Microscopic study of the 7 Li-nucleus potential
Wen-Di Chen, Hai-Rui Guo, Wei-Li Sun, Tao Ye, Yang-Jun Ying, Yin-Lu Han, Qing-Biao Shen
aa r X i v : . [ nu c l - t h ] J a n Microscopic study of the Li-nucleus potential
Wen-Di Chen , Hai-Rui Guo, ∗ , Wei-Li Sun , Tao Ye , Yang-Jun Ying , Yin-Lu Han , andQing-Biao Shen Graduate School of China Academy of Engineering Physics, Beijing 100088, China Institute of Applied Physics and Computational Mathematics, Beijing 100094, China Key Laboratory of Nuclear Data, China Institute of Atomic Energy, Beijing 102413, ChinaJanuary 22, 2020
Abstract
The optical potential without any free parameters for Li-nucleus interaction system is studiedin a microscopic approach. It is obtained by folding the microscopic optical potentials of the constituentnucleons of Li over their density distributions. We employ an isospin-dependent nucleon microscopic opticalpotential, which is based on the Skyrme nucleon-nucleon effective interaction and derived by using the Green’sfunction method, to be the nucleon optical potential. Harmonic oscillator shell model is used to describe theinternal wave function of Li and get the nucleon density distribution. The Li microscopic optical potentialis used to predict the reaction cross sections and elastic scattering angular distributions for target range from Al to
Pb and energy range below 450 MeV. Generally the results can reproduce the measured datareasonably well. In addition, the microscopic optical potential is comparable to a global phenomenologicaloptical potential in fitting the presently existing measured data generally.
Key words: Li microscopic optical potential, Li elastic scattering, folding model
PACS:
Optical potential is an usual and basic tool used in the dynamic analyses of nuclear reactions. Nowadays, mostof the optical potentials are phenomenological. They have some parameters, and are determined by fittingexperimental data. When the experimental data are not sufficient, it is difficult to get reliable phenomenolog-ical optical potential. In contrast, the microscopic optical potential (MOP) is derived from nucleon-nucleoninteraction theoretically, has no free parameters, and does not rely on the experimental data. Therefore, toobtain optical potentials in microscopic approach is a goal of the nuclear physics. It is of great significance forthe analyses of nuclear reactions lacking experimental data.The studies of nuclear reactions involving light-particle projectile or ejectile are an important part of nuclearphysics and very useful for practical applications. Thus, we have already obtained the MOPs for nucleon [1],deuteron [2], triton [3], and , , He [4, 5, 6]. Recent years, the weakly bound Li induced reactions has been asubject of great interest. Breakup, complete and incomplete fusion, and some other reaction mechanisms areconcerned by the experimental and theoretical nuclear physicists [7, 8]. The Li optical potential is requiredin the theoretical analyses.Up to now, there are some Li optical potentials to analyze the experimental data. A semi-microscopicoptical potential, whose real part is generated by double folding model and nucleon-nucleon effective interactionand imaginary part is in the Woods-Saxon form, is given by Woods et al. [9] and used to analyze the elasticscattering data for N and Mg target. Deshmukh et al. [10] provided a Wood-Saxon form optical potential,while it can only be used for
Sn. An optical potential provided by Camacho et al. [11] meets the dispersionrelation of real part and imaginary part, but it is only suitable for Si target. Recently, Xu et al. [12] provideda new global phenomenological optical potential (GOP) based on the presently existing experimental data,which is applicable to a more extensive incident energy and target region. ∗ Corresponding author: guo [email protected] Li scattering data are not sufficient up to now, a Li MOP is obtained in the presentwork by folding the MOPs of its internal nucleons over their density distributions. The isospin-dependentnonrealistic nucleon MOP derived by using the Green’s function method in our previous work [1, 13, 14, 15]is adopted to be the MOP for the constituent nucleons. Shell model is applied to construct the internal wavefunction and generate the nucleon density distributions. The Li elastic-scattering angular distributions andreaction cross sections are calculated by the MOP and compared with the experimental data and the resultscalculated by the GOP [12].This paper is organized as follow: the theoretical model and formulas of the MOP are presented in Sec.2; the calculated results and analysis are provided in Sec. 3; the summary and conclusion are given in Sec. 4finally.
The MOP for Li is generated by the folding model [16] and expressed as U ( ~R ) = Z U n ( ~R + ~r ) ρ n ( ~r ) + U p ( ~R + ~r ) ρ p ( ~r ) d~r, (1)where Z ρ n d~r = N ; Z ρ p d~r = Z. (2) U n and U p represent the MOPs for neutron and proton respectively. ρ n and ρ p are the density distributionsof neutron and proton in the ground state Li respectively. ~R is the relative coordinate between the centers ofmass of the target and Li, and ~r is the internal coordinate of Li.The isospin-dependent nonrealistic nucleon MOP [1, 13, 14, 15] is adopted to be U n and U p and here is abrief introduction for it. From the perspective of many-body theory, the nucleon optical potential is equivalentto the mass operator of the single-particle Green’s function [17]. Based on the Skyrme nucleon-nucleon effectiveinteraction SKC16 [14], which is able to describe the nuclear matter properties, ground state properties andneutron-nucleus scattering well simultaneously, the first- and second-order mass operators of single-particleGreen’s function were derived through the nuclear matter approximation and the local density approximation.The real part of the nucleon MOP was denoted by the first-order mass operator and the imaginary part of thenucleon MOP was denoted by the imaginary part of the second-order mass operator. The incident energy ofnucleon is regarded as one seventh of the incident energy of Li.Shell model is adopted to give an appropriate nucleon density in Li. Since a 1p-shell model space can welldescribe its structure [18] and we only concern the ground-state properties of Li here, harmonic oscillatorpotential is adopted to describe the mean interaction for the nucleons in Li, and the internal Hamiltonian of Li is expressed as H Li = X i =1 T i + X i =1 mω r i , (3)where m is the nucleon mass and r i is coordinate of the i th nucleon in Li relative to the center of mass of Li. T i represents the kinetic energy of the i th nucleon.As the harmonic oscillator potential is used in the shell model, the ground-state wave function of Li isexpressed as Φ g.s. = N A ( ( ~r · ~r ) r Y µ (ˆ r ) exp {− β X i =1 r i } ζ ) , (4)where A is the antisymmetrization operator of the nucleons and N is the normalization factor. ζ representsthe spin and isospin part. Φ g.s. is determined by the parameter β = mω/ ~ , under the conditions of meetingantisymmetrization, spin and parity ( I π = 3 / − ). On base of the constraint condition X i =1 ~r i = 0 , (5)2 set of Jacobi coordinates is used to replace r i and expressed as ~r = 12 ~ξ + 13 ~ξ + 14 ~ξ + 13 ~ξ + 17 ~ξ ,~r = − ~ξ + 13 ~ξ + 14 ~ξ + 13 ~ξ + 17 ~ξ ,~r = − ~ξ + 14 ~ξ + 13 ~ξ + 17 ~ξ ,~r = − ~ξ + 13 ~ξ + 17 ~ξ ,~r = − ~ξ ,~r = 12 ~ξ − ~ξ + 17 ~ξ ,~r = − ~ξ − ~ξ + 17 ~ξ . (6)The value of β is determined by (cid:10) r rms (cid:11) = h Φ g.s. | X i =1 r i | Φ g.s. i , (7)where p h r rms i is the nuclear matter root-mean-square radius of Li and set as 2.50 fm which was obtainedby fitting the reaction cross section in Ref. [19]. It will be convenient to rewrite the Eq. (4) as belowΦ g.s. = N A { φ (1234) φ (5) φ (67) ζ } , (8)where φ (1234) = exp {− β X i =1 r i } ,φ (5) = r Y m (ˆ r ) exp {− β r } ,φ (67) = ( ~r · ~r ) exp {− β r + r ) } . (9)Then we can get a detailed expression for (cid:10) r rms (cid:11) , (cid:10) r rms (cid:11) = A − A − A + 2 A + A − A N − N − N + 2 N + N − N , (10)where A = h φ (1234) φ (5) φ (67) | ˆ O A | φ (1234) φ (5) φ (67) i ξ ,A = h φ (1234) φ (5) φ (67) | ˆ O A | φ (1264) φ (5) φ (37) i ξ ,A = h φ (1234) φ (5) φ (67) | ˆ O A | φ (5234) φ (1) φ (67) i ξ ,A = h φ (1234) φ (5) φ (67) | ˆ O A | φ (5264) φ (1) φ (37) i ξ ,A = h φ (1234) φ (5) φ (67) | ˆ O A | φ (1267) φ (5) φ (34) i ξ ,A = h φ (1234) φ (5) φ (67) | ˆ O A | φ (5267) φ (1) φ (34) i ξ , ˆ O A = 17 X i =1 r i . (11)3 ... i ξ means that the coordinates { ~r i , i = 1 − } are replaced by the Jacobi coordinates { ~ξ i , i = 1 − } . Theformulas for N i are the same as A i while ˆ O A is replaced by ˆ O N = 1. Thus we can get that (cid:10) r rms (cid:11) = 127 β (12)and therefore β =0.2743 fm − . ( f m - ) r (fm) neutron proton Figure 1: Neutron ( ρ n ) and proton ( ρ p ) density distributions. ρ n and ρ p are defined as ρ n ( p ) ( ~r ) = h Φ g.s. | X i =1 δ ( ~r − ~r i ) δ τ n ( p ) ,τ i | Φ g.s. i , (13)where τ i is the isospin of i th nucleon. τ n and τ p are the isospin of neutron and proton respectively. It wouldbe convenient to calculate ρ i firstly, whose formula is the same as Eq. (10) while only ˆ O A is replaced byˆ O ρ,i = δ ( ~r − ~r i ). ρ p = ρ + ρ + ρ , ρ n = ρ + ρ + ρ + ρ , and they have analytical expressions as ρ n ( p ) ( ~r ) = (cid:0) a n ( p ) + b n ( p ) r (cid:1) exp (cid:18) − βr (cid:19) , (14)where a n =0.0921 fm − , a p =0.0621 fm − , b n =0.0081 fm − and b p =0.0076 fm − . The density distributions areplotted in Fig. 1. The MOP for Li+ Ni collision system at incident Li energies of 10 MeV, 100 MeV and 300 MeV is shownin Fig. 2 as an example. The depth of the real part ( V ) decreases with the increase of the radius and energy.However the depth of the imaginary part ( W ) increases a little first and decreases as the radius increases atEL=10, 100 MeV, while it decreases monotonously with the increase of the radius at a higher incident energy,300 MeV. That means the contribution of W changes from the dominant surface absorption to the volumeabsorption as the incident energy increases. The real part of the spin-orbit potential V so ~s · ~l is also obtainedby folding model and V so is shown in Fig. 2, while the imaginary part of the spin-orbit potential is omitted asit is usually very small.The Li elastic-scattering angular distributions and reaction cross sections are predicted by using the MOP.Comparisons with experimental data and the results calculated by the GOP [12] are made.Fig. 3 shows the elastic-scattering angular distribution for Al target at incident energies from 6.0 MeVto 24.0 MeV. The result calculated by the MOP is in good agreement with experimental data [20, 21] except4 Li+ Ni V ( M e V ) EL=300MeV100MeV10MeV (a) W ( M e V ) EL=10MeV100MeV300MeV (b) V S O ( M e V ) R (fm) (c)
Figure 2: The MOP for Li+ Ni system (a) the real part ( V ), (b) the imaginary part ( W ) and (c) the realpart of the spin-orbit potential ( V so ). 5
50 100 15010 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Al( Li,elastic) d / d R c.m. (deg) MOP GOPEL=6.0MeV7.0MeV8.0MeV9.0MeV10.0MeV11.0MeV12.0MeV13.0MeV14.0MeV16.0MeV18.0MeV19.0MeV24.0MeV Figure 3: (color online) Calculated elastic-scattering angular distributions in the Rutherford ratio for Alcompared with experimental data [20, 21]. The solid and dash lines denote the results calculated by the MOPand the GOP [12] respectively. The results from top to bottom are multiplied respectively by 10 , 10 − , 10 − ...6or the underestimation at EL=11.0 MeV for large angles. In addition, the MOP result fits the experimentaldata a little better than the result calculated by the GOP [12] below 14 MeV at larger angles. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Ni( Li,elastic) d / d R c.m. (deg) MOP GOPEL=14.22MeV16.25MeV18.28MeV19.0MeV20.31MeV34.0MeV42.0MeV Figure 4: (color online) Same as Fig. 3 but for Ni. The experimental data are taken from Refs. [22, 23, 24].The results from top to bottom are multiplied respectively by 10 , 10 − , 10 − ...The calculated elastic-scattering angular distribution for Ni target at incident energies from 14.22 MeVto 42.0 MeV is plotted in Fig. 4. The MOP reproduces the experimental data [22, 23, 24] well except theslight underestimation above 70 degrees at 16.25 and 18.28 MeV, where the GOP performs a little better.The elastic-scattering angular distributions for Cu at incident energy 25.0 MeV and Y at incidentenergy 60.0 MeV are shown in Fig. 5. It can be observed for Cu that the theoretical result from the MOPis lower than the measured values [25] above 70 degrees. Reasonable agreement with the experimental data[26] on Y is obtained.Fig. 6 shows the elastic-scattering angular distribution for
Sn target at incident energies from 18.0 MeVto 35.0 MeV. The calculated result from the MOP is in good agreement with experimental data [10] exceptfor those at incident energies 22.0 MeV, 24.0 MeV and 26.0 MeV in large angles. It can be seen that the MOPreproduces the measurements a little better than the GOP does at relatively lower energies.The calculated elastic-scattering angular distribution for
Ba targets is compared with experimental data[27, 28] in Fig. 7. When the scattering angles are less than 80 degrees, good agreement with experimentaldata is obtained for the MOP. The GOP works better at larger angles.In Fig. 8, the calculated elastic-scattering angular distribution for
Pb is shown from 27.0 MeV to 52.0MeV. The MOP result is in satisfying agreement with experimental data [29, 30, 31, 32] and comparable tothe GOP result in fitting the measured data, except for the case at 39 MeV above 70 degrees.Fig. 9 shows the elastic-scattering angular distribution at some specific scattering angles for Al target.The calculated result by the MOP is slightly larger than that by the GOP at incident energies below 15 MeVand has a little better agreement with the measured values [33, 34, 35, 36] when EL ≤ Li induced reactions on C, Al, Si, Zn, nat
Cu,
Sn,
Ba,
Pbare also calculated and shown in Fig. 10 and Fig. 11. Fig. 10 presents the results for C, Si , nat
Cu and
Pb. The theoretical result for C is within the measurement error range [38]. The MOP result for Si is7
50 100 15010 -8 -7 -6 -5 -4 -3 -2 -1 Y( Li,elastic)EL=60.0MeV Cu( Li,elastic)EL=25.0MeV d / d R c.m. (deg) MOP GOP Figure 5: (color online) Same as Fig. 3 but for Cu and Y. The experimental data are taken from Refs.[25, 26]. The data for Y are multiplied by 10 − . -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Sn( Li,elastic) d / d R c.m. (deg) MOP POPEL=18.0MeV19.0MeV20.0MeV21.0MeV22.0MeV23.0MeV24.0MeV26.0MeV30.0MeV35.0MeV Figure 6: (color online) Same as Fig. 3 but for
Sn. The experimental data are taken from Ref. [10]. Theresults from top to bottom are multiplied respectively by 10 , 10 − , 10 − ...8
50 100 15010 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Ba( Li,elastic) d / d R c.m. (deg) MOP GOPEL=21.0MeV22.0MeV23.0MeV24.0MeV28.0MeV30.0MeV32.0MeV52.0MeV Figure 7: (color online) Same as Fig. 3 but for
Ba. The experimental data are taken from Refs. [27, 28].The results from top to bottom are multiplied respectively by 10 , 10 − , 10 − ... -8 -7 -6 -5 -4 -3 -2 -1 Pb( Li,elastic) d / d R c.m. (deg) MOP GOPEL=27.0MeV29.0MeV33.0MeV39.0MeV42.0MeV52.0MeV Figure 8: (color online) Same as Fig. 3 but for
Pb. The experimental data are taken from Refs. [29, 30,31, 32]. The results from top to bottom are multiplied respectively by 10 , 10 − , 10 − ...9 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Al( Li,elastic) d / d R EL (MeV) MOP GOP c.m. =28.0deg135.0deg140.0deg145.0deg150.0deg155.0deg160.0deg165.0deg170.0deg
Figure 9: (color online) Calculated elastic-scattering angular distributions in the Rutherford ratio for Alat some scattering angles compared with experimental data [33, 34, 35, 36, 37]. The solid and dash linesdenote the results calculated by the MOP and the GOP [12] respectively. The results from top to bottom aremultiplied respectively by 10 , 10 − , 10 − ... r ( b ) EL (MeV) MOP GOP Li+ C Si nat Cu Pb Figure 10: (color online) Reaction cross sections calculated by the MOP compared with experimental data for C [38], Si [39, 40, 41, 42, 43], nat
Cu [44] and
Pb [30, 45] and the results calculated by the GOP [12].The data are shifted upwards by adding 0, 2, 4 and 6 b respectively. The solid and dash lines denote theresults calculated by the MOP and the GOP respectively.10
10 20 30 40 500123456789 r ( b ) EL (MeV) MOP GOP Li+ Al Zn Sn Ba Figure 11: (color online) Same as Fig. 10 but for Al, Zn,
Sn and
Ba. The experimental data aretaken from Refs. [21, 46, 47, 10, 28].in good agreement with experimental data [39, 40, 41, 42, 43] below 30 MeV but becomes a little larger from90 MeV to 200 MeV. The reaction cross section for nat
Cu is obtained by averaging the reaction cross sectionsfor Cu and Cu over the natural abundance. It can be seen that the MOP result is in good agreement withexperimental data [44] except for the energy point of 160 MeV. The MOP result for
Pb reproduces theexperimental data [30, 45] reasonably below 70 MeV but gives an underestimation at 300 MeV. In Fig. 11, itcan be observed that the MOP reproduces the experimental data for Al [21, 46], Zn [47],
Sn [10], and
Ba [28] well. The MOP results are comparable to the GOP results in fitting the measured reaction crosssections except for
Pb.The application of the MOP to the prediction of Li elastic scattering from light target nuclei is alsotried. The elastic-scattering angular distribution for O is calculated and compared with experimental data[48, 49, 50, 51] as shown in Fig. 12. The theoretical result from the MOP is only consistent with the magnitudeof measured data in forward angles and gives an overestimation in relative larger angles. Therefore, the MOPis not suitable for the light nuclei. On the one hand, it may be interpreted that the Negele’s nuclear density[52] adopted to calculate the nucleon MOP [1, 13, 14, 15] is not suitable for light nuclei. On the other hand,light nucleus, such as O, has its unique structure characteristics and reaction mechanism [53, 54], which mayalso lead to the discrepancy between the MOP results and measured values.In addition, some discrepancies between the calculated and the measured elastic-scattering angular distri-butions appear at relatively larger angles, such as the case for
Ba target at 28.0 MeV. In order to investigatehow to improve the MOP to give a better global agreement with experimental data, notch perturbation method[55, 56] is employed to analyze the sensitivity of the calculated elastic scattering angular distributions to theoptical potential. The perturbation is performed by setting V , W or V so to 0 in a region of width 0.5 fmcentered at radius R. The scattering sensitivity is assessed by χ /χ , where the χ and χ are the chi-squarescorresponding to the perturbed and original potentials respectively. χ is calculated by χ = 1 N θ N X i =1 (cid:20) σ T ( θ i ) − σ E ( θ i )∆ σ E ( θ i ) (cid:21) , (15)where N θ is the angle numbers of the experimental elastic-scattering angular distributions for Li+
Ba atEL=28.0MeV. σ T ( θ i ), σ E ( θ i ) and ∆ σ E ( θ i ) represent the theoretical value without perturbation, experimentalvalue and experimental error for the i th measured scattering angle respectively. The theoretical value withperturbation, σ T ( θ i ), is used to calculate χ in the same method.Fig. 13 shows the MOP for the Li+
Ba system at EL=28.0 MeV and χ /χ . χ /χ for V so of the MOPalmost remains at unity, so it is acceptable to ignore the impact from changing V so and focus on only V and11
50 100 15010 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 O( Li,elastic) d / d R c.m. (deg) MOP GOPEL=26.0MeV36.0MeV42.0MeV50.0MeV Figure 12: (color online) Same as Fig. 3 but for O. The experimental data are taken from Refs. [48, 49, 50, 51].The results from top to bottom are multiplied respectively by 10 , 10 − , 10 − and 10 − .12 -1 -2 -1 - V ( M e V ) Li+
Ba,EL=28.0MeV (a) - W ( M e V ) (b) / R (fm) V W VSO (c)
Figure 13: (color online) Notch perturbation analysis of the MOP for the Li+
Ba reaction at EL=28.0MeV, (a) V of the MOP; (b) W of the MOP; (c) radial sensitivity of the elastic scattering to the MOP. Thesolid curve, dash curve and dotted curve in (c) represent the results of perturbing V , W and V so respectively. -3 -2 -1 d / d R c.m. (deg) Li+
Ba,EL=28.0MeV MOP GOP N R =0.89,N I =1.00, / =0.618 N R =1.00,N I =1.89, / =0.278 Figure 14: (color online) Calculated elastic-scattering angular distributions in the Rutherford ratio for
Baat EL=28.0 MeV. The solid line, dash line, dotted line, and dash-dotted line denote the results calculatedby the MOP, the GOP [12], the MOP with adjusted V and the MOP with adjusted W respectively. Theadjustment of V and W are made only in the sensitive region 6 fm < R <
12 fm.13 . It can be seen that the peaks of χ /χ locate mainly in the surface interaction region 6 fm < R <
12 fm.We adjust the V and W of the MOP in the sensitive region 6 fm < R <
12 fm by multiplying N R and N I respectively, and calculate the corresponding χ /χ . It can be seen in Fig. 14 that a better agreement withexperimental data at large angles is obtained when N R =0.89 and N I =1.00. This implies that a weaker realpart in the surface region of the MOP may be more suitable for reproducing the measured data. On the otherhand, a smaller χ is gotten when N R =1.00 and N I =1.89, which means that a stronger imaginary part in thesurface region may be better. It is expected that the correction of the MOP results from the breakup effect,because the breakup effect, which is not considered in the folding model, just provides a repulsive contributionto the real part and an absorptive contribution to the imaginary part in the surface region. A Li microscopic optical potential without any free parameter is obtained by folding model. The internal wavefunction of Li is obtained by the shell model, and a nucleon MOP base on Skyrme nucleon-nucleon effectiveinteraction is adopted. The reaction cross sections and elastic-scattering angular distributions for target from Al to
Pb at incident energies below 450 MeV are calculated by the Li microscopic optical potential.Generally, reasonable agreement with the experimental data is obtained, and the MOP is comparable to theGOP in reproducing the measurements in many cases. However, some discrepancies between the calculated andthe measured elastic-scattering angular distributions occur at relatively larger angles. The reason is analyzed,and it is found that the MOP can be improved by adding a repulsive contribution to the real part and anabsorptive contribution to the imaginary part in the surface region, which may be achieved by considering thebreakup effect. That will be our next subject.
Acknowledge
This work is supported by National Natural Science Foundation of China (11705009) and Science ChallengeProject (TZ2018005).