Direct probing of the cluster structure in ${}^{12}$Be via $\alpha$-knockout reaction

TL;DR

This paper combines microscopic structure calculations with reaction modeling to demonstrate how alpha-knockout reactions can directly probe alpha-cluster formations in the neutron-rich nucleus $^{12}$Be.

Contribution

It introduces a new wave function model for $^{12}$Be and integrates it with reaction theory to connect structure predictions with experimental observables.

Findings

Reproduces the low-lying spectrum of $^{12}$Be.

Shows triple differential cross sections are sensitive to alpha-cluster amplitudes.

Demonstrates feasibility of probing clustering states via alpha-knockout reactions.

Abstract

Background: Recent theoretical and experimental researches using proton-induced $\alpha$-knockout reactions provide direct manifestation of $\alpha$-cluster formation in nuclei. In recent and future experiments, $\alpha$-knockout data are available for neutron-rich beryllium isotopes. In $^{12}$Be , rich phenomena are induced by the formation of $\alpha$-clusters surrounded by neutrons, for instance, breaking of the neutron magic number $N=8$. Purpose: Our objective is to provide direct probing of the $\alpha$-cluster formation in the $^{12}$Be target through associating the structure information obtained by a microscopic theory with the experimental observables of $\alpha$-knockout reactions. Method: We formulate a new wave function of the Tohsaki-Horiuchi-Schuck-R{\"o}pke (THSR) type for the structure calculation of ${}^{12}$Be nucleus and integrate it with the distorted wave impulse approximation framework for the $\alpha$-knockout reaction calculation of $^{12}$Be$(p,p\alpha)^{8}$He. Results: We reproduce the low-lying spectrum of the $^{12}$Be nucleus using the THSR wave function and discuss the cluster structure of the ground state. Based on the microscopic wave function, the optical potentials and $\alpha$-cluster wave function are determined and utilized in the calculation of ${}^{12}$Be($p,p\alpha$)${}^{8}$He reaction at 250 MeV. The possibility of probing the clustering state of $^{12}$Be through this reaction is demonstrated by analysis of the triple differential cross sections that are sensitively dependent on the $\alpha$-cluster amplitude at the nuclear surface. Conclusions: This study provides a feasible approach to validate directly the theoretical predictions of clustering features in the $^{12}$Be nucleus through the $\alpha$-knockout reaction.

Tables

Table 1. Table 1: The energy ( E 𝐸 E in MeV) of the ground state and excitation energies ( Δ ​ E Δ 𝐸 \Delta E in MeV) of the 0 2 + subscript superscript 0 2 0^{+}_{2} and 2 1 + subscript superscript 2 1 2^{+}_{1} states in 12 Be calculated with the superposition of the six bases from π 𝜋 \pi -orbit and α + He 8 𝛼 superscript He 8 \alpha+{}^{8}\textrm{He} configurations. The MV1 potential and G3RS potential are used for interactions in the central and spin-orbit channels, respectively. Parameters of these potentials are adopted from Ref.  enyo03 . “THSR” denotes the results calculated in this study using the THSR wave function. “AMD” denotes the results from the AMD investigations using the same interaction as in Ref.  enyo03 . “THSR (weakened V 0 l ​ s superscript subscript 𝑉 0 𝑙 𝑠 V_{0}^{ls} )” denotes the results calculated using the same THSR bases as in Eq. ( 10 ) but with weakened spin-orbit coupling strength V 0 l ​ s superscript subscript 𝑉 0 𝑙 𝑠 V_{0}^{ls} =3000 MeV. “exp.” denotes corresponding experimental values.

12Be	$E$($0_1^{+}$)	$\Delta E$($0_2^{+}$)	$\Delta E$($2_1^{+}$)
THSR	$-$59.5	4.1	2.2
AMD	$-$61.9	3.7	2.1
exp.	$-$68.6	2.3	2.1
THSR (weakened $V_0^{ls}$)	$-$58.0	4.6	3.0

Table 2. Table 2: Parameters of the THSR basis states in the α + He 8 𝛼 superscript He 8 \alpha+{}^{8}\textrm{He} , π 𝜋 \pi -orbit, and σ 𝜎 \sigma -orbit configurations. Detailed explanations of the parameters are given in the main text. All units are in fm.

basis	${\beta }_{\alpha ,xy}$	${\beta }_{\alpha ,z}$	${\beta }_{n,xy}^{9,10}$	${\beta }_{n,z}^{9,10}$	${\beta }_{n,xy}^{11,12}$	${\beta }_{n,z}^{11,12}$
${\Phi }_{\alpha +{}_{}{}^8\text{He}}$	0.1	2.0, 3.0, 4.0	0.1	0.4	0.1	0.4
${\Phi }_{\pi \text{-orbit}}$	0.1	2.0, 3.0, 4.0	1.5	3.0	2.5	3.0
${\Phi }_{\sigma \text{-orbit}}$	0.1	2.0, 3.0, 4.0	1.5	3.0	0.1	2.0

Equations

$$T_{{\mathbf{K}}_0{\mathbf{K}}_1{\mathbf{K}}_2}=$$

$$⟨{\chi }_{1,{\mathbf{K}}_1}^{(-)}({\mathbf{R}}_1){\chi }_{2,{\mathbf{K}}_2}^{(-)}({\mathbf{R}}_2)\left|t_{p\alpha }(\mathbf{s})\right|{\chi }_{0,{\mathbf{K}}_0}^{(+)}({\mathbf{R}}_0){\phi }_{\alpha }({\mathbf{R}}_2)⟩,$$

$$\frac{d^3\sigma }{d{E}_1^{\mathrm{L}}d{\Omega }_1^{\mathrm{L}}d{\Omega }_2^{\mathrm{L}}}=$$

$${\bar{T}}_{{\mathbf{K}}_0{\mathbf{K}}_1{\mathbf{K}}_2}=$$

$$\times {\chi }_{0,{\mathbf{K}}_0}^{(+)}(\mathbf{R}){\phi }_{\alpha }(\mathbf{R})e^{-i{\mathbf{K}}_0\cdot \mathbf{R}{A}_{\alpha }/A},$$

$$\alpha +{}^8\text{He},$$

$$\pi \text{-orbit:}$$

$$\sigma \text{-orbit:}$$

$$|{\Phi }_{\alpha +{}^8\text{He}}⟩=$$

$$\int d\mathbf{R}\mathcal{G}(\mathbf{R},{\mathbf{\beta }}_{\alpha })|\mathcal{A}\{{\Phi }_{\alpha }(-\mathbf{R})\Phi ({}^8\text{He},\mathbf{R})\}⟩,$$

$$|{\Phi }_{\pi \text{-orbit}}⟩=$$

$$\int d\mathbf{R}\mathcal{G}(\mathbf{R},{\mathbf{\beta }}_{\alpha })|\mathcal{A}\{{\Phi }_{\alpha }(-\mathbf{R}){\Phi }_{\alpha }(\mathbf{R}){\varphi }_{10}^{\pi }{\varphi }_{10}^{\pi }{\varphi }_{11}^{\pi \ast }{\varphi }_{12}^{\pi \ast }\}⟩,$$

$$\mathcal{G}(\mathbf{R},\mathbf{\beta })=\exp \left(-\frac{R_x^2+R_y^2}{{\beta }_{xy}^2}-\frac{R_z^2}{{\beta }_z^2}\right).$$

$$|{\Phi }_{\sigma \text{-orbit}}⟩=$$

$$\int d\mathbf{R}\mathcal{G}(\mathbf{R},{\mathbf{\beta }}_{\alpha })|\mathcal{A}\{{\Phi }_{\alpha }(-\mathbf{R}){\Phi }_{\alpha }(\mathbf{R}){\varphi }_9^{\pi }{\varphi }_{10}^{\pi }{\varphi }_{11}^{\sigma }{\varphi }_{12}^{\sigma }\}⟩,$$

$$|{\Psi }^J{(}^{12}\text{Be})⟩=\sum _{m,j}{c}_{m,j}{\hat{P}}_{00}^J{\hat{P}}_{\text{c.o.m}}{\Phi }_m({\beta }_{\alpha ,z;j}),$$

$$\left|ay(a)\right|\approx$$

$$\equiv$$

$$N_c=\frac{1}{\sqrt{2}}{\left(\frac{8\times 4}{12\pi b^2}\right)}^{1/4}$$

$$|{\Phi }_{\text{BB}}^{(0+)}({}^8\text{He},\alpha ,S)⟩$$

$$={\hat{P}}_{00}^0|\varphi (\alpha ,\frac{8}{12}S{\vec{e}}_z)\Phi ({}^8\text{He}(0^{+}),-\frac{4}{12}S{\vec{e}}_z)⟩.$$

$$P_m=\frac{|⟨\Psi {(}^{12}\text{Be})⟩{\Phi }_m^{\prime }{|}^2}{⟨\Psi {(}^{12}\text{Be})⟩\Psi {(}^{12}\text{Be})⟨{\Phi }_m^{\prime }⟩{\Phi }_m^{\prime }},$$

$${\Phi }_m^{\prime }=\sum _{j=1,2,3}{c}_{m,j}{\hat{P}}_{00}^0{\hat{P}}_{\text{c.o.m}}{\Phi }_m({\beta }_{\alpha ,z;j})$$

$$g(\mathbf{r},\mathbf{R})={\left(\frac{1}{\pi b^2}\right)}^{3/4}\exp \left\{-\frac{(\mathbf{r}-\mathbf{R}{)}^2}{2b^2}\right\},$$

$${\varphi }_{\tau s}^s(\mathbf{r},\mathbf{R})=g(\mathbf{r},\mathbf{R})|\tau ,s⟩.$$

$${\Psi }_{\alpha }(\mathbf{R})=\frac{1}{\sqrt{4!}}\mathcal{A}\{{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4\},$$

$${\varphi }_1={\varphi }_{p\uparrow }^s(\mathbf{r},\mathbf{R}),$$

$${\varphi }_3={\varphi }_{n\uparrow }^s(\mathbf{r},\mathbf{R}),$$

$$|\Phi {(}^8\text{He},\mathbf{R})⟩$$

$$=\frac{1}{\sqrt{8!}}|\mathcal{A}\{{\varphi }_{\alpha }(\mathbf{R}){\varphi }_9{\varphi }_{10}{\varphi }_{11}{\varphi }_{12}\}⟩,$$

$${\varphi }_{9,10}(\mathbf{r},\mathbf{R})=$$

$$\times |\tau ,\sigma =\pm 1/2⟩.$$

$${\varphi }_{11}(\mathbf{r},\mathbf{R})$$

$${\varphi }_{12}(\mathbf{r},\mathbf{R})$$

$${\varphi }_{\tau s}^{\pi }(\mathbf{r})=\int d{\mathbf{R}}^{\prime }\mathcal{G}({\mathbf{R}}^{\prime },{\mathbf{\beta }}_{\pi })e^{\pm i{\varphi }_{{\mathbf{R}}^{\prime }}}g({\mathbf{r}}_i,{\mathbf{R}}^{\prime })|\tau ,s=\pm 1/2⟩,$$

$${\varphi }_{\tau s}^{\pi \ast }(\mathbf{r})=\int d{\mathbf{R}}^{\prime }\mathcal{G}({\mathbf{R}}^{\prime },{\mathbf{\beta }}_{\pi }^{\prime })e^{\pm i{\varphi }_{{\mathbf{R}}^{\prime }}}g({\mathbf{r}}_i,{\mathbf{R}}^{\prime })|\tau ,s=\mp 1/2⟩,$$

$${\varphi }_9^{\pi }={\varphi }_{n\uparrow }^{\pi }(\mathbf{r}),$$

Full text

Recent theoretical and experimental researches using proton-induced $\alpha$-knockout reactions provide direct manifestation of $\alpha$-cluster formation in nuclei. In recent and future experiments, $\alpha$-knockout data are available for neutron-rich beryllium isotopes. In 12Be , rich phenomena are induced by the formation of $\alpha$-clusters surrounded by neutrons, for instance, breaking of the neutron magic number $N=8$.

Purpose

Our objective is to provide direct probing of the $\alpha$-cluster formation in the 12Be target through associating the structure information obtained by a microscopic theory with the experimental observables of $\alpha$-knockout reactions.

Method

We formulate a new wave function of the Tohsaki-Horiuchi-Schuck-Röpke (THSR) type for the structure calculation of 12Be nucleus and integrate it with the distorted wave impulse approximation framework for the $\alpha$-knockout reaction calculation of 12Be$(p,p\alpha)^{8}$He.

Results

We reproduce the low-lying spectrum of the 12Be nucleus using the THSR wave function and discuss the cluster structure of the ground state. Based on the microscopic wave function, the optical potentials and $\alpha$-cluster wave function are determined and utilized in the calculation of 12Be($p,p\alpha$)8He reaction at 250 MeV. The possibility of probing the clustering state of 12Be through this reaction is demonstrated by analysis of the triple differential cross sections that are sensitively dependent on the $\alpha$-cluster amplitude at the nuclear surface.

Conclusions

This study provides a feasible approach to validate directly the theoretical predictions of clustering features in the 12Be nucleus through the $\alpha$-knockout reaction.

NITEP 8

February, 2019

I Introduction

In atomic nuclei, the clusters emerge as a result of the competition between the short-range repulsion and the medium-range attraction induced by the Pauli blocking effect and the properties of nuclear forces freer18 . Especially, the $\alpha$-clustering effect is prevalent in nuclear clustering states because of the spin-isospin saturation in the nucleon-nucleon interaction. For the description of $\alpha$-clustering states, various structural theories have been formulated, as introduced in Refs. freer18 ; enyo01 ; itagaki01 ; oertzen06 ; enyo12 ; horiuchi12 ; ito14 ; ren18 and references therein.

In the Hoyle state of 12C, the $\alpha$-cluster formation has been well established and the description of clustering state has been treated elegantly in nuclear theory tohsaki01 . However, in neutron-rich nuclei, the description of the $\alpha$-clustering states is more challenging because of the existence of valence neutrons surrounding $\alpha$-clusters, as shown in the previous studies of Beryllium isotopes Oka77 ; Sey81 ; Des89 ; Oer96 ; Ara96 ; Dot97 ; Kan99 ; Oga00 ; itagaki00 ; Des02 ; ito04 ; ito08 ; ito12 ; enyo03 ; kobayashi12 ; enyo16 . Especially, in 10Be and 12Be isotopes, the nuclear molecular orbit (MO) configuration and the ion-like binary cluster configuration could coexist in clustering states, as predicted by theoretical studies using the generalized two-center cluster model (GTCM) ito04 ; ito08 ; ito12 and antisymmetrized molecular dynamics (AMD) enyo03 ; kobayashi12 ; enyo16 . In the 12Be nucleus, the breaking of the neutron magic number $N$=8 also occurs as a consequence of $\alpha$-cluster formation ito08 ; ito12 ; enyo03 .

In previous decades, the $\alpha$-clustering states in stable nuclei have been investigated through the proton-induced $\alpha$-knockout reactions Roos77 ; Nadasen80 ; Carey84 ; Wang85 ; Nadasen89 ; Mabiala09 ; yoshida16 ; wakasa17 ; yoshida16 ; lyu18 ; yoshida18 . The significant advantage in these studies is that the physical observables are directly connected to the $\alpha$-clusters yoshida16 ; lyu18 ; yoshida18 , and the reaction mechanism is clean as compared to the other direct reactions where $\alpha$-clusters are involved, such as the $\alpha$-transfer reactions Becchetti1978 ; Anantaraman1979 ; Tanabe1981 ; fukui16 . The theoretical description of $\alpha$-knockout reactions has been formulated using the distorted-wave impulse approximation (DWIA) framework Roos77 ; wakasa17 ; yoshida16 ; yoshida18 , and in recent works yoshida16 ; yoshida18 the peripheral property of the $(p,p\alpha)$ reactions has been demonstrated. This is essential for probing $\alpha$-clusters, which are most probably formed in the surface region of nuclei. Recently, there are emerging ($p,p\alpha$) reactions in inverse kinematics for light unstable nuclei including the neutron-rich Be isotopes that is conducted or planned in the Radioactive Isotope Beam Factory (RIBF) yang17 . These experiments provide ideal opportunities to investigate the clustering states of the neutron-rich Be isotopes by comparing theoretical predictions of ($p,p\alpha$) reaction observables and the corresponding experimental results.

In our previous work, we have investigated the 10Be($p,p\alpha$)6He reaction at 250 MeV by integrating the microscopic description of the 10Be target and the 6He residual nuclei into the DWIA framework for $\alpha$-knockout reaction, and predicted the triple differential cross sections (TDX) as a useful observable probing the $\alpha$ clustering in the 10Be nucleus lyu18 . For the structure calculation of the ground state of 10Be and 6He, Tohsaki-Horiuchi-Schuck-Röpke (THSR) wave functions have been formulated based on the previous studies tohsaki01 ; Fun15 ; Fun15a ; Fun16 ; zhou13 ; zhou14 ; zhou16 ; lyu15 ; lyu16 ; zhao18 .

In this work, we further extend the THSR wave function for the 12Be nucleus. The theoretical description of the clustering features of the 12Be nucleus is not so simple as those of the 10Be nucleus because of the coexistence of binary cluster and MO configurations. It is essential to take into account these different cluster configurations for the description of the ground state of 12Be nucleus, in particular, the phenomena of the $N=8$ magic number breaking. In this study, we show that the TDX can be the direct probe for such exotic clustering features in the 12Be nucleus. In addition, this work provides the new formulation of the THSR wave function for neutron-rich nucleus 12Be, which could be utilized in the further studies of other nuclei near the neutron drip line.

This article is organized as follows. In Section II, we recapitulate the DWIA framework for the $\alpha$-knockout reaction and the calculation of triple differential cross sections (TDX). In Section III, we formulate the THSR wave functions for the 12Be target and the 8He residual, and the extraction of the $\alpha$-cluster wave function. Some details of the formulation are given in the appendix. In Section IV, we discuss the numerical results for the nuclear structure of 12Be and the predictions of 12Be($p,p\alpha$)8He reaction observables (TDXs). Last Section V contains the conclusion.

II DWIA framework for the 12Be($p$,$p\alpha$)8He

reaction

We adopt the same DWIA framework as in Refs. yoshida16 ; yoshida18 for the ($p,p\alpha$) reaction. In this section, we introduce briefly the DWIA framework for the 12Be($p$,$p\alpha$)8He reaction. The coordinates for the description of the $\alpha$-knockout reaction are presented in Fig. 1. Here, the normal kinematics is adopted for simplicity. The transition amplitude for the ($p$,$p\alpha$) reaction is given by

[TABLE]

where the $\chi$ with subscripts 0, 1, and 2 denote the distorted wave functions for the incident proton $p$, the outgoing $p$, and the outgoing $\alpha$, respectively. The superscripts $(+)$ and $(-)$ indicate the outgoing and incoming boundary conditions adopted for $\chi$, respectively. The $\varphi_{\alpha}$ is the $\alpha$-cluster wave function inside the target nucleus 12Be, where only the $0^{+}$ channel is included. For each particle $i=0,1,2$, the momentum (wave number) and its solid angle in the center-of-mass frame are denoted by ${\bm{K}}_{i}$ and $\Omega_{i}$, respectively, and the corresponding quantities measured in the laboratory frame are denoted by additional superscript $\rm{L}$. We follow the theoretical approach in Ref. yoshida16 for the numerical calculation of the triple differential cross section (TDX) of the 12Be($p,p\alpha$)8He reaction:

[TABLE]

where $A_{\alpha}=4$ and $A=12$. $F_{\rm kin}$ and $C_{0}$ are kinematical factors, and ${d\sigma_{p\alpha}}/{d\Omega_{p\alpha}}$ is the $p$-$\alpha$ differential cross section at the energy and the scattering angle deduced from the ($p$,$p\alpha$) kinematics. $\bar{T}_{{\bm{K}}_{0}{\bm{K}}_{1}{\bm{K}}_{2}}$ is a reduced transition amplitude obtained by making the factorization approximation to Eq. (1); details can be found in Ref. yoshida16 .

In this calculation, the optical potentials for the $p$-12Be, $p$-8He, and $\alpha$-8He systems and the transition interaction $t_{p\alpha}$ between $p$ and $\alpha$ are determined by the folding model using the Melbourne $G$-matrix interaction Amo00 . The density distributions of the target and residual nuclei are extracted from the THSR wave function, which is formulated in Section III, and the phenomenological density distribution is adopted for the $\alpha$ cluster as introduced in Ref. yoshida16 . The spin-orbit part of each optical potential was disregarded. The distorted wave functions $\chi_{i}$ ($i=1,2,3$) are obtained by solving the corresponding Schrödinger equations using the optical potentials mentioned above. The $\alpha$-cluster wave function $\varphi_{\alpha}$ is extracted from the THSR wave function of 12Be by approximating the reduced width amplitude (RWA), as introduced in Section III.3.

III THSR wave function for the target and residual nuclei

We formulate the THSR wave functions for the target and residual nuclei by extending the microscopic models developed in previous works lyu15 ; lyu16 ; zhao18 . For the target nucleus 12Be, we consider three kinds of cluster configurations suggested in the theoretical works enyo03 ; ito08 as basis states in the THSR framework of nonlocalized cluster motion zhou13 ; zhou14 . One is the binary cluster configuration

[TABLE]

and the other two are the MO configurations

[TABLE]

It has been suggested in Refs. ito08 that the binary cluster configuration $\alpha$+8He dominates in 12Be when the two $\alpha$-clusters are well separated, while the MO configurations contribute when the $\alpha$-$\alpha$ distance is smaller than 6 fm. In what follows, we refer these three configurations as “$\alpha$+8He”, “$\pi$-orbit”, and “$\sigma$-orbit”, respectively.

III.1 The $\alpha$+8He configuration of 12Be

In this work, we describe the $\alpha$+8He configuration of 12Be in the THSR framework, as

[TABLE]

where the formulation of the 8He cluster wave function $\Phi({}^{8}\textrm{He},\bm{R})$ is explained in the Appendix. Similar formulation for the $\alpha$+16O configuration has been proved to be very efficient in describing the ground state of 20Ne in Refs. zhou13 ; zhou14 . We note that in the THSR framework, each basis wave function expresses not localized $\alpha$ and 8He clusters but non-localized clusters with almost free motion, which is different from the basis states used in other models itagaki00 ; ito08 .

III.2 The molecular orbit configurations of 12Be

The $\pi$-orbit configuration of 12Be is written as

[TABLE]

where the function $\mathcal{G}$ is the deformed Gaussian for describing the nonlocalized motion of two $\alpha$-clusters within the 12Be nucleus, which is defined by

[TABLE]

The four valance neutrons occupying the $\pi$-orbits are described by the $\phi_{9,10}^{\pi}$ and $\phi_{11,12}^{\pi*}$ states, which correspond to the parallel and antiparallel spin-orbit couplings, respectively. The formulation of single nucleon states $\phi_{9\mbox{--}12}$ are explained in the Appendix. In Fig. 2 (a), the density distribution is presented for $\phi_{11,12}$, in which the typical structure of the $\pi$-orbit configuration is clearly demonstrated. This $\pi$-orbit configuration goes to the $p$-shell closed configuration in the compact limit of the $\alpha$-cluster and valence neutron motions.

The $\sigma$-orbit configuration of 12Be is formulated as

[TABLE]

where $\phi_{9,10}^{\pi}$ are the same $\pi$-states as in Eq. (LABEL:eq:12Be-pi) and $\phi_{11,12}^{\sigma}$ are states for valance neutrons occupying the $\sigma$-orbits. The formulation of $\phi_{11,12}$ are explained in the Appendix. We also show the density distributions of $\phi_{11,12}^{\sigma}$ in Fig. 2 (b), where the typical nodal structure in the $\sigma$-orbit is reproduced. As seen in Fig. 2 (c), the $\alpha+{}^{8}\textrm{He}$ configuration also has a similar nodal structure of the valence neutrons along the $x=0$ axis as a result of the antisymmetrization effect between neutrons in the $\alpha$ and 8He clusters. In fact, the $\sigma$-orbit configuration is redundant in the present framework as it is already included in model space when the THSR bases of the $\alpha$+8He configuration are superposed, as discussed in the Subsection IV.2.

III.3 Total wave function and $\alpha$-cluster wave function of target

nucleus

The total wave function of 12Be is obtained by superposing the basis states in the three configurations formulated in Eqs. (LABEL:eq:12Be-2cluster), (LABEL:eq:12Be-pi), and (LABEL:eq:12Be-sigma). For each configuration, we formulate basis wave functions with different $\beta_{\alpha,z}$ parameters that manipulate the motion of clusters, and set other parameters to be the variationally optimized values for each configuration. All the parameters in the THSR bases are listed in Table 2 in the Appendix. The translational and rotational projections are performed for the bases to restore corresponding symmetry, as introduced in Ref. lyu15 . With these bases, the total wave function of 12Be can be written as

[TABLE]

where $\Phi_{m}$ labeled by $m$ for cluster configurations are the THSR bases for 12Be and $j$ denotes the choice of the parameter $\beta_{\alpha,z}$. The operators $\hat{P}^{0}_{00}$ and $\hat{P}_{\text{c.o.m}}$ denote angular momentum projection schuck and the projection for center-of-mass motion Oka77 , respectively. The $c_{i,j}$ are superposition coefficients to be obtained by diagonalizing the Hamiltonian matrix.

To extract the $\alpha$-cluster amplitude in the surface region, we approximate RWA $y(a)$ of the $\alpha$-cluster by the overlap of the total wave function of 12Be in Eq. (10) with the $\alpha$+8He cluster wave function as

[TABLE]

where

[TABLE]

with $\Phi_{\textrm{BB}}({}^{8}\textrm{He},\alpha,S)$ being a Brink-Bloch-type wave function Bri66 for the $\alpha$+8He two-body system separated with the relative distance $S$:

[TABLE]

Here $\Phi({}^{8}{\textrm{He}(0^{+})},-\frac{4}{12}S\vec{e}_{z})$ is the wave function of the residual nucleus 8He projected onto the $0^{+}$ state, which is located at $-({4}/{12})S\vec{e}_{z}$. The $y^{\text{app}}(a)$ is found to be a good approximation of the exact $y(a)$ in the surface region enyo14 and applicable to the present case because the observables in knockout reactions are only affected by the $\alpha$-cluster probability at the surface lyu18 .

IV Results

IV.1 Numerical inputs

In this study, we fix the following kinematical conditions for the 12Be($p$,$p\alpha$)8He reaction in the laboratory frame. The kinetic energy for the incident and emitted protons are set to be 250 MeV and 180 MeV, respectively. The emission angle of the outgoing proton is set to be $(\theta_{1}^{\rm{L}},\phi_{1}^{\rm{L}})$= $(60^{\circ},0^{\circ})$. To satisfy the recoilless condition for the 8He residue, the angle $\theta_{2}^{\rm{L}}$ of the emitted $\alpha$-cluster varies around $51^{\circ}$, and the angle $\phi_{2}^{\rm{L}}$ is set as $180^{\circ}$. The relativistic treatment is adopted in all the reaction kinematics in this calculation as well as the kinematics of the $p$-$\alpha$ binary collision. Recently, the importance of the dynamical relativistic corrections to the Coulomb and nuclear interactions has been revealed for the breakup reactions Bertulani05 ; Ogata09 ; Ogata10 ; Long11 . To see the effect of the dynamical relativistic corrections in the $(p,p\alpha)$ knockout reactions will be interesting, but it is beyond the scope of current study.

For the Hamiltonian of 12Be in structural calculation, we adopt the MV1 interaction ando80 of the central force, which includes finite-range two-body term and zero-range three-body terms. The two-body spin-orbit term is adopted from the G3RS interaction Yamaguchi79 . The parameters in these interactions and the width $b$ of the Gaussian wave packet adopted here are those used in Ref. enyo03 , where the energy spectra of low lying states in 11Be and 12Be nuclei are well reproduced by the AMD calculations enyo02 ; enyo03 .

IV.2 Energy spectrum of the 12Be nucleus

We calculate the energy and the wave function of 12Be by diagonalizing the Hamiltonian with respect to the basis states formulated in Section III. First, we discuss the properties of the bases in each cluster configuration, that is $\pi$-orbit, $\sigma$-orbit, or $\alpha+{}^{8}\textrm{He}$. In Fig. 3, the energies are plotted as functions of the parameter $\beta_{\alpha,z}$, which specifies the spatial extent of cluster motion. One sees clear dependence of the energy on $\beta_{\alpha,z}$ for all the three configurations. In the $\pi$-orbit (dashed curve) configuration, the energy minimum locates at about 2 fm, which corresponds to a very compact $\alpha$-clustering structure due to the external bounding from valance neutrons in $\pi$-states. In $\sigma$-orbit (solid curve) and $\alpha+{}^{8}\textrm{He}$ (dotted curve) configurations, the energy minima locate at much larger $\beta_{\alpha,z}=4$ fm, which corresponds to very large spatial distribution of $\alpha$-clusters. In addition, the energies described by the $\pi$-orbit and $\alpha+{}^{8}\textrm{He}$ configurations are found to be comparable with each other, which indicates that the breaking of the neutron-magic number $N=8$ could occur through a strong state mixing between these two configurations in the ground state of 12Be. The bases in the $\sigma$-orbit configuration are energetically unfavored compared with the other two configurations, and could give small contribution to the ground state of 12Be.

In numerical calculations, we prepare the THSR bases with various parameters $\beta_{\alpha,z}$ in the $\pi$-orbit, $\sigma$-orbit and $\alpha+{}^{8}\textrm{He}$ configurations. After the diagonalization of the Hamiltonian matrix, it is found that the total wave function of the ground state of 12Be is efficiently described only by bases of the $\pi$-orbit and $\alpha+{}^{8}\textrm{He}$ configurations. However, the $\sigma$-orbit configuration gives negligible contribution because the $\sigma$-orbit bases have large overlap with the corresponding $\alpha+{}^{8}\textrm{He}$ bases and its contribution to the ground state can be effectively taken into account by the $\alpha+{}^{8}\textrm{He}$ configuration in the THSR framework. Therefore, for the final result of 12Be, we omit the $\sigma$-orbit bases and adopt 6 bases of the $\alpha+{}^{8}\textrm{He}$ and $\pi$-orbit configurations by choosing $\beta_{\alpha,z}=2,3,4$ fm for each configuration in the superposition given by Eq. (10).

The calculated energy spectrum for low lying states, $0^{+}_{1}$, $0^{+}_{2}$, and $2^{+}_{1}$, of 12Be are listed in Table 1. For the ground state, we obtain the binding energy of $-$59.5 MeV, which is somewhat higher than the experimental value but acceptable as our main purpose is to describe correctly the cluster wave functions and to reproduce the energy spectrum, not to precisely reproduce the total energy. Especially, the excitation energy of the $2_{1}^{+}$ state is very sensitive to the moment of inertia determined by the distribution of the $\alpha$-clusters. In this work, we reproduce well the energy gap for the ground band as $\Delta E(2_{1}^{+})$=2.2 MeV, which is consistent with the experimental value 2.1 MeV. As a comparison, we also show the results calculated using the same THSR bases as in Eq. (10) but with a weakened spin-orbit coupling strength $V_{0}^{ls}$=3000 MeV (the default value is 3700 MeV in Ref. enyo03 ). The calculation with the weakened spin-orbit strength shows the higher excitation energy of $\Delta E(2_{1}^{+})$=3.0 MeV than the experimental value and may indicate a weaker $\alpha$-clustering in the ground band. The excitation energy calculated for the $0_{2}^{+}$ state is 4.1 MeV with the default parameter in this study, which is consistent with the corresponding AMD result of 3.7 MeV but still higher than the experimental value.

IV.3 Mixture of configurations in the 12Be nucleus

The mixture of the $\pi$-orbit and $\alpha+{}^{8}\textrm{He}$ configurations is explicitly treated as shown in Eq. (10). In order to discuss the contribution from the cluster configurations to the total wave function of 12Be, we define a probability to find each configuration in the total wave function by

[TABLE]

where $\Psi(^{12}\text{Be})$ is the total wave function of 12Be in Eq. (10) and $\Phi^{\prime}$ is defined by

[TABLE]

with $\beta_{\alpha,z;\{j=1,2,3\}}=\{2,3,4$ fm} and the label $m$ denotes $\pi$-orbit or $\alpha+{}^{8}\textrm{He}$ within the two configurations. Here the coefficients $c_{i}$ are fixed to be the values determined by the full diagonalization for 6 bases.

In Fig. 4, we show the probability of each component in the total wave function of 12Be. The probabilities of the $\pi$-orbit (dashed curve) and $\alpha$+8He (solid curve) components are plotted as function of the spin-orbit coupling strength $V_{0}^{ls}$. A strong dependence on $V_{0}^{ls}$ is observed for the mixing ratio between the two configurations. As the spin-orbit coupling strength increases, the $\alpha$+8He component increases because the $\alpha$+8He configuration comes to the energy relatively lower than the $\pi$-orbit configuration as shown in Fig. 3. As a consequence, the dominant component is changed from the $\pi$-orbit to the $\alpha$+8He configuration, which simulates the gradual transition from the normal state to the intruder state in the ground state wave function. With the default choice of $V_{0}^{ls}$=3700 MeV, the ground state of 12Be contains a 90% $\alpha$+8He component and have a largely developed $\alpha$-clustering. With a weakened spin-orbit coupling strength $V_{0}^{ls}$=3000 MeV, the $\alpha$+8He component reduces significantly to about 60% corresponding to the modest $\alpha$-clustering. It should be noted that the $\pi$-orbit and $\alpha$+8He configurations are not orthogonal to each other and the ground state also has 50% and 80% $\pi$-orbit probabilities for the default and weakened $V_{0}^{ls}$ cases, respectively. Considering that the $V_{0}^{ls}$ and other parameters in the $NN$ interactions are model dependent in different microscopic calculations, the ambiguities are inevitable for the mixing ratios between clustering configurations. Hence, the experimental observables that are directly related to these mixing ratios are essential for the validation of the predictions from the nuclear theories.

IV.4 The $\alpha$-cluster wave function

The $\alpha$-cluster wave function of 12Be can be obtained with the approximated RWA as described in Sec III.3. In Fig. 5, we compare the approximated RWAs for the THSR bases in the $\alpha+{}^{8}\textrm{He}$ (solid curve), $\pi$-orbit (dashed curve), and $\sigma$-orbit (dotted curve) configurations. It is clearly shown that the $\alpha+{}^{8}\textrm{He}$ configuration shows much larger amplitude at the surface region because it describes the enhanced $\alpha$-clustering comparing to in the $\pi$-orbit configurations. We note again that the cross sections of the $\alpha$-knockout reaction are sensitive to the $\alpha$ amplitudes in the surface region but are not affected by the amplitudes in the inner region.

In Fig. 6, we compare the approximated RWAs for the 12Be target with the default interaction (solid curve) and that with the weakened spin-orbit coupling strength (dashed curve). In the surface region, a significant difference is observed for the $\alpha$ amplitudes between the curves. The calculation with the default interaction gives larger surface amplitude than with the weakened spin-orbit coupling strength because of the larger $\alpha$+8He component. As shown in Sec. IV.5, this difference in the $\alpha$ amplitudes in the surface region can be examined by the TDX observables in the $\alpha$-knockout reaction.

IV.5 The triple differential cross sections

In Fig. 7, the TDXs are compared for the THSR bases in the $\alpha+{}^{8}\textrm{He}$ (solid curve), $\sigma$-orbit (dotted curve), and $\pi$-orbit (dashed curve) configurations with $\beta_{\alpha,z}=3$ fm. A prominent TDX is obtained for the solid curve, which is a logical outcome of the strong $\alpha$-clustering in the $\alpha+{}^{8}\textrm{He}$ configuration. On the other hand, the dashed curve has a significantly lower peak height, which is consistent to the weak $\alpha$-clustering in the $\pi$-orbit configuration. The huge difference in the magnitude with a factor of 10 between these two configurations indicates that the ($p,p\alpha$) reactions could be used as a sensitive tool to differentiate the mixing of the strong and week clustering components. For the $\sigma$-orbit configuration, we note that the dashed curve in Fig. 7 shows the TDX with about half magnitude of the solid curve, as expected from the RWA shown in Fig. 5, where a ratio of about 0.5 is obtained for the squared values between the $\sigma$-orbit and the $\alpha+{}^{8}\textrm{He}$ configurations.

In Fig. 8, the theoretical predictions of the TDXs are shown. When default $V_{0}^{ls}$ is adopted in the $NN$ interaction, as shown by the solid curve in Fig. 8, the TDXs are found to be analogous to the values of the $\alpha+{}^{8}\textrm{He}$ configuration in Fig. 7. In this case, the neutron magic number $N=8$ apparently breaks because of the intruder occupation induced by the $\alpha$-cluster formation. The dashed curve in Fig. 8 corresponds to the weakened $V_{0}^{ls}=3000$ MeV, where the intermediate strength of $\alpha$-cluster formation is suggested by the probability calculation in Fig. 4, and we expect weaker breaking of $N=8$ than in the default case. The ratio of about 2 is observed for the TDXs between the default and weakened curves at the zero momentum. We stress again that the TDX curves are sensitive to the $\alpha$-clustering in the wave function. In particular, this difference is much larger than in the RWA curves in Fig. 6.

In both Figs. 7 and 8, the high sensitivities of the ($p,p\alpha$) reaction are established for clarifying the strong and the week $\alpha$-clustering. We credit this superiority to the peripheral property of the ($p,p\alpha$) reaction yoshida16 ; yoshida18 , which allows probing of the $\alpha$-clusters only in the surface region where the probability of cluster formation is the largest. Hence, by comparing the experimental values of TDX with the theoretical predictions in Figs. 7 and 8, we can validate the breaking of $N=8$ by cluster formation. Furthermore, differentiation between the strong and the weak $\alpha$-clustering in the ground state of 12Be will be feasible.

Recently, there have been several works regarding the use of eikonal scattering waves in the DWIA framework for the $(p,2p)$ reactions Aumann13 ; Atar18 ; Liu19 . It will be interesting to evaluate the efficiency of eikonal approximation in the current $(p,p\alpha)$ case, which is expected to be discussed in our future work.

V Summary

We have provided the direct probing for the $\alpha$-clustering structures in the ground state of 12Be nucleus through the $(p,p\alpha)$ reaction at 250 MeV. The target and residual nuclei are described by the new framework of a nonlocalized cluster model with valence neutrons, and the reaction process is treated by the DWIA framework. The rich phenomena in the low-lying states of 12Be target, such as the coexistence of binary cluster and MO configurations, are described by the superposition of bases extending the THSR wave functions. The low-lying energy spectrum and the probabilities of strong and weak clustering components in the ground state of 12Be are obtained by the structural calculations using the newly formulated wave function. It is found that the magic number $N=8$ breaking occurs because of the strong clustering in the ground state of 12Be. The huge difference in the magnitude of TDXs at the zero momentum between the $\pi$-orbit and the $\alpha+{}^{8}\textrm{He}$ configurations shows that the TDX is a good measure for the breaking of $N=8$. In addition, the TDX is found to be highly sensitive to the strong and weak cluster formations, which allows quantitative discussions for the corresponding mixing ratio in the ground state of 12Be. This study provides a feasible approach to probe directly the exotic clustering features in the ground state of 12Be. Furthermore, the new THSR wave function formulated in this work provides new option for the study of neutron rich nuclei near the drip-line.

Acknowledgements.

The authors thank K. Minomo, Y. Neoh, Y. Chazono, and N. Itagaki for valuable discussions. The computation was carried out with the computer facilities at the Research Center for Nuclear Physics, Osaka University. This work was supported in part by Grants-in-Aid of the Japan Society for the Promotion of Science (Grants No. JP16K05352, No. JP15J01392, and No. JP18K03617).

Appendix A Formulations of cluster and molecular orbit states

We prepare the cluster and the molecular orbit states by the new extension of the THSR formulations used in our previous work lyu15 ; lyu16 .

A.1 The $\alpha$-cluster and the 8He-cluster

In order to simplify the discussion, we define the Gaussian wave packet in real space for nucleons as

[TABLE]

and the $s$-states of nucleons are written as the product of the spatial wave packet and the spin-isospin term

[TABLE]

The $\alpha$-clusters are described by the antisymmetrization of four $s$-states with spin-isospin saturation, as

[TABLE]

where

[TABLE]

The 8He-cluster wave function is written as the Slater determinant of eight single nucleon states, including four $s$-states in $\alpha$-cluster and four surrounding $p$-states, as

[TABLE]

where $\phi_{5\mbox{--}8}$ denote the $s$-states and $\phi_{9\mbox{--}12}$ denote the $p$-states. The $\phi_{9,10}$ correspond to the $1P_{3/2,\pm 3/2}$ states with the ring-type distribution on the horizontal plane, and they are simulated by the integration

[TABLE]

In the limit of $\bm{\beta}_{p}\to\bm{0}$ fm, the states $\phi_{9,10}$ converge to the $1P_{3/2,\pm 3/2}$ states of the harmonic oscillators. For the other two $p$-states in the 8He cluster, we project the desired $1P_{3/2,\pm 1/2}$ states from the vertical rotation of states $\phi_{9,10}$, as

[TABLE]

where the Euler angle $\Omega$ is $\{0,\pi/2,0\}$. Because of the total antisymmetrization between the four neutron states $\phi_{9\mbox{--}12}$ in the $1P_{3/2}$ orbits, only the $1P_{3/2,\pm 1/2}$ components of the rotated states $\phi_{11,12}$ contribute to the total cluster wave function of 8He.

A.2 The $\pi$-orbit states

The $\pi$-orbits are written as lyu15

[TABLE]

and

[TABLE]

where the superscripts $\pi$ and $\pi*$ denote the $\pi$-orbits with parallel and antiparallel spin-isospin coupling, respectively. The states $\phi^{\pi}_{9\mbox{--}12}$ of four neutrons occupying $\pi$-orbits in Eqs. (LABEL:eq:12Be-pi) and (LABEL:eq:12Be-sigma) are defined as

[TABLE]

A.3 The $\sigma$-orbit states

The $\sigma$-orbits in 12Be nucleus are formulated with respect to the $\alpha$-clusters, as

[TABLE]

where $\pm\bm{R}$ are the generate coordinates of two $\alpha$-clusters in Eq. (LABEL:eq:12Be-sigma). The factor functions $\mathcal{F}$ are defined by

[TABLE]

In Eq. (LABEL:eq:12Be-sigma), there is integration over the $\alpha$-cluster generate coordinate $\bm{R}$, as

[TABLE]

which numerically describes the single nucleon state in $\sigma$-orbit configuration, as shown by corresponding density distribution in Fig. 2 (b). The states $\phi_{11,12}^{\sigma}$ of two neutrons occupying the $\sigma$ orbits in Eq. (LABEL:eq:12Be-sigma) are defined as

[TABLE]

A.4 Parameters of the THSR bases

We list in Table 2 the parameters of the THSR basis states used in the numerical calculation. In the table, $\beta_{\alpha}$s are parameters for the cluster motion in each configuration as shown in Eqs. (LABEL:eq:12Be-2cluster), (LABEL:eq:12Be-pi) and (LABEL:eq:12Be-sigma). For the $\alpha+{}^{8}\textrm{He}$ configuration, $\beta_{n}$s denote parameters $\beta_{p}$s for the neutrons occupying the $p$-orbits in the 8He cluster, as shown in Eqs. (21) and (22). For the $\pi$-orbit and $\sigma$-orbit configurations, $\beta_{n}$s denote parameters $\beta_{\pi}$s and $\beta_{\sigma}$s respectively for neutrons occupying the MO orbits, as shown in Eqs. (23), (24) and (26). The superscripts $9\mbox{--}12$ denote parameters for corresponding single neutron states $\phi_{9\mbox{--}12}$ in each configuration.