Generalized coherent states satisfying the Pauli principle in a nuclear cluster model

We begin with the harmonic oscillator (HO) basis state $\phi_{n\ell m}(\mbox{\boldmath$r$},\nu)$ with a range $\nu=1/b^{2}$ and a principal quantum number $N=2n+\ell$, where $n$ represents the number of nodes in the radial wave function and $\ell$ is an orbital angular momentum. Using the operators of the creation and annihilation of a quanta $N$, the HO basis state can be written as [13, 14]

where ${\cal Y}_{\ell m}(\mbox{\boldmath$r$})=r^{\ell}Y_{\ell m}(\hat{\mbox{\boldmath$r$}})$ is a solid spherical harmonics and $\phi_{0}$ is a vacuum with $N=0$. This HO basis state can be used to represent the single-nucleon wave function in nuclei and also the relative wave function between nuclear clusters.

In this study, we introduce the following scalar operators $\hat{D}^{\dagger}$ (raising) and $\hat{D}$ (lowering) following Ref. [10] as

These operators belong to the symplectic Sp(3,$R$) Lie algebra of the coherent state of the collective motion and change the quanta of the wave function by two for the radial part without changing the angular momentum. Using the raising operator $\hat{D}^{\dagger}$, we introduce the following new basis state $\phi_{n\ell m}^{\beta}(\mbox{\boldmath$r$},\nu)$, in which $\hat{D}^{\dagger}$ is coherently multiplied by the HO basis state with the weight of the real parameter $\beta$ in the exponential form as

The derivation of this equation is given in Appendix A. This new basis state is a kind of generalized coherent state [9] in terms of $\hat{D}^{\dagger}$ and can be represented by the HO basis state with the same quanta $N$ and the different range parameter $\nu/(1+\beta)$ and multiplying the Gaussian function with the coordinate $r$. This equation plays an essential role in this study. It is noted that the basis state has the following exponential dependence;

This form gives a condition of $|\beta|<1$ to satisfy the asymptotically damping behavior, which is imposed throughout this study. When $n=0$, the basis state $\phi_{0\ell m}^{\beta}(\mbox{\boldmath$r$},\nu)$ becomes the nodeless Gaussian function multiplying $r^{\ell}$, which is often used in the Gaussian expansion technique [15, 16, 17].

From the property of the raising operator $\hat{D}^{\dagger}$, the function $\phi_{n\ell m}^{\beta}(\mbox{\boldmath$r$},\nu)$ includes only the quanta larger than or equal to $N$ of the HO basis states with the range $\nu$. Hence the following orthogonal condition is satisfied;

This property is useful to construct the HO basis states with a quanta $N$ and any values of $\beta$, which are orthogonal to the HO states with a lower quanta $N^{\prime}$. If one regards the HO basis states with the lower quanta $N^{\prime}$ as the occupied states in the nucleus, namely the Pauli-forbidden states, the generalized coherent basis states $\phi_{n\ell m}^{\beta}(\mbox{\boldmath$r$},\nu)$ can be the unoccupied states in the nucleus automatically, and represents the Pauli-allowed states. A specific case of this formulation is introduced in the shell model, in which the HO particle state with a free range parameter is taken to be orthogonal to the HO hole states by adjusting the polynomial in the HO particle state [18, 19].

The parameter $\beta$ controls the spatial range of the generalized coherent basis states. When $\beta$ is close to unity, the basis state has a long tail and is suitable to describe a weak-binding state of nuclei such as a halo structure and the low-energy scattering solution in the nuclear reaction. When $\beta$ is close to $-1$, the basis state becomes a short-range and is suitable to describe the short-range and tensor correlations of nucleons with high momenta in nuclei [18]. From these properties, the parameter $\beta$ plays a role on the radial dilation of the coherent basis state, and then we call $\beta$ “dilation parameter” hereafter.

In the cluster model, the present coherent basis state is useful to describe relative motion between clusters with the orthogonality condition from the Pauli principle for the following two reasons:

1. (i)

   When the cluster wave functions are the HO shell-model ones, the Pauli-forbidden states in relative motion become the HO states with a specific quanta $N_{\rm PF}$. Hence, the coherent basis states with a relative oscillator quanta $N$ and $\beta$ become the Pauli-allowed states that are orthogonal to the Pauli-forbidden states with the condition of $N_{\rm PF}<N$ [20].

2. (ii)

   The relative motion between clusters is solved precisely and the relative wave function is optimized by superposing the coherent basis states $\phi_{n\ell m}^{\beta}(\mbox{\boldmath$r$},\nu)$ with various dilation parameters $\beta$, each of which shows a different spatial distribution.

In the multicluster system, we can prepare the cluster wave function using the coherent basis states in relative motion between every cluster-pair. In this paper, we consider the two-cluster case with clusters C₁ and C₂ and one intercluster motion with the coordinate $r$ in the single channel. We express the total nuclear wave function $\Psi$, in which the relative wave function $\Phi_{\rm rel}(\mbox{\boldmath$r$})$ is in the linear combination form of the coherent basis states $\{\phi^{\beta_{i}}_{n\ell m}(\mbox{\boldmath$r$},\nu_{\rm rel})\}$ with the range parameter $\nu_{\rm rel}$, the set of $\{\beta_{i}\}$ with $i=1,\cdots,N_{\rm base}$, and the condition of $N=2n+\ell$ for Pauli-allowed states, $N>N_{\rm PF}$;

where ${\cal A}$ is the antisymmetrizer of nucleons between different clusters and $\phi_{{\rm C}}$ is the internal wave function of the cluster $\rm C$. Hereafter we omit the notation of the quantum numbers $n$,$\ell$, and $m$ in the basis states for simplicity. It is possible to add the basis states with different $n$ to $\Phi_{\rm rel}(\mbox{\boldmath$r$})$ as well as $\beta_{i}$.

In the present study, we adopt the orthogonality condition model (OCM). The eigenvalue problem of the Hamiltonian $H$ for relative motion is given to obtain the relative energy $E$ between clusters:

where $T_{\rm rel}$ and $V_{{\rm C}_{1}{\rm C}_{2}}$ are the kinetic energy and the potential of relative motion between clusters, respectively. The matrix elements of $H_{ij}$ and $N_{ij}$ are those of the Hamiltonian and norm with the individual $\beta$-values, respectively. In this paper, we call the present framework “coherent basis method”.

In the coherent basis method, the matrix elements can be calculated analytically, and we use the formulas using the HO basis states with the independent range parameters in the bra and ket states [7], characterized by $\beta_{i}$ and $\beta_{j}$ in Eq. (7). For kinetic energy, we give the formula in Appendix B.

We demonstrate the present new scheme in the $\alpha$-$\alpha$ cluster system of ⁸Be. The $\alpha$ cluster is represented by the $(0s)^{4}$ configuration of the HO basis state where the range parameter $\nu$ of the single-nucleon state is taken as 0.535 fm^-2, which corresponds to the length of $b=1.3672$ fm, to reproduce the charge radius of the $\alpha$ particle. We prepare the coherent basis states for the relative wave function of $2\alpha$, the range parameter of which is $\nu_{\rm rel}=2\nu=$ 1.070 fm^-2 corresponding to the length of $b_{\rm rel}=0.9667$ fm. We employ the folding potential between $\alpha$-$\alpha$ with the nucleon-nucleon interaction and the Coulomb interaction using the $\alpha$ cluster wave function. We adopt the Schmid-Wildermuth effective nucleon-nucleon interaction [21], which is often used in the previous studies of the multi-$\alpha$ cluster systems [22, 23, 24, 25]. The form of the $\alpha$-$\alpha$ folding potential $V_{\alpha\alpha}(r)$ is given with nuclear (N) and Coulomb (C) parts as

where $r=|\mbox{\boldmath$r$}|$, $V_{0}=-72.98$ MeV, $\mu=0.46$ fm^-2, $W=M=0.4075$, and $B=H=0.0925$.

The lowest shell-model configuration of the $2\alpha$ system is $(0s)^{4}(0p)^{4}$ in the HO basis state with a total quanta of 4. Hence the Pauli-forbidden HO states, $\omega_{\rm F}(\mbox{\boldmath$r$},\nu_{\rm rel})$, are defined by the condition of quanta $N_{\rm PF}<4$ in the relative motion with the range $\nu_{\rm rel}$. In the coherent basis state, we impose this condition of the Pauli-allowed states and set $N=4$ for the $0^{+}$, $2^{+}$, and $4^{+}$ states and $N=6$ for the $6^{+}$ state in the present study. For the $4^{+}$ and $6^{+}$ states, there is no Pauli-forbidden state.

We take various dilation parameters $\beta_{i}$ in Eq. (6) to optimize the radial wave function. In the present study, we choose the set of $\beta_{i}$ in the form of the geometric progression of the length parameters $b_{i}$ of the HO basis state [15, 16] according to Eq. (4) as

We set $b_{0}=0.2$ fm, $\gamma=1.2$, and $N_{\rm base}=30$ in the present calculation, which are transformed to $\beta_{i}$ in the coherent basis states.

To show the reliability of the coherent basis method, we compare the obtained results with those of the conventional projection operator method (PO) [8]. In PO, one usually adds the pseudo potential of the projection operators with a positive prefactor $\lambda$ to the original Hamiltonian given as;

One uses a large value of $\lambda$ to make the solutions orthogonal to the Pauli-forbidden states $\{\omega_{{\rm F},f}\}$ with the index $f$ and we take $\lambda=10^{6}$ MeV in this study [17]. The number of the Pauli-forbidden states is determined from the condition of of quanta as $N_{\rm PF}=2n_{f}+\ell_{f}<4$. For the basis states of the relative motion in PO, we adopt the nodeless HO basis functions with $n=0$, which are often used in the OCM calculation, as

where $N_{\ell}(b)$ is a normalization factor of the basis state. The choice of the length parameters $\{b_{i}\}$ is the same as those of the coherent basis states in Eq. (9), which is suitable for comparing the solutions.

We describe the resonances and the scattering of the $\alpha$-$\alpha$ system using the complex scaling [12, 17, 26, 27, 28] in both the coherent basis method and the projection operator method. In the complex scaling, the relative coordinate $r$, the relative momentum $p$ in the Hamiltonian $H$, and the relative wave function $\Phi_{\rm rel}(\mbox{\boldmath$r$})$ are transformed using a scaling angle $\theta$ with an operator $U(\theta)$ as

where $\theta$ is a real positive number. The complex-scaled Hamiltonian $H^{\theta}$, the complex-scaled relative wave function $\Phi_{\rm rel}^{\theta}$, and the corresponding energy $E^{\theta}$ are given as

After solving the last equation, $E^{\theta}$ are obtained for bound, resonant, and continuum states on the complex energy plane according to the ABC theorem [29]. The energies of the continuum states start from the $\alpha$+$\alpha$ threshold energy and are obtained along the line rotated down by $2\theta$ from the real energy axis. The energies of the bound and resonant states are independent of $\theta$. The resonance has a complex energy $E_{\rm R}=E_{r}-i\Gamma/2$ with a resonance energy $E_{r}$ and a decay width $\Gamma$. The asymptotic behavior of the resonance wave function becomes a damping form if $2\theta>|\arg(E_{\rm R})|$ [29]. In calculations with a finite number of the basis states, the resonances are identified from the stationary property of $E_{\rm R}$ with respect to $\theta$ on the complex energy plane [17, 26, 27], and the continuum states are discretized with the complex energies. The wave function $\tilde{\Phi}_{\rm rel}^{\theta}(\mbox{\boldmath$r$})$ is the biorthogonal state of $\Phi_{\rm rel}^{\theta}(\mbox{\boldmath$r$})$ [30], and used for the bra state in the complex-scaled matrix elements. One does not take the complex conjugate of the radial part of the bra state in the matrix elements [26, 27].

The Pauli forbidden state $\omega_{\rm F}(\mbox{\boldmath$r$},\nu_{\rm rel})$ is also transformed in the complex scaling as

In the last equation, we use the explicit form of the HO basis state, and the range parameter $\nu_{\rm rel}$ is transformed instead of $r$.

In the projection operator method, the Hamiltonian $H_{\lambda}$ in Eq. (10) is transformed as $H_{\lambda}^{\theta}=U(\theta)H_{\lambda}U^{-1}(\theta)$, in which the Pauli-forbidden states are transformed in the pseudo potential [17]. In analogy with Eq. (11), the complex-scaled wave function is given as

where the $\theta$ dependence is included in $\{\bar{C}^{\theta}_{i}\}$. This expansion is often used in the conventional OCM calculation with complex scaling [17].

In the coherent basis method, the coherent basis state with a dilation parameter $\beta$ in Eq. (6) is transformed because the basis state should be orthogonal to the complex-scaled Pauli-forbidden states as

Hence the relative wave function $\Phi_{\rm rel}^{\theta}(\mbox{\boldmath$r$})$ is expanded in terms of the complex-scaled coherent basis states $\{\phi^{\beta_{i},\theta}(\mbox{\boldmath$r$},\nu_{\rm rel})\}$ with the index $i$ for $\beta_{i}$ as

One solves the following eigenvalue problem of the complex-scaled Hamiltonian matrix and obtains $E^{\theta}$ and $\{C_{i}^{\theta}\}$ for each eigenstate;

Technically, the matrix elements with the complex-scaled coherent basis states are calculated in the following procedure;

Here we insert the the completeness relation consisting of the states with a finite number $N_{p}$ ; $1=\sum_{p=1}^{N_{p}}|\phi_{p}\rangle\,\langle\tilde{\phi}_{p}|$. In this study, we construct the completeness relation in terms of the nodeless HO basis function with $\theta=0$, which are the same as those used in the projection operator method. We use the same set of $\{b_{i}\}$.

We diagonalize the norm matrix of $\phi_{0\ell m}(\mbox{\boldmath$r$},b_{i})$ and construct the orthonormalized basis states $\{\phi_{p}\}$ in the linear combination of $\phi_{0\ell m}(\mbox{\boldmath$r$},b_{i})$ with the coefficients $C_{i,p}$, which nicely describe the completeness relation in the present calculation. The states $\{\phi_{p}\}$ involve the Pauli-forbidden states, which are removed by diagonalizing the norm matrix with the elements of $N_{ij}^{\theta}$ in Eq. (20), because of the overlap with the coherent basis state in $\{D_{p,i}^{\theta}\}$. In the eigenvalue problem in Eq. (18), when one diagonalizes the norm matrix, the eigenstates of the Pauli-forbidden states show the zero-energy eigenvalue, which are removed from the basis states before diagonalizing the Hamiltonian matrix. It is noted that this procedure is used for the calculation of the unbound states with the complex scaling only, and is not necessary for the bound-state calculation with $\theta=0$.

In the complex scaling, the solutions $\{\Phi_{n}^{\theta},\tilde{\Phi}_{n}^{\theta}\}$ construct the completeness relation [30, 31] given as

where $n$ is the state index. Using the energy eigenvalues $\{E_{n}^{\theta}\}$, the complex-scaled Green’s function ${\cal G}^{\theta}(E)$ is expressed as

We calculate the level density $\rho(E)=\sum_{n}\delta(E-E_{n})$ with complex scaling [32, 33, 34]. The complex-scaled level density $\rho^{\theta}(E)$ is given with ${\cal G}^{\theta}(E)$ as

We also consider the asymptotic Hamiltonian $H_{0}^{\theta}$ with the energy eigenvalues $\{E_{0,n}^{\theta}\}$, and define the asymptotic level density $\rho_{0}^{\theta}(E)$ as

One defines the continuum level density $\Delta(E)=\rho^{\theta}(E)-\rho_{0}^{\theta}(E)$, which is related to the scattering matrix $S(E)$ [35]:

In the single channel, $\Delta(E)$ becomes the derivative of the phase shift $\delta(E)$ and the phase shift is obtained as

We define the asymptotic Hamiltonian $H_{0}$ for the $2\alpha$ system as [36]

We omit the nuclear interaction, and replace the Coulomb interaction with the point type. In the asymptotic wave function of $2\alpha$, one omits the antisymmetrization between the nucleons in the different $\alpha$ clusters [36, 37]. This means no Pauli-forbidden state in the relative motion between $2\alpha$ and then we set $N=\ell$ with $n=0$ in the coherent basis method. We also omit the projection operator in $H_{0}$ in the projection operator method.

In this study, we treat the $2\alpha$ system of ⁸Be and discuss the $\alpha$-$\alpha$ resonances. First, we compare the diagonal energies of the basis states in the coherent basis method (CH) and the projection operator method (PO) as functions of the HO length parameter $b$ in the Gaussians using Eq. (9). In two methods, the treatments of Pauli-forbidden states are different, and affect the diagonal energies. We show the results of the $0^{+}$ and $2^{+}$ states in Fig. 1 (top) in a logarithmic scale. We also show the results as functions of the dilation parameter $\beta$ used in the coherent basis states in Fig. 1 (bottom). These figures are useful to understand the treatment of the Pauli principle in the coherent basis method, which leads to the low-energy states in the large value of $b$, namely a large $\alpha$-$\alpha$ distance, and also in the values of $\beta$ close to unity.

In the projection operator method, the basis states in Eq. (11) can involve the Pauli-forbidden states, and then the pseudo potential with the strength of $\lambda=10^{6}$ MeV makes the states have high energies. The HO length of Pauli-forbidden states is $b_{\rm rel}=$0.9667 fm and the maximum energies appear at this length for two spin states. For the $0^{+}$ state, there are two Pauli-forbidden states with $n=0$ and $1$ and then the repulsive effect is distributed in a wider range of $b$ than the results of the $2^{+}$ state, which includes one Pauli-forbidden state with $n=0$. In the projection operator method, the superposition of the basis states makes the Pauli-allowed states with low energies. The comparison of the two methods explains the reasonable treatment of the Pauli-allowed states in the coherent basis method.

Next, we solve the eigenvalue problem of the Hamiltonian matrix. For $0^{+}$ state, there are two Pauli-forbidden states and in the projection operator method, two states are obtained to have the high energies close to $\lambda$. On the other hand in the coherent basis method, the basis states do not involve the Pauli-forbidden states, and all eigenstates are obtained as the Pauli-allowed states.

Before the calculation of resonances, we discuss the reliability of the present coherent basis method for the bound state. For this purpose, we artificially strengthen the $\alpha$-$\alpha$ nuclear potential $V^{\rm N}_{\alpha\alpha}(r)$ to make the $0^{+}$ and $2^{+}$ states of ⁸Be bound. We introduce the enhancement factor $\delta$ in $V^{\rm N}_{\alpha\alpha}(r)$ as

We compare the resulting energies with those obtained in the projection operator method.

In Table 1, we show the energies of ⁸Be ($0^{+}$ and $2^{+}$) measured from the $\alpha$+$\alpha$ threshold energy by changing $\delta$. It is found that the two methods give the same energies of the $0^{+}$ and $2^{+}$ states from weak to strong bindings with various values of $\delta$. These results indicate the reliability of the present coherent basis method.

Next, we keep $\delta=0$ in the $\alpha$-$\alpha$ nuclear interaction and describe the unbound states of ⁸Be in the complex scaling. We solve the complex-scaled eigenvalue problem in Eq. (18) for 2$\alpha$ of ⁸Be ($0^{+}$, $2^{+}$, $4^{+}$, and $6^{+}$). In Fig. 2, we show the energy eigenvalues $\{E_{n}^{\theta}\}$ of four spin states on the complex energy plane. The scaling angle $\theta$ is optimized in each spin state from the stationary condition of the energy eigenvalues of resonances on the complex energy plane with respect to $\theta$. This condition gives $\theta=16^{\circ},18^{\circ},20^{\circ}$, and $25^{\circ}$ for $0^{+},2^{+},4^{+}$, and $6^{+}$, respectively. We show two kinds of solutions obtained in the coherent basis method (CH) and projection operator method (PO) in the $0^{+}$ and $2^{+}$ states. For $4^{+}$ and $6^{+}$, the results obtained in the coherent basis method are shown. The continuum states are discretized along a straight line and we obtain one resonance in each state deviating from the line of the continuum states. In Fig. 2, the discretized continuum states also agree with each other by using the same range parameters in the relative wave function of $2\alpha$.

In Table 2, we list the resonance energies and decay widths of four resonances of ⁸Be obtained in the coherent basis method in comparison with the projection operator method. We also include the experimental data. It is found that resonance energies and decay widths of two states of ⁸Be agree with each other in the two methods. These results mean the reliability of the coherent basis method to describe resonances with complex scaling.

We calculate the eigenstates of the asymptotic Hamiltonian $H_{0}^{\theta}$ of $2\alpha$ using Eq. (28) to obtain the continuum level densities and phase shifts in the coherent basis method. We employ the same set of the dilation parameters $\{\beta_{i}\}$ as used in the calculation with the full Hamiltonian $H^{\theta}$ and set the same scaling angle $\theta$ for each state.

Using the energy eigenvalues $\{E_{n}^{\theta}\}$ and $\{E_{0,n}^{\theta}\}$ of 2$\alpha$, we calculate the continuum level density, $\Delta(E)$, and evaluate the phase shift of the $\alpha$–$\alpha$ scattering by integrating $\Delta(E)$ in Eq. (27). In Fig. 3, we show the phase shifts of the four states obtained, where we put the arrows at the resonance energies of four states shown in Table 2.

The resulting phase shifts with dashed or dotted lines are obtained in the coherent basis method, and they agree with the gray lines obtained in the projection operator method for the $0^{+}$ and $2^{+}$ states. In each state, the energy at the maximum derivative of the phase shift is close to the resonance energy shown in the arrow. From these results, one can apply the present coherent basis method to the scattering problem between various nuclear clusters with complex scaling. One does not need the projection operator to eliminate the Pauli-forbidden states between clusters, which are automatically removed in the coherent basis method.