³He-γ coincidence

${}^{\text{6}}\text{Li}{+}^{89}\text{Y}{\to }^{92}\text{Zr}{+}^3\text{He}+(-90\text{ keV})$ (1) In Fig. 1(b), ³He particles can be clearly distinguished. According to the kinematics calculation for ³H transfer, the energies of the outgoing ³He were approximately 27 MeV, according to Eq. (1). Thus, some particles could pass through the Al absorber. The product of the reaction was ⁹²Zr, and ^91,90Zr could be produced by the evaporation of neutrons. ³He-gated γrays are shown in Fig. 2, where the γ-rays of ^92,91,90Zr are evident, and the counts of each γray are obtained. The normalized intensities of several significant γrays are presented in the third and fourth columns of Table 2, with the γ-ray at 934.5 keV set to a standard reference value of 100. The direct population strengths of the excited states of ⁹²Zr are presented in Fig. 3(a). Further evidence can be obtained from ³H-γ coincidence.

Fig. 2Full-screen View

(Color online) Total γ spectrum of ³He-γ coincidence over all the angles

Fig. 3Full-screen View

(Color online) (a) Residuals at all angles on ³H gating. (b) Typical γrays of ³H-γ coincidence at uncovered angles

³H-γ coincidence

In the ³H coincident γ spectrum in Fig. 3a and Fig. 4, ⁹¹Nb and ⁹²Nb are the primary residuals. In the inset window, the strong γ-rays 237 keV and 274 keV were the characteristic γ-rays of ¹⁹Ne. The counts are significant but disappear at the uncovered angles in Fig. 3(b). ¹⁹Ne from the ³He transfer, as shown in Eq. (2) has already been studied [27]. ³H, which coincides with these γ-rays, indicates a stronger association with the transfer reaction owing to its forward trend. However, ¹⁶O may originate from the support frame of the target or the backing foil. Therefore, the yield of the residues ¹⁹Ne could not be obtained in this experiment.

Fig. 4Full-screen View

(Color online) γ spectrum of ³H-γ coincidence at all angles

To further analyze the main products, ⁹²Nb and ⁹¹Nb, Fig. 3(a) shows the correlations between ³H energies and different γ-rays. The projections of the energy of the γ-rays are shown on the left side, and the counts of the γ-rays of ⁹¹Nb are larger when ³H particles are gated. In the bottom window, the 501-keV (⁹²Nb) γ-ray-gated ³H spectrum (red dots) also shows a higher energy distribution than that from 1790-keV γrays in ⁹¹Nb (blue dots). In terms of energy conservation, when the energy of the gated ³H increased, the excitation energy of ⁹²Nb from ³He transfer reaction decreased. Thus, fewer neutrons evaporated. The normalized intensities of certain significant γ-rays are presented in the second column of Table 2, assuming a standard intensity of 100 for the γ-rays with energy of 150 keV. According to Eq. (3), referring to the ³He-γ coincidence analysis results, ⁹¹Nb and ⁹²Nb originated from ³He transfer process, similar to the reaction in Ref. [27]. In the kinematics calculations, the total energy of ³H was approximately 25 MeV, as shown in Eq. (3) showing that ³H particles can pass through the Al absorber and be detected. ⁹¹Nb, which has a neutron magic number of 50, is the one-neutron evaporation product of ⁹²Nb. ${}^{\text{6}}\text{Li}{+}^{16}\text{O}{\to }^{19}\text{Ne}{+}^3\text{H}+(-7350\text{ keV})$ (2) ${}^{\text{6}}\text{Li}{+}^{89}\text{Y}{\to }^{92}\text{Nb}{+}^3\text{H}+(-2120\text{ keV})$ (3)

Calculation results of direct transfer reactions

One of the most common methods for two-particle transfer calculations is to set the spectroscopic amplitude equal to 1.0 (SA = 1.0) [51]. In this case, the nucleon spins are considered antiparallel, and n=1 and l=0 are assumed as the internal state of the cluster quantum number. In our case, where the cluster is composed of three nucleons, the spin of the transferred cluster is assumed to be equal to the spin of the free nucleus in its ground state. The relevant parameters for defining the cluster wave function are the principal quantum number N and the orbital angular momentum L relative to the core. N and L can be determined from the conservation of the total number of quanta in the transformation of the wavefunction of three independent nucleons into a cluster [52]. ${\displaystyle \sum _{i=1}^3}2\left(n_i-1\right)+l_i=2(N-1)+L+2(n-1)+l\text{,}$ (4) where ni and li (i=1, …, 3) are the quantum numbers of each nucleon. For ³He and ³H clusters, l=1 because the unpaired particle is in the 1p_3/2 orbital. In addition, the spectroscopic amplitudes for the overlaps of the wave functions for both the projectile and target are set to 1.0. However, these spectroscopic amplitudes might be unrealistic. The nuclear structures of the projectile, target, and residuals are ignored.

The level scheme of the nuclei and couplings adopted in the direct transfer calculations is shown in Fig. 5 for both transfer reactions. Here, we are interested in the order of magnitude rather than in a quantitative description of the reaction process. Because the selection rules for the transfer of particles are relevant in these processes, for each case, the states are characterized by the spin, parity, and energy values considered in the calculations according to the total angular momentum J, orbital momentum L, and spin of the transferred cluster. The theoretical results of the direct ³H and ³He transfers for SA = 1.0, are shown in Tabs. 3 and 4, respectively. The transfer cross sections are of the order of mb. However, these results are overestimated because of unrealistic spectroscopic amplitudes.

Fig. 5Full-screen View

(Color online) Coupling scheme considered for the projectile and target overlaps in both direct reactions. The energies are given in MeV. (a) Coupling scheme considered in the ⁸⁹Y(⁶Li, ³He)⁹²Zr reaction. (b) Coupling scheme considered in the ⁸⁹Y(⁶Li, ³H)⁹²Nb reaction

The next step was to calculate realistic spectroscopic amplitudes for the projectile and target overlap. Microscopic spectroscopic amplitudes calculated from the shell model were necessary to obtain these spectroscopic amplitudes. These were obtained by performing shell model calculations using the NuShellX code [53]. For the target overlaps, a closed ⁷⁸Ni core and valence protons in the 1f_5/2, 2p_3/2, 2p_1/2, and 1g_9/2 orbitals and valence neutrons in the 1g_7/2, 2d_5/2, 2d_3/2, 3s_1/2, and 1h_11/2 orbitals were considered. Because of computational limitations, certain restrictions were imposed on the valence proton orbitals. Specifically, within the 1g_9/2 orbit, we determined that a maximum of six protons could occupy this orbital. The n-n, p-p, and n-p effective phenomenological interactions were based on jj45apn interaction [54], in which the two-body matrix elements were determined considering the charge-dependent Bonn potential (CD-Bonn) [55, 56]. In this interaction, the single particle energies for proton model space were set to ${ϵ}_{1f_{5/2}}=-0.7166\text{ MeV}$, ${ϵ}_{2p_{3/2}}=1.1184\text{ MeV}$, ${ϵ}_{2p_{1/2}}=1.1262\text{ MeV}$, and ${ϵ}_{1g_{9/2}}=0.1785\text{ MeV}$. In addition, the single particle energies for the neutron space were set to ${ϵ}_{1g_{7/2}}=5.7402\text{ MeV}$, ${ϵ}_{2d_{5/2}}=2.4422\text{ MeV}$, ${ϵ}_{2d_{3/2}}=2.5148\text{ MeV}$, ${ϵ}_{3s_{1/2}}=2.1738\text{ MeV}$, and ${ϵ}_{1h_{11/2}}=2.6795\text{ MeV}$. In our approach, we modified the single-particle energies for the proton model space, for which the values were obtained from the glbepn interaction [57]. In this context, ${ϵ}_{1f_{5/2}}=-3.706\text{ MeV}$, ${ϵ}_{2p_{3/2}}=-2.133\text{ MeV}$, ${ϵ}_{2p_{1/2}}=-1.101\text{ MeV}$, and ${ϵ}_{1g_{9/2}}=-0.638\text{ MeV}$ were considered. With this modification, the spectra of the ⁸⁹Y, ⁹²Zr, and ⁹²Nb nuclei could be described quite well. In addition, we built an effective phenomenological interaction that could describe the single-particle energies of ⁸⁹Y, ⁹²Zr, and ⁹²Nb nuclei. This new interaction was based on jj45pna and glbepn interactions [54]. For the projectile overlap, 1s_1/2, 1p_3/2, and 1p_1/2 were considered for both the neutron and proton model spaces for which a no-core interaction was used [58]. Single-particle energies and two-body matrix elements were inspired by the Warburton and Brown interactions [59]. The theoretical results for the direct ³H and ³He transfers are shown in Tabs. 3 and 4, respectively.

We observed that the integrated transfer cross sections of both reactions decreased when microscopic spectroscopic amplitudes were used. These results were expected because this cluster configuration should have a low probability in these nuclei. The calculated spectroscopic amplitudes are presented in the Appendixes.

Calculation of sequential transfer reactions

The calculations also considered sequential processes in which the three nucleons are transferred. We performed calculations for multi-step transfer reactions passing through the intermediate partitions. We focused on two two-step sequential transfers, ²H + n and ²H + p because these are the only reactions that pass through partitions with stable projectile-like nuclei. The others occur through unstable nuclei (⁵Li or ⁵He), which is unlikely because they are two-step processes and the unbound particles decay easily. Nevertheless, some tests that included ground states as bounds were also performed. The cross sections were observed to be very small. The cross sections for these transfer reactions were three orders of magnitude lower than those for direct cluster transfer. Therefore, we do not provide details here and consider only two two-step sequential transfer reactions passing through the same intermediate partition (⁴He + ⁹¹Zr). Three excited and ground states were considered for ⁹¹Zr, as shown in Appendix C. These reactions involve only stable nuclei or nuclei with long half-lives. In these two transfer reactions, either a proton is transferred after a ²H-like (a correlated n-p) for the ³He sequential transfer reaction, or a neutron is transferred after a ²H-like (a correlated n-p) for the ³H sequential transfer reaction. Similarly, the São Paulo potential (SPP) was used for the real and imaginary parts of the optical potential. As no couplings were considered in the intermediate partitions, the strength factor of the imaginary part was set to 0.78, as in the outgoing partitions. The Woods–Saxon potential was used to build single-particle wave functions, in which the parameters were varied to reproduce the corresponding experimental binding energies. The ⁹¹Zr states used in the theoretical calculations were obtained according to Brink’s criteria for optimal excitation energies [60]. The spectroscopic amplitude results are presented in the Appendix C.

The level scheme and the couplings adopted in the sequential transfer calculations are shown in Fig. 6 for both the reactions. Theoretical calculations were performed considering SA = 1.0 for the ²H transfer (in the first step), and spectroscopic amplitudes were set equal to 1.0 for proton and neutron transfer (in the second step). The results are summarized in Tables 3 and 4. Comparing the results shown in Tables 3 and 4, we observed that the cross sections of the direct process were three orders of magnitude larger than those of the sequential process; therefore, the sequential process is negligible compared to the direct process.

Fig. 6Full-screen View

(Color online) Coupling scheme considered for the projectile and target overlaps for sequential process. The energies are given in MeV. (a) Coupling scheme considered in the ⁸⁹Y(⁶Li, ³He)⁹²Zr reaction. (b) Coupling scheme considered in the ⁸⁹Y(⁶Li, ³H)⁹²Nb reaction

The microscopic spectroscopic amplitudes calculated from the shell model are shown in a sequential process. The same interaction and model space were used to evaluate the microscopic spectroscopic amplitudes that were used in the direct reaction calculations above. In the first step, a deuterium-like particle was transferred, followed by a proton or neutron when microscopic spectroscopic amplitudes were used. In this case, the independent coordinate model was used [51] in which the coordinates of the nucleons are transformed to the relative coordinates between the neutron and proton and that of its center of mass relative to the α core. The spectroscopic amplitudes were calculated for the correlated n-p, including the projectile-like and target-like overlaps. In the second step, a neutron was transferred to the ³ H-stripping reaction. Similarly, a proton was transferred in the second step of ³He transfer.

The theoretical results for the sequential transfer reactions with microscopic spectroscopic amplitudes are presented in Tables 3 and 4. The integrated transfer cross sections of the 9^- and 9⁺ states for ⁹²Nb are missing. This is because the selection rules prohibit these transitions during the sequential transfer process in the used model space. Comparing the results in Table 3, we can conclude that the sequential process is negligible for most of the studied states when microscopic spectroscopic amplitudes are applied to the ³H transfer reaction. Conversely, sequential transfer is relevant to the ³He transfer reaction when compared with the direct transfer process, as they have cross sections of the same order of magnitude.

These results can be explained by examining the structures of the residual nuclei. The transfer cross section is proportional to the spectroscopic amplitudes of two overlaps: one for the projectile and the second for the target. The projectile overlaps are similar because the transferred neutrons or protons in the second step lie on the same shell. The protons and neutrons transferred to the target-like nucleus (⁹¹Zr) will occupy different orbitals in the residual nuclei. Therefore, the spectroscopic amplitudes of the second step are very different, and consequently, the two-step transfer is very different in the ²H + n and ²H + p transfers.

Comparison with experimental data

The interplay between direct and sequential transfer reactions is discussed in Ref. [61] for deuteron transfer. The relevance of both processes in our experimental data is also discussed. The intensities of some γ-rays are listed in Table 2. Therefore, the relative ratios of some states that are directly populated can be defined to compare the theoretical results in the framework of microscopic spectroscopic amplitude calculations with the experimental data. The 6⁺ state of ⁹²Nb was used as a benchmark, whereas the 2⁺ state was used for ⁹²Zr. The selected states were guided by the fact that they decayed only to the ground state. The results are presented in Fig. 7. Fig. 7(a) shows the ratio of the cross sections of the different excited states of ⁹²Zr to that of the first excited state (2⁺) observed experimentally. The experimental and theoretical ratios for 6⁺ and 7^- states were in good agreement for direct transfer. The theoretical and experimental ratios for the other two states were also of the same order of magnitude for the direct ³H transfer reaction. The sequential transfer cross sections were in good agreement with the experimental ratios for the three lower energy states. As mentioned previously, by comparing the results presented in Table 4, the sequential and direct processes were of the same order of magnitude in the ³He transfer reactions. Therefore, both processes are relevant when microscopic spectroscopic amplitudes are used. The ratios of most states are consistent with the theoretical calculations, as shown in Fig. 7(b). For ⁹²Nb, the errors of states 285.7 keV(3⁺) and 357.4 keV(5⁺) were larger, because the γ-rays 357.4 keV and 194.8 keV also emanate from ⁹¹Nb.

Fig. 7Full-screen View

(Color online) (a) Ratio of cross sections of different excited states of ⁹²Zr to that of the first excited state (2⁺) observed in experiment. (b) Ratio of cross sections of different excited states of ⁹²Nb to that of the state (501 keV 2⁺) observed in the experiment. The MS_Direct means the theoretical results of direct transfer reaction using microscopic spectroscopic amplitudes calculated from shell model, and MS_Sequential represents sequential transfer reaction; Data indicates the experimental data