Processing math: 100%

The angular distributions of elastic scattering of 12,13C+Zr

  • To obtain the neutron spectroscopic amplitudes for 9096Zr overlaps, experimental data of elastic scattering with small experimental errors and precise optical potentials were analyzed. In this study, the elastic scattering angular distributions of 12,13C + AZr (A = 90, 91, 92, 94, 96) were measured using the high-precision Q3D magnetic spectrometer in the Tandem accelerator. The São Paulo potential was used for the optical potential. The optical model and coupled channel calculations were compared with the experimental data. The theoretical results were found to be very close to the experimental data. In addition, the possible effects of the couplings to the inelastic channels of the AZr targets and 12,13C projectiles on the elastic scattering were studied. It was observed that the couplings to the inelastic channels of the 12,13C projectiles could improve the agreement with the experimental data, while the inelastic couplings to the target states are of minor importance. The effect of the one-neutron stripping in the 13C+AZr elastic scattering was also studied. The one-neutron stripping channel in 13C + AZr was found to be not relevant and did not affect the elastic scattering angular distributions. Our results also show that in the reactions with the considered zirconium isotopes, the presence of the extra neutron in 13C does not influence the reaction mechanism, which is governed by the collective excitation of the 12C core.
  • The accurate description of the elastic scattering is sensitive to the structure of the nuclei involved as well as to the nuclear potential [1-4]. In this way, the elastic scattering process comprises an essential part of the overall understanding of heavy-ion reaction dynamics that depend on the structure of the colliding nuclei. Therefore, the elastic scattering of the heavy-ion system is of pronounced importance as it provides information about the interaction potential, and measures the reflection of the flux as it reaches the region of the interaction. It is often described by optical model calculations with an optical model potential (OMP) having some adjustable parameters. Most reaction theories require the knowledge of the OMP derived from the elastic scattering angular distribution of the colliding nuclei involved or from more fundamental criteria, like double folding potentials.

    The OMP can be used to estimate the cross sections of nuclear reactions; not only elastic scattering but also the other nuclear reactions such as breakup, transfer, and fusion, etc [5-7]. The interaction potential consists of nuclear and Coulomb potentials. The imaginary part of the nuclear potential represents the absorption from the elastic channel to other reaction channels, such as inelastic scattering, re-arrangement reactions, breakup channels (when weakly bound nuclei are involved), etc. The Coulomb potential between projectile and target nuclei is well known. The nuclear potential is often parameterized based on the optical potential. The potential parameters, which are related to the nuclear structure, can be extracted by fitting the elastic scattering angular distributions. Many kinds of studies have focused on the optical potential in heavy-ion physics for many years [8-17]. However, large ambiguities always existed for the optical potential parameters. Many studies devoted to deriving optical potentials have been performed in recent years, and several achievements were realized. Chamon et al. derived a global description of the nucleus-nucleus interactions within the double-folding model named the São Paulo potential (SPP), which can be successfully used for stable and unstable nuclei [18]. Xu et al. proposed a global single folding potential based on the elastic scattering angular distributions of 6,7Li on target nuclei with masses larger than 40 for energies ranging from 5 to 40 MeV/u [19]. Wang et al. obtained an energy independent Woods-Saxon potential at energies much higher than the Coulomb barrier [20]. Gan et al. selected several elastic scattering angular distributions of 12C from target nuclei of A39 to extract the Woods-Saxon potential parameters [21]. The derived potential was able to reproduce many elastic scattering angular distributions induced not only by 12C but also by other projectiles. In our work, we considered the double folding São Paulo potential [18], which has an energy dependence, as the nuclear potential.

    In collisions of typical strongly bound nuclei besides the elastic scattering, other relevant reaction channels, such as the inelastic scattering of projectile and target nuclei, are also accompanied. Therefore, the analysis of the elastic scattering, together with other relevant reaction channels, can be treated in the coupled channel (CC) method. The CC method is the most powerful tool to study multichannel scattering [22]. In this paper, the CC method is used to analyze the elastic scattering of 12,13C on zirconium isotopes.

    For stable Zr isotopes, the neutron numbers are at or close to the magic number of 50. Their neutron capture cross sections are relatively low [23, 24] and very scarce. In nuclear astrophysics, Zr isotopes occupy the intersection of the weak and main s-process, and the neutron capture reactions by Zr isotopes are particularly significant [25]. For this reason, researches focused on them have attracted great attention. The valuable information about the astrophysical medium, including the neutron flux density and temperature, can be extracted from the abundances of the Zr isotopes. Therefore, the neutron capture reaction rates of Zr isotopes should be determined with high accuracy. For example, in 89Zr(n,γ)90Zr, the direct neutron capture reaction rate contributes about 13% of the total reaction rate, which is much larger than the required accuracy of 5%. For the 93Zr(n,γ)94Zr reaction rates, an accuracy of 3%-5% is required [25]. From the evaluated data of the National Nuclear Data Center, the direct neutron capture reaction rate contributes about 10% to the total reaction rates. For this reason, it is meaningful to accurately study the direct components of neutron capture reaction with various experiments. The spectroscopic factor can be used to estimate the direct component of the (n,γ) cross section. The values of the spectroscopic factor can be obtained by comparing the experimental cross sections with the predicted ones from reaction models. Experimental measurements of the neutron spectroscopic factors about Zr isotopes have been done for several decades [26-28]. However, the published spectroscopic factors have large differences with one another, especially for 90Zr. The published neutron spectroscopic factors vary from 3.4-10.0 for 90Zr. Such a large difference may cause more than 20% uncertainty to the total reaction rate of 89Zr(n,γ)90Zr. The main shortcoming of the previous studies is the large experimental errors [29, 30] and the neglect of the influence of optical potentials, which gives large uncertainties to the spectroscopic factors. To improve these defects, a more accurate experiment should be carried out. In this study, 12,13C + AZr reaction systems were selected since 12C is a typical stable nucleus, and the 12C heavy-ion beam can be easily obtained and tuned with the accelerator. For the 13C beam, it is used to extract the optical potentials of exit channels for the one-neutron transfer reaction of 12C + Zr isotopes. For Zr isotopes, some unstable isotopes such as 89Zr (t1/2 = 3.3 d), 93Zr (t1/2 = 1.5 ×106 y), and 95Zr (t1/2 = 64.0 d) exist. These Zr isotopes are not available as reaction targets. Thus, in the experiment using 13C + 90,92,94,96Zr reactions can be complemented with 13C + 89,93,95Zr to extract the OMP of the exit channels.

    This paper is organized as follows. In Sec. 2, we describe the experimental apparatus. In Sec. 3, we compare the experimental angular distributions with OMP and CC calculations. In Sec. 4, we study the influence of one-neutron stripping on the 13C+90,91,92,94,96Zr systems on the elastic scattering. The conclusions are presented in Sec. 5.

    The experiment was performed at the HI-13 Tandem accelerator at the China Institute of Atomic Energy (CIAE), Beijing, with the Q3D magnetic spectrometer. The Q3D magnetic spectrometer has a high-energy resolution of approximately 0.02%, and the angular distributions of elastic scattering can be measured with high precision. The beams of 12C at 66.0 MeV and 13C at 64.0 MeV from the accelerator impinged on the carbon-supported zirconium enriched isotope targets of 90,91,92,94,96ZrO2. The abundance of Zr isotopes in 90,91,92,94,96ZrO2 is shown in Table 1. The thicknesses of the 90,91,92,94,96ZrO2 targets were (32.9 ± 2.5), (27.3 ± 1.6), (30.0 ± 2.2), (41.0 ± 2.9), (34.4 ± 2.3) μg/cm2, respectively, which were calibrated by normalizing the elastic scattering cross sections at forward angles to Rutherford's scattering cross sections.

    Table 1

    Table 1.  Isotopic compositions of targets (%).
    target names90Zr91Zr92Zr94Zr96Zr
    90ZrO299.43.240.970.77.19
    91ZrO20.394.590.510.21.46
    92ZrO20.21.6398.060.42.31
    94ZrO20.10.460.4198.60.89
    96ZrO20.040.080.050.185.15
    DownLoad: CSV
    Show Table

    The experimental setup is shown in Fig. 1. The diameter of the target chamber was 479 mm. To provide a better angular resolution, a collimator with a diameter of 5 mm was put at the entrance (located at the diaphragm position of Fig. 1) of the Q3D magnetic spectrometer to certify the angular resolution of differential cross section better than 0.4. Accordingly, the solid angle acceptance of the Q3D magnetic spectrometer was set to be 0.34 ± 0.01 msr for excellent angular resolution. The targets were placed at the the center of the target chamber upstream of the Q3D magnetic spectrometer. A movable Faraday cup was placed behind the target to monitor the beam intensity, which was used for the absolute normalization of the reaction cross sections. A ΔE-E detector telescopic system was set at approximately 23 downstream of the reaction targets for the cross-check of the beam intensity. The reaction products were separated by Q3D and then measured by a 50 mm × 50 mm two-dimensional position-sensitive silicon detector (PSSD) at the focal plane. The PSSD consists of 16 strips at the horizontal and perpendicular directions, respectively. The width of each strip is 3 mm. In this experiment, we only focus on the horizontal position and do not consider the vertical position. The high momentum resolution of Q3D (the momentum dispersion is 0.025 mm/% at the final focal plane) and the position-energy information from PSSD enables us to identify the specific ions from other reaction channels since the horizontal position of PSSD reflects the radius/momentum of deflected ions in the magnetic spectrometer. i.e. the magnetic rigidity.

    Figure 1

    Figure 1.  Experimental setup.

    The typical two-dimensional spectrum of kinetic energy versus the horizontal position for the 96Zr(12C,12C)96Zr reaction at 26 is shown in Fig. 2(a). It can be seen that the object ions (the 12C in this case) from the reactions can be clearly identified via the energy and position information. Thus, the number of object ions can be counted accurately through the position spectrum of object ions, as shown in Fig. 2(b). In Fig. 2(b), the width of the spectrum is determined by the set momentum range of Q3D. The position width is mainly related to three parameters: the energy spread of scattered ions, the characteristics of the magnetic spectrometer, and the position resolution of the PSSD. In the whole experiment, these three parameters were almost kept unchanged. Therefore, the position width was practically not changed. Through rotating the Q3D magnetic spectrometer, the angular distributions of elastic scattering of 12,13C+Zr were measured in the range of 0 - 60. The ratios of the experimentally obtained differential elastic scattering cross sections to Rutherford's differential cross sections with the change of angles are shown in Figs. 3 and 4 by circles for the reactions induced by 12C and 13C, respectively. The experimental errors mainly stem from the statistical error (3%) and the uncertainty of the target thickness (5%).

    Figure 2

    Figure 2.  (a) The two-dimensional spectrum of kinetic energy versus the horizontal position and (b) the horizontal position spectrum of object ions for 96Zr(12C,12C)96Zr at 26.

    Figure 3

    Figure 3.  (color online) Elastic scattering cross sections for the 12C+AZr reactions at 66 MeV.

    Figure 4

    Figure 4.  (color online) Elastic scattering cross sections for the 13C+AZr reactions at 64 MeV.

    Different kinds of calculations can be performed to derive the elastic scattering angular distributions, depending on the goal of these calculations and the nuclei involved in the collision. Among these calculations, we employed the OMP calculation, usually devoted to determine the energy dependence of the optical potential and to derive the reaction cross sections, and the coupled channel calculations. The previous method is commonly applied to study the effect of channel couplings on elastic scattering, fusion, etc. Both methods solve a Shrödinger equation or the system of equations with specific boundary conditions that allow determining different observables. To describe the system of colliding nuclei, one must postulate a Hamiltonian that includes an optical potential (or potentials in the case of nuclei with cluster or halo structures that have low break up thresholds). The São Paulo potential (SPP) [18], which is a double folding potential with systematics for the matter density of the interacting nuclei, has been commonly used for the optical potential. As we are using the SPP in this work, some brief details about this potential will be given below.

    The nuclear interaction part of the double folding potential is given by the following expression:

    VF(R)=V0ρ1(r1)ρ2(r2)δ(Rr1+r2)dr1dr2,

    (1)

    where V0 = − 456 MeV fm3 and ρi(ri) (i=1− projectile and 2-target, respectively) is the matter densities. In the zero-range approach, where the range of the effective nucleon-nucleon interaction is negligible in comparison with the diffuseness of the nuclear densities, the usual M3Y [31] nucleon-nucleon interaction becomes V0δ(Rr1+r2), where ri is the coordinates of the nucleons inside the nuclei, and R is the vector joining the center of mass of the two interacting nuclei. The SPP accounts for the Pauli non-locality, which arises from quantum exchange effects; its local equivalent form is given by the following expression:

    VSPPLE(E,R)=VF(R)e(4v2/c2),

    (2)

    where VF(R) is the double-folding potential of Eq. (1), c stands for the speed of light, and v is the local projectile-target relative velocity obtained from

    v2(R,E)=2μ[EVC(R)VLE(R,E)],

    (3)

    where, VC is the Coulomb potential. Many studies using the SPP as the optical potential to derive the elastic scattering to study its energy dependence and to calculate direct reaction cross sections have been reported (see for examples Refs. [4, 22, 32-37]).

    To describe the scattering cross sections for the 12C+AZr at 66 MeV and 13C+AZr at 64 MeV with A=90,91,92,94 and 96, we first performed optical model calculations considering only the ground state of each nucleus. The double-folding SPP was used in the optical potential in both the real and imaginary parts [U=(NR+iNI)VSPP] with NR=1.0 and NI=0.78. These strength factors for the real and imaginary parts were adopted because we are not considering any coupling to the ground states. Many systems have had the elastic cross section well described by this value of the strength coefficients in a wide energy interval by means of optical model calculations [38]. The conclusion achieved in the mentioned paper was that these systematics are valid when there is no strong coupling of any relevant channel to the elastic scattering.

    Optical model calculations are important for understanding whenever there are strong couplings to the elastic scattering. Thus, to highlight dynamic effects such as strong couplings or static effects like cluster structures, we used the SPP at energies close to the barrier since this potential (in principle) does not take these effects into account in its systematics. Therefore, if the results do not describe the elastic experimental data, this means there are some relevant dynamic or static effects not included in the one-channel or optical model calculation. On the contrary, if the one-channel calculation describes the elastic scattering angular distributions, there are no important couplings to the elastic channel left out in the calculation, or there are polarizations of different signs that cancel each other. In Figs. 3 and 4, the comparison between the optical model calculations and the experimental data for 12,13C+90,91,92,94,96Zr elastic scattering is shown. One can see that the theoretical results have good agreement with the experimental data. However, when the projectile is 12C (see Fig. 3), the theoretical results are slightly above the experimental data, which means there are some couplings with the elastic channels not included in the calculations. For the projectile 13C (see Fig. 4), the results are slightly below the data when the target is 91Zr.

    To explain this small difference between the experimental data and the theoretical elastic scattering, we performed CC calculations, including the inelastic states of the projectile and target, using the FRESCO code [39]. In Table 2, the states included in the coupling scheme are shown. To describe the transitions between the target and projectile states, a model-independent procedure to account for the Coulomb and nuclear deformations was used, and the electromagnetic transition B(E2) for 91Zr and B(E1) for 13C were taken from Ref. [40]. The quadrupole (β2) and octupole (β3) deformation parameters to couple 13C states were assumed to be equal to the ones reported for 12C in Refs. [41] and [42], respectively. For the 90,92,94,96Zr isotopes, a vibrational model was used, and the deformation parameters β2 and β3 were also taken from Refs. [41] and [42]. The values of deformation parameters are shown in Table 3. The Coulomb and nuclear deformations were considered to have the same values, as long as no statistic effect in matter distributions was expected for these nuclei. The transition form-factors were taken as derivatives of the monopole term, following the usual convention.

    Table 2

    Table 2.  Projectile and target states considered in the coupling scheme.
    nucleusJπenergy/MeV
    12C0+0.0
    2+4.440
    13C1/20.0
    1/2+3.089
    3/23.684
    5/2+3.854
    90Zr0+0.0
    2+2.186
    91Zr5/2+0.0
    1/2+1.205
    5/2+1.466
    7/2+1.882
    3/2+2.042
    9/2+2.131
    92Zr0+0.0
    2+0.934
    0+1.383
    4+1.495
    2+1.847
    94Zr0+0.0
    2+0.919
    0+1.300
    4+1.450
    2+1.671
    96Zr0+0.0
    2+1.750
    DownLoad: CSV
    Show Table

    Table 3

    Table 3.  Deformation parameters and reduced electromagnetic transition probabilities considered in the CC calculations [40-42].
    nucleusβ2β3
    12C0.582
    13C0.5820.44
    90Zr0.0894
    92Zr0.10270.18
    94Zr0.090
    96Zr0.0800.27
    nucleusIIB(E1) /w.u
    13C1/2+1/20.039 (4)
    B(E2) /w.u.
    91Zr1/2+5/2+15 (4)
    5/2+5/2+10.7 (10)
    7/2+5/2+7.7 (13)
    3/2+5/2+59 (6)
    9/2+5/2+4.2(6)
    DownLoad: CSV
    Show Table

    The deformation parameters are related to the reduced electromagnetic transition probabilities by

    βλ=4π3ZRλB(Eλ,II)(1)(II+|II|)/2<IKλ0|IK>,

    (4)

    where Z is the nuclear charge, R=r0A1/3, A is the mass number, and r0= 1.06 fm is the reduced radius. λ stands for the multipolarity of the transition. I and I are the spins of initial and final states, respectively, and K is their projection in the quantization axis. For the real part of the optical potential, the SPP was used along with a short-range potential in the imaginary part, which had the Woods-Saxon [43] form with W=50 MeV, ri=1.06 fm, and ai=0.2 fm for the depth, reduced radius, and diffuseness, respectively. This short-range potential is important for accounting for the absorption of flux due to fusion because this process cannot be explicitly included in the calculations.

    The comparisons between the CC calculations and the experimental data for the 12C+AZr systems are shown in Fig. 5. In this figure, the solid blue line stands for no coupling calculation, in which only the ground state of each nucleus is considered. The dashed green line represents the results when only couplings to the inelastic states of the projectile were considered. In the calculations, represented by the dashed dot red curve, we included only the inelastic states of the target in the coupling scheme. The dotted line is the full CC calculation, in which all couplings with inelastic states of the projectile and target were considered.

    Figure 5

    Figure 5.  (color online) Comparison between CC calculations and elastic scattering data for 12C+AZr reactions at 66 MeV.

    One can note that the couplings with the inelastic states of the target (AZr - dashed red dot curve) do not have a significant influence on the elastic cross section. The coupling with the first excited state of 12C or the full CC calculation has led to the results that slightly overestimate the angular distributions at backward angles.

    Similar CC calculations were performed for the reactions induced by 13C (see Fig. 6). Similar to the 12C case, a good agreement of angular distributions is observed when the couplings to inelastic states of the projectile were included in the calculations (dashed green line).

    Figure 6

    Figure 6.  (color online) Comparison between CC calculations and elastic scattering data for 13C+AZr reactions at 64 MeV.

    Nevertheless, the full CC calculation underestimates the angular distribution from approximately 55 for most of the systems. This might be an indication that there could be missing couplings, like the one-neutron transfer.

    To determine whether there were some missing couplings left out in the CC calculations, we performed finite-range Coupled Reaction Channel (CRC) calculations for the one-neutron transfer for some reactions involving the 13C+AZr systems that have positive Qvalue (see Table 4). For the optical potential in the entrance partitions, the same potentials used in the CC calculations were used in the real and imaginary parts. In the final partitions, the SPP was used in both real and imaginary parts. Again, the imaginary part was multiplied for a coefficient NI=0.78 because no couplings were considered. The single-particle wave functions were obtained using Woods-Saxon potentials with diffuseness and a reduced radius equal to 0.65 and 1.25 fm, respectively, for the target and projectile. The depths of the Woods-Saxon potentials were varied to fit the experimental one-neutron binding energy.

    Table 4

    Table 4.  Qvalue for the one-neutron stripping transfer reaction for the 13C+AZr systems.
    reactionQvalue/MeV
    90Zr(13C,12C)91Zr2.248
    91Zr(13C,12C)92Zr3.688
    92Zr(13C,12C)93Zr1.788
    94Zr(13C,12C)95Zr1.516
    96Zr(13C,12C)97Zr0.629
    DownLoad: CSV
    Show Table

    Shell-model calculations were performed to derive the spectroscopic amplitudes for the projectile and target overlaps, using the NuShellX code [44]. For the projectile overlaps, the psdpn model space and the psdmod effective phenomenological interaction [45] were used. This model space assumes the 4He as a closed core with 1p3/2, 1p1/2, 1d3/2, 1d5/2, and 2s1/2 orbitals as valence orbitals for protons and neutrons. For most of the target overlaps, the glekpn model space and effective phenomenological interaction of the same name [46] were used. This model space uses 1f7/2, 1f5/2, 2p3/2, 2p1/2, and 1g9/2 as valence orbitals for protons and 1g9/2, 1g7/2, 2d5/2, 2d3/2,and 3s1/2 for neutrons. Owing to our computational limitations to perform shell-model calculations using that large valence space, it was necessary to introduce some constraints to generate the spectroscopic amplitude. For this, the 86Sr nucleus was considered to be a closed core, and the number of protons in the higher 2p1/2 and 1g9/2 orbitals was reduced. For neutrons, the 2d3/2 and 3s1/2 orbitals were also restricted. For the 96,97Zr nuclei, this model space and interaction was not able to describe their structure characteristics (eigenvalues, spins, and parities). Thus, it was necessary to use the glepn [46] model space and interaction. This model space uses 2p3/2, 1f5/2, 2p1/2, 1g9/2, 3s1/2, 2d5/2 and 2d3/2 as valence orbitals for protons and 2p3/2, 1f5/2,2p1/2, 1g9/2, 3s1/2, 2d5/2, and 2d3/2 for neutrons. The same closed core (86Sr) considered in the previous calculations was used here, and the number of protons in the higher 1g9/2 orbitals was reduced and 2d5/2 and 2d3/2 were closed for protons. The overlap schemes for the projectile and target are shown in Figs. 7 and 8, respectively. The spectroscopic amplitudes for the projectile and target overlaps are given in Tables 5, 6, 7, 8, 9, and 10.

    Table 5

    Table 5.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the psdpn model space with the psdmod effective phenomenological interaction for projectile overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    initial statefinal statenljspect. ampl.
    13Cg.s(1/2)12Cg.s(0+)1p1/2−0.8009
    12C4.439(2+)1p3/20.9946
    13C3.098(1/2+)12Cg.s(0+)2s1/20.8983
    12C4.439(2+)1d3/2−0.0385
    1d5/20.3118
    13C3.684(3/2)12Cg.s(0+)1p3/2−0.3617
    12C4.439(2+)1p1/2−0.8194
    1d3/20.5415
    13C3.854(5/2+)12Cg.s(0+)1d5/20.9108
    12C4.439(2+)2s1/20.1130
    1d3/2−0.0586
    1d5/20.1965
    DownLoad: CSV
    Show Table

    Table 6

    Table 6.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glekpn model space and effective phenomenological interaction for 90Zr|91Zr ovelaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    initial statefinal statenljspect. ampl.
    90Zrg.s(0+)91Zrg.s(5/2+)2d5/2−0.9872
    91Zr1.205(1/2+)3s1/2−0.9158
    91Zr1.466(5/2+)2d5/20.0407
    91Zr1.882(7/2+)1g7/20.1135
    91Zr2.042(3/2+)2d3/20.9650
    91Zr2.131(9/2+)1g9/20.0007
    90Zr2.186(2+)91Zrg.s.(5/2+)3s1/2−0.0095
    2d3/20.0141
    2d5/2−0.0761
    1g7/20.0107
    1g9/20.0101
    91Zr1.205(1/2+)2d3/20.0850
    2d5/20.7814
    91Zr1.466(5/2+)3s1/20.0021
    2d3/20.0545
    2d5/2−0.5699
    1g7/20.0564
    1g9/20.0017
    91Zr1.882(7/2+)2d3/2−0.0583
    2d5/20.9029
    1g7/20.0618
    1g9/20.0012
    91Zr2.042(3/2+)3s1/20.1176
    2d3/20.0549
    2d5/20.1420
    1g7/20.0521
    91Zr2.131(9/2+)2d5/2−0.7541
    1g7/2−0.0536
    1g9/2−0.0007
    DownLoad: CSV
    Show Table

    Figure 7

    Figure 7.  (color online) Coupling scheme for projectile overlaps used in the calculations. The double side arrows mean all the possible couplings between states.

    Table 7

    Table 7.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glekpn model space and effective phenomenological interaction for 91Zr|92Zr overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    initial statefinal statenljspect. ampl.
    92Zr0.934(2+) 92Zrg.s(0+) 2d5/2 1.3174
    91Zrg.s(5/2+) 3s1/2 0.1597
    2d3/2 0.0580
    2d5/2 1.3631
    1g7/2 0.0227
    1g9/2 −0.0564
    92Zr1.383(0+) 2d5/2 0.1529
    92Zr1.495(4+) 2d3/2 −0.1639
    2d5/2 −0.1639
    1g7/2 −0.0210
    1g9/2 0.0171
    92Zr1.847(2+) 3s1/2 −0.0381
    2d3/2 −0.0030
    2d5/2 0.1627
    1g7/2 −0.0269
    1g9/2 −0.0016
    91Zr1.205(1/2+) 92Zrg.s.(0+) 3s1/2 −0.3706
    92Zr0.934(2+) 2d3/2 −0.0728
    2d5/2 −0.1586
    92Zr1.383(0+) 3s1/2 0.2814
    92Zr1.495(4+) 1g7/2 0.0248
    1g9/2 −0.0245
    92Zr1.847(2+) 2d3/2 0.0383
    2d5/2 −0.0083
    91Zrg.s(5/2+) 92Zrg.s.(0+) 2d5/2 0.0893
    92Zr0.934(2+) 3s1/2 0.022
    2d3/2 −0.0038
    2d5/2 0.0204
    1g7/2 0.0122
    1g9/2 0.00004
    92Zr1.383(0+) 2d5/2 −1.2431
    92Zr1.495(4+) 2d3/2 −0.0101
    2d5/2 0.0625
    1g7/2 −0.0050
    1g9/2 0.0009
    92Zr1.847(2+) 3s1/2 −0.0651
    2d3/2 −0.0236
    2d5/2 −0.8603
    1g7/2 −0.0649
    1g9/2 0.0290
    Continued on next page
    DownLoad: CSV
    Show Table

    Figure 8

    Figure 8.  (color online) Coupling scheme for target overlaps used in the calculations. The double sided arrows indicate all the possible couplings between states.

    Table 7

    Table 7-continued from previous page
    initial statefinal statenljspect. ampl.
    91Zr1.992(7/2+) 92Zrg.s.(0+) 1g7/2 0.0354
    92Zr0.934(2+) 2d3/2 0.0185
    2d5/2 −0.1247
    1g7/2 0.0060
    1g9/2 −0.0029
    92Zr1.383(0+) 1g7/2 −0.0762
    92Zr1.495(4+) 3s1/2 0.0175
    2d3/2 −0.0165
    2d5/2 0.0170
    1g7/2 −0.0005
    1g9/2 0.0036
    92Zr1.847(2+) 2d3/2 0.0013
    2d5/2 0.7293
    1g7/2 −0.0082
    1g9/2 0.0171
    91Zr2.042(3/2+) 92Zrg.s.(0+) 2d3/2 −0.2628
    92Zr0.934(2+) 3s1/2 0.0755
    2d3/2 −0.0748
    2d5/2 0.0543
    1g7/2 −0.0327
    92Zr1.383(0+) 2d3/2 0.0025
    92Zr1.495(4+) 2d5/2 −0.1652
    1g7/2 0.0117
    1g9/2 0.0186
    92Zr1.847(2+) 3s1/2 −0.0625
    2d3/2 0.0106
    2d5/2 −0.0201
    1g7/2 0.0162
    91Zr2.131(9/2+) 92Zrg.s.(0+) 1g9/2 −0.0360
    92Zr0.934(2+) 2d5/2 0.0720
    1g7/2 −0.0046
    1g9/2 −0.0006
    92Zr1.383(0+) 1g9/2 0.0547
    92Zr1.495(4+) 3s1/2 −0.0210
    2d3/2 −0.0037
    2d5/2 −0.0681
    1g7/2 −0.0014
    1g9/2 0.0038
    92Zr1.847(2+) 2d5/2 −0.4477
    1g7/2 0.0251
    1g9/2 −0.0097
    DownLoad: CSV
    Show Table

    Table 8

    Table 8.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glekpn model space and effective phenomenological interaction for 92Zr|93Zr overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    initial statefinal statenljspect. ampl.
    92Zrg.s.(0+) 93Zrg.s.(5/2+) 2d5/2 0.8177
    93Zr0.266(3/2+) 2d3/2 0.0522
    93Zr0.947(1/2+) 3s1/2 0.7849
    93Zr0.949(9/2+) 1g9/2 −0.0014
    92Zr0.934(2+) 93Zrg.s.(5/2+) 3s1/2 0.0234
    2d3/2 −0.0806
    2d5/2 −0.8425
    1g7/2 −0.0138
    1g9/2 −0.0291
    93Zr0.266(3/2+) 3s1/2 −0.2468
    2d3/2 0.0387
    2d5/2 1.4099
    1g7/2 0.0025
    93Zr0.947(1/2+) 2d3/2 −0.0795
    2d5/2 −0.1150
    93Zr0.949(9/2+) 2d5/2 −0.7792
    1g7/2 0.0012
    1g9/2 −0.0297
    92Zr1.384(0+) 93Zrg.s.(5/2+) 2d5/2 0.0554
    93Zr0.266(3/2+) 2d3/2 −0.0080
    93Zr0.947(1/2+) 3s1/2 −0.4118
    93Zr0.949(9/2+) 1g9/2 −0.0013
    92Zr1.495(4+) 93Zrg.s.(5/2+) 2d3/2 −0.0127
    2d5/2 −1.1493
    1g7/2 −0.0118
    1g9/2 −0.0244
    93Zr0.266(3/2+) 2d5/2 −0.8747
    1g7/2 −0.0251
    1g9/2 −0.0730
    93Zr0.947(1/2+) 1g7/2 −0.0369
    1g9/2 −0.0150
    93Zr0.949(9/2+) 3s1/2 0.0866
    2d3/2 0.1659
    2d5/2 1.4825
    1g7/2 0.0301
    1g9/2 −0.0232
    92Zr1.847(2+) 93Zrg.s.(5/2+) 3s1/2 0.0039
    2d3/2 0.0135
    2d5/2 −0.1143
    1g7/2 0.0003
    1g9/2 −0.0031
    93Zr0.266(3/2+) 3s1/2 −0.0059
    2d3/2 −0.0140
    2d5/2 0.1175
    1g7/2 −0.0018
    93Zr0.947(1/2+) 2d3/2 0.0740
    2d5/2 −0.0573
    93Zr0.949(9/2+) 2d5/2 −0.0568
    1g7/2 −0.0053
    1g9/2 −0.0065
    DownLoad: CSV
    Show Table

    Table 9

    Table 9.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glekpn model space and effective phenomenological interaction for 94Zr|95Zr overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    initial statefinal statenljspect. ampl.
    94Zrg.s.(0+)95Zrg.s.(5/2+)2d5/2−0.4773
    95Zr0.954(1/2+)3s1/20.8236
    94Zr0.919(2+)95Zrg.s.(5/2+)3s1/2−0.1345
    2d3/20.0302
    2d5/21.1630
    1g7/20.0213
    1g9/20.0322
    95Zr0.954(1/2+)2d3/2−0.0708
    2d5/2−0.1705
    94Zr1.300(0+)95Zrg.s.(5/2+)2d5/20.2090
    95Zr0.954(1/2+)3s1/2−0.3599
    94Zr1.469(4+)95Zrg.s.(5/2+)2d3/2−0.0806
    2d5/2−1.5742
    1g7/2−0.0095
    1g9/2−0.0181
    95Zr0.954(1/2+)1g7/20.0329
    1g9/20.0183
    94Zr1.671(2+)95Zrg.s.(5/2+)3s1/20.0242
    2d3/20.0050
    2d5/2−0.2516
    1g7/2−0.0034
    1g9/2−0.0041
    95Zr0.954(1/2+)2d3/2−0.0228
    2d5/2−0.2061
    DownLoad: CSV
    Show Table

    Table 10

    Table 10.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glepn model space and effective phenomenological interaction for 96Zr|97Zr overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    initial statefinal statenljspect. ampl.
    96Zrg.s.(0+)97Zrg.s.(1/2+)3s1/20.8918
    97Zr1.103(3/2+)2d3/20.9583
    96Zr1.750(2+)97Zrg.s.(1/2+)2d3/20.1571
    2d5/21.0769
    97Zr1.103(3/2+)3s1/2−0.0826
    2d3/2−0.0726
    2d5/20.0338
    97Zr1.264(7/2+)2d3/20.0.0155
    2d5/2−0.0356
    DownLoad: CSV
    Show Table

    From Fig. 9, one notes that the one-neutron transfer (solid purple curve) has a small influence when compared to CC results for the elastic scattering angular distributions for AZr(13C,12C)A+1Zr systems (with A = 90, 91, 92, 94, 96). The two curves, which correspond to CC and CRC calculations, cannot be clearly distinguished.

    Figure 9

    Figure 9.  (color online) Comparison of CC and CRC calculations with the experimental elastic scattering angular distributions for the 13C+AZr system, for A = 90, 91, 92, 94, 96.

    From the present CC and CRC calculations analysis, one can conclude that, indeed, the couplings to direct reaction channels are weak. This is the reason why the [U=(1.0+0.78i)VSPP] works well in potential scattering. In addition, it is evident that the optical potential used in these calculations is appropriate for describing the elastic cross section.

    Spectroscopic factors for the 90Zr|91Zr, 91Zr|92Zr and 92Zr|93Zr overlaps were obtained in previous works [27, 47-50] by means of distorted wave Born approximation (DWBA) calculations. In Tables 11, 12, and 13 the SFs of these works and from shell-model calculations (present work) are compared. It is clear that the SFs for the ground to ground (elastic transfer) transitions agree quite well with each other. For the 92Zr |93Zr overlaps, the SFs are very close for all the overlaps considered in Ref. [50] (see Table 12). In general, the SFs obtained in this study are in agreement with those previously published.

    Table 11

    Table 11.  Comparison between one-neutron spectroscopic factors for 90Zr|91Zr overlaps obtained by shell-model calculations and from (d,p) transfer reaction [27, 47, 48]. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    transitionsnljSF
    present workRef. [27]Ref. [47]Ref. [48]
    90Zrg.s.|91Zrg.s.2d5/20.970.751.040.95
    90Zrg.s.|91Zr1.2053s1/20.840.660.930.66
    90Zrg.s.|91Zr1.4662d5/20.0010.0280.030.024
    90Zrg.s.|91Zr1.8821g7/20.0130.0820.080.13
    90Zrg.s.|91Zr2.0422d3/20.930.560.630.55
    DownLoad: CSV
    Show Table

    Table 12

    Table 12.  Comparison between one-neutron spectroscopic factors for 92Zr|91Zr overlaps obtained by shell-model calculations and from (p,d) transfer reaction [27, 49]. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    transitionsnljSF
    present workRef. [27]Ref. [49]
    92Zrg.s.|91Zrg.s.2d5/21.731.181.86
    92Zrg.s.|91Zr1.2053s1/20.140.0450.06
    92Zrg.s.|91Zr1.4662d5/20.0080.011
    92Zrg.s.|91Zr1.8821g7/20.001
    92Zrg.s.|91Zr2.0422d3/20.0690.0470.07
    DownLoad: CSV
    Show Table

    Table 13

    Table 13.  Comparison between one-neutron spectroscopic factors for 92Zr|93Zr overlaps obtained by shell-model calculations and from (d,p) transfer reaction [50]. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.
    transitionsnljSF
    present workRef. [50]
    92Zrg.s.|93Zrg.s.2d5/20.670.54
    92Zrg.s.|93Zr0.2662d3/20.0030.007
    92Zrg.s.|93Zr0.9473s1/20.620.53
    *The authors consider the first excited state 93Zr0.266 with spin equal to 5/2+.
    DownLoad: CSV
    Show Table

    The main difference is that the spectroscopic amplitudes obtained in this study are derived from shell-model calculations and applied to heavy-ion interactions, where the high orders of the interactions and the transitions from excited states of the projectile or the target might be relevant [34, 51]. These kinds of effects can be masked by the overestimation of the spectroscopic amplitudes from the DWBA fits to the transfer cross sections (this seems not be the case for the present reactions. See discussion below).

    In Fig. 10, the results for 13C + 90,91,92Zr elastic scattering angular distributions from CRC results using different SFs obtained by shell-model calculations and derived from DWBA fits are compared. One can see that the theoretical results are very similar. The advantage of using SFs obtained from microscopic calculations is that these SFs are not fitted, which increases confidence in the results.

    Figure 10

    Figure 10.  (color online) Effect of one-neutron transfer channel on the 13C + 90,91,92Zr elastic angular distributions using different SFs (see text for details).

    Finally, we compared the theoretical CRC and DWBA results for 90Zr(d, p)91Zr and 92Zr(p, d)91Zr one-neutron transfer reactions using the spectroscopic amplitudes derived in this study and those reported in Ref. [27] (see Tables 11 and 12). This comparison is shown in Figs. 11 and 12. It is evident that the transfer angular distributions are reasonably well described using the spectroscopic amplitudes derived microscopically and adjusted by the DWBA calculations. However, for the state at 1.466 MeV of 91Zr the theoretical result using shell-model spectroscopic amplitude underpredict the experimental data. This occurs because the SF for this channel is very small compared to the SF from Ref. [27]. In addition, we should emphasize that our description for the transfer angular distributions is not as good as that reported in Ref. [27]. This is caused by the optical potential used to describe this reaction. We are not using the original optical potential used in Ref. [27], but the São Paulo potential, which might not be suitable for this (p,d) reaction.

    Figure 11

    Figure 11.  (color online) Transfer angular distribution for 90Zr(d, p)91Zr reaction at 15.89 MeV [27], using different reaction models and spectroscopic factors. For details, see the text.

    Figure 12

    Figure 12.  (color online) Transfer angular distribution from 92Zr(p, d)91Zr reaction at 22.11 MeV [27], using different reaction models and spectroscopic factors. For details, see the text.

    From Figs. 11 and 12, one also realizes that the high-order coupling terms are irrelevant for the very forward angles (compare full blue lines with the dash-dot red lines), but they are relevant for the other angular ranges. It is worth mentioning that the calculations performed in the present work do not have any fitted parameter or scaling factor.

    From the present analysis we expected to find some effects of the one-neutron channel in systems involving 13C when compared to the reactions induced by 12C. The ground state Q-value is positive for the reactions with the four zirconium isotopes included in our study (ranging between 0.63 MeV (96Zr) and 3.67 MeV (91Zr)). So, one might think of some effects on the elastic scattering angular distribution that could help to improve the accuracy of the determination of the spectroscopic factors for target overlaps.

    In the case of the fusion cross section, the effect on the neutron transfer channels is clear [52-55]. After the neutrons are transferred, the deformation of the nuclei is the relevant information that decides the dynamic of the reactions. If the deformation increases, the effective Coulomb barrier decreases, and the fusion cross section is enhanced. Concerning the effect on the elastic angular distribution, it is well known that breakup of neutron halo nuclei is usually governed by the dipole polarizability of these nuclei that produces a hindrance of the Fresnel peak and decreases the elastic scattering angular distribution [1, 2]. How does the neutron transfer affect the elastic cross section, and is Q value one of the ingredients that affect it? This is one of the questions that remain open. In this study, we attempted to contribute to this issue and also to improve our knowledge on the spectroscopy of the zirconium isotopes, although we found that the elastic angular distributions were not sensible to the spectroscopic details.

    In this study, elastic scattering angular distributions for the 12,13C+AZr (A = 90, 91, 92, 94, 96) reactions were measured at the HI-13 Tandem accelerator in China Institute of Atomic Energy (CIAE), Beijing. Optical model, coupled channel and coupled reaction channel calculations for 12,13C+AZr were performed and the results were compared with the experimental data.

    The theoretical calculations provided results very close to those obtained experimentally. The optical model calculations using the São Paulo potential described quite well the elastic angular distributions for 12,13C+AZr systems. It was also observed that the couplings to the inelastic states of the AZr did not affect the elastic cross section significantly. Still, the agreement with the experimental data was improved when the coupling to the lower excited states of 12,13C was considered. The one-neutron stripping for 13C+AZr reactions also appeared as a weak channel. The effects of the one-neutron stripping channel (13C,12C) on the elastic scattering angular distributions were found to be negligible in the case of all the zirconium isotopes used as targets in this study, showing that the optical potential used in the calculations is reliable for describing the elastic cross section.

    Spectroscopic information for various overlaps of zirconium isotopes was obtained from shell-model calculations. The spectroscopic amplitudes presented in this study are close to previous ones and described reasonably well the transfer angular distribution for (d,p) and (p,d) reactions without any scaling factors.

    Our results also show that, in the reactions with the considered zirconium isotopes, the presence of the extra neutron in 13C does not strongly affect the reaction mechanism that is governed by the collective excitation of the 12C core. Similar results have been reported recently for the two-neutron transfer reactions 18O + 12,13C at 84 MeV, where the effect of the pairing correlations between the two transferred neutrons in the two-neutron stripping reaction was not affected by the presence of an extra neutron in 13C [56-58].

    We are grateful to the staff of the China Institute of Atomic Energy for providing stable 12,13C beam throughout the experiment.

    [1] M. Cubero, J. P. Fernández-García, M. Rodríguez-Gallardo et al., Phys. Rev. Lett., 109: 262701 (2012) doi: 10.1103/PhysRevLett.109.262701
    [2] A. Di Pietro, G. Randisi, V. Scuderi et al., Phys. Rev. Lett., 105: 022701 (2010) doi: 10.1103/PhysRevLett.105.022701
    [3] M. Mazzocco, N. Keeley, A. Boiano et al., Phys. Rev. C, 100: 024602 (2019) doi: 10.1103/PhysRevC.100.024602
    [4] G. L. Zhang, G. X. Zhang, C. J. Lin et al., Phys. Rev. C, 97: 044618 (2018) doi: 10.1103/PhysRevC.97.044618
    [5] J. F. Liang, J. R. Beenea, H. Esbensenb et al., Phys. Lett. B, 491: 23 (2000) doi: 10.1016/S0370-2693(00)01016-9
    [6] L. Gan, H. B. Sun, Z. H. Li et al., Phys. Rev. C, 97: 064614 (2018) doi: 10.1103/PhysRevC.97.064614
    [7] S. P. Hu, G. L. Zhang, J. C. Yang et al., Phys. Rev. C, 91: 044619 (2015) doi: 10.1103/PhysRevC.91.044619
    [8] B. Sinha, Phys. Rev. C, 11: 1546 (1975) doi: 10.1103/PhysRevC.11.1546
    [9] J. Fleckner and U. Mosel, Nucl. Phys. A, 277: 170 (1977) doi: 10.1016/0375-9474(77)90268-8
    [10] T. Izumoto, S. Krewald, and A. Faessler, Nucl. Phys. A, 341: 319 (1980) doi: 10.1016/0375-9474(80)90316-4
    [11] N. Ohtsuka, R. Linden, A. Faessler et al., Nucl. Phys. A, 465: 550 (1987) doi: 10.1016/0375-9474(87)90364-2
    [12] D. T. Khoa, Nucl. Phys. A, 484: 376 (1988) doi: 10.1016/0375-9474(88)90077-2
    [13] W. Myers and W. Światecki, Phys. Rev. C, 62: 044610 (2000) doi: 10.1103/PhysRevC.62.044610
    [14] V. B. Soubbotin, W. von Oertzen, X. Viñas et al., Phys. Rev. C, 64: 014601 (2001) doi: 10.1103/PhysRevC.64.014601
    [15] V. Y. Denisov, Phys. Lett. B, 526: 315 (2002) doi: 10.1016/S0370-2693(01)01513-1
    [16] W. M. Seif, J. Phys. G-Nucl. Part. Phys., 30: 1231 (2004) doi: 10.1088/0954-3899/30/9/021
    [17] V. Y. Denisov, Phys. Rev. C, 89: 044604 (2014) doi: 10.1103/PhysRevC.89.044604
    [18] L. C. Chamon, B. V. Carlson, L. R. Gasques et al., Phys. Rev. C, 66: 014610 (2002) doi: 10.1103/PhysRevC.66.014610
    [19] Y. P. Xu and D. Y. Pang, Phys. Rev. C, 87: 044605 (2013) doi: 10.1103/PhysRevC.87.044605
    [20] N. Wang and W. Scheid, Phys. Rev. C, 78: 014607 (2008) doi: 10.1103/PhysRevC.78.014607
    [21] L. Gan, Z. H. Li, H. B. Sun et al., Sci. China: Physics, Mechanics and Astronomy, 60: 082013 (2017) doi: 10.1007/s11433-017-9061-5
    [22] L. F. Canto, P. R. S. Gomes, R. Donangelo et al., Phys. Rep., 596: 1 (2015) doi: 10.1016/j.physrep.2015.08.001
    [23] S. Dutta, G. Gangopadhyay, and A. Bhattacharyya, Phys. Rev. C, 94: 024604 (2016) doi: 10.1103/PhysRevC.94.024604
    [24] M. Guttormsen, S. Goriely, A. C. Larsen et al., Phys. Rev. C, 96: 024313 (2017) doi: 10.1103/PhysRevC.96.024313
    [25] G. Tagliente, P. M. Milazzo, K. Fujii et al., Phys. Rev. C, 87: 014622 (2013) doi: 10.1103/PhysRevC.87.014622
    [26] G. M. Crawley, J. Kasagi, S. Gales et al., Phys. Rev. C, 23: 1818 (1981)
    [27] H. Block, L. Hulstman, E. Kaptein et al., Nucl. Phys. A, 273: 142 (1976) doi: 10.1016/0375-9474(76)90305-5
    [28] E. Frota-Pessôa and S. Joffily, Il Nuovo Cimento A (1965-1970) 91, 370 (1986)
    [29] J. Kasagi, G. M. Crawley, E. Kashy et al., Phys. Rev. C, 28: 1065 (1983) doi: 10.1103/PhysRevC.28.1065
    [30] B. L. Cohen and O. V. Chubinsky, Phys. Rev., 131: 2184 (1963) doi: 10.1103/PhysRev.131.2184
    [31] G. Bertsch, J. Borysowicz, H. McManus et al., Nucl. Phys. A, 284: 399 (1977) doi: 10.1016/0375-9474(77)90392-X
    [32] G. L. Zhang, G. X. Zhang, S. P. Hu et al., Phys. Rev. C, 97: 014611 (2018) doi: 10.1103/PhysRevC.97.014611
    [33] E. N. Cardozo, J. Lubian, R. Linares et al., Phys. Rev. C, 97: 064611 (2018) doi: 10.1103/PhysRevC.97.064611
    [34] E. N. Cardozo, M. J. Ermamatov, J. L. Ferreira et al., Eur. Phys. J. A, 54: 150 (2018) doi: 10.1140/epja/i2018-12587-1
    [35] J. R. B. Oliveira, V. Zagatto, D. Pereira et al., EPJ Web of Conferences, 2: 02002 (2010) doi: 10.1051/epjconf/20100202002
    [36] M. A. G. Alvarez, L. C. Chamon, M. S. Hussein et al., Nucl. Phys. A, 723: 93 (2003) doi: 10.1016/S0375-9474(03)01158-8
    [37] M. S. Hussein, P. R. S. Gomes, J. Lubian et al., Phys. Rev. C, 73: 044610 (2006) doi: 10.1103/PhysRevC.73.044610
    [38] L. Gasques, L. Chamon, P. R. S. Gomes et al., Nucl. Phys. A, 764: 135 (2006) doi: 10.1016/j.nuclphysa.2005.09.001
    [39] I. J. Thompson, Comput. Phys. Rep., 7: 167 (1988) doi: 10.1016/0167-7977(88)90005-6
    [40] See http://www-nds.iaea.or.at/nudat
    [41] S. Raman, C. W. Nestorjr, and P. Tikkanen, Atomic Data and Nuclear Data Tables, 78: 1 (2001) doi: 10.1006/adnd.2001.0858
    [42] T. Kibédi and R. H. Spear, Atomic Data and Nuclear Data Tables, 80: 3582 (2002)
    [43] R. A. Broglia and A. Winther, in Heavy-Ion Reactions, Parts I and Ⅱ FIP Lecture Notes Series (Addison-Wesley, New York, 1991)
    [44] W. D. M. Rae, http://www.garsington.eclipse.co.uk (2008)
    [45] Y. Utsuno and S. Chiba, Phys. Rev. C, 83: 021301 (2011) doi: 10.1103/PhysRevC.83.021301
    [46] H. Mach, E. K. Warburton, R. L. Gill et al., Phys. Rev. C, 41: 226 (1990) doi: 10.1103/PhysRevC.41.226
    [47] C. R. Bingham and M. L. Halbert, Phys. Rev. C, 2: 6 (1970)
    [48] A. Graue, L. H. Herland, K. J. Lervik et al., Nucl. Phys. A, l87: 141 (1972)
    [49] J. B. Ball and C. B. Fulmkr, Phys. Rev., 172: 4 (1968)
    [50] N. Baron, C. L. Fink, P. R. Christensen et al., NASA Technical Memorandum NASA TM X- 67993 (1972)
    [51] M. J. Ermamatov, R. Linares, J. Lubian et al., Phys. Rev. C, 96: 044603 (2017) doi: 10.1103/PhysRevC.96.044603
    [52] V. I. Zagrebaev, Phys. Rev. C, 67: 061601(R) (2003)
    [53] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko et al., Phys. Rev. C, 84: 064614 (2011) doi: 10.1103/PhysRevC.84.064614
    [54] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko et al., Phys. Rev. C, 85: 017603 (2012) doi: 10.1103/PhysRevC.85.017603
    [55] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko et al., Phys. Rev. C, 85: 037602 (2012) doi: 10.1103/PhysRevC.85.037602
    [56] M. Cavallaro, F. Cappuzzello, M. Bondì et al., Phys. Rev. C, 88: 054601 (2013) doi: 10.1103/PhysRevC.88.054601
    [57] D. Carbone, J. Ferreira, F. Cappuzzello et al., Phys. Rev. C, 95: 034603 (2017) doi: 10.1103/PhysRevC.95.034603
    [58] F. Cappuzzello, D. Carbone, M. Cavallaro et al., Nat. Commun., 6: 6743 (2015) doi: 10.1038/ncomms7743
  • [1] M. Cubero, J. P. Fernández-García, M. Rodríguez-Gallardo et al., Phys. Rev. Lett., 109: 262701 (2012) doi: 10.1103/PhysRevLett.109.262701
    [2] A. Di Pietro, G. Randisi, V. Scuderi et al., Phys. Rev. Lett., 105: 022701 (2010) doi: 10.1103/PhysRevLett.105.022701
    [3] M. Mazzocco, N. Keeley, A. Boiano et al., Phys. Rev. C, 100: 024602 (2019) doi: 10.1103/PhysRevC.100.024602
    [4] G. L. Zhang, G. X. Zhang, C. J. Lin et al., Phys. Rev. C, 97: 044618 (2018) doi: 10.1103/PhysRevC.97.044618
    [5] J. F. Liang, J. R. Beenea, H. Esbensenb et al., Phys. Lett. B, 491: 23 (2000) doi: 10.1016/S0370-2693(00)01016-9
    [6] L. Gan, H. B. Sun, Z. H. Li et al., Phys. Rev. C, 97: 064614 (2018) doi: 10.1103/PhysRevC.97.064614
    [7] S. P. Hu, G. L. Zhang, J. C. Yang et al., Phys. Rev. C, 91: 044619 (2015) doi: 10.1103/PhysRevC.91.044619
    [8] B. Sinha, Phys. Rev. C, 11: 1546 (1975) doi: 10.1103/PhysRevC.11.1546
    [9] J. Fleckner and U. Mosel, Nucl. Phys. A, 277: 170 (1977) doi: 10.1016/0375-9474(77)90268-8
    [10] T. Izumoto, S. Krewald, and A. Faessler, Nucl. Phys. A, 341: 319 (1980) doi: 10.1016/0375-9474(80)90316-4
    [11] N. Ohtsuka, R. Linden, A. Faessler et al., Nucl. Phys. A, 465: 550 (1987) doi: 10.1016/0375-9474(87)90364-2
    [12] D. T. Khoa, Nucl. Phys. A, 484: 376 (1988) doi: 10.1016/0375-9474(88)90077-2
    [13] W. Myers and W. Światecki, Phys. Rev. C, 62: 044610 (2000) doi: 10.1103/PhysRevC.62.044610
    [14] V. B. Soubbotin, W. von Oertzen, X. Viñas et al., Phys. Rev. C, 64: 014601 (2001) doi: 10.1103/PhysRevC.64.014601
    [15] V. Y. Denisov, Phys. Lett. B, 526: 315 (2002) doi: 10.1016/S0370-2693(01)01513-1
    [16] W. M. Seif, J. Phys. G-Nucl. Part. Phys., 30: 1231 (2004) doi: 10.1088/0954-3899/30/9/021
    [17] V. Y. Denisov, Phys. Rev. C, 89: 044604 (2014) doi: 10.1103/PhysRevC.89.044604
    [18] L. C. Chamon, B. V. Carlson, L. R. Gasques et al., Phys. Rev. C, 66: 014610 (2002) doi: 10.1103/PhysRevC.66.014610
    [19] Y. P. Xu and D. Y. Pang, Phys. Rev. C, 87: 044605 (2013) doi: 10.1103/PhysRevC.87.044605
    [20] N. Wang and W. Scheid, Phys. Rev. C, 78: 014607 (2008) doi: 10.1103/PhysRevC.78.014607
    [21] L. Gan, Z. H. Li, H. B. Sun et al., Sci. China: Physics, Mechanics and Astronomy, 60: 082013 (2017) doi: 10.1007/s11433-017-9061-5
    [22] L. F. Canto, P. R. S. Gomes, R. Donangelo et al., Phys. Rep., 596: 1 (2015) doi: 10.1016/j.physrep.2015.08.001
    [23] S. Dutta, G. Gangopadhyay, and A. Bhattacharyya, Phys. Rev. C, 94: 024604 (2016) doi: 10.1103/PhysRevC.94.024604
    [24] M. Guttormsen, S. Goriely, A. C. Larsen et al., Phys. Rev. C, 96: 024313 (2017) doi: 10.1103/PhysRevC.96.024313
    [25] G. Tagliente, P. M. Milazzo, K. Fujii et al., Phys. Rev. C, 87: 014622 (2013) doi: 10.1103/PhysRevC.87.014622
    [26] G. M. Crawley, J. Kasagi, S. Gales et al., Phys. Rev. C, 23: 1818 (1981)
    [27] H. Block, L. Hulstman, E. Kaptein et al., Nucl. Phys. A, 273: 142 (1976) doi: 10.1016/0375-9474(76)90305-5
    [28] E. Frota-Pessôa and S. Joffily, Il Nuovo Cimento A (1965-1970) 91, 370 (1986)
    [29] J. Kasagi, G. M. Crawley, E. Kashy et al., Phys. Rev. C, 28: 1065 (1983) doi: 10.1103/PhysRevC.28.1065
    [30] B. L. Cohen and O. V. Chubinsky, Phys. Rev., 131: 2184 (1963) doi: 10.1103/PhysRev.131.2184
    [31] G. Bertsch, J. Borysowicz, H. McManus et al., Nucl. Phys. A, 284: 399 (1977) doi: 10.1016/0375-9474(77)90392-X
    [32] G. L. Zhang, G. X. Zhang, S. P. Hu et al., Phys. Rev. C, 97: 014611 (2018) doi: 10.1103/PhysRevC.97.014611
    [33] E. N. Cardozo, J. Lubian, R. Linares et al., Phys. Rev. C, 97: 064611 (2018) doi: 10.1103/PhysRevC.97.064611
    [34] E. N. Cardozo, M. J. Ermamatov, J. L. Ferreira et al., Eur. Phys. J. A, 54: 150 (2018) doi: 10.1140/epja/i2018-12587-1
    [35] J. R. B. Oliveira, V. Zagatto, D. Pereira et al., EPJ Web of Conferences, 2: 02002 (2010) doi: 10.1051/epjconf/20100202002
    [36] M. A. G. Alvarez, L. C. Chamon, M. S. Hussein et al., Nucl. Phys. A, 723: 93 (2003) doi: 10.1016/S0375-9474(03)01158-8
    [37] M. S. Hussein, P. R. S. Gomes, J. Lubian et al., Phys. Rev. C, 73: 044610 (2006) doi: 10.1103/PhysRevC.73.044610
    [38] L. Gasques, L. Chamon, P. R. S. Gomes et al., Nucl. Phys. A, 764: 135 (2006) doi: 10.1016/j.nuclphysa.2005.09.001
    [39] I. J. Thompson, Comput. Phys. Rep., 7: 167 (1988) doi: 10.1016/0167-7977(88)90005-6
    [40] See http://www-nds.iaea.or.at/nudat
    [41] S. Raman, C. W. Nestorjr, and P. Tikkanen, Atomic Data and Nuclear Data Tables, 78: 1 (2001) doi: 10.1006/adnd.2001.0858
    [42] T. Kibédi and R. H. Spear, Atomic Data and Nuclear Data Tables, 80: 3582 (2002)
    [43] R. A. Broglia and A. Winther, in Heavy-Ion Reactions, Parts I and Ⅱ FIP Lecture Notes Series (Addison-Wesley, New York, 1991)
    [44] W. D. M. Rae, http://www.garsington.eclipse.co.uk (2008)
    [45] Y. Utsuno and S. Chiba, Phys. Rev. C, 83: 021301 (2011) doi: 10.1103/PhysRevC.83.021301
    [46] H. Mach, E. K. Warburton, R. L. Gill et al., Phys. Rev. C, 41: 226 (1990) doi: 10.1103/PhysRevC.41.226
    [47] C. R. Bingham and M. L. Halbert, Phys. Rev. C, 2: 6 (1970)
    [48] A. Graue, L. H. Herland, K. J. Lervik et al., Nucl. Phys. A, l87: 141 (1972)
    [49] J. B. Ball and C. B. Fulmkr, Phys. Rev., 172: 4 (1968)
    [50] N. Baron, C. L. Fink, P. R. Christensen et al., NASA Technical Memorandum NASA TM X- 67993 (1972)
    [51] M. J. Ermamatov, R. Linares, J. Lubian et al., Phys. Rev. C, 96: 044603 (2017) doi: 10.1103/PhysRevC.96.044603
    [52] V. I. Zagrebaev, Phys. Rev. C, 67: 061601(R) (2003)
    [53] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko et al., Phys. Rev. C, 84: 064614 (2011) doi: 10.1103/PhysRevC.84.064614
    [54] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko et al., Phys. Rev. C, 85: 017603 (2012) doi: 10.1103/PhysRevC.85.017603
    [55] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko et al., Phys. Rev. C, 85: 037602 (2012) doi: 10.1103/PhysRevC.85.037602
    [56] M. Cavallaro, F. Cappuzzello, M. Bondì et al., Phys. Rev. C, 88: 054601 (2013) doi: 10.1103/PhysRevC.88.054601
    [57] D. Carbone, J. Ferreira, F. Cappuzzello et al., Phys. Rev. C, 95: 034603 (2017) doi: 10.1103/PhysRevC.95.034603
    [58] F. Cappuzzello, D. Carbone, M. Cavallaro et al., Nat. Commun., 6: 6743 (2015) doi: 10.1038/ncomms7743
  • 加载中

Cited by

1. Yang, G., Duan, F.-F., Wang, K. et al. Elastic scattering of 13C and 14C isotopes on a 208Pb target at energies of approximately five times the Coulomb barriers[J]. Chinese Physics C, 2024, 48(3): 034001. doi: 10.1088/1674-1137/ad1678
2. Kumar, C., Gonika, Yashraj, Kalkal, S., Nath, S. Coupled reaction channel analysis of one- and two-nucleon transfer in 28 Si+ 90,94 Zr[J]. European Physical Journal A, 2023, 59(11): 277. doi: 10.1140/epja/s10050-023-01195-1
3. Olorunfunmi, S.D., Adeojo, S.A., Bahini, A. Investigation of elastic scattering angular distributions of 12 , 13 C + 90 , 91 , 92 , 94 , 96 Zr: a comparative analysis of different optical model potentials[J]. Indian Journal of Physics, 2023. doi: 10.1007/s12648-023-03025-y
4. Xu, Y.-L., Han, Y.-L., Su, X.-W. et al. Description of elastic scattering induced by the unstable nuclei 9,10,11,13,14C[J]. Chinese Physics C, 2021, 45(11): 114103. doi: 10.1088/1674-1137/ac1fe1

Figures(12) / Tables(14)

Get Citation
Cui-Hua Rong, Gao-Long Zhang, Lin Gan, Zhi-Hong Li, L. C. Brandão, E. N. Cardozo, M. R. Cortes, Yun-Ju Li, Jun Su, Sheng-Quan Yan, Sheng Zeng, Gang Lian, Bing Guo, You-Bao Wang, Wei-Ping Liu and J. Lubian. The angular distributions of elastic scattering of 12,13C+Zr[J]. Chinese Physics C. doi: 10.1088/1674-1137/abab8d
Cui-Hua Rong, Gao-Long Zhang, Lin Gan, Zhi-Hong Li, L. C. Brandão, E. N. Cardozo, M. R. Cortes, Yun-Ju Li, Jun Su, Sheng-Quan Yan, Sheng Zeng, Gang Lian, Bing Guo, You-Bao Wang, Wei-Ping Liu and J. Lubian. The angular distributions of elastic scattering of 12,13C+Zr[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abab8d shu
Milestone
Received: 2020-05-28
Article Metric

Article Views(1956)
PDF Downloads(38)
Cited by(4)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

The angular distributions of elastic scattering of 12,13C+Zr

    Corresponding author: Gao-Long Zhang, zgl@buaa.edu.cn
  • 1. School of Physics, Beihang University, Beijing 100191, China
  • 2. Beijing Advanced Innovation Center for Big Data-Based Precision Medicine, School of Medicine and Engineering, Beihang University, and Key Laboratory of Big Data-Based Precision Medicine (Beihang University), Ministry of Industry and Information Technology, Beijing 100191, China
  • 3. College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China
  • 4. China Institute of Atomic Energy, Beijing 102413, China
  • 5. Instituto de Física, Universidade Federal Fluminense, 24210-340, Niterói, Rio de Janeiro, Brazil
  • 6. Instituto de Física, Universidade de São Paulo, Brazil

Abstract: To obtain the neutron spectroscopic amplitudes for 9096Zr overlaps, experimental data of elastic scattering with small experimental errors and precise optical potentials were analyzed. In this study, the elastic scattering angular distributions of 12,13C + AZr (A = 90, 91, 92, 94, 96) were measured using the high-precision Q3D magnetic spectrometer in the Tandem accelerator. The São Paulo potential was used for the optical potential. The optical model and coupled channel calculations were compared with the experimental data. The theoretical results were found to be very close to the experimental data. In addition, the possible effects of the couplings to the inelastic channels of the AZr targets and 12,13C projectiles on the elastic scattering were studied. It was observed that the couplings to the inelastic channels of the 12,13C projectiles could improve the agreement with the experimental data, while the inelastic couplings to the target states are of minor importance. The effect of the one-neutron stripping in the 13C+AZr elastic scattering was also studied. The one-neutron stripping channel in 13C + AZr was found to be not relevant and did not affect the elastic scattering angular distributions. Our results also show that in the reactions with the considered zirconium isotopes, the presence of the extra neutron in 13C does not influence the reaction mechanism, which is governed by the collective excitation of the 12C core.

    HTML

    1.   Introduction
    • The accurate description of the elastic scattering is sensitive to the structure of the nuclei involved as well as to the nuclear potential [1-4]. In this way, the elastic scattering process comprises an essential part of the overall understanding of heavy-ion reaction dynamics that depend on the structure of the colliding nuclei. Therefore, the elastic scattering of the heavy-ion system is of pronounced importance as it provides information about the interaction potential, and measures the reflection of the flux as it reaches the region of the interaction. It is often described by optical model calculations with an optical model potential (OMP) having some adjustable parameters. Most reaction theories require the knowledge of the OMP derived from the elastic scattering angular distribution of the colliding nuclei involved or from more fundamental criteria, like double folding potentials.

      The OMP can be used to estimate the cross sections of nuclear reactions; not only elastic scattering but also the other nuclear reactions such as breakup, transfer, and fusion, etc [5-7]. The interaction potential consists of nuclear and Coulomb potentials. The imaginary part of the nuclear potential represents the absorption from the elastic channel to other reaction channels, such as inelastic scattering, re-arrangement reactions, breakup channels (when weakly bound nuclei are involved), etc. The Coulomb potential between projectile and target nuclei is well known. The nuclear potential is often parameterized based on the optical potential. The potential parameters, which are related to the nuclear structure, can be extracted by fitting the elastic scattering angular distributions. Many kinds of studies have focused on the optical potential in heavy-ion physics for many years [8-17]. However, large ambiguities always existed for the optical potential parameters. Many studies devoted to deriving optical potentials have been performed in recent years, and several achievements were realized. Chamon et al. derived a global description of the nucleus-nucleus interactions within the double-folding model named the São Paulo potential (SPP), which can be successfully used for stable and unstable nuclei [18]. Xu et al. proposed a global single folding potential based on the elastic scattering angular distributions of 6,7Li on target nuclei with masses larger than 40 for energies ranging from 5 to 40 MeV/u [19]. Wang et al. obtained an energy independent Woods-Saxon potential at energies much higher than the Coulomb barrier [20]. Gan et al. selected several elastic scattering angular distributions of 12C from target nuclei of A39 to extract the Woods-Saxon potential parameters [21]. The derived potential was able to reproduce many elastic scattering angular distributions induced not only by 12C but also by other projectiles. In our work, we considered the double folding São Paulo potential [18], which has an energy dependence, as the nuclear potential.

      In collisions of typical strongly bound nuclei besides the elastic scattering, other relevant reaction channels, such as the inelastic scattering of projectile and target nuclei, are also accompanied. Therefore, the analysis of the elastic scattering, together with other relevant reaction channels, can be treated in the coupled channel (CC) method. The CC method is the most powerful tool to study multichannel scattering [22]. In this paper, the CC method is used to analyze the elastic scattering of 12,13C on zirconium isotopes.

      For stable Zr isotopes, the neutron numbers are at or close to the magic number of 50. Their neutron capture cross sections are relatively low [23, 24] and very scarce. In nuclear astrophysics, Zr isotopes occupy the intersection of the weak and main s-process, and the neutron capture reactions by Zr isotopes are particularly significant [25]. For this reason, researches focused on them have attracted great attention. The valuable information about the astrophysical medium, including the neutron flux density and temperature, can be extracted from the abundances of the Zr isotopes. Therefore, the neutron capture reaction rates of Zr isotopes should be determined with high accuracy. For example, in 89Zr(n,γ)90Zr, the direct neutron capture reaction rate contributes about 13% of the total reaction rate, which is much larger than the required accuracy of 5%. For the 93Zr(n,γ)94Zr reaction rates, an accuracy of 3%-5% is required [25]. From the evaluated data of the National Nuclear Data Center, the direct neutron capture reaction rate contributes about 10% to the total reaction rates. For this reason, it is meaningful to accurately study the direct components of neutron capture reaction with various experiments. The spectroscopic factor can be used to estimate the direct component of the (n,γ) cross section. The values of the spectroscopic factor can be obtained by comparing the experimental cross sections with the predicted ones from reaction models. Experimental measurements of the neutron spectroscopic factors about Zr isotopes have been done for several decades [26-28]. However, the published spectroscopic factors have large differences with one another, especially for 90Zr. The published neutron spectroscopic factors vary from 3.4-10.0 for 90Zr. Such a large difference may cause more than 20% uncertainty to the total reaction rate of 89Zr(n,γ)90Zr. The main shortcoming of the previous studies is the large experimental errors [29, 30] and the neglect of the influence of optical potentials, which gives large uncertainties to the spectroscopic factors. To improve these defects, a more accurate experiment should be carried out. In this study, 12,13C + AZr reaction systems were selected since 12C is a typical stable nucleus, and the 12C heavy-ion beam can be easily obtained and tuned with the accelerator. For the 13C beam, it is used to extract the optical potentials of exit channels for the one-neutron transfer reaction of 12C + Zr isotopes. For Zr isotopes, some unstable isotopes such as 89Zr (t1/2 = 3.3 d), 93Zr (t1/2 = 1.5 ×106 y), and 95Zr (t1/2 = 64.0 d) exist. These Zr isotopes are not available as reaction targets. Thus, in the experiment using 13C + 90,92,94,96Zr reactions can be complemented with 13C + 89,93,95Zr to extract the OMP of the exit channels.

      This paper is organized as follows. In Sec. 2, we describe the experimental apparatus. In Sec. 3, we compare the experimental angular distributions with OMP and CC calculations. In Sec. 4, we study the influence of one-neutron stripping on the 13C+90,91,92,94,96Zr systems on the elastic scattering. The conclusions are presented in Sec. 5.

    2.   The experimental method
    • The experiment was performed at the HI-13 Tandem accelerator at the China Institute of Atomic Energy (CIAE), Beijing, with the Q3D magnetic spectrometer. The Q3D magnetic spectrometer has a high-energy resolution of approximately 0.02%, and the angular distributions of elastic scattering can be measured with high precision. The beams of 12C at 66.0 MeV and 13C at 64.0 MeV from the accelerator impinged on the carbon-supported zirconium enriched isotope targets of 90,91,92,94,96ZrO2. The abundance of Zr isotopes in 90,91,92,94,96ZrO2 is shown in Table 1. The thicknesses of the 90,91,92,94,96ZrO2 targets were (32.9 ± 2.5), (27.3 ± 1.6), (30.0 ± 2.2), (41.0 ± 2.9), (34.4 ± 2.3) μg/cm2, respectively, which were calibrated by normalizing the elastic scattering cross sections at forward angles to Rutherford's scattering cross sections.

      target names90Zr91Zr92Zr94Zr96Zr
      90ZrO299.43.240.970.77.19
      91ZrO20.394.590.510.21.46
      92ZrO20.21.6398.060.42.31
      94ZrO20.10.460.4198.60.89
      96ZrO20.040.080.050.185.15

      Table 1.  Isotopic compositions of targets (%).

      The experimental setup is shown in Fig. 1. The diameter of the target chamber was 479 mm. To provide a better angular resolution, a collimator with a diameter of 5 mm was put at the entrance (located at the diaphragm position of Fig. 1) of the Q3D magnetic spectrometer to certify the angular resolution of differential cross section better than 0.4. Accordingly, the solid angle acceptance of the Q3D magnetic spectrometer was set to be 0.34 ± 0.01 msr for excellent angular resolution. The targets were placed at the the center of the target chamber upstream of the Q3D magnetic spectrometer. A movable Faraday cup was placed behind the target to monitor the beam intensity, which was used for the absolute normalization of the reaction cross sections. A ΔE-E detector telescopic system was set at approximately 23 downstream of the reaction targets for the cross-check of the beam intensity. The reaction products were separated by Q3D and then measured by a 50 mm × 50 mm two-dimensional position-sensitive silicon detector (PSSD) at the focal plane. The PSSD consists of 16 strips at the horizontal and perpendicular directions, respectively. The width of each strip is 3 mm. In this experiment, we only focus on the horizontal position and do not consider the vertical position. The high momentum resolution of Q3D (the momentum dispersion is 0.025 mm/% at the final focal plane) and the position-energy information from PSSD enables us to identify the specific ions from other reaction channels since the horizontal position of PSSD reflects the radius/momentum of deflected ions in the magnetic spectrometer. i.e. the magnetic rigidity.

      Figure 1.  Experimental setup.

      The typical two-dimensional spectrum of kinetic energy versus the horizontal position for the 96Zr(12C,12C)96Zr reaction at 26 is shown in Fig. 2(a). It can be seen that the object ions (the 12C in this case) from the reactions can be clearly identified via the energy and position information. Thus, the number of object ions can be counted accurately through the position spectrum of object ions, as shown in Fig. 2(b). In Fig. 2(b), the width of the spectrum is determined by the set momentum range of Q3D. The position width is mainly related to three parameters: the energy spread of scattered ions, the characteristics of the magnetic spectrometer, and the position resolution of the PSSD. In the whole experiment, these three parameters were almost kept unchanged. Therefore, the position width was practically not changed. Through rotating the Q3D magnetic spectrometer, the angular distributions of elastic scattering of 12,13C+Zr were measured in the range of 0 - 60. The ratios of the experimentally obtained differential elastic scattering cross sections to Rutherford's differential cross sections with the change of angles are shown in Figs. 3 and 4 by circles for the reactions induced by 12C and 13C, respectively. The experimental errors mainly stem from the statistical error (3%) and the uncertainty of the target thickness (5%).

      Figure 2.  (a) The two-dimensional spectrum of kinetic energy versus the horizontal position and (b) the horizontal position spectrum of object ions for 96Zr(12C,12C)96Zr at 26.

      Figure 3.  (color online) Elastic scattering cross sections for the 12C+AZr reactions at 66 MeV.

      Figure 4.  (color online) Elastic scattering cross sections for the 13C+AZr reactions at 64 MeV.

    3.   Optical model and coupled channel calculations
    • Different kinds of calculations can be performed to derive the elastic scattering angular distributions, depending on the goal of these calculations and the nuclei involved in the collision. Among these calculations, we employed the OMP calculation, usually devoted to determine the energy dependence of the optical potential and to derive the reaction cross sections, and the coupled channel calculations. The previous method is commonly applied to study the effect of channel couplings on elastic scattering, fusion, etc. Both methods solve a Shrödinger equation or the system of equations with specific boundary conditions that allow determining different observables. To describe the system of colliding nuclei, one must postulate a Hamiltonian that includes an optical potential (or potentials in the case of nuclei with cluster or halo structures that have low break up thresholds). The São Paulo potential (SPP) [18], which is a double folding potential with systematics for the matter density of the interacting nuclei, has been commonly used for the optical potential. As we are using the SPP in this work, some brief details about this potential will be given below.

      The nuclear interaction part of the double folding potential is given by the following expression:

      VF(R)=V0ρ1(r1)ρ2(r2)δ(Rr1+r2)dr1dr2,

      (1)

      where V0 = − 456 MeV fm3 and ρi(ri) (i=1− projectile and 2-target, respectively) is the matter densities. In the zero-range approach, where the range of the effective nucleon-nucleon interaction is negligible in comparison with the diffuseness of the nuclear densities, the usual M3Y [31] nucleon-nucleon interaction becomes V0δ(Rr1+r2), where ri is the coordinates of the nucleons inside the nuclei, and R is the vector joining the center of mass of the two interacting nuclei. The SPP accounts for the Pauli non-locality, which arises from quantum exchange effects; its local equivalent form is given by the following expression:

      VSPPLE(E,R)=VF(R)e(4v2/c2),

      (2)

      where VF(R) is the double-folding potential of Eq. (1), c stands for the speed of light, and v is the local projectile-target relative velocity obtained from

      v2(R,E)=2μ[EVC(R)VLE(R,E)],

      (3)

      where, VC is the Coulomb potential. Many studies using the SPP as the optical potential to derive the elastic scattering to study its energy dependence and to calculate direct reaction cross sections have been reported (see for examples Refs. [4, 22, 32-37]).

      To describe the scattering cross sections for the 12C+AZr at 66 MeV and 13C+AZr at 64 MeV with A=90,91,92,94 and 96, we first performed optical model calculations considering only the ground state of each nucleus. The double-folding SPP was used in the optical potential in both the real and imaginary parts [U=(NR+iNI)VSPP] with NR=1.0 and NI=0.78. These strength factors for the real and imaginary parts were adopted because we are not considering any coupling to the ground states. Many systems have had the elastic cross section well described by this value of the strength coefficients in a wide energy interval by means of optical model calculations [38]. The conclusion achieved in the mentioned paper was that these systematics are valid when there is no strong coupling of any relevant channel to the elastic scattering.

      Optical model calculations are important for understanding whenever there are strong couplings to the elastic scattering. Thus, to highlight dynamic effects such as strong couplings or static effects like cluster structures, we used the SPP at energies close to the barrier since this potential (in principle) does not take these effects into account in its systematics. Therefore, if the results do not describe the elastic experimental data, this means there are some relevant dynamic or static effects not included in the one-channel or optical model calculation. On the contrary, if the one-channel calculation describes the elastic scattering angular distributions, there are no important couplings to the elastic channel left out in the calculation, or there are polarizations of different signs that cancel each other. In Figs. 3 and 4, the comparison between the optical model calculations and the experimental data for 12,13C+90,91,92,94,96Zr elastic scattering is shown. One can see that the theoretical results have good agreement with the experimental data. However, when the projectile is 12C (see Fig. 3), the theoretical results are slightly above the experimental data, which means there are some couplings with the elastic channels not included in the calculations. For the projectile 13C (see Fig. 4), the results are slightly below the data when the target is 91Zr.

      To explain this small difference between the experimental data and the theoretical elastic scattering, we performed CC calculations, including the inelastic states of the projectile and target, using the FRESCO code [39]. In Table 2, the states included in the coupling scheme are shown. To describe the transitions between the target and projectile states, a model-independent procedure to account for the Coulomb and nuclear deformations was used, and the electromagnetic transition B(E2) for 91Zr and B(E1) for 13C were taken from Ref. [40]. The quadrupole (β2) and octupole (β3) deformation parameters to couple 13C states were assumed to be equal to the ones reported for 12C in Refs. [41] and [42], respectively. For the 90,92,94,96Zr isotopes, a vibrational model was used, and the deformation parameters β2 and β3 were also taken from Refs. [41] and [42]. The values of deformation parameters are shown in Table 3. The Coulomb and nuclear deformations were considered to have the same values, as long as no statistic effect in matter distributions was expected for these nuclei. The transition form-factors were taken as derivatives of the monopole term, following the usual convention.

      nucleusJπenergy/MeV
      12C0+0.0
      2+4.440
      13C1/20.0
      1/2+3.089
      3/23.684
      5/2+3.854
      90Zr0+0.0
      2+2.186
      91Zr5/2+0.0
      1/2+1.205
      5/2+1.466
      7/2+1.882
      3/2+2.042
      9/2+2.131
      92Zr0+0.0
      2+0.934
      0+1.383
      4+1.495
      2+1.847
      94Zr0+0.0
      2+0.919
      0+1.300
      4+1.450
      2+1.671
      96Zr0+0.0
      2+1.750

      Table 2.  Projectile and target states considered in the coupling scheme.

      nucleusβ2β3
      12C0.582
      13C0.5820.44
      90Zr0.0894
      92Zr0.10270.18
      94Zr0.090
      96Zr0.0800.27
      nucleusIIB(E1) /w.u
      13C1/2+1/20.039 (4)
      B(E2) /w.u.
      91Zr1/2+5/2+15 (4)
      5/2+5/2+10.7 (10)
      7/2+5/2+7.7 (13)
      3/2+5/2+59 (6)
      9/2+5/2+4.2(6)

      Table 3.  Deformation parameters and reduced electromagnetic transition probabilities considered in the CC calculations [40-42].

      The deformation parameters are related to the reduced electromagnetic transition probabilities by

      βλ=4π3ZRλB(Eλ,II)(1)(II+|II|)/2<IKλ0|IK>,

      (4)

      where Z is the nuclear charge, R=r0A1/3, A is the mass number, and r0= 1.06 fm is the reduced radius. λ stands for the multipolarity of the transition. I and I are the spins of initial and final states, respectively, and K is their projection in the quantization axis. For the real part of the optical potential, the SPP was used along with a short-range potential in the imaginary part, which had the Woods-Saxon [43] form with W=50 MeV, ri=1.06 fm, and ai=0.2 fm for the depth, reduced radius, and diffuseness, respectively. This short-range potential is important for accounting for the absorption of flux due to fusion because this process cannot be explicitly included in the calculations.

      The comparisons between the CC calculations and the experimental data for the 12C+AZr systems are shown in Fig. 5. In this figure, the solid blue line stands for no coupling calculation, in which only the ground state of each nucleus is considered. The dashed green line represents the results when only couplings to the inelastic states of the projectile were considered. In the calculations, represented by the dashed dot red curve, we included only the inelastic states of the target in the coupling scheme. The dotted line is the full CC calculation, in which all couplings with inelastic states of the projectile and target were considered.

      Figure 5.  (color online) Comparison between CC calculations and elastic scattering data for 12C+AZr reactions at 66 MeV.

      One can note that the couplings with the inelastic states of the target (AZr - dashed red dot curve) do not have a significant influence on the elastic cross section. The coupling with the first excited state of 12C or the full CC calculation has led to the results that slightly overestimate the angular distributions at backward angles.

      Similar CC calculations were performed for the reactions induced by 13C (see Fig. 6). Similar to the 12C case, a good agreement of angular distributions is observed when the couplings to inelastic states of the projectile were included in the calculations (dashed green line).

      Figure 6.  (color online) Comparison between CC calculations and elastic scattering data for 13C+AZr reactions at 64 MeV.

      Nevertheless, the full CC calculation underestimates the angular distribution from approximately 55 for most of the systems. This might be an indication that there could be missing couplings, like the one-neutron transfer.

    4.   Coupled reaction channel calculations
    • To determine whether there were some missing couplings left out in the CC calculations, we performed finite-range Coupled Reaction Channel (CRC) calculations for the one-neutron transfer for some reactions involving the 13C+AZr systems that have positive Qvalue (see Table 4). For the optical potential in the entrance partitions, the same potentials used in the CC calculations were used in the real and imaginary parts. In the final partitions, the SPP was used in both real and imaginary parts. Again, the imaginary part was multiplied for a coefficient NI=0.78 because no couplings were considered. The single-particle wave functions were obtained using Woods-Saxon potentials with diffuseness and a reduced radius equal to 0.65 and 1.25 fm, respectively, for the target and projectile. The depths of the Woods-Saxon potentials were varied to fit the experimental one-neutron binding energy.

      reactionQvalue/MeV
      90Zr(13C,12C)91Zr2.248
      91Zr(13C,12C)92Zr3.688
      92Zr(13C,12C)93Zr1.788
      94Zr(13C,12C)95Zr1.516
      96Zr(13C,12C)97Zr0.629

      Table 4.  Qvalue for the one-neutron stripping transfer reaction for the 13C+AZr systems.

      Shell-model calculations were performed to derive the spectroscopic amplitudes for the projectile and target overlaps, using the NuShellX code [44]. For the projectile overlaps, the psdpn model space and the psdmod effective phenomenological interaction [45] were used. This model space assumes the 4He as a closed core with 1p3/2, 1p1/2, 1d3/2, 1d5/2, and 2s1/2 orbitals as valence orbitals for protons and neutrons. For most of the target overlaps, the glekpn model space and effective phenomenological interaction of the same name [46] were used. This model space uses 1f7/2, 1f5/2, 2p3/2, 2p1/2, and 1g9/2 as valence orbitals for protons and 1g9/2, 1g7/2, 2d5/2, 2d3/2,and 3s1/2 for neutrons. Owing to our computational limitations to perform shell-model calculations using that large valence space, it was necessary to introduce some constraints to generate the spectroscopic amplitude. For this, the 86Sr nucleus was considered to be a closed core, and the number of protons in the higher 2p1/2 and 1g9/2 orbitals was reduced. For neutrons, the 2d3/2 and 3s1/2 orbitals were also restricted. For the 96,97Zr nuclei, this model space and interaction was not able to describe their structure characteristics (eigenvalues, spins, and parities). Thus, it was necessary to use the glepn [46] model space and interaction. This model space uses 2p3/2, 1f5/2, 2p1/2, 1g9/2, 3s1/2, 2d5/2 and 2d3/2 as valence orbitals for protons and 2p3/2, 1f5/2,2p1/2, 1g9/2, 3s1/2, 2d5/2, and 2d3/2 for neutrons. The same closed core (86Sr) considered in the previous calculations was used here, and the number of protons in the higher 1g9/2 orbitals was reduced and 2d5/2 and 2d3/2 were closed for protons. The overlap schemes for the projectile and target are shown in Figs. 7 and 8, respectively. The spectroscopic amplitudes for the projectile and target overlaps are given in Tables 5, 6, 7, 8, 9, and 10.

      initial statefinal statenljspect. ampl.
      13Cg.s(1/2)12Cg.s(0+)1p1/2−0.8009
      12C4.439(2+)1p3/20.9946
      13C3.098(1/2+)12Cg.s(0+)2s1/20.8983
      12C4.439(2+)1d3/2−0.0385
      1d5/20.3118
      13C3.684(3/2)12Cg.s(0+)1p3/2−0.3617
      12C4.439(2+)1p1/2−0.8194
      1d3/20.5415
      13C3.854(5/2+)12Cg.s(0+)1d5/20.9108
      12C4.439(2+)2s1/20.1130
      1d3/2−0.0586
      1d5/20.1965

      Table 5.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the psdpn model space with the psdmod effective phenomenological interaction for projectile overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      initial statefinal statenljspect. ampl.
      90Zrg.s(0+)91Zrg.s(5/2+)2d5/2−0.9872
      91Zr1.205(1/2+)3s1/2−0.9158
      91Zr1.466(5/2+)2d5/20.0407
      91Zr1.882(7/2+)1g7/20.1135
      91Zr2.042(3/2+)2d3/20.9650
      91Zr2.131(9/2+)1g9/20.0007
      90Zr2.186(2+)91Zrg.s.(5/2+)3s1/2−0.0095
      2d3/20.0141
      2d5/2−0.0761
      1g7/20.0107
      1g9/20.0101
      91Zr1.205(1/2+)2d3/20.0850
      2d5/20.7814
      91Zr1.466(5/2+)3s1/20.0021
      2d3/20.0545
      2d5/2−0.5699
      1g7/20.0564
      1g9/20.0017
      91Zr1.882(7/2+)2d3/2−0.0583
      2d5/20.9029
      1g7/20.0618
      1g9/20.0012
      91Zr2.042(3/2+)3s1/20.1176
      2d3/20.0549
      2d5/20.1420
      1g7/20.0521
      91Zr2.131(9/2+)2d5/2−0.7541
      1g7/2−0.0536
      1g9/2−0.0007

      Table 6.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glekpn model space and effective phenomenological interaction for 90Zr|91Zr ovelaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      Figure 7.  (color online) Coupling scheme for projectile overlaps used in the calculations. The double side arrows mean all the possible couplings between states.

      initial statefinal statenljspect. ampl.
      92Zr0.934(2+) 92Zrg.s(0+) 2d5/2 1.3174
      91Zrg.s(5/2+) 3s1/2 0.1597
      2d3/2 0.0580
      2d5/2 1.3631
      1g7/2 0.0227
      1g9/2 −0.0564
      92Zr1.383(0+) 2d5/2 0.1529
      92Zr1.495(4+) 2d3/2 −0.1639
      2d5/2 −0.1639
      1g7/2 −0.0210
      1g9/2 0.0171
      92Zr1.847(2+) 3s1/2 −0.0381
      2d3/2 −0.0030
      2d5/2 0.1627
      1g7/2 −0.0269
      1g9/2 −0.0016
      91Zr1.205(1/2+) 92Zrg.s.(0+) 3s1/2 −0.3706
      92Zr0.934(2+) 2d3/2 −0.0728
      2d5/2 −0.1586
      92Zr1.383(0+) 3s1/2 0.2814
      92Zr1.495(4+) 1g7/2 0.0248
      1g9/2 −0.0245
      92Zr1.847(2+) 2d3/2 0.0383
      2d5/2 −0.0083
      91Zrg.s(5/2+) 92Zrg.s.(0+) 2d5/2 0.0893
      92Zr0.934(2+) 3s1/2 0.022
      2d3/2 −0.0038
      2d5/2 0.0204
      1g7/2 0.0122
      1g9/2 0.00004
      92Zr1.383(0+) 2d5/2 −1.2431
      92Zr1.495(4+) 2d3/2 −0.0101
      2d5/2 0.0625
      1g7/2 −0.0050
      1g9/2 0.0009
      92Zr1.847(2+) 3s1/2 −0.0651
      2d3/2 −0.0236
      2d5/2 −0.8603
      1g7/2 −0.0649
      1g9/2 0.0290
      Continued on next page

      Table 7.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glekpn model space and effective phenomenological interaction for 91Zr|92Zr overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      Figure 8.  (color online) Coupling scheme for target overlaps used in the calculations. The double sided arrows indicate all the possible couplings between states.

      Table 7-continued from previous page
      initial statefinal statenljspect. ampl.
      91Zr1.992(7/2+) 92Zrg.s.(0+) 1g7/2 0.0354
      92Zr0.934(2+) 2d3/2 0.0185
      2d5/2 −0.1247
      1g7/2 0.0060
      1g9/2 −0.0029
      92Zr1.383(0+) 1g7/2 −0.0762
      92Zr1.495(4+) 3s1/2 0.0175
      2d3/2 −0.0165
      2d5/2 0.0170
      1g7/2 −0.0005
      1g9/2 0.0036
      92Zr1.847(2+) 2d3/2 0.0013
      2d5/2 0.7293
      1g7/2 −0.0082
      1g9/2 0.0171
      91Zr2.042(3/2+) 92Zrg.s.(0+) 2d3/2 −0.2628
      92Zr0.934(2+) 3s1/2 0.0755
      2d3/2 −0.0748
      2d5/2 0.0543
      1g7/2 −0.0327
      92Zr1.383(0+) 2d3/2 0.0025
      92Zr1.495(4+) 2d5/2 −0.1652
      1g7/2 0.0117
      1g9/2 0.0186
      92Zr1.847(2+) 3s1/2 −0.0625
      2d3/2 0.0106
      2d5/2 −0.0201
      1g7/2 0.0162
      91Zr2.131(9/2+) 92Zrg.s.(0+) 1g9/2 −0.0360
      92Zr0.934(2+) 2d5/2 0.0720
      1g7/2 −0.0046
      1g9/2 −0.0006
      92Zr1.383(0+) 1g9/2 0.0547
      92Zr1.495(4+) 3s1/2 −0.0210
      2d3/2 −0.0037
      2d5/2 −0.0681
      1g7/2 −0.0014
      1g9/2 0.0038
      92Zr1.847(2+) 2d5/2 −0.4477
      1g7/2 0.0251
      1g9/2 −0.0097
      initial statefinal statenljspect. ampl.
      92Zrg.s.(0+) 93Zrg.s.(5/2+) 2d5/2 0.8177
      93Zr0.266(3/2+) 2d3/2 0.0522
      93Zr0.947(1/2+) 3s1/2 0.7849
      93Zr0.949(9/2+) 1g9/2 −0.0014
      92Zr0.934(2+) 93Zrg.s.(5/2+) 3s1/2 0.0234
      2d3/2 −0.0806
      2d5/2 −0.8425
      1g7/2 −0.0138
      1g9/2 −0.0291
      93Zr0.266(3/2+) 3s1/2 −0.2468
      2d3/2 0.0387
      2d5/2 1.4099
      1g7/2 0.0025
      93Zr0.947(1/2+) 2d3/2 −0.0795
      2d5/2 −0.1150
      93Zr0.949(9/2+) 2d5/2 −0.7792
      1g7/2 0.0012
      1g9/2 −0.0297
      92Zr1.384(0+) 93Zrg.s.(5/2+) 2d5/2 0.0554
      93Zr0.266(3/2+) 2d3/2 −0.0080
      93Zr0.947(1/2+) 3s1/2 −0.4118
      93Zr0.949(9/2+) 1g9/2 −0.0013
      92Zr1.495(4+) 93Zrg.s.(5/2+) 2d3/2 −0.0127
      2d5/2 −1.1493
      1g7/2 −0.0118
      1g9/2 −0.0244
      93Zr0.266(3/2+) 2d5/2 −0.8747
      1g7/2 −0.0251
      1g9/2 −0.0730
      93Zr0.947(1/2+) 1g7/2 −0.0369
      1g9/2 −0.0150
      93Zr0.949(9/2+) 3s1/2 0.0866
      2d3/2 0.1659
      2d5/2 1.4825
      1g7/2 0.0301
      1g9/2 −0.0232
      92Zr1.847(2+) 93Zrg.s.(5/2+) 3s1/2 0.0039
      2d3/2 0.0135
      2d5/2 −0.1143
      1g7/2 0.0003
      1g9/2 −0.0031
      93Zr0.266(3/2+) 3s1/2 −0.0059
      2d3/2 −0.0140
      2d5/2 0.1175
      1g7/2 −0.0018
      93Zr0.947(1/2+) 2d3/2 0.0740
      2d5/2 −0.0573
      93Zr0.949(9/2+) 2d5/2 −0.0568
      1g7/2 −0.0053
      1g9/2 −0.0065

      Table 8.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glekpn model space and effective phenomenological interaction for 92Zr|93Zr overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      initial statefinal statenljspect. ampl.
      94Zrg.s.(0+)95Zrg.s.(5/2+)2d5/2−0.4773
      95Zr0.954(1/2+)3s1/20.8236
      94Zr0.919(2+)95Zrg.s.(5/2+)3s1/2−0.1345
      2d3/20.0302
      2d5/21.1630
      1g7/20.0213
      1g9/20.0322
      95Zr0.954(1/2+)2d3/2−0.0708
      2d5/2−0.1705
      94Zr1.300(0+)95Zrg.s.(5/2+)2d5/20.2090
      95Zr0.954(1/2+)3s1/2−0.3599
      94Zr1.469(4+)95Zrg.s.(5/2+)2d3/2−0.0806
      2d5/2−1.5742
      1g7/2−0.0095
      1g9/2−0.0181
      95Zr0.954(1/2+)1g7/20.0329
      1g9/20.0183
      94Zr1.671(2+)95Zrg.s.(5/2+)3s1/20.0242
      2d3/20.0050
      2d5/2−0.2516
      1g7/2−0.0034
      1g9/2−0.0041
      95Zr0.954(1/2+)2d3/2−0.0228
      2d5/2−0.2061

      Table 9.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glekpn model space and effective phenomenological interaction for 94Zr|95Zr overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      initial statefinal statenljspect. ampl.
      96Zrg.s.(0+)97Zrg.s.(1/2+)3s1/20.8918
      97Zr1.103(3/2+)2d3/20.9583
      96Zr1.750(2+)97Zrg.s.(1/2+)2d3/20.1571
      2d5/21.0769
      97Zr1.103(3/2+)3s1/2−0.0826
      2d3/2−0.0726
      2d5/20.0338
      97Zr1.264(7/2+)2d3/20.0.0155
      2d5/2−0.0356

      Table 10.  Spectroscopic amplitudes used in the CRC calculations for the one-neutron transfer using the glepn model space and effective phenomenological interaction for 96Zr|97Zr overlaps. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      From Fig. 9, one notes that the one-neutron transfer (solid purple curve) has a small influence when compared to CC results for the elastic scattering angular distributions for AZr(13C,12C)A+1Zr systems (with A = 90, 91, 92, 94, 96). The two curves, which correspond to CC and CRC calculations, cannot be clearly distinguished.

      Figure 9.  (color online) Comparison of CC and CRC calculations with the experimental elastic scattering angular distributions for the 13C+AZr system, for A = 90, 91, 92, 94, 96.

      From the present CC and CRC calculations analysis, one can conclude that, indeed, the couplings to direct reaction channels are weak. This is the reason why the [U=(1.0+0.78i)VSPP] works well in potential scattering. In addition, it is evident that the optical potential used in these calculations is appropriate for describing the elastic cross section.

      Spectroscopic factors for the 90Zr|91Zr, 91Zr|92Zr and 92Zr|93Zr overlaps were obtained in previous works [27, 47-50] by means of distorted wave Born approximation (DWBA) calculations. In Tables 11, 12, and 13 the SFs of these works and from shell-model calculations (present work) are compared. It is clear that the SFs for the ground to ground (elastic transfer) transitions agree quite well with each other. For the 92Zr |93Zr overlaps, the SFs are very close for all the overlaps considered in Ref. [50] (see Table 12). In general, the SFs obtained in this study are in agreement with those previously published.

      transitionsnljSF
      present workRef. [27]Ref. [47]Ref. [48]
      90Zrg.s.|91Zrg.s.2d5/20.970.751.040.95
      90Zrg.s.|91Zr1.2053s1/20.840.660.930.66
      90Zrg.s.|91Zr1.4662d5/20.0010.0280.030.024
      90Zrg.s.|91Zr1.8821g7/20.0130.0820.080.13
      90Zrg.s.|91Zr2.0422d3/20.930.560.630.55

      Table 11.  Comparison between one-neutron spectroscopic factors for 90Zr|91Zr overlaps obtained by shell-model calculations and from (d,p) transfer reaction [27, 47, 48]. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      transitionsnljSF
      present workRef. [27]Ref. [49]
      92Zrg.s.|91Zrg.s.2d5/21.731.181.86
      92Zrg.s.|91Zr1.2053s1/20.140.0450.06
      92Zrg.s.|91Zr1.4662d5/20.0080.011
      92Zrg.s.|91Zr1.8821g7/20.001
      92Zrg.s.|91Zr2.0422d3/20.0690.0470.07

      Table 12.  Comparison between one-neutron spectroscopic factors for 92Zr|91Zr overlaps obtained by shell-model calculations and from (p,d) transfer reaction [27, 49]. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      transitionsnljSF
      present workRef. [50]
      92Zrg.s.|93Zrg.s.2d5/20.670.54
      92Zrg.s.|93Zr0.2662d3/20.0030.007
      92Zrg.s.|93Zr0.9473s1/20.620.53
      *The authors consider the first excited state 93Zr0.266 with spin equal to 5/2+.

      Table 13.  Comparison between one-neutron spectroscopic factors for 92Zr|93Zr overlaps obtained by shell-model calculations and from (d,p) transfer reaction [50]. nlj are the principal quantum number, orbital, and total angular momentum of the neutron state, respectively.

      The main difference is that the spectroscopic amplitudes obtained in this study are derived from shell-model calculations and applied to heavy-ion interactions, where the high orders of the interactions and the transitions from excited states of the projectile or the target might be relevant [34, 51]. These kinds of effects can be masked by the overestimation of the spectroscopic amplitudes from the DWBA fits to the transfer cross sections (this seems not be the case for the present reactions. See discussion below).

      In Fig. 10, the results for 13C + 90,91,92Zr elastic scattering angular distributions from CRC results using different SFs obtained by shell-model calculations and derived from DWBA fits are compared. One can see that the theoretical results are very similar. The advantage of using SFs obtained from microscopic calculations is that these SFs are not fitted, which increases confidence in the results.

      Figure 10.  (color online) Effect of one-neutron transfer channel on the 13C + 90,91,92Zr elastic angular distributions using different SFs (see text for details).

      Finally, we compared the theoretical CRC and DWBA results for 90Zr(d, p)91Zr and 92Zr(p, d)91Zr one-neutron transfer reactions using the spectroscopic amplitudes derived in this study and those reported in Ref. [27] (see Tables 11 and 12). This comparison is shown in Figs. 11 and 12. It is evident that the transfer angular distributions are reasonably well described using the spectroscopic amplitudes derived microscopically and adjusted by the DWBA calculations. However, for the state at 1.466 MeV of 91Zr the theoretical result using shell-model spectroscopic amplitude underpredict the experimental data. This occurs because the SF for this channel is very small compared to the SF from Ref. [27]. In addition, we should emphasize that our description for the transfer angular distributions is not as good as that reported in Ref. [27]. This is caused by the optical potential used to describe this reaction. We are not using the original optical potential used in Ref. [27], but the São Paulo potential, which might not be suitable for this (p,d) reaction.

      Figure 11.  (color online) Transfer angular distribution for 90Zr(d, p)91Zr reaction at 15.89 MeV [27], using different reaction models and spectroscopic factors. For details, see the text.

      Figure 12.  (color online) Transfer angular distribution from 92Zr(p, d)91Zr reaction at 22.11 MeV [27], using different reaction models and spectroscopic factors. For details, see the text.

      From Figs. 11 and 12, one also realizes that the high-order coupling terms are irrelevant for the very forward angles (compare full blue lines with the dash-dot red lines), but they are relevant for the other angular ranges. It is worth mentioning that the calculations performed in the present work do not have any fitted parameter or scaling factor.

      From the present analysis we expected to find some effects of the one-neutron channel in systems involving 13C when compared to the reactions induced by 12C. The ground state Q-value is positive for the reactions with the four zirconium isotopes included in our study (ranging between 0.63 MeV (96Zr) and 3.67 MeV (91Zr)). So, one might think of some effects on the elastic scattering angular distribution that could help to improve the accuracy of the determination of the spectroscopic factors for target overlaps.

      In the case of the fusion cross section, the effect on the neutron transfer channels is clear [52-55]. After the neutrons are transferred, the deformation of the nuclei is the relevant information that decides the dynamic of the reactions. If the deformation increases, the effective Coulomb barrier decreases, and the fusion cross section is enhanced. Concerning the effect on the elastic angular distribution, it is well known that breakup of neutron halo nuclei is usually governed by the dipole polarizability of these nuclei that produces a hindrance of the Fresnel peak and decreases the elastic scattering angular distribution [1, 2]. How does the neutron transfer affect the elastic cross section, and is Q value one of the ingredients that affect it? This is one of the questions that remain open. In this study, we attempted to contribute to this issue and also to improve our knowledge on the spectroscopy of the zirconium isotopes, although we found that the elastic angular distributions were not sensible to the spectroscopic details.

    5.   Conclusions
    • In this study, elastic scattering angular distributions for the 12,13C+AZr (A = 90, 91, 92, 94, 96) reactions were measured at the HI-13 Tandem accelerator in China Institute of Atomic Energy (CIAE), Beijing. Optical model, coupled channel and coupled reaction channel calculations for 12,13C+AZr were performed and the results were compared with the experimental data.

      The theoretical calculations provided results very close to those obtained experimentally. The optical model calculations using the São Paulo potential described quite well the elastic angular distributions for 12,13C+AZr systems. It was also observed that the couplings to the inelastic states of the AZr did not affect the elastic cross section significantly. Still, the agreement with the experimental data was improved when the coupling to the lower excited states of 12,13C was considered. The one-neutron stripping for 13C+AZr reactions also appeared as a weak channel. The effects of the one-neutron stripping channel (13C,12C) on the elastic scattering angular distributions were found to be negligible in the case of all the zirconium isotopes used as targets in this study, showing that the optical potential used in the calculations is reliable for describing the elastic cross section.

      Spectroscopic information for various overlaps of zirconium isotopes was obtained from shell-model calculations. The spectroscopic amplitudes presented in this study are close to previous ones and described reasonably well the transfer angular distribution for (d,p) and (p,d) reactions without any scaling factors.

      Our results also show that, in the reactions with the considered zirconium isotopes, the presence of the extra neutron in 13C does not strongly affect the reaction mechanism that is governed by the collective excitation of the 12C core. Similar results have been reported recently for the two-neutron transfer reactions 18O + 12,13C at 84 MeV, where the effect of the pairing correlations between the two transferred neutrons in the two-neutron stripping reaction was not affected by the presence of an extra neutron in 13C [56-58].

      We are grateful to the staff of the China Institute of Atomic Energy for providing stable 12,13C beam throughout the experiment.

Reference (58)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return