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The study of the alpha formation probability in 10Be and 12Be within the microscopic cluster model

The alpha clustering in the 10Be and 12Be has been studied within the framework of the real-time evolution method (REM). By using the effective interaction tuned to reproduce the charge radii and the threshold energies, we have evaluated the alpha reduced width amplitude (RWA) and spectroscopic factor (S-factor) for the ground and excited states. With several improvements made in this work, comparing with our previous calculations, larger rate of clustering results are obtained for the 10Be and 12Be with correct asymptotic at large distance.


I. INTRODUCTION
In nuclear physics, the clustering phenomena have been intensively investigated by many experimental and theoretical studies [1][2][3]. One of the most important examples is the Hoyle state [4,5] which has clustered structure composed of three α particles, and plays an essential role for the synthesis of carbon in the universe [6]. For decades, the clustering in light stable nuclei has been studied by various experimental methods such as the resonant scattering, transfer reactions and the break-up reactions [7,8]. On the other hand, the clustering in neutron-rich nuclei is not known in as much detail as that of stable nuclei, due to the limited experimental techniques [9,10].
Recently, the proton-induced α knockout reaction (p,pα) has been established as a quantitative measure for the α-cluster formation at nuclear surface [11][12][13][14][15]. The reaction has been used to measure the α clustering in Sn isotope chain [16] and revealed negative correlation between the α-cluster formation and the neutron-skin thickness. This fascinating experiment has opened a gate to explore the α clustering in the light neutron-rich nuclei.
In fact, the (p, pα) experiment is under preparation to measure the α clustering in the Be and C isotopes. Combined with the data obtained from break-up reactions, it will provide comprehensive information on how the clustering in the ground and excited states evolves as a function of neutron number. Therefore, an accurate microscopic calculation is highly desired for the deeper understanding of the α clustering in the light neutron-rich nuclei.
In our previous work, we adopted the antisymmetrized molecular dynamics (AMD) framework to evaluate α formation probability in Be and C isotopes [17,18]. The calculations showed the negative correlation between the α clustering and the neutron-skin thickness in the C isotopes, which is qualitatively consistent with what was observed in the Sn isotopes. However, from a quantitative point of view, those AMD results may underestimate the α formation probability due to the following reasons. Firstly, the Gogny D1S density functional in our previous study cannot describe light s-and p-shell nuclei accurately, because its parameters are optimized for medium-and heavy-mess nuclei [19]. In particular, it does not reproduce the α and 6 He decay threshold energies and radii of He isotopes, both of which strongly affect the cluster formation probability. Secondly, in our AMD+GCM calculation, we have used the quadrupole deformation parameter β as the generator coordinate. However, this choice does not span the model space large enough to describe various cluster motions in nuclei, in particular, that at a large inter-cluster distance. In fact, it has been pointed out that an AMD+GCM calculation undershoots the α cluster formation probability of 48 Ti in order of magnitude [20].
To overcome those defects, we perform alternative theoretical calculations with the following improvements. Firstly, we adopt the Volkov No.2 interaction [21] for the nucleon-nucleon interaction which reproduces the radius of the α particle and the phase shift of the α-α scattering. As for the spin-orbit interaction, we have employed the G3RS interaction. We have tuned the parameter set of these interactions to reproduce the threshold energies. We also note that the combination of those interactions has been adopted by many microscopic calculations to discuss the energy spectrum of light nuclei [22,23]. Secondly, we employ the real-time evolution method (REM) to generate basis wave functions to describe various cluster motions. This method has successfully described the α clustering of Be, and C isotopes [24][25][26]. With these improvements, we have obtained a more reliable cluster formation probability which has the correct asymptotic at a large distance. As we a-priori assume the α + α + xn cluster structure of 10 Be and 12 Be, this work will give the upper limit of α cluster formation probability, whereas our previous study by AMD may give the lower limit. This paper is organized as follows. In the next section, the theoretical framework of the generator coordinates method (GCM) with the real-time evolution method (REM) and the method to evaluate the reduced width amplitude (RWA) are briefly explained. In section III, we present the numerical results and discuss the differences in the results between the AMD and the REM. The final section summarizes this work.

A. The Hamiltonian and the wave function
The Hamiltonian adopted in this study is given aŝ wheret i andT c.m. denote the kinetic energy operators of each nucleon and the center of mass, respectively.v N ,v C ,v LS denote the effective nucleon-nucleon interaction, the Coulomb interaction and the spin-orbit interaction, respectively.
In this work, we use the Volkov No.2 interaction for the central nucleon-nucleon interaction [21], which is expressed aŝ where W , M, B and H denote the Wigner, Majorana, Bartlett, and Heisenberg exchanges, whose strengths are explained in the next section. The other parameters are, V 1 = −60.65 MeV, V 2 = 61.14 MeV, c 1 = 1.80 fm and c 2 = 1.01 fm. As for the spin-orbit interaction, we use the G3RS potential [27,28], HereP 31 projects the two-body system into triplet odd state, which can be expressed aŝ The Gaussian range parameters d 1 and d 2 are set to be 5.0 fm −2 and 2.778 fm −2 , respectively.
We approximate the He and Be isotopes as being composed of α-clusters plus valence nucleons. Thus, the wave functions of these isotopes are written as where Φ α (z α ) is the antisymmetrized wave function of the α-cluster with (0s) 4 configuration oriented at z α . φ(Z) are the single-neutron wave functions. The single-particle wave function φ(r, Z) are expressed in a Gaussian form multiplied by the spin-isospin part χτ as Here Z represents the time-dependent parameters of the wave function, which includes the three-dimensional coordinate z for the spatial part of the wave function as well as the spinor a and b for the spin part χ = a |↑ + b |↓ . The isospin part is τ = {proton or neutron}. The harmonic oscillator parameter is set to b = 1/(2ν) = 1.46 fm for both of α cluster and neutron wave functions, which is same with that used in Refs. [22,29].

B. Real-time evolution method
We use the real-time evolution method (REM) [25,26] to generate the basis wave functions which have various cluster configurations in the phase space. From the time-dependent variational principle: We obtain the equation of the motion (EOM) for all the time-dependent parameters Z(t) as By solving this EOM starting from a certain wave function, a series of basis wave functions for generator coordinate method (GCM) is obtained.
Under the framework of GCM, the basis wave functions with different parameters Z are superposed to describe the total wave function whereP J π M K is the parity and the angular momentum projector. This integral can be approximately discretized as The corresponding coefficients f i,K and the eigen-energy E are obtained by solving the Hill-Wheeler equation.

C. α reduced width amplitude
To evaluate the degree of α clustering, we calculate the α reduced width amplitude (RWA) from the obtained wave functions, which is defined as the overlap amplitude between the A-body wave function of the mother nucleus Ψ and the reference state composed of the clusters with mass numbers C 1 and C 2 , where Ψ C 1 and Ψ C 2 are the ground state wave functions of the two clusters. This equation is calculated by using the Laplace expansion method [30]. The α clustering may be evaluated by the α spectroscopic factor, which is defined as the squared integral of the RWA, It is noted that S α is not normalized unity because of the antisymmetrized effects between the clusters. In addition, we also introduce the root-mean-square radius of RWA, which is a measure for the average distance between clusters.

III. RESULTS
A. Structure properties of 10,12 Be and applied interaction parameters already been adopted in many calculations for this nucleus [22,23]. The obtained numerical results are denoted as "REM (set1)" in Table I, where the numerical results are compared with those obtained by the AMD calculation with the Gogny D1S interaction. The α-decay threshold energy is an important quantity for the discussion of the α-cluster formation.
We see that the AMD fails to reproduce it, while the REM yields a consistent result with the experimental data. The charge radius is another important quantity to discuss the α formation probability. The AMD results overestimate the charge radii of light nuclei. The REM also overestimates them but is smaller than the AMD result. In short, the REM with the Volkov No.2 interaction provides a better description of α threshold energy and the charge radius. We will discuss how these differences affect the RWA in the next subsection.
It has been known that the 8 He+ 4 He and 6 He+ 6 He configurations and even the 7 He+ 5 He configuration play important role for the low-lying states of 12 Be [2, 10, 31, 32]. The ground state of 12 Be is dominated by the configuration with the breaking of N = 8 magic number [23,31,33], and the 0 + 2 state has N = 8 closed shell structure. The numerical results are shown as "REM (set1)" in Table II. We found that REM does not reproduce many properties TABLE I. The numerical results of the 0 + ground states of 10 Be, 6 He, and 4 He. "Exp.", "AMD" and "REM (set1)" denote the results from experimental data, AMD calculations, and REM calculations, respectively. Rc denotes the charge radius which is calculated from the point proton radius. ∆E(α) represents the threshold energy to 6   and V ls = 2800 to reproduce the following properties; the breaking of the magic number in the ground state, the α+ 8 He and 6 He+ 6 He threshold energies, and the excitation energy of the 0 + 2 state. The results obtained by the fitted interaction are denoted by "REM (set2)" in Table II. This guarantees consistency with the observed data.

B. RWA and Whittaker function
Before we discuss the RWA, let us remind the structure of 10 Be and 12 Be. Two valence neutrons of 10 Be occupy the p-shell, whereas two of four valence neutrons in 12 Be are promoted to sd-shell across the N = 8 shell gap. We then compare the RWA calculation results to see the similarities and differences in the α cluster formation probability obtained by AMD and REM frameworks. Fig. 1 shows the RWA of the ground state of 10 Be for the 6 He+ 4 He channel. It shows that the REM provides a much larger amplitude than the AMD calculation at the peak region, which indicates the larger possibility of the α formation in 10 Be. This is due to the α cluster assumption in the REM wave function, while the AMD does not assume it. Therefore, the current REM result provides the upper limit of clustering. In addition to the difference in the amplitude, it can be found that the positions of the nodes of RWA from REM are shifted slightly to the inner part compared with the AMD calculations, which indicates the shorter spatial distribution of the α cluster. This is because that REM gives a smaller radius of 10 Be ground state. Thus, REM yields a more enhanced but narrower distribution of α cluster. An advantage of REM over AMD is that it can describe the correct asymptotics of RWA.
As explained in the appendix, the RWA should be identical to the Whittaker function at a large distance. We compare the logarithmic derivatives of the RWA and the Whittaker function in Fig. 2. We can see that the RWA calculated from the AMD is inconsistent with the Whittaker function, whereas REM yields the correct asymptotics. From this result, we can easily obtain the asymptotic normalization constant (ANC), which is calculated as 5.3.
Unlike 10 Be, the discussion on 12 Be is complicated because of its more exotic structure.
In Fig. 3, we show the RWA of the ground state of 12 Be calculated by using the original and modified interaction parameters. Firstly, we note that the number of the nodes of RWA has an additional node. Secondly, we again see a larger amplitude at the peak value in the modified parameter set calculation than the AMD result, which can also be attributed to the cluster assumption node in the REM calculation. We also see that the RWA from the AMD calculation is much more spreading than the REM (set2) calculation. This is because of the much-overestimated radii by the AMD results.
The comparison between the RWA and the Whittaker function is shown in Fig. 4. The deviation from the Whittaker function are found for both of the AMD and REM (set2) calculations. This result may be explained as follows. The Whittaker function is under the The vertical axis shows the ratio between the derivation of the function as Eq. A6.
assumption of a two-body system. However, as already been discussed in many reference papers [31,34], the ground state of 12 Be is an admixture of the 8 He+ 4 He, 6 He+ 6 He and 5 He+ 7 He channels. This fact can also be found in Fig. 5, where the compatible amplitude of RWAs for 12 Be to 6 He+ 6 He and 8 He+ 4 He channels are shown. Therefore, the ground state of 12 Be can not be simply treated as a two-body system, which causes a departure from the Whittaker function.
In addition to the limitation of the two-body system assumption, the requirement that the cluster is only affected by the Coulomb interaction may also be violated by the RWA to cause the departure from the Whittaker function. As already been shown in Table III, the r.m.s. radii of point neutron are much larger than the point proton for 12 Be especially in the REM calculations. It suggests that the formed cluster in 12 Be may be surrounded by the valence neutrons even in the outside region of the nucleus so that the cluster still feels the residual nuclear force. This residual nuclear force from the neutrons also leads to the departure from the Whittaker function. Another cause can be found from the RWA results of the 0 + 2 state for 12 Be shown in Fig. 6. It shows that the middle region of 12 Be is dominated by the 6 He+ 6 He channel while the outside region is dominated by the 8 He+ 4 He channel, which should satisfy the assumption of two-body system. However, the amplitudes of RWA are only visible within about 5 fm, where the nuclear force can not be neglected.
This narrow distribution of RWA also cause the departure from the Whittaker function as shown in Fig. 7.
Finally, we show the α spectroscopic factors and the r.m.s. radii of RWA in Table IV. It explicitly shows that the AMD result may provide a lower limit of the α clustering in the Be isotopes, while the REM procedure with the Volkov No.2 interaction may provide the upper limit. Both calculations show the suppression of the clustering with increasing neutron number, especially in REM calculation, which satisfies the discussion on the effect of the   The vertical axis shows the ratio between the derivation of the function as Eq. A6.
previous AMD calculations, the new results can reproduce the charge radii and the threshold energies. These quantities could be more important for the description of the α clustering.
The RWA results from these frameworks show that the REM procedure may provide the upper limit of α clustering, while the AMD gives the lower limit. Besides, the suppression in r.m.s. radius of RWA by the neutron skin thickness can be correctly obtained by the REM calculations.
The Whittaker function is expected to be an important criterion for testing the calculation of cluster formation. The comparison with the Whittaker function shows that the current REM result of 10 Be provides the correct asymptotic at large distance, which indicates more reliable description on the α cluster formation.
For the calculation of 12 Be, the breaking of N = 8 magic number is reproduced by slightly modifying the parameters of the interaction. The RWA results of 12 Be suggest the mixing of 8 He+ 4 He channel and 6 He+ 6 He channel in the ground state. This complex structure causes the departure from the Whittaker function. In addition to this, the narrow distribution of the RWA and the influence of the valence neutrons may also affect the asymptotic of RWA.
The behavior of RWA in the nuclei with exotic cluster structures or neutron skin should be further investigated in our future works.
where l is relative angular momentum between two particles and other parameters are summarised as reduced mass : m = m 1 m 2 /(m 1 + m 2 ) wave number : k = −2mE/ 2 dimensionless Sommerfeld parameter : η = Z 1 Z 2 e 2 m/4πǫ 0 2 k . (A3) Among these two solutions, we only adopt the Whittaker W function for the physical meaning.
In this sense, if the RWA calculated from a framework is accurate enough, it should be identical to the Whittaker function starting from a large distance a with an normalization factor C as ry l (r) = CW (r) for r > a .
This factor C is called Asymptotic Normalization Constant (ANC), which is of importance for the study of the astrophysical reactions. In addition, if C is really a constant in the calculation, the derivation of Eq. A4 should still hold as [ry l (r)] ′ = CW ′ (r) for r > a .
(A6) the existence of the constant C can be confirmed if these two ratios are identical. The comparison between RWA and Whittaker equation can be treated as an important criterion of the accuracy of the calculation. In this work, we will compare the RWA obtained from AMD and REM calculations with the Whittaker function to discuss the model dependence of the description of cluster formation.