Systematic study for relation between nuclear structure and reaction in ¹⁰Be nucleus

Here, we show the dependence of $V_{\rm LS}$ in the Hamiltonian of the nuclear structure calculation on the obtained nuclear structure and reaction. The results are essentially the same as in Ref. Furumoto et al. (2021). However, we show the calculation results as a table instead of a figure in this paper.

Table 1 shows the calculated binding energies (BE). The experimental data are -65.07, -61.70, and -59.11 MeV for 0${}^{+}_{1}$, 2${}^{+}_{1}$, and 2${}^{+}_{2}$, respectively. By changing the strength $V_{\rm LS}$ in the range 0–4000 MeV, the binding energies are drastically changed. A large $V_{\rm LS}$ value results in strong binding energy. On the other hand, a small $V_{\rm LS}$ value results in weak binding energy. The calculated values were almost identical to the experimental data around 2500 MeV. However, we have taken 2000 MeV as a criterion for $V_{\rm LS}$, because the effect of the $\alpha+t+t$ cluster structure is not included in this paper. The effect of the $\alpha$ cluster breaking and recombination gives a deeper binding energy Itagaki et al. (2008).

Table 1 also shows the calculated root mean squared radii for the point proton ($r_{p}$), point neutron ($r_{n}$), and point nucleon matter ($r_{m}$) of the 0${}^{+}_{1}$, 2${}^{+}_{1}$, and 2${}^{+}_{2}$ states. The experimental values of the proton and matter radii of the ¹⁰Be ground state are 2.357 $\pm$ 0.018 fm N ortersh auser et al. (2009) and 2.39 $\pm$ 0.02 fm Tanihata et al. (1985), respectively. The calculated radii are very close to the experimental data around $V_{\rm LS}$ = 2500 MeV. The increase in the strength of the spin-orbit interaction is responsible for the small size of the point nucleon-matter, neutron, and proton because of the strong binding. The size of the 0${}^{+}_{1}$ and 2${}^{+}_{1}$ states show consistent behaviors, especially with the strong binding. On the other hand, the 2${}^{+}_{2}$ state, which is larger than the 0${}^{+}_{1}$ and 2${}^{+}_{1}$ states except for very small $V_{\rm LS}$ values, exhibits much gentler shrinkage behavior with increasing binding energy. This result implies that the structure of the 2${}^{+}_{2}$ state is quite different from that of the ground and 2${}^{+}_{1}$ states when strong spin-orbit interaction is applied. Indeed, the 2${}^{+}_{2}$ state with $K=2$ corresponds to the excitation of one neutron from the spin-orbit favored orbit to the unfavored one according to the simple molecular-orbital picture.

In order to gain more precise information, we present the expectation values of $\langle\bm{L}\cdot\bm{S}\rangle$ and $\langle\bm{S}^{2}\rangle$ for the neutron in Table 1. The expectation values of the proton part are 0 because all protons are in the $\alpha$ clusters. The 0${}^{+}_{1}$ and 2${}^{+}_{1}$ states clearly exhibit similar growth of the expectation value with changing the strength of the spin-orbit interaction. Thus, the $0^{+}_{1}$ and $2^{+}_{1}$ states have similar inner structures and undergo similar changes with increasing strength of the spin-orbit interaction. The expectation value of the 2${}^{+}_{2}$ state shows a different trend, unlike the behavior of the $0^{+}_{1}$ and $2^{+}_{1}$ states. This finding implies that the inner structure of the 2${}^{+}_{2}$ state is different from the structures of the 0${}^{+}_{1}$ and 2${}^{+}_{1}$ states when the strong spin-orbit interaction is applied.

Next, we see the quadrupole transition strengths between each state. Table 1 also shows the quadrupole transition strengths ($B$(E2), the neutron part $B$((E2)_n), $B$(IS2), and $B$(IV2)) from the 2${}^{+}_{1}$ state to the 0${}^{+}_{1}$ state, from the 2${}^{+}_{2}$ state to the 0${}^{+}_{1}$ state, and from the 2${}^{+}_{2}$ state to the 2${}^{+}_{1}$ state. Although the radii and expectation values of the 0${}^{+}_{1}$ and 2${}^{+}_{1}$ states show similar tendencies, the change in the transition strengths of $B$(E2) and $B$((E2)_n) from the 2${}^{+}_{1}$ state to the 0${}^{+}_{1}$ state is not simple. Under weak spin-orbit interaction, the strengths of the proton and neutron parts are quite different. With increasing the $V_{\rm LS}$ value, the transition strength of the proton part becomes large. For $V_{\rm LS}$ = 2000–4000 MeV, both the proton and neutron parts decrease together. This behavior is caused by the sign reversal of the transition density for the neutron part as mentioned in Ref. Furumoto et al. (2021). From another perspective, we see IS and IV transition strengths. The obtained IS transition strengths from 2${}^{+}_{1}$ to 0${}^{+}_{1}$ stay in the range of 20–60 fm⁴. The IV transition strengths from 2${}^{+}_{1}$ to 0${}^{+}_{1}$ get gradually small as $V_{\rm LS}$ grows. On the other hand, the IS transition strengths from 2${}^{+}_{2}$ to 0${}^{+}_{1}$ are drastically changed by the $V_{\rm LS}$ value. This effect on the cross section will be discussed later. Here, we also discuss the transition strength from the 2${}^{+}_{2}$ state to the 2${}^{+}_{1}$ state. The $B$(E2: 2${}^{+}_{2}$ $\to$ 2${}^{+}_{1}$) value becomes smaller when the strength of the $V_{\rm LS}$ value becomes larger. In Ref. Itagaki et al. (2002), it is mentioned that the increase of the $B$(E2: 2${}^{+}_{2}$ $\to$ 2${}^{+}_{1}$) value with the decrease of the $V_{\rm LS}$ value supports a triaxial $\alpha$ + $\alpha$ + dineutron clustering configuration. However, the $B$(E2: 2${}^{+}_{2}$ $\to$ 2${}^{+}_{1}$) values with $V_{\rm LS}$ = 0 and 500 MeV are smaller than that with 1000 MeV. This is considered to be caused by the further development of the dineutron cluster. Namely, the coupling pattern of the angular momentum among the two $\alpha$ clusters and the dineutron cluster is different in the 2${}^{+}_{1}$ and 2${}^{+}_{2}$ states. Concretely, when the small $V_{\rm LS}$ value is adopted, it is considered that the two $\alpha$ clusters with the relative angular momentum $L=0$ are synthesized with the dineutron around them with $L=2$ in the 2${}^{+}_{1}$ state. On the other hand, the two $\alpha$ clusters with the relative angular momentum $L=2$ are synthesized with the dineutron moving around them with $L=0$ in the 2${}^{+}_{2}$ state. The decrease of the $B$(E2: 2${}^{+}_{2}$ $\to$ 2${}^{+}_{1}$) value and the increase of the $B$(IV2: 2${}^{+}_{2}$ $\to$ 2${}^{+}_{1}$) value corroborate the situation. We will discuss this situation again in Sec. III.3.

Figure 1 shows the (a) elastic cross section, (b) inelastic cross section for the 2${}^{+}_{1}$ state, and (c) inelastic cross section for the 2${}^{+}_{2}$ state of the ¹⁰Be nucleus by the proton target at $E/A=$ 59.4 MeV. The two-dot-dashed, dot-dashed, dashed, dotted, bold-solid, bold(red)-dotted, bold(red)-dashed, bold(red)-dot-dashed, and bold(red)-two-dot-dashed curves are obtained with $V_{LS}$ = 0, 500, 1000, 1500, 2000, 2500, 3000, 3500, and 4000 MeV, respectively. The experimental data are well reproduced in a wide range of the $V_{\rm LS}$ values. The change in the size of the ground state with varying $V_{\rm LS}$ slightly affects the elastic cross section. The relation among the binding energy, the nuclear size, and the elastic cross section will be discussed in Sec. III.2 in detail. The visible change in the transition strength from the 2${}^{+}_{1}$ state to the 0${}^{+}_{1}$ state for the proton and neutron parts results in a slight change in the inelastic cross section. As mentioned above, the $B$(IS2) values are obtained within 20–60 fm⁴. Therefore, the calculated inelastic cross sections are similar to each other. On the other hand, the calculated inelastic cross sections for the 2${}^{+}_{2}$ state are greatly dependent on the $V_{\rm LS}$ value adopted in the structure calculation. The drastic change in the inelastic cross section is caused by the difference in the transition strengths, as shown in Table 1. As mentioned in Ref. Furumoto et al. (2021), the behavior of the neutron part, in particular, is important and affects the IS transition strength from the 2${}^{+}_{2}$ state to the 0${}^{+}_{1}$ state. Here, we mention that the transition strength from the 2${}^{+}_{2}$ state to the 2${}^{+}_{1}$ state cannot be ignored. Therefore, the multistep reaction effect (0${}^{+}_{1}$ $\to$ 2${}^{+}_{1}$ $\to$ 2${}^{+}_{2}$) also contributes and makes the inelastic cross section for the 2${}^{+}_{2}$ state at $E/A=$ 59.4 MeV complicated. To avoid the multistep reaction effect, we also calculate the elastic and inelastic cross sections at a higher energy at which the multistep reaction effect is considered to be minor. In fact, we confirmed that the multistep reaction effect arising from the transition from the 2${}^{+}_{1}$ state to the 2${}^{+}_{2}$ state is minor at 200 MeV. Figure 2 shows the calculated (a) elastic cross section, (b) inelastic cross section for the 2${}^{+}_{1}$ state, and (c) inelastic cross section for the 2${}^{+}_{2}$ state at $E/A=$ 200 MeV. The inelastic cross section for the 2${}^{+}_{2}$ state is drastically influenced by the choice of the spin-orbit strength for the structure calculation, and this choice affects the inner structure of ¹⁰Be. On the other hand, it is difficult to distinguish the effect of the $V_{\rm LS}$ parameter on the elastic cross section and the inelastic cross section for the 2${}^{+}_{1}$ state.

We also calculate the elastic and inelastic cross sections for the ¹⁰Be + ¹²C system at $E/A=$ 59.4 and 200 MeV to investigate the effect of a different target. Figures 3 and 4 show the (a) elastic cross section, (b) inelastic cross section for the 2${}^{+}_{1}$ state, and (c) inelastic cross section for the 2${}^{+}_{2}$ state of the ¹⁰Be + ¹²C system at $E/A=$ 59.4 and 200 MeV. The difference in the nuclear size has a minor effect on the elastic cross sections for the ¹²C target. The inelastic cross sections for the 2${}^{+}_{1}$ state for the ¹²C target have similar behavior to each other. Also, we can see the drastic change of the inelastic cross sections for the 2${}^{+}_{2}$ state. Here, we note that the calculated inelastic cross sections for the 2${}^{+}_{2}$ state show different behavior in comparison with the proton target, especially for the dotted and bold solid curves. The results are caused by the weak IS transition strength shown in Table 1. In the present calculation for the ¹²C target, the IV component has no effect on the cross section. Comparing the proton target with ¹²C, the effect of IS and IV components can be seen.

Next, we investigate the dependence of the Majorana $(M)$ parameter. The $M$ parameter is responsible for describing the nucleon-nucleon exchange effect and is related to the Wigner ($W$) parameter as $W=1-M$. The values of 0.56–0.63 are widely used in the cluster model calculations. For the ¹⁰Be nucleus, 0.56 or 0.60 is usually adopted. Therefore, we investigate the range of $M$ = 0.52–0.64, which is wider than the usual cases, in this paper.

Table 2 shows calculated values of binding energies, radii, expectation values, and quadrupole transition strengths. Not only the $V_{\rm LS}$ parameter but also the $M$ parameter give drastic changes in the binding energies. With the increase of $M$, the repulsive effect becomes larger. Therefore, the large (small) $M$ value gives weak (strong) binding. However, it is found that $M$ has a minor effect on the excitation energy of the 2${}^{+}_{1}$ state. On the contrary, when the $M$ parameter is large, the difference of the energies between the 2${}^{+}_{1}$ and 2${}^{+}_{2}$ states becomes small. The size of the ground and excited states are also influenced by the change in the binding energy. The strong binding gives the small size for the ground and excited states. Therefore, the trend of the dependence of size on the $M$ parameter is simple. Also, the expectation values show a simple increase or decrease, except for the $\langle\bm{S}^{2}\rangle$ value of 2${}^{+}_{2}$ state, which shows minor dependence on the $M$ value.

Next, we discuss the transition strength. Both of the proton and neutron parts of the transition strengths from the 2${}^{+}_{1}$ state to the 0${}^{+}_{1}$ state simply increase with the $M$ parameter. The increase is concerned with the IS component. For the transition strength from the 2${}^{+}_{2}$ state to the 0${}^{+}_{1}$ state, the proton part increases, but the neutron part decreases with the $M$ parameter. The increase of the proton part and the decrease of the neutron part give complicated transition strength of the IS and IV components. The IV component of the transition strength from the 2${}^{+}_{2}$ state to the 0${}^{+}_{1}$ state is always larger than the IS component. This result is also different in comparison with the $V_{\rm LS}$ case. In addition, the transition strength from the 2${}^{+}_{2}$ state to the 2${}^{+}_{1}$ state gets larger with the $M$ parameter. This indicates that we cannot ignore the multistep reaction effect on the inelastic cross section caused by the strong transition from the 2${}^{+}_{1}$ state to the 2${}^{+}_{2}$ state.

Figures 5 and 6 show the calculated elastic and inelastic cross sections by the proton target with the experimental data. The two-dot-dashed, dot-dashed, dashed, dotted, bold-solid, bold(red)-dotted, and bold(red)-dashed curves are obtained with $M$ = 0.52, 0.54, 0.56, 0.58, 0.60, 0.62, and 0.64, respectively. The effect of changing the nuclear size is barely visible on the elastic cross section. We will discuss the effect of the binding energy, nuclear size, and elastic cross section again in Sec. III.2. The effect of the change of the transition strength from the 2${}^{+}_{1}$ state to the 0${}^{+}_{1}$ state is barely visible on the inelastic cross section for the 2${}^{+}_{1}$ state. On the contrary, we can see the drastic change of the inelastic cross sections for the 2${}^{+}_{2}$ state. In Ref. Furumoto et al. (2021), we expected that the change of the $M$ parameter would not have much impact on the inelastic cross section. However, it indeed does give the change. Here, we compare the inelastic cross sections for the 2${}^{+}_{2}$ state at $E/A=$ 59.4 with 200 MeV. At 59.4 MeV, the multistep reaction effect caused by the strong transition strength from the 2${}^{+}_{2}$ state to the 2${}^{+}_{1}$ state gives complicated angular distribution. On the other hand, we can discuss the effect of changing the $M$ parameter on the inelastic cross section for the 2${}^{+}_{2}$ state at 200 MeV because the multistep reaction effect can be ignored at higher incident energy. The effect of changing $M$ is indeed seen on the inelastic cross section for the 2${}^{+}_{2}$ state at 200 MeV as shown in Fig. 6. Although this is smaller compared with the change of the $V_{\rm LS}$ parameter shown in Fig. 2, we cannot ignore the effect. Therefore, we will discuss the contribution of the $V_{\rm LS}$ and $M$ parameters simultaneously in Sec. III.2.

Figures 7 and 8 show the calculated elastic and inelastic cross sections by the ¹²C target with the experimental data. Again, the change in the nuclear size has a minor effect on the elastic cross sections for the ¹²C target. The inelastic cross sections for the 2${}^{+}_{1}$ state for the ¹²C target show similar behavior to that for the proton target. Also, we can see the drastic change of the inelastic cross sections for the 2${}^{+}_{2}$ state. However, the behavior of the inelastic cross sections obtained with weak IS transition to the 2${}^{+}_{2}$ state is slightly different in comparison with the proton target. This is caused by the small transition strength of the IS component. Again, we note that the IV component has no effect on the present calculation for the ¹²C target.

Next, we investigate the dependence of the $B$ and $H$ parameters. The role of these parameters is to adjust the relative strength of the proton-proton (neutron-neutron) and proton-neutron interactions. The original Volkov interaction has no $B$ and $H$ terms. However, not only deuteron but also two neutrons are bound within $B=H=0$. Therefore, the $B$ and $H$ are often applied to give a realistic condition with the Volkov interaction.

Table 3 shows calculated values of binding energies, radii, expectation values, and quadrupole transition strengths. Almost all the calculated values show a minor dependence on the change of the $B$ and $H$ parameters. The expectation value of $\langle\bm{S}^{2}\rangle$ for the 0${}^{+}_{1}$, 2${}^{+}_{1}$ and 2${}^{+}_{1}$ states and the transition strength from the 2${}^{+}_{2}$ state to the 0${}^{+}_{1}$ state are slightly affected. However, their causal relationship is not clear.

Figures 9 and 10 show the calculated elastic and inelastic cross sections for the proton target with the experimental data. The dashed, dotted, bold-solid, bold(red)-dotted, bold(red)-dashed, and bold(red)-dot-dashed curves are obtained with $B=H$ = 0.00, 0.04, 0.08 0.12, 0.16, and 0.20, respectively. Again, the effect of changing the $B$ and $H$ values is minor in the elastic and inelastic cross sections. In the low-lying states of ¹⁰Be, the $B$ and $H$ parameters are found to play a minor role. Nevertheless, “the bound two neutrons” is unphysical, and these parameters allow the elimination of such conditions; the application of the $B$ and $H$ parameters is to reproduce a realistic physical situation even if the effect is small. Figures 11 and 12 show the calculated elastic and inelastic cross sections for the ¹²C target with the experimental data. Again, we cannot see the effect of changing the $B$ and $H$ parameters in the elastic and inelastic cross sections for the ¹²C target. These results imply that it is difficult to see the difference of the $B$ and $H$ parameters in the elastic and inelastic cross sections.