Persistence of cluster structure in the ground state of ${}^{11}\mathrm{B}$

In our framework, each single particle is described by a Gaussian form as in many other cluster models including the Brink model [32],

where the Gaussian center parameter $\bm{\zeta}$ is related to the expectation value of the position of the nucleon, and $\chi^{\tau,\sigma}$ is the spin-isospin part of the wave function. For the size parameter $\nu$, here we use $\nu=0.23\,\mathrm{fm}^{-2}$, which gives the optimal $0^{+}$ energy of ${}^{12}\mathrm{C}$ within a single AQCM basis state. The Slater determinant is constructed from these single-particle wave functions by antisymmetrizing them.

When four single-particle wave functions with different spin and isospin share a common $\bm{\zeta}$ value, an $\alpha$ cluster is formed. Similarly, when two neutrons with opposite spin orientations and a proton share the same $\bm{\zeta}$, a triton cluster is formed. We prepare 75 basis states with different $\alpha+\alpha+t$ configurations. The distance between two $\alpha$ clusters is changed from $1\,\mathrm{fm}$ to $5\,\mathrm{fm}$ in the step of $1\,\mathrm{fm}$. The triton is set at the places with the distance of $1$–$5\,\mathrm{fm}$ from the center of two $\alpha$ clusters, which is changed in the step of $1\,\mathrm{fm}$, and the axis measured from the $\alpha$–$\alpha$ axis is set to $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$. In all these states, the $\bm{\zeta}$ values in Eq. (1) are real numbers.

This cluster wave function is transformed into $jj$-coupling shell model based on the AQCM, by which the contribution of the spin-orbit interaction due to the breaking of $\alpha$ clusters is included. Here the $\bm{\zeta}$ values in Eq. (1) are changed to complex numbers. When the original value of the Gaussian center parameter $\bm{\zeta}$ is $\bm{R}$, which is real and related to the spatial position of this nucleon, it is transformed by adding the imaginary part as

where $\bm{e}^{\text{spin}}$ is a unit vector for the intrinsic-spin orientation of this nucleon. The control parameter $\Lambda$ is associated with the breaking of the cluster, and two nucleons with opposite spin orientation have $\bm{\zeta}$ values that are complex conjugate to each other. This situation corresponds to the time-reversal motion of two nucleons. After this transformation, the $\alpha$ clusters are called quasi clusters.

In our previous analysis on ${}^{12}\mathrm{C}$, we have prepared three quasi clusters with an equilateral triangular shape. We introduced two parameters, $R$ representing the $\alpha$–$\alpha$ distances and $\Lambda$ for the breaking of the $\alpha$ clusters. The subclosure configuration of $\left(s_{1/2}\right)^{2}\,\left(p_{3/2}\right)^{4}$ of the $jj$-coupling shell model can be obtained at the limit of $R\to 0$ and $\Lambda=1$.

For ${}^{11}\mathrm{B}$, we remove one spin-down proton from a quasi cluster. The $R$ and $\Lambda$ values are taken as $R=1$, $2$, $3$, $4\,\mathrm{fm}$ and $\Lambda=0.1$, $0.2$, $0.3$ ($\Lambda=0$ states are not necessary because they have large overlaps with the $\alpha+\alpha+t$ basis states).

It has been discussed that the cluster model space covers the model space of the shell model, if the wave function is antisymmetrized and we take the small-distance limit between clusters. This statement is correct when the term of the “shell model” is used in the sense of the three-dimensional harmonic oscillator. The cluster-model wave function becomes the three-dimensional harmonic oscillator one at the limit of the small relative distances. However, the three-dimensional harmonic oscillator wave function (corresponding to $\Lambda=0$ in AQCM) is quite different from the $jj$-coupling shell-model one (corresponding to $\Lambda=1$ in AQCM), as we can easily assess using our AQCM. Figure 2(a) shows the squared overlap between the three-dimensional harmonic oscillator (cluster model with small distances) and AQCM as functions of the $\Lambda$ value. The dotted line is for ${}^{12}\mathrm{C}$ with equilateral triangular configuration and very small distances between three quasi clusters. The vertical axis shows the squared overlap between $\Lambda=0$ and finite $\Lambda$ states. The angular momentum and parity are projected to $0^{+}$. The dotted line rapidly drops and the state with $\Lambda=0$ (identical to the three-dimensional harmonic oscillator one) has very small squared overlap with the subclosure configuration of the $jj$-coupling shell model ($\left(s_{1/2}\right)^{4}\,\left(p_{3/2}\right)^{8}$) of about $5\,\%$ at $\Lambda=1$. The solid line is for the $3/2^{-}$ of ${}^{11}\mathrm{B}$. Since one proton is missing compared with ${}^{12}\mathrm{C}$, the squared overlap between the three-dimensional harmonic oscillator and $jj$-coupling shell model increases, but the value is still quite small (slightly above $10\,\%$).

Next, the squared overlaps between a typical three-$\alpha$ cluster state and AQCM basis states are shown in Fig. 2(b). Here, the typical three-$\alpha$ cluster state means an equilateral triangular configuration with the $\alpha$–$\alpha$ distance ($R$) of $4\,\mathrm{fm}$. The solid and dotted lines are for the squared overlap with this typical cluster state and the AQCM basis states with $R=2\,\mathrm{fm}$ and $R=3\,\mathrm{fm}$ as functions of $\Lambda$, respectively. In our previous work, we have discussed that $R=2.0\,\mathrm{fm}$ and $\Lambda=0.2$ is the optimal AQCM basis state, which is found have the squared overlap of $10\,\%$ with the typical cluster state.

To describe the single-particle nature of three nucleons outside two $\alpha$ clusters, we prepare $\alpha+\alpha+3N$ basis states, where $\alpha$–$\alpha$ distance is determined using the random number (between $0$–$5\,\mathrm{fm}$ with equal probability) and positions of three nucleons (a spin-up proton and two neutrons with opposite spin directions) are also determined randomly. We generate 48 basis states and include them in the diagonalization process of the norm and Hamiltonian matrices.

As we have mentioned before, totally we generate 135 intrinsic states. The basis states from 1 to 75 are $\alpha+\alpha+t$ states with various configurations, those from 76 to 87 are AQCM basis states with $R=1$, $2$, $3$, $4\,\mathrm{fm}$ and $\Lambda=0.1,0.2,0.3$, and those from 88 to 135 are $\alpha+\alpha+3N$ configurations with the $\alpha$–$\alpha$ distance of $0$–$5\,\mathrm{fm}$. These 135 basis states are numerically projected to eigenstates of angular momentum and parity. After the angular momentum projection, different $K$ number states are generated from the same intrinsic basis state. Here, the $z$ direction of the intrinsic frame is parallel to the axis of two $\alpha$ (quasi) clusters. These different $K$ states are treated independently when we determine the coefficients for the linear combination of the basis states based on GCM. Therefore, after the angular momentum projection, the number of the basis states increases to 540, the basis states of 1–135, 136–270, 271–405, and 406–540 are $K=1/2$, $K=3/2$, $K=-1/2$, and $K=-3/2$, respectively.

The Hamiltonian consists of the kinetic energy and potential energy terms. For the potential part, the interaction consists of the central, spin-orbit, and Coulomb terms. The interactions are the same as those in our previous analysis on ${}^{12}\mathrm{C}$ [23]. For the central part, the Tohsaki interaction [30] is adopted. This interaction has finite-range three-body terms in addition to two-body terms, which is designed to reproduce both saturation properties and scattering phase shifts of two $\alpha$ clusters. For the spin-orbit part, There is no free parameter left for each nucleus for this interaction. For the spin-orbit part, we use the spin-orbit term of the G3RS interaction [31], which is a realistic interaction originally developed to reproduce the nucleon-nucleon scattering phase shifts. The strength of the spin-orbit interactions is set to $V_{ls}^{1}=V_{ls}^{2}=1800\,\mathrm{MeV}$, which allows consistent description of ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ [18].

We project 135 basis states to the eigenstates of angular momentum and parity, and basis states with $K=1/2$, $3/2$, $-1/2$, and $-3/2$ are independently treated. Thus, based on GCM, we diagonalize the norm and Hamiltonian matrices with the dimension of 540. The obtained energy levels of ${}^{11}\mathrm{B}$ are listed in Fig. 3, where dotted lines are threshold energies of $\alpha+\alpha+t$. The ground $3/2^{-}$ state is obtained at $-73.84\,\mathrm{MeV}$, compared with the experimental value of $-76.20\,\mathrm{MeV}$. Although we have no adjustable parameter for the central interaction and the spin-orbit strength is determined in our previous work for ${}^{12}\mathrm{C}$, the absolute value of the calculated ${}^{11}\mathrm{B}$ energy is quite reasonable.

The theoretical binding energy underestimates the experimental one by $2.4\,\mathrm{MeV}$, but the agreement between theoretical and experimental binding energies becomes even better if we measure them from the $\alpha+\alpha+t$ threshold. This is because the underestimate can be absorbed in the internal energies of the clusters; the theoretical value of $\alpha$ (triton) energy is $-27.31\,\mathrm{MeV}$ ($-7.82\,\mathrm{MeV}$), compared with the experimental value of $-28.29\,\mathrm{MeV}$ ($-8.48\,\mathrm{MeV}$). The ground-state energy of ${}^{11}\mathrm{B}$ is $-11.40\,\mathrm{MeV}$ from the $\alpha+\alpha+t$ threshold compared with the experimental value of $-11.20\,\mathrm{MeV}$.

The energy convergence of the $3/2^{-}$ states of ${}^{11}\mathrm{B}$ is shown in Fig. 4(a). The horizontal axis shows the number of the basis states superposed in the GCM calculation. As mentioned before, we generated 135 basis states, which are projected to the $J^{\pi}$ eigenstates with $K=1/2$, $3/2$, $-1/2$, and $-3/2$, and thus we diagonalize the norm and Hamiltonian with the dimension of $135\times 4=540$. From this figure, the first and third $3/2^{-}$ states are found to have predominantly the $K=3/2$ component, whereas the second $3/2^{-}$ state has mainly the $K=-1/2$ component. Figure 4(b) is the excerpt of the $K=3/2$ part. Here the first 75 basis states (136–210 on the horizontal axis) are basis states with the $\alpha+\alpha+t$ cluster configurations, the next 12 basis states (211–222) are AQCM, and the last 48 basis states (223–270) are $\alpha+\alpha+3N$ configurations.

From these figures, we can confirm that the inclusion of the breaking effect of $\alpha+\alpha+t$ cluster structure contributes to the lowering of the ground-state energy by about 2 MeV (for instance, we can compare the energies at 210 and 270 on the horizontal axis of Fig. 4(b)). The decrease of the energy when cluster breaking is incorporated is about half compared with ${}^{12}\mathrm{C}$, where the spin-orbit effect decreases the ground-state energy by several MeV (the same interaction is applied to ${}^{12}\mathrm{C}$ in Ref. [18]). This is because the spin-orbit acts for a proton in the triton cluster already within the $\alpha+\alpha+t$ model space; the antisymmetrization effect excites the triton cluster with the $\left(0s\right)^{3}$ configuration to the $p$-shell and the angular momentum projection allows the change of this proton wave function to $p_{3/2}$. Anyhow, two $\alpha$ clusters and di-neutron cluster in the triton cluster are free from the spin-orbit interaction, and breaking these clusters based on AQCM has a certain effect of decreasing the ground state of about $2\,\mathrm{MeV}$.

Figure 5(a) shows the squared overlap between the ground $3/2^{-}$ state and each basis state. Here, basis states with $K=1/2$ correspond to 1–135 on the horizontal axis, and $K=3/2$, $K=-1/2$, and $K=-3/2$ correspond to 136–270, 271–405, and 406–540 on the horizontal axis. As expected, the contribution of $K=3/2$ is dominant in the ground state.

Figure 5(b) is the excerpt of the $K=3/2$ part. As in the previous figure, the first 75 basis states (136–210 on the horizontal axis) are basis states with the $\alpha+\alpha+t$ cluster configurations, the next 12 basis states (211–222) are AQCM, and the last 48 basis states (223–270) are $\alpha+\alpha+3N$ configurations. For the $\alpha+\alpha+t$ cluster configurations basis states (136–210), we see five peak structures, and the basis states are classified into five groups with the interval of 15 basis states (136–150, 151–165, 166–180, 181–195, and 196–210). These five groups correspond to the basis states with the $\alpha$–$\alpha$ distances of $1$, $2$, $3$, $4$, and $5\,\mathrm{fm}$, and as we can see, $\alpha$–$\alpha$ distance of 3 fm contributes the most importantly (166–180). These 15 basis states are further classified into five groups with the interval of three (166–168, 169–171, 172–174, 175–177, and 178–180) corresponding to the position of the triton at $1$, $2$, $3$, $4$, and $5\,\mathrm{fm}$ form the center of $\alpha$–$\alpha$. The largest squared overlap of $0.68$ within the $\alpha+\alpha+t$ cluster configurations is given for the basis state 170 on the horizontal axis, which corresponds to the $\alpha$–$\alpha$ distance is $3\,\mathrm{fm}$, and the triton is located at $2\,\mathrm{fm}$ from the center of $\alpha$–$\alpha$ with the angle of $30^{\circ}$. Therefore, the ground state is found to have the $\alpha+\alpha+t$ component of about $70\,\%$.

Furthermore, we can get insight about the overlap with the higher excited cluster states of ${}^{12}\mathrm{C}$ when a proton is added, which is important medically, as explained in the introduction. Such cluster states can be characterized as a state with the three $\alpha$ clusters with the relative distances of $\approx 4\,\mathrm{fm}$, since the $\alpha$–$\alpha$ system has the minimum energy with such relative distance in the free space. In our calculation, the basis states 181–195 correspond to the $\alpha$–$\alpha$ distance of $4\,\mathrm{fm}$, and in the basis states 190–192, the last $\alpha$ cluster is placed at 4 fm from the center of two $\alpha$ clusters. We can confirm that the ground state has the squared overlap of about $50\%$ with those basis states proving it contains enough amount of seeds for the Hoyle state of ${}^{12}\mathrm{C}$.

However, the figures show that the AQCM basis states (211–222) describe the ground state better than the $\alpha+\alpha+t$ cluster configuration and they are more important. These AQCM basis states are classified into three groups with the interval of four, and basis states 211–214 are for $\Lambda=0.1$ (215–218 are for $\Lambda=0.2$ and 219–222 are for $\Lambda=0.3$). The largest squared overlap of 0.81 is given for the basis state 216 on the horizontal line corresponding to the basis states with $\Lambda=0.2$ and $R=2\,\mathrm{fm}$, where $R$ represents the distances between quasi clusters with equilateral triangular shape. The AQCM basis states with finite $\Lambda$ values have larger squared overlaps than the $\alpha+\alpha+t$ cluster configurations, and this means that the breaking of clusters due to the spin-orbit contribution is important. In our previous work on ${}^{12}\mathrm{C}$ with the same interaction, similarly, $\Lambda=0.2$ was found to give the optimal energy, where the $\alpha$–$\alpha$ distance was reduced to $2\,\mathrm{fm}$ owing to the strong attraction of three quasi clusters.

Figure 6(a) shows the squared overlap between the second $3/2^{-}$ state and each basis state. The orders of the basis states are the same as in the previous figures, and it is obvious that $K=-1/2$ is important for this state. In our basis states, the spin direction of a valence proton is set to spin-up, thus $K=-1/2$ means that the orbital angular momentum of the proton and its spin are antiparallel, for which the spin-orbit interaction acts repulsively. Figure 6(b) is the excerpt of the $K=-1/2$ part. The squared overlaps with the $\alpha+\alpha+t$ cluster states (271–345 on the horizontal axis) are increases compared with the previous cases of the ground state, and again, the $\alpha$–$\alpha$ distance of 3 fm is found to be most important among them. However, the squared overlaps with the AQCM basis states (346–357 on the horizontal axis) have even larger overlaps. Contrary to the ground-state case, the basis state with $\Lambda=0.1$ has the largest squared overlap.

Figure 7(a) shows the squared overlap between the third $3/2^{-}$ state, which has been suggested as a candidate for the cluster state both theoretically and experimentally, and each basis state. Here, we find that the contribution of $K=3/2$ is dominant, and Figure 7(b) is the excerpt of the $K=3/2$ part. The squared overlaps with the $\alpha+\alpha+t$ cluster states (136–210 on the horizontal axis) are much larger than those of the AQCM basis states (211–222). We can confirmed that the breaking effect of clusters is small and this state is really a cluster state as expected. The $\alpha$–$\alpha$ distance of $4\,\mathrm{fm}$ is found to have the largest overlap, which can be considered as well developed clustering.

Summarizing, one can find that $K=3/2$ contributes dominantly to the ground $3/2^{-}$, where AQCM states dominates the most, especially with $\Lambda=0.2$, while the $\alpha+\alpha+t$ cluster states, especially with $\alpha$–$\alpha$ distance $3\,\mathrm{fm}$, contributes largely as well; $K=-1/2$ contributes dominantly to the second $3/2^{-}$, where AQCM states dominates the most, especially with $\Lambda=0.1$, while the $\alpha+\alpha+t$ cluster states, especially with $\alpha$–$\alpha$ distance $3\,\mathrm{fm}$, contributes largely as well; $K=3/2$ contributes dominantly to the third $3/2^{-}$, where the $\alpha+\alpha+t$ cluster states, especially with $\alpha$–$\alpha$ distance $4\,\mathrm{fm}$, dominates.

We have shown how the components of the $\alpha+\alpha+t$ configurations are mixed in the ground state of ${}^{11}\mathrm{B}$. Here, finally, we discuss the squared overlap with the three-$\alpha$ cluster configuration when a proton approaches ${}^{11}\mathrm{B}$. We start with the case of single-$\alpha$ formation. The dotted line in Fig. 8 shows the squared overlap between an $\alpha$ cluster and $t+p$ as a function of the distance between the triton cluster and the proton. The total angular momentum is projected to $0^{+}$. Here the spin orientations of the proton and the one in the triton cluster are assumed to be anti-parallel. The squared overlap becomes zero when they are parallel. Therefore, we need an additional factor of $1/2$, which is not considered in Fig. 8, in the real situation of the proton scattering, where spin orientations are not fixed. The squared overlap between $\alpha$ and $t+p$ is unity at zero-distance and gradually drops as the distance increases.

Next, the solid line shows the squared overlap between $p+{}^{11}\mathrm{B}$ and three-$\alpha$-cluster state as a function of the distance between the proton and ${}^{11}\mathrm{B}$. In the real situation of the proton scattering, various three-$\alpha$ cluster states of ${}^{12}\mathrm{C}$ are created when the proton meets ${}^{11}\mathrm{B}$, and in this calculation, such three-$\alpha$-cluster states are represented by a single configuration of an equilateral triangular shape with the relative $\alpha$–$\alpha$ distance of $4\,\mathrm{fm}$ (optimal $\alpha$–$\alpha$ distance of ${}^{8}\mathrm{Be}$ in the free space). The target nucleus, ${}^{11}\mathrm{B}$, is represented by the most important basis state in the previous subsection; AQCM basis state with $R=3\,\mathrm{fm}$ and $\Lambda=0.2$. To compare with the real situation of the proton scattering, the target nucleus of ${}^{11}\mathrm{B}$ should be angular momentum projected to $3/2^{-}$ and the relative angular momentum between $p$ and ${}^{11}\mathrm{B}$ also must be treated as a good quantum number. However, here these are approximated as the total angular momentum projection, and the proton is assumed to approach from the perpendicular direction to the three quasi clusters. The threshold energy of ${}^{11}\mathrm{B}+p$ corresponds to $E_{x}=15.95668\,\mathrm{MeV}$ in ${}^{12}\mathrm{C}$, and the resonances slightly above this energy are clinically important sources of the $\alpha$-cluster emission. According to the database at National Nuclear Data Center, the $2^{+}$ resonance state of ${}^{12}\mathrm{C}$ at $E_{x}=16.10608\,\mathrm{MeV}$ $\alpha$ decays with the branching ratio of $100\,\%$. Therefore, here we project the angular momentum to $2^{+}$, and in this case, the relative wave function between the proton and ${}^{11}\mathrm{B}$ dominantly has the $p$-wave component. Again the spin orientations of the incident proton and that in the target nucleus are assumed to be antiparallel, and thus we need an additional factor of $1/2$, which is not considered in Fig. 8, to compare with the real situation. We can see the squared overlap of $10$–$20\,\%$ at small distances. The value is not very large, but the proton scattering on ${}^{11}\mathrm{B}$ is considered tot have a sizeable branching ratio for creating the three-$\alpha$ cluster states.