.83MeV P 13.0MeV S 21.5MeV P 14.0MeV S 21.7MeV P 14.2MeV S 21.7MeV P 15.0MeV S 25.5MeV P 16.0MeV K 14.07MeV P 17.0MeV Ca 1.175MeV P 18.0MeV Ca 1.579MeV P 19.0MeV Ca 1.88MeV
C.M. (deg.) d / d ( m b / s r) P 20.0MeV Ca 2.08MeV
FIG. 1: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [26][27][38][40][49][54]-[61]. -4 -5 -6 -3 -3 -2 -2 -2 Ca 2.56MeV Ca 2.43MeV Ca 2.83MeV Ca 2.71MeV Ca 9.91MeV Ca 3.55MeV Ca 11.91MeV Ca 6.0MeV Ca 13.9MeV Ca 7.97MeV Ca 14.1MeV Sc 1.6MeV Ca 16.916MeV Sc 1.89MeV Ca 65.0MeV Sc 2.37MeV Ca 75.0MeV Sc 2.86MeV Ca 85.0MeV Sc 3.2MeV Ca 95.0MeV Sc 3.83MeV Ca 107.5MeV Sc 4.5MeV Ca 155.0MeV Sc 5.0MeV Ca 185.0MeV Sc 5.5MeV Ca 225.0MeV
C.M. (deg.) d / d ( m b / s r) Sc 5.9MeV Ca 2.35MeV Sc 6.5MeV
FIG. 2: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [61]-[66][196][197]. Te 0.5MeV
Cs 0.8772MeV
Te 0.7MeV
Ba 3.0MeV
Te 0.9MeV
Ba 4.0MeV
Te 1.2MeV
Ba 5.0MeV
Te 1.97MeV
Ba 6.0MeV
Te 0.3MeV
Ba 7.0MeV
Te 0.5MeV
Ba 8.0MeV
Te 0.7MeV
Ba 9.0MeV
Te 0.9MeV
Ba 10.0MeV
Te 1.2MeV
Ba 11.0MeV
Te 1.97MeV
Ba 12.0MeV
I 0.8893MeV
Ba 13.0MeV
I 2.9MeV
Ba 14.0MeV
I 4.0MeV
Ba 15.0MeV
I 16.1MeV
C.M. (deg.) d / d ( m b / s r) Ba 16.0MeV
FIG. 3: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [152][155]-[157][202][203]. Ba 18.0MeV
Ba 15.0MeV
Ba 20.0MeV
Ba 16.0MeV
Ba 2.0MeV
Ba 18.0MeV
Ba 3.0MeV
Ba 20.0MeV
Ba 4.0MeV
Ba 4.0MeV
Ba 5.0MeV
Ba 5.0MeV
Ba 6.0MeV
Ba 6.0MeV
Ba 7.0MeV
Ba 7.0MeV
Ba 8.0MeV
Ba 8.0MeV
Ba 9.0MeV
Ba 9.0MeV
Ba 10.0MeV
Ba 10.0MeV
Ba 11.0MeV
Ba 11.0MeV
Ba 12.0MeV
Ba 12.0MeV
Ba 13.0MeV
Ba 13.0MeV
Ba 14.0MeV
C.M. (deg.) d / d ( m b / s r) Ba 14.0MeV
FIG. 4: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [202]. Ba 15.0MeV
Ba 14.0MeV
Ba 16.0MeV
Ba 15.0MeV
Ba 18.0MeV
Ba 16.0MeV
Ba 20.0MeV
Ba 18.0MeV
Ba 3.0MeV
Ba 20.0MeV
Ba 4.0MeV
Ba 5.0MeV
Ba 5.0MeV
Ba 6.0MeV
Ba 6.0MeV
Ba 7.0MeV
Ba 7.0MeV
Ba 8.0MeV
Ba 8.0MeV
Ba 9.0MeV
Ba 9.0MeV
Ba 10.0MeV
Ba 10.0MeV
Ba 11.0MeV
Ba 11.0MeV
Ba 12.0MeV
Ba 12.0MeV
Ba 13.0MeV
Ba 13.0MeV
C.M. (deg.) d / d ( m b / s r) Ba 14.0MeV
FIG. 5: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [202]. Ba 15.0MeV
Pr 2.03MeV
Ba 16.0MeV
Pr 2.545MeV
Ba 18.0MeV
Pr 3.07MeV
Ba 20.0MeV
Pr 3.2MeV
La 0.98MeV
Pr 3.578MeV
La 8.0MeV
Pr 5.0MeV
Ce 7.5MeV
Pr 8.0MeV
Ce 14.6MeV
Nd 2.5MeV
Ce 7.5MeV
Nd 7.0MeV
Pr 0.8788MeV
Nd 2.5MeV
Pr 0.98MeV
Nd 7.0MeV
Pr 1.2MeV
Nd 2.5MeV
Pr 1.5MeV
Nd 7.0MeV
Pr 1.7MeV
Nd 2.5MeV
Pr 1.9MeV
C.M. (deg.) d / d ( m b / s r) Nd 7.0MeV
FIG. 6: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [34][44][155][158]-[164][203][204]. Nd 2.5MeV
Ta 0.95MeV
Nd 7.0MeV
Ta 1.1MeV
Sm 2.47MeV
Ta 1.309MeV
Sm 6.25MeV
Ta 1.465MeV
Sm 7.0MeV
Ta 5.19MeV
Sm 2.47MeV
Ta 6.47MeV
Sm 7.0MeV
Ta 7.49MeV
Sm 2.47MeV
Ta 7.94MeV
Sm 7.0MeV
Ta 11.01MeV
Sm 6.25MeV
Ta 14.1MeV
Sm 7.0MeV
Ta 14.6MeV
Ta 0.323MeV
Ta 14.8MeV
Ta 0.47MeV
W 1.5MeV
Ta 0.65MeV
W 1.7MeV
Ta 0.77MeV
C.M. (deg.) d / d ( m b / s r) W 1.9MeV
FIG. 7: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [44][78][138][163]-[167][205]-[209]. W 2.1MeV
W 1.9MeV
W 2.3MeV
W 2.1MeV
W 2.5MeV
W 2.3MeV
W 3.9MeV
W 2.5MeV
W 4.87MeV
W 3.95MeV
W 1.5MeV
Os 2.5MeV
W 1.7MeV
Os 4.0MeV
W 1.9MeV
Os 1.6MeV
W 2.1MeV
Os 2.52MeV
W 2.3MeV
Os 3.94MeV
W 2.5MeV
Pt 2.5MeV
W 3.9MeV
Pt 4.55MeV
W 4.84MeV
Pt 2.53MeV
W 1.5MeV
Pt 4.64MeV
C.M. (deg.) d / d ( m b / s r) W 1.7MeV
Au 0.134MeV
FIG. 8: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [168][169][209]-[214]. -6 -1 -2 -3 -8 -1 -1 -3 -4 -5 -6 Pb 11.0MeV
Pb 107.5MeV
Pb 13.7MeV
Pb 127.5MeV
Pb 13.9MeV
Pb 155.0MeV
Pb 14.6MeV
Pb 185.0MeV
Pb 16.9MeV
Pb 225.0MeV
Pb 20.0MeV
Bi 2.0MeV
Pb 22.0MeV
Bi 2.5MeV
Pb 24.0MeV
Bi 4.5MeV
Pb 26.0MeV
Bi 5.0MeV
Pb 30.3MeV
Bi 5.5MeV
Pb 40.0MeV
Bi 6.0MeV
Pb 65.0MeV
Bi 6.5MeV
Pb 75.0MeV
Bi 7.0MeV
Pb 85.0MeV
C.M. (deg.) d / d ( m b / s r) Bi 7.5MeV
Pb 95.0MeV
Bi 8.0MeV
FIG. 9: Comparisons of the experimental angular distributions of elastic scattering (dots) with thecalculated results from our global potential parameters (red solid lines) and those of A. J. Koningand J. P. Delaroche (black dashed lines) in the center of mass frame. The experimental data aretaken from Refs. [44][65][178][182][185][187]-[192]. Bi 8.4MeV
Th 2.0MeV
Bi 9.0MeV
Th 2.5MeV
Bi 9.5MeV
Th 3.1MeV
Bi 10.0MeV
Th 3.4MeV
Bi 10.4MeV
Th 4.5MeV
Bi 11.0MeV
Th 5.0MeV
Bi 12.0MeV
Th 5.5MeV
Bi 15.2MeV
Th 6.5MeV
Bi 20.0MeV
Th 7.14MeV
Bi 21.6MeV
Th 7.5MeV
Bi 24.0MeV
Th 8.03MeV
Th 0.144MeV
Th 8.4MeV
Th 0.55MeV
Th 9.06MeV
Th 1.0MeV
C.M. (deg.) d / d ( m b / s r) Th 9.5MeV
Th 1.5MeV
Th 9.999MeV
FIG. 10: Comparisons of the experimental angular distributions of elastic scattering (dots) withthe calculated results from our global potential parameters (red solid lines) and those of A. J.Koning and J. P. Delaroche (black dashed lines) in the center of mass frame. The experimentaldata are taken from Refs. [47][90][181][191]-[194][217]-[221]. Th 14.1MeV
U 5.5MeV
U 0.7MeV
U 0.055MeV
U 1.5MeV
U 0.55MeV
U 0.185MeV
U 1.1MeV
U 0.5MeV
U 1.5MeV
U 1.0MeV
U 1.9MeV
U 1.5MeV
U 2.0MeV
U 1.9MeV
U 2.3MeV
U 2.0MeV
U 2.5MeV
U 2.3MeV
U 3.0MeV
U 3.0MeV
U 3.4MeV
U 3.4MeV
U 4.0MeV
U 4.0MeV
U 4.5MeV
U 4.5MeV
U 5.0MeV
U 5.0MeV
C.M. (deg.) d / d ( m b / s r) U 5.5MeV
FIG. 11: Comparisons of the experimental angular distributions of elastic scattering (dots) withthe calculated results from our global potential parameters (red solid lines) and those of A. J.Koning and J. P. Delaroche (black dashed lines) in the center of mass frame. The experimentaldata are taken from Refs. [181][219][222]-[231]. U 7.0MeV
Pu 3.4MeV
U 9.0MeV
Pu 4.0MeV
U 9.6MeV
Pu 4.5MeV
U 11.0MeV
Pu 5.0MeV
U 12.0MeV
Pu 5.5MeV
U 14.0MeV
Pu 14.1MeV
U 14.7MeV
Pu 0.4MeV
U 15.0MeV
Pu 0.4MeV
Pu 0.149MeV
Pu 0.8MeV
Pu 0.5MeV
Pu 1.2MeV
Pu 1.9MeV
Pu 0.57MeV
Pu 2.3MeV
C.M. (deg.) d / d ( m b / s r) Pu 1.5MeV